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+{"text":"\\section{Background}\nOgorodnikov (1958) and \nVorontsov-Vel'yaminov (1967) were among \nthe first to draw attention to a subset of\nedge-on spiral galaxies with  disk axial\nratios as extreme as 30:1\\footnote{These disk axial ratios were\nmeasured on blue photographic survey plates; however, we find\n the axial ratios tend to \nbecome less extreme when  measured at a fixed isophote on \ndeeper, red-sensitive CCD images. For this reason, measurements of \nvertical disk scale heights offer\n a more robust means of quantifying\nthe thickness of edge-on galaxies.}   and having little or no\ndiscernable bulge\ncomponent. Goad \\& Roberts (1981) dubbed these galaxies ``superthins'' \nand recognized\nthat spiral galaxies selected on the basis of their thin stellar disks and \nextreme axial ratios also tend to\npossess a variety of other intriguing properties. \n\nFor example, Goad \\& Roberts (1981) discovered that superthins have low\nemission-line intensity ratios ([\\mbox{N\\,{\\sc ii}}]\/H$\\alpha$ and [\\mbox{N\\,{\\sc ii}}]\/[\\mbox{S\\,{\\sc ii}}]),\nwhich in a modern perspective suggests that superthins are moderate\nionization, low metallicity galaxies more like irregulars than spirals\nin terms of their \\mbox{H\\,{\\sc ii}}\\ region properties (see also Bergvall \\&\nR\\\"onnback 1995). Similar trends were confirmed for larger samples by\nKarachentsev \\& Xu (1991) and Karachentsev (1991).\n\nIn spite of having ``classic'' double-horned global \\mbox{H\\,{\\sc i}}\\ profiles\n(e.g., Matthews \\& van Driel 1999), often superthins exhibit\nslowly-rising, solid-body rotation curves throughout most or all of\ntheir stellar disks (Goad \\& Roberts 1981; Karachentsev \\& Xu\n1991; Karachentsev 1991; Cox {\\it et al.} 1996; Makarov {\\it et al.}\\ 1997,1998;\nAbe {\\it et al.} 1999). \nSuch rotation curves are generally characteristic of\nlate-type dwarf and irregular \ngalaxies rather than normal spirals (cf. Casertano \\&\nvan Gorkom 1991).\n\nNot only are superthins fascinating objects in their own right, but \nthe very nature of these galaxies makes them extremely valuable objects\nfor exploring a variety of fundamental astrophysical problems,  ranging from \ndisk formation and evolution scenarios, to the interpretation of\nhigh-redshift galaxy populations, to constraining the stability\nand dark matter contents of disks (e.g., Karachentsev 1989; Zasov\n{\\it et al.}\\ 1991; Dalcanton \\& Schectman 1996). In spite of this,\nto date only a handful of \ndetailed studies of individual superthins have\nbeen undertaken (e.g., Gallagher \\& Hudson 1976; \nBergvall \\& R\\\"onnback 1995; Cox {\\it et al.}\\ 1996;\nMatthews 1998; Abe {\\it et al.} 1999).\n\nMeasurements of the vertical structure and\ndisk color gradients are of particular interest in the case of superthin \nspirals, as these can provide important\nclues to their evolutionary history.\nVertical structure measurements provide vital constraints on disk\nstability, dynamical heating mechanisms, and dark matter contents\n(Matthews 1998; Matthews 1999, hereafter Paper~II), \nand when combined with measures of global properties\nand disk color gradients, information on the star-formation\nhistories and dynamical evolutionary histories of the galaxies can be gleaned.\n\nUGC~7321 is a superthin galaxy ideally suited to such investigations. Since\nit is nearby (D$\\sim$10~Mpc; Matthews {\\it et al.} 1999b), \nits structure can be reasonably resolved from the ground.\nIn addition, UGC~7321 appears to suffer little from \ninternal extinction (see below), and it is nearly\nexactly edge-on ($i\\approx$88$^{\\circ}$), hence projection effects\nwill have a minimal influence on derived vertical structural parameters \n(e.g., de Grijs, Peletier, \\& van der Kruit 1997). Finally, the analysis\nof UGC~7321 is simplified compared with other edge-on spirals\ndue to the absence of a bulge component. \n\nHere we present\nmeasurements of the global  properties of UGC~7321 based on\nnew multiwavelength observations\nincluding $B$- and $R$-band imaging and photometry, narrow-band H$\\alpha$\nimaging, near-infrared $H$-band imaging, and \\mbox{H\\,{\\sc i}}\\ 21-cm pencil beam\nspectroscopy.  We also examine \\mbox{H\\,{\\sc i}}\\ aperture synthesis data \nobtained by Rots, Roberts, \\& Goad (private communication).\nWe use our datasets to measure the radial and vertical\ncolor gradients in the\ndisk of UGC~7321 (Sect.~3) and briefly discuss what the color\ngradients reveal about the \nevolutionary history \nof this galaxy. \nIn a second paper (Paper~II), we present\nmeasurements of the vertical scale height of the UGC~7321 disk. There we\ncombine vertical structural measurements\nwith  results from the present\nwork in order to further\nconstrain the stability and dynamical \nhistory of UGC~7321.\n\n\\section{Imaging Data}\n\\subsection{Near-Infrared Imaging}\n\nWe obtained near-infrared (NIR) $H$-band imaging observations of\nUGC~7321 in May 1997 using IRIM on the 2.1-m telescope at the Kitt Peak\nNational Observatory\\footnote{Kitt Peak National Observatory\nis operated by the Association of Universities for Research\nin Astronomy, Inc. under\ncontract with the National Science Foundation.}. IRIM employs\na 256$\\times$256 HgCdTe NICMOS3\ndetector with\n\\as{1}{09} pixels, yielding a field-of-view of  \\am{4}{6} per side.\nThe detector\nhas a gain of 10.5~$e^{-}$\/ADU and a readnoise of  $\\sim$37~$e^{-}$ per\npixel rms. The observations were taken in the presence of \nmoderate cirrus, hence\nour $H$-band data cannot be reliably photometrically calibrated.\n\nWe obtained sets of 9 co-added, 20-$s$ exposures\nof UGC~7321 at each of 35 telescope pointings. To\ninsure accurate flatfielding and sky subtraction, we shifted the\ngalaxy to a different position on the detector before each series of\nexposures.\nIn addition, interspersed throughout the target observations,\nwe obtained several series of ``blank''\nsky  exposures at regions  adjacent to the\ngalaxy. Linearization corrections to the data\nwere made\nautomatically during data acquisition.\n\nData reduction was accomplished using standard IRAF\\footnote{IRAF is\ndistributed by the National Optical Astronomy Observatories, which is\noperated by the Associated Universities for Research in Astronomy,\nInc. under cooperative agreement with the National Science Foundation.} tasks.\nTo remove the sky background, a\nmedian  was made of\nseveral groups of temporally consecutive object and blank\nsky exposures. From these, a   median-averaged  combination\n of 25 dark frames was subtracted  to remove the structure\npresent in the detector dark current. Finally, the dark-corrected\nsky frame was\nsubtracted from all dark-corrected object frames.\n\nFlatfielding was accomplished by dividing by a night sky flat composed\nof the median of all dark-corrected sky and object exposures. Lastly,\nthe flatfielded object exposures were aligned and co-added.\nOur resulting $H$-band image of UGC~7321 is shown in Fig.~1. Note the\nstriking thinness of the stellar disk and the complete absence of \na bulge component even in the NIR. At the disk center, the vertical\nexponential scale\nheight is only $\\sim$140~pc (Paper~II)! \n\nFig.~2 shows\nthe $H$-band data in the form of a contour map. Because the disk is so\nthin, the contours have been\nstretched in the $z$-direction for clarity. Note that the contours are\nquite regular, even near the galaxy center, suggesting that\nthere is not  significant dust absorption in the $H$-band.\n\nIn Fig.~3 we plot a major axis profile extracted from the $H$-band\ndata. From this profile, we see that the disk \nbrightness of UGC~7321 \nfalls off somewhat more steeply on the east side (the righthand side\nof Fig.~3) than on the west side. It is unlikely this can be\nattributed to internal extinction effects (which should be small \nat NIR wavelengths), or to\nflatfield errors (which are quite small in our $H$-band images), hence this\nsuggests a slight lopsidedness in the stellar disk of UGC~7321.\n\nArrows overplotted on Fig.~3 denote ``zones'' between which the slope of the\nbrightness profile change fairly abruptly.  \nThese zones can also be seem from visual inspection of\nboth our NIR and the optical images and are an indication that the\ndisk surface brightness of UGC~7321 deviates somewhat from a smooth,\nexponential distribution (see also Sect.~2.2.3). This makes it\ndifficult to characterize the $H$-band major axis profile in terms of\na single exponential scale length. \nAs shown by Matthews \\& Gallagher (1997),\nthis phenomenon is common among extreme late-type spiral galaxies,\nalthough the effect is generally lost if the light profile is\nazimuthally averaged. Matthews \\& Gallagher (1997) suggested these\nsurface brightness zones \nmay be a signature of limited dynamical evolution in some of the\nsmallest and faintest spirals\n(see also Sect.~2.3.3). \n\n\\subsection{Optical CCD Imaging and Photometry}\n\n\\subsubsection{Data Acquisition}\nWe obtained photometrically calibrated $B$- and $R$-band\nCCD imaging observations of UGC~7321 in June 1997 using the 3.5-m\nWIYN\\footnote{The WIYN Observatory is a joint facility of the\nUniversity of\nWisconsin-Madison, Indiana University, Yale University, and the\nNational Optical Astronomy Observatories.} telescope\nat Kitt Peak. In addition, we obtained\nnarrow-band H$\\alpha$ and uncalibrated $B$-and $R$-band observations of the\ngalaxy with the WIYN telescope in April 1997.\n\nThe direct imaging camera on the WIYN telescope employs a thinned STIS\n2048$\\times$2048 CCD with \\as{0}{2} pixels, yielding a field-of-view of\n$\\sim$7$'$ per side. The CCD gain was 2.8~$e^{-}$ per ADU, and\nreadnoise was $\\sim$8~$e^{-}$\nper pixel rms. Exposure times were 900$s$ in $B$, 600$s$ in $R$, \nand 1500$s$ in H$\\alpha$. Seeing during both the April and\nJune observations\nwas $\\sim$\\as{0}{6} FWHM. The WIYN telescope takes advantage of the\nbest seeing on Kitt Peak through a combination of thermal controls,\nactive optics, and dome ventilation systems. This,\ncombined with the  small pixel scale of the WIYN CCD\nimaging camera, makes WIYN an ideal facility\nfor obtaining the data necessary for detailed studies of disk \nstructure and color gradients of galaxies.   \n\nOur CCD images were reduced using standard\nIRAF tasks. Individual frames were overscan corrected\nand bias subtracted in the usual manner. Flatfielding was accomplished\nby dividing by a mean of 5 dome flats taken in the appropriate\nfilter. The WIYN\nCCD contains a bad column which was corrected in our images by\ninterpolation from the two adjacent columns.\n\nBoth the April and the June observations were obtained during dark time.\nPhotometric calibration  for the June run was accomplished by\nmonitoring two standard star fields from Landolt (1992) at two\ndifferent air masses and then applying standard color term corrections\nto convert from WIYN natural magnitude system to the Landolt\nsystem. The total formal dispersion in our WIYN magnitude zero points\nis $\\pm$0.03 magnitudes; the validity of our transformations\nwas  checked via comparisons with\nother recent WIYN photometric solutions. Errors in the colors are\napproximately $\\pm$0.02 magnitudes. Sky brightnesses during our\nobservations were $\\mu_{B,sky}\\sim$22.1~mag~arcsec$^{-2}$ and \n$\\mu_{R,sky}\\sim$19.8~mag~arcsec$^{-2}$.\n\nOur $R$-band image of UGC~7321 is shown in Fig.~4 \\& 5. Note the very\ndiffuse, low\nsurface brightness (LSB) appearance of the disk in spite of its edge-on\ngeometry. Several background\ngalaxies are clearly visible through the disk (e.g., Fig.~6), implying\ninternal extinction is low.\n\n\\subsubsection{H$\\alpha$ Emission}\nOur continuum-subtracted H$\\alpha$ image of UGC~7321 is shown in\nFig.~7.  Due to cirrus, we were unable\nto obtain a flux calibration for this frame. \nHowever, qualitatively, this image shows that H$\\alpha$ emission is\npresent out to radii of at least $r=\\pm$\\am{2}{5} (where $r$ is the\ndistance from the galaxy center measured along the disk major axis in\nthe plane of the sky).\n\nWe see in Fig.~7 that the bulk of the \nH$\\alpha$ emission in UGC~7321 is\nconfined to a region near the disk midplane roughly 4$''$ thick,\nalthough diffuse emission is clearly visible at $z$-heights of up to\n$\\pm8''$ at galactocentric radii $|r|\\le$\\am{1}{0}.\nWe note however that there is a relative\ndearth of emission within a region offset 8$''$ \nto the east of the disk center (Fig.~8).\nThis ``gap'' is $\\sim 8''$ wide and contains faint plumes\nof emission extending\nto $z\\ge\\pm$7$''$ out of the galaxy plane. We also see marginal\nevidence of several other \nthin filaments which are more extended in $z$-height.\nDeeper narrow-band \nimages are clearly needed to establish whether the central\nH$\\alpha$ feature could be a\nsignature of blowout from the disk, and whether UGC~7321 may indeed contain a\ndiffuse, ionized medium extended to large $z$-heights, analogous \nto that seen in\nmore luminous edge-on spirals (cf. Dettmar 1995 and references therein). \n\n\n\\subsubsection{Surface Photometry}\nWe performed surface photometry on UGC~7321 using routines from\nthe IRAF STSDAS analysis package. With the ``ellipse'' program,\nwe fit a series\nof 9 concentric ellipses to the UGC~7321 images. Position angle\nand ellipticity of the ellipses were determined from the outermost\ngalaxy isophotes and were kept fixed throughout the\nfitting. The galaxy center was chosen to be the position of  peak\nbrightness; this location was identical in the $B$ and $R$ (and $H$)\nframes.\nForeground stars, cosmic rays, and background galaxies \nwithin the aperture were removed via background interpolation \nusing the IRAF task\n``imedit''. Sky values were determined by measuring the sky counts\nin each of several rectangular apertures at various locations on the image\n(see Matthews \\& Gallagher 1997).\n\nBecause UGC~7321 is so close to edge-on, elliptical apertures are not\nideal fits to the galaxy isophotes. Nonetheless, these apertures allow us\nto measure  aperture magnitudes and colors for UGC~7321, and\npermit an estimate of the mean exponential scale length of its\ndisk.  Although the\nellipticity of the fitted isophotes is an unreliable method of\ndetermining the\ninclination of near edge-on galaxies, our derived value\n($i\\approx88^{\\circ}$)\nis consistent with the inclination we\nestimate from \nthe slight asymmetry of the dust features along the galaxy major axis.\n\nOur derived photometric parameters for UGC~7321\nare given in Table~1. Errors for the  aperture magnitude \nmeasurements were computed\nfollowing Matthews \\& Gallagher (1997) and take into account sky,\nflatfield and Poisson errors, as well as the scatter \nin the photometric solution. Large-scale \nflatfield errors are the dominant source\nof uncertainty; the maximum amplitude of the flatfield variations was\n$\\sim$1\\% of sky in both $B$ and $R$.  \nOur aperture magnitudes are in good\nagreement with previous photoelectric values reported by Tully (1988;\n$m_{B}$=13.86) and de Vaucouleurs \\& Longo (1988; $m_{R}$=12.99).\n\nFig.~9 shows our\nazimuthally averaged $B$-band radial surface brightness profile of \nUGC~7321. Although\nwe noted in Sect.~2.1 that the disk of UGC~7321 is slightly lopsided\nand shows deviations from a perfect exponential, an\nazimuthally-averaged brightness profile permits determination of a\n``mean'' scale length which is useful for offering a global\ncharacterization of the size scale of the disk.\n\nBecause UGC~7321 is viewed close to edge-on, one must take into account\nprojection effects in analyzing the projected light profile and in\nderiving an exponential scale length.\nFor a disk with an exponential radial brightness distribution viewed\nexactly edge-on ($i=90^{\\circ}$), \nthe projected radial brightness profile\nalong the galaxy midplane ($z$=0) is\nexpressed as \n\\begin{equation}\nL(r)=L_{0}(r\/h_{r})K_{1}(r\/h_{r})\n\\end{equation}\n\\noindent where $K_{1}$ is the\nmodified Bessel function of first order \n(e.g., van der Kruit \\& Searle 1981). At\nsmall radii ($r\/h_{r}<<1$), this expression can be approximated as\n\\begin{equation}\nL(r)\\approx L_{0}[1 + (r^{2}\/2h_{r}^{2}){\\rm ln}(r\/2h_{r})].\n\\end{equation}\n\\noindent Note this implies  a slight\nflattening of the light profile will be observed at small radii\ncompared with a simple, unprojected $e^{-r\/h_{r}}$ function.\nAt large radii ($r\/h_{r}>>1$), one can write\n\\begin{equation}\nL(r)\\approx\nL_{0}(\\pi r\/2h_{r})^{1\/2}{\\rm exp}(-r\/h_{r})\n\\end{equation}\n\\noindent (see van der Kruit \\& Searle 1981). Because of the\n$\\sqrt{r}$ term, this produces a slightly less steep light profile\nthan a pure exponential function with the same scale length. As a result, a\nscale length derived from fitting a simple, unprojected \nexponential profile to an edge-on disk will be \noverestimated. At $r=2$-$3h_{r}$ this effect is $\\sim$10\\% (e.g., van\nder Kruit \\& Searle 1981).\n\nTo derive the scale length for UGC~7321, we have fitted a model\nprojected exponential light profile to our data. Our\nmodel is overplotted on Fig.~9. For the purpose of deriving a scale length,\nthe slight deviation of UGC~7321 from the edge-on ($i=90^{\\circ}$) case is not\nsignificant. From our fit we derive a scale length of\n$h_{r,B}$=44$''\\pm2$ \n(compared with $h_{r,B}$=51$''\\pm5''$ we derive\nfrom simply fitting the function\n$L(r)=L_{0}e^{-r\/h_{r}}$ to the disk at intermediate $r$).\n\nA comparison between the data and our model (Fig.~9) shows that in spite of\nlacking a bulge component, UGC~7321 exhibits\na light excess at small radii compared with the prediction\nof an exponential disk. \nThe $B$-band {\\it measured} central surface brightness of UGC~7321\n(before correction to a face-on value) is 21.6~mag~arcsec$^{-2}$,\nwhile that predicted from extrapolation of the exponential fit to\nsmall $r$ is $\\sim$0.35 magnitudes fainter.  Interestingly, this\nexcess is even more pronounced in the $R$-band and \nappears to correspond to a distinct region in the color map of\nUGC~7321 (see Sect.~3).\nThis strengthens the suggestion that \nUGC~7321 appears to have a\nmulti-component disk that is more complex than a simple,  pure\nexponential (see also Sect.~2.1).\nIn addition, we note that \nat $r>120''$ our observed light profile falls off faster than the\nprojected exponential \nmodel, suggesting the stellar disk of UGC~7321 \nmay be truncated (cf. Barteldrees\n\\& Dettmar 1994). However, this latter trend should be confirmed with deeper\nobservations on a wide-field CCD where sky subtraction and\nflatfielding can be accomplished with greater accuracy.\n\n\n\\subsubsection{Discussion: The LSB Nature of UGC~7321}\nFor an optically thin \ndisk which is exponential in both the $r$ and $z$ directions,\nand which is observed inclined at 90$^{\\circ}$,  \nthe face-on central surface brightness will be\n$h_{z}\/h_{r}$ times the observed edge-on value. Here  $h_{z}$ is the\nexponential scale height (cf., van der Kruit \\& Searle\n1981). Transformations from observed surface brightness values to face-on\nvalues at other radii may be derived from Equations 1-3\nabove. \nUsing a\nthin-disk approximation (see the Appendix) we have derived the additional\ncorrections\nto these values required for\nthe case where a is disk observed at an inclination slightly less than\n90$^{\\circ}$. \nAfter also taking into account internal\nextinction corrections (see Sect.~3.2.2), we then find that if\nprojected to face-on, the disk of UGC~7321 would have an {\\it observed} \n$B$-band central surface brightness of $\\sim$23.4~mag~arcsec$^{-2}$, \nan extrapolated central surface\nbrightness of $\\sim$23.8~mag~arcsec$^{-2}$, and a mean total disk surface\nbrightness $\\bar\\mu_{B}\\sim$27.6~mag~arcsec$^{-2}$.  \nThus UGC~7321 is a very LSB galaxy, and much of its disk \nwould likely be nearly invisible if viewed\ncloser to face-on. As shown below, the internal extinction in UGC~7321\nappears to be quite low, thus its LSB appearance must\nresult from a rather low current star formation rate.\n\nFurther\nevidence that the seemingly ``anemic'' nature of UGC~7321 is due to\nminimal current star\nformation comes from comparing its blue luminosity to its\nfar-infrared \nluminosity. Using the {\\it IRAS} 60$\\mu$m and 100$\\mu$m fluxes for\nUGC~7321 from\nthe NED database and Sage (1993), respectively, \nand following the prescription of\nRice {\\it et al.}\\ (1988), we derive $L_{FIR}$=7.8$\\times10^{7}$\\mbox{${\\cal L}_\\odot$}, a\nvalue nearly two orders of magnitude fainter than the mean for Scd\/Sd\nspirals in the UGC catalogue (Roberts \\& Haynes 1994). Using our blue\nluminosity (corrected for internal extinction), we find that \n$L_{FIR}\/L_{B}\\approx$0.08 for UGC~7321. A comparison with Rice {\\it et al.}\\ (1988)\nreveals that such low $L_{FIR}\/L_{B}$ ratios are\ncommonly found only in two classes of galaxies: (1) \nold, red early-type spirals, dE's and S0's with few young stars; (2) \ngas-rich, extreme late-type spirals with \ndiffuse, extremely LSB disks. \nExamples of the latter group of objects include the\nnearby galaxies NGC~4395 (Sd~IV), NGC~45 (Sdm~IV), and IC~2574\n(Sm~IV-V). All three of these galaxies have $L_{FIR}\/L_{B}\\le$0.10,\n$\\frac{{\\cal M}_{HI}}{L_{B}}$$>$1, blue optical colors, and highly transparent disks (e.g.,\nSandage 1961; Matthews {\\it et al.}\\ 1999a). \n\n\n\\section{Color Maps}\nUsing our $B$- and $R$-band WIYN data, we have produced a $B-R$ color\nmap of UGC~7321 (Fig.~10). A variety of structure is evident in this\nmap, including significant \n$B-R$ color gradients in both the vertical and radial\ndirections. We discuss these features in detail below.\n\nThe global $B-R$ color of UGC~7321 is 0.99, as \nmeasured within the outermost observed\nisophote \n(25.2~mag~arcsec$^{-2}$ in $B$ before correction for inclination),\nand after correction  for Galactic foreground extinction. \nThis $B-R$ color\nis a typical value for a normal, late-type spiral (cf.\nLauberts \\& Valentijn 1988; de Blok, van der Hulst, \\& Bothun 1995).\nThis suggests that in spite of its edge-on geometry, the disk of\nUGC~7321 does not suffer severely\nfrom internal reddening due to dust. We return to \nthis issue below.\n\n\\subsection{The Nuclear Region of UGC~7321}\n\nNear the\ncenter of the UGC~7321 disk, our color map reveals  a small, very red\n($B-R\\approx$1.5)\nnuclear feature, only a few arcseconds across (Fig.~10). This feature\nis offset $\\sim$5$''$\nto the east of the disk center as determined from the brightness peak\nof the $R$- and $B$-band images. One possibility is \nthat we are seeing the signature of an embedded nucleus. However,\nalthough compact star cluster nuclei are common in late-type\nlow-luminosity spirals with diffuse, LSB disks (Matthews \\& Gallagher 1997;\nMatthews {\\it et al.}\\ 1999a),\nwe do not see any direct evidence of such a feature in UGC~7321\nin either our WIYN images\nor in images obtained with the {\\it Hubble Space Telescope}\n(Matthews {\\it et al.}\\ 1999b). \nNonetheless, \nthe location of the compact red feature in the\n$B-R$ color map is near to that of \nthe peculiar H$\\alpha$ emission features shown in Fig.~8.\nThe optical longslit spectrum of UGC~7321 obtained by\nGoad \\& Roberts (1981) also shows a possible kinematic disturbance in \nthe H$\\alpha$\nemission at this location. These are hints \nthat  some interesting physical processes\nare at work near the center of the UGC~7321 disk. Further\ninvestigation of \nthis region  via high resolution\nspectroscopy, deep H$\\alpha$ imaging, and NIR colors \nmay be fruitful.\n\nSurrounding the compact red feature, our color maps reveal a more\nextended red region ($B-R\\sim$1.2), visible to $r\\approx\\pm20''$ on\neither side of the disk center and showing a rather abrupt\nboundary (see also below). Intriguingly, the extent of this region \ncorresponds very\nclosely to the region over which we observe a \nlight excess over a pure exponential disk fit\n(Sect.~2.2.3). This raises the possibility that this red central region\nmight possibly represent an\nancient central starburst,  the core\nof the original protogalaxy, or perhaps even a \nkinematically distinct disk subsystem analogous to\nthe bulge of normal spirals.\n\n\n\\subsection{Radial Color Gradients}\n\n\\subsubsection{General Trends}\n\nCutting into the red central region discussed above are thin blue bands of\nstars visible along the  midplane\nof the galaxy. These bands grow both thicker and bluer with increasing\ndistance from the galaxy center.\nAt $|r|=$20$''$, this layer has $B-R\\approx$1.05, \nreaching $B-R\\approx$0.85 at $|r|$=\\am{1}{0}, \n$B-R\\approx$0.80 at $|r|=$\\am{1}{5},\n$B-R\\approx$0.45$\\pm0.10$ at $r$=\\am{2}{7}, and $B-R\\approx$0.55$\\pm$0.10\nat $r=-$\\am{2}{7}. Thus, from the nuclear region (where $B-R\\approx\n1.5$) to the outer disk edge we\nsee a total\n$B-R$ color change of up to 1.05 magnitudes along the major\naxis of UGC~7321. This is illustrated in Fig.~11$a-d$,\nwhich shows $B-R$ color\nprofiles of UGC~7321 extracted along the major axis, as well as\n\\as{1}{6} north (through the red nuclear feature),\n\\as{1}{2} south, and \\as{4}{6} south  \nrelative to \nthe major axis, respectively. The extracted profiles were averaged\nover  12-pixel-wide\nstrips and then smoothed, hence the red color of the compact red\ncentral disk feature\nis slightly subdued in the profiles shown in Fig.~11. Because  we are\ninterested in accurately measuring colors in the faintest regions of the outer\ndisk, we extracted our color profiles from our\nour uncalibrated $B$ and\n$R$ images, which have slightly superior flatfields \nto our photometrically-calibrated  data; maximum amplitude \nvariations of the flatfields are\n$\\sim$1\\% of the sky in\n$B$ and $\\sim$0.5\\% of the sky in $R$. We then  used comparisons\nbetween high\nsignal-to-noise inner disk regions\nto calibrate the absolute colors from  the\nphotometrically-calibrated images. The maximum \nuncertainties expected from the\ncombination of sky subtraction and flatfield errors are overplotted as\ndotted lines on Fig.~11.\nAlthough signal-to-noise in the outer disk is low, the profiles\nextracted at all four positions parallel to the major axis\nshow regions with $B-R\\le$0.60, suggesting these blue outer disk\ncolors are real.\n\nFinally, we draw attention to the presence of a faint, thicker, but highly \nflattened disk\nof unresolved stars visible\nin our color map surrounding the UGC~7321 disk at\n$|r|\\le$\\am{2}{0}. This component has $B-R\\approx$1.1 and \nshows little change in\ncolor with galactocentric distance (Fig.~12). Based upon the\nobservations of Galactic globular clusters (see Secker 1995), old,\nmetal-poor stellar populations are expected to have $B-R>$0.8, hence\nthe highest $z$-height stars in UGC~7321 appear to represent an ``old disk''\npopulation (see also below).\n\n\\subsubsection{Gauging the Effects of Dust}\n\nBluing as a\nfunction of increasing galactocentric distance is a well-known feature of\nspiral galaxy disks (e.g., de Jong 1996). De Jong\n(1996) has  demonstrated through Monte Carlo simulations\nthat in face-on spirals, this observed radial bluing  generally\ncannot be accounted for by dust. He shows that for\nrealistic models, dust creates color gradients  of less than 0.3\nmagnitudes in $B-R$, and argues  instead\nthat the observed radial color changes in spirals\nare indicative of star formation progressing radially outwards\nin disks over time, leading to stellar age and metallicity gradients. \nThis picture is consistent with a number of \nsemi-analytic galaxy formation models where galaxy disks are built ``from the\ninside out'' (e.g., White \\& Frenk 1991;\nMo {\\it et al.}\\ 1998). Nonetheless, the radial color gradient we observe in\nUGC~7321 ($\\Delta B-R\\sim1$) is significantly larger than typical\ngradients in the de Jong sample (where typically $\\Delta B-R\\sim$0.6\nmagnitudes, even when bulge light is included). Moreover, in most\nspirals observed\nedge-on, radial color gradients are generally  found to be small\n(e.g., Sasaki 1987; Aoki {\\it et al.}\\ 1991; Wainscoat, Freeman, \\& Hyland \n1989; Bergvall\n\\& R\\\"onnback 1995) or negligible (e.g., Hamabe {\\it et al.}\\ 1980; Jensen \\&\nThuan 1982; de Grijs 1998).\n\nUnfortunately, in edge-on spiral galaxies, \ncolor gradients become more difficult to interpret physically,\nparticularly since internal\nextinction can be quite significant near the galactic plane. The\nnet effect  can be both an alteration in the galaxy luminosity profile,\nand a reddening of the observed optical \ncolors which can vary as a function of $r$ and $z$. Therefore in order\nto determine what fraction of the color gradients\nwe observe in UGC~7321  are due to true\nage and\/or metallicity\ngradients, we must include an assessment \nof  the role of dust in this galaxy.\n\nIn\npractice,  accurately quantifying the effects of dust on the observed\ncolors and luminosity distribution in a galaxy is a complicated\nproblem (e.g., de Jong 1996; \nXilouris {\\it et al.}\\ 1999). \nMoreover, standard assumptions about the nature and\ndistribution of dust in normal giant spirals are unlikely to be applicable to \nlow-metallicity LSB galaxies  (cf. Han 1992).  As a result, the effects\nof internal extinction in a given galaxy are ideally derived \nusing a combination of sophisticated\nradiative transfer models and empirical measurements (e.g., Xilouris\n{\\it et al.}\\ 1999).  \nSuch modelling beyond the scope of the\npresent work,  so here we derive some fundamental constraints on the\neffects of dust in UGC~7321 using a very simple model.\n\nAlthough some authors have argued that the internal extinction in LSB\ngalaxies may be almost negligible (e.g., McGaugh 1994;\nTully {\\it et al.}\\ 1998), these\nclaims have only rarely been tested in edge-on systems (e.g., Goad \\&\nRoberts 1981; Kodaira \\&\nYamashita 1996; Bergvall \\& R\\\"onnback 1995; Karachentsev 1999). \nWe have already mentioned several lines of evidence\nthat the internal extinction in UGC~7321 is low (regularity of the \n$H$-band\nisophotes; visibility of background galaxies through the disk; low\n$L_{FIR}\/L_{B}$ ratio; correspondance between the galaxy center in $H$-\nand $B$-band). Nonetheless, visible inspection of our WIYN images shows\nthe galaxy is clearly not completely devoid of dust, and a number of individual\nclumps (possibly molecule-rich dark clouds)\ncan be seen in our optical images (e.g., Fig.~5; see also Matthews\n{\\it et al.}\\ 1999b). In spite of this, \nthe fact that we resolve many of these individual dust clumps instead of\nseeing a uniform dust lane \nimplies that  the disk of UGC~7321 is not optically thick.\n\nAnother test of the optical thickness of the UGC~7321 disk comes from\na comparison between\nits optical rotation curve (derived from longslit spectroscopic\nmeasurements of H$\\alpha$ emission from \\mbox{H\\,{\\sc ii}}\\ regions), with a\nrotation curve derived \nfrom \\mbox{H\\,{\\sc i}}\\ measurements. From optical longslit measurements,\nGoad \\& Roberts (1981) found UGC~7321 \nto have a slowly rising, solid-body rotation curve throughout much of\nits stellar disk. If a galaxy disk is optically thick,\nthe\nrotation curve may appear solid body regardless of its intrinsic shape,\nas an artifact of one's inability to observe \\mbox{H\\,{\\sc ii}}\\ regions at small\ngalactocentric radii\n (Goad \\& Roberts 1981; Byun 1993). However, a rotation curve of\nUGC~7321 derived from \\mbox{H\\,{\\sc i}}\\ aperture synthesis measurements (Sect.~4.2)\nconfirms that the slow rise of the rotation curve throughout\nthe stellar disk \nis indeed intrinsic (see Fig.~19, discussed below). \nOnce again we conclude the disk of UGC~7321 is not optically thick.\n\nDust affects the color and luminosity profiles of galaxies through\nboth the scattering and absorption of optical photons. \nHowever, the maximum color change (i.e., maximum reddening) occurs\nin the pure extinction case. In addition, Byun, Freeman, \\& Kylafis \n(1994) have\nshown that scattering effects become decreasingly important with\nincreasing inclination (see also Bianchi, Ferrara, \\& Giovanardi\n1996). \nSince here we are primarily\ninterested in estimating the effects of dust on the color\ngradient, and since we are\nin the optically thin regime, we can simplify the problem by\nconsidering  a  model of extinction due to a foreground dust\nscreen (see Disney, Davies, \\& Phillips 1989). \nFor a given amount of dust, the Screen model produces\nmore reddening than  a more realistic model of stars mixed with\ndust, hence it establishes an upper limit to $E(B-R)$ as a\nfunction of radius.\n\nTo evaluate $E(B-R)$, we begin by attempting to attribute \nas much of our observed color\ngradient as possible to dust.\nWe further assume that, to first order,\ndust extinction in the $H$-band is negligible (e.g., Fig.~2). In the\ncase of zero intrinsic radial color gradient,\na dust-free version of UGC~7321 would\nexhibit no radial color gradient in $R-H$, hence we can use the\nchanges we do observe in the $R-H$ color to estimate the amount of\nextinction at a given radius.\nWe do not attempt to reproduce the  clumpy nature of the dust in our\nmodel (which again, would decrease any reddening effects), \nbut assume the dust distribution is roughly exponential in\nthe radial and vertical directions (e.g., Wainscoat, Freeman, \\&\nHyland 1989; Xilouris {\\it et al.}\\ 1999). In this case the\nextinction along an interval from $r$ to $\\delta r$ can be\nexpressed as\n\\begin{equation}\n\\delta A_{\\lambda,i}(r,z)=\\lbrace\\begin{array}{r}\nA_{\\lambda,0}e^{(-r\/h_{r,d}-|z|\/h_{z,d})}\\delta r~~~~~~~~~~r\\le R_{max} \\\\ \n0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~r>R_{max} \n\\end{array} \n\\end{equation}\n\\noindent where $A_{\\lambda,0}$ is the absorption in magnitudes per\nunit length at a\ngiven wavelength, $r=R_{max}$ corresponds to the edge of the\nstellar disk, $h_{r,d}$ is the scale length of the dust, and $h_{z,d}$\nis the scale height of the dust (see also Wainscoat, Hyland, \\&\nFreeman 1989).\n \nFig.~13 shows a plot of the ``pseudo'' $R-H$ color profile of UGC~7321\n along the major axis interval $r$=0--120$''$. Here the $R$-band data were\nsmoothed to match the resolution of the $H$-band observations.\nBecause our $H$-band data are not photometrically-calibrated, \nwe have chosen an arbitrary\nnormalization such  that our ``pseudo'' $R-H$ color is 0 near\n$r$=80$''$.  In Fig.