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+{"text":"\\section{Introduction}\n The archetypical ultracompact \\hbox{H{\\sc ii}}\\ region (UC \\hbox{H{\\sc ii}}) W3(OH) has been\n the topic of numerous studies targeted at understanding the phenomena\n involved in the formation of massive stars and the interaction of\n these objects with their environment.  In particular, interferometric\n observations at radio and millimeter wavelengths have revealed an\n ever more detailed picture of the dense gas associated with W3(OH)\n (e.g., Baudry et al.\\ 1993; Bloemhof et al.\\ 1992; Reid et al.\\ 1995;\n Turner \\& Welch 1984; Wilner et al.\\ 1995; Wink et al.\\ 1994;\n Wyrowski et al.\\ 1997).\n \n The observations suggest that there are several sites of very recent\n and on-going star formation: the most prominent of these are the UC\n \\hbox{H{\\sc ii}}\\ itself, which is ionized by a young O star, and a region\n showing hot molecular line and dust emission that is associated with\n strong water maser emission. The latter region is located at a\n projected distance of $\\approx 0.06$~pc to the east of the UC \\hbox{H{\\sc ii}},\n assuming a distance of 2.2 kpc for W3(OH).  Most of the dense neutral\n gas in the W3(OH) complex appears to be associated with the \\WAT\\ \n maser\/``hot core'' source, whereas the most luminous young star in\n the region seems to be the exciting star of the UC \\hbox{H{\\sc ii}}.\n \n There are several lines of evidence suggesting that there are one or\n more young stars embedded in the dense molecular gas near the \\WAT\\ \n maser source: Turner \\&\\ Welch (1984) found a compact source of\n emission in the HCN $J = 1-0$ transition toward the \\WAT\\ maser\n position (in the following referred to as W3(OH)-TW).  The HCN line\n showed broad wings suggestive of mass loss from a young embedded\n star.  Alcolea et al.\\ (1992) measured the \\WAT\\ maser proper motions\n and found them to be consistent with a bipolar outflow along an E-W\n axis in the plane of the sky.  Mauersberger et al.\\ (1986a,b) used\n their ammonia observations to demonstrate the presence of dense, hot\n ($> 160$~K) gas close to the water masers and this was confirmed by\n the studies of Wink et al.\\ (1994) and Helmich et al.\\ (1994).\n However, the most surprising result was perhaps that of Reid et al.\\ \n (1995), who studied weak elongated centimeter continuum emission\n toward the TW source. They concluded that there was strong evidence\n for this being synchrotron emission from a ``jet-like'' structure\n aligned with the outflow indicated by the proper motions of the \\WAT\\ \n masers associated with W3OH-TW.  Moreover, the synchrotron emission\n centroid was found to be coincident with the center of expansion of\n the \\WAT\\ outflow as determined by Alcolea et al.\\ (1992).  All of\n these facts suggest the presence of a relatively luminous embedded\n star which drives a collimated outflow.\n \n In this paper, we present sensitive, sub-arcsecond resolution\n interferometric observations of the W3(OH) complex at a frequency of\n 220 GHz; results from simultaneous observations at 107 GHz will be\n presented elsewhere. We find an excellent coincidence between the\n structures seen at millimeter wavelengths in thermal dust emission\n and those seen in non-thermal emission (see Wilner et al.\\ 1998) at\n centimeter wavelengths.\n\n \\section{Observations}\n \n Our observations were made with the 5 element Plateau de Bure\n Interferometer in three configurations: B1-N13 on 1998 January 6\/7,\n A1 on February 7, and B2 on February 17. The phase center was\n $\\alpha(J2000)$ = $02^{\\rm h}27^{\\rm m}04{\\rlap.}{^{\\rm s}}284$,\n $\\delta(J2000)$ = +$61^{\\circ}52'24{\\rlap.}{''}55$, which is between\n the positions of W3(OH) and W3(OH)-TW. The total observing time on\n source was 17.1 hours covering a baseline range from 30 to 410~m. We\n used the dual-frequency receiver systems to simultaneously observe\n the \\METH\\ $J_k = 3_1-4_0A^+$ and \\CEIO\\ ($J = 2-1$) lines.  Due to\n good winter weather conditions, the 220 GHz system temperature was in\n the range 200 to 400~K and the radio seeing on all days better than\n 0\\farcs5. On-source integrations of 20~min were interspersed with\n phase calibrator observations on the quasar 0224+671.  For bandpass\n and flux density calibration, the sources 3C454.3, 3C273 and 3C84\n were used and the absolute flux density scale was established by\n observing MWC~349, for which a flux density of 1.66~Jy was assumed at\n 220 GHz.  From the day-to-day variance in W3(OH)'s continuum flux\n density, we estimate that our 220 GHz flux density scale is accurate\n to within 20\\%.\n \n The data were processed using the GILDAS software package.  To remove\n the effects of short-term atmospheric fluctuations on the\n interferometer phases, phase corrections derived from 220 GHz total\n power measurements were applied to the 107 and 220 GHz data (see\n Bremer et al.\\ 1996).  Then, correcting for phase drifts on long\n timescales was done in the standard manner by observing a calibrator\n source. The phase solutions for the 107~GHz data were subtracted,\n appropriately scaled, from the 220~GHz data.  The RMS noise of the\n fits to the residual 220 GHz phase was found to be only 10\\arcdeg\\ on\n average.  After gridding and Fast Fourier Transform of the $uv-$data,\n a 220 GHz beam size of 0\\farcs83$\\times$0\\farcs55 (FWHM) was\n determined for natural weighting with a position angle of\n 100$\\arcdeg$. From all correlator units, spectral line data cubes\n were built and checked for frequency ranges free of line emission in\n order to produce a continuum map. All data were deconvolved using the\n CLEAN algorithm and the resulting dynamical range limited RMS noise\n in the 220 GHz continuum maps is 20~mJy.\n\n \\section{Results}\n\n  \\subsection{220 GHz continuum measurements}\n  \n  In Fig.~\\ref{cmap}, we present our 220 GHz continuum map of W3(OH)\n  superimposed upon the 8~GHz VLA map of Wilner et al.\\ (1998).  For a\n  better comparison with the latter, our 220 GHz map was restored with\n  a circular beam of 0\\farcs5 diameter (FWHM) and is thus slightly\n  superresolved. On both maps, the main features are the UC \\hbox{H{\\sc ii}}\\ to\n  the west and the water maser source 6\\arcsec \\ to the east.  We\n  checked the alignment of the maps by computing their correlation and\n  found them to coincide within 0\\farcs05. Toward the \\hbox{H{\\sc ii}}\\ region,\n  the main emission mechanism even at 220 GHz is free-free emission\n  from the ionized gas. In fact, in agreement with the measurements of\n  Wyrowski et al.\\ (1997), we observe an integrated flux density of\n  $3.1 \\pm 0.5$~Jy at 220~GHz toward the \\hbox{H{\\sc ii}}\\ region as compared to a\n  value of $3.0 \\pm 0.15$~Jy at 107~GHz.  Combining the 3~mm\n  measurements with results from the literature allows us to\n  extrapolate the flux density to 220~GHz assuming optically thin\n  free--free emission and we find $3.0 \\pm 0.3$~Jy.  This value is\n  consistent with the actually measured flux density at 220~GHz and we\n  place a conservative upper limit of 0.5 Jy on any contribution from\n  emission from dust associated with the UC \\hbox{H{\\sc ii}}\\ region.\n  \n  Most strikingly, Fig.~\\ref{cmap} reveals the small-scale spatial\n  coincidence of dust emission features (shown in grey-scale) with\n  radio continuum sources detected toward the TW-object in the 3.6~cm\n  map of Wilner et al.\\ (1998, shown as contours). As seen on the\n  inset to Fig.~\\ref{cmap}, the easternmost component A is coincident\n  with the centroid of the elongated structure seen with the VLA.\n  There is also a rough correspondence between components B and C of\n  the 220 GHz map with features seen at 3.6~cm.  In particular, it\n  seems clear that dust emission source A is associated with the\n  non-thermal radio source detected at centimeter wavelengths.\n  \n  We note here that there is also evidence in Fig.~\\ref{cmap} for\n  extended emission surrounding components A, B, and C.  Our\n  observations are not sensitive to structures extended over size\n  scales larger than about 5\\arcsec\\ and hence it is likely that we\n  are missing flux.  We consider therefore that the total integrated\n  flux density measured toward the water maser position (1.6~Jy from\n  Fig.~\\ref{cmap}) is a lower limit.  Estimates of the integrated\n  fluxes in components A, B, and C are rendered difficult due to,\n  both, the presence of the extended emission and blending obvious on\n  Fig.~\\ref{cmap}. Consequently, we only quote peak flux densities of\n  components A, B, and C measured in our 0\\farcs5 beam in\n  Table~\\ref{peaks}.\n\n  \\subsection{Line measurements near 220 GHz}\n  \n  Compared to previous, similar, observations, our data have much\n  improved sensitivity and angular resolution and thus allow a new\n  discussion of the molecular line emission distribution in the W3(OH)\n  region.  The observations discussed in Wyrowski (1997) clearly\n  showed that there is a dichotomy between oxygen-containing molecules\n  related to methanol (essentially methyl formate and dimethyl ether)\n  and nitrogen-containing species such as ethyl and vinyl cyanide.\n  The former are seen both toward the UC \\hbox{H{\\sc ii}}\\ and the water masers\n  whereas the latter are {\\it only} seen toward the water masers.\n  This is shown with improved angular resolution in\n  Fig.~\\ref{line-overlays} where we compare naturally weighted\n  integrated intensity maps of lines detected near 220 GHz from a\n  variety of species.  We note in particular the fact that the\n  $24_{2,22}-23_{2,21}$ transition of \\ETHCN \\ and the $v_7=2$,\n  $J=24-23$, transition of \\CYAC\\ (respectively 135~K and 772~K above\n  ground) peak precisely in the direction of continuum component C and\n  are not detected toward the \\hbox{H{\\sc ii}}\\ region. In contrast, the\n  transitions which we detect from \\METH, \\MEFORM, and \\FORM \\ are\n  seen both toward the ionized gas and toward the water maser\n  concentration. The similarity of the intensity distributions in all\n  three cases suggests a chemical link between these molecules as\n  indeed discussed by Blake et al.\\ (1987) in the context of Orion.\n  Finally, we note the curious behavior of the $22_{2,20}-22_{1,21}$\n  transition of \\SOTW \\ which, in contrast to e.g. \\METH, is seen\n  toward the eastern rather than the western border of the \\hbox{H{\\sc ii}}\\ \n  region.\n  \n  We also detected the $K_{a}=0,2,3,4$ components of the $J=10-9$ HNCO\n  transitions at excitations of 50--750~K above ground.  The relative\n  populations of these levels is thought to be determined by a\n  combination of collisions and radiative transitions induced by the\n  FIR radiation field within the hot cores (Churchwell et al.\\ 1986).\n  We have attempted to use the relative intensities of these lines as\n  a ``thermometer'' measuring the temperature of the hot core by\n  assuming, first, that all transitions are optically thin (which is\n  justified by the fact that the $K_{a}=0$ line has a brightness\n  temperature of less than 20~K) and, second, that the level\n  populations are maintained by the radiation temperature of the dust.\n  Since the dust is optically thick at the HNCO FIR pump wavelengths\n  of 50--330~$\\mu\\hbox{m}$, the radiation temperature equals the dust\n  temperature.  We thus estimate rotational temperatures as a function\n  of position over the water maser source.  In practise, the derived\n  level column densities are consistent with a single rotational\n  temperature at each point and allow us (assuming LTE) to infer the\n  dust (and gas) temperature.  The resulting temperature map is shown\n  in Fig.~\\ref{hnco-results}, where one sees that there are two 200~K\n  peaks roughly coincident with our continuum peaks A and C.  Between\n  these there is a ``plateau'' where the temperature is of order\n  150~K. We interpret this as evidence that there are two energy\n  sources (young proto-B stars) embedded in the molecular gas whose\n  positions are roughly defined by the temperature peaks and which are\n  responsible for the cm-emission.\n  \n  Moreover, we can use the temperature distribution thus derived to\n  deduce the dust and, thus, the hydrogen column density distribution\n  from our continuum map.  The resulting hydrogen column density map\n  is also shown on Fig.~\\ref{hnco-results} where the resulting gas\n  distribution appears to be {\\it single-peaked} and extended with\n  dimensions of $0.02\\times 0.01$~pc. Using the formula given by\n  Mezger et al.\\ (1990) we calculate a total mass of 15~$\\hbox{M}_\\odot$\\ for\n  this structure. Thus, the triple-peaked appearance of our continuum\n  map (Fig.~\\ref{cmap}) is caused by an interplay of the temperature\n  and column density distributions.  This interpretation is supported\n  by our \\CEIO\\ $(2-1)$ integrated intensity distribution\n  (Fig.~\\ref{hnco-results}) which has the same general form as our\n  hydrogen column density map but is more sensitive to cooler material\n  further from the embedded stars.\n\n \\section{Discussion and Conclusions}\n \n Our new 220 GHz interferometer data suggest the presence of two\n embedded young stars lying slightly offset from the center of the gas\n clump associated with the water maser complex close to W3(OH). The\n mass of high temperature gas in the region is of order 15~$\\hbox{M}_\\odot$,\n although our \\CEIO \\ map makes it clear that there is more cooler gas\n at larger distances from the protostars.  The luminosity of the\n hypothesised embedded stars coincident with positions A and C can be\n crudely inferred from the observed temperature distribution\n (Fig.~\\ref{hnco-results}) by applying the Stefan-Boltzmann law. This\n way we find $3\\times 10^4$~\\solum\\ for the gas clump with a factor of\n 3 uncertainty due to temperature errors.  This (Panagia 1973)\n suggests embedded B0 stars.  It is of course likely that there are\n other lower mass objects associated with the complex and it is\n possible that there is more mass in stars than in gas.\n \n Finally, we note that perhaps the most remarkable result of this\n study has been the coincidence of continuum source A with the\n ``synchrotron jet'' of Wilner et al.\\ (1998). What is the\n significance of this?  It seems reasonable that the dense hot core\n gas plays a role in confining the ``jet''. The magnetic pressure of\n the medium in which the relativistic electrons radiate ( B$^{2}\/(8\\pi) \n \\sim \\, 4\\, 10^{-6}$ erg \\percc, Reid et al.\\ 1995) is of the same\n order as the thermal pressure in the molecular gas for a density of\n $10^8$ \\percc \\ and thus such confinement seems feasible.  The origin\n of the relativistic electrons is unclear but there are analogous\n cases known. An example is the the jet in Cepheus A (Garay et al.\\ \n 1996), whose radio emission is thought to be produced in shocks\n resulting from the interaction of the jet with the confining medium.\n The explanation in the case of W3(OH) may be similar.\n\n \\acknowledgments\n \n We thank D. Wilner and M. Reid for providing the VLA data in advance\n of publication and an anonymous referee for helpful comments.  CMW\n acknowledges travel support from CNR grant 97.00018.CT02 and ASI\n grant ARS-96-66. He would also like to thank the Max Planck Institut\n f{\\\"u}r Radioastronomie for hospitality during the course of this\n work.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nString theory is a promising candidate for a consistent quantum theory of gravity and needless to say that the characteristics of black holes (BH) in string theory would be utmost interest. The predictions of string theory differ from those of general relativity and one of the reasons for this difference is the presence of a scalar field called dilaton that can change the properties of the BH geometries.\n\nThe spherically symmetric static charged BH solution in low energy heterotic string theory in four dimension was found by Gibbons and Maeda~[1] and independently by Garfinkle, Horowitz, Strominger~[2], which from now on will be referred to as the Gibbons-Maeda-Garfinkle-Horowitz-Strominger (GMGHS) solution. These works generated enormous interest in the dilatonic charged BHs (see, e.g., [3-16] and references therein). In particular, the GMGHS BH spacetime can be described either in the Einstein frame (EF) or in the conformally related string frame (SF). (Unless the frame is specifically mentioned, the solution will be understood to be in EF). In EF, the action is in the form of the Einstein-Hilbert action, while in the SF strings directly couple to the metric as $e^{2\\phi }g_{\\mu \\nu }$, where $\\phi $ is the dilaton and $g_{\\mu \\nu }$ is the EF metric. Even though the solutions in the two frames are related by a conformal transformation so that they are mathematically isomorphic to each other~[3], there are differences in some of the physical properties of the BH solutions in these two frames~[4]. For instance, Sagnac delay for rotating Kerr-Sen metric of heterotic string theory was studied and an estimate of assumed terrestrial dilatonic charge was given in [5]. Strong gravitational lensing by charged stringy BHs do not produce any significant string effect on the Schwarzshild BH as was shown by Bhadra [6]. Timelike geodesics of particles around GMGHS BH were investigated in [7]. The magnetically charged GMGHS interior spacetime was studied in [8]. Particle collision near the horizon of GMGHS BH was studied in [9]. Quasinormal mode frequencies in the string BH were evaluated by using WKB approximation with P\\\"{o}schl-Teller potential in [10]. Some notable works on the implications of electric charge on various relativistic observables in the context of general relativistic BHs can be found in [17-21].\n\nIn addition to the above studies, which are by no means exhaustive, the accretion disk properties could be yet another diagnostic for distinguishing various types of objects. The first comprehensive study of accretion disks using a Newtonian approach was made in [22]. Later a general relativistic model of thin accretion disk was developed in three seminal papers, by Novikov and Thorne [23], Page and Thorne [24] and Thorne [25] under the assumption that the disk is in a steady-state, that is, the mass accretion rate $\\dot{M}_{0}$ is constant in time and does \\textit{not} depend of the radius of the disk. The disk is further supposed to be in hydrodynamic and thermodynamic equilibrium, which ensure a black body electromagnetic spectrum and properties of emitted radiation. The thin accretion disk model further assumes that individual particles are moving on Keplerian orbits, but for this to be true the central object should have weak magnetic field, otherwise the orbits in the inner edge of the disk will be deformed. More recently, the properties of the accretion disk around exotic central objects, such as wormholes (WH) [26,27], and non-rotating or rotating quark, boson or fermion stars, brane-world BHs, gravastars or naked singularities (NS) [28-44], $f(R)$ modified gravity models [44-46] have been studied. One of the most promising method to distinguish the types of astrophysical objects and their spin through their accretion disk properties is the profile analysis of iron line for different spacetimes [47-52]. While various accreting objects have been considered, the study of emissivity properties of the GMGHS objects, for which the celebrated Page-Thorne model is the most ideal one, have somehow been left out in the literature, to our knowledge.