data_all_eng_slimpj/shuffled/split2/finalzpoc

{"text":"\\section{Introduction}\nBlazars are a sub-class of active galactic nuclei (AGN) with their relativistic jet oriented  towards the observer's line of sight \\citep{bland1979, Urry1995} leading to the Doppler boosted emission from the jet. They show extreme variability in their brightness and polarization over the time scale of  minutes to tens of years. Owing to these properties, their study serves as a tool to probe deeper into the central engine to understand the structure and emission processes in AGN. The continuum spectral energy distribution (SED) of blazars is dominated by the non-thermal emission with two broad peaks covering entire electromagnetic spectrum (EMS), ranging from radio to high energy $\\gamma$-rays. The first peak  in the SED lies in the sub-mm to X-ray region and is known to be due to synchrotron process in which the relativistic electrons gyrate in a strong magnetic field present inside the jet \\citep{urrymush1982} and radiate by cooling. The second, high energy peak, is understood to be due to inverse Compton scattering of low energy photons, the origin of which is not understood well.  Under the leptonic scenario  \\citep[see,][ for a review]{bottcher2007}, inverse Compton scattering of the low energy photons by the relativistic electrons, which gave rise to the synchrotron emission, is responsible for the high energy peak. The seed photons which are up-scattered, could either be synchrotron photons (Self Synchrotron Compton; SSC) or external photons from the accretion disk, the broad-line region, molecular torus, cosmic microwave background etc. (external Compton; EC), or a combination of both \\citep[][ and references there-in]{maraschi1994}. The exact source of these seed photons is  still an open question.  As an alternative approach, hadronic models \\citep{mannheim1989} are also used to explain the high energy component in the SED \\citep{Zdziarski-Bottcher2015}.\n\\smallskip\n\nBlazars consist  of flat spectrum radio quasars (FSRQ) and BL Lac objects, with FSRQs differentiated from BL Lacs by the presence of broad emission lines in their spectrum (with equivalent width, EW $> 5~ \\AA$; \\citep{Urry1995, Laurent-Muehleisen1999}). Depending upon the frequency of the synchrotron peak in their SED, BL Lacs are further classified into three categories \\citep{abdo2010} - low, intermediate and high energy peaked BL-Lacs, abbreviated as LBL, IBL and HBL, respectively. The synchrotron peak frequency, $\\nu^p_{sync}$ for LBL lies below $10^{14}$Hz; for IBL, between  $10^{14}$Hz and  $10^{15}$ Hz; while for HBL $\\nu^p_{sync}$ $>$ $10^{15}$ Hz.    \\citet{Fossati1998} found an anti-correlation between the synchrotron peak frequency and the synchrotron peak luminosity in the blazar. Also, the Compton dominance parameter, which is the ratio of  inverse Compton peak luminosity to the synchrotron peak luminosity, decreases from the high-luminosity (FSRQs) to low-luminosity blazars (BL-Lacs). This could be due to the presence of external seed photons, from BLR or torus, leading to higher inverse-Compton luminosity \\citep{Sikora-begelman-rees1994}. It was, therefore, noticed that the luminosity, degree of polarization and the $\\gamma-ray$ dominance decrease from FSRQ to LBL, IBL,  and  HBL while ratio of non-thermal to thermal component and the synchrotron peak frequency increase, indicating   a blazar  sequence \\citep{maraschi1994, Fossati1998, ghis-tavec2009}. \n\\smallskip\n\nSince AGN are not resolvable by any existing telescope facility, understanding their structure and emission mechanisms pose a big challenge. Blazars, which are  variable over diverse time-scales across the whole spectrum, provide a viable tool as their variability time-scales, correlated variations among multi-frequency  light-curves, color variations and SEDs  are used as probes\\citep[][ and references there-in]{marscher2008, marscher2010, jorstad2010, dai2015, ciprini2007}.  \nThe temporal variability in blazars has been classified into three categories, namely, long-term variability (LTV) - months to years \\citep{Fan2005, fan2009}, short-term variability (STV) - a few days to months, and intra-night variability (INV) or micro-variability -  a few minutes to several hours within a night \\citep{wagner1995, kaur-3c2017}. Though the mechanisms responsible for  the variability remain largely unclear, long-term variability could be due to the disk perturbation\/instability or structural changes in the jet, e.g., precession, bending of jet \\citep{MarscherGear1985, kawaguchi1998, nottale1986, nair2005}. The STV in optical flux, including inter-night variability, could be caused by intrinsic and extrinsic processes, e.g., injection of fresh plasma in the jet, shock moving down the turbulent jet, changes in the boosting factor due to change in the viewing angle, gravitational micro-lensing etc, and sometimes results in the spectral changes \\citep{ghis1997, villata2002, Hong2017}. The INV, also known as microvariability, could be due to shock compression of the plasma in the jet, shock interacting with local inhomogeneities, blob passing through quasi-stationary core, changes in the viewing angle in a jet-in-jet scenario\\citep{Narayan2012} or other processes causing small scale jet turbulence \\citep{MarscherGear1985, marscher2008, chandra2011,kaur-3c2017}. However, the exact processes responsible for variability, in particular INV, are not well understood and significant amount of work is required to have a better understanding of this complex phenomenon. \n\\smallskip \n\n\nThe intermediate BL Lac object S5 0716+714 is one of the most active blazars and makes a perfect candidate for variability study on the blazars at diverse time scales \\citep{aliu2012}. It is available in the sky for a  longer time during the night (due to its high declination), is almost always active, fairly bright, and hence can be observed with moderate facilities.  It was discovered by \\citet{kuhr1981} in NRAO 5 GHz radio survey with flux larger than 1 Jy\n \\footnote{1 Jy = 10$^{-23}$ erg cm$^{-2}$ s$^{-1}$ Hz$^{-1}$} \nand due to its featureless spectra \\citep{bierman1981}, was categorized as a BL-Lac source. \\citet{nilson2008} derived a redshift of 0.31$\\pm$0.08 by taking the host galaxy as a standard candle, but recently \\citet{danforth2013} put a statistical upper limit of z $<$ 0.322 (with 99$\\%$ confidence) on its redshift.  The source S5 0716+714 has been observed across the EMS, including its discovery as a TeV candidate in 2008 by MAGIC collaboration \\citep{Anderhub2009}, when a strong optical and $\\gamma-$ray correlated activity was noticed.\n\nS5 0716+714 shows high duty cycle of variation (DCV) as reported by \\citet[][ and references there-in]{chandra2011}.  Due to all these properties,  it has been the target of several multi-wavelength campaigns around the globe \\citep{wagner1995, villata2002, raiteri2005, nesci2005, montagni2006, gupta2008, dai2015}, focusing on INV and STV. After being reported in its high phase, the object was followed  by \\citet{bachev2012} who claimed  historical  maxima and minima of 12.08 (MJD 56194) and 13.32 (MJD 56195) in R-band, respectively. \\citet{rani2013} found the $\\gamma-$ray emission to be correlated with optical and radio, supporting SSC mechanism responsible for the high energy emission. However, an orphan flare in X-rays indicated to the limitation of such simple scenario. \n\nInvestigating the long-term variability trend, \\citet[][ and references there-in]{nesci2005} reported a decreasing average brightness of the source during 1961-1983 followed by an increasing one upto 2003, superposed with short term flares. They extracted source brightness data from photographic plates obtained  from the Asiago Observatory, POSS1 and Quick V surveys dating back to 1953 to generate long-term light curves. It underlined the importance of the astronomical data, even if taken for some other purpose.  Based on these data, they even predicted a decrease in the mean brightness of the source during the next 10 years, i.e., after 2003.  Indeed,  the source was inferred, from the 2003 to 2014 optical data, to  be in decreasing brightness phase  by \\citet{Chandra2013th, Baliyan2016} and the present work,  suggesting a precessing jet with increasing viewing angle.