~13, we see that near $r$=0, \nthe $R$-band light is depressed by roughly 40\\%\nrelative to the $H$-band, corresponding to an $R$-band extinction\n$A_{R,0}\\approx$0.55 magnitudes per kiloparsec. \nAssuming the extinction law of Bouchet\n{\\it et al.}\\ (1985), we derive\n$A_{B,0}$=0.89 magnitudes per kiloparsec. \n\nFrom Fig.~13 we  estimate a \nscale length of the dust distribution of $h_{r,d}\\approx$40$''$, which\nappears consistent with the observed dust distribution in our images. \nAlthough in giant spirals, it is\noften assumed the dust has a similar scale length to the old stellar\ndisk (e.g., Kylafis \\& Bahcall 1987), or even extends beyond it (e.g.,\nXilouris {\\it et al.}\\ 1999),\nthis does not appear to hold in UGC~7321. We see an abrupt end to the\npresence of resolved dust clumps beyond radii $|r|\\approx 80''$ in both\nour WIYN images, and in {\\it Hubble Space Telescope} images obtained\nby Matthews\n{\\it et al.}\\ (1999b). Moreover, background galaxies are readily visible through the\nUGC~7321 disk beyond these radii (Matthews {\\it et al.}\\ 1999b), implying\nextinction has dropped appreciably. For the vertical scale height of\nthe dust, we adopt $h_{z,d}$=\\as{2}{3}, which is half the scale height\nof the old stellar disk (Sect.~3.3; see also Paper~II).\n\nUsing Equation 4, we can now derive an\nextinction-corrected $B-R$ \ncolor profile for UGC~7321 along its major axis (Fig~14$a$) and offset\n\\as{1}{6} to the north (Fig~14$b$).  We see that even  the extinction-corrected\nradial color gradient remains pronounced ($\\Delta B-R\\sim0.80$ mag). This  \ndemonstrates that {\\it a significant fraction of the observed\nradial color gradient in UGC~7321 appears to be due to stellar population\nand\/or metallicity gradients.}\n\n\\subsubsection{Discussion and Interpretation}\n\nThe presence of the observed radial disk \ncolor gradients in UGC~7321 has several\nimplications. Regions with $B-R<0.8$ are too blue to be accounted for\nby old, metal-poor stellar populations (e.g., Secker 1995), but\ninstead must result from a mixture of ages. Without resolving these\npopulations, we cannot uniquely\nconstrain the mix of stars present. However, the bluest\ncolors we reliably measure in UGC~7321 ($B-R<0.6$) are comparable, for\nexample,  to\nyoung regions in the moderate metallicity giant Magellanic galaxy\nNGC~4449 (Bothun 1986) and to the predicted colors of the  \nyounger outer regions of a smoothly evolving galaxy with a high halo spin\nparameter and an age of $\\sim$10~Gyr, as in the models of \nJimenez {\\it et al.}\\ (1998). In either case, it is likely that youthful stars are a\nsignificant contributor to the blue colors of the outer disk of\nUGC~7321 (see also Bell {\\it et al.}\\ 1999). This assertion is strengthened\nby the presence of H$\\alpha$ emission in the outer disk (Fig.~7) and\nthe existence of resolved supergiant stars at these radii (Matthews\n{\\it et al.}\\ 1999b).\n\nWe note that edge-on, superthin LSB galaxies like UGC~7321\nare especially valuable for studies of the outer regions of disks, as\nthey may contain some of the most pristine, unevolved galaxian disk\nenvironments observable in the nearby universe. However, because these disk  \nregions are so optically thin, they \nwould become very difficult to detect in galaxies viewed at\nlower inclinations. For example, if viewed near face-on,\nthe portions of the  UGC~7321 disk where $B-R\\le$0.6  would\nhave an observed $B$-band surface brightnesses of only \n$\\ge$27.6~mag~arcsec$^{-2}$.  Such faintness levels are only rarely\nreliably achieved in  galaxy imaging surveys \ndue to combinations of short integration times,\nscattered light, limited detector fields-of-view, and flatfield\nuncertainties (cf. Morrison, Boronson, \\& Harding 1994; Lequeux {\\it et al.}\\ 1996). \n\nIn spite of the potentially very young outer disk \nregions of UGC~7321, its moderately red central disk (where\n$B-R\\approx$1.2) and its somewhat thicker ``old disk'' \n(where $B-R\\approx$1.1) both indicate that\nUGC~7321 is likely to contain \nstellar populations with ages in excess of 10~Gyr\n(see Jimenez {\\it et al.}\\ 1998). As demonstrated above, accounting for \na reasonable amount of reddening due to dust does\nnot affect this conclusion. The red stellar components in \nUGC~7321 imply that this is not a young \ngalaxy \npresently undergoing its first epoch of star formation. \n\nOur observed colors and color gradients in UGC~7321 are similar to\nthose found in other LSB spirals by de Blok, van der Hulst, \\& Bothun\n(1995; see also Impey \\& Bothun 1997). A consistent model for UGC~7321\nis then that of a disk galaxy which has made stars slowly and is now\nseen to be under-evolved relative to typical giant spirals (see also\nO'Neil {\\it et al.}\\ 1997; Jimenez\n{\\it et al.}\\ 1998). This would account for the low gas metallicity of\nUGC~7321 (Goad \\&\nRoberts 1981),\nthe large \\mbox{H\\,{\\sc i}}\\ mass fraction (Sect.~4), and the low density of stars\n(i.e., the\nlow optical surface brightness disk).\nThus while LSB disks like UGC~7321 may be {\\it\nunevolved} galaxies, we concur with the suggestion of Jimenez {\\it\net al.} (1998) that {\\it LSB galaxies are not\nnecessarily young.}\n\nIn order to preserve its relatively \nstrong radial color gradients over many Gyr, it appears that\nviscous evolution in the disk of UGC~7321 has been minimal.  In giant\nspirals, viscous evolution (which redistributes angular\nmomentum in the disk) is often argued to be responsible for the \nrelatively smooth exponential luminosity profiles of most disk galaxies\n(e.g., Lin \\& Pringle 1987). This process is also expected to\npartially smooth color gradients due to stellar age and metallicity \nchanges with \nradius (e.g., Firmani, Hern\\'andez, \\& Gallagher 1996). \nHowever, such a mechanism may not be able to\nwork efficiently in galaxies like UGC~7321, due to both the low disk surface\ndensity (which considerably extends dynamical timescales) and the \nslowly rising rotation curve [which produces limited shear in the\ninner disk regions\n(Lin \\& Pringle 1987; see also Matthews \\& Gallagher 1997)]. The \ngradients in the $H$-band major-axis profile of the UGC~7321 disk \n(Sect.~2.1) and\nthe deviation of the azimuthally-averaged profile from a pure\nexponential disk at small $r$ (Sect.~2.2.3) may be\nadditional\nsigns that  viscous evolution has been limited in this galaxy compared\nwith normal spirals.\n\nFinally, we note that at intermediate galactic radii, \nthe eastern side of UGC~7321 has a slightly bluer mean color than the\nwestern\nside.\nThis result is insensitive to reasonable sky subtraction and\nflatfield errors. One explanation could be that we \nare seeing the projection of  a spiral arm(s), although it remains\nuncertain how much spiral structure a thin\ndisk like UGC~7321 might actually sustain, since spiral structure is\nbelieved to be an efficient disk heating mechanism (e.g., Lacey 1991\nand references therein). Moreover, at least some self-gravity\nis needed for spiral arm formation, but if the disk of UGC~7321 were\ncompletely self-gravitating, it would be highly unstable to ``firehose''\ninstabilities and hence to \nvertical thickening (Zasov, Makarov, \\& Mikailova 1991; Matthews 1998).\nAn alternative is that\nthere are  asymmetries in the stellar distribution of UGC~7321\ndue to patchy star-formation\n(see also Gerritsen \\& de Blok 1999). \n\n\\subsection{Vertical Color Gradients}\nThe presence of vertical color gradients in galaxies offers\nan important clue toward the dynamical evolution of galaxian disks, since such\ngradients are predicted to occur as a consequence of dynamical heating\nprocesses  (e.g., Just,\nFuchs, \\& Wielen\n1996). Unfortunately, vertical color gradients are very difficult to\nmeasure in most edge-on spirals due to dust, contamination from\nthe bulge component, and the effects of atmospheric \nseeing. To date, only a few\nanalyses of edge-on spirals have uncovered non-negligible vertical\ncolor gradients (e.g., Wainscoat, Freeman, \\& Hyland 1989; Bergvall \\&\nR\\\"onnback 1995), while many workers  have reported such gradients to\nbe very small or negligible  (e.g., Jensen \\& Thuan 1982; de Grijs,\nPeletier, \\& van der Kruit 1997).\n\nFig.~15$a$-$h$ illustrates  the vertical color profiles of the UGC~7321 disk\nat various galactocentric\ndistances.\nAt $r=0$, we see\nvery little $z$ color gradient: $B-R\\sim$1.2 at a range of $z$ values\n(Fig.~15$a$), although a slight asymmetry\nis visible.\nAt $r=6''$ (Fig~15$b$), we  see the addition of a  \npeak with $B-R\\sim$1.45. Note its displacement from the disk\nmidplane; this peak is in \nthe vicinity of the red nuclear feature discussed in Sect.~3.1. \nAt $r$=10$''$ (Fig~15$c$), the vertical color gradient\nis again fairly flat ($B-R\\sim$1.2), with  a very slight bluing visible\nnear $z$=\\as{1}{5}.\nAt $r=$\\am{0}{5} (Fig~15$d$),\nthe high $z$ regions of the disk have nearly the same\ncolor as those at $r$=0 (this is due to the ``old disk'' \nof red stars\ndiscussed above), but at\nsmall $z$ values, the disk has become much bluer than at\nsmaller galactocentric radii.\nFinally, with further increasing radius, while the color at high $z$ continues\nto stay nearly constant, the color at small $z$ grows increasingly\nbluer (Fig.~15$e$-$h$),\nconsistent with the radial bluing observed along the major axis of the\ngalaxy in Fig.~11.\n\nVertical color gradients of the type we observe over much of the disk\nof  UGC~7321 (i.e. redder with\nincreasing $z$ height)\nare predicted to occur due to dynamical\ndisk evolution in which older stellar populations acquire higher velocity\ndispersions over time due to heating processes  (e.g., Fuchs \\& Wielen 1987;\nJust, Fuchs, \\& Wielen 1996).\nUGC~7321 represents one of the\nfew examples  of this\ntype of vertical color structure being directly \nobserved in a galaxy disk. Thus even the most dynamically cold\nexamples of nearby galaxy disks appear to have undergone some dynamical\nheating. Nonetheless, the exact mechanism by\nwhich older stars are dynamically heated and how they are \nredistributed as a function time\nremain uncertain (see review by Lacey 1991) and different mechanisms\nmay operate in different galaxies, depending upon factors like\nenvironment, degree of self-gravity, and the size and number of\nmolecular clouds.\nThe relative simplicity of UGC~7321, and\nits status as a relatively\nunevolved galaxy make this an ideal system to help\nplace important constraints on these issues. We explore this \nfurther in Paper~II.\n\nUsing the extinction parameters derived in Sect.~3.2.2, \nwe can also estimate the effects of dust reddening on our observed\nvertical color gradients. We adopt for a scale height of the dust half\nthe scale height of the old stellar disk (Xilouris {\\it et al.}\\ 1999). \nUsing the stellar scale height of the old disk\nderived in Paper~II, this yields $h_{z,d}$=\\as{2}{3}.\nFig.~16  shows an  example of one of our vertical color profiles\ncorrected for dust reddening. We see that at the galactocentric radii\nwhere the vertical color gradients become the strongest, correction\nfor dust reddening acts  to slightly increase $\\Delta(B-R)$ along\nthe vertical direction.\n\n\\section{HI Observations}\n\\subsection{HI Pencil Beam Mapping}\nIn June 1997 we used\nthe Nan\\c{c}ay\\ Decimetric Radio Telescope to obtain a\n7-point pencil-beam map of UGC~7321 in the  21-cm line of neutral\nhydrogen (\\mbox{H\\,{\\sc i}}). At the declination of\nUGC~7321, the Nan\\c{c}ay\\ Radio Telescope has a FWHM beam size of\napproximately 4$'$E-W$\\times$23$'$N-S. Other\ndetails regarding the  Nan\\c{c}ay\\\ntelescope may be found in, e.g., Matthews, van Driel, \\& Gallagher\n(1998).\n\nUGC~7321 was observed  at 7 different telescope pointings, including a\nposition corresponding to the optical center of the galaxy, and\npositions offset 2$'$, 4$'$, and 6$'$ east and west of center, respectively.\nThe observations\nwere obtained in total\npower (position-switching) mode using consecutive pairs\nof two-minute on- and two-minute\noff-source integrations. Total integration times were 2-3 hours\nat each pointing. For these observations, the\nautocorrelator was divided into two pairs\nof cross-polarized receiver banks, each with 512 channels and a 6.4~MHz\nbandpass. This yielded a channel spacing of 2.6~\\mbox{km s$^{-1}$}, for an effective\nvelocity resolution of $\\sim$3.3~\\mbox{km s$^{-1}$}.\n\n\nFortuitously, UGC~7321 is oriented along nearly an E-W line, so the\nNan\\c{c}ay\\ telescope provides sufficient spatial resolution along\nthis direction to obtain\ncrude \ninformation about the \\mbox{H\\,{\\sc i}}\\ distribution and kinematics of the\ngalaxy.\nThe individual spectra obtained at each telescope pointing are shown in\nFig.~17 and\nthe  resulting global spectrum in Fig.~18. \nNote the global profile appears quite symmetric. Table~1 \ngives our derived global\n\\mbox{H\\,{\\sc i}}\\ parameters. Errors were computed following Matthews, van Driel,\n\\& Gallagher (1998). \nOur values are in good agreement\nwith the recent single-dish \\mbox{H\\,{\\sc i}}\\ observations of UGC~7321 \nobtained by Haynes {\\it et al.}\\ (1998)\nusing the Green Bank 140-ft telescope.\n\nOur Nan\\c{c}ay\\ observations show that\nthe \\mbox{H\\,{\\sc i}}\\ gas in UGC~7321 clearly extends beyond the\noptical galaxy, confirming the results  of Hewitt, Haynes, \\& Giovanelli\n(1983). Significant flux is detected at the pointings both 4$'$E\nand 4$'$W of the galaxy, giving a lower limit for the for \\mbox{H\\,{\\sc i}}-to-optical\ndiameter ratio of $D_{HI}\/D_{opt}\\ge$1.25. In addition, we detect a\nsmall amount of \\mbox{H\\,{\\sc i}}\\ flux\n($\\sim$1.1~Jy~\\mbox{km s$^{-1}$}) in our observation 6$'$W of the galaxy\ncenter. From the Nan\\c{c}ay\\ data alone, it\nis difficult to assess whether this flux is due to a sidelobe\ncontamination or real extended emission, \nsince the strength of the Nan\\c{c}ay\\ telescope sidelobes varies\nsignificantly with the\nhour angle of the source (Guibert 1973).\n\n\\subsection{HI Aperture Synthesis Data}\nAfter our Nan\\c{c}ay\\ observations were obtained, we had the opportunity to\nexamine \\mbox{H\\,{\\sc i}}\\ aperture synthesis data\nof  UGC~7321 obtained by Rots, Roberts, \\& Goad\n(private communication) using the Very Large Array (VLA).  These \nobservations were obtained in C array in 1981 using 18 antennas,\nhence they do not achieve the same sensitivity limits as \nmodern VLA data, but they still offer a useful complement to \nour Nan\\c{c}ay\\\ndataset. Thirty-one independent velocity channels were used for the\nobservations, with a velocity resolution of $\\sim$10.3~\\mbox{km s$^{-1}$}\\ per\nchannel. The\nFWHM\nbeamwidth was $\\sim$12$''$. \n\nUsing the VLA data, we derived a\nposition-velocity diagram for  UGC~7321. By assuming  that at\neach point along the major axis the maximum velocity traces the\nrotation, we have derived the rotation curve shown in Fig.~19. \nThis figure confirms\nthe slowly-rising, solid body nature of the rotation of UGC~7321\nthroughout\nits stellar disk, as was first seen in the optical rotation curve of\nGoad \\& Roberts (1981). Although beam-smearing may be\nexpected to slightly decrease the amplitude of our derived \\mbox{H\\,{\\sc i}}\\ rotational\nvelocities, this effect is not expected to exceed a few kilometers per second,\nor to be significant beyond 2--2.5 beam diameters from the disk center\n(e.g., Swaters 1999), \nhence it cannot explain the shallow shape of the inferred rotation curve.\nFinally, we note that Fig.~19 suggests that the rotation curve\nof UGC~7321 does not begin to flatten until near the edge of the\nstellar disk. \nThis is consistent\nwith UGC~7321 being a galaxy whose dynamics are \ndominated by a dark halo even at small galactocentric radii \n(e.g., Matthews 1998). However,\ndeeper, more sensitive \\mbox{H\\,{\\sc i}}\\ aperture synthesis measurements would be\nvaluable for further constraining the detailed shape of the outer\nrotation curve.\n\nThe integrated \\mbox{H\\,{\\sc i}}\\ map of UGC~7321 derived by Rots, Roberts, \\& Goad \n(private communication)\nconfirms a slight extension of the \\mbox{H\\,{\\sc i}}\\ on the west side of\nthe galaxy compared with the east side, consistent with our Nan\\c{c}ay\\\ndata.\nIt also permitted us to\nmeasure an \\mbox{H\\,{\\sc i}}\\ diameter of $D_{HI}\\sim$\\am{7}{1} for UGC~7321 at the\nlimiting  \\mbox{H\\,{\\sc i}}\\ column density  of the observations ($N_{HI}\\sim\n7\\times10^{20}$~atoms~cm$^{-2}$). Thus\n$D_{HI}\/D_{opt}\\ge$1.2, in agreement with our Nan\\c{c}ay\\ lower limit.\n\n\\subsection{Discussion}\nIt is somewhat difficult to accurately\ncompare the \\mbox{H\\,{\\sc i}}\\ extent of UGC~7321 with other\nspirals since the commonly-quoted\n$D_{HI}\/D_{25}$ ratio (where $D_{25}$ is measured at a\nface-on-corrected isophote) in not particularly meaningful for a galaxy like\nUGC~7321, whose face-on central surface brightness is only\n$\\sim 23.4$~mag~arcsec$^{-2}$. Moreover, we cannot unambiguously\ntranslate our limiting observed \\mbox{H\\,{\\sc i}}\\ column density into an \\mbox{H\\,{\\sc i}}\\\nsurface density due to projection effects, thus it is difficult to\naccurately measure $D_{HI}$ at some canonical surface density (e.g.,\n1~\\mbox{${\\cal M}_\\odot$}~pc$^{-2}$). \n\nIn terms of optical scale length,\nthe \\mbox{H\\,{\\sc i}}\\ in UGC~7321 extends to at least 5.5$h_{r}$.  \nAlthough this value may increase  with more sensitive VLA\nobservations, such \nan extent is still quite normal for both high and\nlow surface brightness late-type spirals\nhaving similar rotational velocities to UGC~7321 \n(see Fig.~8 of de Blok, McGaugh, \\& van der Hulst 1996).\n\nIf we assume both the stellar and the \\mbox{H\\,{\\sc i}}\\ disks of UGC~7321 are\noptically thin, and hence both similarly enhanced in surface\nbrightness due to their edge-on projection, then we see that the\nmaximum \\mbox{H\\,{\\sc i}}\\ extent of the disk relative to the maximum optical extent\nis typical for Sd spirals (e.g., Hewitt, Haynes, \\& Giovanelli\n1983).  We can more directly compare with two other Sd superthins\npreviously mapped\nin \\mbox{H\\,{\\sc i}}:\n$D_{HI}\/D_{opt}\\sim$1.2 was also found by Cox {\\it et al.}\\ (1996) for the\nsuperthin spiral UGC~7170 and by Abe {\\it et al.}\\ (1999) for the superthin\nIC~5249, both at similar limiting \\mbox{H\\,{\\sc i}}\\ column densities and optical\nsurface brightnesses. \n\nIgnoring internal extinction effects, \nthe $\\frac{{\\cal M}_{HI}}{L_{B}}$\\ ratios of UGC~7321, UGC~7170, and IC5249 are\n2.6, 3.4, and 2.3 (in solar units), respectively, and the superthin\nspiral ESO~146-014 studied by Bergvall \\& R\\\"onnback (1995) has\n$\\frac{{\\cal M}_{HI}}{L_{B}}$$\\sim$2.8. If we assume all have internal extinctions similar to\nUGC~7321 ($A_{B,i}\\sim0.89$ mags; Sect.~3.2.2), the implied $\\frac{{\\cal M}_{HI}}{L_{B}}$\\\nratios are still significantly\nlarger than typical values for normal Sd spirals derived by Roberts \\&\nHaynes (1994; $\\frac{{\\cal M}_{HI}}{L_{B}}$$\\sim$0.63),\nalthough not as extreme as some of the unevolved dwarf galaxies\nstudied by van Zee {\\it et al.}\\ (1995)\nor some of the extreme late-type LSB spirals in the sample of  Matthews\n\\& Gallagher (1997) which have $\\frac{{\\cal M}_{HI}}{L_{V}}$\\ as high as 10 \n(see also Salzer {\\it et al.}\\ 1991; \nMatthews, van Driel, \\& Gallagher 1998).\n\nVan der Hulst {\\it et al.}\\ (1993) suggested that low gas surface densities\nare responsible for the low star-formation efficiencies in LSB\ngalaxies (see also de Blok, McGaugh, \\& van der Hulst 1996; Gerritsen\n\\& de Blok 1999). Although we cannot unambiguously recover the radial \n\\mbox{H\\,{\\sc i}}\\ surface\ndensity distribution in UGC~7321 due to its edge-on geometry\n(cf. Olling 1996), we can compute a mean \\mbox{H\\,{\\sc i}}\\ surface density within\nthe stellar disk. Adopting the definition $\\overline{\\Sigma}_{HI}=$\\mbox{${\\cal M}_{HI}$}$\/\\pi\nR^{2}_{opt}$, where $R_{opt}$ is the linear optical diameter of the disk\n(e.g., Roberts \\& Haynes 1994) we find\n$\\overline{\\Sigma}_{HI}$=5.3~\\mbox{${\\cal M}_\\odot$}~pc$^{-2}$. This value is considerably\nlower than the median of this quantity found by Roberts \\& Haynes\n(1994) for Scd\/Sd spirals in the\nUGC catalogue: $\\overline{\\Sigma}_{HI}$=9.80~\\mbox{${\\cal M}_\\odot$}~pc$^{-2}$, but is\nconsistent with the \n$\\overline{\\Sigma}_{HI}$ ratios of several of \nthe LSB galaxies in the sample of\nde Blok, McGaugh,\n\\& van der Hulst (1996). \n\n\n\\section{Summary}\nWe have presented $B$- and $R$-band imaging and photometry,\nnear-infrared $H$-band imaging, narrow-band H$\\alpha$ imaging, and\n\\mbox{H\\,{\\sc i}}\\ 21-cm line measurements of the nearby, edge-on  Sd spiral galaxy\nUGC~7321.\n\nUGC~7321 is a ``superthin'' galaxy, with an extremely\ndynamically cold stellar disk and no discernible bulge component, even\nin the near-infrared. In spite of its\nedge-on orientation, UGC~7321 is visibly quite diffuse and it is clear\nthat its intrinsic optical surface brightness is \nquite low. The dust content of UGC~7321 also appears to \nbe small, hence we argue that the ``anemic'' appearance of the galaxy\nresults from\nthe  low-level of current star formation rather than severe\ninternal extinction.\n\nUGC~7321 exhibits significant  $B-R$ color gradients in the radial\ndirection:\nmeasured from the disk center to its edges\nedge, $\\Delta (B-R)\\sim$1.05 magnitudes along the galaxy major\naxis. Dust alone cannot explain the large gradient. Using a simple\nextinction model we find\n$\\Delta (B-R)_{cor}\\sim$0.80 after correction for internal reddening.\nThis is somewhat larger\nthan the color gradients typically observed in normal giant spirals\n(cf. de Jong 1996), and suggests significant stellar population\ngradients in the disk of UGC~7321.\n\nThe outermost disk regions of UGC~7321 have $B-R\\le$0.6, suggesting\nthey are composed of stellar populations that include a significant\nfraction of young stars. \nHowever, UGC~7321 also contains a\npopulation of old stars with $B-R\\ge$1.1, indicating it is not a\nyoung or recently-formed  galaxy. The rather strong radial\nsegregation of these populations suggests that the galaxy has evolved\nquite slowly and perhaps that viscous\nevolution has not operated efficiently in this system.\n\nUGC~7321 also exhibits appreciable vertical color gradients:\n$\\Delta B-R$ as large as 0.45 magnitudes was measured parallel to the\nminor axis. This is a\nreflection of a concentration of young blue stars along the galaxy\nmidplane, and a population of the older, red stars at larger scale heights. \nThis type of age segregation is predicted to occur due to\ndynamical heating processes in spiral disks, \nbut UGC~7321 represents one of the few examples of it being directly observed\nin an external galaxy. This  implies that even\ndynamically cold disks like UGC~7321 have undergone some dynamical heating.\n\nThe stellar light distribution of the UGC~7321 disk cannot be characterized by\na single exponential function. A light excess over the prediction of an\nexponential model is seen at small radii, and in both optical and NIR\nwavelengths, the disk appears to\ncontain distinct surface brightness ``zones'' between which the slope\nof the brightness profile changes. This may be an additional signature\nof minimal viscous evolution. It also brings into question the\nprediction of some semi-analytic galaxy models (e.g., Dalcanton,\nSpergel, \\& Summers 1997) that exponential stellar disks are a natural\nproduct of the disk formation process.\n\nIn spite of its regular, organized disk,\nmany of the global properties of UGC~7321, including its luminosity,\n\\mbox{H\\,{\\sc i}}\\ content, rotational velocity, and $\\frac{{\\cal M}_{HI}}{L_{B}}$\\ ratio \n(Table~1), as well as its slowly-rising, solid-body \nrotation curve  (Fig.~19)\nare more typical\nof a dwarf irregular galaxy than a normal Sd spiral. The origin of such\nvastly different disk morphologies in an otherwise similar\nphysical parameter space remains unclear, but it may place an\nimportant constraint on galaxy formation and evolution models.\n\nTogether the properties of UGC~7321 suggest that it is an under-evolved\ngalaxy in both a dynamical and in a star-formation sense. Nonetheless,\nthis galaxy clearly demonstrates that\neven seemingly simple ``pure disk'' galaxies like the superthins are\nhighly complex systems. \n\n\\acknowledgements\n{We are grateful Arnold Rots and Mort Roberts \nfor providing us with the fully-calibrated\nVLA \\mbox{H\\,{\\sc i}}\\ data of UGC~7321. We also thank\nMike Merrill for assistance with the IRIM observations\nat Kitt Peak and Wanda Ashman for creating the artwork for the Appendix. \nThis research was partially funded by the Wide Field and \nPlanetary Camera 2 (WFPC2) Investigation Definition Team,  which is\nsupported at the Jet Propulsion Laboratory (JPL) via the National\nAeronautics and Space Administration (NASA) under contract No.\nNAS7-1260.  The Nan\\c{c}ay\\ Radio Observatory is the department {\\it Unit\\'e\nScientifique Nan\\c{c}ay} of the {\\it Observatoire de Paris} and is associated with \nthe French {\\it Centre National de Recherche  Scientifique} (CNRS) as \nthe {\\it Unit\\'e de Service et de Recherche} (USR), No. B704. The Nan\\c{c}ay\\\nObservatory also gratefully acknowledges the financial support of the \n{\\it R\\'egion Centre} in France.  This research made use of the\nNASA\/IPAC Extragalactic Database (NED), which is operated by the Jet\nPropulsion Laboratory, California Institute of Technology, under\ncontract with the National Aeronautics and Space Administration.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the last years ab initio approaches have gained an ever increasing importance in the calculation \nof bound-state and reaction observables of nuclear systems with a nucleon number $A \\ge 4$ \n(for a comparison of various techniques see~\\cite{LeO13,CaD14}).\nA particularly challenging aspect in such calculations is the proper treatment of resonances. \nRecently, via the EIHH expansion technique~\\cite{BaL00}, the Lorentz integral transform (LIT) method \n\\cite{EfL94,EfL07} was applied to calculate  the\nisoscalar monopole strength of the $^4$He$(e,e')$ reaction with realistic nuclear forces\npaying special attention to the role of the low-lying $0^+$ resonance of $^4$He~\\cite{BaB13}. While\nthe resonance strength could be obtained, it was not possible to determine\nthe width of the resonance. As it was explained the origin of this deficiency arises from the too\nlow density of LIT states in the resonance region, \nbut it was not really clear why the density happens to be so low.\nOne may think of various possible reasons: (i) the LIT method is based on bound-state techniques\nand might be incapable to describe the shape of narrow resonances, (ii) the used expansion in hyperspherical\nharmonics (HH) was not extended to a sufficiently large basis, or (iii) the HH expansion itself is not able\nto have a sufficient density of LIT states in the resonance region. The first possibility\ncan be safely ruled out. In fact in \\cite{Le08} it was shown that for a fictitious\ntwo-nucleon system with a narrow resonance in the $^3P_1$ partial wave exact information about\nthe shape of the resonance could be obtained with the LIT method. Thus the aim of the present paper\nis to study which of the two remaining reasons is responsible for the partial failure of the LIT\nmethod in \\cite{BaB13} and how one can improve the calculation for having a precise information\non the resonance width.\n\nTo clarify the matter first a classical two-body problem, namely deuteron\nphotodisintegration in unretarded dipole approximation, is discussed. It is shown how the \ndensity of  LIT states can be increased in this case. In a next step it is considered how one can\nachieve such an increase also for the above mentioned isoscalar monopole strength of the $^4$He$(e,e')$ \nreaction. In order to do so a central NN potential model (TN potential) is used that had been previously \nemployed in the very first LIT applications for the calculation of the electromagnetic break-up of $^4$He by \nelectrons and real photons~\\cite{EfL97a,EfL97b}. It is interesting to note that in \\cite{EfL97a} one already \nfinds a clear signal of the $0^+$ resonance in the $^4$He longitudinal response function, however, at\nthat early stage of LIT applications the resonance was not studied in greater detail. \n\nThe paper is organized as follows. Section~II contains a brief description of the LIT method.\nIn section~III the question of the density of the  LIT states \nis addressed and illustrated for the above mentioned two-body case. A LIT calculation for the isoscalar \nmonopole strength of the $^4$He$(e,e')$ is discussed in section~IV, where it will be shown how an HH\ncalculation can be modified in order to determine the width of narrow resonances. Finally, a summary\nis given in Section~V.\n\n\n\\section{Lorentz Integral Transform (LIT)}\n\nOver the years the LIT approach~\\cite{EfL94} has been applied to a variety of inelastic electroweak reactions.\nA rather large number of applications can be found in the review articles \\cite{EfL07,LeO13}. \n\nThe LIT of a function $R(E)$ is defined as follows\n\\begin{equation}\nL(\\sigma) = \\int dE \\, {\\cal L}(E,\\sigma) \\, R(E) \\,, \n\\end{equation}\nwhere the kernel ${\\cal L}$ is a Lorentzian,\n\\begin{equation}\n{\\cal L}(E,\\sigma) = {\\frac {1}{(E-\\sigma_R)^2 + \\sigma_I^2}}\n\\end{equation}\n($\\sigma = \\sigma_R + i \\sigma_I$); the parameter $\\sigma_I$\ncontrols the width of the Lorentzian. Because of the adjustable width, and different from \nmany other integral transforms, the LIT is a transform with a controlled\nresolution. However, aiming at a\nhigher resolution by reducing $\\sigma_I$ might make\nit necessary to increase the precision of the calculation. This point will be discussed in greater detail \nin section~III.\n\nFor inclusive reactions the LIT $L(\\sigma)$  is calculated by solving an equation of the form\n\\begin{equation} \n\\label{eqLIT}\n(H-\\sigma) \\, \\tilde\\Psi = S \\,,\n\\end{equation}\nwhere $H$ is the Hamiltonian of the system under consideration and $S$ is an\nasymptotically vanishing source term related to the operator inducing the  specific reaction. The solution $\\tilde\\Psi$ is localized.\nThis a very important property, since it allows to determine $\\tilde\\Psi$ with bound-state methods,\neven for reactions where the many-body continuum is involved. \n\nHaving calculated $\\tilde\\Psi$ one obtains the LIT from the following expression\n\\begin{equation}\nL(\\sigma) = \\langle \\tilde\\Psi | \\tilde\\Psi \\rangle \\,.\n\\end{equation}\nThe response function $R(E)$ is determined from the calculated $L(\\sigma)$ by inverting \nthe transform. A general discussion of the inversion and details about various inversion methods\nare given in \\cite{EfL07,Le08,AnL05,BaE10}.\n\nAn alternative way to express the LIT is given by\n\\begin{equation}\n\\label{LITeq}\nL(\\sigma) = - {\\frac {1}{\\sigma_I} } \n Im \\Big(\\langle S | {\\frac {1}{\\sigma_R + i \\sigma_I - H} } | S \\rangle \\Big) \\,.\n\\end{equation}\nThis reformulation is useful since it allows a direct application of the Lanczos\nalgorithm for the determination of $L(\\sigma)$ \\cite{MaB03}.\nIn fact the calculations discussed in the following sections are performed \nusing expansions on basis sets with a subsequent use of the Lanczos technique.\n\nIn order to calculate a specific reaction one has to specify the source term $S$ in Eqs.~(\\ref{eqLIT}) and\n(\\ref{LITeq}). In case of an inclusive electroweak reaction response functions have the general form\n\\begin{equation}\n\\label{response}\nR(E_f) = \\int dE_f |\\langle f| \\theta | 0 \\rangle|^2 \\delta(E_f-E_0-\\omega) \\,,\n\\end{equation} \nwhere $| 0 \\rangle$ and $| f \\rangle$ are ground-state and final-state wave functions of the system under\nconsideration, while $E_0$ and $E_f$ are the corresponding eigenenergies and $\\omega=E_f-E_0$ is\nthe energy transferred to the system, finally, $\\theta$ is the specific \ntransition operator that induces the reaction. Note that different from the normal convention here the\nargument of the response function $R$ is $E_f$ and not $\\omega$.\n\nThe source term $S$ for inclusive reactions has the following form\n\\begin{equation}\n| S \\rangle = \\theta |0 \\rangle \\,.\n\\end{equation}\n \nA solution of the LIT equation~(\\ref{LITeq})\nvia an expansion on a basis with $N$ basis functions can be understood as follows. One determines the spectrum\nof the Hamiltonian for this basis thus finding $N$ eigenstates $\\phi_n$ with eigenenergies $E_n$. \nThe energies $E_n$ define the positions of the above mentioned LIT  states. Furthermore, the LIT solution\nassigns to any eigenenergy a Lorentzian with strength $S_n$ and width $\\sigma_I$. It should\nbe noticed that the source term $|S\\rangle$ solely affects the strength leading to \n\\begin{equation}\nS_n = |\\langle \\phi_n| \\theta | 0 \\rangle |^2 \\,. \n\\end{equation}\nThe LIT result then reads\n\\begin{equation}\n\\label{LIT_En}\n L(\\sigma) = \\sum_{i=1}^N {\\frac {S_n}{(\\sigma_R-E_n)^2 + \\sigma_I^2}} \\,.\n\\end{equation}\nNote that this result is related to the so-called Lanczos response $R_{\\rm Lnczs}$ by\n\\begin{equation}\n R_{\\rm Lnczs}(\\omega,\\sigma_I) = {\\frac {\\sigma_I}{\\pi} } L(\\sigma_R=E_0+\\omega,\\sigma_I) \\,. \n\\end{equation}\nIn the limit $\\sigma_I \\rightarrow 0$ the Lanczos response is equal to the true response function $R$.\nHowever, one often calculates $R_{\\rm Lnczs}$ for a small but finite $\\sigma_I$ value and identifies\nthe Lanczos response with the true response, which in general is an uncontrolled approximation.\nIn the LIT approach one does not make such an identification of transform and response\nfunction. A proper treatment requires an inversion (see discussion in \\cite{EfL07}). \n\nIt is important to note that the definition of the LIT in Eq.~(1)\ncontains the full response function $R$ with all break-up channels. \nFor the calculation of the LIT one may use any complete localized $A$-body basis set. \nAutomatically for any such set strength from all break-up channels is contained in the LIT.\nHowever, in principle, it can happen that in a specific energy interval of a given reaction\none basis set is more advantageous than another one. \nIn fact, such a case is discussed in section~IV.