\n\nOne recent work by Bahamonde and Jamil [11] pertains to fluid motion (as opposed to particle motion) in different spherically symmetric spacetimes, where the authors focused on the radial variation of the fluid velocity, the density and the accretion rate\\ $\\dot{M}_{0}$ of the fluid to the GMGHS BH. The work in [11], though useful in its own right, is distinct from the Page-Thorne emission model since it studied only the non-emissivity aspects of the fluid flow with the predicted mass accretion rate $\\dot{M}_{0}$ depending on the radius of the disk. Also the critical radius $r_{\\textmd{\\scriptsize{c}}}$ used in [11] is not the marginally stable radius $r_{\\textmd{\\scriptsize{ms}}}$ required in the Page-Thorne model. On the other hand, we are motivated by the understanding that genuine observable signatures of the accretion disk should be provided by an analysis of the properties of radiation emerging from the surface of the disk for which the Page-Thorne model is most suitable.\n\nThe present paper is thus devoted to studying the kinematic and emissivity properties of a central object represented by stringy GMGHS solutions (not necessarily BHs) in the EF and SF using the Page-Thorne model. We shall analyze the luminosity spectra, flux of radiation, temperature profile, efficiency etc. In particular, we wish to see how the kinematic and accretion features compare between EF and SF including their extreme counterparts and with similar features from Reissner-Nordstr\\\"{o}m and Schwarzschild BH of general relativity. We shall assume for numerical illustration a toy model of a central object with mass $15M_{\\odot }$ and accretion rate $\\dot{M}_{0}=10^{18}$ gm.sec$^{-1}$, which could be a BH, a WH or a NS.\n\nThe paper is organized as follows: In Sec.2, we give a brief preview of the GMGHS solutions and in Sec.3, summarize the main formulas relating to the accretion phenomenon to be used in the paper. In Sec.4, we present the kinematic and accretion properties in two frames and compare how they differ from those for the Reissner-Nordstr\\\"{o}m and Schwarzschild BH. In Sec.5, the obtained results are summarized. We take units such that $8\\pi G=c=1$, metric signature ($-,+,+,+$) and greek indices run from $0$ to $3$.\n\n\n\n\n\\section{GMGHS solutions}\n\nIn this section, a brief preview of the action and the static spherically symmetric dilaton BHs are given. In the EF, the GMGHS action is [1,2]\n\\begin{equation}\nS_{\\textmd{\\scriptsize{EF}}}=\\int d^{4}x\\sqrt{-g}\\left[ R_{(g)}-2(\\bigtriangledown \\phi\n)^{2}-e^{-2\\phi }F_{\\mu \\nu }F^{\\mu \\nu }\\right],\n\\end{equation}%\nwhere $\\phi $ is a dilaton, $R_{(g)}$ is the scalar curvature related to $g_{\\mu \\nu }$, and $F_{\\mu \\nu }$ is the Maxwell field. The line element\nrepresenting a 4-dimensional charged dilatonic GMGHS BH in the EF is given by\n\\begin{eqnarray}\nd\\tau _{\\textmd{\\scriptsize{Mag,EF}}}^{2}&=&-\\left( 1-\\frac{2M}{r}\\right) dt^{2}+\\left( 1-\\frac{2M}{r}\\right)^{-1}dr^{2} \\nonumber\\\\\n&&+r\\left( r-\\frac{Q^{2}}{M}\\right) (d\\theta^{2}+ \\sin^{2}{\\theta} d\\varphi^{2}),\n\\end{eqnarray}%\nwhere $M$ is the mass and $Q$ is the magnetic charge. The Maxwell field is given by\n\\begin{equation}\nF=Q\\sin {\\theta }d\\theta \\wedge d\\varphi\n\\end{equation}%\nand the dilaton field $\\phi $ is defined as\n\\begin{equation}\ne^{-2\\phi }=e^{-2\\phi _{0}}\\left( 1-\\frac{Q^{2}}{Mr}\\right) ,\n\\end{equation}%\nwhere $\\phi _{0}$ is the asymptotic value of the dilaton. As we consider only asymptotically flat cases, we will assume $\\phi _{0}\\equiv 0$. Solution (2) describes BHs of mass $M$ and charge $Q$ when $Q\/M$ is sufficiently small. For $Q=\\sqrt{2}M$, the event horizon $r=2M$ becomes singular. (Figs.1,4,5 show the properties of this singularity at $r=2M$: potential $V_{\\textmd{\\scriptsize{eff}}}$, Flux of radiation and Temperature diverge).\n\nSince the metric, for fixed $\\theta $ and $\\varphi $, is the same as that of Schwarzschild, $r=2M$ is a regular event horizon only when $Q<\\sqrt{2}M$. Note that the area goes to zero at $r=Q^{2}\/M$ $<2M$ causing this surface to be singular since the Kretschmann scalar and the dilaton $\\phi $ diverge there. However, the surface is hidden under $r=2M$ and no information can emerge from it to outside observers (All observable quantities also diverge on that surface). The dilatonic charge for the charged BH (2) is\n\\begin{equation}\nD=-\\frac{Q^{2}}{2M},\n\\end{equation}%\nwhere $D$ is not a new free parameter in (2) since once the asymptotic value of $\\phi $ is fixed, it is determined by $M$ and $Q$, and is always negative. The dilaton charge is also responsible for a long-range, attractive force between BHs.\n\nElectrically charged solutions may be obtained by a duality rotation defined by\n\\begin{equation}\n\\tilde{F}_{\\mu \\nu }=\\frac{1}{2}e^{-2\\phi }\\epsilon _{\\mu \\nu }^{\\lambda\n\\rho }F_{\\lambda \\rho }.\n\\end{equation}%\nThe equations of motion (2) are invariant under $F\\rightarrow \\tilde{F}$ and $\\phi \\rightarrow -\\phi $. Such solutions can therefore be obtained by simply changing the sign of $\\phi $ while keeping the metric fixed. This implies that the dilaton charge is $D$ is positive for electrically charged BHs in the EF.\n\nThe effective action and equations of motion can be further modified by applying the conformal transformation $\\widetilde{g}_{\\mu \\nu }= e^{2\\phi}g_{\\mu\\nu }$ thus obtaining the action in the SF\n\\begin{equation}\nS_{\\textmd{\\scriptsize{SF}}}=\\int d^{4}x\\sqrt{-\\widetilde{g}}e^{-2\\phi }\\left[ R_{(\\widetilde{g})}-4(\\bigtriangledown \\phi )^{2}- F_{\\mu\\nu}F^{\\mu\\nu}\\right],\n\\end{equation}%\nin which the space-time coordinates have been left unchanged and $R_{(\\widetilde{g})}$ is the Ricci curvature from $\\widetilde{g}_{\\mu \\nu }$. Upon transforming to SF one obtains magnetically charged GMGHS BH metric is given by [15,16]:\n\\begin{eqnarray}\nd\\tau _{\\textmd{\\scriptsize{Mag,SF}}}^{2}&=&-\\frac{\\left( 1-\\frac{2M}{r}\\right) }{\\left( 1-\\frac{Q^{2}}{Mr}\\right) }dt^{2}+\\frac{dr^{2}}{\\left(1- \\frac{2M}{r}\\right)\\left(1-\\frac{Q^{2}}{Mr}\\right)} \\nonumber\\\\\n&&+r^{2}(d\\theta ^{2}+\\sin^{2}{\\theta} d\\varphi ^{2}).\n\\end{eqnarray}%\n\nThis is the metric that appears in the string $\\sigma $ model. The Kretschmann scalar diverges at $r=Q^{2}\/M$, hence it is a singular surface. For SF, the statement that the horizon is singular when $Q^{2}=2M^{2}$ is actually irrelevant. This is because strings do not couple to the metric $g_{\\mu \\nu }$ but rather to $e^{2\\phi }g_{\\mu \\nu }$. For $Q^{2}<2M^{2}$, this again describes a BH with an event horizon at $r_{\\textmd{\\scriptsize{eh}}}=2M$. We have simply rescaled the metric by a conformal factor, which is finite everywhere outside (and on) the horizon. However, at the extremal value $Q^{2}=2M^{2}$, the metric becomes\n\\begin{eqnarray}\nd\\tau _{\\textmd{\\scriptsize{WH,SF}}}^{2}&=&-dt^{2}+\\left( 1-2M\/r\\right)^{-2}dr^{2} \\nonumber\\\\\n&&+r^{2}(d\\theta ^{2}+\\sin^{2}{\\theta} d\\varphi ^{2}).\n\\end{eqnarray}%\nThe geometry of a $t=$ const. surface in this spacetime is identical to that\nof a static slice in the extreme Reissner-Nordstr\\\"{o}m metric. But the\nhorizon, along with the singularity inside it, have completely disappeared.\nIn its place have appeared a WH. This metric, with $r>2M$, is globally\nstatic and geodesically complete and has all the properties of a traversable\nMorris-Thorne WH with a regular throat at $r_{\\textmd{\\scriptsize{th}}}=2M$ with redshift\nfunction $\\Phi =0$ and a shape function $b(r)=4M\\left( 1-\\frac{M}{r}\\right) $\nhaving an imbedding surface%\n\\begin{eqnarray}\nz(r)&=&4\\sqrt{M}\\left[ \\sqrt{r-M}\\right. \\nonumber\\\\\n&&\\left.-\\sqrt{M}\\textmd{arctanh}\\left\\{ \\sqrt{\\frac{r}{%\nM}-1}\\right\\} \\right].\n\\end{eqnarray}%\n\nThe solution of the electrically charged GMGHS solution in the SF is given\nby [15,16]\n\\begin{eqnarray}\nd\\tau _{\\textmd{\\scriptsize{Elec,SF}}}^{2}&=&-\\frac{\\left( 1+\\frac{Q^{2}-2M^{2}}{Mr}\\right) }{%\n\\left( 1+\\frac{Q^{2}}{Mr}\\right) ^{2}}dt^{2}+\\frac{dr^{2}}{\\left( 1+\\frac{Q^{2}-2M^{2}}{Mr}\\right) } \\nonumber\\\\\n&&+r^{2}(d\\theta ^{2}+\\sin^{2}{\\theta} d\\varphi ^{2}).\n\\end{eqnarray}%\nThis solution describes a BH, when $Q^{2}<2M^{2}$, but in the extremal case (%\n$Q^{2}=2M^{2}$) the resulting solution is\n\\begin{eqnarray}\nd\\tau _{\\textmd{\\scriptsize{NS,SF}}}^{2}&=&-\\left( 1+2M\/r\\right)\n^{-2}dt^{2}+dr^{2} \\nonumber\\\\\n&&+r^{2}(d\\theta ^{2}+\\sin^{2}{\\theta} d\\varphi ^{2}).\n\\end{eqnarray}%\nwhich describes a singularity at $r=0$ since the Kretschmann scalar diverges\nthere but this divergence is not covered by an event horizon, and so the\npoint $r=0$ represents a NS.\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Veff1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Veff2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Veff3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{VeffRN}\n\\end{center}\n\\caption{The effective potential $V_{\\textmd{\\scriptsize{eff}}}(r)$ for a GMGHS BH in the EF (top left hand), magnetically charged one in SF (top right hand), electrically charged one in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand). The specific angular momentum of the orbiting particle is chosen to be $\\widetilde{L}=4M$. The potentials for $Q<\\protect\\sqrt{2}M$ do not show appreciable difference with those of the Reissner-Nordstr\\\"{o}m and Schwarzschild BH (Fig.1d). At the extreme limit, $Q=\\protect\\sqrt{2}M$, the potential diverges at the NS radii (Figs.1a,c) and at the WH throat (Fig.1b). The potential coincides with that of the Schwarzschild asymptotically.}\n\\label{Veff}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpAngulMom1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpAngulMom2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpAngulMom3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpAngulMomRN}\n\\end{center}\n\\caption{The specific angular momentum $\\tilde{L}(r)$ of the orbiting particle as a function of the radial coordinate $r$ (in cm) for a GMGHS BH in EF (top left hand), magnetically charged in the SF (top right hand), electrically charged in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand) plotted for different values of $Q$. We see that $\\tilde{L}\\rightarrow 0$ at the NS radii $r\\rightarrow 2M$ (Fig.2a), $r\\rightarrow 0$ (Fig.2c), but $\\tilde{L}\\rightarrow 0$ at the WH throat (Fig.2b). Fig.2d displays the behavior only for BHs. All plots show no appreciable difference with those for Schwarzshild BH either in the far field of the accreting object.}\n\\label{SpAngulMom}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpEnergy1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpEnergy2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpEnergy3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{SpEnergyRN}\n\\end{center}\n\\caption{The specific energy $\\tilde{E}(r)$ of the orbiting particles as a function of the radial coordinate $r$ (in cm) for a GMGHS BH in EF (top left hand), magnetically charged in the SF (top right hand), electrically charged in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand) plotted for different values of $Q$. We see that $\\tilde{E}\\rightarrow 0$ at the NS radii $r\\rightarrow 2M$ (Fig.3a), $r\\rightarrow 0$ (Fig.3c), but $\\tilde{E}$ assumes a constant value at the WH throat (Fig.3b). Fig.3d displays the behavior only for BHs, and there are no appreciable differences in the far field of observation.}\n\\label{SpEnergy}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Flux1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Flux2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Flux3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{FluxRN}\n\\end{center}\n\\caption{The time averaged radiation flux $F(r)$ as a function of the radial coordinate $r$ (in cm) radiated by the disk around a GMGHS BH in EF (top left hand), magnetically charged in the SF (top right hand), electrically charged in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand) plotted for different values of $Q$. We see that $F(r)\\rightarrow \\infty $ at NS radii $2M$ coinciding with $r_{\\textmd{\\scriptsize{ms}}}$ (Fig.4a), $r=r_{\\textmd{\\scriptsize{ms}}}\\rightarrow 0$ (Fig.4c), but $F(r)$ assumes a finite value at the WH throat (Fig.4b). Fig.4d displays the behavior only for BHs. All plots except that of WH show no appreciable differences at the asymptotic limit.}\n\\label{Flux}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Temp1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Temp2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Temp3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{TempRN}\n\\end{center}\n\\caption{Temperature distribution $T(r)$ of of the accretion disk for a GMGHS BH in EF (top left hand), magnetically charged in the SF (top right hand), electrically charged in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand) plotted for different values of $Q$. We see that $T(r)\\rightarrow \\infty $ at NS radii $2M$ coinciding with $r_{\\textmd{\\scriptsize{ms}}}$ (Fig.5a), $r=r_{\\textmd{\\scriptsize{ms}}}\\rightarrow 0$ (Fig.5c), but $T(r)$ assumes a finite value at the WH throat (Fig.5b). Fig.5d displays the behavior only for BHs. All plots except that of WH show no appreciable differences at asymptotic distances of observation.}\n\\label{Temp}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{center}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Lum1}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Lum2} \\\\\n\\vspace{0.5cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{Lum3}\n\\hspace{0.3cm}\n\\includegraphics[type=pdf,ext=.pdf,read=.pdf,width=8.5cm]{LumRN}\n\\end{center}\n\\caption{The emission spectra $\\protect\\nu L(\\protect\\nu )$ vs frequency ($\\protect\\nu $ in Hz) of the accretion disk with inclination $i=0^{0}$ for a GMGHS BH in EF (top left hand), magnetically charged in the SF (top right hand), electrically charged in the SF (bottom left hand) and Reissner-Nordstr\\\"{o}m BH (bottom right hand) plotted for different values of $Q$. \\ Fig. 6a shows that, at $Q\\leq \\protect\\sqrt{2}M$, $\\protect\\nu L(\\protect\\nu )$ behaves like a BH of general relativity at the low frequency range but assumes a steady value until $\\protect\\nu \\sim 10^{19}$ Hz, when it begins to decline to the Schwarzschild profile (not shown). Similar behavior is seen in Fig.6c. Fig.6b shows the emission spectra of the disk around WH throat. At low $\\protect\\nu $, the features are almost indistinguishable from those of BHs but at higher frequency, the spectra diminishes considerably. Fig.6d displays the behavior only for BHs. These features are consistent with efficiency of conversion (see summary).}\n\\label{Lum}\n\\end{figure*}\n\n\n\n\n\n\\section{Thin accretion disk}\n\nThe accretion disk is formed by particles moving in circular orbits around a\ncompact object, with the geodesics determined by the space-time geometry\naround the object, be it a WH, BH or NS. For a static and spherically\nsymmetric geometry the metric is generically given by\n\n\\begin{equation}\nd\\tau ^{2}=-g_{tt}dt^{2}+g_{rr}dr^{2}+g_{\\theta \\theta }d\\theta\n^{2}+g_{\\varphi \\varphi }d\\varphi ^{2}.\n\\end{equation}%\nAt and around the equator, i.e., when $\\left\\vert \\theta -\\pi \/2\\right\\vert\n\\ll 1,$we assume, with Harko \\textit{et al.} [27], that the metric functions\n$g_{tt},g_{rr},g_{\\theta \\theta }$ and $g_{\\varphi \\varphi }$ depend only on\nthe radial coordinate $r$. The radial dependence of the angular velocity $%\n\\Omega $, of the specific energy $\\widetilde{E}$, and of the specific\nangular momentum $\\widetilde{L}$ of particles moving in circular orbits in\nthe above geometry are given by\n\\begin{eqnarray}\n&&\\frac{dt}{d\\tau }=\\frac{\\widetilde{E}}{g_{tt}}, \\\\\n&&\\frac{d\\varphi }{d\\tau }=\\frac{\\widetilde{L}}{g_{\\varphi \\varphi }}, \\\\\n&&g_{tt}g_{rr}\\left( \\frac{dr}{d\\tau }\\right) ^{2}+V_{\\textmd{\\scriptsize{eff}}%\n}\\left( r\\right) =\\widetilde{E}^{2}.\n\\end{eqnarray}%\nFrom the last equation, the effective potential $V_{\\textmd{\\scriptsize{eff}}}(r)$\ncan be obtained in the form\n\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}\\left( r\\right) =g_{tt}\\left( 1+\\frac{\\widetilde{L}^{2}}{g_{\\varphi \\varphi }}\\right).\n\\end{equation}%\nExistence of circular orbits at any arbitrary radius $r$ in the equatorial\nplane demands that $V_{\\textmd{\\scriptsize{eff}}}\\left( r\\right) =0$ and $dV_{%\n\\textmd{\\scriptsize{eff}}}\/dr=0$. These conditions allow us to write\n\\begin{eqnarray}\n&&\\widetilde{E}=\\frac{g_{tt}}{\\sqrt{g_{tt}-g_{\\varphi \\varphi }\\Omega ^{2}}},\n\\\\\n&&\\widetilde{L}=\\frac{g_{\\varphi \\varphi }\\Omega }{\\sqrt{g_{tt}-g_{\\varphi\n\\varphi }\\Omega ^{2}}}, \\\\\n&&\\Omega =\\frac{d\\varphi }{dt}=\\sqrt{\\frac{g_{tt,r}}{g_{\\varphi \\varphi ,r}}}.\n\\end{eqnarray}\n\nStability of orbits depend on the signs of $d^{2}V_{\\textmd{\\scriptsize{eff}}}\/dr^{2}$, while the condition $d^{2}V_{\\textmd{\\scriptsize{eff}}}\/dr^{2}=0$ gives the inflection point or \\textit{marginally stable} (ms) orbit or innermost stable circular orbit (ISCO) at $r=r_{\\textmd{\\scriptsize{ms}}}$. For the solutions under consideration, we explicitly find the $r_{\\text{ms}}$ as under:\n\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,EF}}}= - \\frac{(r-2M)^{2}(2Mr-Q^{2})}{%\nrJ},\n\\end{equation}%\n\\begin{eqnarray}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,EF}}\\;\\prime\\prime} &=&\n-\\frac{4M^{3}(r-6M)}{r(Mr-Q^{2})J} \\nonumber\\\\\n&&-\\frac{8M^{2}Q^{2}(3Mr-Q^{2})}{r^{3}(Mr-Q^{2})J},\n\\end{eqnarray}%\nwhere\n\\begin{eqnarray}\nJ=2Mr(r-3M)-Q^{2}(r-4M). \\nonumber\n\\end{eqnarray}\nSolving $V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,EF}}\\;\\prime\\prime}=0$, we find\n\\begin{eqnarray}\nr_{\\textmd{\\scriptsize{ms}}}^{\\textmd{\\scriptsize{Mag,EF}}}&=&2M+\\left( \\frac{2}{M}\\right) ^{\\frac{1}{3}%\n}(2M^{2}-Q^{2})^{\\frac{2}{3}} \\nonumber \\\\\n&&+2^{\\frac{2}{3}}\\{M(2M^{2}-Q^{2})\\}^{\\frac{1}{3}%\n}.\n\\end{eqnarray}\n\nSimilarly, the effective potential $V_{\\textmd{\\scriptsize{eff}}}(r)$ and $V_{\\textmd{\\scriptsize{eff}}}^{\\prime \\prime}(r)$ for magnetic GMGHS spacetime in SF is\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,SF}}}= -\\frac{2M(r-2M)^{2}}{J},\n\\end{equation}%\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,SF}}\\;\\prime\\prime}=-\\frac{%\n2M(r-6M)(2M^{2}-Q^{2})}{r(Mr-Q^{2})J}.\n\\end{equation}%\nSolving $V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Mag,SF}}\\;\\prime\\prime}=0$, we find\n\\begin{equation}\nr_{\\textmd{\\scriptsize{ms}}}^{\\textmd{\\scriptsize{Mag,SF}}}=6M.\n\\end{equation}\n\nThe effective potential $V_{\\textmd{\\scriptsize{eff}}}(r)$ and $V_{\\textmd{\\scriptsize{eff}}}^{\\prime\n\\prime }(r)$ for electric GMGHS spacetime in SF\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Elec,SF}}}=-\\frac{2Mr\\{M(r-2M)+Q^{2}\\}^{2}}{(Q^{2}+Mr)P},\n\\end{equation}%\n\\begin{equation}\nV_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Elec,SF}}\\;\\prime\\prime}=-\\frac{2MN}{%\nr(Mr+Q^{2})^{3}P}.