\n\nThe blazar S5 0716+714 has undergone several optical outbursts in the past, superposed on the mean decreasing or increasing long-term trends as  reported by many workers \\citep{Raiteri2003, nesci2005, gupta2008, larionov2013}. Micro variability (INV) on the time-scales of a few hours to 15 minutes is reported  \\citep[][ and references there-in]{chandra2011, rani2013, man2016} with S5 0716+714 showing bluer-when brighter (BWB) behaviour in general. On the other hand, \\citet{Raiteri2003} found a weak correlation with color, while  others did not find any  correlation between color and brightness\\citep{stalin2009, agarwal2016, wu2005}. The blazar S5 0716+714 has also been reported to show (quasi-) periodic variations (QPV) in optical at several epochs and at many time scales ranging from sub-hours to years \\citep{Raiteri2003, gupta2008}. However,  \\citet{bhatta2016}  did not find  3 and 5 hr QPV as genuine. Recently, \\citet{Hong2018} reported 50 min QPV when the source was relatively fainter during 2005 - 2012, ascribing it to the activity in the innermost orbit of the accretion disk.\n\n\nThe blazar  S5 0716+714 was reported achieving new historical brightness levels (11.68 in R-band) in optical on 2015 January 18 by \\citet{atel2015, chandra2015}, reassuring that it will  never stop to surprise us. It, therefore, justifies a  continuous coverage of the source to help us understand the nature of blazars in general and S5 0716+714 in particular. Keeping this objective in mind and to understand the  variability characteristics, chromatic behaviour and relationship between variability amplitude and brightness of the source, here we present our results obtained from the observations during January, 2013 to June, 2015. Section 2 describes the observations and data analysis; section 3 presents the results and discussions while section 4 summarizes the work. \n\n\\section{Observations and Data Reduction\/Analysis}\nTo investigate intra-night and inter-night variability in BL-Lac source S5 0716+714, we carried out  optical observations using the 1.2m telescope of the  Mount Abu Infrared Observatory(MIRO), operated by the Physical Research Laboratory, Ahmedabad. The observatory is located at Gurushikhar mountain peak, about 1680 m above the sea-level, in Mount Abu (Rajasthan), India, with  a typical  seeing of 1.2 arcsec.  \nThe observations were taken  with liquid-nitrogen cooled Pixcellent CCD camera as the backend instrument, equipped with Johnson-Cousins optical BVRI filter set. The dimension of the CCD array is 1296 x 1152 pixels of size 22 $\\mu$m each. The field of view (FOV) is about 6.5 x 5.5 arcmin$^{2}$ with a plate scale of 0.29 arcsec\/pixel. The CCD readout time is 13 seconds with readout noise of four electrons and negligible dark current when cooled to a temperature of about -120$^{\\circ}$.  \n\n\\smallskip\nIn order to study INV (microvariability), as a strategy, we monitored the source for a minimum of two hours in the Johnsons R-band with a high temporal resolution (less than a minute) to resolve any rapid flare, while for STV and LTV in the source brightness and color, 4\/5 images were taken in B, V and  I-bands everyday during the campaign period.  The source and its comparison\/control stars, as they appear in the finding chart available at the web-page of the Heidelberg University\\footnote{\\url{http:\/\/www.lsw.uni-heidelberg.de\/projects\/extragalactic\/charts\/0716+714.html}} \\label{comp},-- having brightness close to that of the source  \\citep{howell1986}  were kept in the same observed frame. Differential photometry was performed to minimize the effect of non-photometric conditions (however,  majority of the observations were made during photometric nights), like minor fluctuations due to turbulent sky and other seeing effects. The exposure time was decided by keeping the counts   well below the saturation limit and  in the linear regime of our CCD \\citep{cellone2008}. Several twilight flat field images  and bias images were taken on each observation night to calibrate the science images. By following the above mentioned strategy, a total of 6256, 159, 214 and 177 images in R, B, V and I-band, respectively, were obtained and subjected to analysis.\n\nThe observed data were checked for  spurious features, if any, and reduced using IRAF\\footnote{IRAF-Image Reduction and Analysis Facility is data reduction and analysis package by NOAO, Tuscon, Arizona operated by AURA, under agreement with NSF.} standard tasks- bias subtraction, flat fielding, cosmic ray treatment etc. The comparison stars 5 and 6\\footnote{Stars taken from the sequence A,B,C,D by \\citet{ghis1997} and corresponding sequence,  2, 3, 5, 6 by \\citet{villata1998}}, present in the source field  were chosen to perform differential photometry . Other stars (stars 2 and 3) in the field were too bright to be used for differential photometry as they could introduce errors  (from differential photon statistics and random noise, like sky)  \\citep{hwm1988}. An optimum aperture size, three times the FWHM, was used based on the prescription by \\citet{cellone2008}, as a smaller apertures can give better Signal-to-Noise(SNR), but might lead to  spurious variations if the seeing was not good, while  a larger aperture  would have significant contribution from the host galaxy  thermal emission\\citep{Cellone2000} and might suppress the genuine variations in the blazar flux. Aperture photometry on the blazar S5 0716+714 and comparison stars 5 \\& 6, using the same aperture size, was performed using \\textit{DAOPHOT} package in IRAF on photometric nights.\n\nThe aperture photometry technique was employed on a total of 6806 images in BVRI-bands and the source magnitudes were calibrated with the average magnitude of the comparison stars 5 and 6, which were also used to check for the stability of  sky during  observations, as described in equation 1 and equation 2. No correction for host galaxy of S5 0716+714 was applied as the host galaxy is much  fainter  with R-band $>$20 mag \\citep{montagni2006}) than the central bright source. The differential light curves ($LC$s) were constructed to detect  INV, while BVI \\& R band long-term $LC$s were generated from daily averaged values in each band. To quantify the INV nights, we applied several statistical tests, for example confidence parameter test (C$-$test), amplitude of variability ($A_{var}$) test, as discussed in the next section.\n\n\\section{Results and discussion}\nAs already mentioned earlier, the  photometric data obtained after the aperture photometry were used to plot the intra-night and inter-night light curves. Though the lightcurves themselves are not sufficient to reveal the complexities of the variability and blazar phenomena, they are good indicators of the emission mechanism and can help put constraints on various models. The nature of most of the light curves  differs from one night to the other,  indicating to the  emission from random and turbulent process in the jet. Since the physical mechanisms which trigger blazar variability, especially on intra-night time scales, are still debatable, any detailed study of $LC$s should add to our understanding.\n\n In order to identify and characterize  the nights showing INV, we performed  variability amplitude  and  confidence parameter (C-test)  tests. In the following we also discuss STV,  LTV  and  color behaviour of the source during the period of our observing campaign.\n\n\\subsection{Intra-night variability}\nBlazars show rapid variability which can sample very compact sizes of their emission regions. To determine the number of  INV nights, we first excluded the  not-so-photometric nights when sky conditions were changing drastically, and those with less than two hours of monitoring. We were left with  29 nights that qualified this criterion during 2013 January - 2015 June. \n The  $LC$s  for S5 0716+714, being very complex  with a number of features, made it very difficult to infer INV from just visual inspection, barring a few clear cases.  To resolve this problem, following statistical methods are used to quantify the INV. \n\n\\noindent\n\\subparagraph{\\bf Confidence parameter test (C$-$test):}\n\\smallskip\nThe C-test was first introduced by \\citet{jang1997} and further generalized by \\citet{Romero1999}. It is basically a ratio between calibrated source magnitudes and the differential magnitude of the comparison stars, given as,\n\n\\begin{equation}\n\\label{eq3}\nC= \\frac{\\sigma_{S-C_{5,6}}}{\\sigma_{C6-C5}}\n\\end{equation}\nwhere, $C_{5,6}$ is the average of difference in instrumental and standard magnitudes of stars 5 and 6, $\\sigma_{S-C_{5,6}}$ and $\\sigma_{C6-C5}$ are the standard deviations of differential LCs.  We consider the source to be variable when confidence  parameter is greater than 2.57 (i.e., C $>$ 2.57) for more than 3$\\sigma$ confidence (or 99 $\\%$ confidence level)\\citep{jang1997}. The standard deviation $\\sigma$ for differential light curves is given by,\n\n\\begin{equation}\n\\label{eq3a}\n\\sigma = \\sqrt{\\frac {\\sum{(m_i-\\overline{m})^2}}{N-1}}\n\\end{equation}\nwhere, $m_i = (m_2-m_1)_i$ is the differential magnitude of the two objects, $\\overline{m} = \\overline{m_2-m_1}$ represents differential magnitude averaged over the night and N is the total number of data points.