\n\n\n\n\\section{A simple two-body problem}\n\n\nTo better illustrate the LIT energy resolution \na simple two-body case is discussed, namely deuteron photodisintegration in unretarded dipole approximation. \nThe corresponding cross section is given by\n\\begin{equation}\\label{unret}\n \\sigma_{\\rm unret}(\\omega) = 4 \\pi^2 \\, \\alpha \\, \\omega \\, R_{\\rm unret}(E_f=E_0+\\omega) \\,, \n\\end{equation}\nwhere $\\omega$ denotes the photon energy and $\\alpha$ is the fine structure constant.\nThe transition operator for the response function $R_{\\rm unret}$\nis the dipole operator,\n\\begin{equation} \n\\theta = \\sum_{i=1}^2 z_i \\, {\\frac {(1+\\tau_{i,z})}{2}} \\,,\n\\end{equation}\n where $z_i$ and $\\tau_{i,z}$ are the z-components\nof the position vector and of the isospin operator of the ith nucleon, respectively. \nIn case of the deuteron the dipole operator induces only transitions to the $np$ final\nstates $^3P_0,$ $^3P_1$, and $^3P_2$-$^3F_2$. For simplicity in the following example\nonly transitions to the $^3P_1$ partial wave are considered.\nThe ansatz for the corresponding $\\tilde\\Psi$ reads\n\\begin{equation}\\label{3p1_a}\n|\\tilde\\Psi\\rangle = R(r) \\,\\,|(l=1,S=1)j=1\\rangle \\,|T=1\\rangle\\,,\n\\end{equation}\nwhere $r$, $l$, $S$, $j$, and $T=1$ is the relative distance, orbital angular momentum, total spin,\ntotal angular momentum, and isospin of the $np$ pair, respectively.\nThe resulting LIT equation \ncan be easily solved by direct numerical methods or by expansions\nof $R(r)$ on a complete set. \nFor nuclei with $A>2$ very often \nHH expansions are used with separate hyperspherical and hyperradial parts, where the latter is usually expanded\nin Laguerre polynomials $L_n^{(m)}$  times an exponential fall-off.\nTherefore a corresponding ansatz is made here for R(r),\n\\begin{equation}\n\\label{3p1}\nR(r) = r \\sum_{n=0}^N c_{n} \\,  L_n^{(1)}(r\/b) \\, \\exp\\left(-{\\frac{r}{2b}}\\right) \\,,\n\\end{equation}\nwhere $c_{n}$ are the expansion coefficients and $b$ is a parameter regulating\nthe spatial extension of the basis. \n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.48\\textwidth]{L_Res_fig1.eps}}\n\\caption{(Color online) Spectrum of the Hamiltonian eigenenergies for the $^3P_1$ NN channel with the AV18 NN\npotential for the basis of\nEqs.~(\\ref{3p1_a},\\ref{3p1}) with various values of $N$ and $b$: $N=10$ with $b=1$ fm, 0.5 fm,\nand 0.25 fm in (a) and $b=0.5$ fm with $N=10$, 30, and 50 in (b).} \n\\end{figure}\nFor the following LIT results of the $^3P_1$ channel \nthe AV18 NN potential~\\cite{AV18} is used. First the Hamiltonian eigenvalues $E_n$,\nentering in Eq.~(\\ref{LIT_En}), are studied for various basis sets. In the upper panel of Fig.~1 \nresults are shown with 11 basis functions ($N=10$) and different values for the extension parameter $b$.\nOne sees that a greater spatial extension of the basis functions leads to a shift of the spectrum\nto lower energies. As lowest (highest) eigenenergies one finds 5.17 MeV (7424 MeV), 1.27 MeV (2689 MeV),\nand 0.32 MeV (790 MeV) for $b=0.25$, 0.5, and 1 fm, respectively. To obtain a higher density of\nLIT states one has to increase the number of the basis states $N$. This is illustrated in the \nlower panel of Fig.~1, where the cases with $N=10$, 30, and 50 are shown for $b=0.5$ fm. \nIt is evident that the increase of basis functions does not only lead to a higher density of LIT states, but also\nto an extension of the eigenvalues $E_n$ both to lower and higher energies. In fact as lowest (highest)\nvalues one has now 1.27 MeV (2689 MeV), 0.19 MeV (11056 MeV), and 0.076 MeV (25457 MeV) for $N=10$, 30, and 50,\nrespectively. If one chooses the energy range up to pion threshold one finds that in all three cases\nabout two-thirds of the $N$ LIT states are located therein.\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.48\\textwidth]{L_Res_fig2.eps}}\n\\caption{(Color online) LIT $L(\\sigma)$ of the $^3P_1$ channel with $\\sigma_I=10$ MeV and $b=0.5$ fm. (a): $N=10$ and 50; \n(b): the ratio $r$ of the LIT with $N=10$, 20, 30, and 40 to the LIT with $N=50$ \n(the values 0.995 and 1.005\nare illustrated by full lines).} \n\\end{figure}\nAfter having discussed the energy distribution of the LIT states we come to the transform $L(\\sigma)$ itself.\nFirst, in Fig.~2, the case with $\\sigma_I=10$ MeV and $b=0.5$ fm is illustrated. The upper panel of the figure\nshows that an increase of the number of basis functions from 11 ($N=10$) to 51 ($N$=50) does\nnot lead to noticeable differences. In other words, in this case, the LIT is already quite well\nconverged with a rather small basis. This nice agreement of both results\nis not obvious, since Fig.~1 exhibits rather different positions and  densities of the LIT states for both cases.\nThis has to be interpreted as follows. A discretization of the continuum has no direct physical meaning\nand leads in principle to random results. On the contrary one can use the discretization to calculate integral\ntransforms, as for example the LIT, which lead in convergence to a unique result despite of the randomness of the discrete\neigenenergies.\n\nIn order to find differences between the two results of Fig.~2a one has to present them in a different way, as is done\nin Fig.~2b. There one sees that the agreement is extremely good up to about 30 MeV and thus one can consider the LIT\nfor $\\sigma_I=10$ MeV to be already converged in this energy range with only very few basis functions.\nBeyond 30 MeV the LIT is not yet converged with  $N=10$ and starts oscillating about the LIT with $N=50$ with differences \nbecoming even greater than 5\\%. Because of this regular oscillations about the more precise result it should even not lead \nto serious inversion problems as long the response does not contain any specific structure at higher energies.\nIf one wants to check the existence of such structures one should improve the precision of the calculation\nby enlarging the basis.\nIn fact with larger $N$ values of 20, 30, and 40 one finds an increased high-precision range with relative differences \ncompared to the case with $N=50$ of less than 0.5\\% up to about 50, 60, and 85 MeV, respectively.  \nHowever, in order to search for possible structures it is not the proper strategy to just enlarge the basis.\nOne has to consider that a resonance with a width considerably smaller than $\\sigma_I$ \nis smoothed out in the LIT such that the details of the shape are hidden in tiny contributions to the transform.\nTo disentangle the details one should reduce $\\sigma_I$ increasing in this way the energy resolution of the LIT.\n \nThe situation for smaller values of $\\sigma_I$ is illustrated in Fig.~3, where\nLIT results are shown up to about pion threshold for $\\sigma_I=0.1$, 1, 2.5, and 10 MeV\nand for various values of $N$. For the smallest value, $N=10$ (Fig.~3a), one finds isolated Lorentzian peaks. In case of\n$\\sigma_I=0.1$ MeV they appear already at low energy, for $\\sigma_I=1$ MeV at somewhat higher energy, and\nfor $\\sigma_I=2.5$ MeV at even higher energy. Since the density of LIT states grows with growing $N$\n(see other panels of Fig.~3) isolated peaks are pushed to higher and higher energies if $N$ is increased. \nIn addition one notes that any decrease \nof $\\sigma_I$, $i.e.$ any increase of the resolution, shrinks the convergence range for a given value of \n$N$. For the smallest $\\sigma_I$ value of 0.1 MeV a rather strong oscillatory behavior is still present at very low \nenergies even with $N=50$. In order to get a smooth and converged LIT also in this case one would need to increase $N$ considerably.\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.48\\textwidth]{L_Res_fig3.eps}}\n\\caption{(Color online) LIT $L(\\sigma)$ of the $^3P_1$ channel with $\\sigma_I=0.1$, 1, 2.5, and 10 MeV in\nall four panels ($b=0.5$ fm), but different $N$ values in the various panels: 10 (a), 20 (b), 30 (c), and 50 (d).} \n\\end{figure}\nFor a reliable inversion of $L(\\sigma)$ the transform has to be sufficiently converged for a given $\\sigma_I$. In \nparticular isolated peaks of single Lorentzians should not appear, $i.e.$ for any $\\sigma_R$ value one should \nhave a significant contribution from more than one Lorentzian. \nOf course, for the present two-body\ncase one could increase $N$ further without greater problems, but in a general many-body calculation\nthis might not be easily possible. On the other hand it is not necessary to work with a single\n$\\sigma_I$ value only. Considering again the LIT results of Fig.~3 with $N=50$ one sees that for $\\sigma_I=1$ MeV\na rather converged result is obtained up to about 10 MeV. Thus structures of\na relatively small width would leave a visible signal in the transform in this energy range.\nIn fact one can use this LIT in the low-energy range and combine it with a LIT with a larger $\\sigma_I$ for higher energies.\nIn general one can proceed as follows, one defines a new\ntransform $T$ in an interval $[\\sigma_0,\\sigma_M]$ by combining $M$ transforms $L_m=L(\\sigma_R,\\sigma_{I,m})$ such that only the \nLIT with the specific  $\\sigma_I=\\sigma_{I,m}$ enters significantly in the energy range $[\\sigma_{R,m-1},\\sigma_{R,m}]$: \n\\begin{equation}\\label{LIT_mult}\nT = L_1 f_1 \n  + \\sum_{m=2}^{M-1} a_m L_m (1-f_{m-1})) f_m \n+ a_M L_M (1-f_{M-1}) \\,.\n\\end{equation}\nThe function $f_m$ in Eq.~(\\ref{LIT_mult}) is a smooth cutoff from 1 to 0 at $\\sigma_R = \\sigma_{R,m}$,\nfor example $f_m(\\sigma_R \\le \\sigma_{R,m})=1$ and $f_m(\\sigma_R > \\sigma_{R,m})= \\exp(-((\\sigma_R-\\sigma_{R,m})\/\\Delta)^{2n})$,\nwhere one has to take reasonable values for the parameters $\\Delta$ and $n$. The coefficients $a_m$ in Eq.~(\\ref{LIT_mult})\ncan be chosen such that size of the transform does not change too drastically from one energy range to the next.\nThus, for the case of the LITs of Fig.~3 with $N=50$, one could set $\\sigma_{I,1}=1$, $\\sigma_{I,2}=2.5$, and \n$\\sigma_{I,3}=10$ MeV with $\\sigma_{R,1}=10$ and $\\sigma_{R,2}=30$ MeV. In order to improve the precision\nin the threshold region even further one could include the case with \n$\\sigma_I=0.1$ MeV, but one would need to further increase $N$.\nAlternatively, keeping $N=50$,\none could check the convergence behavior of a LIT with a somewhat larger $\\sigma_I$, for example $\\sigma_I=0.25$\nMeV.\n\nThe discussion above shows that the LIT is an approach with a controlled resolution.\nIn an actual calculation one should check which is the lowest $\\sigma_I$ that leads in a specific\nenergy range to a sufficiently converged and smooth LIT without that a single LIT state sticks out. \nStructures which are considerably smaller than such a $\\sigma_I$ value cannot be resolved by the inversion. \nA helpful criterion is given in \\cite{Le08} (see discussion of Fig.~7 in \\cite{Le08}). \n\n\n\n\n\\section{$^4$He Isoscalar monopole response function}  \n\nThe isoscalar monopole response function $M(q,E_f=E_0+\\omega)$ can be determined in inclusive\nelectron scattering ($q$ and $\\omega$ represent momentum and energy transferred by\nthe virtual photon to the nucleus). In this case the transition operator $\\theta$ of Eq.~(\\ref{response}) \nbecomes $q$-dependent and takes the form\n\\begin{equation}\\label{monopole}\n \\theta(q) =\\frac{G_E^s(q^2)}{2} \\sum_{i=1}^A \\, j_0(q r_i)\\,,\n\\end{equation}\nwhere $G_E^s(q^2)$ is the nucleon isoscalar electric form factor, \n$\\bs{r}_i$ is the position of nucleon $i$,\nand  $j_0$ is the spherical Bessel function of 0$^{th}$ order.\n\nExperimental investigations of the $^4$He$(e,e')$ reaction in \\cite{Wa70,Fr65,Ko83} revealed a \n$0^+$ resonance located less than 1 MeV above the $^4$He break-up threshold with quite a narrow \nwidth of about 250 keV.  It is interesting to note that\nvery recently a EIHH-LIT calculation was carried out, where it is pointed out that the resonance might be \ninterpreted as a breathing mode \\cite{BaB15}. Unfortunately, this and the preceding EIHH-LIT calculations~\\cite{BaB13}\nwere only able to determine a resonance strength but not a resonance width  \nsince the density of LIT states in the resonance region was too low.\nOn the other hand, with the experience made in the previous section it seems to be easy to \nincrease the density of LIT states also in the region of the $^4$He isoscalar monopole resonance, namely \nby increasing the number of HH basis states. Unfortunately with an HH expansion this does not work very well\nin the energy region below the three-body break-up threshold. To illustrate this, \nthe first LIT application~\\cite{EfL97a}, already mentioned in the introduction, is considered in the following.\nThe calculation was carried out for inclusive electron scattering off $^4$He and used a correlated HH (CHH) basis,\nwhere NN short range correlations are introduced to accelerate the convergence of the HH expansion. \nThe NN interaction (TN potential, Coulomb force included) consisted in a central spin-dependent \ninteraction active only in even NN partial waves. In Fig.~4 we show an unpublished\nresult for the LIT of the response function $M$ from this calculation.\nOne sees that there is only an isolated LIT state in the resonance region. In this specific case\none finds the next LIT state only about 0.1 MeV below the four-body break-up threshold at $\\sigma_R=0$ MeV. On the contrary for\npositive values of $\\sigma_R$ one has quite a high density of LIT states. An increase of the number\nof hyperspherical and\/or hyperradial basis states does not change the general picture of having\nonly very few LIT states for negative $\\sigma_R$ values, as in fact it was the case in the LIT-EIHH calculations~\\cite{BaB13,BaB15}. \nIn addition it was checked for the present work that an increase of the parameter $b$ in the hyperradial \nfunctions ($\\sim L_n^{(8)}(\\rho\/b) \\, \\exp(-\\rho\/2b))$ in the four-body CHH calculation of \\cite{EfL97a}\ndoes not lead to any significant change concerning the low-energy density of LIT states.\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.4\\textwidth]{L_Res_fig4.eps}}\n\\caption{LIT $L(\\sigma)$ of $M(q,\\omega)$ at $q=300$ MeV\/c (TN potential) of $^4$He with four-body CHH basis from~\\cite{EfL97a}.} \n\\end{figure}\n\nAt first sight it is not understandable why an increase of the HH basis states does not have\na significant effect on the density of LIT states in the energy range below the three-body break-up\nthreshold. On the other hand one has to consider that the dynamical variable for a two-body break-up, \n$i.e.$ the relative vector ${\\bf r}^\\prime_4 = {\\bf r}_4 - \\sum_{i=1}^3 {\\bf r}_i\/3$ between the free nucleon and the bound\nthree-body system, does not appear explicitly\nin the HH formalism. Therefore it might be helpful to use a different basis, where ${\\bf r}^\\prime_4$ \nis taken into account as variable explicitly. In the following, the four-body calculation is switched to such a basis.\nThe new basis states consist in a product of a three-body CHH basis state for the first three nucleons \ntimes a single-particle basis state for the fourth nucleon. The CHH basis state is then again a product of a hyperspherical basis state\n${\\cal Y}_{[K\\alpha]}(\\Omega)$ times a hyperradial basis state $R_n(\\rho)$, where the former\ndepends on the grand-angle $\\Omega$ and is characterized by the grand-angular quantum number $K$ and \na set of other quantum numbers $[\\alpha]$, whereas the latter is proportional to $L_n^{(5)}(\\rho\/b) \\, exp(-\\rho\/2b)$\n(for a detailed definition of the HH basis see for example \\cite{BaL00}).\n\nThe single-particle states are defined as follows \n\\begin{equation} \\label{basis}\n|\\Phi({\\bf r}^\\prime_4)\\rangle = \\sum_{l_4 m_{s_4} m_{t_4}} \\phi_{l_4}(r^\\prime_4) \\,|l_4 m_{l_4}\\rangle \\, \n |s_4 m_{s_4}\\rangle\\,\n |t_4 m_{t_4}\\rangle\\,,\n\\end{equation}\nwhere $l_4$, $s_4=1\/2$ and $t_4=1\/2$ are the orbital angular momentum, spin, and isospin quantum numbers, respectively,\nand $m_{l_4}$, $m_{s_4}$ and $m_{t_4}$ denote the corresponding projections. The radial wave function $\\phi_{l_4}(r^\\prime_4)$\nis given by\n\\begin{equation}\\label{rel_r}\n\\phi_{l_4}(r^\\prime_4) = (r^\\prime_4)^{l_4} \\prod_{i=1}^3 f(r_{i4})  \\sum_{n_4=0}^{N_4-1} c_{n_4} \nL_{n_4}^{(2)}(r^\\prime_4\/b_4) \\exp\\left({\\frac{-r^\\prime_4}{2b_4}}\\right),\n\\end{equation}\nwhere $f(r_{i4}) = f(|{\\bf r}_i - {\\bf r}_4|)$ is the NN correlation function (same correlation as in CHH basis).\nThe CHH basis provides antisymmetric\nstates for the first three nucleons, whereas the product of the CHH basis states and the single-particle states\nhas to be antisymmetrized in order to have totally antisymmetric basis states for all four particles.\nIn order to calculate the $^4$He ground state or the transitions induced by the action of the operator $\\theta(q)$ of \nEq.~(\\ref{monopole}) on the $^4$He ground state one has to couple the CHH basis states with those of Eq.~(\\ref{basis}) \nto a total angular momentum $L$ equal to zero, and also the total spin (isospin) wave function of the four nucleons\nhas to be coupled to a total spin $S$ (isospin $T$) equal to zero. \n\nThe calculations described in the following proceed in the same way as it was carried out in \\cite{EfL97a,EfL97b},\nnamely by using a nine-dimensional Monte Carlo integration for the evaluation of the various Hamiltonian\nand norm matrix elements. Such an approach leads to reliable results (see benchmark test~\\cite{BaF01}). \n\n\\begin{table}\n\\caption{Convergence of $^4$He binding energy BE with the TN potential (Coulomb force included): $l_4$ denotes the orbital angular momentum of\nthe single-particle motion (see Eqs.~(\\ref{basis},\\ref{rel_r})), $K_3$ is the grand-angular quantum number of the three-body CHH states,\n``sym'' and ``mixed'' indicate that the CHH state is symmetric and mixed symmetric, respectively \n(note that for $l_4>0$ two symmetric CHH states\nand only the lowest mixed symmetric state are taken).}\\label{tab:BE1}\n\\begin{center}\n\\begin{tabular}{c|c|c|c|c} \\hline\\hline\n$l_4$ & $K_3$ \\,\\,\\, & \\,\\,\\,CHH symmetry \\,\\,\\, &  \\# CHH states & BE [MeV]   \\\\\n\\hline\n  0 &  0  &  sym  & 1 &  28.66  \\\\\n  0 &  4  &  sym  & 1 &  30.11  \\\\\n  0 &  6  &  sym  & 1 &  30.56  \\\\\n  0 &  8  &  sym  & 1 &  30.67  \\\\\n  0 &  10  &  sym  & 1 &  30.77  \\\\\n  0 &  12  &  sym  & 2 &  30.82  \\\\  \n  0 &  14  &  sym  & 1 &  30.85  \\\\\n  0 &  2  &  mixed      & 1 &  31.28  \\\\\n  0 &  4  &  mixed      & 1 &  31.32 \\\\\n  1 & 1,3,5 & sym and mixed  & 3 & 31.39 \\\\\n  2 & 2,4 & sym and mixed  & 3 & 31.41 \\\\\n\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nIn Table~\\ref{tab:BE1} the convergence of the $^4$He binding energy is illustrated.\nIt is evident that the dominant contribution comes from the basis functions where the three-body CHH basis\nis in a symmetric state and the relative angular momentum $l_4$ of the single-particle motion is equal to zero. \nMixed symmetric CHH states together with $l_4=0$ enhance the binding energy by about 0.5 MeV. The contribution\nof states with $l_4>0$ is very small and leads to a further increase of about 0.1 MeV\n(antisymmetric CHH states have been neglected).\nThe final result of 31.41(5) MeV agrees very well with the converged result of the four-body CHH calculation of 31.40(5) MeV.\n\n\nFor the calculation of the LIT of the $^4$He isoscalar monopole response function $M(q,\\omega)$\nhyperradial and radial basis states are chosen with a rather large extension in space. This should lead to a sufficiently\nhigh density of LIT states at low energy (see Fig.~1). The following choice is made: $b=4$ fm (hyperradial CHH states) \nand $b_4=1.33$ fm (radial single-particle states). \nFor the hyperradial part 15 basis functions are taken, whereas the number of the radial single-particle basis states is \nkept variable and denoted by $N_4$.\nSince the main interest of this investigation is concentrated on the low-energy part of $M(q,\\omega)$  only\nthose three-body CHH basis states which lead to a significant contribution to the \n$^4$He binding energy (see Table~\\ref{tab:BE1}) are taken into account: symmetric CHH states up to $K_3 = 6$ and the mixed symmetric CHH state \nwith $K_3 = 2$. All the remaining states contribute only with 0.42 MeV to the $^4$He binding energy. Thus\nit is reasonable to expect also a shift of the position of the $^4$He $0^+$ resonance by about 0.5 MeV towards higher\nenergies with respect to the four-body CHH calculation of Fig.~4 leading to a value of about -5.9 MeV.\nTo take into account only four hyperspherical three-body CHH states reduces the numerical effort of the \ncalculation quite a bit, but even with such a reduced\nbasis the calculation requests a considerable numerical effort (note that due to the rather weak fall-off of the hyperradial\/radial\nbasis functions the relevant integration volume becomes much larger than for the bound-state calculation). \n\n \n\\begin{figure}\n\\centerline{\\includegraphics[width=0.4\\textwidth]{L_Res_fig5.eps}}\n\\caption{(Color online) Spectrum of the Hamiltonian eigenenergies for the $^4$He$(J^\\pi=0^+)$ states with the TN NN\npotential for the basis described in the text with $N_4$ basis functions for the radial single-particle basis.} \n\\end{figure}\n\nIn Fig.~5 the energy distribution of LIT states is shown up to the four-body break-up threshold\nwith three different $N_4$ values. As it has been anticipated\nthe density of LIT states increases with a growing number of radial single-particle states. \nDue to the interplay of CHH basis states and radial single-particle\nbasis states the pattern is not as regular as in the two-body example\nof Fig~1. \n\nIt is obvious that a further increase of $N_4$ would lead to an even higher density of low-energy \nLIT states, but, unfortunately, the precision of the nine-dimensional Monte Carlo integration is not sufficiently high and\na solution with $N_4>20$ becomes problematic. In this respect it is a drawback to work with a correlated\nHH basis because one looses the orthogonality. Nonetheless, as shown in the following,\nthe present calculation allows a determination of the width of the $^4$He isoscalar monopole resonance.\n\nIn Fig.~6 the LIT of $M(q,\\omega)$ is shown for $N_4=20$ and $N_4=22$. In the latter case\nthe basis function with $n_4=20$ (see Eq.~(\\ref{rel_r})) is dropped because it makes the numerical solution\nof the corresponding eigenvalue problem problematic (precision of Hamiltonian and norm matrices is not sufficiently high). \nThe figure illustrates that with increasing resolution,\n$i.e.$ with decreasing $\\sigma_I$ the resonance becomes more and more pronounced against the background. One also finds\nthat the anticipated peak position of about -5.9 MeV is roughly confirmed. \n\nIn order to determine a resonance width the LIT has to be inverted. The procedure of the\ninversion in presence of a resonance is described in \\cite{Le08}.\nAs rule of thumb one can say that the chosen $\\sigma_I$ should not be much larger than the resonance width. However, in principle,\nif the LIT is calculated with a very high precision one can choose also a considerably larger $\\sigma_I$. In fact in the model study\n\\cite{Le08} it is nicely demonstrated that the obtained width is independent from the used $\\sigma_I$ over a very large range,\neven with $\\sigma_I=5$ MeV the shape of a resonance with a width of 270 keV was exactly determined. In the present \ncase the precision of the calculated LIT is not as high as in \\cite{Le08}. The actual inversion was made with two different\n$\\sigma_I$ values (see discussion of transform $T$ introduced in Eq.~(\\ref{LIT_mult})), \n$\\sigma_{I,2}=5$ MeV beyond -4 MeV, whereas $\\sigma_{I,1}$ was varied in the range\nfrom 0.1 to 0.5 MeV in the low-energy region. The various results for the width were quite stable (maximal difference: 0.01 MeV) and lead to the\nfollowing values: 120(10) keV ($N_4=20$) and\n240 keV ($N_4=22)$, the mean value amounts to 180(70) keV. The result lies in the same ballpark\nas the experimental value of 270(50) keV \\cite{Wa70}. One could try to increase the theoretical precision of the\ndetermination of the width by a further increase of $N_4$. Because of the problem just mentioned above this would require\na non negligible effort in the present calculation. On the other hand it is much more desirable to make\nsuch a calculation with a realistic nuclear force instead of having a very precise result for a simple NN interaction like\nthe TN potential model. \n\n\n\nFor the inversion of the LIT it is assumed that there is only a single peak in the resonance region.\nIn principle from an increase of the precision of the calculation one could also find out that the resonance has a more \ncomplicated structure, for example a double peak. It is evident from Fig.~6 that in the present calculation one controls \nthe resonance with a resolution of about 100 keV, thus it is not possible to resolve structures with an even smaller width.\nThe situation is similar to experiment, where one works with a given resolution of the experimental\napparatus.\n\n\\begin{figure}\n\\centerline{\\includegraphics[width=0.45\\textwidth]{L_Res_fig6.eps}}\n\\caption{(Color online) LIT $L(\\sigma)$ of $M(q,\\omega)$ at $q=250$ MeV\/c (TN potential) of $^4$He with $\\sigma_I=0.01$, 0.1, 0.25, 1, and 5\nMeV calculated with the new basis states (three-body CHH basis times\nsingle-particle basis of Eqs.~(\\ref{basis},\\ref{rel_r})); in (a) the single-particle basis with 20 and in (b) with 21 states (see text).} \n\\end{figure}\n\n\\section{Summary}\n\nIn this work first a brief outline of the LIT approach is given, a method that allows to calculate continuum observables \nwith bound-state techniques. In a next step the energy resolution that can be obtained with the method is discussed. \nTo this end a specific channel of the final state in deuteron photodisintegration is studied\nin unretarded dipole approximation. In order to do so the LIT equation is solved using\na bound-state basis to calculate the relevant quantities. It is shown how the spatial extension\nand the number of basis states affects the calculation. The role of the width parameter $\\sigma_I$\nof the LIT is discussed, especially which $\\sigma_I$ values should be chosen\nin a specific situation. The discussion makes clear that the LIT constitutes an\napproach with a controlled resolution. In particular it allows to determine the width\nof narrow resonances in the continuum. However, for LIT calculation with an arbitrary many-body bound-state basis\nthis is not always guaranteed. Usually narrow resonances of an $A$-body system in the continuum\nare located below the three-body break-up threshold.\nIf one intends to study the resonance in a scattering state calculation it is mandatory\nto take into account the relevant {\\it dynamical} variable,\n$i.e.$ the relative vector between the two fragments.\nIn spite of the LIT bound-state character this {\\it dynamical} variable should also appear explicitly in a LIT calculation. \nIt guarantees that the density of LIT states can be enhanced in the resonance region by increasing the number of \nthose basis states that directly depend on the {\\it dynamical} variable.\nOn the contrary, if this variable is not included explicitly, as for example in an $A$-body HH calculation, \nit is difficult, maybe even impossible, to obtain a detailed information about the resonance.\n\nIn a general case of a resonance with various open channels the resonance can have\na {\\it partial} decay width to all these channels, which then results in a {\\it total} decay width.\nIn this case it is sufficient to choose just one of the various possible {\\it dynamical} variables.\nThe only aim which has to be fulfilled is a sufficient density of LIT states.\nThe LIT then by definition collects strength from all open channels and a determination of the\nwidth via inversion leads to the {\\it total} width.\n\nIn order to illustrate the situation in greater detail the isoscalar monopole response function $M$ of $^4$He is considered\nas test case using a simple spin-dependent central NN interaction (Coulomb force is included). \nIn fact, here one finds a narrow resonance in the continuum below the three-body break-up threshold.\nObviously, in this case the {\\it dynamical} variable is given by the relative vector of the free nucleon\nand the bound three-nucleon system. \nUsing a proper basis as described above it is indeed found that the density of LIT states\ngrows in the resonance region if the number of single-particle basis states is enhanced. \nIt is shown that the LIT state density becomes sufficiently high to determinate the resonance width\nvia an inversion of the transform.\nIn fact, a width of 180(70) keV is found, a result that is not too far from the experimental\nvalue of 270(50) keV. It would be very interesting to perform such a calculation also\nwith modern realistic nuclear forces as have been used in the LIT-EIHH calculations of the response function $M$\\cite{BaB13,BaB15},\nhowever the effective interaction approach has to be a bit redesigned, since one would not have any more a pure HH basis.  \n \nThe present approach is not only advantageous in case of\ncross sections with narrow resonances, but also for non-resonant two-body break-up cross sections at very low energies.\nThe possibility to increase the LIT state density allows to work with smaller $\\sigma_I$ values thus\nenhancing the energy resolution of the calculation. This might be particularly interesting in case of\nastrophysical reactions.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe best planets for detailed characterization are transiting planets, first and foremost\nbecause they allow for the possibility of unambiguous mass measurement \n\\citep[e.g., HD209458b][]{Charbonneau2000, Mazeh2000, Henry2000}.\nFrom the Doppler effect we can determine the minimum\nmass of the planet ($m \\sin i$), and from the transit light curve we can determine\nthe planetary radius and the orbital inclination ($i$), thus yielding a measurement of \nthe planet's mass. Moreover, from this combination we can calculate the planet's mean\ndensity, and shed light on its internal structure by comparison with models containing\ndifferent amount of iron, silicates, water, hydrogen, and helium. Furthermore, transiting\nplanets are unique because of the  feasibility to characterize the upper atmosphere by\nspectroscopy during transits and occultations of a significant number of planets.  \nThe forthcoming \\textit{James Webb Space Telescope} \\cite[JWST,][]{Gardner2006} and the\n\\textit{Extremely Large Telescope} \\citep[ELT,][]{deZeeuw2014} will have unprecedented\ncapabilities for detailed studies of the atmospheres of terrestrial planets, and the\ninterpretation of the results will require an accurate mass measurement (e.g., \\citealt{Batalha:2019}).\n\nThe Transiting Exoplanet Survey Satellite (TESS) \\citep{Ricker2015} started scientific\noperations in July 2018, aiming to detect transiting planets around bright and nearby stars\n-- bright enough for Doppler mass measurement to be feasible.\nFor this task, TESS surveys about 85\\% of the sky during the Prime Mission.\nThe survey covering the southern ecliptic hemisphere is now complete,\nand the northern survey is underway.  Each hemisphere is divided into\n13 rectangular sectors of $96\\degree \\times 24\\degree$ each. Each sector is\ncontinuously observed for an interval of 27-days, with a cadence of 2 minutes for several hundred\nthousand pre-selected stars deemed best suited for planet searching.\nAdditionally, during the TESS Prime Mission, the Full Frame\nImages -- the full set of all science and collateral pixels across all CCDs of a given \ncamera -- are available with a cadence of 30 minutes.  M dwarfs are of special interest \nbecause the transit and radial-velocity signals of a given type of planet are larger for\nthese low-mass stars than they are for Sun-like stars. In addition, compared to hotter stars\nM dwarfs present better conditions for the detection of planets orbiting the circumstellar\nhabitable zone:\nless time consuming, larger Doppler signals, and an increased transit probability.\n\\citet{Sullivan2015} anticipated that from 556 small ($<$2R$_\\oplus$) transiting planets\ndiscovered by TESS, 23\\% of them will be detected orbiting bright (K$_S<$9) stars, and\nthat 75\\% of small planets will be found around M dwarfs.\n\nThis paper reports the discovery of a small planet orbiting the star L~168-9 (TOI-134),\nbased on TESS data. The host star is a bright M dwarf. An intense precise radial-velocity\ncampaign with HARPS and the Planet Finder Spectrograph (PFS) revealed the terrestrial \nnature of the newly detected world. This work is presented as follows:\nSection~\\ref{sec:L168-9} describes the host star properties. Sections~\\ref{sec:phot}\nand~\\ref{sec:rv} describe the photometric and radial-velocity observations.\nSection~\\ref{sec:analysis} \npresents an analysis of all the data, including the study of\nstellar activity.  Finally, Section~\\ref{sec:conclusion} places L~168-9~b within the larger\ncontext of the sample of detected planets.\n\n\\section{L~168-9}\n\\label{sec:L168-9}\n\n\\begin{table}[t]\n  \\caption{L~168-9 (TIC~234994474) properties. Superscripts indicate the reference.}\n  \\label{tab:stellarprop}\n  \\centering\n  \\small\n  \\begin{tabular}{p{0.2\\linewidth}l c c c}  \n    \\hline\\noalign{\\smallskip}\n    Parameter & Units & Value & Reference \\\\\n    \\noalign{\\smallskip}\n    \\hline\\noalign{\\medskip}\n    R.A. & [J2000] & $23^h 20^m 07.52^s$  & Gaia2018\\\\\n    Decl. & [J2000] & $-60\\degree 03^\\prime 54.64^{\\prime\\prime}$ &  Gaia2018\\\\\n    Spectral type    &  &  M1V & Ga2014\\\\\n    B & [mag] & 12.45$\\pm$0.19 & Ho2000\\\\\n    V & [mag] & 11.02$\\pm$0.06 & Ho2000\\\\\n    B$_A$ & [mag] & 12.460$\\pm$0.025 & He2016\\\\\n    V$_A$ & [mag] & 11.005$\\pm$0.018 & He2016\\\\\n    g$_A$ & [mag] & 11.752$\\pm$0.032 & He2016\\\\\n    r$_A$ & [mag] & 10.416$\\pm$0.028 & He2016\\\\\n    i$_A$ & [mag] & 9.675 & He2016\\\\\n    W$_1$ & [mag] & 6.928$\\pm$0.060 & Cu2013\\\\\n    W$_2$ & [mag] & 6.984$\\pm$0.020 & Cu2013\\\\\n    W$_3$ & [mag] & 6.906$\\pm$0.016 & Cu2013\\\\\n    W$_4$ & [mag] & 6.897$\\pm$0.074 & Cu2013\\\\\n    T && 9.2298$\\pm$0.0073 & St2018\\\\\n    J & &7.941$\\pm$0.019 & Cu2003\\\\         \n    H & &7.320$\\pm$0.053 & Cu2003\\\\         \n    K$_s$ & &7.082$\\pm$0.031 & Cu2003\\\\\n    B$_p$ & &11.2811$\\pm$0.0016 & Gaia2018\\\\\n    G & & 10.2316$\\pm$0.0008 & Gaia2018\\\\\n    R$_p$ & &9.2523$\\pm$0.0011 & Gaia2018\\\\\n    $\\pi$ &  [mas]   &  39.762$\\pm$0.038 & Gaia2018\\\\\n    Distance & [pc] & 25.150 $\\pm$ 0.024 & Gaia2018\\\\\n    $\\mu_\\alpha$ & [mas\/yr] & -319.96$\\pm$0.10 & Gaia2018\\\\\n    $\\mu_\\delta$ & [mas\/yr] &  -127.78$\\pm$0.12 & Gaia2018\\\\\n    $dv_r\/dt$ &[m\/s\/yr] & 0.06865$\\pm$0.00011 & this work\\\\\n    M$_s$ & [M$_\\odot$] & 0.62$\\pm$0.03 & Ma2019\\\\\n    R$_s$ & [R$_\\odot$] & 0.600 $\\pm$0.022 & Sect.