\n\\end{equation}%\nSolving $V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Elec,SF}}\\;\\prime\\prime}=0$, we find the\nmarginally stable orbit\n\\begin{eqnarray}\nr_{\\textmd{\\scriptsize{ms}}}^{\\textmd{\\scriptsize{Elec,SF}}}&=&\\frac{2M^{2}-Q^{2}}{M}+2^{\\frac{4}{3}%\n}M^{2}(2M^{2}-Q^{2})K^{-\\frac{1}{3}} \\nonumber \\\\\n&&+\\frac{2^{\\frac{2}{3}}K^{\\frac{1}{3}}}{%\n2M^{2}+Q^{2}},\n\\end{eqnarray}%\nwhere\n\\begin{eqnarray}\nP &=&2M^{2}r(r-3M)-Q^{2}(2M^{2}-3Mr-Q^{2}), \\\\\nN &=&2M^{5}r^{2}(r-6M)+M^{3}Q^{2}r^{3}+M^{2}Q^{4}\\left(4M^{2} \\right.  \\nonumber \\\\\n&& \\left.-3Mr+3r^{2}\\right)-Q^{6}(4M^{2}-3Mr-Q^{2}), \\\\\nK &=&16M^{9}-8M^{5}Q^{4}+MQ^{8}+MQ^{2}(Q^{4}-4M^{4})^{\\frac{3}{2}}.\n\\end{eqnarray}\n\nWe assume geometrically thin accretion disk, which means that the disk height $H$\nabove the equator is much smaller than the characteristic radius $R$ of the\ndisk, $H\\ll R$. The disk is assumed to be in hydrodynamical equilibrium\nstabilizing its vertical size, with the pressure and vertical entropy\ngradient being negligible. An efficient cooling mechanism via heat loss by\nradiation over the disk surface is assumed to be functioning in the disk,\nwhich prevents the disk from collecting the heat generated by stresses and\ndynamical friction. The thin disk has an inner edge defined by the \\textit{%\nmarginally stable} circular radius $r_{\\textmd{\\scriptsize{ms}}}$, while the\norbits at higher radii are Keplerian.\n\nIn the above approximation, Page and Thorne [24], using the rest mass\nconservation law, showed that the time averaged rate of rest mass accretion $%\ndM_{0}\/dt$ is independent of the radius: $\\dot{M}_{0}\\equiv dM_{0}\/dt=-2\\pi\nru^{r}\\Sigma =$ \\textmd{const}. (Here $t$ and $r$ are the coordinate time\nand radial coordinates respectively, $u^{r}$ is the radial component of the\nfour velocity $u^{\\mu }$ of the accreting particles and $\\Sigma $ is the\naveraged surface density of the disk). In the steady-state thin disk model,\nthe orbiting particles have $\\Omega $ , $\\widetilde{E}$ and $\\widetilde{L}$\nthat depend only on the radii of the orbits. We omit other technical details\nhere (see [27]), but quote only the relevant formulas below.\n\nThe flux $F$ of the radiant energy over the disk can be expressed in terms\nof $\\Omega $ , $\\widetilde{E}$ and $\\widetilde{L}$ as [22-24]\n\n\\begin{equation}\nF(r)= -\\frac{\\dot{M}_{0}}{4\\pi \\sqrt{-g}}\\frac{\\Omega _{,r}}{%\n\\left( \\widetilde{E}-\\Omega \\widetilde{L}\\right) ^{2}}\\int_{r_{\\textmd{\\scriptsize{ms}}%\n}}^{r}\\left( \\widetilde{E}-\\Omega \\widetilde{L}\\right) \\widetilde{L}_{,r}dr.\n\\end{equation}\nThe disc is supposed to be in thermodynamical equilibrium, so the radiation\nflux emitted by the disk surface will follow Stefan-Boltzmann law:%\n\\begin{equation}\nF\\left( r\\right) =\\sigma T^{4}\\left( r\\right) ,\n\\end{equation}%\nwhere $\\sigma $ is the Stefan-Boltzmann constant. The observed luminosity $%\nL\\left( \\nu \\right) $ has a redshifted black body spectrum [36]\n\n$$L_{\\nu }=4\\pi \\textmd{d}^{2}I(\\nu )=\\frac{8\\pi h\\cos {i}}{c^{2}}\\int_{r_{\\textmd{\\scriptsize{in}}}}^{r_{\\textmd{\\scriptsize{f}}}}\\int_{0}^{2\\pi }\\frac{\\nu _{e}^{3}rdrd\\varphi }{\\textmd{Exp}\\left[ \\frac{h\\nu _{e}}{k_{B}T}\\right] -1},$$\nwhere $i$ is the disk inclination angle, d is the distance between the\nobserver and the center of the disk, $r_{\\textmd{\\scriptsize{in}}}$ and $r_{\\textmd{\\scriptsize{f}}}$ are\nthe inner and outer radii of the disc, $h$ is the Planck constant, $\\nu _{e}$\nis the emission frequency, $I(\\nu )$ is the Planck distribution, and $k_{B}$\nis the Boltzmann constant. The observed photons are redshifted and their\nfrequency $\\nu $ is related to the emitted ones in the following way $\\nu\n_{e}=(1+z)\\nu $. The redshift factor $(1+z)$ has the form [27]:\n\n\\begin{equation}\n(1+z)=\\frac{1+\\Omega r\\sin {\\varphi }\\sin {i}}{\\sqrt{g_{tt}-\\Omega\n^{2}g_{\\varphi \\varphi }}},\n\\end{equation}%\nwhere the light bending effect is neglected.\n\nAnother important characteristic of the accretion disk is its efficiency $%\n\\epsilon$, which quantifies the ability with which the central body\nconverts the accreting mass into radiation. The efficiency is measured at\ninfinity and it is defined as the ratio of two rates: the rate of energy of\nthe photons emitted from the disk surface and the rate with which the\nmass-energy is transported to the central body. If all photons reach\ninfinity, an estimate of the efficiency is given by the specific energy of\nthe accreting particles measured at the marginally stable orbit [25]:\n\n\\begin{equation}\n\\epsilon =1-\\widetilde{E}\\left( r_{\\textmd{\\scriptsize{ms}}}\\right) .\n\\end{equation}%\nThe Eqs.(18-20,33-36) are valid for any static spherically symmetric spacetime and\nhence valid both in the EF and SF since spherical symmetry is preserved\nunder conformal transfomation.\n\n\n\n\n\n\\section{GMGHS solutions: kinematic and accretion features}\n\nWe shall consider for illustration a central compact object of mass $%\nM=15M_{\\odot }$ with an accretion rate $\\dot{M_{0}}=10^{18}$gm.sec$^{-1}$\nand assume that it possesses magnetic charge $Q$ with the spacetime\ndescribed by stringy BH solutions (2), (8) and (11).\n\nWe examine how kinematic features of the concerned solutions differ from\nthose of Schwarzschild BH. First, the plot for $V_{\\textmd{\\scriptsize{eff}}}$ are shown in\nFigs.1a-1c, for $r\\in \\lbrack r_{\\textmd{\\scriptsize{eh}}}$,$\\infty )$, while the parameter\n$Q$ is taken to assume values $0.5M$, $M$, $1.2M$ and $\\sqrt{2}M$. The plots\nreveal that all potentials show finite maxima for $Q^{2}<2M^{2}$ but diverge\nat $r_{\\textmd{\\scriptsize{eh}}}^{\\textmd{\\scriptsize{Mag,EF}}}=r_{\\textmd{\\scriptsize{eh}}}^{%\n\\textmd{\\scriptsize{Mag,SF}}}=2M$ in the extreme case $Q^{2}=2M^{2}$ and at $r_{%\n\\textmd{\\scriptsize{eh}}}^{\\textmd{\\scriptsize{Elec,SF}}}=0$ corresponding to a NS, see\nafter Eq.(12). The effective potentials of the Schwarzschild and\nReissner-Nordstr\\\"{o}m solution are plotted in Fig.1d for comparison. An\ninteresting feature is that all the potentials in the EF and SF show smooth\nasymptotic fall-off approaching the Schwarzschild curve from above. In\ncontrast, the potential $V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Elec,SF}}}$, after reaching a\nlocal maximum at $r_{\\textmd{\\scriptsize{ms}}}$, dips below the Schwarzschild curve\nintersecting it at a radius given by (Fig.1c):\n\\begin{equation}\nr|_{V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Elec,SF}}}=V_{\\textmd{\\scriptsize{eff}}}^{\\textmd{\\scriptsize{Sch}}}}=\\frac{%\n4M^{2}-Q^{2}+\\sqrt{16M^{4}+Q^{4}}}{2M}.\n\\end{equation}\n\nSecond, we can observe similar features in Figs.2a-2d demonstrating the\neffect of $Q$ on specific angular momenta $\\widetilde{L}$. Note that, in all\nthese plots, the distance scale is $r\\in \\lbrack r_{\\textmd{\\scriptsize{eh}}}$,$\\infty )$\nexcept for $Q^{2}=2M^{2}$, when $r_{\\textmd{\\scriptsize{eh}}}=0$ (singularity). However, in\ncase of $\\widetilde{L}^{\\textmd{\\scriptsize{Elec,SF}}}$, its intersection point\nwith that of the Schwarzschild BH differs from the intersection point of\neffective potential and is given by\n\\begin{equation}\nr|_{\\widetilde{L}^{\\textmd{\\scriptsize{Elec,SF}}}=\\widetilde{L}^{\\textmd{\\scriptsize{Sch}%\n}}}=\\frac{8M^{2}-Q^{2}+\\sqrt{32M^{4}+Q^{4}}}{2M}.\n\\end{equation}\n\nThird, Figs.3a-3d show influence of $Q$ on the specific orbital energies.\nFig.3a shows that energies of a particle orbiting a Schwarzschild BH is\nhigher than that orbiting a magnetic\\ GMGHS BH at the same radius in the EF.\nA similar behavior is seen for orbits in the electric\\ GMGHS BH in the SF\n(Fig.3c). This behavior is similar to that in the Reissner-Nordstr\\\"{o}m\ncase (Fig.3d). However, exactly the opposite behavior is shown by the orbits\nin the magnetic\\ GMGHS BH in the SF, where orbital energies for particles in\nthe Schwarzschild BH spacetime are at the lowest (Fig.3b).\n\nFinally, we focus on the observable emissivity features such as the\nradiation flux, luminosity, temperature and conversion efficiency. Figs.4a-d\ndisplay the flux of radiation $F\\left( r\\right) $ emitted by the disk\nbetween $r_{\\textmd{\\scriptsize{ms}}}$ and received at an arbitrary radius $r$ [Eq.(33)],\nFigs 5a-d show variation of temperature over the disk from $r_{\\textmd{\\scriptsize{ms}}}$\nto an arbitrary radius and Figs.6a-d show observed luminosity variations on\na logarithmic scale over different frequency ranges. The conversion\nefficiency $\\epsilon $ of the accreting mass into radiation, measured at\ninfinity is given by Eq.(36). In Tab.1, we show the radiation properties of\nthe accretion disks using marginally stable orbits $r_{\\textmd{\\scriptsize{ms}}}$ and $%\n\\epsilon $ with the parameter $Q$ in the range used in the previous plots.\n\n\\begin{table*}[!ht]\n\\caption{The $r_{\\textmd{\\scriptsize{ms}}}$ and the efficiency $\\epsilon $ for GMGHS and Reissner-Nordstr\\\"{o}m BHs, all having a mass $M=15M_{\\odot }$\n         with the accretion rate $\\dot{M}_{0}=10^{18}$ gm.sec$^{-1}$. The general relativistic Schwarzschild BH corresponds to $Q=0$.}\n  \\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\n  \\hline\n  $Q$ & \\multicolumn{2}{c|}{Magnetic, EF} & \\multicolumn{2}{c|}{Magnetic, SF}\n  & \\multicolumn{2}{c|}{Electric, SF} & \\multicolumn{2}{c|}{Reissner-Nordstr\\\"{o}m BH} \\\\\n  \\cline{2-9}\n     & $r_{\\textmd{\\scriptsize{ms}}}$ [$M$] & $\\epsilon $ & $r_{\\textmd{\\scriptsize{ms}}}$ [$M$] & $\\epsilon $\n  & $r_{\\textmd{\\scriptsize{ms}}}$ [$M$] & $\\epsilon $ & $r_{\\textmd{\\scriptsize{ms}}}$ [$M$] & $\\epsilon $ \\\\\n  \\hline\n  & \\multicolumn{8}{c|}{BH} \\\\\n  $0$ (Sch) & 6.0000 & 0.0572 & \\multicolumn{6}{c|}{} \\\\\n  $0.5M$ & 5.7426 & 0.0607 & 6.0000 & 0.0506 & 5.2746 & 0.0715 & 5.6066 &\n  0.0608 (BH) \\\\\n  $M$ & 4.8473 & 0.0768 & 6.0000 & 0.0298 & 3.2743 & 0.1361 & 4.0000 & 0.0814\n  (extr. BH) \\\\\n  $1.2M$ & 4.1644 & 0.0950 & 6.0000 & 0.0170 & 2.1163 & 0.2080 & 2.4548 &\n  0.1149 (NS) \\\\\n  - & \\multicolumn{2}{c|}{NS} & \\multicolumn{2}{c|}{WH}\n  & \\multicolumn{2}{c|}{NS} &  &  \\\\\n  $\\sqrt{2}M$ & 2.0000 & 1.0000 & 6.0000 & 0 & 0 & 1.0000 & 2.3129 & 0.0773\n  (NS) \\\\ \\hline\n  \\end{tabular}\n\\end{table*}\n\nTable 1 demonstrates the variation in the location of the inner disk edge\nwith the changing charge $Q$. For GMGHS BH in EF, we notice that the higher\nvalues of charge $Q$ are, the closer are the marginally stable orbits to the\ncenter. However, for magnetically charged GMGHS BH in SF, we see that its\nconversion efficiency for $Q\\sim 0.3M$ mimics that of the Schwarzschild BH,\nboth being $0.0572.$ For electrically charged GMGHS BH in SF, the efficiency\nincreases to over $20\\%$, whereas, interestingly, that for magnetically\ncharged GMGHS BH in SF the efficiency decreases to lower than $2.45\\%$.\nThese features are characteristic of the frames chosen for describing BHs.\n\n\\begin{table*}[!ht]\n \\caption{Comparison with BH of luminosity spectra from accretion disk around\n  different extreme GMGHS central objects.}\n \\begin{tabular}{|c|c|c|c|c|}\n  \\hline\n  $\\nu$[1\/s] & \\multicolumn{4}{c|}{$\\nu L(\\nu)$[erg s$^{-1}$]} \\\\\n  \\cline{2-5}\n            & \\multicolumn{2}{c|}{Naked singularity} & Wormhole & Schwarzschild \\\\\n  \\cline{2-4}\n             & Magnetic, EF    & Electric, SF          & Magnetic, SF & black hole\\\\\n  \\hline\n  $10^{12}$  & $1.15\\times 10^{24}$   & $1.63\\times 10^{24}$  & $8.75\\times 10^{22}$   & $1.15\\times 10^{24}$   \\\\\n  $10^{13}$  & $1.13\\times 10^{26}$   & $1.60\\times 10^{26}$  & $8.20\\times 10^{24}$   & $1.11\\times 10^{26}$   \\\\\n  $10^{14}$  & $1.07\\times 10^{28}$   & $1.51\\times 10^{28}$  & $6.21\\times 10^{26}$   & $9.84\\times 10^{27}$   \\\\\n  $10^{15}$  & $9.08\\times 10^{29}$   & $1.31\\times 10^{30}$  & $8.95\\times 10^{27}$   & $5.57\\times 10^{29}$   \\\\\n  $10^{16}$  & $4.57\\times 10^{31}$   & $1.02\\times 10^{32}$  & $1.13\\times 10^{19}$   & $1.90\\times 10^{29}$   \\\\\n  $10^{17}$  & $2.13\\times 10^{31}$   & $8.43\\times 10^{33}$  & $\\sim 0$               & $1.01\\times 10^{0}$    \\\\\n  $10^{18}$  & $3.34\\times 10^{15}$   & $6.77\\times 10^{35}$  & $\\sim 0$               & $\\sim 0$               \\\\\n  $10^{19}$  & $\\sim 0$               & $8.31\\times 10^{36}$  & $\\sim 0$               & $\\sim 0$               \\\\\n  $10^{20}$  & $\\sim 0$               & $1.10\\times 10^{27}$  & $\\sim 0$               & $\\sim 0$               \\\\\n  $10^{21}$  & $\\sim 0$               & $\\sim 0$              & $\\sim 0$               & $\\sim 0$               \\\\\n  \\hline\n\\end{tabular}\n\\end{table*}\n\nThe Figs.1-6 show different kinematic and emissivity parameters for\ndifferent values of $Q$ related to dilatonic charge $D$ [Eq.(5)]. We have\nassumed the central mass to be $M=15M_{\\odot },$ and mass accretion rate $%\n\\dot{M}_{0}=10^{18}$ gm$.$sec$^{-1}$.\n\n\n\n\n\n\\section{Summary}\n\nThe spacetime structure of GMGHS BHs differ from that of the Schwarzschild BH in many important ways, which are expected to show up in their kinematic and accretion disk properties. In the present paper, we analyzed thin accretion disk properties around magnetically and electrically charged GMGHS BHs in EF and in SF. The physical parameters describing the disk such as the effective potential, radiation flux, temperature and emissivity profiles have been explicitly obtained for several values of the parameter $Q$, that in turn correspond to several values for dilationic charge $D$ for a given mass $M$. All the astrophysical quantities related to the observable properties of the accretion disk in the two frames have been compared with those for the Schwarzschild BH of the same mass. We considered as a toy model a stellar sized compact object of mass $M=15M_{\\odot}$ with an accretion rate $\\dot{M}_{0}=10^{18}$ gm.sec$^{-1}$ and assumed that its accretion properties can be described by the those of the GMGHS spacetimes including their extreme limits of NS and WHs. Our aim has been to examine whether the kinematic and emissivity properties significantly change when the central object changes.\n\nThe main conclusions of our analyses are as follows: Kinematic properties were already analyzed in Sec.4 with corresponding figures and need not be repeated here. Suffice it to say that at the NS radius, all kinematic quantities diverge, as expected. The emissivity properties of GMGHS BHs do not appreciably differ from those of the Schwarzschild or Reissner-Nordstr\\\"{o}m BH for $Q<\\sqrt{2}M$. This conclusion is in accord with the strong lensing properties of GMGHS BH studied by Bhadra [6]. However, in the extreme limit $Q=\\sqrt{2}M$, the GMGHS BH in the EF yields NS. In the SF, there are two GMGHS solutions, one is electrically charged and the other is magnetically charged. In the extreme limit the former yields NS and the latter yields WHs. These latter types of geometries are interesting in their own right since the existence of NS is associated with the no-hair theorem and a recent work includes accretion properties in the JNW NS [42,43]. Also, the horizonless WHs are seriously considered as candidates for mimicking initial post-merger ring down signals characteristic of BH horizon [55-60]. While the emissivity properties of ordinary GMGHS BHs in EF and in SF do not differ appreciably from those of the Schwarzschild BH, their extreme counterparts studied here show that they differ quite significantly from those of the Schwarzschild BH. These differences provide yet another avenue to distinguish between ordinary and extremal objects.\n\nA very interesting result, qualitatively similar to the one obtained by Torres [36] for boson stars, is presented in Table 2 and correspondingly in Figs.6a-d for our toy model. The table shows the difference with Schwarzschild BH, when NS and WH are concerned. From these, we see that for BHs ($Q<\\sqrt{2}M$), and WHs in the SF ($Q=\\sqrt{2}M$) the spectra decay rapidly for $\\nu >10^{16}$ Hz but for NS ($Q=\\sqrt{2}M$), the luminosity (Figs. 6a,6c) does not decay until a frequency $\\nu >10^{18}$ Hz (for magnetic NS in EF) beyond which it becomes nearly invisible. The same thing happens for electric NS in SF beyond $\\nu>10^{20}$ Hz. In either case, there is almost an \\textit{infinite} increase in observed luminosity compared to nearly invisible WH and BH at the same frequency. Thus NS should be the brightest objects in the sky (like QSOs). This conclusion is supported by the efficiency $\\epsilon =1$ for NS in Table 1. It thus seems that, at the singular radius $r=r_{\\textmd{\\scriptsize{ms}}}=2M$ (first column of the Table, see Figs. 6a,c), all infalling matter is minced into radiation all of which then escape to us.\n\nIt would be of interest to examine the physical reason as to why such rapid decays in the luminosity spectra occur at higher frequencies but it is evident that observable distinctions exist among different objects described by GMGHS. Understandably, a more adequate and realistic model for the central object should include spin. A choice for this purpose could be the spinning Kerr-Sen solution [61] but it is expected that spin might not drastically alter the main conclusions derived here. We keep it as an open problem for future work.\n\n\n\n\n\n\n\n\n\\begin{acknowledgments}\nWe thank an anonymous referee for useful suggestions. The reported study was funded by RFBR according to the research project No. 18-32-00377.\n\\end{acknowledgments}\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThere has been much hope that one might be able to use  AdS\/CFT\n\\cite{AdS\/CFT} to describe the real systems after certain amount of\ndeformations. In fact it has been suggested that the fireball in\nRelativistic Heavy Ion Collision (RHIC)\ncan be explained from dual gravity point of view\n\\cite{SZ,Nastase,SSZ,Aharony}, since the quark-gluon plasma (QGP)\ncreated there are in the strong coupling region \\cite{RHIC-1,RHIC-2}.\nAlthough the YM theory described by the standard AdS\/CFT\nis large-$N_{c}$ ${\\cal N}=4$ SYM theory,\nthere are many attempts to construct models closer to QCD\n\\cite{AdS\/QCD}.\nSUSY is not very relevant in the finite temperature context\nbecause it is broken completely.\n\n\nSince the RHIC fireball is expanding, we need to understand AdS\/CFT\nin time dependent situations. Recently, Janik and Peschanski\n\\cite{Janik, Janik-2} discussed this problem in non-viscous cases.\nThey use the conservation law and conformal invariance together with\nthe holographic renormalization \\cite{HSS,Skend} to express the bulk\ngeometry from the given boundary data. As a result, the bulk\ngeometry reproduces the basic features of Bjorken theory\n\\cite{Bjorken}. Theses results were generalized  to the cases where\nshear viscosity is included \\cite{sin-shin}. Indeed, it had been\npointed out that inclusion of shear viscosity is very important in\nthe analysis of real RHIC physics since it plays an essential role\nin the elliptic flow (see for example,\n\\cite{KH,bulk-zero,Teaney,Kolb,Hirano-Heinz,Zhang}). The shear\nviscosity at the strong coupling limit was calculated for the ${\\cal\nN}=4$ SYM systems in \\cite{PSS-1} using AdS\/CFT, which fits the\nperfect fluidity of RHIC QGP (see \\cite{RHIC-2,bulk-zero,Hirano} and\nthe references therein).