\n\n\n\\subparagraph{\\bf F-test:} \n\\smallskip\n\nF-test, also known as Fisher-Snedecor distribution test, measures the sample variances of two quantities i.e, variance of calibrated source magnitudes and that of the differential magnitudes of the comparison stars. To test the significance of variability during each night, it is written as,\n\\begin{equation}\n \\label{eq.ftest}\n F = \\frac {{{\\sigma}^2}_{B}}{{{\\sigma}^2}_{CC}}\n\\end{equation}\nwhere ${{\\sigma}^2}_{B}$, ${{\\sigma}^2}_{CC}$ are the variances in the blazar magnitudes and differential magnitudes of the standard stars for nightly observations, respectively. An F-value of $\\geq 3$ implies variability with a significance of more than 90\\% while an  F-value of $\\geq 5$ corresponds to 99\\% significance level.\n\n\\noindent\n\\subparagraph{\\bf Amplitude of variability ($A_{var}$):} \n\\smallskip\n\nThe  intra-night amplitude of variability in the source is calculated  by using the expression given by \\citet{heidt1996},\n\\begin{equation}\n\\label{eq4}\nA_{var} = \\sqrt{(A_{max}-A_{min})^{2}-2 \\sigma^{2}}\n\\end{equation}\nwhere, $A_{max}$ and $A_{min}$ are the maximum and minimum magnitudes in the intra-night calibrated  light curve of the source and $\\sigma$ is the standard deviation in the measurement. For a night to be considered as variable, $A_{var}$ should be more than 5\\%.\n\n\\subsubsection{\\bf INV light curves and duty cycle of variation (DCV)}\n After performing statistical tests on the entire data set, we get 9 out of a total of 29 nights which are found to be variable based on all the above mentioned criteria i.e., C$\\geq$ 2.57,  F $\\geq$ 5 for more than 99$\\%$ confidence level and $A_{var} \\geq$ 0.05 mag.   Figure 1 shows the light curves for these INV  nights where time in Modified Julian Date (MJD) is plotted along X-axis and the brightness magnitude in R-band along Y-axis. Lower curve is the differential $LC$ for the two comparisons (5 and 6), to check the stability of that particular night, thus providing extent of uncertainty in the source values. The $rms$ values of these differential lightcurves for comparison stars are a measure of accuracy in our magnitude measurements. Upper curve (solid circles) shows the  calibrated  brightness magnitudes for the source. The plotted photometric errors  are of the order of a few milli-magnitudes. \n\n\n\\begin{figure*}\n\n\\includegraphics[width = 0.35\\textwidth]{IDV_S5_12-02-2013_BW.pdf} \n\\includegraphics[width = 0.35\\textwidth]{IDV_S5_06-03-2013_BW.pdf}\n\\includegraphics[width = 0.35\\textwidth]{IDV_S5_07-03-2013_BW.pdf}\n\\includegraphics[width = 0.35\\textwidth]{IDV_S5_12-03-2013_BW.pdf}\n\\includegraphics[width = 0.35\\textwidth]{IDV_S5_11-11-2013_BW.pdf} \n\\includegraphics[width = 0.35\\textwidth]{IDV_S5_28-12-2013_BW.pdf} \n\\includegraphics[width = 0.35\\textwidth]{IDV_S5_30-12-2013_BW.pdf}\n\\includegraphics[width = 0.35\\textwidth]{IDV_S5_02-12-2014_BW.pdf}\n\\includegraphics[width = 0.35\\textwidth]{IDV_S5_03-12-2014_BW.pdf} \n\n\\caption{Intra-night light curves for the source S5 0716+714 on various nights during January, 2013 to June, 2015.}\n\\label{idv}\n\\end{figure*}\n\n\n The INV light-curves  (Figure \\ref{idv}) feature monotonic  rise or fall,  slow rise or decay with  rapid fluctuations superimposed on them, alongwith a few $LC$s indicating to a possibility of  quasi-periodic oscillations with short timescale. It can be noted that the shapes of  most of the nightly lightcurves are different, as also reported by several other authors \\citep[][ and references there-in]{chandra2011, kaur-3c2017, Hong2018} indicating that the emission processes are stochastic and complex in nature. A symmetric flare in a lightcurve would mean the cooling timescale is much shorter than the light-crossing timescale. On the night of 2013 February 12 (Fig. \\ref{idv}), the brightness decays slowly with no distinct peak, with total change in the amplitude of variation by about 7.5\\%. In the same figure, a slow increase in flux by about 0.07 mag in about 2.6 hr, with several rapid fluctuations superimposed (including one with 0.04 mag  in about 30 min) is noticed on 2013 March 6.  Next day $LC$  starts with slight decreasing trend but begins brightening up at MJD 56358.88, with a rapid increase after 1.44 hr leading to 0.06 mag (\\textgreater  2 $\\sigma$). The flux decreases up to MJD 56607.0 and then remains stable within errors on 2013 November 11. The INV $LC$ on 2013 March 12 shows interesting features with a brightening by 0.11 mag in about 30 min, followed by a decay of about 0.17 mag in about one hour. It starts increasing again reaching initial level of about 13.41 mag.   A slow decrease in flux and then relatively faster increase by about 0.07 mag within about 70 min characterizes the $LC $ on 2013 December 28(Fig. \\ref{idv}). A significant increase in flux by 0.13 mag within about 2.9 hr is noticed on 2013 December 30, while on 2014 December 02 night, brightness decreases continuously, with no peak. On 2014 December 3, flux rises by 0.08 mag within 2.4 hr during the total monitoring time of about 6 hrs. \n \n However, it is difficult to determine variability time scales accurately only from visual inspection of $LC$s  and therefore, in the next section we introduce and use structure functions and later analyze them to estimate required parameters.\n\n\\subparagraph{\\bf Duty Cycle of variation:}\n\\smallskip\nMost of the blazars show very high probability of variation even on intra-night time scales with an amplitude of variation of a few tenths of magnitude,  for example, CTA 102: \\citet{Bachev2017}, 3C 66A: \\citet[][ and references there-in]{kaur-3c2017}, S5 0716+714: \\citet{chandra2011}. In order to quantify the probability of variation in a source, duty cycle of variation (DCV) is often used. The DCV is defined as the fraction of  total number of nights the source is monitored for,  which are found variable \\citep{Romero1999}.  An expression to estimate DCV  is given by,\n\n\\begin{equation}\nDCV = 100 \\frac{\\Sigma_{i=1}^n (N_i \/ \\Delta t_i)}{\\Sigma_{i=1}^n (1 \/ \\Delta t_i)}  \\%\n\\end{equation}\nwhere, $\\Delta t_i$ = $\\Delta t_{i,obs} (1+z)^{-1} $ is duration of monitoring in rest frame of the source, \nand $N_i$ is $0$ or $1$ depending on whether the source is non-variable or variable, respectively. \n\n\\smallskip\nSeveral authors \\citep[][ and references there-in]{wagner1995, chandra2011, dai2015} have reported INV $DCV$ for  S5 0716+714 ranging from 40\\% to 100\\% during their observations, which indicates that the source is almost always active. In our case, 9-nights out of a total of 29 nights monitored for more than two hours, are detected as confirmed variable ones. Thus based on  our observations during 2013-2015,  we get a value of  31\\% as duty cycle,  which is on the lower side. Reasons could be that we monitored the source, by chance,  when it did not show much activity or our  duration of monitoring may not be sufficient.  \\citet{Hong2018} monitored the source for less than one hour and reported a DCV of 19.57\\% and, in another study done over 13 nights during 2012 January-February,  a value of 44\\% \\citep{Hong2017} was estimated, when the source was monitored for about 5-hours. In order to check for any connection between the INV shown by the source and the duration over which it was monitored, we calculated the duty cycle  with more than one hour  and two hour monitoring period. \n\nOut of the total 46 nights of observation during 2013-2015, we find 35 nights and 29 nights monitored for a minimum of  one hour and two hours, respectively. Based on these, we obtained  INV duty cycle values for the S5 0716+714 as 26$\\%$ and 31$\\%$, respectively, in two cases. It, therefore,  indicates that longer the duration of monitoring, higher will be the probability of finding a source variable, i.e., a higher DCV. \n\n\\noindent\n\\subparagraph{\\bf Rise \\&  fall rates of variation in INV lightcurves:} \n\\smallskip\nTo investigate the extent of the  intra-night variability  of the source, we determined rate of change in magnitude (rise\/fall) on each INV night for S5 0716+714 by fitting a line segment to light curves.  