~\\ref{subsec:stellarparam}\\\\\n    T$_{\\rm eff}$ & [K] & 3800$\\pm$70 & Sect.~\\ref{subsec:stellarparam}\\\\\n    L$_s$ & [L$_{\\odot}$] & 0.0673$\\pm$0.0024 & Sect.~\\ref{subsec:stellarparam}\\\\\n    ${log(g)}$ & [g\/cm$^{-3}$]  &4.04$\\pm$0.49 & Sect.~\\ref{subsec:stellarparam}\\\\\n    $[\\mathrm{Fe\/H}]$ &  & 0.04$\\pm$0.17 & Ne2014\\\\\n    ${log(R^\\prime_{HK})}$ &  & -4.562$\\pm$0.043 & As2017A\\\\\n    P$_{\\text{rot}}$ & [days] & $29.8\\pm1.3$ & Sect.~\\ref{subsec:rotation}\\\\\n    \\noalign{\\smallskip}\\hline\n  \\end{tabular}\n\n  \\begin{list}{}{}\n  \\item[Reference notes: ] Gaia2018~--~\\citet{Gaia2018}; \n  Ga2014~--~\\citet{Gaidos2014};\n  Ho2000~--~\\citet{Tycho-2};\n  H22016~--~\\citet{apass};\n  Cu2013~--~\\citet{wise};\n  St2018~--~\\citet{StassunTIC:2018};\n  Cu2003~--~\\citet{Cutri2003}; \n  Ma2019~--~\\citet{Mann:2019};\n  Ne2014~--~\\citet{Neves2014};\n  As2017a~--~\\citet{Astudillo2017a}\n  \\end{list}\n\\end{table}\n\n\nL~168-9, also known as CD-60~8051, HIP~115211, 2MASS~J23200751-6003545, with \nthe entry 234994474 of the TESS Input Catalog (TIC) or 134 of the Tess Object of Interest (TOI) list, \nis a red dwarf of spectral type M1V.\nIt appears in the southern sky, and resides at a distance of 25.150$\\pm$0.024 $pc$ from the\nSun \\citep{Gaidos2014, Gaia2018}. \nTable~\\ref{tab:stellarprop} lists the key parameters of the star, namely \nits position, visual and near-infrared apparent magnitudes, parallax, proper motion, \nsecular acceleration, and its essential physical properties.\n\n\n\\subsection{Derived stellar properties}\n\\label{subsec:stellarparam}\n\nWe performed an analysis of the broadband spectral energy distribution (SED) \ntogether with the {\\it Gaia\\\/} DR2 parallax in order to determine an empirical \nmeasurement of the stellar radius, following the procedures described by\n\\citet{Stassun:2016,Stassun:2017,Stassun:2018}. We took the $B_T V_T$ \nmagnitudes from {\\it Tycho-2}, the $BVgri$ magnitudes from APASS, \nthe $JHK_S$ magnitudes from {\\it 2MASS}, the W1--W4 magnitudes from \n{\\it WISE}, and the $G$ magnitude from {\\it Gaia}. Together, the \navailable photometry spans the full stellar SED over the wavelength \nrange 0.35--22~$\\mu$m (see Figure~\\ref{fig:sed}). \n\n\nWe performed a fit using NextGen stellar atmosphere models \\citep{Hauschildt1999},\nwith the effective temperature ($T_{\\rm eff}$) and surface gravity ($\\log g$) constrained \non the ranges reported in the TESS Input Catalog \\citep{StassunTIC:2018}, while the metallicity\n[Fe\/H] was fixed to a typical M-dwarf metallicity of -0.5. \nWe fixed the extinction ($A_V$) to be zero, considering proximity of the star, the\ndegrees-of-freedom of the fit is 10. \nThe resulting fit (Figure~\\ref{fig:sed}) has a $\\chi^2$ of 42.3 ($\\chi^2_{red}$=4.2), with \n$T_{\\rm eff} = 3800 \\pm 70$~K. The relatively high $\\chi^2$ is likely due to systematics, \nas the stellar atmosphere model is not perfect. We artificially increased\nthe observational uncertainty estimates until $\\chi^2_{red}=1$ was achieved.\nIntegrating the (unreddened) model \nSED gives the bolometric flux at Earth of $F_{\\rm bol} = 3.41 \\pm 0.12 \\times 10^{-9}$\nerg~s$^{-1}$~cm$^{-2}$. \nTaking the $F_{\\rm bol}$ and $T_{\\rm eff}$ together with the {\\it Gaia\\\/} DR2 parallax, \nadjusted by $+0.08$~mas to account for the systematic offset reported by \n\\citet{StassunTorres:2018}, gives the stellar radius as $R = 0.600 \\pm 0.022$~R$_\\odot$. \nFinally, estimating the stellar mass from the empirical relations of \\citet{Mann:2019} \ngives $M = 0.62 \\pm 0.03 M_\\odot$. With these values of the mass and radius,\nthe stellar mean density is $\\rho = 4.04 \\pm 0.49$ g~cm$^{-3}$. We also tested to fit \nthe SED using BT-Settl theoretical grid of stellar model \\citep{Allard2014}, where we \nobtained a consistent result.\n\n\nWe searched for infrared (IR) excess in {\\it WISE} data using the Virtual Observatory SED\nAnalyser \\citep[VOSA,][]{Bayo:2008}, which could point for the presence of debris disks. \nFor that we computed the excess significance parameter\n\\citep[$\\chi_\\lambda$,][]{Beichman:2006,Moor:2006}, where $\\chi_\\lambda \\ge 3$ \nrepresents a robust detection of IR excess. We obtained  $\\chi_\\lambda = 0.70$ in the\nW$_3$ band, ruling out the presence of a debris disk around L~168-9.\n\n\n\\begin{figure}\n\\centering\n    \\includegraphics[clip, trim=3.6cm 2.6cm 3cm 3cm,scale=0.4]{figures\/gj4332_sed.pdf}\n    \\caption{Spectral energy distribution (SEDs) of L~168-9. Red error bars represent the \nobserved photometric measurements, where the horizontal bars represent the effective \nwidth of the passband. Blue circles are the model fluxes from the best-fit NextGen \natmosphere model (black). \n    \\label{fig:sed}}\n\\end{figure}\n\n\n\\section{Observations}\n\nThe first hint of a planetary companion orbiting L~168-9 came from analyzing TESS data.\nAfter the Data Validation Report was released to the community, a follow-up campaign started\nwith several instruments and by different teams to check on whether the transit-like signal\nseen by TESS originated from a planet, as opposed to a stellar binary or other source.\nThe follow-up observations included supplementary time series photometry aiming to detect \nadditional transits, seeing-limited and high-resolution imaging to analyze the possibility \nthat the signal comes from a star on a nearby sightline, and precise radial-velocity monitoring\nto measure the companion's mass.\n\n\\subsection{Photometry}\n\\label{sec:phot}\n\n\\subsubsection{TESS}\n\nTESS observed Sector 1 from the 25th of July to the 22nd of August 2018\n\\footnote{The Sector 1 pointing direction was $RA(J2000):+352.68\\degree,\\\n  Dec(J2000):-64.85\\degree,\\ Roll:-137.85\\degree$.},\na 27.4-day interval that is typical of each sector. L~168-9 is listed in the Cool Dwarf Catalog \nthat gathered the known properties of as many dwarf stars as possible with $V-J>2.7$ and effective\ntemperatures lower than 4\\,000 K \\citep{Muirhead2018}.\nThe predicted TESS-band apparent magnitudes are also given in this catalog.\nL~168-9 was chosen for 2-min time sampling as part of the TESS Candidate Target List\n\\citep[CTL,][]{StassunTIC:2018}, consisting of a subset of the TESS Input Catalog (TIC)\nidentified as high-priority stars in the search for small transiting\nplanets. Time series observations of L~168-9 were made with CCD 2 of Camera 2.\n\nThe TESS Science Processing Operations Center (SPOC) at the NASA Ames Research Center\nperformed the basic calibration, reduction, and de-trending of the time series, and also\nperformed the search for transit-like signals \\citep{Jenkins2016}. The light curves were\nderived by the SPOC pipeline and consist of a time series based on Simple Aperture \nPhotometry (SAP), as well as a corrected time series based on Pre-search Data Conditioning\n\\citep[PDC,][]{Smith2012,Stumpe2014} referred to as PDCSAP (as detailed by Tenenbaum and\nJenkins 2018\\footnote{\\url{https:\/\/archive.stsci.edu\/missions\/tess\/doc\/EXP-TESS-ARC-ICD-TM-0014.pdf}}).\nThis work made use of the PDCSAP time series available on the Mikulski Archive for Space\nTelescopes\n(MAST\\footnote{\\url{https:\/\/mast.stsci.edu\/portal\/Mashup\/Clients\/Mast\/Portal.html}}).\nThe TESS photometry is presented in the upper panel in Figure~\\ref{fig:tess_phot}.\n\nA Data Validation Report for L~168-9 was released to the community as part of the MIT TESS\nAlerts\\footnote{\\url{https:\/\/tev.mit.edu\/toi\/alerts\/}}. The report includes \nseveral validation tests to assess the probability that the signal is a false positive:\neclipsing-binary discrimination tests, a statistical bootstrap test, a ghost diagnostic\ntest, and difference-image centroid offset tests. These are described by\n\\citet{Twicken2018}. All the tests were passed successfully. The formal false-alarm \nprobability of the planet candidate was reported to be 5.85$\\times 10^{-37}$.\n\nThe TESS time series covers 19 transits of what was originally deemed a planet candidate\n(L~168-9\\,b or TOI-134.01), and reported on the TESS exoplanet Follow-up Observing Program\n(TFOP) website\\footnote{\\url{https:\/\/exofop.ipac.caltech.edu\/tess\/}}. According to the \nData Validation Report the orbital period is P~$=1.401461\\pm0.000137$ days and\nthe transit depth is $\\Delta F\/F_0=566\\pm38\\ ppm$, which translates into a planetary radius \nof R$_p=1.58\\pm0.36$ R$_\\oplus$ (Sect.~\\ref{subsec:stellarparam} describes how we determined\nthe stellar radius, which is the same value as the used in the Validation Report). \nThe time of mid-transit at an arbitrarily chosen reference epoch is (BJD)\nT$_c=2458326.0332\\pm0.0015$.\n\n\\subsubsection{LCOGT, MKO, and SSO T17}\n\\label{subsubsec:groundphot}\n\nWe acquired ground-based time series photometric follow-up of L~168-9 and the nearby field\nstars as part of the TESS Follow-up Observing Program (TFOP) to attempt to rule out nearby\neclipsing binaries (NEBs) in all stars that are bright enough to cause the TESS detection\nand that could be blended in the TESS aperture. We used the \n{\\tt TESS Transit Finder}\\footnote{\\url{https:\/\/astro.swarthmore.edu\/telescope\/tess-secure\/find\\_tess\\_transits.cgi}}, \nwhich is a customized version of the {\\tt Tapir} software package \\citep{Jensen:2013}, to schedule our\ntransit observations.\n\nWe observed one full transit simultaneously using three 1-meter telescopes at the Las Cumbres\nObservatory Global Telescope (LCOGT) \\citep{Brown2013} South Africa Astronomical Observatory\nnode on 21 September 2018 in $i'$-band. The Sinistro detectors consist of \n4K$\\times$4K 15-$\\mu m$ pixels with an image scale of $0\\farcs389$  pixel$^{-1}$, \nresulting in a field-of-view of $26\\farcm5 \\times 26\\farcm5$.\nThe images were calibrated by the standard LCOGT BANZAI pipeline. \n\nWe observed a full transit from the Mount Kent Observatory (MKO) 0.7-meter telescope near\nToowoomba, Australia on 23 September 2018 in $r'$-band. The Apogee U16 detector consists of\n4K$\\times$4K 9-$\\mu m$ pixels with an image scale of $0\\farcs41$  pixel$^{-1}$, resulting in\na field-of-view of $27\\arcmin\\times27\\arcmin$. The images were calibrated using \n{\\tt AstroImageJ} ({\\tt AIJ}) software package \\citep{Collins:2017}.\n\nWe observed a full transit from the Siding Spring Observatory, Australia, iTelescope T17\n0.43-meter telescope on 27 September 2018 with no filter. The FLI ProLine PL4710 detector\nconsists of 1K$\\times$1K pixels with an image scale of $0\\farcs92$  pixel$^{-1}$, \nresulting in a field-of-view of $15\\farcm5\\times15\\farcm5$. The images were calibrated using \n{\\tt AstroImageJ}.\n\nWe used the {\\tt AstroImageJ} to extract differential light curves of L~168-9 and all known\nstars within $2\\farcm5$ of the target star that are bright enough to have possibly produced the\nshallow TESS detection, which includes 11 neighbors brighter than TESS-band = 17.9 mag. \nThis allows an extra 0.5 in delta magnitude relative to L~168-9 to account for any inaccuracies\nin the TESS band reported magnitudes in the TICv8. The L~168-9 light curves in all five\nphotometric data sets show no significant detection of the shallow TESS detected event, as\nexpected from our lower precision ground-based photometry. Considering a combination of all five\nphotometric data sets, we exclude all 11 known neighbors that are close enough and bright enough\nto L~168-9 to have possibly caused the TESS detection as potential sources of the TESS\ndetection. \n\n\\subsubsection{WASP}\n\nWASP-South, located in Sutherland, South Africa, is the southern station of the \n\\textit{Wide Angle Search for Planets} \\citep[WASP,][]{Pollacco2006}.  It consists \nof an array of 8 cameras each backed by a 2048x2048 CCD. Observations in 2010 and 2011 \n(season A) used 200mm, f\/1.8 lenses with a broadband filter spanning $400-700$ nm and a \nplate scale of $13.7\\arcsec$\/pixel. Then, from 2012 to 2014 (season B), WASP-South used \n85mm, f\/1.2 lenses with a Sloan r' filter and a  plate scale of $32\\arcsec$\/pixel. \nThe array rastered a number of fields each clear night at typically 10-min cadence.\n\nL~168-9 was monitored for four consecutive years, from to May 20, 2010 to December 12, 2014;\ntypically covering 150 days in each year. In one campaign two cameras with overlapping\nfields observed the star, giving a total of 27\\,300 data points; in another campaign,\nL~168-9 was observed by three cameras with overlapping fields, totalling 170\\,000 data\npoints. The photometry has a dispersion of 0.027 $\\delta$mag and average uncertainty of \n0.024 $\\delta$mag, presenting clear signs of variability, as shown below in\nSec.~\\ref{subsec:rotation}.\n\n\n\\subsection{High-resolution Imaging}\n\\label{sec:image}\nThe relatively large 21\\arcsec pixels of TESS can lead to photometric contamination from \nnearby sources. These must be accounted for to rule out astrophysical false positives, \nsuch as background eclipsing binaries, and to correct the estimated planetary radius, \ninitially derived from the diluted transit in a blended light curve \\citep{ziegler18}. \nWithout this correction, the interpreted planet radius can be underestimated \n\\citep[e.g.,][]{ciardi2015, Teske2018}. \n\n\n\\subsubsection{SOAR}\n\\label{subsubsec:soar}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=0.5]{figures\/TIC234994474I-cz20180925_2.pdf}\n\\caption{SOAR speckle results of L~168-9. Black points represent the I-band contrast \nobtained at a given separation of the star. The solid black line shows the $5 \\sigma$\ndetection limit curve.}\n\\label{fig:soar}\n\\end{figure}\n\n We searched for close companions to L~168-9 with speckle imaging on the \n 4.1-m Southern Astrophysical Research telescope \\citep[SOAR,][]{tokovinin18} installed \n in Cerro Pach\\'on, Chile, on 2018 September 25 UT using the I-band \n ($\\lambda_{cen}=824\\ nm$, full width at half maximum$=170\\ nm$) centered  approximately on \n the TESS passband. Further details of the TESS SOAR survey are published in \\cite{Ziegler2019}.\n \n Figure \\ref{fig:soar} shows the 5$\\sigma$ detection sensitivity. No nearby stars \n ($\\rho < 1 \\farcs 6$) to L~168-9 were detected within the sensitivity\n limits of SOAR.\n\n\n\\subsubsection{Gemini-South}\n\\label{subsubsec:gemini}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[scale=1.8]{figures\/TOI134_f1ab.png}\n\\caption{DSSI\/Gemini-S detection limit curves of L~168-9 in the 692 nm (top) and 880 nm \n(bottom) filters. Squares and points on left panel represent local maxima and local minima, respectively.\n}\n\\label{fig:speckle}\n\\end{figure}\n\n Observations of L~168-9 were conducted with the Differential Speckle Survey Instrument \n (DSSI; \\citealt{Horch2009}) on Gemini South, Chile, on UT 28 October 2018 under program \n GS-2018B-LP-101 (PI: I. Crossfield). The usual 692 nm and 880 nm filters on DSSI were used, \n and three sequences of 60 ms\/frame$\\times$1000 frames were taken. The total time on target, \n including readout overhead, was six minutes. \\cite{Howell2011}, \\cite{Horch2011a}, \n and \\cite{Horch2012} detail the speckle observing and data reduction procedures. \n The detection limit curves are shown in Figure~\\ref{fig:speckle}. \n While the 692 nm image was taken at too low of gain, leading to a shallow detection limit, \n the 880 nm detection limit curve ($b$) in Figure~\\ref{fig:speckle} indicates that \n L~168-9 lacks any companions of $\\Delta m \\sim$5.0 mag beyond 0.1\\arcsec \n and any companions of $\\Delta m \\sim$5.5 mag beyond 0.2\\arcsec. Gemini-South and \n SOAR result (Sect.~\\ref{subsubsec:soar}) mean the transit signal is likely to be \n associated with L~168-9.\n\nThus, we conclude that there is no significant contamination of the TESS photometric\naperture that would bias the determination of the planet radius. \nThe Sinistro data (Sect.~\\ref{subsubsec:groundphot}) rule out surrounding stars as \npotential sources of the TESS detection, so we can assume the planet orbits L~168-9. \nGiven the limits placed on nearby companions by the Gemini-South data, if there were \na close companion it would have to be fainter by $\\sim 5$ magnitudes than the primary star,\nmeaning the planet radius correction factor would be at most\n\\begin{equation}\n    R_{p,corr} = R_{p}\\times \\sqrt{1 + 10^{-0.4\\Delta m}} = R_{p}\\times1.005.\n\\end{equation}\n\n\\noindent This is smaller than the derived planet radius uncertainty (Table~\\ref{tab:results}).\n\n\n\\subsection{Radial velocity}\n\\label{sec:rv}\n\n\\subsubsection{HARPS}\nThe \\textit{High Accuracy Radial velocity Planet Searcher} \\citep[HARPS][]{Mayor2003} \nis an echelle spectrograph mounted on the 3.6m telescope at La Silla Observatory, Chile.\nThe light is spread over two CCDs (pixel size 15 $\\mu m$) by a science fiber and a\ncalibration fiber. The calibration fiber can be illuminated with the calibration\nlamp for the best radial velocity precision, or it can be placed on sky for moderate\nprecision. HARPS is stabilized in pressure and temperature, and has a resolving\npower of 115{,}000.  The achievable precision in radial velocity of better than 1~$m\/s$.\n\nWe began monitoring L~168-9 with HARPS on September 29, 2018, soon after the\nTESS alert. We elected not to use the simultaneous wavelength calibration \n(i.e.\\ the on-sky calibration fiber) to ensure that the bluer spectral regions \nwould not be contaminated by the calibration lamp, that provides a much stronger flux \nfor any instrumental setup.. The exposure time was set to 900 s for ESO programs \n198.C-0838 and 1102.C-0339, and to 1{,}200 s for ESO program 0101.C-0510, translating \nin a median signal-to-noise ratio per spectral pixel of 51 and 70 at 650 nm, respectively. \nA single spectrum on October 1, 2018, had an exposure time of 609 s for an unknown reason. \nA total of 47 HARPS spectra were collected, ending with observations on December 19, 2018.\n\nHARPS spectra were acquired in roughly three packs of data separated in time by about \n40 days. This sampling is reflected in the window function presented in\nFigure~\\ref{fig:GLSP}. Two archival spectra of L~168-9 are available at the ESO database. \nHowever, a radial velocity offset was introduced on May 2015 because the vacuum vessel was\nopened during a fiber upgrade \\citep{LoCurto2015}. As this offset is not yet well\ncharacterized for M dwarfs, we decided to disregard those two points (from July 2008 and\nJune 2009) in our subsequent analysis.\n\nThe HARPS Data Reduction Software \\citep{LovisPepe2007} computes radial velocities\nby a cross-correlation function technique \\citep[e.g.,]{Baranne:1996}. Nevertheless \nwe derived radial velocities by a different approach to exploit as much as possible the\nDoppler information of spectra \\citep[e.g.,][]{Anglada2012}.\nWe performed a maximum likelihood analysis between a stellar template and each individual\nspectrum following the procedure presented in \\citet{Astudillo-Defru2017b}. The stellar\ntemplate corresponds to a true stellar spectrum of the star, enhanced in signal-to-noise. \nIt was made from the median of all the spectra, after shifting them into a common\nbarycentric frame. The resulting template is Doppler shifted by a range of trial radial\nvelocities to construct the maximum likelihood function, from which we derived the HARPS\nradial velocity used in the subsequent analysis. The obtained radial velocities -- listed \nin Table~\\ref{tab:rvHARPS} -- present a dispersion of 4.01 $m\/s$ and a median photon \nuncertainty of 1.71 $m\/s$. Figure~\\ref{fig:phasedRV} shows the radial velocities \nfolded to the orbital period.\n\n\n\\subsubsection{PFS}\n\nThe Planet Finder Spectrograph is an iodine-calibrated, environmentally controlled high resolution \nPRV spectrograph \\citep{Crane:2006,Crane:2008,Crane:2010}. Since first light \nin October 2009, PFS has been running a long-term survey program to search for \nplanets around nearby stars (e.g., ~\\citealt{Teske:2016}). In January 2018, \nPFS was upgraded with a new large format CCD with $9\\mu m$ pixels and switched \nto a narrower slit for its regular operation mode to boost the resolution from \n80{,}000 to 130{,}000. The PFS spectra are reduced and analyzed with \na custom IDL pipeline that is capable of delivering RVs with $<$1~m\/s precision \n\\citep{Butler:1996}.\n\nWe followed up L~168-9 with PFS on the 6.5~m Magellan II Clay telescope \nat Las Campanas Observatory in Chile from October 13--26, and then \non December 16 and 21 in 2018. Observations were conducted on 15 nights,\nwith multiple exposures per night. There were a total of 76 exposures\nof 20 minutes each. We typically took 2--6 exposures per night over a range of timescales,\nto increase the total SNR per epoch and also to average \nout the stellar and instrumental jitter. Each exposure had a typical \nSNR of 28 per pixel near the peak of the blaze function, or 56 per resolution element. \nThe radial velocity dispersion is 4.61 m\/s and the median RV uncertainty per exposure is \nabout 1.8~m\/s. Five consecutive 20-minute iodine-free exposures were obtained to allow for \nthe construction of a stellar spectral template in order to extract the RVs. These were\nbracketed with spectra of rapidly rotating B stars taken through the iodine cell, for\nreconstruction of the spectral line spread function and wavelength calibration for the \ntemplate observations.\n\nThe PFS observations of L~168-9 presented here are part of the Magellan \nTESS Survey (MTS) that will follow up $\\sim$30 super-Earths and sub-Neptunes \ndiscovered by TESS in the next three years using PFS (Teske et al.~in prep.). \nThe goal of MTS is to conduct a statistically robust survey to understand the \nformation and evolution of super-Earths and sub-Neptunes. The observation \nschedules of all MTS targets, including L~168-9, can be found on the \nExoFOP-TESS website.\\footnote{\\url{https:\/\/exofop.ipac.caltech.edu\/tess\/}}\n\n\n\\section{Analysis}\n\\label{sec:analysis}\n\n\\begin{figure*}\n    \\sidecaption\n   \n    \\includegraphics[scale=0.5]{figures\/WASP_gpv2.png}\n    \\caption{The WASP photometry. \\emph{Left column}: The photometry time series; the blue\n    shaded regions depict $\\pm 1\\sigma$ about the mean GP regression model of the binned\n    photometry. \\emph{Right column}: the generalized Lomb-Scargle periodogram of each season\n    highlights the prevalence of photometric variations close to the measured rotation\n    period and\/or its first and second harmonics (vertical dashed lines).}\n    \\label{fig:WASPphot}\n\\end{figure*}\n\n\n\\subsection{Photometric rotation period}\n\\label{subsec:rotation}\n\n\\begin{figure}\n    \\centering\n    \\includegraphics[width=0.98\\hsize]{figures\/periodogramsv2.png}\n    \\caption{\\emph{Left column}: Generalized Lomb-Scargle periodograms of \n    the HARPS and PFS RV time series, window functions, and the S-index, \n    $H\\alpha$, $H\\beta$, $H\\gamma$, and sodium doublet activity indicators. \n    The vertical dotted lines highlight the locations of the L~168-9~b orbital \n    period, the stellar rotation period, and its first two harmonics. \n    \\emph{Right column}: the false alarm probabilities computed \n    from bootstrapping with replacement.}\n    \\label{fig:GLSP}\n\\end{figure}\n\nKnowledge of the stellar rotation period helps to disentangle\nspurious RV signals arising from rotation and true RV signals due to orbital motion\n\\citep[e.g.,][]{Queloz2001,Cloutier2017b}. L~168-9 was photometrically monitored by WASP \nbetween May 2010 and December 2014 within five observing seasons each lasting approximately \n200 days in duration. The photometric precision, observing cadence, and baselines within \nthe WASP fields are sufficient to detect quasi-periodic (QP) photometric variations of\nL~168-9 due to active regions on the stellar surface rotating in and out of view at the\nstellar rotation period $P_{\\text{rot}}${.} The WASP photometry of L~168-9 is shown in \nFig.~\\ref{fig:WASPphot} along with the generalized Lomb-Scargle periodogram \n\\citep[GLSP;][]{Zechmeister2009} of the photometry in each of the five WASP observing \nseasons. It is clear from the GLSPs that a strong periodicity exists within the data \nwhose timescale is often $\\sim 30$ days except for within the second WASP season wherein \nthe dominant periodicity appears at the first harmonic of $P_{\\text{rot}}${} $\\sim 15$ days.\n\nGiven periodicities significantly detected in the WASP photometry, we proceeded to measure \nthe photometric rotation period of L~168-9 $P_{\\text{rot}}${} with each WASP camera and in each \nWASP field\\footnote{At times, L~168-9 appeared within the fields-of-view of multiple \nWASP cameras.} in which L~168-9 was observed. As the photometric variations appear to \nvary nearly periodically, we modeled the photometry with a Gaussian process (GP) regression \nmodel and adopted a QP covariance kernel (see Eq.~\\ref{eq:covariance})\n\\citep{Angus2018}. \nThe covariance function's periodic timescale was a free parameter $P_{\\text{rot}}${} for which the\nposterior probability density function (PDF) was sampled using a Markov Chain \nMonte Carlo (MCMC) method (see Appendix~\\ref{sec:GP}). We modeled each binned WASP light \ncurve with a GP. We adopted a bin size of 1 day to reduce the computation time. \nIn preliminary analyses, we also tested bin sizes of 0.25, 0.5, and 2 days \nand found that, probably due to the very large number of points, the recovered values of \n$P_{\\text{rot}}${} were not very sensitive to this choice. \n\nAfter sampling the posterior PDFs of the GP hyperparameters, we arrived at point estimates \nof each parameter value based on the maximum a-posteriori values and 68 percent confidence\nintervals. The resulting mean GP model of the data from each WASP observing season is\ndepicted in Fig.~\\ref{fig:WASPphot} along with the corresponding $1\\sigma$ confidence \ninterval. Over the five observing seasons, the measured (median) rotation period \nof L~168-9 was $P_{\\text{rot}}${} $=29.8\\pm 1.3$ days. \n\nIn principle, the WASP signal could have arisen from any star within the 48\" extraction\naperture. However, L~168-9 is by far the brightest star in the aperture. Another concern\nwith any photometric signal with a period near 30 days is whether it was affected by\nmoonlight. To check on this possibility, we searched for modulations in the WASP data of\nseveral stars of similar brightness within the surrounding 10 arcmin field, but did not find\nany 30-d signals similar to the one that was seen for L~168-9. In any case, the star\nlocation is far from the ecliptic, and moonlight contamination is not expected.\nFurthermore, the modulation was sometimes seen at the 15-d first harmonic, which would not\nbe expected for moonlight. We can therefore be confident that the 30-d periodicity in the\nWASP data arises from L~168-9. The $R^\\prime_{HK}$ from HARPS spectra supports the \nobtained photometric rotation period, as $log(R^\\prime_{HK})=-4.562\\pm0.043$ (active star)\ntranslates into $P_{\\text{rot}}${} $=22\\pm 2$ days using the $R^\\prime_{HK}$vs.$P_{\\text{rot}}${} relationship\nfrom \\cite{Astudillo2017a}.\n\n\\begin{figure*}[t]\n\\centering\n\\includegraphics[scale=0.5]{figures\/TESS_phot_detrend_small.pdf}\n\\caption{The time series of TESS data for L~168-9. One fifth of original data are \n    plotted for visualization purposes. The de-trended TESS light curve is \n    shown in the \\textit{upper panel}. Red vertical bars represent the transits \n    of the planet candidate. \\textit{Bottom panel:} the phase folded normalized photometry.\n    Red points correspond to binned data for illustrative purposes. The whole orbital phase \n    is shown to the left, while the right presents the transit phase, as well\n    as the best transit model whose parameters come from the Table~\\ref{tab:results}.}\n\\label{fig:tess_phot}\n\\end{figure*}\n\n\\subsection{Radial Velocity Periodogram Analysis}\n\\label{subsec:kep}\n\nA first identification of significant periodicities in the HARPS and PFS RV time series is \nrequired in order to develop an accurate model of the observed RV variations. In a manner\nsimilar to our analysis of the WASP photometry, we computed the GLSP of the following \nHARPS and PFS spectroscopic time series: the RVs, the window functions (WF), and the\nS-index, $H\\alpha$, $H\\beta$, $H\\gamma$, and the sodium doublet NaD activity indicators.\n\\cite{Astudillo-Defru2017b} details how these spectroscopic activity indicators were\nderived. Each GLSP is shown in Fig.~\\ref{fig:GLSP} along with a false alarm probability\n(FAP) that was computed via bootstrapping with replacement using\n$10^4$ iterations and normalizing each periodogram by its standard deviation.\n\nEach of the GLSPs of the HARPS and PFS RV time series is dominated \nby noise and aliases arising from the respective WF. For example, the HARPS WF contains \na forest of peaks with comparable FAP from $\\sim 8$ days and extending out \ntowards long periodicities. Similarly the GLSP of the PFS WF reveals a series \nof broad peaks for periodicities $\\gtrsim 1$ day. These features, particularly \nthose from the PFS WF, have clear manifestations in the GLSPs of their respective \nRV and activity indicator time series, thereby complicating the robust identification of \nperiodocities in the data. However, strong peaks at the orbital period \nof L~168-9~b ($\\sim 1.4$ days) are discernible in both RV time series at \nFAP $\\sim 0.4\\%$ and $\\sim 0.3\\%$ with HARPS and PFS respectively. This periodicity \nis not apparent in any of the ancillary activity indicators time series as expected \nfor a signal originating from an orbiting planet. \n\nIn addition to the signal from L~168-9~b, the HARPS RVs exhibit some power close \nto $P_{\\text{rot}}${} and its second harmonic $P_{\\text{rot}}${}$\/3$. Although the FAPs of these \nperiodicities over the full frequency domain are large, they each appear \nlocally as strong periodicities as most of the power in the HARPS RVs exists\nat $\\lesssim 5$ days. The GLSP of the PFS RVs is much more difficult to interpret\nat periodicities in the vicinity of $P_{\\text{rot}}${} and its first and second harmonics \ndue to strong aliases from the PFS WF. Due to these effects it is difficult to \ndiscern from the available PFS activity indices (i.e. S-index and $H\\alpha$) \nwhether or not a coherent activity signal is seen with PFS. Although each of \nthe HARPS and PFS RV time series are significantly affected by sampling aliases, \nwe do see evidence for L~168-9~b and rotationally modulated stellar activity \nin the RVs we endeavor to mitigate with our adopted model discussed \nin Sect.~\\ref{subsec:model}. \n\n\n\\subsection{Radial velocity + transit model}\n\\label{subsec:model}\n\nGuided by the periodicities in the HARPS and PFS RV time series, we proceeded \nto fit a model to the RVs including the effects of both stellar activity and the planet.\nFollowing numerous successful applications on both Sun-like \n\\citep[e.g.,][]{Haywood2014,Grunblatt2015,Faria2016,Lopezmorales2016,Mortier2016} \nand M dwarf stars \\citep[e.g.,][]{Astudillo-Defru2017c,Bonfils2018,Cloutier2019a,Ment2019}, \nwe adopted a QP kernel for the GP as a non-parametric model of the physical processes \nresulting in stellar activity. When used to model RV stellar activity, the QP \ncovariance kernel is often interpreted as modelling the rotational component of \nstellar activity from active regions on the rotating stellar surface whose lifetimes \ntypically exceed many rotation cycles on M dwarfs \\citep{Giles2017} plus the \nevolutionary time scale of the active regions. The corresponding GP hyperparameters \nare described in detail in Appendix~\\ref{sec:GP} and include each spectrograph's \ncovariance amplitudes $a_{\\text{HARPS}}$, $a_{\\text{PFS}}$, the common exponential \ntimescale $\\lambda_{\\text{RV}}$, the common coherence parameter $\\Gamma_{\\text{RV}}$, \nand the common periodic timescale $P_{\\text{RV}}$ equal to the stellar rotation period $P_{\\text{rot}}${.}\n\nThe planetary component attributed to the transiting planet L~168-9~b was fitted  \nto the de-trended light curve with a \\cite{MandelAgol2002} planetary transit model. \nThe de-trended TESS light curve was produced by adjusting a QP GP systematic \nmodel to the photometry alone and with all the transits previously removed. \nThe best QP GP model was subtracted to the entire TESS data set. The planetary component is \nmodelled by a Keplerian solution parameterized by the planet's orbital period $P_b$, \ntime of mid-transit $T_0$, RV semi-amplitude $K$, and the orbital parameters \n$h=\\sqrt{e}\\cos{\\omega}$ and $k=\\sqrt{e}\\sin{\\omega}$ where $e$ and $\\omega$ \nare the planet's orbital eccentricity and argument of periastron respectively. \nIn addition, our RV model contains each spectrograph's zero point velocity \n$\\gamma_{\\text{HARPS}}$, $\\gamma_{\\text{PFS}}$ and an additive scalar jitter \n$s_{\\text{HARPS}}$, $s_{\\text{PFS}}$ is account for any residual jitter that, \nunlike the stellar activity signal, is not temporally correlated. The complete \nRV model therefore contains fourteen model parameters.\n\nTo ensure self-consistent planet solutions between the available TESS transit \ndata and the RV observations, we simultaneously fitted the de-trended light curve \nand the RVs. The common planetary parameters between these two data sets \nare $P_b$, $T_0$, $h$, and $k$.\nThe additional model parameters required to model the TESS transit light curve \nincluded an additive scalar jitter $s_{\\text{TESS}}$, \nthe baseline flux $\\gamma_{\\text{TESS}}$, the scaled semi-major axis $a\/R_s$, \nthe planet-star radius ratio $r_p\/R_s$, the orbital inclination $i$, and the \nnearly-uncorrelated parameters $q_1$ and $q_2$ which are related to the \nquadratic limb darkening coefficients $u_1$ and $u_2$ via\n\\begin{align}\n    q_1 &= (u_1+u_2)^2 \\\\\n    q_2 &= \\frac{u_1}{2(u_1+u_2)}.\n    \\label{eq:LDCs}\n\\end{align}\n\n\\noindent \\citep{Kipping2013}. Thus we required eleven model parameters to describe the TESS \ntransit light curve and a total of twenty-one model parameters of the joint RV + light curve \ndata set: $\\boldsymbol{\\Theta}=\\{a_{\\text{HARPS}}, a_{\\text{PFS}}, \\lambda_{\\text{RV}}, \n\\Gamma_{\\text{RV}}, P_{\\text{RV}}, s_{\\text{TESS}}, s_{\\text{HARPS}}, s_{\\text{PFS}}, \\gamma_{\\text{TESS}}, \\gamma_{\\text{HARPS}}$, $\\gamma_{\\text{PFS}}, \nP_b, T_0, K, h, k, a\/R_s, r_p\/R_s, i, q_1, q_2 \\}$.