\n\n\nIn this paper, we consider a holographic dual of strongly interacting\n${\\cal N}=4$ large-$N_{c}$ SYM fluid with non-isotropic\nthree-dimensional expansion which is relevant to\n``Little Bang'' of RHIC.\nWe will first make a simplest generalization of the Bjorken's\none-dimensional expansion to three-dimensional cases.\nThe resulting local rest frame of the fluid is described by\nthe Kasner metric\nwhose extreme limit reproduces the Bjorken hydrodynamics.\nInterestingly, the hydrodynamic equation and the\nequation of state will be shown to be independent of\nthe non-isotropy parameters.\n\nFurther more, using holographic\nrenormalization, we will establish the holographic dual\ngeometry of the anisotropically expanding fluid in the Kasner\nspacetime and discuss its physical consequences by extending\nthe work of \\cite{Janik,sin-shin}.\nInterestingly, the gravity dual carries more information than\nthe hydrodynamics which is our input.\nThe integration constants coming from\nthe hydrodynamic equation cannot be determined by the hydrodynamics\nitself. However, we will show that such quantities can be determined\nby considering the dual geometry. Therefore it provides a method to\nextract useful information on the macroscopic properties of the\nexpanding fluid in terms of microscopic data.\n\nKasner spacetime is a curved spacetime in general.\nHowever, we find that the Kasner spacetime\ncan be a well-controled approximation of a local rest frame\nof an anisotropically expanding fluid on Minkowski spacetime.\nTherefore, the Kasner spacetime provides a useful framework\nfor analysing the three-dimensional expansion because of\nthe simplification of the hydrodynamic equations.\n\n\nThe organization of the paper is the following. In Section\n\\ref{hydro},\nwe introduce Kasner spacetime as a local rest frame of a fluid\nunder three-dimensional anisotropic expansion.\nWe also establish the hydrodynamics based on that local rest\nframe. Section \\ref{gravity} gives analyses in the gravity\ndual in the late time regime.\nWe extend the results of \\cite{Janik,sin-shin} to the\ncase of our interest: the three-dimensional anisotropic expansion\nwith shear viscosity.\nWe show that the dual geometry determines some of the hydrodynamic\nquantities in terms of the initial condition and the fundamental\nconstants. In Section \\ref{flow} we show that the hydrodynamics\non the Kasner spacetime\n(and hence its gravity dual) describes\nan elliptic flow on the flat spacetime\nwithin a well-controlled approximation.\nWe analyze the properties of the flow.\nWe conclude in the final section.\nThe definition of the approximation and its justification are\ngiven in appendix.\n\n\n\\section{Hydrodynamics in Kasner space and anisotropic expansion \\label{hydro}}\n\nThe remarkable success of Bjorken's hydrodynamics needs to be\nextended for more realistic cases of\nthree-dimensional expansions. The Bjorken's basic assumption is\nexistence of the so-called ``central rapidity region'' (CPR). The\nlocal rest frame of the fluid can be given by\npropertime($\\tau$)-rapidity($y$), whose relationship with the\ncartesian coordinate is\n$(X^{0},X^{1},X^{2},X^{3})=(\\tau \\cosh y,\\tau \\sinh y,X^{2},X^{3})$.\nWe have chosen the collision axis to\nbe in the $x^{1}$ direction. The Minkowski metric in this coordinate\nhas the form\n\\begin{eqnarray}\nds^{2}=-d\\tau^{2}+\\tau^{2}dy^{2}+(dX^{2})^{2}+(dX^{3})^{2}.\n\\label{g00}\n\\end{eqnarray}\nOur starting point is an observation that the above metric describes\na one-dimensional Hubble expansion, i.e., an expansion of the\nuniverse instead of that of the fluid, in which the rapidity plays\nthe role of the co-moving coordinate. The hydrodynamic equations can\nbe derived from the covariant conservation of the energy-momentum\ntensor with the above metric.\n\nThe real expansion in RHIC is not one-dimensional but a\nthree-dimensional one, although the dominant flow is along the\ncollision axis. The idea is that since the one-dimensional\napproximation of RHIC fireball expansion is seen as a Hubble flow, a\nthree-dimensional Hubble flow may describe the RHIC plasma better.\nLet us begin with an ansatz for a local rest frame given by\n\\begin{eqnarray}\nds^{2}=-d\\tau^{2}+\\tau^{2a}(dx^{1})^{2}\n+\\tau^{2b}(dx^{2})^{2}+\\tau^{2c}(dx^{3})^{2}.\n\\label{kasner}\n\\end{eqnarray}\nHere $x^{i},i=1, 2, 3$ denote the co-moving coordinates.\nThe $a, b$ and $c$ are arbitrary constants for a moment. However, as\nwe will show in Section \\ref{gravity}, they satisfy \\begin{equation}\na+b+c=1,\\quad a^2+b^2+c^2=1, \\label{bdeq} \\end{equation} if we impose conformal\ninvariance of the fluid. Under the above conditions, the metric is\ncalled Kasner metric describing a homogeneous but anisotropic\nexpansion of the Universe. In short, we identify the ``Little Bang''\nin ``SYM version of RHIC'' with a Big Bang with the homogeneous but\nanisotropic expansion described by the Kasner metric.\nWe use this metric in the late time regime\nbecause we describe only the late time\nevolution of the fluid where hydrodynamics is valid.\nTherefore the initial\nsingularity of the metric is not a relevant feature for us. Since we\ntake $x^1$ as the longitudinal direction, a realistic set up is to\nchoose\n$a\\sim 1$ and $b,c \\sim 0$ (see section \\ref{flow}).\n\nWe first establish the hydrodynamics in the Kasner metric.%\n\\footnote{We use the late time approximation, which is employed in\n\\cite{sin-shin}, where the macroscopic quantities are assumed to\nevolve sufficiently slowly. } We assume that the expansion is\nanisotropic but homogeneous, and the physical quantities depend only\non $\\tau$. Using the fact that the energy-momentum tensor is\ndiagonal on the local rest frame, and using the symmetry in transverse\ncoordinates, we can write \\begin{equation} T^{\\mu}_{~~\\nu}={\\rm\ndiag}(-\\rho,f_1,f_2,f_3), \\label{general} \\end{equation} where $\\rho$ is the\nenergy density of the fluid. By use of the conservation law $\n\\nabla_{\\mu}T^{\\mu\\nu}=0$ we get \\begin{equation} {\\dot \\rho}+\\rho\/\\tau+\\sum_i\na_if_i\/\\tau=0,\\label{consv} \\end{equation} and from the conformal invariance\n$T^{\\mu}_{\\mu}=0$, we get \\begin{equation} -\\rho+\\sum f_i=0. \\label{cfinv} \\end{equation} To\nget the above results, we have used the following non-zero\n$\\Gamma^{\\mu}_{\\alpha\\beta}$'s: \\begin{equation} \\Gamma^{\\tau}_{11}=a\\tau^{2a-1},\n\\quad \\Gamma^{\\tau}_{22}=b\\tau^{2b-1},\\quad\n\\Gamma^{\\tau}_{33}=c\\tau^{2c-1} , \\quad \\Gamma^{1}_{1\\tau}=a\/\\tau,\n\\quad \\Gamma^{2}_{2\\tau}=b\/\\tau, \\quad \\Gamma^{1}_{3\\tau}=c\/\\tau.\n\\end{equation}\n\nOn the other hand, the energy-momentum tensor in the framework of relativistic hydrodynamics is known to be\n\\begin{eqnarray}\nT^{\\mu\\nu}=(\\rho+P)u^{\\mu}u^{\\nu}+Pg^{\\mu\\nu}+\\tau^{\\mu\\nu},\n\\label{T-ideal}\n\\end{eqnarray}\nwhere $P$ is the pressure of the fluid, $u^{\\mu}=(\\gamma, \\gamma\n\\vec{v})$ is the four-velocity field in terms of the local fluid\nvelocity $\\vec{v}$, and  $\\tau^{\\mu\\nu}$ is the dissipative term. In\na frame where the energy three-flux vanishes,\n$\\tau^{\\mu\\nu}$ is given in terms of the bulk viscosity\n$\\xi$ and the shear viscosity $\\eta$ by\n\\begin{eqnarray}\n\\tau^{\\mu\\nu}\n=-\\eta\n(\\bigtriangleup^{\\mu\\lambda}\\nabla_{\\lambda}u^{\\nu}\n+\\bigtriangleup^{\\nu\\lambda}\\nabla_{\\lambda}u^{\\mu}\n-\\frac{2}{3}\\bigtriangleup^{\\mu\\nu}\\nabla_{\\lambda}u^{\\lambda})\n-\\xi \\bigtriangleup^{\\mu\\nu}\\nabla_{\\lambda}u^{\\lambda},\n\\label{T-dissp}\n\\end{eqnarray}\nunder the assumption that $\\tau^{\\mu\\nu}$ is of first order in\ngradients. We have defined the three-frame projection tensor as\n$\\bigtriangleup^{\\mu\\nu}=g^{\\mu\\nu}+u^{\\mu}u^{\\nu}$.\nFor the conformal invariance, we set $\\xi=0$.\nNotice that the bulk viscosity in the realistic RHIC setup is\nalso negligible. (See for example, Ref. \\cite{bulk-zero}.)\n\nThe four-velocity of the fluid at any point in  the local\nrest frame is\n$u^{\\mu}=(1,0,0,0)$, and this makes the energy-momentum tensor\ndiagonal. Using $\\Delta^{\\mu\\nu}={\\rm\ndiag}(0,\\tau^{-2a},\\tau^{-2b},\\tau^{-2c})$, $\\nabla_\\lambda\nu^\\nu=\\Gamma^\\nu_{\\lambda 0}={\\rm diag}(0,a\/\\tau,b\/\\tau,c\/\\tau)$ and\n$\\nabla_\\nu u^\\nu =(a+b+c)\/\\tau$, we get the mixed energy-momentum\ntensor:\n\\begin{eqnarray}\nT^{\\mu}_{~~\\nu}\n=\n\\left(\n  \\begin{array}{cccc}\n  - \\rho&   0&   0&   0\\\\\n      0&  P-\\frac{2}{3}(3a-1)\\frac{\\eta}{\\tau}\n      &   0&   0\\\\\n      0&   0&\n  P-\\frac{2}{3}(3b-1)\\frac{\\eta}{\\tau} &   0\\\\\n      0&   0&   0&\n       P-\\frac{2}{3}(3c-1)\\frac{\\eta}{\\tau} \\\\\n  \\end{array}\n\\right). \\label{T-diag}\n\\end{eqnarray}\nBy identifying (\\ref{T-diag}) with (\\ref{general}), we obtain\n\\begin{equation}\nf_i= p-(a_i-{1\\over3})\\frac{2\\eta}{\\tau},\\end{equation}\nwhere $a_i=a,b,c$ for $i=1,2,3$ respectively.\nInserting these into (\\ref{consv}) and (\\ref{cfinv}),\nwe get\n\\begin{equation}\n{\\dot \\rho}+{1\\over\\tau}\\rho+\n\\sum_i{a_i\\over\\tau}\\left(p-(a_i-{1\\over 3}){2\\eta\\over\\tau}\\right)=0,\n\\label{consv-2}\n\\end{equation}\nand\n\\begin{equation}\n-\\rho+3p-2\\eta\\left({{\\sum_ia_i-1}\\over\\tau}\\right)=0.\n\\label{cfinv-2}\n\\end{equation}\nUsing the second equation, we may write the first equation in terms\nof the energy density as\n\\begin{equation} {\\dot\n\\rho}+\\left(1+\\frac{1}{3}\\sum_i{a_i} \\right){\\rho\\over\\tau} =\n\\left(\\sum_i a_i^2-\\frac{1}{3}(\\sum_ia_i)^2\\right)\n{2\\eta\\over\\tau^2} . \\end{equation}\nAs we will discuss in Section \\ref{gravity},\n{\\it if we impose the conformal invariance}, we get the conditions\n\\begin{equation}\n\\sum_i a_i=1, \\;\\;\\; \\sum_ia_i^2=1,\n\\label{abc-cond}\n\\end{equation}\nunder which\nthe equation of state and the conservation law become: \\begin{eqnarray}\np&=&{\\rho\\over 3},\\nonumber \\\\\n\\frac{d\\rho}{d\\tau}+\\frac{4}{3}\\frac{\\rho}{\\tau}&=&\\frac{4}{3}\\frac{\\eta\n}{\\tau^2}. \\label{consv2}\\end{eqnarray} Remarkably, the equations for fluid\ndynamics are completely {\\it independent of the parameters $a,b,c$}\nin this case. In fact the dynamical law should not depend on the\ninitial conditions ($a,b$ and $c$) that reflects the initial\ncollision geometry. In this respect, (\\ref{abc-cond})\nleads us a satisfactory consequence.\n\nNotice that both of $\\rho$ and $\\eta$ depend on the proper time\n$\\tau$ in general.\nLet's assume that the shear viscosity evolves by\n$\\eta= {\\eta_{0}}\/{\\tau^{\\beta}},$\nwhere $\\eta_{0}$ is a positive constant.\nThe solution of (\\ref{consv2}) is then given by\n\\begin{eqnarray}\n\\rho(\\tau) &=&\\frac{\\rho_{0}}{\\tau^{4\/3}}\n+\\frac{4\/3}{1\/3-\\beta} \\frac{\\eta_{0}}{\\tau^{1+\\beta}}\n \\:\\:\\:\\:({\\rm for}\\:\\: \\beta\\neq 1\/3),\n \\label{solution}\n \\\\ \\nonumber\n &=&\\frac{\\rho_{0}}{\\tau^{4\/3}}\n+\\frac{4\\eta_{0}}{3} \\frac{\\ln\\tau}{\\tau^{4\/3}}\n\\:\\:\\:\\:({\\rm for}\\:\\: \\beta=1\/3),\n\\end{eqnarray}\nwhere $\\rho_{0}$ is a positive constant.\nFor $\\beta \\leq 1\/3$ case,\nthe viscous corrections in the hydrodynamic quantities become\ndominant in the late time, which invalidates the hydrodynamic\ndescription. If $\\beta > 1\/3$, the shear viscosity term\nis subleading in the late time as we expect.\nTherefore we will consider only the latter case from now on.\n\nThe proper-time dependence of the temperature $T$\ncan be read off by assuming\n the Stefan-Boltzmann's law $\\rho \\propto T^{4}$:\n\\begin{eqnarray}\nT= T_{0}\\left( \\frac{1}{\\tau^{1\/3}} +\n\\frac{1\/3}{1\/3-\\beta}\\frac{\\eta_{0}}{\\rho_{0}}\n\\frac{1}{\\tau^{\\beta}} +\\cdots \\right).\n\\label{T-general}\n\\end{eqnarray}\nIn the {\\em static} finite temperature system of strongly coupled\n${\\cal N}=4$ SYM theory, it is known that $\\eta \\propto T^{3}$\n\\cite{PSS-1}. Let us assume that the result is valid\n in the slowly varying non-static cases.\nThen we can set $\\beta=1$ hence\n\\begin{eqnarray}\n \\eta=\\frac{\\eta_{0}}{\\tau}.\n\\label{beta1}\n\\end{eqnarray}\nWe know $\\rho\\sim T^4$ and\n$\\eta\\sim T^{3}$ can   be consistent only if there is an additional\nterm in (\\ref{beta1}), but the correction term is negligible in our\ncase.\nThe temperature behavior is then given by\n\\begin{eqnarray}\nT=T_{0}\\left( \\frac{1}{\\tau^{1\/3}} -\n\\frac{1}{2}\\frac{\\eta_{0}}{\\rho_{0}} \\frac{1}{\\tau}\n+\\cdots \\right).\n\\end{eqnarray}\n\nWe can evaluate the entropy change in the presence of shear\nviscosity by using hydrodynamics.\nThe conservation of energy-momentum tensor can be rewritten as\n\\begin{eqnarray}\n \\frac{d(\\sqrt{g}\\rho)}{d\\tau}+ \\frac{d\\sqrt{g}}{d\\tau}P =\n \\frac{4}{3}\\frac{\\sqrt{g}\\eta}{\\tau^2}\n, \\label{3deom}\n\\end{eqnarray}\nwhere $\\sqrt{g}=\\tau$ is the volume element in the co-moving coordinate.\nBy integrating over the unit volume in the co-moving coordinate,\nand using the thermodynamic relation between the entropy and energy-work,\n\\begin{equation}\ndE+PdV=TdS,\n\\end{equation}\n(\\ref{3deom}) can be written as\n\\begin{eqnarray}\nT\\frac{d(\\sqrt{g} s)}{d\\tau}=\\frac{4}{3}\\frac{\n\\eta\\sqrt{g}}{\\tau^2},\n\\end{eqnarray}\nwhere $s$ denotes the entropy density and $\\sqrt{g} s \\equiv S$\nis the entropy per unit  co-moving volume.\nNotice that in the absence of viscosity, the entropy per unit\nco-moving volume is constant.\nNow, the entropy per unit co-moving volume has the time dependence:\n\\begin{eqnarray}\nS(\\tau) &=& S_0+ \\frac{4}{3} \\int_0^\\tau d\\tau \\frac{ \\eta\\sqrt{g}}{\n\\tau^{2} T}\n\\nonumber \\\\\n&=& S_\\infty - 2\\frac{\\eta_{0}}{T_{0}}\\tau^{-2\/3} + \\cdots,\n\\label{entropy}\n\\end{eqnarray}\nwhere $S_\\infty = S(\\infty)$. The dissipation creates entropy but\nits rate slows down with time. Notice that all these arguments are\ncompletely in parallel with the case of the one-dimensional\nexpansion.\n\nThe hydrodynamics does not calculate the constants $S_\\infty$ and\n$T_0$ in terms of the initial condition. It is  important to  point\nout that by embedding the hydrodynamics into AdS\/CFT, we can\ndetermine these parameters in terms of the initial condition\n$\\rho_{0}$ and the fundamental constants of the theory.\n\n\\section{Holographic dual of anisotropic expansion }\n\\label{gravity}\n\n\nIn this section, we will find a five-dimensional metric that is dual\nto the hydrodynamic description of the YM fluid in the previous\nsection. Some of the parameters of hydrodynamics will be determined\nas a consequence. The basic strategy is to use the Einstein's\nequation together with the boundary condition given by the\nfour-dimensional energy-momentum tensor \\cite{HSS,Skend,Janik}.\nWe consider general asymptotically AdS metrics in the\nFefferman-Graham coordinate:\n\\begin{eqnarray}\nds^{2}=\nr_{0}^{2}\n\\frac{g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}+dz^{2}}{z^{2}},\n\\label{FG-metric}\n\\end{eqnarray}\nwhere $x^{\\mu}=(\\tau,x^1,x^{2},x^{3})$ in our case.\n$r_{0}\\equiv (4\\pi g_{s} N_{c} \\alpha'^{2})^{1\/4}$\nis the length scale given by the string coupling $g_{s}$ and\nthe number of the colors $N_{c}$.\nThe four-dimensional metric $g_{\\mu\\nu}$ is expanded\nwith respect to $z$ in the following form \\cite{HSS,Skend}:\n\\begin{eqnarray}\ng_{\\mu\\nu}(\\tau, z)=\ng^{(0)}_{\\mu\\nu}(\\tau)+z^{2}g^{(2)}_{\\mu\\nu}(\\tau)\n+z^{4}g^{(4)}_{\\mu\\nu}(\\tau)\n+z^{6}g^{(6)}_{\\mu\\nu}(\\tau)+\\cdots .\n\\label{expansion}\n\\end{eqnarray}\nHere $g^{(0)}_{\\mu\\nu}$ is the physical four-dimensional metric for\nthe gauge theory on the boundary, which is given by (\\ref{kasner}) in\nthe present case. The $g_{\\mu\\nu}^{(n)}$'s depend only on $\\tau$\nbecause our physical quantities are assumed to depend only on\n$\\tau$.\n\n$g^{(2)}_{\\mu\\nu}$ is related to the conformal anomaly of the\nYM theory in the following way \\cite{HSS}:\n\\begin{eqnarray}\n\\langle T^{\\mu}_{\\mu} \\rangle\n=-\\frac{1}{16\\pi G_{5}}\n\\left[\n({\\rm Tr}g^{(2)})^{2}-{\\rm Tr}(g^{(2)})^{2}\n\\right],\n\\label{anomaly}\n\\end{eqnarray}\nwhere $G_{5}$ is the 5d Newton's constant given by\n$G_{5}=8\\pi^{3}\\alpha'^{4}g_{s}^{2}\/r_{0}^{5}$ so that\n\\begin{eqnarray}\n\\frac{4\\pi G_{5}}{r_{0}^{3}}=\\frac{2\\pi^2}{N_c^2}\n\\end{eqnarray}\nin our notation. Since we are dealing with ${\\cal N}=4$ SYM theory\non $R^{1,3}$, we should require the conformal invariance: $\\langle\nT^{\\mu}_{\\mu} \\rangle=0$. The most natural choice is given by\n\\begin{eqnarray}\ng^{(2)}_{\\mu\\nu}=0.\n\\label{g2}\n\\end{eqnarray}\nThis is equivalent to the Ricci flat condition for the\nfour-dimensional metric:\n\\begin{eqnarray}\nR_{\\mu\\nu}=0,\n\\label{ricciflat}\n\\end{eqnarray}\nthrough the relationship \\cite{HSS}\n\\begin{eqnarray}\ng^{(2)}_{\\mu\\nu}\n=\\frac{1}{2}\n\\left(\nR_{\\mu\\nu}-\\frac{1}{6}R g^{(0)}_{\\mu\\nu}\n\\right).\n\\label{g2-g0}\n\\end{eqnarray}\nSince\n\\begin{equation}\nR_{00}=(\\sum_i a_i-\\sum_i a_i^2)\/\\tau^2, \\quad \\quad\nR_{ii}=a_i(\\sum_j a_j-1)\\tau^{2a_i-2},\n\\end{equation}\nthe Ricci flat condition (\\ref{ricciflat}) gives\nthe Kasner condition (\\ref{bdeq}).\n\nOne should notice that we are {\\em not} solving the four-dimensional\nEinstein's equation in the presence of $T_{\\mu\\nu}$. The Kasner\nmetric is not a consequence of the gravitational effect of\n$T_{\\mu\\nu}$, but an effective description of the spacetime\nexpansion satisfying the conformal invariance of strong gauge theory\ninteraction.\n\n\nWe can identify the first non-trivial data in (\\ref{expansion}),\n$g^{(4)}_{\\mu\\nu}$, with the energy-momentum tensor at the boundary\n\\cite{HSS}:\n\\begin{eqnarray}\ng^{(4)}_{\\mu\\nu}\n=\\frac{4\\pi G_{5}}{r_{0}^{3}} \\langle T_{\\mu\\nu} \\rangle,\n\\end{eqnarray}\nin our notation.\nFor the time being, we set\n$4\\pi G_{5}=1$ and $r_{0}=1$.\nThe higher-order terms in (\\ref{expansion}) are determined by\nsolving the Einstein's equation with the negative cosmological\nconstant $\\Lambda=-6$ \\cite{HSS} (see also \\cite{Janik}):\n\\begin{eqnarray}\nR_{MN}-\\frac{1}{2}G_{MN}R-6G_{MN}=0,\n\\label{Eeq}\n\\end{eqnarray}\nwhere the metric and the curvature tensor are the five-dimensional\nones of (\\ref{FG-metric}). $g_{\\mu\\nu}^{(2n)}$ is described by\n$g_{\\mu\\nu}^{(2n-2)}, g_{\\mu\\nu}^{(2n-4)}, \\cdots, g_{\\mu\\nu}^{(0)}$\nthrough the Einstein's equation in five dimensions. In other words,\nwe can obtain the higher-order terms in (\\ref{expansion})\nrecursively by starting with the initial data $g_{\\mu\\nu}^{(0)}$\n($\\sim$Kasner) and $g_{\\mu\\nu}^{(4)}$ ($\\sim T_{\\mu\\nu}$).\n\nLet us come back to our main interest  to obtain\nthe bulk geometry in the presence of shear viscosity.\nThe energy-momentum tensor for $\\beta=1$ is\nwritten by using  (\\ref{solution}) as\n\\begin{eqnarray}\nT^{\\mu}_{\\nu}=\\left(\n  \\begin{array}{cccc}\n    -\\frac{\\rho_{0}}{\\tau^{4\/3}}\n    +\\frac{2\\eta_{0}}{\\tau^{2}}\n       & 0  & 0  & 0  \\\\\n     0 &   \\frac{\\rho_{0}}{3\\tau^{4\/3}}\n    -\\frac{2a\\eta_{0}}{\\tau^{2}}   & 0  & 0  \\\\\n     0 & 0  &  \\frac{\\rho_{0}}{3\\tau^{4\/3}}\n    -\\frac{2b\\eta_{0}}{\\tau^{2}}  & 0  \\\\\n     0 & 0  & 0  &  \\frac{\\rho_{0}}{3\\tau^{4\/3}}\n    -\\frac{2c\\eta_{0}}{\\tau^{2}} \\\\\n  \\end{array}\n\\right).