These rates of variation are given in  Table \\ref{t_risefall}. \n\n\n\\begin{table*}\n\\centering\n\\caption{Details of the rates of rise\/fall in the magnitude for  INV nights. $\\Delta{m}_{+}$,  $\\Delta{m}_{-}$ represent the source brightening or  dimming, respectively. }\n\\begin{tabular}{cllrc}\n\\hline\n\\hline\nDate &Trend &Rise\/Fall mag & Duration  & Rate=$\\Delta$m\/$\\Delta$t\\\\\n& &($\\Delta{m}_{+}$\/$\\Delta{m}_{-}$) &(in minutes) &(mag$\/$hr) \\\\\n\\hline\n12-02-2013 &Fall&0.05 ($\\Delta{m}_{-}$)& $~$ 198 \t&0.015 \\\\\n06-03-2013&Flickering over&0.08 ($\\Delta{m}_{+}$)&$>$150  & 0.02\\\\\n&a monotonic rise& & &\\\\\n07-03-2013\t&Fall  \t\t& 0.05\t($\\Delta{m}_{-}$)\t\t&$~$ 72 \t&0.03\\\\\n\t\t\t&Rise &\t0.12 ($\\Delta{m}_{+}$)\t\t& $~$144  \t&\\\\\n12-03-2013\t&Sine like \t&0.10 ($\\Delta{m}_{+}$)\t\t\t& 30  \t\t\t& 0.05\\\\\n\t\t\t&pattern\t\t\t&0.20 ($\\Delta{m}_{-}$)\t\t\t& $~$72 \t&\\\\\n\t\t\t&\t\t\t\t&0.20 ($\\Delta{m}_{+}$)\t\t\t& $~$72 \t&\\\\\t\n11-11-2013\t&Fall \t&0.08 ($\\Delta{m}_{-}$)\t\t& $>$ 72 \t\t&0.04\\\\ \n28-12-2013\t&Fall with\t\t& 0.08 ($\\Delta{m}_{-}$)\t\t&72  \t\t\t&0.38\\\\\n\t\t\t&flickering\t\t& 0.02\t($\\Delta{m}_{-}$)\t\t\t& $>$ 20 &\\\\\n30-12-2013\t&Monotonic rise\t\t&0.14 ($\\Delta{m}_{+}$)\t\t\t& 144  \t&0.07\t\\\\ \n\t\t\t\n02-12-2014\t&Monotonic fall\t&0.02 ($\\Delta{m}_{-}$)\t\t&216 \t\t&0.05\\\\\n03-12-2014\t&Sine like\t\t&0.08 ($\\Delta{m}_{-}$)\t\t&288 \t\t&0.02\\\\\n\\hline\n\\label{t_risefall}\t\t\n\\end{tabular} \n\\end{table*}\n\n                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           During our observations, 2013 February 12 and 2013 December 28 represent the nights with minimum and maximum rates of  change in the  magnitudes of the  source  with 0.015 mag hr$^{-1}$ and 0.381 mag hr$^{-1}$  (cf,Table \\ref{t_risefall}), respectively. The rate of brightness change on 2013 December 28 happens to be one of the fastest for this source. Earlier,  \\citet{chandra2011, man2016} have reported 0.38 mag\/hr  \\& 0.35 mag\/hr rates, respectively. Rate of change in the brightness magnitude as high as 0.43\/hr has been reported for the  PKS 2155-304 \\citep{Sandrinelli2014}. The source showed smooth decline in its brightness  by 0.05 mag on February 12 with 7.60 \\% amplitude of variability. On 2013 March 6, S5 0716+71 became brighter by 0.08 mag in 3 hours with rapid fluctuations (few tens of minutes duration) superposed over the day-long trend showing 7.58$\\%$ amplitude of variation in the light curve. On 2013 March 7, brightness decreases from 13.90 mag to almost 13.95 mag within an hour, after which source brightened by more than 0.1 mag in next 3 hours with a rate of change of 0.03 mag hr$^{-1}$ as mentioned in the Table \\ref{t_risefall}. On 2013 March 12, light-curve shows a sine like feature, with rising (0.1 mag in 30 min) - declining (0.2 mag in $~$72 mins) - rising ($>$0.2 in about 72 min approx.) trends in brightness over the duration of more than three hours.\n\n The light curve on 2013 December 28  showed sharp \nrise\/fall magnitudes over two peaks and again showed a rising trend with overall change in magnitude by 0.38 mag hr$^{-1}$  (see Table 1).  However, the features in the light curves are asymmetric in nature, which rules out variation being caused by extrinsic\/geometric mechanisms. The variability in blazars is stochastic in nature at almost all timescales. The flares, therefore, appear to be produced independently and any similarity or difference might reflect different scales of particle acceleration and energy dissipation \\citep{Nalewajko2015}. The variations in blazars are caused largely in the jet but it is difficult to ascertain whether these are intrinsic or geometric in nature. Intrinsic variations are dissipative and  irreversible in time. Hence they should cause asymmetric flares. The geometric variations, on the other hand, are symmetric in time \\citep{bachev2012} and achromatic in nature. The intrinsic variability could be due to fast injection of relativistic electrons and radiative cooling and\/or escape of the particles or radiation from the emission zone.  The symmetric flares, however, might result if cooling time scale is much shorter than the light crossing time \\citep{chatterjee2012}.  \n\n\n The  INV $LC$s are, in general,  asymmetric and complex indicating the random\/turbulent nature of the flow inside the jet. Based on the visual inspection of these curves, we identify three observed trends:\n \na) Rapid intra-night changes in the source flux, indicating to the violent, evolving nature of the shock formed in the jet. It might be either  due to the presence of oblique shocks or instabilities in the jet.\n\\smallskip\nb) The steady rise or fall in the light-curve during a night indicates to the light crossing timescale to be shorter than the cooling time scale of the shocked region. It is when data series is shorter than characteristic time scale of variability. The cooling times shorter than light crossing time would have resulted in symmetric light-curves \\citep{Chiaberg_Ghis1999, chatterjee2012}.\n\\smallskip\nc) The small-amplitude rapid fluctuations (asymmetric in shape) superimposed over slowly varying  light curve suggest small scale perturbations in the shock front or  oscillations in the hot-spots downstream the jet and may not be associated with size of the emission regions. \n\n \n\n\\begin{table*}\n\\textwidth=7.0in\n\\textheight=10.0in\n\\vspace*{0.5in}\n\\noindent\n\\caption{Details of the INV nights for  the  source S5 0716+714 during 2013-2015. Column 1 to 11 present; date, MJD, observation start time, duration, no. of images, average magnitude with error, test parameter C, amplitude of variation, variability time scale, SF parameters;  k \\& $\\beta$. }\n\\begin{tabular}{ccccccrrrcc}\n\\hline\n\\hline \nDate of \t   &MJD & $T_{start}$ &  Duration &  N\\footnote{Number of data points}    &  $\\bar{m}$ $\\pm$ $\\sigma$ & C  &  $A_{var}$  & $t_{var}$ &k &$\\beta$\\\\\n observation\t& \t& (hh:mm:ss)   & (hrs)\t &   &  & \t& ($\\%$)& \t\\\\\t\t\n\\hline\n\n 12-02-2013   & 56335.84038 & 20:10:09    & 3.28  & 195 & 13.94 $\\pm$ 0.02 & 2.63   & 7.60  & \\textgreater 2.36 hr &0.31 &4.82\\\\\n 06-03-2013  & 56357.96250 & 23:06:00    & 3.46  & 237 & 14.10 $\\pm$ 0.02 & 2.72   & 7.58& \\textgreater 1.68 hr &0.99 &1.67\\\\\n\n 07-03-2013 \t & 56358.82424\t& 19:46:54     & 3.71  & 203 &  13.90 $\\pm$ 0.03\t&  4.46 & 11.38&  2.04 hr &1.71 &1.81\\\\\n\n 12-03-2013\t& 56363.84308  & 20:14:02\t  & 2.83 \t\t& 229 & 13.50 $\\pm$ 0.05\t\t& 8.90\t& 15.61 &  \n1.11 hr &0.08 &1.21 \\\\\n 11-11-2013\t\t&  56607.03115\t&  00:44:51\t&  1.88 \t& 139  &  14.08 $\\pm$ 0.02\t&  4.62\t&  9.75 &  \n0.96 hr  &3.61 &1.33\\\\\n 28-12-2013\t& 56654.88889\t& 21:20:00\t & 2.48 \t& 284 & 14.72 $\\pm$ 0.02\t\t& 2.65\t& 11.89 &  \n0.76 hr  &1.14 &0.97 \\\\\n 30-12-2013\t\t& 56656.90536 & 21:43:43\t & 2.26\t & 350 & 14.30 $\\pm$0.05\t\t\t& 7.55\t& 15.38 &  3.1 hr &0.88 &1.31\\\\\n 02-12-2014\t\t& 56993.05133  & 01:13:05\t & 4.58\t& 284 & 13.41 $\\pm$0.06\t\t& 12.37\t& 20.50 \n&  3.54 hr \t&2.29 &2.34\\\\\n 03-12-2014\t\t& 56994.98155  & 01:13:55\t & 5.97\t& 454 & 13.27 $\\pm$0.03\t\t& 5.23\t& 10.07&  3.89 hr &2.32 &2.41\t\\\\\n&   &    &   &  &    &    &  &  & & \\\\\n\\hline\t\n\\label{t_idv}\t\t\n\\end{tabular}\n\\end{table*}\n\n\n\n\\subsubsection{\\bf Variability timescale, size of emission region and black hole mass}\n\nIt is important to know the characteristic timescale of intra-night variability which can constrain the emission size and structures of the blazar emission zones.  If we consider the shape of the jet as conical close to its origin, the opening angle and the extent of vertical expansion of the jet can provide us a rough estimate of the location of emission region with respect to the supermassive black hole \\citep{ahnen2017, nav1es2017}. \n\nThe rapid variations with duration of a few hours originate, perhaps, in the close vicinity of the central engine where jets are launched and might be caused by a combination of accretion disk instability, shock propagating within the jet, and\/or particle acceleration and consequent radiative cooling near the base of  jet \\citep{ulrich1997}. \nThis assumption is also used to estimate the mass of the black hole, which is difficult to determine otherwise as BL Lacs do not show emission lines.  Since the INV light curves are complex, we  use statistical tools, described here, to  discuss features in the intra-night light curves and estimate INV timescales and any possible quasi-periodicity.\n\n\\subparagraph{\\bf Structure function:} \n\\smallskip\n The structure function (SF) described by \\citet{simonetti1985, gliozzi2001} provides   information about characteristic timescale ofvariability for flat- and steep spectrum radio sources by analyzing their light curves. In order to estimate the characteristic variability timescale, we used first order structure function for a magnitude data series, defined as,\n\\begin{equation}\n\\label{eq5}\nSF(\\tau_{i}) = \\frac{\\Sigma{[M(t + \\tau_{i}) - M(t)]^{2}}}{N}\n\\end{equation}\nwhere, M(t) is the magnitude at time $t$ and $\\tau_{i}$ is the time lag. \nThe chi-square method is used to fit the structure function where from minimum variability timescale and corresponding errors are estimated \\citep{zhang2012}. The SF reveals extent of changes in the magnitude as a function of time between two observations. In this curve of growth of variability with time lag, a plateau (change of slope or saturation of SF) might indicate presence of a characteristic time.