\n\nWe sampled the posterior PDF of this 21-dimensional parameter space using an MCMC sampler. \nDetails on the sampler and the adopted prior distributions on each model parameter are \ngiven in Appendix~\\ref{sec:GP} and Table~\\ref{tab:priors}.\n\n\n\\section{Results}\n\\label{sec:results}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=1.0\\hsize]{figures\/RVphased.png}\n    \\caption{Phase folded radial velocity acquired HARPS (blue points) and PFS (red points)\n    where the best GP model was subtracted. The black curve represents the maximum a-posteriori\n    model adjusted to the data set.}\n    \\label{fig:phasedRV}\n\\end{figure}\n\n\nIn our analysis of the light curve and radial velocity time series of L~168-9, we evaluated \nthe model presented in Sect.~\\ref{subsec:model} on the separate RV data sets obtained with \nHARPS and PFS as well as the combined time series. Subtracting the model to the 46 HARPS \nradial velocity points reduces the dispersion to 3.37 m\/s (equivalent to $\\chi^2_{red}=3.5$),\nwhile the dispersion of the 76 PFS residual points gives 4.05 m\/s (translating into\n$\\chi^2_{red}=1.5$). The dispersion obtained from the 122 HARPS+PFS residual points is \n3.80 m\/s ($\\chi^2_{red}=2.2$).\n\nFrom point estimates of the model parameters \n$\\boldsymbol{\\Theta}$ from our joint RV plus transit analysis with each of these \ninput data sets we retrieved that L~168-9~b has a radius of 1.39$\\pm$0.09 R$_\\oplus$ and \na mass of 4.60$\\pm$0.56 M$_\\oplus$, translating into a bulk mean density of \n$9.6^{+2.4}_{-1.8}$ g cm$^{-3}$. The planet is orbiting at 0.02091$\\pm$0.00024 AU from \nthe parent star, therefore the hot terrestrial planet has an equilibrium temperature between \n668 K and 965 K assuming a Venus-like and zero bond albedo, respectively.\nResults for the entire set of parameters are reported in Table~\\ref{tab:results}. \nExplicitly, we report the maximum a-posteriori value of each parameter along with its\n16$^{th}$ and 84$^{th}$ percentiles, corresponding to a $1 \\sigma$ confidence interval. \nWe check for consistency of our joint analysis by performing the analysis for each instrument\nindependently. Table~\\ref{tab:results} details the results from this test. We note that HARPS\nand PFS results are in agreement within their uncertainties, translating into a robust detection\nof the planetary signal in radial velocity data.\n\nFigure~\\ref{fig:tess_phot} show the TESS photometry and adjusted transit model and\nFigure~\\ref{fig:phasedRV} the phase folded radial velocity with the model that best \nfits the data.\n\n\n\\begin{table*}[t]\n  \\caption{Measured transit and RV model parameters of the L~168-9 planetary system.}\n  \\label{tab:results}\n  \\centering\n  \\begin{tabular}{lccc}\n    \\hline\\noalign{\\smallskip}\n    Measured transit model parameters &  & TESS &  \\\\\n    \\noalign{\\smallskip}\n    \\hline\\noalign{\\medskip}\n    Baseline flux, $\\gamma_{\\text{TESS}}$ & \\multicolumn{3}{c}{$0.00001\\pm0.00029$}  \\\\\n    \\noalign{\\smallskip}\n    Orbital period, $P_b$ [days] & \\multicolumn{3}{c}{$1.40150\\pm 0.00018$}   \\\\\n    \\noalign{\\smallskip}\n    Time of mid-transit, $T_0$ [BJD-2,457,000] & \\multicolumn{3}{c}{$1340.04781^{+0.00088}_{-0.00122}$}  \\\\\n    \\noalign{\\smallskip}\n    Scaled semi-major axis, $a\/R_s$ & \\multicolumn{3}{c}{$7.61\\pm 0.31$}   \\\\\n    \\noalign{\\smallskip}\n    Planet-star radius ratio, $r_p\/R_s$ & \\multicolumn{3}{c}{$0.0212\\pm 0.001$}   \\\\\n    \\noalign{\\smallskip}\n    Orbital inclination, $i$ [deg] & \\multicolumn{3}{c}{$85.5^{+0.8}_{-0.7}$}  \\\\\n    \\noalign{\\smallskip}\n    Linear limb darkening coefficient, $q_1$ & \\multicolumn{3}{c}{$0.397^{+0.125}_{-0.111}$}   \\\\\n    \\noalign{\\smallskip}\n    Quadratic limb darkening coefficient, $q_2$ & \\multicolumn{3}{c}{$0.189^{+0.058}_{-0.056}$}   \\\\\n    TESS additive jitter, $s_{\\text{TESS}}$ & \\multicolumn{3}{c}{$0.00003^{+0.00011}_{-0.00003}$}  \\\\\n    \\noalign{\\smallskip}    \n    \\noalign{\\medskip}    \n    \\hline\\noalign{\\smallskip}\n    Radial velocity GP hyperparameters & HARPS+PFS & HARPS & PFS \\\\\n    \\noalign{\\smallskip}\n    \\hline\\noalign{\\medskip}\n    ln HARPS covariance amplitude,  $\\ln{(a_{\\text{HARPS}}\/\\text{m s}^{-1})}$ & $3.27^{+1.38}_{-1.03}$ & $3.24^{+1.24}_{-1.19}$ & - \\\\\n    \\noalign{\\smallskip}\n    ln PFS covariance amplitude, $\\ln{(a_{\\text{PFS}}\/\\text{m s}^{-1})}$ & $3.63^{+1.24}_{-1.08}$ & - & $4.01^{+1.23}_{-0.97}$ \\\\\n    \\noalign{\\smallskip} \n    ln RV exponential timescale,\n    $\\ln{(\\lambda_{\\text{RV}}\/\\text{day})}$ & $11.90^{+2.81}_{-1.99}$ & $12.25^{+3.16}_{-2.18}$ & $13.3^{+2.02}_{-1.44}$\\\\\n     \\noalign{\\smallskip} \n    ln RV coherence, $\\ln{(\\Gamma_{\\text{RV}})}$ & $-0.09^{+0.20}_{-0.24}$ & $-0.42^{+0.41}_{-0.55}$ & $-0.22^{+0.39}_{-0.44}$ \\\\\n     \\noalign{\\smallskip}\n    ln RV periodic timescale,\n    $\\ln{(P_{\\text{RV}}\/\\text{day})}$ & $3.47^{+0.02}_{-0.03}$ & $3.47^{+0.03}_{-0.03}$ & $3.48^{+0.04}_{-0.03}$ \\\\\n    \\noalign{\\smallskip} \n    HARPS additive jitter, $s_{\\text{HARPS}}$ [m s$^{-1}${]} & $0.10^{+0.20}_{-0.09}$ & $0.86^{+0.98}_{-0.86}$ & - \\\\\n    \\noalign{\\smallskip} \n    PFS additive jitter, $s_{\\text{PFS}}$ [m s$^{-1}${]} & $2.82\\pm 0.42$ & - & $2.78\\pm 0.30$  \\\\\n    \\noalign{\\medskip}\n    \\hline\\noalign{\\smallskip}\n    Measured RV model parameters & HARPS+PFS & HARPS & PFS \\\\\n    \\noalign{\\smallskip}\n    \\hline\\noalign{\\medskip}\n    HARPS zero point velocity, $\\gamma_{\\text{HARPS}}$ [km s$^{-1}$] & $29.7687\\pm 0.0013$ & $29.7692\\pm 0.0013$ & -  \\\\\n    \\noalign{\\smallskip}\n    PFS zero point velocity, $\\gamma_{\\text{PFS}}$ [km s$^{-1}$] & $0.00077 \\pm 0.00186$ & - & $0.00063\\pm 0.00142$  \\\\\n    \\noalign{\\smallskip}\n    Semi-amplitude, $K$ [m s$^{-1}${]} & $3.66^{+0.47}_{-0.46}$ & $3.74^{+0.54}_{-0.59}$ & $3.26^{+0.52}_{-0.70}$  \\\\\n    \\noalign{\\smallskip}\n    $h=\\sqrt{e}\\cos{\\omega}$ & $-0.10^{+0.17}_{-0.12}$ & $0.04^{+0.15}_{-0.18}$ & $-0.01^{+0.18}_{-0.21}$  \\\\\n    \\noalign{\\smallskip}\n    $k=\\sqrt{e}\\sin{\\omega}$ & $0.00^{+0.17}_{-0.16}$ & $-0.09^{+0.22}_{-0.23}$ & $0.01^{+0.23}_{-0.30}$  \\\\\n    \\noalign{\\medskip}\n    \\hline\\noalign{\\smallskip}\n    Derived L~168-9~b parameters & HARPS+PFS+TESS & HARPS+TESS & PFS+TESS \\\\\n    \\noalign{\\smallskip}\n    \\hline\\noalign{\\medskip}\n    Semi-major axis, $a$ [AU] & $0.02091\\pm 0.00024$ && \\\\\n    Equilibrium temperature, $T_{\\text{eq}}$ [K] &&& \\\\\n    \\hspace{10pt} Zero bond albedo & $965\\pm 20$ && \\\\\n    \\hspace{10pt} Venus-like bond albedo = 0.77 & $668\\pm 14$ && \\\\\n    Planetary radius, $R_p$ [R$_{\\oplus}$] & $1.39\\pm 0.09$ && \\\\\n    Planetary mass, $M_p$ [M$_{\\oplus}$] & $4.60\\pm 0.56$ & $4.74^{+0.71}_{-0.75}$ & $4.08^{+0.70}_{-0.90}$ \\\\\n    Planetary bulk density, $\\rho_p$ [g cm$^{-3}$] & $9.6^{+2.4}_{-1.8}$ & $10.0^{+2.5}_{-1.9}$ & $8.7^{+2.3}_{-2.1}$ \\\\\n    Planetary surface gravity, $g_p$ [m s$^{-2}$] & $23.9^{+4.5}_{-3.9}$ & $24.4^{+4.8}_{-4.2}$ & $21.4^{+4.4}_{-4.8}$ \\\\\n    Planetary escape velocity, $v_{\\text{esc}}$ [km s$^{-1}$] & $20.5^{+1.4}_{-1.4}$ & $20.8^{+1.6}_{-1.7}$ & $19.4^{+1.6}_{-2.2}$ \\\\\n    Orbital eccentricity, $e^\\dagger$ & $< 0.21$ & $< 0.25$ & $< 0.26$ \\\\\n    \\noalign{\\smallskip}\\hline\n  \\end{tabular}\n  \n  \\begin{list}{}{}\n  \\item[$^\\dagger$] 95\\% confidence interval.\n  \\end{list}\n\\end{table*}\n\n\n\\section{Discussion \\& Conclusions}\n\\label{sec:conclusion}\n\nL~168-9~b adds to the family of small ($<2R_\\oplus$) transiting planets around bright \n($J<$ 8 mag) stars with mass measurements and contributes to the completion of the \nTESS Level One Science Requirement to detect and measure the masses of 50 small planets.\nIn particular, L~168-9~b is one of fourteen\\footnote{L~98-59~bc \\citep{Cloutier2019b}, GJ~357~b \\citep{Luque2019}, HD~15337~b \\citep{Dumusque2019}, HD~213885~b \\citep{Espinoza2019}, GJ~9827~b \\citep{Rice2019}, K2-265~b \\citep{Lam2018}, K2-141~b \\citep{Barragan2018}, K2-229~b \\citep{Santerne2018}, HD~3167~b \\citep{Gandolfi2017}, K2-106~b \\citep{Guenther2017}, TRAPPIST-1~fh \\citep{Wang2017}, HD~219134~b \\citep{Motalebi2015}} \nlikely rocky planets without primordial hydrogen-helium envelopes \n-- that is,  with a radius $<1.8R_\\oplus$ -- \nfor which the mass has been measured with an uncertainty smaller than 33\\%.\nThus, our result represents progress toward the understanding of the \ntransition between super-Earths and mini-Neptunes previously reported in the radii of\nplanets \\citep[e.g.][]{Fulton2017,Cloutier2019c} but, here, including the information on mass.\n\nFigure~\\ref{fig:MR} shows the mass~-~radius diagram centered in the sub-Earth to\nmini-Neptune regime. With about twice the Earth average density, L~168-9~b bulk density \nis compatible with a terrestrial planet with an iron core (50\\%) surrounded by a mantle of\nsilicates (50\\%). In this diagram the detected planet is located in an interesting place:\nfor masses lower than that of L~168-9~b the great majority of planets are consistent with a \n50\\% Fe--50\\% MgSiO3 or 100\\% MgSiO3 bulk composition, while for higher planetary masses\nthere is a great diversity of density. Being one of the densest planets for masses greater\nthan 4M$_\\oplus$, L~168-9~b can help to define the mass limits of the rocky planets\npopulation. \n\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[scale=0.6]{figures\/MR.pdf}\n    \\caption{The mass-radius diagram showing L~168-9~b (red circle) in the context of known\n    exoplanets. The transparency of each point is proportional to its associated mass\n    uncertainty. Error bars correspond to $1 \\sigma$ uncertainties. Different models for \n    the bulk composition are plotted where the legend details the fraction of iron,\n    silicates, and\/or water for each color-coded curve.}\n    \\label{fig:MR}\n\\end{figure}\n\nGood targets for atmospheric characterization with transmission\/emission spectroscopy are \nthose transiting nearby, bright stars ($J<$ 8 mag). There are currently 11 small planets\ndetected transiting a bright star, according to the\nNASA Exoplanet Archive\\footnote{\\url{https:\/\/exoplanetarchive.ipac.caltech.edu\/}}, two of\nthem orbit M~dwarfs. Overall, there would not be a large number of small, transiting\nplanets around nearby, bright M~dwarfs. There are roughly 200 M~dwarfs within the 25~pc\nsolar neighborhood with $J<$ 8 mag, about 2\/3 of which are single stars\n\\citep{Winters:2015,Winters:2019}. Considering the occurrence rate of small planets with\norbital period smaller than 10 days from \\cite{Dressing:2015} and combining with the transit\nprobability of such planets (about a couple \\% to $\\sim$20\\%), there would be a few to up to\nabout twenty such planets.\n\nThe measured properties of L~168-9~b and its host star make it a promising target for \nthe atmospheric characterization of a terrestrial planet via either transmission \nor emission spectroscopy measurements with JWST \\citep{Morley2017,Kempton2018} and\/or\nthermal phase curve analysis to infer the absence of a thick atmosphere\n\\citep[e.g.,][]{Seager2009}. \nIts transmission and emission spectroscopy metrics from \\cite{Kempton2018} \nare reported in Table~\\ref{tab:atmosphere} and compared to other confirmed transiting \nterrestrial planets with known masses that are of interest for atmospheric characterization.\nBased on this assessment, L~168-9~b is an excellent candidate for emission spectroscopy or\nfor detecting the planetary day-side phase curve as recently done for the similar planet,\nLHS~3844~b \\citep{Kreidberg2019}.\n\n\n\\begin{table*}[t]\n  \\caption{Prospects of atmospheric characterization of confirmed terrestrial planets including L~168-9~b. \n  TSM and ESM correspond to transmission and emission spectroscopy metrics, respectively.}\n  \\label{tab:atmosphere}\n  \\centering\n  \\small\n  \\begin{tabular}{lcccccccccccccc}  \n    \\hline\\noalign{\\smallskip}\n    Star ID  & R$_{p}$ & M$_{p}$ & P & a& T$_{eff}$ & T$_{eq}$ & T$_{day}$ & J & K$_{\\rm s}$  & R$_\\star$ & M$_\\star$ & TSM & ESM & Ref.\\\\\n    units  & [R$_\\oplus$] & [M$_\\oplus$] & [days] & [AU]& [K] & [K] & [K] & [mag] & [mag]  & [R$_\\odot$] & [M$_\\odot$] &  &  & \\\\\n    \\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n    L~168-9~b   & 1.39 & 4.39 & 1.401 & 0.021 & 3743 & 963.01 & 1059.31& 7.941 & 7.0819 & 0.60 & 0.62 & 8.025  & 9.692& \\\\\n    LHS~3844~b   & 1.30 & --   & 0.463 & 0.006 & 3036 & 805.90 & 886.49 & 10.046& 9.145  & 0.19 & 0.15 &     -- & 29.004 & Kr2019\\\\\n    GJ~1132~b    & 1.13 & 1.66 & 1.629 & 0.015 & 3270 & 590.61 & 649.67 & 9.245 & 8.322  & 0.21 & 0.18 & 31.166 & 9.872 & Bo2018\\\\\n    L~98-59~c    & 1.35 & 2.17 & 3.690 & 0.032 & 3412 & 515.31 & 566.84 & 7.933 & 7.101  & 0.31 & 0.31 & 29.168 & 6.696 & Cl2019b\\\\\n    LTT~1445A~b  & 1.38 & 2.20 & 5.359 & 0.038 & 3335 & 433.34 & 476.68 & 7.29  & 6.5    & 0.28 & 0.26 & 44.976 & 6.382 & Wi2019\\\\\n    TRAPPIST-1~b & 1.09 & 1.02 & 1.511 & 0.011 & 2559 & 402.38 & 442.62 & 11.4  & 10.3   & 0.12 & 0.08 & 36.914 & 4.007 & Gi2017\\\\\n    LHS~1140~c   & 1.28 & 1.81 & 3.778 & 0.027 & 3216 & 436.43 & 480.07 & 9.612 & 8.821  & 0.21 & 0.18 & 25.225 & 3.401 & Me2019\\\\\n    \\noalign{\\smallskip}\\hline\n  \\end{tabular}\n\n  \\begin{list}{}{}\n  \\item Reference notes: Kr2019~--~\\citep{Kreidberg2019}; Bo2018~--~\\citet{Bonfils2018}; Cl2019b~--~\\citet{Cloutier2019b}; Wi2019~--~\\citet{Winters2019}; Gi2017~--~\\citet{Gillon2017}; Me2019~--~\\citet{Ment2019}; \n  \\end{list}\n\\end{table*}\n\n\\begin{acknowledgements}\n  N. A.-D. acknowledges the support of FONDECYT project 3180063.\n  J.K.T. acknowledges that support for this work was provided by NASA through Hubble\n  Fellowship grant HST-HF2-51399.001 awarded by the Space Telescope Science Institute, \n  which is operated by the Association of Universities for Research in Astronomy, Inc., \n  for NASA, under contract NAS5-26555.\n  R.B.\\ acknowledges support from FONDECYT Post-doctoral Fellowship Project 3180246, and\n  from the Millennium Institute of Astrophysics (MAS).\n  X.B. and J.M-A. acknowledge funding from the European Research Council\n  under the ERC Grant Agreement n. 337591-ExTrA.\n  X.D.; X.B.; T.F.; et L.M. acknowledge the  support by the French National Research Agency\n  in the  framework of the Investissements d'Avenir program (ANR-15-IDEX-02), through the\n  funding of the \"Origin of Life\" project of the Univ. Grenoble-Alpes.\"\n  LM acknowedge the support of the Labex OSUG@2020 (Investissements d'avenir -- ANR10\n  LABX56).\n  JRM acknowledges CAPES, CNPq and FAPERN brazilian agencies.\n  This work was supported by FCT\/MCTES through national funds and by FEDER - \n  Fundo Europeu de Desenvolvimento Regional through COMPETE2020 - Programa Operacional\n  Competitividade e Internacionaliza\u00e7\u00e3o by these grants: UID\/FIS\/04434\/2019;\n  PTDC\/FIS-AST\/32113\/2017 \\& POCI-01-0145-FEDER-032113; PTDC\/FIS-AST\/28953\/2017 \\&\n  POCI-01-0145-FEDER-028953.\n  T.H. acknowledges support from the European Research Council under the Horizon 2020\n  Framework Program via the ERC Advanced Grant Origins 83 24 28.\n  REM acknowledges support by the BASAL Centro de Astrof\\'isica y Tecnolog\\'ias Afines (CATA) and FONDECYT\n  1190621.\n  A.J.\\ acknowledges support from FONDECYT project 1171208 and by the Ministry for the Economy, Development, and Tourism's Programa Iniciativa Cient\\'{i}fica Milenio through grant IC\\,120009, awarded to the Millennium Institute of Astrophysics (MAS).\n  JGW is supported by a grant from the John Templeton Foundation. The opinions expressed here are\n  those of the authors and do not necessarily reflect the views of the John Templeton Foundation.\n  Work J.N.W.\\ was partly funded by the Heising-Simons Foundation.\n  The authors would like to acknowledge Zachary Hartman for his help conducting the \n  Gemini-South\/DSSI observations. Some of the work here is based on observations \n  obtained at the Gemini Observatory, which is operated by the Association of Universities \n  for Research in Astronomy, Inc., under a cooperative agreement with the NSF on \n  behalf of the Gemini partnership: the National Science Foundation (United States), \n  National Research Council (Canada), CONICYT (Chile), Ministerio de Ciencia, \n  Tecnolog\\'{i}a e Innovaci\\'{o}n Productiva (Argentina), \n  Minist\\'{e}rio da Ci\\^{e}ncia, Tecnologia e Inova\\c{c}\\~{a}o (Brazil), and Korea \n  Astronomy and Space Science Institute (Republic of Korea). \n  This work makes use of observations from the LCOGT network.\n  Funding for the TESS mission is provided by NASA's Science Mission directorate.\n  We acknowledge the use of public TESS Alert data from pipelines at the TESS \n  Science Office and at the TESS Science Processing Operations Center. \n  This research has made use of the Exoplanet Follow-up Observation Program website, \n  which is operated by the California Institute of Technology, under contract with \n  the National Aeronautics and Space Administration under the Exoplanet Exploration Program.\n  Resources supporting this work were provided by the NASA High-End Computing (HEC) Program\n  through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for the\n  production of the SPOC data products.\n  This paper includes data collected by the TESS mission, which are publicly available \n  from the Mikulski Archive for Space Telescopes (MAST).\n  This research has made use of the NASA Exoplanet Archive, which is operated by the\n  California Institute of Technology, under contract with the National Aeronautics and \n  Space Administration under the Exoplanet Exploration Program.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLow to intermediate mass stars (0.8 $\\lesssim M_* \\lesssim 8$~M$_{\\odot}$) will \nultimately end their lifes on the Asymptotic Giant Branch (AGB)\n\\citep{vassi93,habing1996}. \nTwo striking characteristics of the AGB are the variability of the stars and\nthe mass loss. Different types of large amplitude variables are classified on\nthe \nbasis of the amplitude: Semi-Regular Variables, Miras and OH\/IR stars, where the\nlatter have the largest amplitudes (1 magnitude bolometric) and periods of several\nhundred days. In the final phases on the AGB, the mass loss is the dominant \nprocess which will determine the AGB lifetime and its ultimate luminosity.  The \nmass-loss rates (MLRs) range approximately from $10^{-8}$  up to \n$10^{-4}$ M$_\\odot$\/yr.\nAlthough the mass loss for these stars is already well known for many years \nthere is still no firm understanding of what triggers the mass loss.\nIt is believed that, through large amplitude variability, the outer parts of the \natmosphere are cool and the density is high enough to start dust formation. \nRadiation pressure on the grains drives these outwards, dragging with them the \ngas creating a slow ($\\approx 15$ \\ks) but strong stellar wind\n\\citep{Goldreich1976}. Through their mass loss, these stars provide a significant \ncontribution to the gas and dust mass returned to the interstellar medium.  \n\nWhen nearing the tip of the AGB, stars will start experiencing thermal pulses \n(a.k.a. helium shell flashes). \nThe thermal pulses can lead\nto the change of chemical type from the originally oxygen-rich star to either a \nS-type (C\/O $\\sim 1$) or carbon star (C\/O $> 1$) through dredge-up of nuclear \nprocessed material to the surface \\citep{Iben1975}. The change of chemical type \nis metallicity and stellar mass dependent. Stars with approximately solar \nmetallicity and below two\nor above four \\msol\\ are expected to remain oxygen-rich \\citep{Marigo2013}. \n\nThe OH\/IR stars are the subset of AGB stars with the highest MLRs $> 10^{-5}$\nM$_\\odot$\/yr observed \\citep{Baud1983}. Such high MLR are significantly higher \nthan the stellar \nmass loss description by \\cite{Reimers1975} and are often called a {\\it\nsuperwind}, a term introduced by \\cite{Renzini1981} to describe the MLR needed\nto explain the characteristics of Planetary Nebulae.\n\nThe dust formed in the circumstellar shell completely obscures\nthe photospheric radiation and re-radiates it at infrared wavelengths\n\\citep{bedijn87}. The OH part \nof the name comes from the fact that in most cases of these infrared stars OH maser \nemisson, originating in a circumstellar thin shell, is detected. OH\/IR stars are \nmostly found through either ``blind'' OH surveys that searched the galactic plane at the 18 cm \nradio line \\citep[e.g.,][]{baud81,Sevenster1997} \nor through a dedicated search on cool infrared sources with colours typical \nfor a few 100 K temperature dust shell \\citep{telintel1991}. A recent database\nof circumstellar OH masers can be found in \\citet{Engels2015}.\n\nIn the literature, OH\/IR stars are often associated with more massive AGB \nstars. Well known studied examples of these are the OH maser sources near the \ngalactic plane, \nlike OH~26.5$+0.6$ (e.g. \\cite{vanlangevelde90}). These stars have high \nluminosities well above 10,000~$\\lsol$  and periods larger than a 1,000 days. \nStudies of OH\/IR stars in the Bulge \\citep{WilHarm1990,JEE2015} and the IRAS \nbased study by \\cite{Habing1988} of galactic disk OH\/IR stars, find however \nluminosity distributions peaking at approximately 5,000~\\lsol, expected to \nhave relatively low mass progenitors below 2~\\msol. \n\nSelecting Bulge stars provides the advantage of a relatively well known \ndistance within our Galaxy. Generally the Bulge stellar population is considered \nto be\nold \\citep{Renzini1994, Zoccali2003, evelien2009}, however several studies also\nindicate the presence of intermediate age stars \\citep{vLoon2003, GroenBlom2005}. \nThe question on the nature of the Bulge OH\/IR stars \nis part of our analysis and will be discussed in Section~\\ref{sec:population}.\n\nThe mass loss in AGB stars is studied by several means \\citep{veenolofsson1990, \nOlofsson2003} of which infrared  studies of the circumstellar dust and the \n(sub-) millimetre detection of CO transitions are the most frequently used. \nIn this study we combine the two techniques allowing to compare two independent\ntechniques and to study the gas-to-dust ratio, which is expected to be \nmetallicity dependent. Earlier efforts to observe CO emission from AGB stars in \nthe inner Galaxy had only limited success because of the interference\nof interstellar CO emission along the galactic plane \\citep{Winnberg2009, \nSargent2013}, even though they used interferometric techniques. \nThey selected OH\/IR stars close to the galactic centre and plane respectively, \nwhich have different star formation histories than the Bulge\n\\citep{Launhardt2002, GenzelGC2010}. To avoid the galactic plane ISM \ninterference we selected a population of OH\/IR stars from the Bulge  \nat higher latitudes. The sample selection of our paper is described in the \nfollowing Section. We then continue with a description of the CO observations and \ndata in Section~\\ref{sec: data}. The results from the modelling of the IR and CO \ndata are given in the\n``Analysis'' Section~\\ref{sec: analysis} \\& \\ref{sec:comparison}. The resulting characteristics are \ndescribed in the Section~\\ref{sec: characteristics}, followed by discussions on \nthe Bulge\npopulation of OH\/IR stars and the superwind MLR in Section~\\ref{sec:population}.\n\n\n\\section{Sample selection and description} \n\\label{sec: sample}\n\nThe eight sources in this study are taken from a larger sample of fifty-three Galactic \nBulge AGB stars which were selected to study the dust formation in the circumstellar\nshell of oxygen-rich AGB stars \\citep{Blommaert2007}.\nThe stars in the original sample were selected on the basis of infrared colours \n(observed with the ISO and\/or IRAS satellites) to represent the whole range in MLR \nobserved on the AGB, from naked stars with no observed mass loss up to OH\/IR stars \nwith MLRs in the order of $10^{-4}\\,{\\rm M}_{\\odot} \/ {\\rm yr}$.\nDifferent studies of this sample were performed and presented in several papers: the \ndust content through Spitzer-IRS spectroscopy \\citep{vanhollebeke2007, golriz14}, \ngroundbased spectroscopy and photometry, including a monitoring programme to determine \nthe variability \\citep{vanhollebeke2007}(vH2007 from now on) \nand a high resolution near-infrared spectroscopic study of the abundances \n\\citep{Uttenthaler2015}.  \n\nFrom this sample we selected those with the reddest colours and thus also likely \nthe stars with the \nhighest MLRs ($\\sim 10^{-4}\\,{\\rm M}_{\\odot} \/ {\\rm yr}$). The stars were\ndetected in the IRAS survey and originally studied in \\citet{WilHarm1990}. \n\nWe searched for counterparts of our sources in the OH maser database created by\n\\citet{Engels2015} which is considered complete for the published 1612 MHz maser\ndetections until the end of 2014.  \nSeven out of our eight sources were searched for the OH (1612~MHz) maser \nemission and were detected by \\citet{telintel1991, David1993, Sevenster1997}. \nThe velocities of the OH maser \nemission peaks are given in Table~\\ref{tab: sources}. For IRAS 17251--2821 two possible \nOH maser sources were detected and we give the observed velocities for both. \nThe stars have absolute galactic \nlatitudes above 2 degrees (except IRAS 17382-2830 with $b = 1.01^\\circ$), which limits\nthe interference by interstellar CO and increases the chance to detect the \ncircumstellar CO emission. Two sources are also detected in the infrared ISOGAL \nsurvey \\citep{Omont2003}. \nAs in the other papers on the Bulge sample, they are refered to with their ISOGAL name, \nthese names have also been included in Table~\\ref{tab: sources}.\n\n\\begin{table*}\n\n\\caption{Target list. } \n\n\\label{tab: sources}\n\n \\begin{tabular}{  l c c c c c c c}\n  \\hline\n & & & & & & & \\\\\n  IRAS name   &  Right Ascension & Declination   & $l$ & $b$     & OH (peak velocities)   &  A$_V$ & ISOGAL name\\\\\n               & (J2000)         & (J2000)       & (deg) & (deg) & LSR (\\ks)             &  mag    &      \\\\\n  \\hline \n & & & & & & & \\\\\n17251-2821  & 17 28 18.60 & -28 24 00.4  & 358.41 & 3.49 & $-$181.0, $-$149.0$^a$ & 3.17 & \\\\\n\t\t&\t\t&\t\t&\t&    &  $-$246.3, $-$227.5 & & \\\\\n17276-2846  & 17 30 48.29 & -28 49 01.7 & 358.41 & 2.80 & $-$71.2, $-$40.4$^c$ &  4.27 & \\\\\n17323-2424   & 17 35 25.92 & -24 26 30.5 & 2.61 & 4.31 & +70.0, +41.8$^c$ & 4.95 & \\\\\n17347-2319  & 17 37 46.28 & -23 20 53.4  & 3.83 & 4.44 & +74.1, +90.1$^a$ &  4.29 & \\\\\n17382-2830 & 17 41 22.59 & -28 31 48.0 & 359.86 & 1.01 & $-$68.7,   $-$33.7$^c$ & 5.19 & J174122.7-283146 \\\\\n17413-3531  & 17 44 43.46 & -35 32 34.1 & 354.26 & $-$3.28 & --$^a$  & 2.44 & \\\\\n17521-2938 & 17 55 21.80 & -29 39 12.9 & 0.47 & $-$2.19 & $-$88.5, $-$56.0$^b$ & 2.60 & J175521.7-293912 \\\\\n18042-2905  & 18 07 24.39 & -29 04 48.0 & 2.27 & $-$4.19 &+39.2,  +69.1$^a$ & 1.48 & \\\\\n\\hline\n\\end{tabular}\n\nNotes:  Positions taken from AllWISE \\citep{cutri2013}. \nVelocities of the OH maser peak emission given with respect to the Local Standard of Rest (LSR). \na: \\cite{telintel1991}, b: \\cite{David1993}, c: \\cite{Sevenster1997}. \nVisual extinctions, A$_V$, are taken from \\cite{vanhollebeke2007}, ISOGAL names are from \\cite{Omont2003}.\n\n\\end{table*}\n\n\\begin{figure*}\n\\label{fig: seds}\n\n\\begin{minipage}{0.42\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{iras17251_sed.ps}}\n\\end{minipage}\n\\begin{minipage}{0.42\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{iras17276_sed.ps}}\n\\end{minipage}\n\n\n\\begin{minipage}{0.42\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{iras17323_sed.ps}}\n\\end{minipage}\n\\begin{minipage}{0.42\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{iras17347_sed.ps}}\n\\end{minipage}\n\n\\begin{minipage}{0.42\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{iras17382_sed.ps}}\n\\end{minipage}\n\\begin{minipage}{0.42\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{iras17413_sed.ps}}\n\\end{minipage}\n\n\\caption[]{\nPhotometry and Spitzer-IRS spectra with the model fits (black full line vs \nphotometry points and dashed blue vs IRS spectra), see\nSections~\\ref{sec: irdata} \\& \\ref{sec:IRmodel}. The horizontal lines indicate\nwavelength ranges with forsterite bands, which were discarded in\nthe modelling.\n}\n\\end{figure*}\n \n\\renewcommand{\\thefigure}{\\arabic{figure} (Cont.)}\n\\addtocounter{figure}{-1}\n\\begin{figure*}\n\\begin{minipage}{0.42\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{iras17521_sed.ps}}\n\\end{minipage}\n\\begin{minipage}{0.42\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{iras18042_sed.ps}}\n\\end{minipage}\n\\caption[]{\n}\n\\label{Fig-SEDcont}\n\\end{figure*}\n\\renewcommand{\\thefigure}{\\arabic{figure}}\n\t    \n\\cite{JEE2015} (JEE15 from now onwards) selected a sample of thirty-seven\nBulge IRAS \nsources with OH\/IR star-like IRAS colours and \nmodelled the spectral energy distributions (SED). We will compare our analysis \nwith theirs in Section~\\ref{sec:comparison}. A difference in the selection of\nour sample, based on \\citet{Blommaert2007}, with JEE15 is that all our sources\nhave IRAS 12~$\\mu$m flux densities below 10~Jy. This limit was imposed as\n\\citet{WilHarm1990} considered these to be most likely Bulge members and not of the\ngalactic disk. JEE15 did not impose such a flux criterion.\n\n\n\n\\section{Data and observations description}\n\\label{sec: data}\n\n\n\\subsection{Spectral Energy Distribution data}\n\\label{sec: irdata}\nWe made use of VizieR \\citep{VizieR} and data in the literature to collect photometric data at the \nposition of the OH\/IR stars to create their SEDs.\nTwo important aspects about our sources need to\nbe considered. One is the fact that our stars are highly variable and that the\ndata obtained from different catalogues and publications have not been observed \nat the same single epoch. Second, the Bulge is a high source-density area, so that \nconfusion with nearby sources is a risk. The strategy followed was to start \nsearching in the VizieR database \nfor the nearest AllWISE counterpart \\citep{cutri2013} of the IRAS position. The\nWISE and IRAS surveys overlap in the mid-infrared wavelength regime, where \nfewer sources are detected, limiting the chance of a wrong association. \nThe WISE sources selected in this manner have a [W3] $-$ [W4] colour of about 2\nmagnitudes, consistent with the IRAS colours.\nThe AllWISE positions are accurate up to 50~mas. In a second step, a search \narea of $ 4^{\\prime\\prime}$ radius around the position was used to search \nin other infrared catalogues. In case of finding more than one source, the \nreddest object is selected. The data is given in Table~\\ref{tab: photom}.\n\n\\begin{table*}\n\n\\centering\n\n \\caption{ Photometric data of our targets.