\n\\label{Tij-viscous}\n\\end{eqnarray}\nWe find that the metric components,\n$g_{\\tau\\tau}$, $g_{yy}\/\\tau^{2}$, $g_{xx}$\nhave the following structure by solving the Einstein's equation\nrecursively:\n\\begin{eqnarray}\nf^{(1)}(v)+\\eta_{0}h^{(1)}(v)\/\\tau^{2\/3}+\n{\\tilde f}^{(2)}(v)\/\\tau^{4\/3}+\\cdots ,\n\\label{expansion2}\n\\end{eqnarray}\nNote that {\\it the viscosity dependent terms exist at the order of\n$\\tau^{-2\/3}$ and these are more important than the higher-order\nterms neglected in} \\cite{Janik}. We are considering the late time\nregion $\\tau \\gg 1$. But to see the effects of the viscosity, we\nneed to keep the terms at least to the order of $\\tau^{-2\/3}$. In\nthis paper we consider the viscosity effects to the minimal order.\n\nNow we solve the Einstein's equation recursively.\nThe power series that appear in the solution\ncan be re-summed to give a compact form of the metric.\nAfter some hard work,\nwe  can get the late time 5d bulk geometry given by%\n\\footnote{ One should keep in mind that we are\nlooking for the late time geometry; the metric (\\ref{our-geometry})\nis correct only to the order of $\\gamma$ and the $O(\\gamma^2)$\ncontributions are not unambiguously determined. The representation\nof (\\ref{our-geometry}) is chosen since it makes the volume of the\nhorizon finite.}\n\\begin{eqnarray}\nds^{2}\n&=&\n\\frac{1}{z^{2}}\n\\left\\{\n-\\frac{(1-\\frac{\\rho z^{4}}{3})^{2}}{1+\\frac{\\rho z^{4}}{3}}\nd\\tau^{2}\n+\n\\left(1+\\frac{\\rho z^{4}}{3}\\right)\n\\sum_{i=1}^3\n\\left(\\frac{1+\\frac{\\rho z^{4}}{3}}{1-\\frac{\\rho z^{4}}{3}}\\right)^{(1-3a_i)\\gamma}\n\\tau^{2a_i}(dx^i)^{2}\n\\right\\}+\\frac{dz^{2}}{z^{2}},\n\\label{our-geometry}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\gamma \\equiv \\frac{\\eta_{0}}{\\rho_{0}\\tau^{2\/3}}\n ~~{\\rm and}~~\n\\rho=\n\\frac{\\rho_{0}}{\\tau^{4\/3}}\n-\n\\frac{2\\eta_{0}}{\\tau^{2}}.\n\\label{def-gamma-1}\n\\end{eqnarray}\nNotice that the energy-momentum tensor (\\ref{Tij-viscous}) can NOT\nbe written in terms of the whole $\\rho(\\tau)$. It is an amusing\nsurprise to see that the final metric nevertheless can be written in\nterms of $\\rho(\\tau)$  (apart from the powers) in a compact form.\nThis implies that the position of the horizon can be determined\nsolely by the energy density.\n\n\\subsection{Macroscopic Parameters from Gravity Dual }\nThe Hawking temperature in the adiabatic approximation\nis given by\n\\begin{equation}\nT(\\tau)=\\sqrt{2}\/(\\pi z_{0}(\\tau)),\n\\end{equation}\nwhere\n\\begin{equation}\nz_{0}(\\tau)=[3\/\\rho(\\tau)]^{1\/4}\n\\end{equation}\n is the time-dependent position of the horizon.\n Using  these, we obtain\n\\begin{eqnarray}\n\\rho=\n\\frac{3}{8}\\pi^{2}N_{c}^{2}T^{4}(\\tau),\n\\end{eqnarray}\nby restoring the normalization of $\\rho$: $\\rho\\to 4\\pi G_{5}\/r_0^3 \\cdot \\rho$.\nThe entropy per unit co-moving volume\nis given by\n\\begin{eqnarray}\nS\n&=& \\lim_{z\\to z_0(\\tau)} \\frac{1}{4G_{5}} \\sqrt{\\prod_{i=1}^3\\left(1+\\frac{\\rho z^{4}}{3}\\right)\n\\left(\\frac{1+\\frac{\\rho z^{4}}{3}}{1-\\frac{\\rho z^{4}}{3}}\\right)^{(1-3a_i)\\gamma}\n\\tau^{2a_i} }\\nonumber \\\\\n&=&\\frac{1}{4G_{5}} \\frac{2\\sqrt{2}\\tau r_{0}^{3}}{z_{0}^{3}(\\tau)}\n\\nonumber \\\\\n&=&\nN_c^2 \\frac{\\sqrt{2}}{\\pi} \\left(\\frac{2\\pi^2}{N_{c}^{2}}\\frac{\\rho_0}{3}\\right)^{3\/4}\n\\left(\n1-\\frac{3}{2} \\frac{\\eta_{0}}{\\rho_{0}\\tau^{2\/3}}\n+O(\\tau^{-4\/3})\n\\right).\n\\label{s-evolve-ads}\n\\end{eqnarray}\nNotice that {\\it the entropy is completely independent of geometric\n parameters (i.e., $a_i$).}  Without conformal invariance this result is not guaranteed.\nAs we emphasized in \\cite{sin-shin}, the value of $S$ at\n$\\tau=\\infty$, which cannot be determined by hydrodynamics alone, is\nprecisely determined to be\n\\begin{eqnarray}\nS_{\\infty}\n=N_c^2 \\frac{\\sqrt{2}}{\\pi} \\left(\\frac{2\\pi^2}{N_{c}^{2}}\\frac{\\rho_0}{3}\\right)^{3\/4}\n\\label{Sinfty}\n\\end{eqnarray}\nin terms of the initial condition $\\rho_{0}$ and the microscopic\ngauge theory parameter $N_c$. Similarly, \\begin{equation}\nT(\\tau)=\\frac{T_0}{\\tau^{1\/3}}(1-2\\gamma(\\tau))^{1\/4}, \\quad {\\rm\nwith} ~~\nT_0=\\frac{\\sqrt{2}}{\\pi}\\left(\\frac{2\\pi^2}{N_{c}^{2}}\\frac{\\rho_0}{3}\\right)^{1\/4}.\n\\label{T0} \\end{equation} These parameters $S_\\infty, T_0$  are precisely the\nquantities used in macroscopic theory (hydrodynamics) which should\nbe provided by a microscopic theory like QCD. What we are showing\nhere is that by considering the AdS\/CFT dual of hydrodynamics, one\ncan determine such quantities.\n\nLet us check consistency of (\\ref{s-evolve-ads}) and (\\ref{entropy}).\nApparent time dependence agrees in the leading order.\nIn fact one can do more.\nThe normalized entropy-creation rate from the gravity dual result (\\ref{s-evolve-ads})\n is given by\n\\begin{eqnarray}\n\\frac{1}{S}\\frac{dS}{d\\tau}\n=\\frac{\\eta_{0}}{\\rho_{0}\\tau^{5\/3}}+O(\\tau^{-7\/3})\n\\label{rate-ads},\n\\end{eqnarray}\n and that from hydrodynamics result (\\ref{entropy}) is\n\\begin{eqnarray}\n\\frac{1}{S}\\frac{dS}{d\\tau}\n=\\frac{4}{3}\\frac{\\eta_{0}}{T_{0}S_{\\infty}\\tau^{5\/3}}\n+O(\\tau^{-7\/3}).\n\\label{rate-hydro}\n\\end{eqnarray}\nComparing\n(\\ref{rate-ads}) and (\\ref{rate-hydro}), we obtain\n\\begin{eqnarray}\nS_{\\infty}=\\frac{4}{3}\\frac{\\rho_{0}}{T_{0}}\n=\n\\left. \\frac{4}{3}\\frac{\\rho \\tau}{T}\\right|_{\\tau=\\infty}. \\label{consistency}\n\\end{eqnarray}\nThis is  the  consistency condition that is required to be checked.%\n\\footnote{In fact this is the relationship among the entropy, the\nenergy (per unit co-moving volume) and the temperature obtained by\nthermodynamics at $\\tau=\\infty$ where the system reaches thermal\nequilibrium.} With  use of $S_\\infty$ and $T_0$ given in eqn's\n(\\ref{Sinfty}) and (\\ref{T0}) we can check that the consistency\ncondition (\\ref{consistency}) is indeed satisfied.\n\n\n\\section{Flow of RHIC fireball and Kasner spacetime \\label{flow}}\n\n\nSo far, we have established the gravity dual of\nthe Yang-Mills system in Kasner spacetime.\nNow we would like to suggest a relevance of\nour model to description of the elliptic flow in RHIC experiments.\nWe have seen in section \\ref{hydro} that the hydrodynamic\ndescription of the three-dimensional expansion in the\nKasner spacetime is as simple as that of the Bjorken expansion\nin the flat spacetime.\nTherefore, it is great if we can apply such a simple formalism\nto anisotropically expanding RHIC fireball.\nHowever, from  the realistic point of view, we sacrificed the flatness of\n4d spacetime for the simplicity of the fluid dynamics of 3 dimensional\nexpansion.\nOne immediate question is when and under which condition we can justify it.\nThe hydrodynamics\non the Kasner spacetime provides a well-approximated\n description of a three-dimensional expansion in the flat\nspacetime  if curvature\nwhich is small enough. Notice also that\nthe spacetime symmetry crucial to our problem, which\nis uniformity of the spacetime, is maintained in Kasner spacetime.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=flow1.eps,width=6.0cm}}\n\\caption{\\small The available region of $(b,c)$ is on the ellipse\nbetween $(0,0)$ and $(1,0)$. $(b,c)=(0,0)$ is   Bjorken point}\n\\label{sus}\n\\end{figure}\n\nIn figure 1, we show the allowed region of the anisotropic\nparameters.\n$(b,c)=(0,0)$ corresponds to the Bjorken expansion.\nThe non-zero components of the Riemann tensor of the Kasner\nspacetime are\n\\begin{equation}\nR_{0i0i}=(1-a_i)a_i \\tau^{2a_i-2}, \\quad\nR_{ijij}=a_ia_j\\tau^{2a_i+2a_j-2},\n\\end{equation}\nand the non-flatness is directly related to the distance\nfrom the Bjorken point $(a,b,c)=(1,0,0)$ on the parameter space.\nIn order to keep the deviation from the flat spacetime small,\nwe restrict ourselves within the vicinity of the Bjorken point,\n\\begin{equation}\na\\simeq 1, \\quad b\\simeq 0, \\quad c\\simeq 0,\n\\end{equation}\nthat corresponds to almost central collisions in RHIC.\n\nIn appendix, we show\n what approximation is necessary to reach kasner space\n starting from a flat spacetime.\nThe conditions are:\n1) the fluid is produced by almost central collision (small $b$),\n2) we consider only  the central part of the fluid (small $x^\\perp$), and\n3) we consider only  the late time regime.\n For more detail, see appendix \\ref{appendix}.\n\nWith these limitations in mind,\nlet us consider how the elliptic flow can be described\nwithin our framework.\nWe can choose $b \\ge c$ without any loss of generality.\nBy considering the intersection of the plane and the unit\nsphere in the $a, b, c$ space\nof (\\ref{bdeq}), we can see that\n\\begin{equation}\na>b>-c>0.\n\\end{equation}\nThis means that one of the transverse directions must\n{\\it contract} and the others\nexpand so that the expansion is elliptical whose eccentricity\ngrows and eventually saturate to 1.\n\\footnote{\nThere is no contraction in the realistic RHIC QGP.\nThe contraction in the present model is due to the conformal\ninvariance which is unavoidable for ${\\cal N}=4$ SYM theory.\nNotice that this volume-preserving nature indicates our\n${\\cal N}=4$ SYM plasma is more ``liquid-like'' than the\nRHIC QGP.\n}\n\nThe transverse expansion has a natural interpretation as\nthe elliptic flow, one of the most concrete evidence for the\nstrong nature of the interaction.\nLet $\\varepsilon$ be the eccentricity defined by%\n\\footnote{ The $\\varepsilon$ defined here contains an extra minus\nsign comparing to other literature such as \\cite{KH,bulk-zero}. We\ndefine its positivity by the direction of the $v_2$ evolution. }\n\\begin{equation}\n\\varepsilon=\\frac{\\langle X_2^2-X_3^2 \\rangle_{X}} {\\langle\nX_2^2+X_3^2 \\rangle_{X}},\n\\label{epsion-def}\n\\end{equation} where $X^{i}$ is\nthe cartesian coordinate and\n$\\langle\\cdots\\rangle_{X} \\equiv \\int \\cdots \\rho \\:dX_{2}dX_{3}$.\nOn the other hand, $v_2$, the quantity experimentally characterizing\nthe elliptic flow, is\n defined by (see for example, \\cite{KH,bulk-zero,Zhang})\n\\begin{eqnarray}\n\\frac{1}{N}\\frac{dN}{d\\phi}\n=\n\\frac{1}{2\\pi}\n\\left(\nv_{0}+2v_{2}\\cos(2\\phi)+2v_{4}\\cos(4\\phi)+\\cdots\n\\right),\n\\end{eqnarray}\nwhere $N$ is the number of the partons and $\\phi$ is the angular\ncoordinate on the transverse momentum plane. It can be calculated\nfrom the following identification,\n\\begin{eqnarray}\nv_2=\n\\frac{\\int d^{2}P_{T}\n\\left(\\frac{P_2^2-P_3^2}{P_2^2+P_3^2 }\\right)\n\\frac{dN}{d^{2}P_{T}}}{\\int d^{2}P_{T}\\frac{d N}{d^{2}P_{T}}},\\label{v2def}\n\\end{eqnarray}\nwhere $P^{i}$ is the momentum of the fluid in the $X^{i}$ coordinate\nand $d^{2}P_{T}=dP^{2}dP^{3}$.\nTo consider $v_2$ in the present model, we introduce a\ncoarse-grained (i.e., averaged over a small volume)\nmomentum flow,\n\\begin{equation} P^i=K^i(x,\\tau). \\end{equation} At each fixed time $\\tau$, this can be\nconsidered as a mapping from the $P^i$-space to the\n$x^i$-space.\\footnote{ In the co-moving coordinate, where the fluid is\nat rest, we do not have any flow. So $P^i(x)$ is the coarse-grained\nmomentum field in Minkowski space written as a function of co-moving\ncoordinate. } Now $v_2$ can be expressed as an integral over the\nco-moving coordinates:\n\\begin{equation} {  v}_2= \\frac{\\int dx^{2}dx^{3}\n\\left(\\frac{P_2^2-P_3^2}{P_2^2+P_3^2 }\\right)_x \\rho_N(x)}{\\int\ndx^{2}dx^{3}\\rho_N(x)},\n\\label{v2-comp}\n\\end{equation}\nwhere $\\rho_N(x)=\\frac{dN}{dx^2 dx^3}$ is the particle density\nin the transverse space.\nNotice that the Jacobians cancel out.\n\nTo proceed, we need an explicit expression for $K(x)$. We first\nrelate the coordinates of the Kasner spacetime and the usual\nMinkowski spacetime.\nWe can identify the flat space variable $X^i$ for $i=2, 3$ by\n\\begin{eqnarray}\nX^i \\simeq \\tau^{a_i}x^i ~~{\\rm near}\n~~x^i=0,\n\\label{x-trans}\n\\end{eqnarray}\nwhich is nothing but (\\ref{transform}) under (\\ref{approximation}).\nThen the fluid momentum in the flat space is given by \\begin{equation}\nP^{i} =\\rho  {dX^i\\over d\\tau}=\\rho  \\:a_i\\tau^{a_i-1} x^{i}. \\end{equation}\nHere $\\rho =m\\rho_N$ is a mass density with some proper mass\nparameter $m$. Since we treat it as a constant from now on, it is\nirrelevant in the calculation below.\n\nA few technical remarks are in order:\n\\begin{enumerate}\n \\item One should notice that the fluid momentum\n$P^i(x)$ is different from the individual particle momentum. By\nreplacing the momentum by fluid momentum, we expect a small\ndeviation from the original $v_2$. So one may want to call the final\nexpression by $\\bar  v_2$. However, this is precisely what we should\nhave when we describe the system by hydrodynamics where everything\nis to be defined in the coarse-grained level.\n  \\item Since our approximation is valid in the small $x^{i}$ region,\nthe integrals in (\\ref{v2-comp}) are now defined within\nthe small $x^{i}$ region by introducing a cut off radius.\n\\end{enumerate}\n\nThe resulting $v_2$ can be calculated in our model to yield\n\\begin{equation} {\nv}_2=\\frac{b\\tau^{b}+ c\\tau^{c} }{b\\tau^{b}- c\\tau^{c}},\n\\label{v2}\n\\end{equation}\nfor $b>0, c<0$. Comparing this with the result for the\neccentricity in the small $x^{i}$ region\n\\begin{equation} \\varepsilon=\\frac{\n\\tau^{2b} -\\tau^{2c} }{ \\tau^{2b}+ \\tau^{2c}},\n\\label{ecc}\n\\end{equation}\nwe plot the time evolution of $v_2$ in figure 2.\n\nNotice that the hydrodynamics describes relatively late time\nregime and we do not consider the time region where $v_2$ is\nnegative.\nAt the time of  zero $v_2$, $\\varepsilon $ is negative\nand its absolute value decreases\nto zero. That is, the fireball core becomes more round in the\ntransverse space.\nAfter that initial period, $\\varepsilon$ becomes positive\nand grows in the same direction of growing $v_2$.\nThis is qualitatively the same behaviour as those given in\n\\cite{KH,bulk-zero} and \\cite{Zhang}.\nNotice that there is only one independent parameter $b$\nthat can be used to parameterize the initial eccentricity.\n\n\\begin{figure}\n\\centerline{\\epsfig{file=flow2.eps,width=6.0cm}}\n\\caption{\\small Time evolution of the elliptic flow parameters.\nDotted line is the eccentricity and the solid line is the $v_2$ for\n$b=0.1$.}\n\\label{sus}\n\\end{figure}\n\n\\section{Discussion}\nIn this paper, we extended the Bjorken hydrodynamics to the case of\nanisotropic three-dimensional expansion. The Ricci flat condition\nsuggested by the conformal invariance imposes the condition on the\nanisotropy parameters. As a consequence, the four-dimensional\nboundary metric becomes precisely that of Kasner universe.\nOur hydrodynamic equation of motion is independent\nof the anisotropic parameters, so that it is the\nsame as that obtained by Bjorken.\n\nAlthough the Kasner spacetime is a curved spacetime, we found\nthat it gives a well-approximated local rest frame\nof an anisotropically expanding fluid on the flat spacetime.\nWe obtained the eccentricity and $v_{2}$ of the elliptic flow\nof the fluid.\nThe expansion in this setting\nhas a deviation from that of realistic RHIC fireballs,\nsince one of the\ntransverse direction contracts as a consequence of the\nKasner condition. This deviation is essentially due to\nthe conformal invariance. Whether one can lessen the condition\nof the conformal invariance is an important issue,\nwhich we will treat in other publication.\n\n\nWe also extended the ``falling horizon solution'' obtained in\n\\cite{Janik,sin-shin} to the case of three-dimensionally expanding\nfluid. It is very important to figure out how to use the bulk metric\napart from reading off the horizon location, which gives the cooling\nrate and the entropy creation rate. It is also interesting to\nworkout the details of the Hawking evaporation and Wilson line\ncalculations for the external quark-antiquark in our time-dependent\nmetric. We will come back to these issues in later publication.\n\n\n\\noindent\n{\\large\\bf Acknowledgments}\\\\\nWe thank T. Hirano and E. Shuryak for valuable comments\n on the manuscript and interesting suggestions.\nS.N. thanks the Yukawa Institute for Theoretical Physics at Kyoto University.\nDiscussions during the YITP workshop YITP-W-99-99 on\n``thermal quantum field theories and their applications''\nwere useful to improve the section 4.\nThis work was supported by\nthe SRC Program of the KOSEF through the Center for Quantum\nSpace-time(CQUeST) of Sogang University with grant number R11 - 2005\n-021 and also by KOSEF Grant R01-2004-000-10520-0.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe high-sensitivity data provided by visible-light space-based transit surveys, designed to detect the minute decrease in flux as a planet transits across its host star, can be used to look for variability along the entire orbital phase of a star-planet system. Beyond the transit, the phase curve includes the secondary eclipse, when the planet's day-side hemisphere is occulted by the host star, and sinusoidal brightness modulations across the orbital phase. \n\nWhile the transit depth is sensitive primarily to the planet-star radius ratio, the secondary eclipse depth is determined by the planet's thermal emission and the geometric albedo in the observed bandpass. Modulations along the orbital phase, in visible light, are shaped by the gravitational interaction between the star and planet, as well as the longitudinal variations of the planet's brightness. \n\nMore specifically, the shape of the measured phase curve is a superposition of the effects of four main processes, described briefly and somewhat simplistically below, where we assume the transit to be at zero orbital phase, the orbit to be circular, and the planet's rotation to be synchronized with the orbit (i.e., tidally locked), as expected for short period systems \\citep{mazeh2008}:\n(1) Beaming, or Doppler boosting, where the periodic red- and blue-shifting of the host star's spectrum observed in the bandpass follows its orbital motion around the system's center of mass \\citep[e.g.,][]{shakura1987, loeb2003, zucker2007, shporer2010}. The shape of the photometric variability reflects the orbital radial velocity (RV) curve, albeit with the opposite sign, and therefore can be described as a sine at the orbital period.\n(2) Tidal distortion of the host star by the planet \\citep[e.g.,][]{morris1985, morris1993, pfahl2008, jackson2012}, which leads to a cosine modulation at the first harmonic of the orbital period, commonly referred to as ellipsoidal modulation.\n(3) Thermal emission from the planet's atmosphere, where a tidally-locked planet's day-side hemisphere (facing the star) is hotter than the planet's night-side hemisphere (facing away from the star), resulting in a cosine modulation at the orbital period. This process dominates the phase curve modulations in the near-infrared \\citep[e.g.,][]{knutson2012, wong2015, wong2016}, but for highly irradiated planets it can also be detected at visible wavelengths \\citep{snellen2009}.