\n\n\n\\begin{equation}\n  SF(\\tau)= \\left \\{\n    \\begin{array}{c l}\n    k{\\tau}^{\\beta},   & \\tau\\underline{<}\\tau_o , \\hfill \\\\\n    C,   & \\tau>\\tau_o.\n  \\end{array}\\right\\}\n\\end{equation}\n\nwhere, $\\tau_{o}$ is characteristic timescale with 1$\\sigma$ uncertainty and $\\beta$ = $\\frac{dlog(F)}{dlog(\\tau)}$ is logarithmic slope in $\\tau-SF$ plane characterizing the nature of the variability and physical  processes. If the value of $\\beta$ is close to 0, it indicates flickering noise, while $\\beta$ $\\geq$ 1 indicates turbulent process in jet (or shot-noise) responsible for the changes. \n\n\n\\begin{figure*}\n\\includegraphics[width = 0.3\\textwidth]{sf120213.PDF} \n\\includegraphics[width = 0.3\\textwidth]{sf060313.PDF} \n\\includegraphics[width = 0.3\\textwidth]{sf070313.PDF}\n\\includegraphics[width = 0.3\\textwidth]{sf060313.PDF}\n\\includegraphics[width = 0.3\\textwidth]{sf111113.PDF}\n\\includegraphics[width = 0.3\\textwidth]{sf281213.PDF}\n\\includegraphics[width = 0.3\\textwidth]{sf301213.PDF} \n\\includegraphics[width = 0.3\\textwidth]{sf021214.PDF} \n\n\n\\caption{Structure functions for the INV nights are plotted for the source S5 0716+714 during 2013-2015. X-axis represents time lag in hours.}\\label{sf}\n\\end{figure*}\n\n\\smallskip  \nFigure 2 shows the structure functions for  the INV nights. It is seen that first order SF for several nights does not show any plateau, that means that the characteristic timescale of variability is  longer than the length of the observational data \\citep{dai2015}.  The local maximum following the smooth rise in the SF-$\\tau$ plane reveals time-scale of variability introduced by the presence of   minimum and maximum or vice-versa in the curve. If the SF consists of more than one  plateau with  slopes ($\\beta$) following a power-law trend, presence of multiple timescales is inferred.  If periodicity is present, it will be seen as local minima in SF after the occurence of a  local maximum. The difference between two minima gives the time period. \n\nOn several nights (2014-12-02, 2013-03-07, 2013-11-11,  \\& 2013-02-13), SF shows continuous increase with no or a feeble plateau, indicating that characteristic time scale of variability is longer than the monitoring period, giving only a lower limit of the variability time scale. While $LC$ for the night 2013 March 12 shows several features, SF shows only one plateau and then a dip at about 2.2 hr. On December 13, 2013, SF shows a discernible peak with a time scale of 3.11 hr, followed by a rise. SF for INV night of 2014 December 14 shows a plateau at 2.89 hr. \nAs can be seen, INV night 2013 December 28 shows a plateau in its SF giving the shortest characteristic timescale of variability of about 45.6 minutes during our observing  campaign. \n\n The quantitative values of various parameters related to INV nights, including SF parameters,  are given in Table 2.  In this table,  the column 1 and 2 represent the date of observation (in dd-mm-yyyy and MJD format, respectively), column 3 represents the start time of the observations, the duration of monitoring and the number of data points (images) are given in column 4 and 5, column 6 presents the average magnitude and the associated errors in the source brightness, column 7 and 8 contain the values of the  statistical parameters, i.e., C and $A_{var}$, respectively.  The variability timescales for INV nights are shown in column 9 along with the structure function parameters, normalization constant(k) and logarithmic slope($\\beta$) in columns 10 $\\&$ 11, respectively.\n \nThe INV lightcurves feature several complete events with specific time scales. Applying light travel time arguments, these time scales can be used to put  limits on the size of the emission regions responsible for the variation in flux. The shortest characteristic timescale puts constraint on the size of emission region \\citep{elliot1974}. Using the characteristic timescales obtained from light curve and SF analysis, the size of the emission region,\n\n\\begin{equation}\n\\label{R}\nR \\leq \\frac{\\delta c \\Delta{t_{var}}}{(1+z)}\n\\end{equation}\n\nwhere, $c$ is speed of light, $\\Delta{t_{var}}$ is minimum timescale of variability, $\\delta$ is the Doppler factor and $z$ is the redshift of the source (z$=$0.32).  When considering long-term behaviour, various authors have used different values of the Doppler factor, $\\delta$.  \\citet{Bach2005} use Doppler factors 13 to 25 when viewing angle changes from 5$^\\circ$ to 0$^\\circ$.5. \\citet{nesci2005}  adapted  a value of 20, while \\citet{fuhrman2008}  apply a range 5 to 15 for the Doppler factor. We have used a value of 15 in this work, taking into account  the brightness of the source during the observed period. Thus, using $\\Delta{t_{var}}$= 45.6 minutes as the characteristic time scale of variability, the estimated size of the emission region  is of the order of $\\approx$ $10^{15}$cm. Apart from this shortest time scale of variability, other time scales estimated on  other nights indicate different sizes of emission regions in the jet. The longest time scale of variability detected in the present study is 3.89 hr which corresponds to a size of  4.8 $\\times 10^{15}$ cm in the source frame. All these emitting regions are very compact and close to the black hole, within the BLR region.\n\nMass is one of the most important properties of a black hole. There are two categories of methods to determine the mass of a black hole in AGN; primary and secondary.  While there are direct, primary  methods applicable to nearby black hole systems  where motion of the surrounding  stars and gas under the influence of black hole, are traceable \\citep{vestergaard2004}, it is very difficult to have an estimate of their masses at high redshifts. In the secondary methods, mass of the black hole is estimated by resorting to  approximations, e.g., using a parameter with which black hole mass is correlated. There are several methods which fall into this category. However, for the sources  which  do not show any emission line and whose host galaxy is also weak\/non-detectable, which is the case for BL Lac type sources, it becomes extremely difficult to estimate the mass of black hole. For such systems, the variability time scale can provide a rough estimate of the black hole mass,  assuming that the shortest time scale of variation is governed by the orbital period of the inner most stable orbit around a Kerr (maximally rotating) black hole. \\citet{miller1989} claim the origin of microvariability to arise from a location very close to the central engine based on the fast variability time scales, while \\citet{marscher1992} associate their location somewhere down the jet and perhaps near the sub-mm core , caused by turbulence.  \n\nMany authors have estimated masses of black holes residing  in the BL Lac sources following the earlier \\citep{miller1989} approach using shortest variability time scales \\citep{Fan2005,gupta2008, Rani2010, chandra2011, kaur-3c2017,  Xie2002, liang2003,   dai2015}. \nHere we use this method to estimate the mass of  a Kerr black hole at the center of S5 0716+71 using the expression \\citep{abram1982, Xie2002}\n\n\n\\begin{equation}\nM =1.62 \\times {10}^4 \\frac{\\delta \\Delta t_{min}}{1+z} {M}_{\\odot},\n\\end{equation}\n\nwhere,  $c$ is the speed of light, z, the redshift and $\\delta$ is the Doppler factor. Taking the shortest variability time scale, $\\Delta{t_{min}}$= 45.6 min and Doppler factor as 15,  we estimate the  mass of the Kerr black hole to be 5.6 $\\times 10^{8}\\, M_{\\odot}$,  which is in close agreement with other values including a value of $1.25\\times 10^8 M_{\\odot}$  \\citep{liang2003} obtained by using optical luminosity. \\citet{bhatta2016} linked plateau in the $LC$ to the characteristic timescale for developing outflow within the jet base, equivalent to the innermost stable orbit and obtained the value of black hole mass as $4\\times 10^9$ (maximally spinning BH) and   $3\\times 10^8\\, M_\\odot$ (lowest spin BH). \\citet{agarwal2016} obtained a value of 2.42$\\times10^9 M_{\\odot}$ for the black hole mass in S5 0716+714. \\citet{Hong2018} estimated mass of the black hole as $5\\times10^6\\, M_{\\odot}$ using 50 min QPV originated from the inner most orbit of the accretion disk.