}\n \n\\scriptsize\n\\begin{tabular}{ c c c c c c c c c}\n  \\hline\n & & & & & & & & \\\\\n  Filter   &  17251--2821 & 17276--2846 & 17323--2424 & 17347--2319 & 17382--2830 & 17413--3531 & 17521--2938 & 18042--2905 \\\\\n & & & & & & & & \\\\\n  \\hline \n\\underline{CASPIR (mag)}  & & & & & & & & \\\\\nH  &                  & & & 13.78 $\\pm$ 0.01 &            & 12.85 $\\pm$ 0.01 & &  \\\\\nK  & 14.02 $\\pm$ 0.01 & & & 10.86 $\\pm$ 0.01 & 10.33 $\\pm$ 0.01 &  8.99 $\\pm$ 0.01 & & 10.09 $\\pm$ 0.01 \\\\ \nK  & 14.47 $\\pm$ 0.01 & & & 11.09 $\\pm$ 0.01 & 11.56 $\\pm$ 0.01 & 10.09 $\\pm$ 0.01 & & 10.87 $\\pm$ 0.01 \\\\ \nK  & 14.92 $\\pm$ 0.01 & & & 11.21 $\\pm$ 0.01 & 12.79 $\\pm$ 0.01 & 11.10 $\\pm$ 0.01 & & 11.65 $\\pm$ 0.01 \\\\ \nnbL & 8.11 $\\pm$ 0.02 & & &  7.02 $\\pm$ 0.02 &  6.80 $\\pm$ 0.02 &            & &         \\\\\n\\underline{ESO (mag)}  & & & & & & & & \\\\\nK & 12.27 $\\pm$ 0.08 &           & 15.0  $\\pm$ 0.2  & 10.6  $\\pm$ 0.1  & &  8.69 $\\pm$ 0.07 & & 12.5 $\\pm$ 0.9  \\\\\nK &                  &           &            & 10.98 $\\pm$ 0.03 & & 10.06 $\\pm$ 0.02 & & 12.2 $\\pm$ 0.2  \\\\\nK &                  &           &            & 11.1  $\\pm$ 0.7  & &  8.33 $\\pm$ 0.03 & &  \\\\\nL &  6.53 $\\pm$ 0.04 & 8.05 $\\pm$ 0.02 &  7.36 $\\pm$ 0.03 &  5.94 $\\pm$ 0.02 & & 5.21 $\\pm$ 0.04 & & 6.38 $\\pm$ 0.02  \\\\\nL &  8.62 $\\pm$ 0.03 &           &  8.94 $\\pm$ 0.05 &  6.36 $\\pm$ 0.1  & & 6.37 $\\pm$ 0.01 & & 6.73 $\\pm$ 0.03  \\\\\nL &                  &           &            &  6.53 $\\pm$ 0.03 & & 5.10 $\\pm$ 0.02 & & \\\\\nM &  4.77 $\\pm$ 0.05 & 6.59 $\\pm$ 0.08 &  5.33 $\\pm$ 0.04 &  4.81 $\\pm$ 0.04 & & 4.30 $\\pm$ 0.04 & & 5.23 $\\pm$ 0.07 \\\\\nM &  6.53 $\\pm$ 0.07 &           &  6.56 $\\pm$ 0.08 &  5.15 $\\pm$ 0.03 & & 5.51 $\\pm$ 0.04 & & 5.56 $\\pm$ 0.05 \\\\\nM &                  &           &            &  5.46 $\\pm$ 0.05 & & 4.26 $\\pm$ 0.06 & & \\\\\nN1 & 2.24 $\\pm$ 0.03 & 3.5 $\\pm$ 0.1 & 2.46 $\\pm$ 0.05 & 2.80 $\\pm$ 0.03 & 2.59 $\\pm$ 0.03 & & & 3.02 $\\pm$ 0.03 \\\\\nN1 & 3.4  $\\pm$ 0.1  & 4.9 $\\pm$ 0.7 & 3.19 $\\pm$ 0.08 & 2.99 $\\pm$ 0.07 & 3.6  $\\pm$ 0.01 & & & 3.4  $\\pm$ 0.1 \\\\\nN1 &                 &         &           & 3.43 $\\pm$ 0.09 & 2.70 $\\pm$ 0.07 & & & \\\\\nN2 & 2.95 $\\pm$ 0.05 & 2.4 $\\pm$ 0.3 & 3.3 $\\pm$ 0.1   & 3.42 $\\pm$ 0.03 & 2.44 $\\pm$ 0.06 & & & 3.39 $\\pm$ 0.05  \\\\\nN2 & 4.9  $\\pm$ 0.4  &         & 5.0 $\\pm$ 0.6   & 3.7  $\\pm$ 0.1  & 3.9  $\\pm$ 0.2  & & & 4.0  $\\pm$ 0.2 \\\\\nN2 &                 &         &           & 4.0  $\\pm$ 0.4  & 2.54 $\\pm$ 0.07 & & &\\\\\nN3 & 1.24 $\\pm$ 0.05 &         & 1.48 $\\pm$ 0.09 & 1.77 $\\pm$ 0.04 & 1.7 $\\pm$ 0.1 & & & 1.93 $\\pm$ 0.07 \\\\\nN3 & 2.3  $\\pm$ 0.3  &         & 1.9  $\\pm$ 0.2  & 1.8  $\\pm$ 0.1  & 2.5 $\\pm$ 0.3 & & & 2.0  $\\pm$ 0.1 \\\\\nN3 &                 &         &           & 2.0  $\\pm$ 0.2  & 1.8 $\\pm$ 0.1 & & &  \\\\\n\\underline{2MASS (mag)}  & & & & & & & & \\\\\nJ          &  & &  &  & 16.57 $\\pm$ 0.04 & & &  \\\\\nH          &  & &  &  & 14.59 $\\pm$ 0.02 & & &  \\\\\nK$_{\\rm s}$ &  & &  &  & 10.72 $\\pm$ 0.01 & 9.06 $\\pm$ 0.02 & & 10.59 $\\pm$ 0.02 \\\\\n\\underline{VISTA (mag)}  & & & & & & & & \\\\\nJ         &                  &     &                 &                 &                  & 16.75 $\\pm$ 0.09 &                & 17.77 $\\pm$ 0.30 \\\\\nH         &                  &     &                 & 15.4  $\\pm$ 0.1 &                  & 13.20 $\\pm$ 0.01 &                & 13.51 $\\pm$ 0.01 \\\\\nK$_{\\rm s}$ & 16.13 $\\pm$ 0.15 &     & 18.3 $\\pm$ 0.3 & 11.70 $\\pm$ 0.01 & 11.92 $\\pm$ 0.01 & 11.08 $\\pm$ 0.01 & 17.0 $\\pm$ 0.5 & 12.01 $\\pm$ 0.01 \\\\\n\\underline{GLIMPSE (mag)}  & & & & & & & & \\\\\nirac36 & 7.91 $\\pm$ 0.03 & 8.27 $\\pm$ 0.03 & & & 6.77 $\\pm$ 0.06 &  &  9.96 $\\pm$ 0.05 & 6.55 $\\pm$ 0.05 \\\\\nirac36 &                 &           & & &                 &  & 10.22 $\\pm$ 0.06       & \\\\\nirac45 & 5.80 $\\pm$ 0.05 & 7.10 $\\pm$ 0.03 & & & 4.91 $\\pm$ 0.05 &  & 7.06  $\\pm$ 0.04 & 5.37 $\\pm$ 0.08 \\\\\nirac45 &                 &           & & &                 &  & 7.31 $\\pm$ 0.04       & \\\\\nirac58 & 4.10 $\\pm$ 0.02 & 4.81 $\\pm$ 0.02 & & & 3.89 $\\pm$ 0.02 &  & 4.87  $\\pm$ 0.02 & 4.35 $\\pm$ 0.04 \\\\\nirac58 &                 &           & & &                 &  & 5.05 $\\pm$ 0.03       & \\\\\nirac80 & 3.00 $\\pm$ 0.02 & 3.20 $\\pm$ 0.06 & & &                 &  &                  & 4.55 $\\pm$ 0.15 \\\\\n\\underline{AllWISE (mag)}  & & & & & & & & \\\\\nWISE1 (3.4~$\\mu$m) & 9.73 $\\pm$ 0.03 & 8.35 $\\pm$ 0.02 & 10.20 $\\pm$ 0.03 & 7.35 $\\pm$ 0.03 & 6.98 $\\pm$ 0.03 & 7.29 $\\pm$ 0.03 & 11.43 $\\pm$ 0.08 & 7.02 $\\pm$ 0.03 \\\\\nWISE2 (4.6~$\\mu$m) & 6.63 $\\pm$ 0.02 & 6.86 $\\pm$ 0.02 &  6.42 $\\pm$ 0.02 & 5.15 $\\pm$ 0.06 & 4.62 $\\pm$ 0.04 & 5.07 $\\pm$ 0.04 &  7.24 $\\pm$ 0.02 & 5.19 $\\pm$ 0.08 \\\\\nWISE3 (12~$\\mu$m) & 2.75 $\\pm$ 0.01 & 2.21 $\\pm$ 0.01 &  2.38 $\\pm$ 0.01 & 2.70 $\\pm$ 0.01 & 2.48 $\\pm$ 0.02 & 3.35 $\\pm$ 0.01 &  3.10 $\\pm$ 0.01 & 2.09 $\\pm$ 0.01 \\\\\nWISE4 (22~$\\mu$m) & 0.34 $\\pm$ 0.02 & 0.06 $\\pm$ 0.01 &  0.18 $\\pm$ 0.02 & 0.65 $\\pm$ 0.01 & 0.53 $\\pm$ 0.02 & 1.23 $\\pm$ 0.02 &  0.52 $\\pm$ 0.02 & 0.12 $\\pm$ 0.01 \\\\\n\\underline{ISOGAL (mag)}   & & & & & & & & \\\\\nLW2 (7~$\\mu$m)  & & & & & 3.47 $\\pm$ 0.01 & & & \\\\\nLW3 (15~$\\mu$m) & & & & & 1.54 $\\pm$ 0.03 & & & \\\\\n\\underline{IRAS (Jy)}   & & & & & & & & \\\\\nF12 & 3.6 $\\pm$ 0.4 & 2.5 $\\pm$ 0.3 & 3.4 $\\pm$ 0.3 & 3.6 $\\pm$ 0.4 & 2.0 $\\pm$ 0.2 & 5.0 $\\pm$ 0.5 & 3.1 $\\pm$ 0.3 & 4.7 $\\pm$ 0.5 \\\\\nF25 & 8.5 $\\pm$ 0.9 & 7.9 $\\pm$ 0.8 & 8.5 $\\pm$ 0.9 & 5.9 $\\pm$ 0.6 & 4.3 $\\pm$ 0.4 & 6.8 $\\pm$ 0.7 & 8.2 $\\pm$ 0.8 & 8.4 $\\pm$ 0.8 \\\\\nF60 & 4.3 $\\pm$ 0.4 & 7.1 $\\pm$ 0.7 & 4.4 $\\pm$ 0.4 & 1.5 $\\pm$ 0.2 &               & 2.2 $\\pm$ 0.2 & 7.1 $\\pm$ 0.7 & 2.6 $\\pm$ 0.3 \\\\\n\\underline{MSX (Jy)}  & & & & & & & & \\\\\nA (8.28)  & 2.1 $\\pm$ 0.2 & 2.1 $\\pm$ 0.2 & 2.3 $\\pm$ 0.2 & 2.8 $\\pm$ 0.3 & 2.1 $\\pm$ 0.2 & 2.3 $\\pm$ 0.2 & 1.5 $\\pm$ 0.5 & 2.9 $\\pm$ 0.3 \\\\\nC (12.13) & 3.7 $\\pm$ 0.4 & 3.4 $\\pm$ 0.3 & 2.9 $\\pm$ 0.3 & 4.9 $\\pm$ 0.5 & 2.7 $\\pm$ 0.3 & 2.9 $\\pm$ 0.3 & 2.4 $\\pm$ 0.2 & 4.0 $\\pm$ 0.4 \\\\\nD (14.65) & 4.9 $\\pm$ 0.5 & 5.8 $\\pm$ 0.6 & 5.5 $\\pm$ 0.6 & 5.2 $\\pm$ 0.5 & 3.3 $\\pm$ 0.3 & 3.1 $\\pm$ 0.3 & 4.3 $\\pm$ 0.4 & 5.0 $\\pm$ 0.5 \\\\\nE (21.34) & 5.8 $\\pm$ 0.6 & 5.4 $\\pm$ 0.5 & 7.1 $\\pm$ 0.7 & 4.7 $\\pm$ 0.5 & 4.0 $\\pm$ 0.4 & 3.0 $\\pm$ 0.5 & 5.5 $\\pm$ 0.6 & 6.2 $\\pm$ 0.6 \\\\\n\\underline{MIPS (mag)}  & & & & & & & & \\\\\nmips24 & & 0.58 $\\pm$ 0.02 & & & 1.05 $\\pm$ 0.02 & & & \\\\\n\\underline{AKARI (Jy)}   & & & & & & & & \\\\\nS9  &               &         &               &                 &  & 2.7  $\\pm$ 0.6  & 2.59 $\\pm$ 0.60 & 2.47 $\\pm$ 0.40 \\\\\nS18 & 4.7 $\\pm$ 0.9 & 5.9 $\\pm$ 0.4 & 4.3 $\\pm$ 0.2 & 3.79 $\\pm$ 0.01 &  & 2.47 $\\pm$ 0.02 & 6.94 $\\pm$ 1.27 & 4.89 $\\pm$ 0.62 \\\\\nS65 &               &               &               &                 &  &                 &                 & 2.09 $\\pm$ 0.22 \\\\\nS90 & 2.2 $\\pm$ 0.2 &               & 1.9 $\\pm$ 0.4 &                 &  &                 &                 & 1.78 $\\pm$ 0.08 \\\\\n\\underline{PACS (Jy)}   & & & & & & & & \\\\\n 70 &  2.1 $\\pm$ 0.3 & 6.4 $\\pm$ 1.3 & 3.3 $\\pm$ 0.7 &  & & & 5.3 $\\pm$ 1.1 & \\\\\n140 &  0.4 $\\pm$ 0.1 & 1.1 $\\pm$ 0.2 & 0.4 $\\pm$ 0.1 &  & & & 0.9 $\\pm$ 0.2 & \\\\\n\\hline\n\\end{tabular}\n\n\\label{tab: photom}\n\n\n\\end{table*}\n\n\nWe have complemented the VizieR data with the $J, H, K$, nb$L$ averaged photometry \nfrom the vH2007 monitoring programme  \nobtained at the Mount Stromlo observatory. We also included the ESO photometric \n2 -- 13~$\\mu$m data obtained by \\citet{WilHarm1990}. \nFinally, public DR4 data from the VVV Survey \\citep{Minniti2010} was included\\footnote{see \\url{http:\/\/horus.roe.ac.uk\/vsa\/index.html}}.\nIRAS17521 and IRAS17323 are not listed in the source catalog, but are visible on the $K$-band image \nand the magnitudes have been estimated by scaling the flux (minus background) in a 3$\\times$3 pixel region \nto that of a nearby catalogue star.\n\nFinally half of our sample were also observed with the Herschel PACS spectrometer\n\\citep{pilbratt10,poglitsch10} in the open time programme ``Study of the cool \nforsterite\ndust around evolved stars'' (OT2\\_jblommae\\_2). The flux densities given in \nTable~\\ref{tab: photom} are the continuum levels at 70 and 140~$\\mu$m of the \ncentral spaxel as obtained from archive pipeline product v14. The flux densities \nwere corrected for the missing part of the point spread function (PSF). The formal\nuncertainties of the PACS spectrometer flux calibration is 15\\%. \n\nFigure~1\nshows the obtained SEDs and the model  fits obtained (see Section\n\\ref{sec:IRmodel}). The figure also includes the Spitzer-IRS spectrum covering \nthe 5-37~$\\mu$m wavelength range and which are taken from \\cite{golriz14}. All sources \nshow typical SEDs for OH\/IR stars, i.e. an optically thick silicate-rich dust \nshell with strong absorption features at 9.7 and 18~$\\mu$m.\n\n\n\n\\subsection{CO Observation and data reduction} \n\\label{sec: COdata}\nThe CO J= 2-1 and J= 3-2 transitions were observed with the APEX telescope located \nin the Atacama dessert in Chile \\citep{APEX}.\nThe observations were obtained in service mode on \nSeptember 11, 12, 13, November 10, 11, 12, 2011 (I17276, I17323, I17521, I18042) \nand\nSeptember 26, 27, 29, 30, 2012 (I17251, I17347, I17413, I17382).\nWeather conditions varied but most observations were taken with a precipitable water \nvapour (PWV) between 0.7 and 1.3 mm for the J= 3-2, and between 1 and 2 mm for \nthe J= 2-1 transition.\nThe APEX-1 and APEX-2 receivers of the Swedish Heterodyne Facility Instrument \n(SHeFI)\\footnote{http:\/\/gard04.rss.chalmers.se\/APEX\\_Web\/SHeFI.html}\n\\citep{Belitsky2006, Vassilev2008} were tuned to the CO J= 2-1 and 3-2 line, respectively. \nThe beam size and the main-beam efficiency at these frequencies are 26.4\\arcsec\\ \n(FWHM), $\\eta_{\\rm mb}$= 0.73,\nrespectively, 17.3\\arcsec, 0.75.\nThe XFFTS (eXtended bandwidth Fast Fourier Transform Spectrometer) backend \n(see \\cite{Klein2012}) was connected to the receivers.\nWobbler switching was used with a throw of 50\\arcsec. Regular observations of bright \nsources were performed to check the pointing and calibration. \n\nThe data were reduced in CLASS\\footnote{http:\/\/www.iram.fr\/IRAMFR\/GILDAS\/}. \nLinear baselines were subtracted avoiding regions that were affected by \ninterstellar contamination and the location\nof the CO detection (or using the velocity range suggested by the OH maser \nemission line in case of a CO non-detection). Typical total integration times per\nsource were 40-50 minutes for the J=2-1 and 100-130 minutes for the J=3-2\ntransitions, leading to a RMS of $\\approx 10$ mK for both transitions at a velocity\nresolution of 1 \\ks.\n\n\\begin{figure*}\n\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17251_CO21_F.ps}}\n\\end{minipage}\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17251_CO32_F.ps}}\n\\end{minipage}\n\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17276_CO21_F.ps}}\n\\end{minipage}\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17276_CO32_F.ps}}\n\\end{minipage}\n\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17323_CO21_F.ps}}\n\\end{minipage}\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17323_CO32_F.ps}}\n\\end{minipage}\n\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17347_CO21_F.ps}}\n\\end{minipage}\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17347_CO32_F.ps}}\n\\end{minipage}\n\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17382_CO21_F.ps}}\n\\end{minipage}\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17382_CO32_F.ps}}\n\\end{minipage}\n\n\\caption[]{\nThe APEX CO (2-1) and (3-2) line spectra, together with the line fits (in black)\nand the model predictions (dashed blue). For a description, see Sections~\\ref{sec: irdata} \\& \n\\ref{sec:COmodel}.\n}\n\\label{Fig-CO}\n\\end{figure*}\n\n\\renewcommand{\\thefigure}{\\arabic{figure} (Cont.)}\n\\addtocounter{figure}{-1}\n\n\\begin{figure*}\n\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17413_CO21_F.ps}}\n\\end{minipage}\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17413_CO32_F.ps}}\n\\end{minipage}\n\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17521_CO21_F.ps}}\n\\end{minipage}\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I17521_CO32_F.ps}}\n\\end{minipage}\n\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I18042_CO21_F.ps}}\n\\end{minipage}\n\\begin{minipage}{0.45\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{profilefit_I18042_CO32_F.ps}}\n\\end{minipage}\n\n\\caption[]{\nNo detection for IRAS~17413--3531, the blue line indicates\nthe predicted line strength from our dynamical modelling at a LSR velocity of 0\nkm $s^{-1}$. \n}\n\\label{Fig-COcont}\n\\end{figure*}\n\\renewcommand{\\thefigure}{\\arabic{figure}}\n\nThe resulting profiles are shown in Figure~\\ref{Fig-CO} plotting main-beam temperatures \nagainst velocity (using the LSR as the velocity reference).\n\nThe profiles were fitted with our own Fortan version of the \"Shell\" profile \navailable within CLASS\/GILDAS software \npackage\\footnote{http:\/\/www.iram.fr\/IRAMFR\/GILDAS\/doc\/html\/class-html\/node38.html},\n\\begin{equation}\n   P(V) = \\frac{I}{\\Delta V \\; (1 + H \/3)} \\; \\left(1 + 4 H \\; \\left(\\frac{V - V_0}{\\Delta V}\\right)^2\\right),\n\\end{equation}\nwhere $V_0$ is the stellar velocity (in \\ks), $I$ is the integrated intensity (in \n\\kks), $\\Delta V$ the full-width at zero intensity (in \\ks, and the expansion \nvelocity v$_{\\rm exp}$ is taken as half that value), and $H$ the horn-to-center \nparameter. This parameter described the shape of the profile, with\n$-1$ for a parabolic profile, $0$ for a flat-topped one, and $>0$ for a \ndouble-peaked profile.\nIn the fitting below, a parabolic profile was assumed for all cases. From the \nfitting we obtain the stellar radial and expansion velocities. However as the\nCO profiles are relatively weak, we have chosen to use the OH velocity information\nand keep the stellar and expansion velocities fixed. \\cite{groen98} compared \nexpansion velocities derived from OH and CO observations and found that generally\nthe CO \tprofiles are 12\\% wider and we applied this correction for our fitting.\n\nThe results are listed in Table~\\ref{tab:CO_results}.\nThe errors in the parameters were estimated by a Monte Carlo simulation where the\nintensity in every channel was varied according to a Gaussian with the observed \nrms noise (assuming the channels are independent), and the profile \nrefitted. For sources where we could obtain an independent \nfitting of the CO profile, without using the OH\nvelocities we give the obtained velocities in the same Table. The \"CO\" and \"modified OH\"\nvelocities generally agree within the errors.\n\n \\begin{table*} \n\\setlength{\\tabcolsep}{0.7mm}\n\\footnotesize\n\\centering\n\\caption{CO data fitting results. }\n\\begin{tabular}{lcccccccccc}\n\\hline \nIRAS  & $V_{\\rm LSR}$  & $\\Delta V$ & $T_{2-1,{\\rm peak}} $ & $I_{2-1}$ &\n            $T_{3-2,{\\rm peak}} $ & $I_{3-2}$ & $V_{\\rm LSR}(2-1)$  & $\\Delta V (2-1)$ & $V_{\\rm LSR}(3-2)$  & $\\Delta V (3-2)$  \\\\\nname      & (\\ks)  & (\\ks) & (K) & (K~\\ks) & (K)  & (K~\\ks) & (\\ks)  & (\\ks) &  (\\ks)  & (\\ks) \\\\\n\\hline\n  &  & & &  &  & & &  &  & \\\\\n17251-2821  & -165.0 & 35.8 & 0.027 $\\pm$ 0.002 & 0.637 $\\pm$ 0.042 & 0.014 $\\pm$ 0.002 & 0.327 $\\pm$ 0.047 & -162.3 $\\pm$ 0.9 & 37.1 $\\pm$ 2.4 & -160.0 $\\pm$ 0.5 & 28.3 $\\pm$ 3.3 \\\\\n17276-2846  &  -55.8 & 34.5 & 0.013 $\\pm$ 0.004 & 0.305 $\\pm$ 0.090 & 0.024 $\\pm$ 0.004 & 0.55  $\\pm$ 0.10  &  -55.7 $\\pm$ 2.0 & 31.7 $\\pm$ 6.1 &  -53.6 $\\pm$ 2.0 & 36.7 $\\pm$ 5.9 \\\\\n17323-2424  &   55.9 & 31.6 & 0.016 $\\pm$ 0.005 & 0.336 $\\pm$ 0.096 & -                 & -                 &   58.9 $\\pm$ 1.5 & 28.0 $\\pm$ 4.0 &         & \\\\\n17347-2319  &   82.0 & 17.9 & 0.017 $\\pm$ 0.003 & 0.198 $\\pm$ 0.039 & 0.005 $\\pm$ 0.004 & 0.065 $\\pm$ 0.046 &       &  &   & \\\\\n17382-2830  &  -51.2 & 39.2 & -                 & -                 & 0.012 $\\pm$ 0.003 & 0.312 $\\pm$ 0.086 &       &  &    & \\\\\n17413-3531  &  - & - & - & - & - & - &   &  &    & \\\\\n17521-2938  &  -72.3 & 36.4 & 0.017 $\\pm$ 0.003 & 0.416 $\\pm$ 0.067 & 0.014 $\\pm$ 0.003 & 0.323 $\\pm$ 0.075 &  -73.3 $\\pm$ 2.2 & 47.3 $\\pm$ 6.7 &  -75.0 $\\pm$ 3.0 & 40.9 $\\pm$ 8.0 \\\\\n18042-2905  &   54.2 & 33.5 & 0.015 $\\pm$ 0.002 & 0.338 $\\pm$ 0.043 & 0.011 $\\pm$ 0.003 & 0.236 $\\pm$ 0.076 &   55.3 $\\pm$ 1.3 & 28.9 $\\pm$ 2.9 &   51.6 $\\pm$ 3.9 & 28.5 $\\pm$ 7.0 \\\\\n\\hline\n\\end{tabular}\n\\label{tab:CO_results}\n\\\\\nNotes: \nThe $V_{\\rm LSR}$ and $\\Delta V$ are taken from the OH observations. For the\nsources where an independent fit of the CO profile was possible \n(see Section~\\ref{sec: COdata}), the velocities are given\nin the last 4 columns. \n\n \\end{table*} \n\n\n\n\\section{Analysis}\n\\label{sec: analysis}\n\n\\subsection{Modelling of the IR data}\n\\label{sec:IRmodel}\n\n\nThe models are based on the \"More of DUSTY\" (MoD) code \\citep{groenewegen12} which uses a\nslightly updated and modified version of the {\\em DUSTY} dust radiative transfer (RT) code \\citep{Dusty1999}\nas a subroutine within a minimization code.\nThe code determines the best-fitting dust optical depth, luminosity, dust temperature\nat the inner radius and the slope, $p$, of the density distribution ($\\rho \\sim r^{-p}$)\nby fitting photometric data and spectra (and visibility data and 1D intensity profiles,\nbut these data are not available for the sample considered here).\nThe code minimizes a $\\chi^2$ based on every available photometric and spectroscopic datapoint, but also\ncalculates the $chi^2$ for the photometric and spectroscopic datapoints seperately.\nThis allows the user to weigh the spectroscopic data relative to the photometric data.\nIn practise the errorbars on the spectroscopic dataset are scaled (typically by a factor of order 0.2)\nso that photometry and spectroscopy give roughly equal weight to the overall fit.\nIn the present model the dust temperature at the inner radius has been fixed to 1000~K, and we\nassume a $r^{-2}$ density law, only fitting for the luminosity and dust optical depth (at 0.55~$\\mu$m).\nThe outer radius is set to a few thousand times the inner radius, to where the \ndust temperature has reached 20~K, typical of the ISM.\nMoD does not take into account the actual heating of the dust grains by \nthe ISM and so this is an approximation.\nBecause of interaction of the expanding AGB wind with the ISM there can also \nbe deviations for a $r^{-2}$ density law.\nThese approximations have no impact on the results as there is no far-IR \ndata available for our sample that could constrain these values.\nThe longest wavelength data available for some stars is the PACS data at \n140~$\\mu$m.\nSome test calculations indicate the the flux in this filter is reduced by less \nthat 10\\% if the outer radius\nwere reduced by a factor of $\\sim4$ below 1000 times the inner radius to where \nthe dust temperature is about 30-35~K.\n\nSeveral combinations of dust species have been tried to obtain a best fit.\nThey were olivine (amorphous MgFeSiO$_4$, optical constants from \n\\citet{dorschner95}),\ncompact amorphous aluminum oxide \\citep{begeman97},\nand metallic iron \\citep{Pollack1994}. The resulting abundance ratios for\neach source are given in Table~\\ref{tab:sed_model}.\n\nAstronomical grains are not solid spheres and to mimic this the absorption and \nscattering\ncoefficients have been calculated assuming a \"distribution of hollow spheres\" \n(DHS, \\citealt{Min03}) with a\nmaximum vacuum volume fraction of 0.7, that is appropriate for interstellar\nsilicate dust grains \\citep{Min07_ISMgrains}.\nAn advantage of a DHS is that the absorption and scattering coefficients can be\ncalculated exactly for arbitrary grain sizes.\nSingle sized grains of 0.1, 0.2 and 0.5 $\\mu$m have been considered.\nThe largest grain size used is inspired by recent observations of dust\naround O-rich stars \\citep{norris12, Scicluna2015, Ohnaka2016}.\n\n\nThe stellar photosphere was represented as a\nMARCS model atmosphere\\footnote{http:\/\/marcs.astro.uu.se\/} \\citep{Gustafsson2008}\nof 2600~K (and $\\log g$= 0.0, 2 \\msol, and solar metallicity).\nAs shown below, all stars are so dust enshrouded that the SED fitting is insensitive to the input model atmosphere.\n\nA canonical distance of 8 kpc has been assumed, slightly smaller than the value\nquoted in the recent review by \\citet{degrijsbono2016}, 8.3 $\\pm$ 0.2 (statistical) $\\pm$ 0.4\n(systematic) kpc and which is based on an analysis of the up-to-date most complete\ndatabase of Galactic Centre distances.\n\nThe reddening law used in MoD is described in \\citet{groenewegen12}.\nThe interstellar reddening $A_{\\rm V}$ is taken from vH2007\n(see Table~\\ref{tab: sources}).\n\nThe model fits are shown in Figure~\\ref{fig: seds} and the resulting parameters\n(L$_*$, R$_{\\rm in}$, $\\dot{M_d}$, dust optical depth ${\\tau}_{\\rm V}$, grain size,\ngrain density $\\rho$, and flux-weighted dust extinction coefficient $<Q>$)\nin Table~\\ref{tab:sed_model}.\n\nError bars are not listed explicitly as they are difficult to estimate.\nThe fitting returns the error on the parameters (luminosity and optical depth in this case).\nThese are typically very small as the resulting $\\chi^2$ are large (reduced $\\chi^2$ in the range 40-600).\nThis is related to the fact that the stars are variable and the SED is constructed by combining multi-epoch data,\nwithout any attempt to average data in similar filters.\nAs the amplitude of the variability is (much) larger than the error on a single measurement this implies that the \n$\\chi^2$ is typically always large.\nOne estimate for the error in luminosity and optical depth (hence dust MLR) comes from the internal error\nscaled to give a reduced $\\chi^2$ of unity.\nA second estimate for the error comes from the values of the parameters in a range of $\\chi^2$ above the best-fitting value.\nThis is required in any case, as for example\nthe absorption and scattering coefficients are external to the code, and the model is only run on a grid\nwith discrete values of the parameters (grain sizes and dust composition in this case).\n\nBased on the above considerations, our best estimate for the 1~$\\sigma$ error on the luminosity is 10\\%, but that does not include the spread in distances\nbecause of the depth of the Bulge ($\\pm 1.4$~kpc, which gives a possible deviation of $\\pm 35 \\%$ in $L$).\nThe MLR scales linearly with the adopted distance.\n\nAs stated above, the errorbars on the spectroscopic datapoints is reduced by a certain factor as to give\nall spectroscopic datapoints about equal weight in the fitting as all the photometry points.\nChanging this scaling factor by a factor of two leads to a change of less than 10\\% in the MLR and less than 1\\% in luminosity.\n\n\nThe error on the optical depth is also of order 10\\%, but the error on the dust MLR is larger.\nThis is related to the derived inner radius. The error on that quantity is 5\\%, but there is a much larger error involved due\nto the unknown effective temperature and dust temperature at the inner radius (both are hard to determine and have been fixed).\nA realistic error on the inner radius would be a factor of 2, and this is then also a realistic error on the derived\ndust MLRs.\n\nThe best fitting grain size is given (out of the considered values of 0.1, 0.2, and 0.5~$\\mu$m), but none of the values\ncan be excluded. A larger grain size will lead to a higher dust extinction (less flux at shorter wavelengths), \nwhich could also be mimicked by a larger interstellar extinction. \nThe values of the flux-weighted extinction coefficient scale with the adopted grain size.\nTo evaluate the impact we redid the SED modelling of IRAS 17251--2821, assuming \na 0.1~$\\mu$m grain (c.f. 0.5~$\\mu$m grains in our best fit model). The \nluminosity remained unchanged, but the inner dust radius\ndecreases from 12.2 to 9.1 $R_*$ and the dust mass loss rate increases by 40\\%.\n\n\n \\begin{table*} \n\\centering\n\\caption{SED modelling parameters. }\n\\begin{tabular}{lccccccccc}\n\\hline \n  &  & & &  &  &  & & \\\\\nIRAS & L$_*$  & R$_{\\rm in}$  & $\\dot{M}_{\\rm d}$ & grain size & dust mix & $<Q>$ & $\\rho$ & ${\\tau}_{\\rm V}$ \\\\\nname & ($L_\\odot$) & ($R_*$) & ($10^{-8}$ M$_{\\odot}$ yr$^{-1}$) & ($\\mu$m) &  & & (g cm$^{-3}$) & (at 0.55~$\\mu$m) \\\\\n\\hline \n  &  & & &  &  &  &  & \\\\\n17251-2821  & 4780 & 12.2 &  9.8 & 0.50 & MgFeSiO$_4$:Al$_2$O$_3$:Fe $=$ 90:10:10 & 0.25  & 2.70 &  64.5 \\\\\n17276-2846  & 5120 & 13.6 & 20.6 & 0.50 & MgFeSiO$_4$:Fe $=$ 100:10 & 0.20  & 2.65 & 119.0 \\\\\n17323-2424  & 4960 & 12.1 & 13.3 & 0.50 & MgFeSiO$_4$:Fe $=$ 100:30 & 0.32  & 3.45 &  90.8 \\\\\n17347-2319  & 3880 &  9.2 &  6.6 & 0.20 & MgFeSiO$_4$:Al$_2$O$_3$:Fe $=$ 95:5:10 & 0.10  & 2.68 & 106.0 \\\\\n17382-2830  & 5460 & 13.5 &  7.7 & 0.50 & MgFeSiO$_4$:Al$_2$O$_3$:Fe $=$ 90:10:3 & 0.23  & 2.35 &  41.7 \\\\\n17413-3531  & 4920 &  9.0 &  2.5 & 0.50 & MgFeSiO$_4$:Al$_2$O$_3$:Fe $=$ 80:20:30 & 0.50  & 3.52 &  23.0 \\\\\n17521-2938  & 4110 & 13.8 & 20.0 & 0.50 & MgFeSiO$_4$:Fe $=$ 100:10 & 0.20  & 2.65 & 127.0 \\\\\n18042-2905  & 4600 &  7.9 &  4.3 & 0.10 & MgFeSiO$_4$:Fe $=$ 100:30 & 0.075 & 3.45 &  37.7 \\\\\n\\hline\n\\end{tabular}\n\\vfill\nNotes: The dust MLR $\\dot{M_d}$ is \ndetermined for a 10~\\ks\\ expansion velocity. $<Q>$ is the flux-weighted\nextinction coefficient and $\\rho$, the grain density. \n\\label{tab:sed_model}\n \\end{table*} \n \\begin{table*} \n\\centering\n\\caption{CO modelling parameters. }\n\\begin{tabular}{lccccccc}\n\\hline\n &  & &  & & &  \\\\\nIRAS & $v_{e}$       & $v_{\\rm drift}$  & $\\dot{M_{\\rm d}}$      & $r_{\\rm gd}$ & $\\dot{M}$            & $\\epsilon$ \\\\\nname & (km s$^{-1}$) & (km s$^{-1}$)  & ($10^{-7}$ M$_{\\odot}$ yr$^{-1}$) &   \n          & ($10^{-5}$ M$_{\\odot}$ yr$^{-1}$) & \\\\\n\\hline \n &  & &  & & &  \\\\\n17251-2821  & 17.9 & 6.3 & 2.4 & 167 & 4.0 & 0.72 \\\\\n17276-2846  & 17.2 & 4.4 & 4.5 & 195 & 8.7 & 0.75 \\\\\n17323-2424  & 15.8 & 5.5 & 2.9 & 324 & 9.5 & 0.75 \\\\\n17347-2319  &  9.0 & 2.8 & 7.8 & 385 & 3.0 & 0.75 \\\\\n17382-2830  & 19.6 & 7.8 & 2.1 & 106 & 2.2 & 0.75 \\\\\n17521-2938  & 18.2 & 4.5 & 4.6 & 154 & 7.0 & 0.75 \\\\\n18042-2905  & 16.7 & 3.9 & 0.9 & 366 & 3.2 & 0.75 \\\\\n\\hline\n\\end{tabular}\n\\vfill\nNotes: The dust MLR is corrected for the \nobtained dust velocity (see Section~\\ref{sec:COmodel}). The $\\epsilon$ \nparameter is the slope of the gas temperature power law, see Eq.~\\ref{Eq-gas}.\n\\label{tab:co_model}\n \\end{table*} \n\n\\subsection{Modelling of the CO data}\n\\label{sec:COmodel}\n\nTo derive the gas MLR, we assume that the dust \nis driven by the radiation pressure and that the gas is driven by collisions\nwith the dust particles. For this, we solve the\nequation of motion for dust-gas interaction based on \\citet{Goldreich1976}. \nThe dust MLR\nand stellar parameters as well as the dust properties are taken\nfrom the SED modelling in section \\ref{sec:IRmodel}. We assumed that the \nstellar mass for these Bulge OH\/IR stars is 2~M$_{\\odot}$. The initial masses \nof the OH\/IR stars will be further discussed in Section~\\ref{sec:population}. \nIn order to drive the gas to the observed gas terminal velocity, we input\nthe gas-to-dust mass ratio. Hence, we obtain the dynamical gas MLR for each \nobject. At the same time, we calculate the dust drift velocity, $v_{\\rm drift}$, \nvia\n\\begin{equation}\n v_{\\rm drift}^{2} = \\frac{<Q> \\, L_*\\, v_{e}}{\\dot{M}\\, c}\n\\end{equation}\nwhere $\\dot{M}$ is the total MLR\nand\n$v_{e}$ the gas velocity which is measured from the OH maser observations\n(Table~\\ref{tab: sources}) and corrected to the terminal velocity hence the gas \nvelocity \nis 1.12$\\times v_{\\rm OH}$ (as described in \\ref{sec: COdata}). The dust \nvelocity is simply $v_{e} + v_{\\rm drift}$. \nThe dust mass loss required to fit the SED is then modified by the\nderived dust velocity, $v_{\\rm dust}$. The SED fitting measures the dust \ncolumn density hence keeping $\\dot{M}_{\\rm dust}\/v$ will maintain the \noverall SED fit. The modified dust MLR is again used as an input in the \ndynamical calculation in the iterative process to \ncalculate the gas velocity and the dust velocity by modifying the dust-to-gas \nmass ratio.\n\nThe new value of the dust velocity is then used to calculate an updated dust\nMLR (keeping the  $\\dot{M}_{\\rm dust}\/v_{\\rm dust}$ constant).\n\nThis \niterative process is said to be converged when the values of the successive \ndust velocities agree to better than 1\\%. Table~\\ref{tab:co_model} lists the\nparameters derived from the dynamical calculations.\n\nIn general, we can use the velocity profile to\nprobe the formation of lines with different excitation but in this study, \nCO J=2-1 and 3-2 arise in the region where the wind has reached its\nfinal velocity and hence do not probe the acceleration zone.\n\nWe assume that the metallicity of the stars in the\nGalactic Bulge is approximately solar \\citep{Uttenthaler2015} and thus take a \ncosmic abundance of carbon and oxygen hence the CO abundance, CO\/H$_{2}$, \nis 3$\\times 10^{-4}$. We used the molecular radiative transfer code based on \nworks by \\citet{SchoenHemp} and \\citet{Kay2004} to simulate the\nCO lines.\nWe took into account up to J=30 levels for both the ground and first\nvibrational states of CO. The collisional rates for between the rotational\nstates in both v=0 and 1 are taken from \\cite{Yang2010}.\n\nWe assume a gas temperature law in a form\n\\begin{equation}\n T_{\\rm g}(r) = T_{\\rm eff}\/r^{\\epsilon}\n\\label{Eq-gas}\n\\end{equation}\nwhere $T_{\\rm eff}$ is the effective temperature of the star and $\\epsilon$\nis a gas temperature exponent between 0.7 - 0.75\n(see Table~\\ref{tab:co_model})\nwhich gives the best result for to the observed CO profiles.\nWe take into account the infrared pumping by the dust and assume a dust \ntemperature in a form of\n\n\\begin{equation}\n T_{\\rm d}(r) = T_{\\rm con}\/r^{\\eta}\n\\label{Eq-dust}\n\\end{equation}\nwhere $T_{\\rm con}$ is the dust condensation temperature, 1000~K  and $\\eta$ is\na dust temperature exponent of 0.45 - a slope derived from a single power\nlaw from the dust SED modelling. We set the CO outer radius to be at\n1.5 times the CO photodissociation radius set to be where the CO\nabundance drops to half its initial value \\citep{Mamon1988}. Since the time this\nanalysis was performed, one of our co-authors presented a new paper on the\ncalculation of the CO photodissociation radius \\citep{Groenewegen2017}, based on \nimproved numerical method and updated H$_2$ and CO shielding functions. Taking the\nresulting radius for a star with a MLR of $5 \\times 10^{-5} M_{\\odot}$ yr$^{-1}$ \nfrom his Table~1, as a representative case for our sample, we find a\nradius which is 10\\% smaller than what we used, on basis of \\cite{Mamon1988},\nwell within the uncertainties.\n\n\nThe CO line intensities derived from our dynamical modelling are plotted\ntogether with our CO data in Figure~\\ref{Fig-CO}. It should be mentioned that at\nthese high MLR ($> 10^{-5} M_{\\odot}$ yr$^{-1}$) the optically thick spectral \nlines become saturated \\citep{Ramstedt2008}, so that the line intensities become \nless dependent to the MLR. This has no effect on our derived MLR as this is\nobtained from the dynamical modelling.\nThe calculated CO lines are more sensitive to the gas kinetic temperature\n  which is described by eqn.~\\ref{Eq-gas}.\n  \n\nThe dynamical MLR \nis derived from the assumption that the dust driven wind varies smoothly as \n1\/$r^2$ for a spherical symmetric wind outside the acceleration zone where it \nhas reached a constant terminal velocity.\nContrary to the modelling of OH\/IR stars by \\citet{just96},\nthere is no need to shorten the CO outer radius. For the sample\nof our study no significant change in the MLR \nis required to fit the CO profile. We will further discuss this issue\nin Section~\\ref{sec: superwind}. \n\n\n\nThe derived gas-to-dust mass ratios range from about 100 to 400 and  \nreflect a large range seen in galactic objects \\citep{Kay2006}.\nThe derived MLRs are reasonably moderate for OH\/IR stars and lower \nthan those derived from galactic extreme OH\/IR stars\nwhich show MLRs in excess of 10$^{-4}$ M$_{\\odot}$\nyr$^{-1}$. The latter stars are thought to be intermediate-mass stars with \ninitial masses M$_{\\rm init} > 5$ M$_{\\odot}$ based on their low $^{18}$O\/$^{17}$O ratios \n\\citep{Justtanont2015}.\n\nIn order to check how the input parameters affect the outcome, we changed the \nvelocity by $\\pm$15\\% and calculate the resulting dust and gas mass loss rates. \nChanging the gas velocity by 15\\% changes the dust mass loss rate and the dust \n(gas $+$ drift) velocity by the same amount but affects the gas-to-dust by 25\\%. \nThe combined changes result in a change in the derived gas mass loss rate by \n$\\sim$ 10\\%. \n\nJust like in Section~\\ref{sec:IRmodel} we investigate here the effect of using \na smaller 0.1~$\\mu$m grain in our\nmodelling for IRAS 17251--2821. With the smaller grain size we find a smaller\n$v_{\\rm drift} = 2.2$~km s$^{-1}$ in the dynamical modelling and hence obtain a\ndifference of only 15\\% in $\\dot{M_{\\rm d}}$, rather than the 40\\% we obtained\nin Section \\ref{sec:IRmodel}. The total MLR becomes $5.2\\, 10^{-5} \n{\\rm M}_{\\odot} {\\rm yr}^{-1}$, i.e. 30\\% higher than in the 0.5~$\\mu$m grain case and \nthe $r_{\\rm gd}$ becomes 190 vs 167 (see Table~\\ref{tab:co_model}).\n\n\\subsection{Periods of the variables}\n\\label{sec:Periods}\nAmongst the Long Periodic Variable stars, OH\/IR stars are known to have the largest\namplitudes ($\\sim$ 1 magnitude bolometric) and the longest periods (several \nhundreds up to more than a thousand days).  vH2007\nmonitored the stars in near-infrared bands ($J,H,K$ and $L$). For four sources the \nperiod of variability could be established which are indicated in \nTable~\\ref{tab: periods}. The other sources showed either variability, but no \nperiod could be established or were not detected in the K band. More recently, \nmulti-epoch\nobservations from the VVV survey  in the K-band (we used public data from DR4) and the \n{\\it AllWISE Multi epoch Photometry Table} and the\n{\\it Single Exposure (L1b) Source Table} from the NEOWISE reactivation mission\n\\citep{Mainzer2014} became available. For the latter we used data in the W2 filter at \n4.6~$\\mu$m where the OH\/IR stars stand out as bright stars with respect to the\nsurrounding stars. We did not use the longer wavelength filters W3 and W4, as  \nthe PSF becomes larger and increases the risk of crowding. We only used \ndata with individual error bars less than 0.04 mag. \n\nThe K- and W2-band data were investigated\nseparately to determine the periods, amplitudes and mean magnitudes using the \nprogram  {\\it Period04} \\citep{Lenz2005}.\n\nThe periods, averaged magnitude over the light\ncurve and the amplitude are shown in Table~\\ref{tab: periods}.\nFor IRAS~17251--2821, which is very weak at K (14.5 mag), we were able to \nestablish a period on the basis of the K-band VVV survey data where previously \nvH2007 could\nonly establish that the source was variable but could not determine a period. For the \nfour sources with periods determined in vH2007\nand now from \nthe VVV-survey we find similar periods, only deviating by a few percent, except \nfor IRAS~17347--2319 (see Section~\\ref{sec:Indiv}). The vH2007 monitoring \nperiod took place in 2004 -- 2006 and the VVV data covers 2011 -- 2013, so that \nslight changes might be real.\nThe average $K$ magnitudes are generally fainter for the VVV survey than in the\nvH2007 result, which can be explained by the difference in the\nfilter profile of the MSSO K band \\citep{McGregor94} and the K$_{\\rm s}$ band used in the\nVISTA system \\citep{Minniti2010}, combined with the very red SEDs of our sources. \nIRAS~17347--2319 and 18042--2905 show consistent K-band amplitudes, 17382--2830 and \n17413--3531 show much smaller amplitudes in our new fitting. For the latter source \nthis might be related to the lower quality of the VVV photometry and \nsubsequently of our fit.\n\nFor all our sources we were able to determine periods from the WISE survey. The\nfact that our sources are brighter at 4.6~$\\mu$m and suffer less of source\nconfusion is likely to explain this higher success rate. The periods are \nconsistent with what is derived from the VVV K-band survey. \nIn case of the WISE data the period from vH2007 or the period determined from analysing \nthe VVV data was used as a first guess and the program was allowed to converge.\nIn the cases that there was only WISE data available, several periods were tried. \nFor the further analysis, we adopt one period per source. In case of 2 K-band periods, \nwe use the average value, for IRAS 17251--2821 we adopt the VVV derived period \nand in all other cases we take the WISE band derived value. \nThe adopted periods are given in the last column of Table~\\ref{tab: periods}.\n\n\n \\begin{table*} \n\\centering\n \\caption{ Variability parameters.