\n(4) Stellar light reflected by the planet's atmosphere, which due to the geometric configuration of the system reaches maximum at the phase of secondary eclipse and minimum at mid-transit, producing a cosine variation at the orbital period \\citep[e.g.,][]{jenkins2003, shporer2015}. Both the thermal emission and reflected light modulations (processes 3 and 4 above) are expected to have the same schematic shape, but a different amplitude. Hence, when combined they are commonly referred to as the atmospheric phase component \\citep{parmentier2017}. \n\nThe summary above shows that the orbital phase curve is sensitive to the star-planet mass ratio and characteristics of the planet's atmosphere, including the geometric albedo and thermal emission, along with their longitudinal distribution. For a review of visible-light orbital phase curves, see \\citealt{shporer2017} and references therein. \n\nWe present here our analysis of the {\\it TESS}\\ orbital phase curve of WASP-18b\\ (TIC  100100827, TOI 185, \\citealt{hellier2009, southworth2009}). This massive 10.4~\\ensuremath{M_{\\rm Jup}}\\ gas giant planet orbits its host star at a very short orbital period of 0.94 days, which establishes favorable conditions for a strong phase curve signal at visible wavelengths and allows us to study the atmosphere of a gas giant planet with high surface gravity subjected to high stellar irradiation. Our analysis of the visible-light orbital phase curve adds to previous measurements in the near-infrared of the phase curve \\citep{maxted2013} and the secondary eclipse \\citep{nymeyer2011, maxted2013, sheppard2017, arcangeli2018}.\n\nThe {\\it TESS}\\ observations are described in \\secr{obs}, and our data analysis is described in \\secr{dataanal}. We present our results in \\secr{res} and discuss their implications in \\secr{dis}. We conclude with a brief summary in \\secr{sum}.\n\n\n\\section{Observations}\n\\label{sec:obs}\n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{stamps.eps}\n\\caption{Stacked {\\it TESS}\\ images of the 11$\\times$11~pixel stamps centered around WASP-18 (red cross) for Sectors 2 and 3. The photometric extraction aperture used by the SPOC pipeline is outlined in red. The positions of two nearby $T\\sim 12.5$ mag stars, which have been deblended by the SPOC pipeline, are denoted with blue crosses: TIC 100100823 (A) and TIC 100100829 (B).\n}\n\\label{fig:stamp}\n\\end{figure}\n\nThe WASP-18 system was observed by Camera 2 of the {\\it TESS}\\ spacecraft during Sector 2 (from 2018 August 22 to 2018 September 20) and Sector 3 (from 2018 September 20 to 2018 October 18). WASP-18 is listed in the {\\it TESS}\\ input catalog (TIC; \\citealt{stassun2018}) as ID 100100827 and included in the list of pre-selected target stars, which are observed with a 2-minute cadence using an 11$\\times$11 pixel subarray centered on the target. The photometric data were processed through the Science Processing Operations Center (SPOC) pipeline \\citep{jenkins2016}, hosted at the NASA Ames Research Center, which is largely based on the predecessor \\textit{Kepler} mission pipeline \\citep{jenkins2010, jenkins2017}. The stacked {\\it TESS}\\ subarrays produced by the SPOC pipeline are shown in Figure~\\ref{fig:stamp}. Outlined in red are the optimal apertures used to extract the WASP-18 light curve in each sector.\n\nFor the results presented in this paper, we use the Presearch Data Conditioning (PDC, \\citealt{smith2012, stumpe2014}) light curves from the SPOC pipeline. The data files include quality flags that indicate when photometric measurements may have been affected by non-nominal operating conditions on the spacecraft or may otherwise yield unreliable flux values. Most of the flagged points occur in the vicinity of momentum dumps, when the spacecraft thrusters are engaged to reset the onboard reaction wheels. Momentum dumps occurred every 2--2.5 days and lasted up to about half an hour. Data taken during or near these momentum dumps typically display anomalous fluxes in the raw photometric time series. We also note that a large portion of data taken during Sector 3 observations suffered from poor pointing and other non-nominal instrumental behavior and were assigned NaN flux values by the SPOC pipeline; these points were removed, resulting in shorter segments of usable data in the two physical orbits of Sector 3.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\linewidth]{plot1.pdf}\n\\caption{Plot of the median-normalized and outlier-removed Presearch Data Conditioning (PDC) Sector 2 and 3 light curves of the WASP-18 system. The gaps in the middle of each sector's time series separate the two physical orbits of the {\\it TESS}\\ spacecraft contained within the sector. We have removed 30 minutes of data from the start of each orbit's light curve. The points highlighted in red show severe residual uncorrected systematics in the fits and are removed in the joint analysis presented in this paper. The phase curve variation of the system is clearly visible, as are residual long-term trends in the data. Those trends can include long-term photometric variability of the target or systematics introduced by the instrument and\/or analysis.}\n\\label{fig:lightcurves}\n\\end{figure*}\n\nWe remove all flagged data points from the light curves. Then, we apply a moving median filter to the photometric time series with a width of 16 data points and remove \\sig{3} outliers, while masking out the transits and secondary eclipses. We also remove the first 30 minutes of data from each orbit's light curve as these segments display residual ramp-like systematic artifacts. The resulting median-normalized light curves are shown in Figure~\\ref{fig:lightcurves}. Even without detrending, the orbital phase variation is clearly discernible, as are the transits and, upon closer inspection, the secondary eclipses. The gaps in the middle of each sector's time series is due to the data downlink, separating the two physical orbits within each {\\it TESS}\\ Sector.\n\nWhen fitting each orbit's light curve separately, several sections of the Sector 3 data display obvious features in the residuals that are not well-corrected by the systematics model we use (Section~\\ref{subsec:systematics}). These sections occur before momentum dumps and at the end of an orbit's light curve. In the final light curve fits presented in this paper, we have carefully inspected the individual orbit light curve fits and removed regions with strong discernible uncorrected systematics. In the first orbit of Sector 3, we remove all points after BJD = 2,458,392, while in the second orbit, we remove 0.75~days worth of data before the last three momentum dumps and 0.5~days worth of data from the end of the time series. These regions are highlighted in red in Figure~\\ref{fig:lightcurves}.\n\nFor Sector 2 data, trimming and outlier filtering remove 1.3\\% and 1.6\\% of the data from the two orbits, respectively. The additional removal of regions in Sector 3 data with uncorrected residuals entails a significantly higher percentage of removed points in the final Sector 3 light curves used in the joint analysis: 37\\% and 29\\% for the two orbits, respectively.\n\nIn addition to the phase curve modulation, there are clear residual long-term trends in the light curves at the level of several hundreds of parts per million (ppm). Previous analyses of {\\it TESS}\\ transit light curves have corrected for these and other flux variations by fitting a basis spline across the out-of-occultation (transit and secondary eclipse) light curve \\citep{huang2018,vanderspek2019} or using a Gaussian Process (GP) model \\citep{wang2019}, thereby removing all non-transit variability. Since such methods would remove the astrophysical phase variations of interest here, we do not utilize them and instead define a detrending model in our fits (Section~\\ref{subsec:systematics}).\n\nWe have also carried out a parallel analysis using the Simple Aperture Photometry (SAP) light curves, which are not corrected for systematics by the SPOC pipeline. One particular characteristic of these data that is not manifested in the PDC light curves is significant flux ramps and periods of increased photometric scatter lasting up to a day at the start of each orbit's photometric time series and preceding each momentum dump. These features are not adequately detrended by the polynomial model we use in this work (Section~\\ref{subsec:systematics}), so we choose to trim one day worth of data prior to each momentum dump, as well as the first day of data for each orbit. All in all, this removes almost 40\\% of the SAP time series from the phase curve analysis. \n\nThe best-fit astrophysical parameters from our analysis of SAP light curves are consistent with the results from the PDC light curves at much better than the \\sig{1} level, with most parameters lying within 0.1-0.2$\\sigma$. Meanwhile, the data trimming and larger red noise in the SAP light curves lead to parameter estimate uncertainties that are as much as 100\\% larger than those derived from fitting the PDC light curves. Given the demonstrated consistency between the PDC and SAP light curve analyses and the poorer quality of the latter, we have decided to present the results from our PDC light curve analysis in this paper.\n\n\n\\section{Data Analysis}\n\\label{sec:dataanal}\n\nIn this work, we utilize the ExoTEP pipeline to analyze the {\\it TESS}\\ PDC light curves for the WASP-18 system. ExoTEP is a highly-modular Python-based tool in development for extracting and analyzing all types of time series photometric datasets of relevance in exoplanet science --- primary transit and secondary eclipse light curves and full-orbit phase curves. The pipeline allows the user to execute joint fits of datasets from multiple instruments (e.g., \\textit{Kepler}, \\textit{Hubble}, \\textit{Spitzer}, and {\\it TESS}) and customize the handling of limb-darkening and systematics models in a self-consistent way. So far, the main application of the ExoTEP pipeline has been in transmission spectroscopy \\citep[e.g.,][see first reference for a detailed technical description]{wong2018,benneke2018,chachan2018}. This work is the first application of the ExoTEP pipeline to time series photometry that includes orbital phase curves.\n\n\\subsection{Transit and eclipse model}\n\\label{subsec:transit}\n\nExoTEP models both transits and secondary eclipses using the BATMAN package \\citep{kreidberg2015}. In our analysis, we fit for the planet-star radius ratio $R_{p}\/R_{s}$ and the relative brightness of the planet's day-side hemisphere $f_{p}$, which determine the planetary transit and secondary eclipse depths, respectively. In order to obtain updated values, we allow the transit geometry parameters --- impact parameter $b$ and scaled orbital semi-major axis $a\/R_{s}$ --- to vary and fit for a new transit ephemeris --- specific mid-transit time $T_{0}$ and orbital period $P$. The zeroth epoch, to which we assign $T_{0}$, is designated to be the transit event closest to the center of the combined time series.\n\nFor the transit model, we use a standard quadratic limb-darkening law and fix the coefficients to values calculated for the {\\it TESS}\\ bandpass by \\citet{claret2017}. Assuming the stellar properties for WASP-18 listed in \\citet{stassun2017} and \\citet{torres2012} ($T_{\\mathrm{eff}}=6431\\pm48$~K, $\\log g=4.47\\pm0.13$, $\\mathrm{[Fe\/H]}=0.11\\pm0.08$; see also Table~\\ref{tab:knownparams}), we take the coefficient values tabulated for the nearest-neighbor set of stellar properties ($T_{\\mathrm{eff}}=6500$~K, $\\log g=4.50$, $\\mathrm{[Fe\/H]}=0.1$): $u_{1}=0.2192$ and $u_{2}=0.3127$. \nUtilizing coefficients for other similar combinations of stellar properties does not significantly affect the results of our fit, yielding changes to the fitted parameter values of at most \\sig{0.2}.\nWhen experimenting with fitting for the quadratic limb-darkening coefficients, we obtain modestly constrained estimates that differ from the \\citet{claret2017} coefficient values at the 3.1--3.4$\\sigma$ level, while likewise maintaining the other fitted astrophysical parameter estimates at statistically consistent values. On the other hand, fitting for a single coefficient assuming a linear limb-darkening law yields estimates that differ significantly from the modeled values in \\citet{claret2017} and introduces strong correlations between the limb-darkening coefficient and the transit depth and transit geometry parameters.\n\nIn the fits presented here, we fix the orbital eccentricity $e$ and argument of periastron $\\omega$ to the values obtained by \\citet{nymeyer2011}: $e=0.0091$ and $\\omega=269^{\\circ}$. For further discussion of orbital eccentricity, see Section~\\ref{sec:res}.\n\n\n\n\\begin{deluxetable}{lllc}\n\\tablewidth{0pc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{\n    Known Parameters\n    \\label{tab:knownparams}\n}\n\\tablehead{\n    \\multicolumn{1}{c}{~~~~~~~~Parameter~~~~~~~~} &\n    \\multicolumn{1}{c}{Value}                     &\n    \\multicolumn{1}{c}{Error}    &\n    \\multicolumn{1}{c}{Source}    \n}\n\\startdata\n$\\ensuremath{T_{\\rm eff}}$ (K)  \\dotfill  & 6431 & 48 & \\cite{stassun2017} \\\\\n$\\ensuremath{R_s}$ (\\ensuremath{R_\\sun}) \\dotfill  & 1.26 & 0.04 & Stassun et~al., in prep.\\tablenotemark{a}\\\\\n$\\ensuremath{M_s}$ (\\ensuremath{M_\\sun}) \\dotfill  & 1.46 & 0.29 & \\cite{stassun2017} \\\\\n$\\log g$ \\dotfill & 4.47 & 0.13 & \\cite{stassun2017} \\\\\n${\\rm [Fe\/H]}$ \\dotfill & 0.11 & 0.08 & \\cite{torres2012} \\\\\n$K_{RV}$ (\\ensuremath{\\rm m\\,s^{-1}})   \\dotfill  & 1816.6 & $_{-6.3}^{+6.1}$ & \\cite{knutson2014}  \n\\enddata\n\\tablenotetext{a}{Stellar radius is derived using Gaia DR2 data.}\n\\end{deluxetable}\n\n\\subsection{Phase curve model}\n\\label{subsec:phase}\n\nWe model the out-of-occultation variation of the system brightness as a third-order harmonic series in phase \\citep[e.g.,][]{carter2011}:\n\\begin{equation}\\label{eq:prefull}\n\\psi'(t) = 1+\\bar{f_{p}}+\\sum\\limits_{k=1}^{3}A_{k}\\sin(k\\phi(t))+\\sum\\limits_{k=1}^{3}B_{k}\\cos(k\\phi(t)).\\end{equation}\nHere, we have normalized the flux such that the average brightness of the star alone is unity. The baseline relative planetary brightness $\\bar{f_{p}}$ is the average of the planet's apparent flux across its orbit. The phase function $\\phi(t)$ is derived from the time series via the relation $\\phi(t) = 2\\pi(t-T_{0})\/P$, where $t$ is time.\n\nSeveral of the harmonic terms in the phase curve model are attributed to various physical processes on the star or planet. The star's brightness is modulated by the beaming effect and ellipsoidal variation. These two processes produce phase curve signals at the fundamental of the sine ($A_{1}$) and the first harmonic of the cosine ($B_{2}$), respectively. A tidally-locked hot Jupiter has a fixed day-side hemisphere facing the star, which produces a variation in its apparent brightness due to the changing viewing geometry. This atmospheric brightness component produces a signal at the fundamental of the cosine ($B_{1}$). See the discussion in Section~\\ref{sec:dis} for more details concerning the astrophysical implications of the phase curve terms.\n\nWe can separate the total system phase curve model into terms describing the star's brightness variation $\\psi_{*}(t)$ and terms describing the planet's brightness variation $\\psi_{p}(t)$:\n\\begin{align}\n\\psi_{p}(t) & = f_{p}-|B_{1}|+B_{1}\\cos(\\phi(t)), \\\\\n\\psi_{*}(t) & = 1 + \\sum\\limits_{k=1}^{3}A_{k}\\sin(k\\phi(t))+\\sum\\limits_{k=2}^{3}B_{k}\\cos(k\\phi(t)).\n\\end{align}\nWe have assigned all terms without a direct corresponding physical process to the star's phase modulation. These other terms can become significant if there are discernible phase shifts in the aforementioned modulation signals. We have also used the fact that the average brightness of the planet is the maximum measured brightness, which occurs at secondary eclipse, minus the semi-amplitude of the atmospheric variation: $\\bar{f_{p}}\\equiv f_{p}-|B_{1}|$. \n\nThe separation of stellar and planetary phase curve terms is important when including the transit and eclipse light curves $\\lambda_{t}(t)$ and $\\lambda_{e}(t)$, since only the brightness modulation of the occulted region on the star or planet is removed from the total system flux. This correction is particularly consequential during secondary eclipse, when the atmospheric brightness component is completely blocked by the star ($\\lambda_{e}=0$), while the ellipsoidal and beaming modulations on the star are unaffected ($\\lambda_{t}=1$). From here, we can write down the full phase curve model, including eclipses (transit and secondary eclipse) and re-normalized such that the average out-of-eclipses flux is unity:\n\\begin{equation}\\label{full}\n\\psi(t) = \\frac{\\lambda_{t}(t)\\psi_{*}(t)+\\lambda_{e}(t)\\psi_{p}(t)}{1+f_{p}-|B_{1}|}.\n\\end{equation}\n\nWhen compared to the standard approach in the literature, which simply multiplies the system phase curve model in \\eqr{prefull} and the occultation light curves together \\citep[e.g.,][]{carter2011}, this more detailed and physical model deviates most significantly during the ingress and egress of secondary eclipse, where the discrepancy for the WASP-18 system can be as large as several tens of ppm. While this level of model discrepancy is well within the noise of the light curves analyzed in this work, studies with higher signal-to-noise photometry would benefit from our more careful treatment of the phase curve model.\n\n\\subsection{Modeling long-term trends}\n\\label{subsec:systematics}\n\nAs can be seen in Figure~\\ref{fig:lightcurves}, the {\\it TESS}\\ PDC light curves from the SPOC pipeline show long-term trends at the level of several hundreds of ppm. Those can be caused by low-frequency residual systematics and\/or long-term stellar variability. \n \nFor the detrending model, we use a polynomial in time of the form\n\\begin{equation} \nS^{\\lbrace i\\rbrace}_{n}(t) = \\sum\\limits_{k=0}^{n}c^{\\lbrace i\\rbrace}_{k}(t-t_{0})^{k},\n\\end{equation}\nwhere $t_{0}$ is the first time stamp in the light curve from orbit $i$, and $n$ is the order of the polynomial model. In the following, we assign $i=1,2$ to the two orbits of the {\\it TESS}\\ spacecraft that comprise the full Sector 2 data (those correspond to physical orbits 11 and 12, where the numbering started during commissioning) and $i=3,4$ to the two orbits contained in Sector 3 data (corresponding to physical orbits 13 and 14). The complete phase curve model is therefore\n\\begin{equation}\nf(t) = S^{\\lbrace i\\rbrace}_{n}(t)|_{i=1,2,3,4} \\times \\psi(t).\n\\end{equation}\n\nTo determine the optimal polynomial order for each orbit, we carry out phase curve fits of individual orbit light curves and choose the order that minimizes the Bayesian Information Criterion (BIC), which is defined as $\\mathrm{BIC} \\equiv k\\log n-2\\log L$, where $k$ is the number of free parameters in the fit, $n$ is the number of data points, and $L$ is the maximum log-likelihood.\n\nFor the first orbit, we find that a 7th-order polynomial model minimizes the BIC, while for the second orbit, we use a 9th-order polynomial. For the two shorter Sector 3 orbits light curves, we use a 5th-order and 3rd-order polynomial, respectively. \n\nThe use of a time-dependent detrending model when fitting for a time-varying astrophysical signal can sometimes introduce artificial biases into the parameter estimates. When selecting polynomials of similar order, the best-fit astrophysical parameters do not vary by more than 0.2$\\sigma$. To further assess the effects of our use of polynomial detrending, we carry out a special joint fit of all four orbital light curves without using any detrending model, instead using a simple multiplicative normalization factor for each orbit's light curve. The resultant estimates of the transit\/eclipse depths and phase curve amplitudes are consistent with the results of the full joint fit with detrending models (\\secr{res} and \\tabr{modelparams}) at better than the \\sig{1} level. This test demonstrates that the results presented in this paper are highly robust to the particular choice of detrending model.