\n\\smallskip\nUsing  the black hole mass, M$_{BH}$, estimated here, the Eddington luminosity can be estimated from the following expression given by \\citet{witta1985},\n\\begin{equation}\nL_{Edd} = 1.3\\times 10^{38} (M\/M_{\\odot})  erg\\, s^{-1}\n\\end{equation}\n which, in case of the S5 0716+71 comes out to be about  7.28$\\times 10^{46} erg\\, s^{-1}$. \n\n\\subparagraph{\\bf  Quasi-periodic variability:} \n\\smallskip\nAnother very interesting albeit highly debatable issue is the possible presence of periodicity in the blazar light curves. Claims for their existence have been made in optical bands \\citep{lainela1999, fan2000}. A few INV $LC$s indicate  the presence of  possible quasi-periodic variations, also noticed in this source by \\citet{wu2005, gupta2008, Rani2010, poon2009, man2016}, with periods varying from 15 minutes to 1.8 hr. The presence of such features, if genuine,  can be explained by the light-house effect \\citep{Camenzind1992}, plasma moving in a helical magnetic field or micro-lensing effect etc. Recently \\citet{Hong2018} obtained 50 min QPV from the observations during 2005 - 2012 when S5 0716+71 was in a relatively fainter state. They opined that the QPV is caused by the activity in the inner most orbit of the accretion disk.   In the present case, the variations seen on timescales of a few hours with asymmetric profiles rule out the possibility of the micro-lensing as the  mechanism.  Here, flares could be  caused by a sweeping beam whose direction changes with time due to helical motion. To estimate variability timescales and\/or periodicity (if present) in our $LC$s, we use structure function and periodogram analysis.\n\nThe SF for the night of 2013-03-12 shows only one minima at about 2.2 hr while that for 2013-12-28 gives two minima at 1.25 hr and 2.3 hr, giving a possible period of about 1.2 hr.\n\n However, since these periods are either closer to the length of the data series and\/or flux enhancements are less than 3$\\sigma$, existence of periodicity is doubtful. Also, these features may not significantly represent departure from a pure red-noise power spectrum. Though the quality of the light curve presented here, in particular its dense sampling, is  good enough for a search of hour-long QPVs,  the fact that we did not find such QPV at a significantly high level to claim the detection, is  meaningful in itself. It implies that no persistent periodic signal exists in the source within the analyzed variability timescale domain. \n\n\n\\subsubsection{\\bf Variability amplitude ($A_{var}$) and the brightness state of source} \n\\smallskip\nIn order to find out whether extent of variability has any dependence on the brightness of the source, we calculated the amplitude of variability ($A_{var}$)  in R-band for all the nights monitored for long enough time to show a minimum of 3\\% amplitude variation (see Equation~ \\ref{eq4}).  The values of  $A_{var}$  are plotted against nightly averaged brightness magnitudes in R-band (Figure 3) for the duration of 2013 January to 2015 June. We notice larger variability amplitudes when the source was brighter. In blazars, the $A_{var}$  is indicative of the environment where turbulent plasma in the jet interacts with frequent shock formations where relativistic electrons are accelerated in the magnetic field which then cools down  leading to synchrotron radiation.  During this period (2013-2015), the source was in a relatively more active phase showing average R-band magnitude of 13.22 $\\pm$ 0.01mag (historical average R = 14.0) and therefore one would expect larger amplitude of variation in the active jet. When the source is relatively faint, thermal emission from the host galaxy is expected to dilute the intrinsic variation in the  jet emission,  resulting in smaller $A_{var}$. Several authors have reported a  similar behaviour to what we have noticed. \\citet{agarwal2016} and \\citet{yuan2017} notice a very mild trend of larger $A_{var}$ when the source was brighter. \\citet{montagni2006} estimated rates of magnitude  variation for 102 nights during 1996--2003 for S5 0716+71 and found faster ($\\sim 0.08\/h$) rates when source was brighter ($R < 13.4$), though the dependence was weak, compared to average rate of change ($\\sim 0.027 mag\/h$) irrespective of the state of the source brightness.  \n\n However, just the opposite  behaviour has been detected  by \\citet{kaur-3c2017} in another IBL, 3C66A,  i.e., larger amplitude of variability when the source was relatively fainter. Similar trend was reported by \\citet{Chandra2013th, Baliyan2016} in a long-term (2005- 2012) study on the blazar S5 0716+714, reporting larger amplitude of variation when source was relatively fainter. Perhaps more extensive study on several blazars is needed to address this issue.\n\nThe behaviour of amplitude of variation as a function of the source brightness  also provides a clue to how the LTV and INV could be related. When the source is bright, it indicates that the  relativistic shock is propagating through the larger scale jet leading to enhanced flux at the longer time scale (LTV) \\citet{Romero1999}. The interaction of the shock with local inhomogeneities (small scale particle or magnetic field irregularities) or turbulence interacting with the shock \\citep{marscher1992} is perhaps giving rise to the intra-night variations (INV). Since we notice an increase in the amplitude of  INV with an increase in the  mean brightness of the source, later being seen due to LTV, there is perhaps a relation between LTV and INV.  A statistical study on a number of sources with good quality long-term data showing INV, STV and LTV on a large number of nights would, perhaps reveal whether INV amplitudes really have any correlation with the long and short term variability amplitudes. Certainly, S5 0716+714 would qualify as one such candidate for the study. \n\n\\begin{figure}\n\\includegraphics[width = 0.4\\textwidth]{Avar_mag_S5716_2013-15.pdf}\n\\caption{The amplitude of variability as a function of the average R-band brightness during 2013-2015.}\n\\label{ampvar}\n\\end{figure}\n\n\n\\subsection{Long-term variability}\nThe long-term optical light curve constructed for the period 2013 January-2015 June for the S5 0716+714 is shown in Figure 4, with time in MJD and B,V,R \\& I brightness in magnitude.  A total of  46 nights  with  6256, 159, 214, \\& 177  data points in R, B, V, \\& I-bands, respectively, are used in generating these $LC$s. The source was in its faintest state with 14.85(0.06) mag in R-band on MJD 56663.02 (2014 Jan 6)  and in its brightest state with 11.68(0.05) mag, almost one year later on MJD 57040.90 (2015 January 18). \n S5 0716+71 has undergone several outbursts and flares during its two \\& half year journey with two major outbursts peaking in 2013 March and 2015 January, having a duration of about 350 \\& 510 days, respectively (see Figure \\ref{long}).\n An outburst here is defined as a significant (more than one magnitude) enhancement in the flux over a considerable duration- tens of days to a few months or longer. In our case, limited by the observational data (ours and those from Steward Observatory), we have estimated the outburst duration, looking at the trend of long term variation, based on the above criterion. These outbursts are superposed by a number of fast flares.\nWe estimate long-term variability (LTV) amplitudes of  about 2.5 mag and 3.45 mag  with time scales of 250 and 360 days, respectively, during these two outbursts. These LTV time scales are estimated with respect to the minimum and maximum brightness values of the source during the two outbursts. During the 2015 January  outburst, S5 0716+714  reached its unprecedented brightness level \\citep{atel2015, chandra2015}.  Using multi wavelength data from Fermi-LAT, Swift-XRT, Swift UVOT, MIRO (optical R-band), Steward optical R-band and polarization data, we \\citep{chandra2015} detected two sub-flares contributing to this major 2015 January outburst. In optical, the source brightened by 0.8 mag in 6 days (MJD 57035--57041) and, post flare,  decayed over next four days at a rate of ~0.13 mag per day. Very sharp drop in brightness within a day (MJD 57040--57041) and subsequent rise in brightness the very next day (MJD 57042) indicated to the  presence of two sub-flares with almost same peak flux during the outburst. A rapid swing in the position angle of polarization indicated to the magnetic reconnection \\citep{zhang2012} in the emission region, causing the outburst. \n \nIn the long term, the source became fainter within a year from its average brightness, R = 13.5 mag in 2013 to R $\\sim$ 15 mag in 2014 January. It was then in the brightening phase  during  2014 to 2015 with intermittent  flaring activity. S5 0716+71 attained brightest flux value in 2015 January and started its journey towards fainter side later as reflected in all the B, V, R, and I-band (from R = 11.68 mag to 13.20 mag, a 1.52 mag decay in five months, c.f., Figure \\ref{long}) $LC$s.  In addition to major outbursts, there are at least 9-flares with their duration ranging from 20 days to 30 days leading to changes in brightness of the source  from a few tenths of magnitude to as much as more than 1.5 magnitude in R-band. The frequent large gaps in the data restrict us from appropriately characterizing these flares which indicate that the source remains almost always active with substantial brightness changes.  