}\n\\begin{tabular}{  l c c c c c c c c c c}\n  \\hline\n & & & & & & & & & & \\\\\n   & vH2007 &     &  & VVV &  & & WISE &    &   \\\\\n  IRAS name   &  Period &  < K >  & $\\Delta$ K & Period & < K > &\n  $\\Delta$ K & Period &  < W2 > & $\\Delta$ W2 &  adopted \\\\\n              &  (days)        &  (mag)    &  (mag)     &  (days)     &  (mag) &  (mag) &  (days)    &  (mag)        &  (mag)     &   P (days)   \\\\\n  \\hline \n & & & & & & & & & & \\\\\n17251-2821  & -- & 14.47 &  & 693 (7) & 16.07 (0.11) & 0.98 (0.02) & 681 (3) & 5.74 (0.03) & 1.10 (0.05) & 693 \\\\\n17276-2846  & -- & &  &  & & & 488 (15) &  7.35 (0.02) & 0.72 (0.03) & 488 \\\\\n17323-2424  & -- & & &  & &  & 552 (5) & 6.53 (0.05) & 0.94 (0.06) & 552 \\\\\n17347-2319  & 355 & 11.09 & 0.23 & 290 (3) & 11.64 (0.02) & 0.26 (0.02) & 292 (1) & 4.99 (0.04) & 0.86 (0.05) & 323 \\\\\n17382-2830  & 594 & 11.56 & 1.23 & 629 (9) & 12.2 (0.2) & 0.78 (0.03) & 625 (6) & 4.38 (0.07) & 1.09 (0.07) & 611  \\\\\n17413-3531  & 624 & 10.09 & 1.10 &  664 (18) & 11.22 (0.05) & 0.68 (0.06) &\n639 (9) & 5.38 (0.03) & 1.03 (0.04) & 644  \\\\\n17521-2938  & --  &  &    &  &   &      & 562 (3) & 7.20 (0.01) & 0.51 (0.02) & 562 \\\\\n18042-2905  & 594 & 10.87 & 0.78 & 574 (5) & 11.71 (0.04) & 0.80 (0.04) & 556 (3) & 5.28 (0.03) & 0.86 (0.04) & 584 \\\\\n\\hline\n\\end{tabular}\n\\label{tab: periods}\n\\\\\nNotes: Periods and semi-amplitudes taken from \\cite{vanhollebeke2007}  and \nnewly determined on basis of VVV and WISE survey data (see text). Uncertainties\nare given between brackets, but are not available in vH2007.\nThe last column  gives the period that is further used in the analysis.\n \\end{table*} \n\n\\subsection{Comments on individual sources}\n\\label{sec:Indiv}\n\\begin{itemize}\n\\item IRAS 17382-2830: Only source where we could only detect the CO (3-2) \ntransition, and not the (2-1). The overall noise of the CO (2-1) measurement\nis not different from the other measurements but this source is significantly \ncloser to the galactic plane (latitude approximately 1 degree). The background\nsubtraction is more problematic because of the interference from the \ninterstellar CO gas \\citep{Sargent2013}. The stellar velocity taken from the OH \nis at $-$51.2~\\ks (LSR) which overlaps with a region designated by\n\\cite{Dame2001} as the Nuclear Disk, which may cause the stronger fluctuations\nin the baseline between -70 and +10 \\ks. \nThis would also explain the fact that whereas the other stars have stronger CO\n(2-1) than (3-2) detection, we here detect only the 3-2 transition, which is\nless hampered by the ISM. \n\\item IRAS 17413-3531: No CO emission was detected. It is the bluest source in \nour sample with the 9.7~$\\mu$m band still partially in emission and has the \nlowest $\\dot{M}_{\\rm d}$ (Table~\\ref{tab:sed_model}). The CO emission may \nbe too weak for a detection in our survey.\n\\item IRAS 17347-2319: Only a weak CO detection but the star has a very red \nSED with a high $\\tau_V$ and a strong silicate absorption band at 9.7~$\\mu$m, \nindicating a high MLR. The star has only a relatively short \nperiod (355 days in vH2007 and 290 days in our analysis). This star will be \nfurther discussed in Section~\\ref{sec:devper}.\n\\item IRAS 17276-2846: This is one out of three sources in common with the\nsample studied by JEE15. They find a double-peaked SED for this source, where\nthe 'blue' peak below 2~$\\mu$m is believed to correspond to the stellar\nphotosphere and the red part to the mass loss during the AGB. The star would\nhave now ended the AGB phase and has become a\nso-called proto-planetary nebula. We do not follow this interpretation. Both the\nmodelling of the SED and the CO line strengths point to a present high MLR. \nThe IRS spectrum still shows a very strong 9.7~$\\mu$m absorption band,\nwhich would disappear rapidly after the mass loss has stopped \\citep{Kay1992}. \nOn the basis of the WISE data we also find that the star is\nvariable with a large amplitude ($\\Delta$W2 $= 0.72$ mag) and so likely still \non the AGB. \nWe believe that the 'blue' counterpart is not associated with the OH\/IR star, \nbut a nearby confusing source and was thus not further considered for our analysis. \n\\end{itemize}\n\n\n\\section{Comparison with JEE15 on L$_*$, $\\dot{M}$ and r$_{gd}$}\n\\label{sec:comparison}\n\nAs described in Sections \\ref{sec:IRmodel} and \\ref{sec:COmodel}, the modelling \nof the observed SED and the CO \nmeasurements is a two-step process where first the\ninfrared observations are fitted. The resulting dust MLR is\nused as an input to derive the gas-to-dust ratio, and hence the gas MLR and CO\ndensity leading to CO (2-1) and (3-2) transition line strengths.  \nThe MLRs ranging from 10$^{-5}$ to 10$^{-4}$ M$_{\\odot}$ are \ntypically what is expected from OH\/IR stars and are not extremely high as MLRs in \nexcess of 10$^{-4}$ M$_{\\odot}$ yr$^{-1}$ have also been found \\citep{Justtanont2015}. \nComparison of MLRs with other studies needs to be done with care as different modelling methods\nand assumptions can lead to different estimates of the mass loss and gas-to-dust\nratios. Also, in many studies the modelling is done on either only\nobserved SEDs or have only CO measurements available. Here we \ncompare our results with those of JEE15 on a larger sample of Bulge OH\/IR stars.\n\nThe JEE15 modelling is restricted to the SED fitting, making use of the OH maser\nobservations to have an estimate of the expansion velocity. They use the DUSTY \nradiative transfer code \\citep{Dusty1999} to determine the luminosities and MLRs. \nWe have three stars in common in our samples. \nFor IRAS~17251--2821 they give a range of luminosity 3~100 - 7~200~L$_{\\odot}$ \nwhere we find 4~780~L$_{\\odot}$. The MLR ranges from 1.7 to 3.4~$10^{-5}$ \\msolyr\\ \nin JEE15 versus our slightly higher value of 4.8~$10^{-5}$ \\msolyr. \nFor IRAS~17322--2424 we find a larger difference in the MLR: 4.2~$10^{-5}$ vs. our 9.5~$10^{-5}$ \\msolyr\\ and\ncomparable luminosities: 4~200 and 4~960~L$_{\\odot}$, respectively. JEE15 quote \nan uncertainty of a factor of 2 for the MLRs of the individual sources. \nFor IRAS~17276--2846, the third source that is in common in our samples, they do not \ngive model results, as they believe that the source has left the AGB (see Section \\ref{sec:Indiv}).\n\nTo get a broader comparison with their results, we also compare their average\nvalues for their larger sample with ours. As mentioned in Section~\\ref{sec: sample}, \nJEE15 selected a sample which includes brighter\nIRAS sources than we have. They divide their sample into low- and\nhigh-luminosity groups, where their division lies at 7,000~L$_\\odot$. As our\nstars all belong to the first group, we will only compare with \nthe average values for the MLRs of the so-called low-luminosity group. \nThe thirteen 'low-luminosity' sources in JEE15 have an \naverage MLR of 2.7~$10^{-5}$~\\msolyr\\ with a standard deviation \nof 1.6~$10^{-5}$~\\msolyr\\ versus our $(5.4 \\pm 3.0)$ $10^{-5}$~\\msolyr. \nJEE15 also used their DUSTY results to derive the gas-to-dust \nratio, based on the SED fit and the expansion velocity, when known from the OH maser profile. \nThis results in a value of r$_{gd} = 44 \\pm 20$, which is considerably lower than\nthe average value that we find of $242 \\pm 113$.\n\nThe differences in total MLRs and gas-to-dust ratios between JEE15\nand ours may be explained by the different assumptions and inputs \nused in the modelling of the SED. JEE15 make use of optical constants \nfor amorphous cold silicates from \\citet{Ossenkopf1992} and \nthe standard MRN \\citet{Mathis1977} dust size distribution with\n$n(a) \\propto a^{-3.5}$, where $n$ is the number density and a is the size of\nthe grains. The grain sizes were limited to 0.005 $\\leq a \\leq 0.25~\\mu$m. Our\nassumptions are described in Section~\\ref{sec:IRmodel}. We make use of a\ncombination of dust species and a single grain size, selecting the\nbest fitting one from 0.1, 0.2 and 0.5~$\\mu$m respectively. Six\nout of eight sources gave a best fit with a grain size of 0.5~$\\mu$m, so larger\nthan what was used in the JEE15 modelling. \n\nTo illustrate the impact of the assumed dust properties used, we redid the \nmodelling of IRAS~17347--2319, \nusing the silicates from \\citet{KayXander1992} with a grain size of \n0.2~$\\mu$m and a specific density of 3.3 g\\,cm$^{-3}$. \nThe resulting total $\\dot{M} = 1.60$ $10^{-5}$~\\msolyr and a \nr$_{gd} = 202$ are both a factor of approximately two lower than the values \ngiven in Table~\\ref{tab:co_model}. JEE15 made a comparison of their gas-to-dust ratio\nwith the modelling by \\cite{Kay2006} of OH\/IR stars in the Galactic Disk. They\nconclude that the Bulge OH\/IR stars are on the low side in comparison to the\nvalues given by \\cite{Kay2006} which range from 50 to 180. Assuming that the\nfactor 2 difference in the gas-to-dust ratio for IRAS~17347--2319 that we find \nbetween our present modelling and the modelling using the input from \n\\cite{Kay2006}, applies to our entire sample, we find a\nsimilar range of r$_{gd}$ as what was found for the Disk stars. \n\n\\section{The characteristics of the observed sample}\n\\label{sec: characteristics}\nIn this Section, we will investigate what we can learn about important\nparameters for the understanding of the AGB evolution, taking advantage of\nhaving a group of stars at relatively well known distance and originating of\nthe same stellar population. In the next Section we will then discuss what we \ncan learn about this population of stars in the Bulge.\n \n\\subsection{Luminosity and Period distribution}\n\\label{sec: lumdist}\nThe luminosities obtained from the SED modelling range from approximately 4~000\ntot 5~500~L$_\\odot$ and are on average 4729 $\\pm$ 521 L$_\\odot$, assuming that all\nsources are at the distance of the Galactic Centre, taken at 8~kpc. \n \n        \\begin{figure}\n        \\resizebox{\\hsize}{!}{\\includegraphics{P_L_OHIR_fina.eps}}\n        \\caption{  Period-Luminosity diagram for the OH\/IR stars with\n\tthe PL relation taken from \\protect\\cite{Whitelock1991}. The error bars \n\tindicate the\n\tuncertainty from the SED modelling. The point at log P $=2.9$ gives the spread\n\tin luminosity because of the depth of the Bulge}\n        \\label{fig: PL}\n        \\end{figure}    \n\n\nThe average luminosity of the sample agrees well with the peak in the \nluminosity distributions found by \\cite{WilHarm1990} (5~000-5~500~L$_\\odot$, for\na distance of 8.05~kpc) and by JEE15 ($\\approx 4~500$~L$_\\odot$, for a distance of 8~kpc). \nHere we want to point out that the well known and best studied OH\/IR stars often have\nluminosities well above 10~000~L$_\\odot$ (e.g. \\cite{beck10}), but that\n\\citet{Habing1988} in his analysis\nof the galactic distribution of IRAS sources with OH\/IR-like colours found a\nluminosity distribution peaking at 5~000~L$_\\odot$. The Bulge OH\/IR stars are\nthus not of exceptionally low luminosity.\n\nThe sample of OH\/IR stars in JEE15 also contains stars with luminosities above\n10~000~L$_\\odot$ (their so-called high-luminosity group). We do not have these\nbecause our selection of candidate OH\/IR stars is based on the\n\\cite{WilHarm1990} sample as described in Section~\\ref{sec: sample}. In the \nlatter analysis of the flux distribution of IRAS\nsources it is found that the stars with \"apparant\" luminosities above \n10~000~L$_\\odot$ (F$_{12 \\mu m} > 10$~Jy) are likely stars from the Galactic\nDisk population. However, because of the lower number density, it could not be \nexcluded that the Bulge also contained higher luminosity OH\/IR stars. In our \nselection of stars to observe CO emission, we chose to select sources with the \nhighest probability to be genuine Bulge stars and thus only selected stars with \nF$_{12 \\mu m} < 10$~Jy. \n\nMost sources have periods in the range of 500 -- 700 days. This distribution\ncorresponds to the longest periods of the Bulge IRAS sources period distribution \nas determined by \\cite{Whitelock1991}. As the OH\/IR sources are the most extreme\nAGB stars, this is no surprise. However, the periods are certainly not as extreme \nas several OH\/IR stars in the Galactic Disk which have periods well \nabove thousand days (e.g. \\citet{vanlangevelde90}).\n\nIRAS~17347--2319 has a clearly deviant period (P $= 323$~days) in comparison \nto the other OH\/IR\nstars and will be discussed in the Section~\\ref{sec:devper}. \n\n\\subsection{Period-Luminosity comparison}\n\nFigure~\\ref{fig: PL} shows the position of our stars in the so-called period-luminosity (PL) diagram. \nThe full line shows the period-luminosity relation as derived by \\cite{Whitelock1991} \nbased on LMC oxygen-rich Mira variables with P$< 420$ days \\citep{Feast1989} \nand Galactic Disk OH\/IR stars with phase-lag distances \\citep{vanlangevelde90}. \nThis is the only PL relation that combines shorter period Miras with the \nlonger period OH\/IR stars that we are aware of. The distances \ntowards the OH\/IR stars in \\cite{vanlangevelde90} were determined with the \nso-called phase-lag method \nand the overall uncertainty in $M_{\\rm bol}$ is still between 0.5 and 1.0 mag. \nClearly our OH\/IR stars fall well below the relation as was also found for a \nsample of OH\/IR stars in the Galactic Centre \\citep{Blommaert1998} and \nwe refer to the discussion in that paper on the PL relation for OH\/IR stars. \nIt should be noted that using an extrapolation of the PL relation of the oxygen-rich Miras in the LMC \\citep{Feast1989} \nwould only show an even larger deviation with our\nOH\/IR stars than with the PL relation used in this analysis.  \n \nRather than considering the OH\/IR stars as an extension of the Miras \ntowards higher masses, we believe that the OH\/IR stars in this sample are to be seen as an\nextension of the Miras towards a further evolved phase, as will be discussed in\nthe next session. The PL relation is also used to derive distances to the OH\/IR\nstars, for instance in \\cite{beck10}. Although usage of the PL relation\nis often the only way to get an estimate of the OH\/IR star's luminosity, our \nresult shows that this can lead to significant and systematic overestimation of the luminosity.  \n\n\n\\subsection{Mass-loss rates versus Luminosity}\n\nFigure~\\ref{fig: MdotvsL} shows the MLRs of our stars versus the \nluminosities. In this\nradiatively driven wind these quantities are not independent. The relation for\nthe so-called classical limit, i.e. only allowing one single scattering event \nper photon ($\\dot M_{\\rm classic} = L_* \/ (v_{\\rm exp} . c)$), is shown in the \nfigure.\n\\cite{vanLoon1999}  \nshowed that for a sample of AGB stars in the LMC several sources surpassed this\nlimit demonstrating that multiple scattering happens in dusty circumstellar\nenvelopes \\citep{GailSed1986}. \\cite{vanLoon1999} suggested a new empirical \nupper limit which is also indicated in Figure~\\ref{fig: MdotvsL}.\nClearly all our sources surpass the classical limit, and three even surpass the\nlimit suggested by van Loon. \nThe three sources with the highest MLRs also have the highest \noptical depths (Table~\\ref{tab:sed_model}), where multiple scattering is likely \nto become increasingly important. Whether or not our stars indeed surpass the \nempirical limit suggested by \\cite{vanLoon1999} is more difficult to answer, considering\nthe uncertainties one needs to take into account when comparing MLRs\nderived from differerent methods (see Section~\\ref{sec:comparison}). Lowering\nthe MLRs by a factor of two would bring the highest MLRs\njust above the empirical relation given by \\cite{vanLoon1999}. On the other\nhand, the number of sources with such high MLRs in their paper is small and the\nlimit may be uncertain because of this. They also show that the optical depth \nby the\ncircumstellar shell is related to the $K-L$ colour. Our sources have redder $K-L$\ncolours than the oxygen-rich stars in their sample and thus may indeed be more \nextreme than the sample of LMC stars studied by \\cite{vanLoon1999}.  This\ndifference in optical depth may be related to the likely higher metallicity of\nthe Bulge OH\/IR stars in comparison to the LMC stars. And could indicate that\ndifference in MLR observed is real and related to the different populations.\n\n        \\begin{figure}\n        \\resizebox{\\hsize}{!}{\\includegraphics{MdotvsL.eps}}\n        \\caption{   Mass-loss rate versus bolometric luminosity. The full\n\tline shows the classical limit of a single scattering event per photon. \n\tThe dashed line shows the empirical limit of MLRs that was suggested by\n\t\\protect\\cite{vanLoon1999} for a sample of oxygen- and carbon-rich stars in the\n\tLMC.}\n        \\label{fig: MdotvsL}\n        \\end{figure}    \n\n\\subsection{Mass-loss rates versus Period}\n\nA first condition to start an efficient dust-driven wind is the levitation of \nthe gas, caused by large amplitude pulsations, to regions above the photosphere \nwhere grains can form. Earlier studies on the mass loss showed the dependency of\nMLRs and the pulsation period of the AGB star (\\cite{vassi93} and \nreferences therein) \\cite{SchOlofsson2001} and \\cite{beck10}). In agreement \nwith \\citet{vassi93}, \\citet{beck10} find an\nexponentially increasing MLR with period, until a maximum level is\nreached where the MLR no longer increases. \\citet{vassi93} see the \nleveling off occuring at a period of 500\ndays, whereas \\citet{beck10} find that the MLR remains constant  \nfrom approximately 850 days onward (at ${\\rm\nlog}(\\dot{M}) = -4.46$, with  $\\dot{M}$ in units of \\msolyr). We will come back to the comparison with the \n\\citet{vassi93} result in the next section. Applying the relation provided by \n\\citet{beck10} for periods shorter than 850 days:\n\n\\begin{equation}\n{\\rm log}(\\dot{M}) = -7.37 + 3.42 \\times 10^{-3} \\times P\n\\label{Eq-MP}\n\\end{equation}\n\ngives MLRs significantly lower than our values by a factor ranging\nfrom 4 to 44 with a mean of 18 (we have excluded IRAS~17347-2319 because of its \nvery short variability period, see Section~\\ref{sec:devper}). On the other\nhand, the scatter around the relation in \\cite{beck10} is quite large  (up to a\nfactor ten below and above the relation), so that \nour MLRs are not entirely inconsistent with the MLRs \nobtained in their analysis. We conclude however that the relation given in \neqn.~\\ref{Eq-MP} for periods below 850 days, is not applicable to our stars, \nbut that they have MLRs which agree with the 'plateau' value of\n$\\dot{M} \\simeq 3.4 \\times 10^{-5}$~\\msolyr, the region associated with the \n{\\it superwind} by \\cite{vassi93}. \n\n\n        \\begin{figure}\n        \\resizebox{\\hsize}{!}{\\includegraphics{MdotvsP.eps}}\n        \\caption{  Mass-loss rate versus the variability period of our sources. \n\tThe full line shows the fit by \\protect\\cite{beck10} to the MLRs in \n\ttheir sample for periods below 850 days. The MLR remains constant for\n\tlonger periods. The dashed lines indicate a spread of a factor 10 around\n\tthe fits, as is indicated in their Figure 14.}\n\n\n        \\label{fig: MdotvsP}\n        \\end{figure}    \n\n\n\\subsection{The deviant variability behaviour of IRAS~17347--2319}\n\\label{sec:devper}\nIRAS 17347--2319 is standing out in the P-L diagram because of its short period \nin respect to the other OH\/IR stars. The average period of\nthe other stars is 589 days, versus the adopted period of 323 we find for this\nstar. The full amplitude in the K-band is small (0.48 and 0.52 according to \nvH2007 and our own analysis, respectively) only just sufficient to be \nclassified as a Mira variable (full $\\Delta$~K $> 0.4$~mag, \\citet{Feast1982}). \nApart from the short period and small amplitude, this is also the only source \nwhich shows a deviating period between what is found by vH2007 and our own \nanalysis (see Section~\\ref{sec:Periods}). Where vH2007 finds a period of 355 days, \nwe find consistently a period of 290 days from the VVV and WISE data. The data \nused in VH2007 was taken from  mid-July 2004 until November 2006, whereas for the VVV\nand WISE survey we have data starting in April 2010 and ending in September 2013\nfor VVV and end of August 2015 for WISE respectively. A change in variability \nis noticed in\nabout one percent of the Mira variables \\citep{Zijlstra2002,Templeton2005}.\n\\cite{Zijlstra2002} define three classes: \"continuous change\", \"sudden change\"\nand \"meandering change\". In the first class, a continuous increase or decrease \nof the order of 15~\\% occurs over a period of 100 years, whereas for the second,\nsuch a change occurs ten times faster. In the third class, a\nchange of about 10~\\% in the period duration is seen to happen over several\ndecades, followed by a return to the original period. The rapid change by 18\\%\nin less than a decade would place IRAS 17347--2319 in the second class, which\ncontains Mira variables like BH Cru, RU Vul and T Umi \\citep{Uttenthaler2011}.\nThe occurance of a thermal pulse, when a helium-burning shell takes over from\nthe hydrogen-burning shell as the main energy source on the AGB, \nwas suggested to explain the rapidly changing period \\citep{Wood1981, \nUttenthaler2011}. This could occur either in the build-up towards \nor in the aftermath of the thermal pulse. The strongly changing radius and\ntemperature of the star during such a thermal pulse will lead to a change in \nperiod ($P \\propto R^{1.94} \/ M^{0.9}$), luminosity and expansion velocity \n\\citep{vassi93}. An alternative explanation that has been suggested for a rapid \ndecrease is a pulsation mode switch from the fundamental (assumed\nto be the case for Mira's and OH\/IR stars) to a low overtone mode (like in Semi\nRegular Variables) \\citep{LebWood2005}. \n\nBecause of the large amplitude variability of this type of stars, it is not\npossible to find evidence of a changing average luminosity of IRAS~17347--2319\nover the time for which we have photometry available. However, the star has \nthe lowest luminosity in our sample and is about 20\\% lower than the average \nvalue of our sample. Also the expansion velocity significantly deviates from \nthe average velocity found for the other stars: 9.0 vs 17.6 $\\pm$ 1.3~km\ns$^{-1}$. \nIf  IRAS~17347--2319 is in a post-thermal pulse phase, one could also expect a\ndecrease in the MLR to occur \\citep{vassi93} as is also observed in R\nHya, a Mira variable which has decreased its period from\n500 to 385 days over a time period of about 300 years \\citep{Zijlstra2002}. \nSuch a MLR change is not clear from\nour analysis. The SED shows a very strong obscuration in the visible and\nnear-infrared wavelengths, together with a strong silicate absorption band \nindicating a high present dust MLR. In the case of a rapidly\ndecreasing MLR, the stellar source would re-appear rapidly and the\nsilicate band would go into emission \\citep{Kay1992}. \nWe have much less information for\nIRAS~17347--2319 than for BH Cru, RU Vul and T Umi, which have been monitored\nfor decades, to confirm that it is \nundergoing a 'sudden change' in variabilty. Further follow-up of the variability\nand its SED would be highly desirable as this may be the first OH\/IR type for\nwhich such a behaviour has been observed.  \n\n\\section{the Bulge OH\/IR population}\n\\label{sec:population}\n\nGenerally OH\/IR stars are associated with stars of a few solar masses. Typical \nsuch examples are stars like OH26.5$+$0.6 which are very bright and have been \nstudied in considerable detail \\citep[e.g.,][]{Kay2006, Justtanont2015, \ngroenewegen12}.\nHowever, the OH\/IR stars in our sample show much lower luminosities, typically in \nthe range of 2000 - 7000 $L_\\odot$ and periods below 700 days, whereas the more\nluminous OH\/IR stars reach periods well above a thousand days and luminosities\nof several tens of thousands times the luminosity of the Sun. In contrast to\na number of extreme OH\/IR stars, which are associated with active\nstar-forming regions in the galactic plane like the Molecular Ring, \nthe OH\/IR stars in the Bulge likely evolved from lower initial masses and are \nolder, but still of intermediate age (1--3 Gyr). To\ninvestigate this claim we will now make a comparison with the \\cite{vassi93}\n(hereafter VW93) evolutionary tracks.\n\n \\begin{figure*}\n\\begin{minipage}{0.7\\textwidth}\n\\resizebox{\\hsize}{!}{\\includegraphics{P_L_mira_OHIR_VW_fin.eps}}\n\\end{minipage}\n        \\caption{ The PL diagram for Bulge Miras (taken from \\protect\\cite{GroenBlom2005}) \n\tand OH\/IR stars, together with the PL relation \n\t(\\protect\\cite{Whitelock1991}; full line) and the evolutionary\n\ttrack from VW93 for a 1.5~M$_\\odot$ star with a solar metallicity\n\t(H-burning phase of the last three thermal pulse cycles; dotted line).\n\t }\n        \\label{fig: PL_VW}\n        \\end{figure*}    \n\n\\subsection{Comparison with VW93}\n\\label{sec:VW93}\n\n\n\\cite{GroenBlom2005} studied the Galactic Bulge Mira variables on basis\nof the OGLE-II survey and near-infrared photometry from the DENIS and 2MASS\nall-sky databases. They show that the period distribution for stars within\nlatitudes ranging from -1.2$^\\circ$ to -5.8$^\\circ$ are indistinguishable and \ncan be explained by a population with initial masses of 1.5 - 2~M$_\\odot$, \ncorresponding to ages of 1 to 3 Gyr. This result was based on synthetic AGB \nevolutionary models\nwhere the synthetic AGB code of \\cite{WagenGroen1998} was fine-tuned to\nreproduce the models of VW93. VW93 provide calculations for a range of\nmetallicities, where \\cite{GroenBlom2005} selected the Z = 0.016 model, for a \nsolar mix. Studies of the metallicity of non-variable M giant stars in the Bulge\ngive a\nslightly sub-solar value \\citep{RichOriglia2005,Rich2007,Rich2012}. The same\nmetallicity was found for a sample of variable AGB stars in the Bulge by\n\\cite{Uttenthaler2015}. There are no direct measurements of the metallicities of\nthe OH\/IR stars, but assuming that they originate from the same population as\nthe Bulge M giants and AGB stars, the solar mix is indeed the most appropriate.\nThe other values included in the VW93 models: Z = 0.008, 0.004 and 0.001, \ncorrespond to  the LMC, SMC and lower metallicity populations.\n\nAccording to VW93 models, the maximum $M_{\\rm bol}$ at the Thermal Pulsing AGB \nduring H-burning are $-4.03$, $-4.52$, $-4.90$~mag for stars with solar metallicity and initial masses \nof 1.0, 1.5 and 2.0 M$_\\odot$ respectively. The model for the 1.5 M$_\\odot$ \nagrees closely to the Bulge OH\/IR stars luminosity (Section~\\ref{sec: lumdist}). In \nFigure~\\ref{fig: PL_VW} we show the VW93 track (dashed line) for a star with \nM$_{init} = 1.5 M_\\odot$ and\nsolar abundance in the PL diagram. The track covers the three last thermal pulse cycles before terminating\nthe AGB (with a duration $\\approx 300,000$ years). \nWe only include the part of track where the luminosity is produced by hydrogen \nburning, excluding the thermal pulses. Also shown are the \nMiras used in the \\cite{GroenBlom2005} analysis and our OH\/IR stars. The\nlarge spread in the Miras can be explained by a larger spread in distance\n(including fore- and background sources) and the single-epoch K band photometry\nwhich was not corrected for variability. The bolometric magnitudes for the \nMira stars were determined using\nrelation B of \\cite{Kerschbaum2010}. The track follows the PL relation for a\nlarge fraction where we also find the Mira stars. However, due to the changing \nmass, when the stars enter the so-called superwind phase, the period \n($P \\propto R^{1.94} \/ M^{0.9}$) keeps increasing while the luminosity (related \nto the core mass) stays almost constant. The track then overlaps with the\nposition of the OH\/IR stars in our sample.\n\nAssuming that a star with $M_{init} = 1.5$ $M_\\odot$ still needs to lose \napproximately 1~$M_\\odot$ of material \nin its final phase before ending as a white dwarf of about 0.5~$M_\\odot$ \nmeans that on average this OH\/IR phase will last 20,000 years. \nVW93 model predicts 98,000 yrs duration of the superwind.\nThe longer duration predicted by VW93 is possibly connected to the single\nscattering 'classical limit' they impose on the superwind MLR. As \nwas discussed in Section~\\ref{sec:comparison}, the MLRs we find \nsurpass the classical limit by at least a factor of two.\nVW93 predicts a ratio of the optical visible thermal pulsing AGB over the superwind phase of \n0.135. \\cite{Blommaert1992} compared the numbers of IRAS sources with optical \nMiras in the Bulge. For the sources with OH\/IR like colours the ratio is 0.02, \nsignificantly lower than predicted in VW93, but in agreement with the shorter\nsuperwind duration of 20,000 years. \n\nIn Figure~\\ref{fig: PL_VW} it can also be seen that the VW tracks allow longer\nperiods (even above thousand days) than what we find in our sample. This is \nhowever, only true for a short\nphase near the end of the AGB, which is even more reduced in time when allowing\nhigher MLRs, as we find is the case for the OH\/IR stars. \n \n\\subsubsection{Link between OH\/IR stars and Miras in the Bulge} \nFurther evidence for the connection between the Mira variables and the OH\/IR \nstars is found in their distribution in the Bulge. \\cite{Whitelock1992, \nGroenBlom2005} find that Mira stars follow a ''Bar'' structure with a viewing\nangle of approximately 4$5^\\circ$. Based on a dynamical modelling,\n\\cite{Sevensteretal1999} find that the OH\/IR stars in the inner Galaxy are \nmembers of the Galactic Bar with a\nviewing angle of 43$^\\circ$ agreeing with the distribution of the Mira stars.\n\nRecently, \\cite{Harm2016} suggested that the galactic bar OH\/IR stars are formed\nin the Molecular Ring, an active star forming region at $\\approx$ 4~kpc from the \ngalactic centre (l = $\\pm 25^\\circ$) which connects to the end \nof the galactic bar \\citep{Blommaert1994,Hammersley1994}. The stellar kinematics of the stars at the\ntips of the bar are equal to those of the star-forming regions at these\nlocations, indicating that stars formed in the molecular ring can easily become\npart of the galactic bar structure. Such a scenario agrees with the fact that \nwe find stars of intermediate age and with the fact that gas-to-dust ratios of \nour OH\/IR stars show the same range as was found by \\cite{Kay2006} for a sample \nof OH\/IR stars, situated predominately in the Molecular Ring.\nThe gas-to-dust ratio is\nbelieved to be inversely related to the metallicity of the stars\n\\citep{Habing1994},\nso that we conclude that the metallicities of the Bulge OH\/IR stars are similar\nto the selection of OH\/IR stars in the Disk studied by \\cite{Kay2006}.   \n\nOne final remark about the ages of the AGB stars in the Bulge. In the last\ndecades, there was often much debate on how the Bulge could contain AGB stars\nlike Miras and OH\/IR stars of intermediate age in a galactic \ncomponent which is believed to contain only an old stellar population \n\\citep{Renzini1994, KuijkenRich2002, Zoccali2003}. This led to\nsuggestions that the Mira population was the result of merged binaries\n(descendents of blue straglers, \\cite{Renzini1990}). In the last decade\nthere has however been growing evidence that at least a (small) fraction of the\nBulge stars is of intermediate age as was shown by \\cite{Gesicki2014} on basis \nof planetary nebula, by \\cite{Bensby2013} for metal-rich dwarf stars.\nThe appearance of Miras and OH\/IR stars in the Bulge is not so controversial in \nview of these recent results.\n\n\n\\subsubsection{Lack of carbon-rich stars in the Galactic Bulge}\n\nFinally we return to the fact that JEE15 also have higher luminosity ($> 10~000 L_\\odot$) OH\/IR stars \nin their Bulge sample. As stated in Section~\\ref{sec: sample} we have not selected such stars for our sample as\ntheir 'true' Bulge membership is uncertain.  \\cite{WilHarm1990} in their \nanalysis of the luminosty distribution of IRAS stars with OH\/IR colours assume\nthat all stars with IRAS $F_{12} > 10$~Jy are disk stars. They cannot exclude \nthat OH\/IR stars with luminosities above 10~000~$L_\\odot$ exist, but that these\nform at most 2\\% of the population. JEE15's analysis of the high-luminosity \ngroup, comparing the luminosities with predictions from stellar evolution\nmodels, shows that these stars have evolved from stars with $M_i \\approx 2.0 - 6.0 M_\\odot$. \nJEE15 indicate that the lack of carbon stars \nin the Bulge region (\\citep{Blanco1989}) or very low number as indicated in the\nrecent paper by \\cite{Matsunaga2017}, imposes problems with \nthe stellar evolution  models.  Stars with initial masses above 4~$M_\\odot$ can remain\noxygen-rich because of hot-bottom burning (HBB), but stars in the mass range between \n2 and 4~$M_\\odot$ are expected to convert to C stars because of the third \ndredge-up when carbon is brought from the nuclear burning region up to the \nphotosphere via convection (\\cite{MarigoGirardi2007,KarakasLatt2014}). If the \nmass range of AGB stars in the Bulge is indeed limited to less\nthan 2~$M_\\odot$, it would solve the problem of non-occurance of C stars in\nthe Bulge region. We repeat that \\cite{GroenBlom2005} do not find\nevidence of  stars with $M_i > 2 M_\\odot$ in fields with galactic latitudes\nabove 1.2 degrees. The comparison field at $l=b=-0.05^\\circ$ indicated the\npresence of a younger population with $M \\approx 2.5 - 3 M_\\odot$ and ages\nbelow 1 Gyr. This field is however much closer the Galactic Centre and in a \nregion called the Nuclear Bulge, which is believed to be still active in star\nformation \\citep{Launhardt2002}.  \n\n\n\\section{The duration of the superwind}\n\\label{sec: superwind}\n\nOur combined SED and CO modelling does not impose any limit on the duration of\nthe superwind (Section~\\ref{sec:COmodel}). The outer radius of the CO shell is \ntaken at 1.5 times the radius of the CO photodissociation \nthrough interstellar UV radiation field. This is in contrast to what is found\nfor other OH\/IR stars like OH~26.5$+$0.6 where the MLR derived from \nfitting the SED and solving the dynamical equation of the dust driven wind give \na high MLR which overestimated the observed low-J CO lines by an \norder of magnitude. A way to reconcile the derived dynamical MLR \nand CO observations is that the current MLR (measured by the warm \ndust) is higher than in the past (as seen in J=2-1 CO line). High-J CO lines \nobserved with {\\it Herschel} are consistent with a sudden increase in MLR in the past \ncouple of hundred years \\citep{Justtanont2013}. This result, based on CO \nobservations, is confirmed by an independent study of the fortserite dust \n69~$\\mu$m band of which the shape and peak wavelength are \nvery temperature sensitive \\citep{koike03, suto06}. \\cite{devries14} studied a sample of \nextreme OH\/IR stars, including OH~26.5$+$0.6 and confirms the short duration \nof the superwind of less than a thousand years. As is stated in \\cite{devries14}, \nsuch a short duration is problematic as the stars cannot lose sufficient mass, \nfor instance in the case of a star like OH~26.5$+$0.6 this would be a couple \nof solar masses. The superwind would need to be followed by a phase of even\nhigher MLRs \\citep{devries15}. An alternative scenario would be one\nwhere several phases of a few hundred years occur in which the MLR increases \nto values above 10$^{-5}$~\\msolyr. Such a timescale hints to a\nconnection to the thermal pulse, which is the only event on the AGB with such a\nduration. The so-called 'detached shells' around carbon stars are believed to be\nthe result of interaction of a high and faster moving wind (10$^{-5}$~\\msolyr) with a slower one with a  2 orders of magnitude lower MLR \n\\citep{Olofsson2000, Schoier2005}.