\n\n\\subsection{Contaminating sources}\n\\label{subsec:contamination}\n\nSince the primary objective of the {\\it TESS}\\ mission is to carry out a survey of nearby, bright stars in search of transiting planets, the camera focus is set so as to spread a point source's flux over several pixels to adequately sample the point spread function and achieve high photometric precision. As a benchmark, approximately 50\\% and 90\\% of a star's flux is contained within a 1$\\times$1 and 4$\\times$4~pixel region around the centroid, respectively \\citep{ricker2015}. Given the pixel scale of $21''$, this indicates that a star's pixel response function (PRF) occupies a significant on-sky area, raising the possibility of contaminating sources overlapping with the target PRF and blending with the extracted photometry.\n\nThere are two moderately bright stars within the {\\it TESS}\\ input catalog that are in the vicinity of WASP-18 --- TIC 100100823 and TIC 100100829. These two nearby sources lie $73''$ and $83''$ away from the target and have {\\it TESS}\\ magnitudes of $T=12.65$ mag and $T=12.50$ mag, respectively. When compared to WASP-18 ($T=8.83$ mag) the nearby sources are 34 and 29 times fainter, respectively.\n\nThe optimal apertures selected by the SPOC pipeline for photometric extraction are shown in Figure~\\ref{fig:stamp}. The locations of the two nearby stars are also indicated. The SPOC pipeline uses a model PRF derived from commissioning data to remove the flux from neighboring sources located on each target's subarray. The relative uncertainty of the deblending process due to imperfections in the PRF model and intrinsic variations in the PRFs of different sources across the detector is estimated to be at the level of a few percent \\citep{jenkins2010}. \n\nIn the Sector 2 pixel stamp, TIC 100100823 is 0.66~pixels away from the edge of the aperture at its closest point. Using the aforementioned benchmark estimates of a point source's flux distribution on the detector, we determine that the central 50\\% of its undeblended flux (from the 1$\\times$1~pixel region around its centroid) was not included in the aperture. Another 40\\% of its flux was distributed in the remaining 15 pixels in the surrounding 4$\\times$4~pixel region. Tracing a 4$\\times$4~pixel region centered on the location of TIC 100100823, with edges aligned with the pixel boundaries, we can estimate that the fractional area lying within the extraction aperture is $\\sim$0.25. Assuming that the flux was evenly distributed across the 15-pixel region to obtain a generous upper limit, we predict that at most $0.25\\times 40\\% = 10\\%$ of the star's flux lay within the science aperture prior to deblending. Multiplying this upper limit with the estimated uncertainty in the deblending process and the flux of the star relative to WASP-18 gives a relative deblending contamination contribution of roughly 0.01\\%. The other similarly-bright nearby star, TIC 100100829, is further away from the edge of the aperture, so its level of residual contamination in the extracted photometry is smaller. For Sector 3, the nearby stars are situated farther away from the edge of the optimal aperture, and thus the expected deblending contamination contribution is even more negligible.\n\nAs detailed in the following section, the relative uncertainties we obtain for the fitted astrophysical parameters that would be directly affected by such contamination  --- phase curve amplitudes and transit\/eclipse depth --- lie above the 0.13\\% level (in fact, for all parameters except for transit depth, the relative uncertainties are greater than 3\\%). This means that any effect stemming from uncorrected contamination is overshadowed by the much larger intrinsic uncertainties in the parameter estimates, given the sensitivity of the data. Therefore, we do not consider contamination in our phase curve analysis.\n\n\n\n\\begin{deluxetable}{lll}\n\\tablewidth{0pc}\n\\tabletypesize{\\scriptsize}\n\\tablecaption{\n    Model Parameters\n    \\label{tab:modelparams}\n}\n\\tablehead{\n    \\multicolumn{1}{c}{~~~~~~~~~~Parameter~~~~~~~~~~} &\n    \\multicolumn{1}{l}{Value}                     &\n    \\multicolumn{1}{l}{Error}    \n}\n\\startdata\n\\sidehead{\\textit{Fixed Parameters}}\n$e$\\tablenotemark{a}   \\dotfill  & 0.0091 & --- \\\\\n$\\omega$\\tablenotemark{a} ($^{\\circ}$) \\dotfill  & 269 & --- \\\\\n$u_{1}$\\tablenotemark{b} \\dotfill & 0.2192 & --- \\\\\n$u_{2}$\\tablenotemark{b} \\dotfill & 0.3127 & --- \\\\ \n\n\\sidehead{\\textit{Fitted Parameters}}\n$R_p\/R_s$   \\dotfill  & 0.09716 & $_{-0.00013}^{+0.00014}$ \\\\\n$f_p$ (ppm) \\dotfill  & 341 & $_{-18}^{+17}$ \\\\\n$T_0$ (BJD$_{\\mathrm{TDB}}$)      \\dotfill  & 2458375.169883 & $_{-0.000025}^{+0.000026}$ \\\\\n$P$ (days)  \\dotfill  & 0.9414526 & $_{-0.0000015}^{+0.0000016}$\\\\\n$b$         \\dotfill  & 0.318 & $_{-0.019}^{+0.018}$\\\\\n$a\/R_s$     \\dotfill  & 3.562 & $_{-0.023}^{+0.022}$\\\\\n$A_1$ (ppm) \\dotfill  & 21.0  & 4.5 \\\\\n$A_2$ (ppm) \\dotfill  & -4.0   & 4.6 \\\\\n$A_3$ (ppm) \\dotfill  & -14.0  & 4.6 \\\\\n$B_1$ (ppm) \\dotfill  & -174.4 & $_{-6.2}^{+6.4}$ \\\\\n$B_2$ (ppm) \\dotfill  & -190.5 & $_{-5.9}^{+5.8}$ \\\\\n$B_3$ (ppm) \\dotfill  & -3.9  & 6.1\\\\\n$\\sigma_{1}$ (ppm) \\dotfill  & 527.6 & $_{-4.1}^{+3.9}$ \\\\\n$\\sigma_{2}$ (ppm) \\dotfill  & 528.6 &  3.8\\\\\n$\\sigma_{3}$ (ppm) \\dotfill  & 526.5 &  5.6\\\\\n$\\sigma_{4}$ (ppm) \\dotfill  & 521.4 &  $_{-5.2}^{+5.3}$\\\\\n\\sidehead{\\textit{Derived Parameters}} \nTransit depth\\tablenotemark{c} (ppm)  \\dotfill  & 9439 & $_{-26}^{+27}$\\\\\n$i$ ($^{\\circ}$)    \\dotfill  & 84.88 & 0.33 \\\\\n$R_p$ ($R_{\\mathrm{Jup}}$)\\dotfill & 1.191 & 0.038\\\\\n$a$ (AU)\\dotfill & 0.02087 & 0.00068\n\\enddata\n\\tablenotetext{a}{Best-fit values from \\citet{nymeyer2011}.}\n\\tablenotetext{b}{Tabulated in \\citet{claret2017}.}\n\\tablenotetext{c}{Calculated as $(R_p\/R_s)^2$.}\n\\end{deluxetable}\n\n\\begin{figure*}\n\\includegraphics[width=\\linewidth]{plot2.pdf}\n\\caption{Top panel: The phase-folded light curve, after correcting for long-term trends, binned in 5-minute intervals (black points), along with the best-fit full phase curve model from our joint analysis (red line). Middle panel: same as top panel, but with an expanded vertical axis to detail the fitted phase curve modulation and secondary eclipse signal. The contributions to the phase curve from ellipsoidal ($B_{2}$), atmospheric brightness ($B_{1}$), and beaming ($A_{1}$) modulations are plotted individually with black solid, dot-dash, and dotted lines. Bottom panel: plot of the corresponding residuals from the best-fit model.}\n\\label{fig:bestfit}\n\\end{figure*}\n\n\\subsection{Model fit}\n\\label{sec:fit}\n\nWe carry out a joint fit of the WASP-18 light curves from Sectors 2 and 3. We fit for $R_p\/R_s$, $f_{p}$, $T_{0}$, $P$, $b$, $a\/R_{*}$, $A_{1-3}$, $B_{1-3}$, $c^{\\lbrace 1\\rbrace}_{0-7}$, $c^{\\lbrace 2\\rbrace}_{0-9}$, $c^{\\lbrace 3\\rbrace}_{0-5}$, and $c^{\\lbrace 4\\rbrace}_{0-3}$. In addition, we fit for a uniform per-point uncertainty for each orbit --- $\\sigma_{1-4}$ --- which ensures that the resultant reduced $\\chi^2$ value is near unity and self-consistently produces realistic uncertainties on the other parameters given the intrinsic scatter of the light curves. The total number of free astrophysical, long-term trend, and noise parameters is 44. \n\nThe ExoTEP pipeline simultaneously computes the best-fit values and posterior distributions for all parameters using the affine-invariant Markov Chain Monte Carlo (MCMC) ensemble sampler \\texttt{emcee} \\citep{emcee}. We set the number of walkers to four times the number of free parameters: 176. To facilitate the convergence of the chains we initialize the walkers with values close to those of the best fit to light curves of individual physical orbits. The length of each walker's chain is 100,000 steps, and we discard the first 80\\% of each chain before calculating posterior distributions. The chains are plotted and visually inspected to ensure convergence. We also re-run the entire process, to confirm that the resultant parameter estimates are consistent to well within $0.1\\sigma$.\n\n\\section{Results}\n\\label{sec:res}\n\nThe results of our joint analysis are listed in Table~\\ref{tab:modelparams}, where we include astrophysical and noise parameters. The median and \\sig{1} uncertainties derived from the posterior distributions are given. The combined phased light curve with long-term trends removed is plotted in Figure~\\ref{fig:bestfit} along with the best-fit full phase curve model.\n\nComparing with other published values in the literature \\citep{hellier2009, southworth2009, nymeyer2011}, we find that our results generally lie in good agreement with previous estimates. The calculated values of the orbital parameters $i$ and $a\/R_{*}$ are the most precise to date and are well within \\sig{1} of the values presented in the discovery paper \\citep{hellier2009} and follow-up photometric studies \\citep{southworth2009}. We obtain an updated and refined mid-transit time with a precision of 2.2~seconds and a period estimate that is consistent at better than the \\sig{0.2} level with the results of \\citet{hellier2009} as well as recent studies that fitted ephemerides across all previously measured transit times \\citep{wilkins2017,mcdonald2018}.\n\nThe measured transit depth $(R_{p}\/R_{*})^{2}$ of $9439_{-26}^{+27}$~ppm is significantly (\\sig{3.3}) larger than the estimate of \\citet{hellier2009} of $8750\\pm 210$~ppm. A separate analysis of the photometry and radial velocity (RV) data obtained by \\citet{hellier2009} incorporating additional RV measurements yielded a somewhat larger transit depth estimate of $9160^{+200}_{-120}$~ppm \\citep{triaud2010}, which is more consistent (\\sig{1.4}) with the value derived here. \\citet{southworth2009} carried out a thorough analysis of WASP-18b transit light curves using a variety of limb-darkening laws and fitting methodologies and obtained a wide range of transit depths (9300--9800~ppm) consistent with the estimate from our analysis.\n\nThe most notable results from our phase curve analysis are the $19\\sigma$ detection of a secondary eclipse in the {\\it TESS}\\ bandpass and the robust detection of phase curve variations corresponding to beaming ($A_{1}$), atmospheric brightness ($B_{1}$), and ellipsoidal ($B_{2}$) modulations (See table~\\ref{tab:modelparams}). These three leading phase curve harmonics are plotted individually in the middle panel of Figure~\\ref{fig:bestfit}.\n\nIn our analysis, we have fixed the orbital eccentricity $e$ and argument of periastron $\\omega$ to the most recent literature values \\citep{nymeyer2011}. Since the orbit is very nearly circular, we have experimented with carrying out fits where we fix $e=0$, as well as fits where we allow $e$ and $\\omega$ to vary freely. In both cases, we obtain parameter estimates that are consistent with the best-fit values listed in Table~\\ref{tab:modelparams} at better than the $0.9\\sigma$ level. \n\nIn the free-eccentricity fits, we obtain relatively weak eccentricity constraints: $e=0.0015_{-0.0011}^{+0.0038}$ and $\\omega=156_{-69}^{+115}$~deg. These estimates are consistent with those published in \\citet{nymeyer2011} at better than the \\sig{2} level. In their analysis of high signal-to-noise thermal infrared secondary eclipse light curves from \\textit{Spitzer}, they obtain orbital eccentricity estimates that are much more precise than what we can constrain from our analysis. Therefore, we have decided to fix $e$ and $\\omega$ to their best-fit values in the joint fits described in Section~\\ref{sec:res}.\n\nThe fitted per-point uncertainties for the light curves $\\sigma_{1-4}$ are nearly identical at 520--530~ppm, indicating that the noise level in the data is consistent across the four orbits. The standard deviations of the residuals binned at 1-hour intervals in time are 130, 143, 142, and 158~ppm for the four orbits, respectively, which are generally consistent with the benchmark prediction of 123~ppm calculated for a $T=8.83$~mag target based on simulated photometry for the {\\it TESS}\\ mission \\citep{sullivan2015,stassun2017}. The larger binned residuals in the Sector 3 light curves (particularly in the last orbit) indicate suboptimal photometric performance, as discussed in \\secr{obs}.\n\n\n\\section{Discussion}\n\\label{sec:dis}\n\nAmong the sinusoidal phase curve coefficients fitted in our model, only three have a high statistical significance (with a signal to noise ratio of well over 3, see \\tabr{modelparams}) --- those corresponding to the fundamental, $A_1$ and $B_1$, and the first harmonic of the cosine, $B_2$. As discussed in Section~\\ref{subsec:phase}, these terms are attributed to the beaming, atmospheric, and ellipsoidal components, respectively. Each component is discussed in Sections~\\ref{sec:beam}--\\ref{sec:atm} below. \n \nFor the beaming and ellipsoidal components we show in Sections~\\ref{sec:beam} and \\ref{sec:tide} that their amplitudes and signs are consistent with theoretical expectations. While that consistency can be expected, there are systems where that is not the case for one of those phase components and cases where the mass ratios derived from the two amplitudes do not agree \\citep[e.g.,][see also \\citealt{shporer2017} Section 3.4]{vankerkwijk2010, carter2011, shporer2011, barclay2012, bloemen2012, esteves2013, faigler2015, rappaport2015}. One possible explanation for that disagreement is a phase shift in the atmospheric component \\citep[e.g.,][]{faigler2015, shporer2015}. Another possible explanation is poor modeling of the tidal distortions of hot stars, given their fast rotation and lack of convective zone in their atmospheres \\citep{pfahl2008}.\n\nOur phase curve model includes three other components --- the first harmonic of the sine, $A_2$, and the second harmonic terms, $A_3$ and $B_3$ (see \\tabr{modelparams}).\nThe first harmonic of the sine, $A_2$, is not statistically significant which is consistent with theoretical expectations as we are not aware of any astrophysical process associated with that coefficient \\citep[e.g.,][]{faigler2011, shporer2017}.  \nThe second harmonic of the cosine ($B_3$) is expected as a higher-order term of the ellipsoidal distortion modulation, and the fitted amplitude, although not statistically significant, is in agreement with expectations (see more details in \\secr{tide}).\nFinally, the sine amplitude of the second harmonic, $A_3$, is $-14.0 \\pm 4.6$ ppm, at the \\sig{3} level. While it is smaller in amplitude and lower in statistical significance than the coefficients associated with the beaming, ellipsoidal, and atmospheric components ($A_1$, $B_2$, and $B_1$, respectively, see \\tabr{modelparams}), the astrophysical origin of that coefficient is not immediately clear. It is interesting to note that a similar phase component was measured at a statistically significant level for Kepler-13Ab \\citep{esteves2013, shporer2014}, although that host star is a hot A-type star \\citep{shporer2014} while WASP-18 is a mid F-type star.\n\nWe have experimented with fitting a simplified phase curve model without the second harmonic terms, i.e., including only the fundamental ($A_{1}$, $B_{1}$) and first harmonic ($A_{2}$, $B_{2}$) terms. We do not find that any of the fitted astrophysical parameter estimates change by more than $0.3\\sigma$. This indicates that our phase curve analysis is not significantly affected by the inclusion of the second harmonic terms.\n\n\\subsection{Beaming}\n\\label{sec:beam}\n\nThe sine component of the fundamental, $A_1$, is expected to result from the beaming modulation. We derive the expected amplitude using:\n\\begin{equation}\n    A_{\\rm beam} = \\alpha_{\\rm beam} 4 \\frac{K}{c} \\ ,\n    \\label{eq:beam}\n\\end{equation}\nwhere $K$ is the orbital RV semi-amplitude, and $c$ the speed of light. The $\\alpha_{\\rm beam}$ coefficient is of order unity and depends on the target's spectrum in the observed bandpass (for a more detailed description of the nature of this coefficient, see \\citealt{shporer2017}). \n\nAssuming the target is a blackbody and integrating across the {\\it TESS}\\ bandpass, we derive an expected beaming modulation amplitude of $A_{\\rm beam}$ = $18 \\pm 2$~ppm, using the known parameters of the system (see \\tabr{knownparams}). This is consistent with the measured value of $21.0 \\pm 4.5$~ppm. Examining the shape of the phase curve in Figure~\\ref{fig:bestfit}, the presence of a significant beaming modulation signal is seen as the difference between the two brightness maxima within the orbit.\n\n\n\\subsection{Tidal interaction}\n\\label{sec:tide}\n\n\nThe ellipsoidal modulation measured in the phase curve is the result of tidal interaction between the planet and the star. To estimate the expected value of that photometric modulation amplitude, we use the following approximate equation \\citep{morris1985, morris1993}: \n\\begin{equation}\n    A_{\\rm ellip} = \\alpha_{\\rm ellip} \\frac{M_p \\sin^2 i}{M_s}\\left(\\frac{R_s}{a}\\right)^3  \\ \\ ,\n    \\label{eq:aellip}\n\\end{equation}\n    where $M_s$ and $R_s$ are the host star mass and radius, respectively, $M_p$ the planet mass, $a$ the orbital semi-major axis, and $i$ the orbital inclination angle. The $\\alpha_{\\rm ellip}$ coefficient is derived from the linear limb darkening coefficient $u$ and gravity darkening coefficient $g$:\n\\begin{equation}\n    \\alpha_{\\rm ellip} = 0.15 \\frac{(15+u)(1+g)}{3-u} \\  \\ .\n\\end{equation}\nWe estimate $u$ and $g$ using the known parameters of the host star (see \\tabr{knownparams}) and the tables of \\cite{claret2017} to arrive at $\\alpha_{\\rm ellip} = 1.10 \\pm 0.05$. This gives an expected amplitude of $A_{\\rm ellip} = 172.5 \\pm 14.6$~ppm (uncertainty derived from uncertainty of known parameters), in good agreement with the measured amplitude of $|B_2|=190.5_{-5.8}^{+5.9}$ ppm.\n\nAs noted earlier, \\eqr{aellip} is an approximation, as it is the leading term in a Fourier series \\citep{morris1985, morris1993}. We have calculated the next term in the series, the coefficient of the cosine of the 2nd harmonic of the orbital period \\citep{morris1985}, to be $-11.8 \\pm 1.8$ ppm. This is in agreement with the measured amplitude of $B_3 = -3.9 \\pm 6.1$ ppm (see \\tabr{modelparams}), although the latter is not statistically significant.\n\n\\subsection{Atmospheric characterization}\n\\label{sec:atm}\n\nThe drop in flux during secondary eclipse, as the planet is occulted by its host star, is due to the blocking of (1) starlight reflected by the planet's atmosphere and (2) thermal emission from the planet's atmosphere, since, given the strong stellar irradiation, the planet's thermal emission is expected to be significant at visible wavelengths. To estimate the planet's thermal emission in the {\\it TESS}\\ band, we use the atmospheric model of \\cite{arcangeli2018}, derived by fitting to the measured secondary eclipse depths in the four Spitzer\/IRAC bands (centered at 3.6, 4.5, 5.8, and 8.0~\\ensuremath{\\mu \\rm m}) and HST\/WFC3 (1.1--1.7~\\ensuremath{\\mu \\rm m}). Integrating that atmospheric model across the {\\it TESS}\\ bandpass gives an expected thermal emission of 327~ppm. It is difficult to accurately quantify the uncertainty on the expected thermal emission and we estimate it to be at the few percents level (the uncertainty on the star-planet radii ratio derived here, which contributes to the thermal emission uncertainty, is only 0.14\\%).\n\nWe measure a secondary eclipse depth of $341_{-18}^{+17}$~ppm. While this depth is significantly shallower than the $z'$ band eclipse depth of $682 \\pm 99$ ppm measured by \\cite{kedziora2019}, it is consistent with the expected thermal emission predicted by the atmospheric models of \\cite{arcangeli2018}.\n\nSince the difference between the measured secondary eclipse depth and the expected thermal emission in the {\\it TESS}\\ bandpass is only at a \\sig{0.8} significance, we cannot claim a detection of reflected light, and therefore place a \\sig{2} upper limit of 35~ppm on the relative contribution of reflected light to the atmospheric brightness. This upper limit is consistent with the upper limit on reflected polarized light from the planet measured by \\cite{bott2018}.\n\nThe upper limit on the reflected light translates to an upper limit on the geometric albedo in the {\\it TESS}\\ bandpass of $A_g <$ 0.048 (\\sig{2}), using the model parameters measured here (specifically $R_p\/R_s$ and $a\/R_s$). This upper limit is consistent with low visible-light geometric albedo measured for other short period gas giant planets, like WASP-12b ($A_g < $ 0.064 at 97.5\\% confidence, \\citealt{bell2017}), HD~209458b ($A_g = 0.038 \\pm 0.045$, \\citealt{rowe2008}), TrES-2b ($A_g = 0.0253 \\pm 0.0072$, \\citealt{kipping2011}), Qatar-2b ($A_g < 0.06$ at \\sig{2}, \\citealt{dai2017}), and others \\citep[e.g.,][]{heng2013, esteves2015, angerhausen2015}.\n\nThe small geometric albedo suggests that the bond albedo is also small since the {\\it TESS}\\ bandpass is close to the wavelength region where the host star is brightest. The low albedo is consistent with a correlation between decreasing albedo and increasing planet mass, suggested by \\citet{zhang2018}, although there is currently no theoretical mechanism to explain such a correlation.\n\n\nThe measurements of the secondary eclipse depth and the atmospheric phase component amplitude allow us to estimate the flux from the planet's night side --- the hemisphere facing away from the star, visible to the observer during transit. The night-side flux is the difference between the secondary eclipse depth $f_p$ and the full amplitude of the atmospheric brightness modulation $2 \\times |B_{1}|$, which is $-8\\pm 22$ ppm (see \\tabr{modelparams}). Therefore, we do not measure statistically significant flux from the planet's night side, and place a \\sig{2} upper limit of 43~ppm on the night side's brightness in the {\\it TESS}\\ bandpass, which is 13\\% of the day-side brightness. Measurements of the orbital phase curve in the near-infrared at 3.6 and 4.5~\\ensuremath{\\mu \\rm m}\\ were also unable to detect flux from the night side \\citep{maxted2013}. This points to very low efficiency of longitudinal heat distribution from the day side to the night side, as suggested by other authors \\citep{nymeyer2011,maxted2013} and consistent with similar findings for other highly irradiated hot Jupiters \\citep[e.g.,][]{wong2015,wong2016} and theoretical models \\citep{perez2013, komacek2017}.\n\n\nWe have assumed here that the atmospheric phase variability is a sinusoidal modulation, where the maximum is coincident with the phase of secondary eclipse, and the minimum occurs at mid-transit. Deviations from this simplistic model have been observed, where the planet's surface brightness distribution is such that the brightest region is shifted away from the sub-stellar point, leading to a shift between the phase of maximum light and the center of secondary eclipse \\citep[e.g.,][]{demory2013, shporer2015, hu2015, faigler2015, parmentier2016}. In our phase curve analysis a phase shift of the atmospheric phase component will manifest itself as a deviation of the beaming phase component amplitude from the theoretically predicted value, since the beaming and atmospheric components are the sine and cosine of the fundamental (the orbital period). The beaming amplitude is consistent with the predicted amplitude (see \\secr{beam}), and we therefore place an upper limit on a phase shift of 2.9~deg (\\sig{2}), derived from the fitted values and uncertainties of $A_1$ and $B_1$ (see \\tabr{modelparams}). \n\nIn principle, since the observed modulation is a superposition of modulations due to reflected light and thermal emission, each of the two processes may have a phase shift that is canceled out in the combined light. However, phase curves in the near-infrared do not show a phase shift of the thermal emission, down to 5--10~deg \\citep{maxted2013}. Hence, we can rule out a phase shift of the reflected light modulation in the optical. The lack of a phase shift is consistent with the inefficient heat redistribution from the day-side to the night-side hemisphere.\n\n\n\n\n\n\\subsection{Detecting non-transiting WASP-18b-like objects}\n\\label{sec:nontransit}\n\nThe clear sinusoidal modulations seen along the orbital phase (see \\figr{lightcurves}) suggests that they could be identified even if the system were not in a transiting configuration, i.e., with a smaller orbital inclination angle. If so, this in turn raises the possibility of detecting non-transiting but otherwise similar systems, as explored by \\citet[][see also \\citealt{faigler2012,faigler2013,talor2015, millholland2017}]{faigler2011} using a phase curve analysis.\n\nTo test that possibility, we have removed all data points within the transit and secondary eclipse and have carried out a period analysis of the remaining light curve, similar to the analysis done in \\citet[][see also \\citealt{shporer2014}]{shporer2011} for the Kepler-13 system. \\figr{ls} shows both the Lomb-Scargle periodogram \\citep{lomb1976, scargle1982} and the double-harmonic periodogram (following \\citealt{faigler2011}). Both periodograms are dominated by variability at the orbital period. This demonstrates that variability at the orbital period can be clearly detected for non-transiting but otherwise similar systems. This also further confirms the prediction of \\citet[][see their Figure 10]{shporer2017} that the orbital phase curve variability of systems such as WASP-18, containing massive planets on short periods, can be detected in {\\it TESS}\\ data.\n\nIt is important to note that the detection of periodic variability does not uniquely reveal the nature of the system as a star-planet system, since similar variability can be induced by a stellar-mass companion, stellar pulsations, and\/or the combination of stellar activity (in the form of starspots) and stellar rotation. However, a star rotating at a period as short as the WASP-18b orbital period would show large rotational broadening of the spectral lines, with a rotation velocity over 50~\\ensuremath{\\rm km\\,s^{-1}}, which can be measured with a single stellar spectrum. Furthermore, as shown by \\citet{faigler2011}, the measured amplitudes can be compared with the expected amplitudes based on the stellar parameters.\n\n\n\\begin{figure}\n\\begin{centering}\n\\includegraphics[width=0.4\\textwidth]{ls.eps}\n\\includegraphics[width=0.4\\textwidth]{beer.eps}\n\\caption{Lomb-Scargle (top) and double-harmonic (bottom) periodograms of the lightcurves where data within the transits and secondary eclipse were removed, and normalized by their standard deviation (estimated through the median absolute deviation, or MAD). Both periodograms are dominated by the orbital frequency and its harmonics and sub-harmonic, marked by short vertical green lines on the x-axis. \n\\label{fig:ls}}\n\\end{centering}\n\\end{figure}\n\n\\subsection{TTV analysis}\n\\label{sec:ttv}\n\nTo complement our analysis of the phase curve we have also analyzed the individual transit times in order to look for transit timing variations (TTV). That analysis was divided into long term TTV, using all available transit times in the literature spanning about a decade, and short term TTV, within the time scale of the {\\it TESS}\\ Sector 2 and 3 data. Both analyses are described below.\n\n\\subsubsection{Long term TTV}\n\\label{sec:ttvlong}\n\nMost hot Jupiters are vulnerable to tidal orbital decay, and their\norbits should be shrinking \\citep[e.g.,][]{levrard2009,matsumura2010}. However, directly observing tidal inspiral has proven an obstinate challenge; there have not yet been any unambiguous detections of orbital decay due to tides (see \\citealt{maciejewski2016,patra2017,bailey2019}\nfor discussion of the most promising case yet).\nWASP-18b is a promising target in the search for tidal orbital decay.\nCompared to other hot Jupiters, it has a particularly high planet to\nstar mass ratio, and an exceptionally small separation from its host\nstar. It was realized quite early that if the stellar tidal dissipation rates inferred from binary star systems were applicable in WASP-18, the system would undergo orbital decay on a timescale of only megayears (\\citealt{hellier2009} and references therein). \\citet{wilkins2017} recently searched for orbital decay of WASP-18b, concatenating previous observations with new data. Within the context of the ``constant phase lag'' model for tidal interaction \\citep{zahn1977}, they reported a limit on the modified stellar tidal quality factor $Q_\\star' \\geq 1\\times10^6$, at 95\\% confidence.\n\nThe {\\it TESS}\\ observations provide new transit times that let us extend WASP-18b\norbital decay search. \\tabr{tt} lists the transit and occultation times we used in our long-term timing analysis, where most of the archival times were already compiled by \\citet{wilkins2017}. We require that each archival time (i) originate from the peer-reviewed literature, and (ii) be based on observations of a single transit or occultation observed in its entirety. We adopt the methods and equations described by \\cite{bouma2019}, who performed a similar study in the context of WASP-4b. To measure the {\\it TESS}\\ transit times, each individual transit is isolated to a window of $\\pm 3$ transit durations. Each transit window is then fitted simultaneously for a local linear trend in relative flux, the mid-transit time, and the depth. The remaining transit parameters are fixed to those found in \\tabr{modelparams}. Given the abundance of transits, we omitted three {\\it TESS}\\ transits that had significant gaps due to momentum wheel dumps, close to BJDs of 2,458,359.2, 2,458,376.1, and 2,458,390.2.\nAfter deriving the new transit times and uncertainties, we fitted the full timing dataset with two competing models: a linear ephemeris model and a quadratic ephemeris model. \\figr{O_minus_C} left panel shows the times and best-fitting models where the linear ephemeris was subtracted. The difference in the Bayesian Information Criterion between the best-fitting linear and quadratic models is $-3.8$, so there is no evidence to prefer the quadratic model over the linear model. The relevant limits from the quadratic model, at 95\\% confidence, can be expressed as:\n\\begin{align}\n  -8.78 < \\dot{P}  &< 4.98 \\ \\mathrm{ms}\\,\\mathrm{year}^{-1}, \\label{eq:pdot} \\\\\n  Q_\\star' &> 1.73 \\times 10^6. \\label{eq:q}\n\\end{align}\nAs shown in \\eqr{pdot} our analysis does not find a long term period derivative. And, as seen in \\eqr{q} we do not find the stellar quality factor of $Q\\star' \\sim 5\\times10^5$ suggested by \\citet{mcdonald2018}, while our result is a modest improvement relative to the results of \\citet{wilkins2017}.\n\nThe linear ephemeris we derived by fitting the decade-long transit timings data set is:\n\\begin{align}\n& P = 0.941452419 \\pm 0.000000021 \\, {\\rm day}, \\\\\n& T_0 \\, ({\\rm BJD_{TDB}}) = 2458002.354726 \\pm 0.000023.\n\\end{align}\n\n\n\n\\begin{figure*}[t]\n\t\\begin{center}\n\t\t\\leavevmode\n\t\t\\includegraphics[width=0.47\\textwidth]{data_bestfit_OminusC_occ-tra.pdf}\n\t\t\\includegraphics[width=0.47\\textwidth]{modl_sinu.pdf}\n\t\\end{center}\n\t\\vspace{-0.7cm}\n\t\\caption{Long-term (left panel) and short-term (right panel) TTV search. Both panels show the residual timing after subtracting the linear ephemeris fitted to the decade-long transit timing data set listed in \\tabr{tt}. In the left panel, the best-fit constant-period model (blue) and constant period derivative model (orange) provide comparable fits to the data, but the latter model has one extra free parameter. In the right panel, the blue curves are fair samples drawn from the posterior of a quasi-periodic model used to fit the measured timing residuals. \n    }\n\t\\label{fig:O_minus_C}\n\\end{figure*}\n\n\\input{WASP-18b_transit_time_table.tex}\n\n\\subsubsection{Short term TTV}\n\\label{sec:ttvshort}\n\nThe existence of a third body in a star-planet system such as WASP-18 may introduce deviations of the observed transit timings from the expected timings assuming a Keplerian orbit \\citep[e.g.,][]{agol2005, holman2005}. Generally known as TTVs, these data features can allow one to probe the system for planetary companions. While these variations can have a wide range of dependencies on the orbital and planetary parameters, the expected generic shape of the TTVs is quasi-sinusoidal.\n\nWe analyzed the 44 transit timing inferred from {\\it TESS}\\ Sectors 2 and 3 data, listed in \\tabr{tt}, to test whether the observed timings contain any evidence for sinusoidal variations. Towards this purpose, we subtract from observed timings the expected timings based on the linear ephemeris derived in \\secr{ttvlong}. Those timing residuals are plotted in \\figr{O_minus_C} right panel and we attempt to model those data by sampling from the posterior probability distribution of two TTV models. The first model (the null model) has only a single parameter which is an offset fitted to the timing residuals. The second model has four parameters where in addition to the timing offset it includes the period, phase, and amplitude of the sinusoidal TTV. \n\nIn order to perform the sampling and calculate the associated Bayesian evidences, we use \\texttt{dynesty}\\footnote{\\url{https:\/\/github.com\/joshspeagle\/dynesty}}. The resulting Bayesian evidence of the sinusoidal TTV model relative to that of the null model is -3, indicating that the null hypothesis should be readily preferred. Furthermore, the posterior-mean reduced $\\chi^2$ (i.e., root mean square of the residuals per degree of freedom) of the null and alternative models are 1.18 and 1.22, respectively. Hence, we conclude that there is no evidence for TTVs in the {\\it TESS}\\ Sectors 2 and 3 data of WASP-18b.\n\n\n\\section{Summary}\n\\label{sec:sum}\n\nWe have presented here an analysis of the full orbital phase curve of the WASP-18 system measured by {\\it TESS}\\ in Sectors 2 and 3. The per-point residual scatter of the 2-minute data is 520--530~ppm, yielding a binned scatter of 130--160~ppm per 1-hour exposure. This noise level is consistent with the expected noise level of {\\it TESS}\\ data for this $T$=8.83~mag target.\n\nWe detect at high significance beaming and ellipsoidal modulations that are consistent with theoretical predictions. We robustly measure a secondary eclipse depth of $341_{-18}^{+17}$ ppm, which when combined with the expected thermal emission in the {\\it TESS}\\ bandpass leads to a null detection of reflected light and a \\sig{2} upper limit of 0.048 on the geometric albedo in the {\\it TESS}\\ bandpass. The low optical geometric albedo is consistent with that of other hot Jupiters in similar wavelength range, especially highly-irradiated hot Jupiters \\citep{schwartz2015}.\n\nWe do not detect a phase shift in the atmospheric phase curve component, with an upper limit of 2.9~deg (\\sig{2}), indicating that the phase of maximum light is well-aligned with the phase of secondary eclipse. In addition, we do not detect light from the planet's night-side hemisphere, with an upper limit of 43~ppm (\\sig{2}), or 13\\% of the day-side brightness. These findings indicate very inefficient distribution of incident energy from the day-side hemisphere to the night-side hemisphere and are consistent with results based on previously published phase curve measurements (both in the near-infrared and in the optical) of similarly highly-irradiated hot Jupiters \\citep[e.g.,][]{wong2015,wong2016} and with theoretical expectations \\citep{perez2013,komacek2017}.\n\nThe clear detection of the WASP-18 phase curve modulations demonstrate that {\\it TESS}\\ data are sensitive to the photometric variations of systems with massive short period planets, a sensitivity that increases when data from several {\\it TESS}\\ sectors is combined. For such objects, the {\\it TESS}\\ phase curve can be used for atmospheric characterization, as shown here, as well as independent mass estimates from the constraints derived from RV analyses.\n\nTo complement our study of the phase curve we have also searched for TTVs, both long term using all available measured transit times spanning about a decade,  and also short term within the time span of the {\\it TESS}\\ Sectors 2 and 3 data. In both cases we do not find a statistically significant deviation of the transit timings from a linear ephemeris.\n\n\\acknowledgments\n\nWe acknowledge the use of {\\it TESS}\\ Alert data, which is currently in a beta test phase. These data are derived from pipelines at the {\\it TESS}\\ Science Office and at the {\\it TESS}\\ Science Processing Operations Center.\nFunding for the {\\it TESS}\\ mission is provided by NASA's Science Mission directorate.\nThis paper includes data collected by the {\\it TESS}\\ mission, which are publicly available from the Mikulski Archive for Space Telescopes (MAST).\nResources supporting this work were provided by the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center.\nThis research has made use of the NASA Exoplanet Archive, which is operated by the California Institute of Technology, under contract with NASA under the Exoplanet Exploration Program.\nI.W.~is supported by a Heising-Simons \\textit{51 Pegasi b} postdoctoral fellowship.\nC.X.H. and M.N.G.~acknowledge support from MIT's Kavli Institute as a Torres postdoctoral fellow.\nT.D.~acknowledges support from MIT's Kavli Institute as a Kavli postdoctoral fellow.\n\n{\\it Facilities:} \n\\facility{\\textit{TESS}}\n\n{\\it Software:} \n\\texttt{astrobase} \\citep{bhatti_astrobase_2018},\n\\texttt{astropy} \\citep{astropy2018},\n\\texttt{BATMAN} \\citep{kreidberg_batman_2015},\n{\\scshape dynesty} (\\url{https:\/\/github.com\/joshspeagle\/dynesty}),\n\\texttt{emcee} \\citep{emcee},\n\\texttt{numpy} \\citep{vanderwalt2011}, \n\\texttt{matplotlib} \\citep{hunter2007}, \n\\texttt{pandas} \\citep{mckinney-proc-scipy-2010},\n\\texttt{scipy} \\citep{jones_scipy_2001}.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\subsection*{Acknowledgements}\n\nWe gratefully acknowledge financial support from Trinity College, Cambridge to A.L.G., from the EPSRC (UK) to A.L.G.\\ and M.J.C., and from the US NSF to D.A.D.\\ under grant DMR 09-03225. We thank D.\\ A.\\ Keen (Rutherford Appleton Laboratory) for useful discussions, and acknowledge the University of Cambridge's CamGrid infrastructure for computational resources.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}