\nStrangely, no oxygen-rich AGB stars are known with detached shells although it\nis expected that thermal pulses would increase the MLR in a similar way.  \n\nAn alternative explanation for the above described CO lines' behaviour \ncould be a higher impact of the interstellar UV radiation than is assumed \nin the radiative-transfer modelling (see Section~\\ref{sec:COmodel}). \nThe outer radius of the CO gas is determined by the photodissociation and is \nbased on the work by \\cite{Mamon1988}. \nIf the interstellar radiation field is underestimated in the modelling, the \nouter radii of the gas\nwill be smaller, increasingly so for lower J-transitions, as is observed in \nthe case of OH26.5$+$0.6. This interpretation may also explain why we do not \nneed to limit the CO outer radii for our Bulge OH\/IR stars. \nRecent work by \\citet{Groenewegen2017} demonstrate the effect of the ISRF and show that on average \na factor of 15 increase in the ISRF will lead to a three times smaller photodissociation radius.\nIt can be expected that at high latitudes in the Galactic Bulge, where our\nOH\/IR stars are situated, the UV radiation field is much weaker than\ncompared to active star forming regions where higher mass stars like \nOH~26.5$+$0.6 are situated. Clearly this alternative interpretation does \nnot explain the spatial distribution of the forsterite dust \\citep{devries14}. \n\n\n\\section{Conclusions}\n\nWe have presented the succesful detection of the CO (3-2) and (2-1) transition\nlines for a sample of OH\/IR stars in the Bulge. On basis of our modelling of the\nobserved SED and CO lines, we find that the stars have an average \nluminosity of 4729 $\\pm$ 521 L$_\\odot$ and the average MLR is $(5.4 \\pm 3.0)$ \n$10^{-5}$~\\msolyr. \nSuch MLR is well above the classical limit, with a single scattering event per\nphoton, for the luminosities in our sample. The \nvariability periods of our OH\/IR stars  are below 700 days and \ndo not follow the Mira-OH\/IR PL relation \\citep{Whitelock1991}.  This result \nshows that usage of the PL relation for the OH\/IR stars can lead to significant\nerrors in the luminosity determination. In comparison \nwith the VW93 evolutionary tracks, we find that the stars have initial masses \nof approximately 1.5~M$_\\odot$, which corresponds well with the findings of\n\\citet{GroenBlom2005} for the Bulge \nMira variables, confirming the connection between the two groups of stars. If\nmore massive OH\/IR stars are rare in the Bulge this may explain the scarcity of \nBulge carbon stars. \nWe find that the gas-to-dust ratio ranges between 100 and 400 and is similar to \nwhat is found for galactic disk OH\/IR stars.\nContrary to findings of bright OH\/IR stars in the Disk, our modelling does not \nimpose a limit to the duration of the superwind below a thousand years. \nIRAS~17347--2319 has a short period of about 300 days which may be further\ndecreasing. Rapid changes in the variability behaviour have been observed for\nMiras and may be connected to the occurance of a thermal pulse. It would be\nthe first time that such behaviour is observed in an OH\/IR star.\n\n\n\n\\section*{Acknowledgements}\nBased on observations with the Atacama\nPathfinder EXperiment (APEX) telescope (Programmes 088.F-9315(A) and \n090.F-9310(A)). APEX is a collaboration between the Max Planck Institute for \nRadio Astronomy, the European Southern Observatory, and the Onsala Space \nObservatory.\nThis research has made use of the VizieR catalogue access tool, CDS,\nStrasbourg, France. The original description of the VizieR service was\npublished in A\\&AS 143, 23. This publication makes use of data products from the \nWide-field Infrared Survey Explorer, which is a joint project of the University \nof California, Los Angeles, and the Jet Propulsion Laboratory\/California \nInstitute of Technology, funded by the National Aeronautics and Space \nAdministration. \nKJ acknowledges the support from the Swedish Nation Space Board. LD acknowledges\nsupport from the ERC consolidator grant 646758 AEROSOL. We thank the referee\nfor a careful review of our manuscript which improved the quality of this paper.\n\n\n\n\\bibliographystyle{mnras}\n   ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAs it has became increasingly difficult to explain cosmological\nobservations in the context of the standard cold dark matter (CDM)\nmodel, the cold+hot dark matter (CDM+HDM) models have emerged as one\nof the promising alternatives that require only moderate modifications\nof the CDM model.  The excess small-scale power relative to the\nlarge-scale power in the standard CDM model (Davis et al. 1985, 1992a;\nGelb, Gradw\\\"ohl, \\& Frieman 1993; Gelb \\& Bertschinger 1993) motivates\nthe addition of a hot component of massive neutrinos to the total mass\ndensity of the universe.  The free streaming of the neutrinos suppresses\nthe growth of small-scale perturbations while leaving the growth of\nlarge-scale perturbations unimpeded, and may therefore alleviate some\nof the problems of the standard CDM model.  Several recent linear\ncalculations (Schaefer, Shafi, \\& Stecker 1989; van Dalen \\& Schaefer\n1992; Taylor \\& Rowan-Robinson 1992; Holtzman \\& Primack 1993) and\n$N$-body simulations (Davis, Summers, \\& Schlegel 1992b; Klypin et al.\n1993) have found a better match of observations with the CDM+HDM\nmodels than with the CDM models, although a fair comparison between the\nmodels and the galactic scale data such as the epoch of galaxy formation\nand galaxy pairwise velocities awaits results from higher resolution\n$N$-body simulations in a large volume.  Most workers (including ourselves)\nhave assumed that the mass density fraction contributed by HDM is\n$\\Omega_{\\rm hdm} \\sim 0.3$ for an $\\Omega_{\\rm total}=1$ universe,\ncorresponding to a neutrino mass of $m_\\nu \\sim 7$ eV, although\nPogosyan \\& Starobinsky (1993) favor $0.17 \\le \\Omega_{\\rm hdm} \\le 0.28$\nfor $H_0=50$ km s$^{-1}$ Mpc$^{-1}$.\n\nThe introduction of HDM into the theory brings about one complication\ndue to the different behavior of the neutrinos and CDM.  In the linear\nregime CDM behaves as a pressureless perfect fluid, but the neutrinos\ncan be appropriately described only by their full phase space distribution\nobeying the Boltzmann equation.  None of the earlier studies of HDM\nmodels of which we are aware has taken account of the full phase space\ninformation of the neutrinos.\n\nIn the particle-particle\/particle-mesh (P$^3$M) simulation performed\nby Davis et al. (1992b), the HDM particles were placed initially on a\ngrid without perturbations at $1+z=20$, based on the argument that\nthe neutrino Jeans length at this redshift is comparable to their\nsimulation box size of 14 Mpc.  The initial conditions for the CDM\nparticles were generated with the Zel'dovich approximation from the\n{\\it pure} CDM spectrum and scaled to the normalization $\\sigma_8 \\sim\n0.45$.  To simulate the thermal motion of the particles, a velocity\ndrawn randomly from the Fermi-Dirac distribution was given to each\nHDM particle.  The cosmological parameters $\\Omega_{\\rm cdm}=0.7$,\n$\\Omega_{\\rm hdm}=0.3$, and $H_0=50$ km s$^{-1}$ Mpc$^{-1}$ were used,\nwith $32^3$ CDM and $32^3$ HDM particles.\n\nIn the particle-mesh (PM) simulations of Klypin et al. (1993), the\ninitial conditions for CDM and HDM were generated with the Zel'dovich\nmethod from individual CDM and HDM power spectra, which differ\nsignificantly on scales smaller than the neutrino free-streaming\ndistance since the growth of perturbations in HDM is suppressed.\nA thermal velocity drawn from the Fermi-Dirac distribution was\nadded to each HDM particle.  The simulations were performed in 14, 50,\nand 200 Mpc boxes with $128^3$ CDM and $6\\times 128^3$ HDM particles\nstarting at redshift $1+z =15$.  The parameters $\\Omega_{\\rm\ncdm}=0.6$, $\\Omega_{\\rm hdm}=0.3$, $\\Omega_{\\rm baryon}=0.1$, and\n$H_0 = 50$ km s$^{-1}$ Mpc$^{-1}$ were used.\n\nIn both groups' simulations, the initial neutrino momenta were\ndrawn from the Fermi-Dirac distribution independently of the neutrino\npositions.  In general, however, the neutrino phase space distribution\nis a complicated function of positions, momenta (or velocities), and\ntime, with the Fermi-Dirac distribution being only the zeroth-order\nterm.  Velocity-position correlations in the neutrinos can arise from\nperturbations to the Fermi-Dirac distribution.\nAlthough the actual HDM correlations are initially small\nin the linear regime, they can play an important role in the nonlinear\nstage of evolution, and in the linear theory should be treated as being\nof the same order as all other perturbations.\n\nIn this paper, we obtain the neutrino initial conditions from the full\nphase space distribution in the linear theory of gravitational\nperturbations.  Since the full phase space distribution depends on 6\ncanonical variables, it is numerically impractical to sample\nthe full neutrino phase space at the start of $N$-body simulations.\nWe resolve this difficulty by sampling the neutrino phase space with\nneutrino particles at a very early time of $z \\sim 10^9$ when the spatial\ndistribution of neutrinos is nearly uniform.  At this time the phase\nspace distribution is Fermi-Dirac to a very good approximation, and\nthe neutrinos can be placed on a grid with momenta drawn from the\nFermi-Dirac distribution.  Since the neutrinos have already decoupled\nfrom other species ($z_{\\nu, {\\rm dec}} \\sim 10^{10}$), their\ntrajectories simply follow geodesics in the perturbed Robertson-Walker\nspacetime.\n\nOur strategy will be to first calculate the metric perturbations by\nintegrating the coupled, linearized Einstein, Boltzmann, and fluid\nequations that govern the evolution of the metric and density\nperturbations of all particle species (CDM, photons, baryons, massless\nneutrinos, and massive neutrinos).  Then we integrate the linearized\ngeodesic equations for each neutrino from $z \\sim 10^9$ until\n$z = 13.55$, after which we will switch to a fully nonlinear Newtonian\nintegration.  The high-redshift approach can be described as a\n``general-relativistic cosmological $N$-body integration'' valid in\nthe linear theory.  It differs from the conventional $N$-body technique\nin that the gravitational forces are precomputed from the metric\nperturbations of the background spacetime rather than directly from\nthe particles.  The configuration of neutrinos at $z = 13.55$ will\nthen represent a fair sample of the full phase space distribution at\nthat redshift, and can be used directly as the HDM initial conditions\nfor subsequent $N$-body simulations.\n\nWe leave the discussion of the first stage of our calculation on the\nEinstein, Boltzmann, and fluid equations to a separate paper (Ma and\nBertschinger 1993).  In the present paper we focus on the geodesic\nintegration assuming the metric perturbations have been computed.  We\nfind the conformal Newtonian gauge a very convenient choice for this\npart of the calculation.  We derive the linearized geodesic equations\nin this gauge in Section 2 and discuss the integration method in\nSection 3.  We report the integration results in Section 4 where\nwe show the effect of the perturbations in the neutrino\nphase space on the correlation between the HDM momenta and the\ndensity contrast.  Section 5 includes a summary and a discussion of\nwork in progress.\n\n\\section{Geodesic Equations in Conformal Newtonian Gauge}\n\\label{section:geodesic}\nAlthough many calculations of the general-relativistic\nlinear perturbation theory have been carried out in the\nsynchronous gauge, we find it most convenient to compute the\ntrajectories of the neutrinos in the conformal Newtonian gauge\n(Mukhanov, Feldman, \\& Brandenberger 1992).\nThe conformal Newtonian gauge has the advantage that spurious coordinate\nsingularities do not arise, and the geodesic equations have simple forms\nwhich are easy to integrate.  The metric in the conformal Newtonian gauge\nis given by\n\\begin{equation}\n    ds^2 = a^2(\\tau)\\left[ -(1+2\\psi)d\\tau^2 +\n        (1-2\\phi)\\gamma_{ij}dx^i dx^j \\right] \\ ,\n\\label{conformal}\n\\end{equation}\nwhere the scalar potentials $\\psi$ and $\\phi$ characterize\nthe perturbations about a flat Robertson-Walker spacetime.\nWe use Cartesian coordinates so that the 3-metric of $\\tau=\\hbox{constant}$\nhypersurfaces is $\\gamma_{ij}=\\delta_{ij}$.\nIt should be emphasized that $\\phi$ and $\\psi$ describe only the\nscalar mode of the metric perturbations.  We do not consider the\nvector and the tensor modes in this paper.\n\nThe geodesic equations for a neutrino of mass $m_\\nu$ can be derived by\nminimizing the action\n\\begin{equation}\n        S = \\int d\\tau L = -m_\\nu \\int (-ds^2)^{1\/2}\\ ,\n\\end{equation}\nwhere $L$ is the Lagrangian and the metric in the conformal Newtonian\ngauge is given by Eq.~(\\ref{conformal}).\nTo linear order in the potentials, the Lagrangian is\n\\begin{equation}\n        L = -m_\\nu a \\sqrt{1-u^2} \\left( 1 +\n        {\\psi+u^2\\phi \\over 1-u^2} \\right) \\ ,\n\\end{equation}\nwhere $\\vec{u} = d\\vec{x}\/d\\tau$ is the coordinate velocity and\n$u^2=\\gamma_{ij}u^iu^j$.  The conjugate momentum $q_i$ is given by\n\\begin{equation}\n      q_i \\equiv {\\partial L \\over \\partial u^i}\n        = {m_\\nu a\\gamma_{ij}u^j \\over \\sqrt{1-u^2}} \\left(\n        1-2\\phi-{\\psi+u^2\\phi \\over 1-u^2}\n        \\right) \\ ,\n\\label{q}\n\\end{equation}\nwhich can be inverted to give, to first order in $\\psi$ and $\\phi$,\n\\begin{equation}\n   u^i = {dx^i \\over d\\tau} = {\\gamma^{ij}q_j \\over \\epsilon(q,\\tau)}\n    \\left\\{ 1 + \\psi(\\vec{x},\\tau) + \\left[2-{q^2 \\over\n        \\epsilon^2(q,\\tau)}\\right] \\phi(\\vec{x},\\tau) \\right\\}\n\\label{geox}\n\\end{equation}\nwith $\\epsilon(q,\\tau)=\\sqrt{q^2+m^2_\\nu a^2}$.\nThe Euler-Lagrange equation of motion gives\n\\begin{equation}\n        { dq_i \\over d\\tau} = -{m_\\nu a \\over \\sqrt{1-u^2}}\n        \\left( \\partial_i{\\psi} + u^2 \\partial_i{\\phi}\n        \\right) \\ .\n\\end{equation}\nReplacing $\\vec{u}$ on the right-hand side with Eq.~(\\ref{geox}),\nwe obtain\n\\begin{equation}\n   {dq_i \\over d\\tau} = -\\epsilon(q,\\tau) \\left[\n        \\partial_i \\psi(\\vec{x},\\tau) + {q^2 \\over \\epsilon^2(q,\\tau)}\n        \\partial_i \\phi(\\vec{x},\\tau) \\right]\\,.\n\\label{geoq}\n\\end{equation}\nEqs.~(\\ref{geox}) and (\\ref{geoq}) give the linearized geodesic\nequations for a particle moving in the perturbed spacetime characterized\nby scalar metric perturbations $\\psi$ and $\\phi$.  In the weak-field,\nnonrelativistic ($q^2\\ll\\epsilon^2$) limit they reduce to the standard\nNewtonian equations.\n\n\\section{Integration of Geodesic Equations}\nTo sample the neutrino phase space as accurately as possible, we\nintegrate the geodesic equations (\\ref{geox}) and (\\ref{geoq}) for\n$10\\times 128^3$ ($\\sim$ 21 million) massive neutrino particles.  A\ncubic simulation box with sides 100 Mpc is used.  The cosmological\nparameters are taken to be $\\Omega_{\\rm cdm}=0.65$, $\\Omega_{\\rm\nhdm}=0.3$, $\\Omega_{\\rm baryon}=0.05$, and $H_0 = 50$ km s$^{-1}$\nMpc$^{-1}$.  We start the integration shortly after neutrino\ndecoupling at redshift $z\\sim 10^9$ when perturbations in the neutrino\ndensity and momenta can be safely ignored.  The neutrinos are\nplaced initially on a $128^3$ grid, 10 per grid point, with the\nneutrino momenta drawn randomly from the Fermi-Dirac distribution\n\\begin{equation}\n        f(\\vec q\\,)\\,d^3q \\propto {d^3q\\over e^{qc \/ kT_{\\nu,0}} + 1} \\ ,\n\\label{fd}\n\\end{equation}\nwhere $T_{\\nu\\,,0}=(4\/11)^{1\/3}\\,T_{\\gamma\\,,0}$ is the neutrino\ntemperature today with $T_{\\gamma\\,,0}=2.735$ K.  We performed test\nruns with the same set of momenta but randomly generated initial\npositions and found no statistically significant difference at the\nend of the integration depending on whether the neutrinos were\ninitially placed at random or on a grid.\n\nWe also tested the momentum pairing scheme used by Klypin et al.\n(1993) in their initial conditions.  For every momentum drawn from the\nFermi-Dirac distribution, they assigned an equal but opposite momentum\nto a second neutrino at the same grid point to preserve the local\ncenter of momentum.  We performed two test runs, with the initial neutrino\nmomenta drawn randomly in one run and paired up in opposite directions\nwith the same magnitude in the other run.  We found no statistically\nsignificant difference in the power spectrum at the end of the\ngeodesic integration.  We thus adopted the simpler scheme without\npairing.\n\nWe integrated the geodesic equations from conformal time\n$\\tau_i=3\\times 10^{-4}$ Mpc ($z \\sim 10^9$) to $\\tau_f=3\\times 10^3$\nMpc ($z = 13.55$), using 701 time steps with stepsize\n$\\Delta(\\log_{10}\\tau) = 0.01$.  The initial $\\tau_i$ is chosen so\nthat the largest $k$ in the simulation box is well outside the horizon\n($k\\tau\\ll1$) at the onset of the integration.  The integration was\nstopped when the fluctuations were still in the linear regime.  We\nused a leap-frog integration scheme in which the positions and momenta\nwere advanced half a timestep out of phase to give a second-order accuracy\nin timestep size.\n\nThe evolution of the metric perturbations $\\psi$ and $\\phi$ in\nEqs.~(\\ref{geox}) and (\\ref{geoq}) were precomputed from the coupled,\nlinearized Einstein, Boltzmann, and fluid equations for all particle\nspecies including massive neutrinos (Ma and Bertschinger 1993).  The\nresulting transfer functions were saved on a grid of 41 $k$- and 701\n$\\tau$- values.  For the geodesic integration the initial $\\psi$ and\n$\\phi$ were generated as Gaussian random variables in $k$-space with\nthe scale-invariant power spectrum $P_\\psi\\propto k^{-3}$ predicted\nby the simplest inflationary cosmology models.  For later times, the\nFourier components of $\\psi$ and $\\phi$ simply scale according to our\nlinear theory computation.  We normalized the amplitude to the COBE rms\nquadrupole fluctuation $Q_{\\rm rms-PS} = 14\\times 10^{-6}$ K (Seljak \\&\nBertschinger 1993; Wright et al. 1992; Smoot et al. 1992) assuming the\nSachs-Wolfe formula for a scale-invariant spectrum and $T_{0\\,,\\gamma}\n=2.735$ K:\n\\begin{equation}\n\t{Q_{\\rm rms-PS}^2 \\over T_{0\\,,\\gamma}^2}\n\t= {5\\over 108}\\,\\left[4\\pi k^3P_\\psi(k,\\tau_{\\rm rec})\n\t\\right]_{k\\to0}\n\t\\ .\n\\label{SW}\n\\end{equation}\nThe gradients of $\\psi$ and $\\phi$ in Eqs.~(\\ref{geox}) and (\\ref{geoq})\nwere first computed on a grid in $k$-space and then Fourier transformed\nto a grid in real space.  The second-order Triangular-Shaped Cloud (TSC)\ninterpolation scheme was then used to interpolate the gradients from\nthe grid to the particle positions (see Ma 1993 for more details).\n\nTest runs were performed with $N_{\\rm grid}=32^3$ using $ N_{\\rm\npart}=10\\times 32^3$, $40\\times 32^3$, and $80\\times 32^3$, and $N_{\\rm\ngrid}=64^3$ using $N_{\\rm part}=10\\times 64^3$ and $40\\times 64^3$.\n(The first factor in $N_{\\rm part}$ gives the number of samples of the\nmomentum space at each initial position.)  We tested the accuracy of\nthe time integration using 351 and 701 timesteps respectively and\nfound little difference in the final positions and velocities,\nindicating that 701 timesteps are sufficient.  We also generated\nrealizations of the potentials and the initial neutrino momenta using\nthree different random number generators.  No correlations in the\nrandom numbers were detected.  Our large production run had $N_{\\rm\ngrid}=128^3$ and $N_{\\rm part}=10\\times 128^3$ ($\\sim$ 21 million).\nThe geodesic integration required a total of $\\sim 1.5$ Gbytes of memory\nand $\\sim 140$ CPU hours on the Convex C3880 supercomputer at the\nNational Center for Supercomputing Applications.\n\n\\section{Numerical Results}\n\nAn image of an intermediate output (timestep 351) from one of the\n$N_{\\rm part}=10\\times 32^3$ test runs is shown in Fig.~\\ref{hc.32.3}.\nThe corresponding redshift is $z \\sim 4.9 \\times 10^5$.  Each side in\nthe figure is 100 Mpc comoving, and the particles in the simulation\nbox have been projected onto the $x-y$ plane.  At the starting $z \\sim\n10^9$, the particles were placed on a $32^3$ grid, 10 per grid point,\nand were given momenta drawn randomly from the Fermi-Dirac distribution.\nIn this figure, one sees that the neutrinos have begun to spread out\nfrom the grid points.  In fact, the size of each ``ball'' is\napproximately the comoving horizon distance $c\\tau \\sim 0.95$ Mpc at\nthis moment since the neutrinos are still relativistic.\n\nFig.~\\ref{hc.128.7} shows the same projection in a 100 Mpc box of the\nlast output (timestep 701) from the $N_{\\rm part} = 10\\times 128^3$\nrun.  The corresponding redshift is $z = 13.55$.  As one can see,\nsmall perturbations are developing in the otherwise uniform\ndistribution of the neutrinos.  We present quantitative analyses of\nthis output below.\n\nTo check the integration results, we computed the HDM power spectrum\nfrom the final output of the $N_{\\rm part} = 10\\times 128^3$ run and\nmade comparison with the prediction from the linear theory.  The\ndensity field $\\delta$ was first computed on a $128^3$ spatial grid\nfrom the positions of the neutrinos by the TSC interpolation scheme,\nand then Fourier transformed into $k$-space.  We calculated the power\nper $\\ln k$ at a given $k$, $4\\pi k^3 P(k)$, by taking a spherical\nshell of radius $k$ and thickness $\\Delta k$ centered at the origin of\nthe $k$-grid and averaging the contribution to $4\\pi k^3 P(k)$ from\nthe grid points that lie within the shell.  The shot noise due to the\nfinite number of particles was subtracted and the TSC window was\ndeconvolved.  The result is shown in Fig.~\\ref{fig:p128} and is\ncompared with the linear theory predictions for the CDM and HDM power\nspectra.  The agreement provides an important check of the accuracy of\nour geodesic integration code.  From smaller simulations we conclude\nthat the deviations from the ensemble-average power HDM spectrum are\ndue to sampling fluctuations.\n\nThe output shown in Fig.~\\ref{hc.128.7} is used as the initial\nconditions for the HDM particles in our $N$-body simulations of\nstructure formation in this CDM+HDM model.  To compare our initial\nconditions to those of others, we recall that the initial positions\nand velocities of the particles in $N$-body simulations are\nconventionally generated from the power spectrum using the\nZel'dovich (1970) approximation.  In this procedure, the positions\nof the particles are displaced from a regular grid:\n\\begin{equation}\n        \\vec{x}(\\tau) = \\vec{x}_0 + \\vec{\\epsilon}\\,(\\vec{x}_0,\\tau)\\ ,\n\\end{equation}\nwhere $\\vec{x}_0$ gives the position of the grid and\n$\\vec{\\epsilon}\\,(\\vec{x}_0,\\tau)$ is the displacement field.  The\ndisplacements are computed from the density perturbation field by\nsolving $\\vec{\\nabla}\\cdot\\vec{\\epsilon} = - \\delta \\,$.  For small\ndisplacements, $\\vec{\\epsilon}\\,(\\vec{x}_0,\\tau)$ is approximated by\n$D_+(\\tau)\\,\\vec{\\epsilon}\\,(\\vec{x}_0)$, where $D_+(\\tau)$ denotes\nthe growth factor of the perturbations, and the velocities of particles\nare given by\n\\begin{equation}\n \\vec{v} \\equiv {d\\vec{x}\\over d\\ln a} = {d\\ln D_+\\over d\\ln a}\n        \\,\\vec{\\epsilon}\\,(\\vec{x}_0,\\tau)\\,.\n\\end{equation}\n\nFor the standard CDM model with $\\Omega =1$, the growth factor in the\nmatter-dominated era is equal to the expansion factor, and $f(\\Omega)\n\\equiv d\\ln D_+\/d\\ln a =1$. The growth rate, however, does not behave\nso simply in models such as the CDM+HDM models where more than one\nparticle species contributes to $\\Omega$.  This is illustrated by the\npower spectra shown in Fig.~\\ref{fig:powspec} for the standard CDM model\nand our CDM+HDM model.  The growth rate of CDM in the CDM+HDM model\nmatches the growth rate in the standard CDM model only at small $k$.\nAt large $k$ where neutrino free-streaming is important, we have\n$\\delta_{\\rm hdm} \\ll \\delta_{\\rm cdm}$, and $f(\\Omega,k) = d\\ln D_+\n\/d\\ln a < 1$ and is $k$-dependent.  We calculated $f(\\Omega,k)$ for CDM\nat $z=13.55$ in our CDM+HDM model from the output of the linear theory\nintegration; the result is shown in Fig.~\\ref{fomega}.  In the limit\n$\\delta_{\\rm cdm} = \\delta_{\\rm baryon}$ and $\\delta_{\\rm hdm}=0$, the\ngrowth rate can be computed analytically to be $f = (\\sqrt{1+24\n\\Omega_{\\rm c}}-1)\/4$ where $\\Omega_{\\rm c}=\\Omega_{\\rm cdm+baryon}$\n(Bond, Efstathiou, \\& Silk 1980).  For our parameters, $f=0.805$.\nAs one can see in Fig.~\\ref{fomega}, $f$ is indeed approaching this\nvalue at high $k$.\n\nIf one did not take into consideration the $k$-dependence of the\ngrowth rate and instead used $f(\\Omega,k)=1$ to obtain the CDM initial\nvelocities in CDM+HDM models, one would give the CDM particles\nexcessive initial velocities on small scales, leading to earlier\ngravitational collapses in the simulations.  We estimated this effect\nin the linear theory on scales below the free-streaming distance.  We\nfind that using $f=1$ instead of $f=0.805$ gives an initial amplitude in\nthe linear growing mode that is too large by a factor $1.093$.  Since\ngalaxies form later in CDM+HDM models than in the standard CDM model\ndue to neutrino free-streaming, the epoch of structure formation is\none of the crucial factors that will determine the fate of the model.\nBy overestimating $f(\\Omega,k)$, one underestimates the severity of\nthe problem with late galaxy formation in CDM+HDM models.\nKlypin et al. (1993) set $f=1$ for the CDM.  However, this error was\ncancelled by an opposite effect (Primack, private communication): they\nused the baryon transfer function for the CDM.  The baryonic perturbations\nare smaller than the CDM by up to 15\\%.  As a result, the CDM particles\nwere given less power, which counteracted their excessive velocities\nso that the two errors essentially cancelled.\n\nIn the simulations by Davis et al. (1992b), the initial HDM momenta\nwere drawn randomly from the Fermi-Dirac distribution.  In Klypin et\nal. (1993), this thermal velocity was added to the velocity arising\nfrom the Zel'dovich approximation for each HDM particle.  At their\nstarting $z \\sim 15$, the thermal component was about a factor of 4\nlarger.  Neither group included the actual correlations between\nneutrino positions and momenta that develop through the Boltzmann\nequation in the linear regime.  To incorporate these first-order\neffects, we retained the full phase space information by sampling the\nphase space at $z\\sim 10^9$ with 21 million neutrino particles and\nfollowing their trajectories in the perturbed background spacetime\nuntil low redshifts.\n\nTo estimate the importance of these neutrino phase space perturbations,\nwe mimicked the approach of Klypin et al. to generate an ``equivalent''\nset of initial positions and momenta at $z = 13.55$ for $10\\times\n128^3$ neutrinos, using the same realization for $\\delta$ as for\n$\\phi$ and $\\psi$ in our geodesic integrations.  A randomly-drawn\nthermal velocity was also added to the Zel'dovich velocity for each\nneutrino particle.\n\nWe calculated $\\delta_{\\rm hdm}$ from the particle positions and\nexamined the correlation between the rms neutrino velocities and\n$\\delta_{\\rm hdm}$ in the two cases.  If the neutrinos obeyed the\nzeroth-order Fermi-Dirac distribution, the neutrino velocities should\nbe uncorrelated with $\\delta_{\\rm hdm}\\,$; any correlation would\nindicate deviations from the Fermi-Dirac distribution.  Our results\nare plotted in Fig.~\\ref{vdel}.  We see that correlations in the\nvelocities and the density perturbations have developed in the\nBoltzmann integration case between $z\\sim 10^9$ and $z = 13.55$.\nThe more clustered neutrinos appear to have higher rms velocities\nand therefore higher temperature, possibly resulting from the increase\nin the kinetic energy when the neutrinos fall into the CDM potential\nwells.  This correlation is absent when the initial conditions are\ngenerated with the conventional Zel'dovich approach because the dominant\nthermal contribution to the neutrino velocities is drawn from the\nzeroth-order Fermi-Dirac distribution with constant temperature.\n\nTo test whether gravitational infall is responsible for the density-velocity\ncorrelation, we also computed for each neutrino particle the velocity\ncomponents parallel and perpendicular to the gravitational acceleration\n$\\vec{g} = -\\vec{\\nabla}\\phi$ at the location of the neutrino:\n$v_\\parallel = \\vec{v}\\cdot\\hat{g}$ and $v_\\perp = \\sqrt{v^2 -\nv_\\parallel^2}$.  Then we calculated the conditional rms $v_\\parallel$\nand $v_\\perp$ for a given $\\delta_{\\rm hdm}$.  The results are shown in\nFig.~\\ref{vpara}, where the solid curve represents $\\langle\nv_\\parallel^2 \\rangle^{1\/2}$ and the dashed curve represents\n$\\langle v_\\perp^2 \\rangle^{1\/2}\/\\sqrt{2}$.  The component along the\ngravitational acceleration is larger than the orthogonal components by\n$\\sim 2\\%$.  Thus, the velocity-density correlation is not simply due\nto a uniform gravitational infall.  This is because the velocity\ndispersion (temperature) of the neutrinos and not just the bulk (fluid)\nvelocity is higher in the denser regions.\n\nFor comparison, Fig.~\\ref{vcdm} shows the rms velocity versus the\ndensity for our CDM particles at redshift $z = 13.55$.  The\npositions and the velocities were generated using the method described\nearlier.  As one can see from Figs.~\\ref{vdel} and \\ref{vcdm}, CDM is\nmore clustered than HDM (the range of $\\delta$ is larger), and the rms\nvelocities of the CDM particles are $\\sim 30$ km s$^{-1}$ compared to\n$\\sim 95 - 105$ km s$^{-1}$ for the HDM particles.  The CDM velocities\nshow no significant correlation with $\\delta_{\\rm cdm}$.  No correlation\nis expected in linear theory.\n\nFig.~\\ref{contour} is a contour plot of neutrino density in the velocity\ncomponent-$\\delta_{\\rm hdm}$ plane from the last output of the $N_{\\rm\npart} = 10\\times 128^3$ geodesic integrations.  The rapid decline with\n$v$ is due to the (approximately) Fermi-Dirac distribution (with rms\n$v\\sim55$ km s$^{-1}$) and the decline with $\\vert\\delta\\vert$ is due to\nthe Gaussian distribution of the potential.  However, the contours are\nasymmetric about $\\delta_{\\rm hdm} = 0$, showing a positive correlation\nin the velocities and the density perturbations in the HDM component.\nOne sees that the overdense regions contain hotter neutrinos than the\nunderdense regions.\n\n\\section{Conclusion and Work in Progress}\nMotivated by the CDM+HDM models, we have presented a\ngeneral-relativistic $N$-body technique that provides an accurate\nsampling of the full neutrino phase space at all times when the linear\nperturbation theory is valid.  Although the evolution of the neutrino\nphase space distribution can be solved from the Boltzmann equation, we\nknow of no practical scheme for computing and sampling the final\ndistribution except for the Monte Carlo method we have employed.\nIn this method we first compute the metric perturbations about a\nRobertson-Walker spacetime by integrating the coupled, linearized\nBoltzmann, Einstein, and fluid equations for all particle species,\nincluding the massive neutrinos.  Then we sample the massive neutrino\nphase space right after neutrino decoupling at $z\\sim 10^9$ when the\ndistribution is Fermi-Dirac to a very good approximation.  We subsequently\nintegrate the linearized geodesic equations for individual neutrinos to\nobtain their trajectories in the perturbed background spacetime described\nby the metric perturbations found in the previous calculation.  This\ntechnique is valid only in the linear regime.  It differs from the\nconventional $N$-body simulation method in that the gravitational forces\nare precomputed from the metric perturbations of the background spacetime\nusing continuum linear theory rather than from the particles directly.\nThe resulting neutrino positions and velocities can be used as the HDM\ninitial conditions for subsequent $N$-body simulations of the nonlinear\nevolution of structures in the CDM+HDM models.\n\nThe same method could be used to generate initial conditions for the pure\nHDM model.  Although these would differ from what previous workers have\nassumed, because of the very large damping of small-scale fluctuations\nwe are doubtful that there would be significant differences in one's\nconclusions about the model.\n\nWe are currently performing a high resolution particle-particle\nparticle-mesh (P$^3$M) $N$-body simulation of the nonlinear\nevolution of the density perturbations, using the positions and\nvelocities from the last output of our large geodesic integration as\nthe HDM initial conditions.  The initial conditions for the CDM\nparticles were generated from the CDM power spectrum with a modified\nform of the Zel'dovich approximation taking into account the\nwavenumber-dependence of the growth rate $f=d\\ln D_+\/d\\ln a$.\nA total of $10\\times 128^3$ HDM and $128^3$ CDM particles are used\nin a 100 Mpc comoving box starting at $z = 13.55$.  If computer time\npermits, we will also perform an ``equivalent'' simulation with the\nsame parameters but with the Zel'dovich initial conditions adopted\nby other groups.  Our simulation box will be large enough to include\nmost of the important long-wavelength power absent in smaller boxes,\nand the P$^3$M force calculation will give us much higher resolution\nthan particle-mesh (PM) simulations.  In addition, we will be able\nto make a fair comparison of the two different treatments of the\ninitial conditions and examine the importance of correlations in the\nneutrino phase space.  Until that time it would be premature for us\njudge the merits of the approximate methods used by previous workers.\n\n\\acknowledgments\n\nThis work was supported by NSF grant AST90-01762 and DOE grant\nDE-AC02-76ER03069.  Supercomputer time\nwas provided by the National Center for Supercomputing Applications.\nWe appreciate the advice and comments of Alan Guth and Jim Frederic.\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}