During our observation period, S5 0716+714 was brightest on 2015 January 18 (MJD= 57040.90) with a value R= 11.68$\\pm0.05$ and faintest on 2014 January 06 (MJD=56663.02) with a value R=14.85$\\pm 0.06$.\n\n\nThere are several approaches to explain the variation at various time scales- long as well as short. The intrinsic variations could be caused by the instability \\& hot spots in the disk or its outflow \\citep{kawaguchi1998, Chakra-wiita1993}, and activities in the relativistic jet\\citep{MarscherGear1985, marscher2008}. Variations could also be caused by the processes extrinsic to the source, e.g., interstellar scintillation- which are highly frequency dependent and normally affect long wavelength radio observations, gravitational microlensing - might cause long-term variations in some source but will result in achromatic, symmetric lightcurves. The later is less likely to cause INV \\citep{wagner1995}. Since S5 0716+714 was in a relatively bright phase and emission is strongly jet dominated, most probable source of variation should be processes in the jet. The shock-in-jet model \\citep{MarscherGear1985, marscher2008} is normally able to explain a variety of variability events with some modifications \\citep{zhang2015, Camenzind1992}, where a shock propagates down the jet interacting with a number of particle over-densities or stationary shocks\/cores distributed randomly in the parsec scale jet. Such standing shocks are formed due to pressure imbalance between jet plasma and inter-stellar medium (ISM). In trying to maintain a balance, an oblique shock is created perpendicular to the jet axis. The relativistic shocks interacting or passing through such regions energize the particles in the presence of magnetic field, which then radiate synchrotron emission while cooling. Either the jet moves in a helical motion or the blob moves in a helical magnetic field,  causing a change in the viewing angle, thus changing the Doppler factor which significantly enhances\/reduces the intrinsic flux variation, depending upon the decrease\/increase in the angle. The model can explain rapid variations by resorting to  a jet-in-jet scenario\\citep{zhang2015, Camenzind1992}.  The small flares before the outburst indicate  acceleration\/cooling of the relativistic particles due to plasma blobs interacting with shock front and an ongoing activity in the jet due to the superposition of all the events leading to long-term variability in the source. In our study, after the second outburst, the source enters into a faint state again.\n\nSeveral studies have been carried out to address the long-term behaviour of the source. \\citep{Raiteri2003, nesci2005} used the historical data, including from the literature, during 1953--2003 and noticed alternate trends of decreasing and increasing mean brightness on a tentative period of about 10 years, claiming precession of the jet to be  responsible for them. It should be noted that even during these slow trends of increasing\/decreasing mean brightness levels, source was very active with a large number of flares superposed on the longer trends. Again, a decreasing trend was noticed by \\citet{Chandra2013th} beginning 2003 which continued upto 2012, they also predicted an increase in average brightness after 2014.  \\citet{agarwal2016} observed the source for 23 nights during 2014 November-- 2015 March and found the source in bright state, showing INV for 7 out of 8 nights and STV with 1.9 mag change during about 28 days (MJD 57013.86 -- 57041.34). In their long-term work for the period 2004 -- 2012, \\citet{dai2015} reported STV at 10 days time scale, 11 INV nights out of 72 nights observed with an average magnitude of R=13.25 and an overall change by 2.14 magnitude.\n\n\n\n\\begin{figure*}\n\n\\includegraphics[width = 0.8\\textwidth]{longterm_MIRO_Stew_s5716.pdf}\n\\caption{Long-term B, V, R \\& I band light curves of S5 0716+714 for the duration 2013 January - 2015 June. Data used are from MIRO and Steward observatory. The source has undergone in the brightest and the faintest phases, during 2013$--$ 2015, exhibiting R-band magnitude of 11.68(0.05) and 14.85(0.06) respectively.}\n\\label{long}\n\\end{figure*}\n\nA complete observation log along with daily averaged R band photometric magnitudes for S5 0716+714  and nature of the night, are provided in Table \\ref{t-comp}.\n\n\n\n\n\\begin{table*}\n\\caption{Observation log and photometric results for  S5 0716$+$714 in R-band during 2013 January-2015 June. Columns are: Date of observation, Time (UT) \\& MJD, No. of data points, average magnitude with standard deviation  $\\&$ Photometric errors, variable (Y\/N)}\n\\textwidth=7.0in\n\\textheight=10.0in\n\\vspace*{0.5in}\n\\noindent\n\\begin{tabular}{cccrcrrc} \n\\hline \n\\hline    \\nonumber\nDate\t & $T_{start}$ & MJD   & N\\footnote{Number of data frames} & $\\bar{m}$  & $m(\\sigma)$ & $E_{phot}$   &Variable  \\\\\n(dd-mm-yyyy) & (hh:mm:ss) &   &  &(Avg mag) &  mag& mag &(Y$\/$N)\\\\\n\\hline\n14-01-2013  \t&21:30:14   &56306.89600\t &785  &13.478  \t&0.01\t&0.01  & N \\\\\n12-02-2013 \t&20:10:09\t&56335.84038\t&195\t  &13.943  \t&0.02\t&0.01\t & Y\\\\\n06-03-2013  \t&23:06:00\t&56357.96250\t&237\t  &14.007\t&0.10\t&0.03  & Y\\\\\n07-03-2013  &00:03:50\t&56358.00266\t\t&203  &13.996  &0.10    &0.01  &Y\\\\\n10-03-2013  &19:37:50   &56361.81794 \t       &231  &13.356   &0.01\t&0.003  &N \\\\\n12-03-2013  &20:14:02\t&56363.84308    \t&229\t  &13.505\t&0.05\t &0.004  &Y\\\\\n13-03-2013  \t&23:38:51 \t&56364.98531\t&160  &13.660\t&0.01\t &0.005  &N\\\\\n11-04-2013  &19:49:09\t&56393.82580\t\t   &182  &12.563\t&0.03\t &0.02  &N\\\\\t\n12-04-2013  &19:46:43\t&56394.82411\t\t   &5  &12.780\t&0.02\t &0.02  &N\\\\\t\n11-11-2013 \t&00:44:51\t&56607.03115\t&139\t  &14.078\t&0.02\t &0.003  &Y\\\\\n26-11-2013 \t&03:20:25\t&56622.13918\t&13 \t  &14.310\t&0.01\t &0.003  &N\\\\\n27-11-2013  &01:48:39\t&56623.07545\t\t&25\t  &14.259\t&0.01\t &0.01  &N\\\\\n28-11-2013  &01:13:25 \t&56624.05098\t\t&169  &14.275\t&0.01 \t&0.01   &N\\\\\n29-11-2013 \t&01:07:09\t&56625.04663\t& 247 &14.337\t&0.01\t &0.01  &N\\\\\n30-11-2013  &02:29:12 \t&56626.10361 \t\t&107\t  &14.357\t&0.01\t &0.008  &N\\\\\n01-12-2013  &03:08:06\t&56627.13063\t\t&112  &14.485\t&0.01\t &0.02  &N\\\\\n02-12-2013  &02:32:13\t&56628.10571\t\t&200  &14.417\t&0.01\t &0.04  &N\\\\ \n03-12-2013  &02:12:03   &56629.09170\t\t&242\t  &14.463\t&0.01\t &0.03  &N\\\\\n05-12-2013  &00:19:24\t&56631.01347\t\t&230  &14.011\t&0.03\t &0.02  &N\\\\\n28-12-2013  &21:20:00\t&56654.88889\t\t&284\t  &14.718\t&0.02\t &0.007  &Y\\\\\n30-12-2013  &21:43:43\t&56656.90536\t\t&350\t  &14.304\t&0.04\t &0.005  &Y\\\\\n01-01-2014  &00:18:37\t&56658.01293\t\t&183\t  &14.423\t&0.01\t &0.004  &N\\\\\n05-01-2014 \t&02:10:57\t&56662.09094\t&50\t  &14.816\t&0.01\t &0.005 &N\\\\\n06-01-2014  &00:43:03\t&56663.02990\t\t&349\t  &14.855\t&0.06  &0.006   &N\\\\\n26-04-2014  &20:26:39   &56773.85184\t\t&10\t  &13.924\t&0.05\t &0.006 &N\\\\    \n27-04-2014  &20:08:22\t&56774.83914\t\t&05\t  &13.969\t&0.01\t &0.008 &N\\\\\n22-11-2014\t&20:08:34   &56983.83929\t&49\t  &13.143\t&0.02\t &0.002  &N\\\\\n23-11-2014  &18:47:49\t&56984.78321\t\t&207  &13.212\t&0.06\t &0.005 &N\\\\\n02-12-2014  &01:13:55\t&56993.05133\t&284  &13.404\t&0.06\t &0.01 &Y\\\\\n03-12-2014  &01:16:34\t&56994.05317\t\t&454  &13.276\t&0.04\t&0.006  &Y\\\\\n22-12-2014  &01:41:07\t&57013.07022\t&579\t  &13.456\t&0.08\t &0.01 &N\\\\\n18-01-2015  &18:15:04     &57040.90131  &105\t &11.681  &0.05  &0.05 &N\\\\\n19-01-2015  &16:32:33  \t&57041.68927\t&442\t  &12.114\t&0.02\t &0.006\t &N\\\\\n20-01-2015  &18:49:11\t&57042.78417\t\t&06\t  &12.087\t&0.04\t &0.002  &N\\\\\n22-01-2015  &14:42:59\t&57044.61319\t&100  &12.063  &0.02\t &0.004 &N\\\\\n23-01-2015  &16:37:47\t&57045.69292\t&934\t  &11.776  &0.03\t &0.004  &N\\\\\n24-01-2015  &22:06:24\t&57046.92112\t&240\t  &11.727\t&0.02\t &0.01 &N\\\\\n28-01-2015  &17:19:42\t&57050.72202\t\t&557\t  &12.398\t&0.02\t &0.002 &N\\\\\n29-01-2015  &19:03:20\t&57051.79399\t&401\t  &12.518\t&0.01\t &0.01 &N\\\\\n30-01-2015  &15:40:12\t&57052.65292\t&513\t  &12.416\t&0.02\t &0.01 &N\\\\\n31-01-2015  &16:35:46\t&57053.69152\t&727\t  &12.726\t&0.05\t &0.01 &N\\\\\n01-02-2015  &20:38:28\t&57054.86005 \t&44\t  &12.837   &0.02\t &0.01 &N\\\\\n25-05-2015  &21:49:50\t&57167.90961\t&02\t  &12.979\t &0.07\t&0.005 &N\\\\\n27-05-2015  &20:35:16\t&57169.85782\t\t&05\t  &12.626\t &0.01\t&0.02 &N\\\\\n30-05-2015  &20:38:38\t&57172.86016\t&10\t  &13.449\t &0.04\t&0.01 &N\\\\\n31-05-2015  &20:22:47\t&57173.84916\t\t&15\t  &13.315\t &0.01\t&0.003 &N\\\\\n01-06-2015  &20:13:27\t&57174.84267\t\t&18\t  &13.171\t &0.05\t&0.003 &N\\\\\n\\hline\n\\end{tabular} \n\\label{t-comp}\n\\noindent\n\\end{table*}\n\n\n \n\n\\subsubsection{Spectral behavior of S5 0716+71}\n\nThe variation of color with the brightness of the source provides useful clues to constrain the blazar emission models \\citep{hao2010}. To investigate the spectral behavior of  S5 0716+714 over long timescale i.e., from 2013 to 2015, the color-magnitude diagrams,  (B-R) v\/s R,  (B-V) v\/s V and (V-R) v\/s R, are plotted using  nightly averaged magnitudes in B, V and R bands, respectively. The minimum and maximum values of the colour indices for better sampled case of B-R v\/s (B+R)\/2 are, 0.40 and 1.3, respectively, while the color average is $<B-R>$ = 0.6 mag, with standard deviation $\\sigma$=0.14 mag.   \n\\smallskip\n The Figure 5 shows the spectral behavior of the source with the brightness  during 2013$--$2015. The first panel (from top) shows the (B-R) spectral color versus its average magnitude. Similarly, the middle and the bottom panels display the (B-V) and (V-R) spectral behavior with their average brightness magnitudes, respectively. \nTo quantitatively determine the correlation between the color index  with brightness in Figure 5 (Color v\/s magnitude),  we performed regression analysis by fitting a straight line, y $=$ mx $+$ c (y = color index, x = average magnitude) using linear model in R software package and extracted various parameters, such as,  intercept(c), slope(m), correlation coefficient(r), p-value etc. The values for these parameters are given in Table \\ref{tabcol}. \n\n\\begin{table*}\n\\centering\n\\caption{The values of the regression parameters for color indices  as a function of brightness for S5 0716+714 during 2013-15.}\n\\begin{tabular}{cccccc}\n\\hline\n\\hline\nColor Index\t& m \t& c\t&$r^2$\t&r &p  \\\\\n\\hline\nB-R &0.08 $\\pm$ 0.03\t\t&-0.22 $\\pm$ 0.45\t&0.26\t&0.51\t& 0.02     \\\\\nB-V\t&0.02 $\\pm$ 0.02\t\t&0.16 $\\pm$ 0.34\t\t&0.06\t&0.25\t& 0.29     \\\\\nV-R\t&0.05 $\\pm$0.03\t\t&-0.37 $\\pm$ 0.48\t&0.12\t&0.35\t& 0.13     \\\\\n\\hline    \n\\end{tabular}\n\\label{tabcol}\n\\noindent \n\nm = Slope of regression line, $r^2$ = square of Pearson correlation coefficient, p = Probability for null hypothesis.\n\\end{table*}\n\n\nIt is clear from the Figure \\ref{allcol} and values of various parameters obtained from regression analysis (see Table \\ref{tabcol}) that the source showed weak positive correlation for B-V and V-R color indices  plotted against brightness magnitudes,  with Pearson correlation coefficient (r) of 0.25 and 0.35 along with p-value of 0.29 and 0.13 respectively. However, comparatively stronger positive correlation for B-R color index versus average magnitude of source with Pearson coefficient $r$ as 0.5 and null hypothesis probability, $p$ = 0.02 are noticed. Thus present study suggests a bluer-when-brighter (BWB) color for S5 0716+71 (cf.,  Figure \\ref{allcol})  as also reported by many workers \\citep{poon2009, chandra2011, wu2009, man2016}. \\citet{li2017} statistically studied the data for S5\\, 0716+71 during 1995-2015 and addressed the issue of long-term, short-term and INV behaviour of the symmetry in flares and color and found flares as asymmetric in general  and BWB color on all the time scales considered. The spectral changes in S5 0716+714, and blazars in general,  are complicated and difficult to explain. The source was reported with strong BWB trend over long-timescales \\citep{poon2009} and during its flaring phase \\citep{ghis1997, wu2005, wu2009, Gu2006}.  \\citet{wu2005} and \\citet{agarwal2016} discussed color trends in their studies but did not find any change on  the intra-night or long-term timescales.  \\citet{Raiteri2003}, on the other hand noticed all possible scenarios, i.e., BWB, RWB and no trend at all,  in their studies. \\citet{stalin2009} found source showing no color dependence with brightness on both - the long and short time scales, albeit a BWB color on intra- and inter-night timescales was noticed.  The fresh injection of high energy particles in the emission region inside the jet might lead to BWB behaviour \\citep{ghis1997, Raiteri2003, Gu2006}. The BWB behaviour can be explained by  the shock-in-jet model \\citep{MarscherGear1985, marscher2008}  where the propagation of a disturbance downstream the jet gives rise to the shock formation and the lag between the emissions at different wavelengths provides information on their relative spatial separations. In the case of  BL Lacs, the higher frequency electrons close to the shock front undergo faster radiation losses than the low-frequency ones. The BWB behavior basically means that the flux enhancements are produced either during the episodes of  intense particle acceleration or, alternatively, by the fluctuating magnetic field superimposed on the local, steady electron energy distribution. The redder when fainter trend indicates that when the jet is not dominant, the contribution from the  disk emission or the host galaxy becomes relevant. These cases show the complex color behaviour of the source with  brightness. However, in case of the S5 0716+714 where host galaxy is several magnitudes fainter,  R \\textgreater  20 mag \\citep{montagni2006}, thermal contribution from the host is negligible. \n\n\\begin{figure}\n\n\\includegraphics[width = 0.35\\textwidth]{BR_color.pdf}\n\\includegraphics[width = 0.35\\textwidth]{BV_color.pdf}\n\\includegraphics[width = 0.35\\textwidth]{VR_color.pdf}\n\\caption{Color-magnitude plot for the source S5 0716+714 during 2013-2015, showing bluer when brighter trend. The fit is obtained by performing the linear regression analysis and values for the parameters are given Table \\ref{tabcol}.}\n\\label{allcol}\n\\end{figure}\n\n\n\\subparagraph{\\bf Spectral variation with time:} \n \\smallskip\nThe long-term optical light curves of blazars  manifest significant details on the nature of the source as these contain various phases in their brightness, color changes during flares, outbursts  and fainter states. Several authors have looked at spectral variations with time on intra-night and inter-night timescales for blazars \\citep{Raiteri2003, wu2005, stalin2009, gaur2012, Rani2010, agarwal2016} and reported a mixed behaviour- some sources showing color dependence while others showed no change in color over considered timescales. \\citet{agarwal2016}, during their 130 day study, found a change of about 0.3 mag in spectral color with no significant dependence on the time or brightness phase.  \\citet{yuan2017} reported a complex pattern for spectral index with time without any specific trend during the period  2000-2014. The color variations are caused by differential cooling of energetic electrons behind the shock front. The relativistic shock moving down the jet accelerates electrons to high energies at the sites of high magnetic field or electron density giving rise to emission at diverse frequencies. In BL Lacs higher energy electrons cool faster with larger change in  flux with time during a flare.  Since the regions of the plasma over-densities or quasi-stationary shocks are randomly distributed in the jet, the interaction of the relativistically moving knot with existing features in the large scale jet gives rise to multiple outbursts which evolve individually and perhaps differently. The processes involved give rise to  changes in the spectral behaviour with time. \n\n In order to understand the  spectral behaviour of   S5 0716+714 with time (2013 January to 2015 June), we plot color index (B-R), (B-V) and (V-R) against the  time in MJD for this period in Figure \\ref{coltime}. To get the correlation between the color index and  time (MJD),  we also performed regression analysis by fitting a straight line, y $=$ mx $+$ c (y = color index, x = Time in MJD ) using linear regression software package and extracted various parameters, such as,  intercept(c), slope(m), correlation coefficient(r) and p-value.  A nicely sampled lightcurve in different optical bands should give a clear picture of the temporal evolution of S5 0716+714. However, our data suffer from substantial gaps and the observations in different bands are not truly simultaneous.   \n\n\\begin{figure}\n\\includegraphics[width = 0.35\\textwidth]{BR_color_time.pdf}\n\\includegraphics[width = 0.35\\textwidth]{BV_color_time.pdf}\n\\includegraphics[width = 0.35\\textwidth]{VR_color_time.pdf}\n\\caption{The color of the source S5 0716+714 plotted as a function of  time (MJD) during 2013-2015.  The continuous line is the best fit obtained using linear regression analysis of the data.}\n\\label{coltime}\n\\end{figure}\n\n \n  The Figure \\ref{coltime} shows a mixed behaviour. While we notice a significant bluer spectral  behavior with time in (B-V) color v\/s time plot, indicating  the source getting brighter at higher frequencies during this period of two and half years, we see very mild bluer trend in (B-R) v\/s Time.  However, the color index  $(V-R)$, shows a mild redder color with time. We, therefore, conclude, based on our data during 2013 January and 2015 June, that the source S5 0716+714 does not show any strong chromatic behaviour, barring (B-V) showing a  bluer behaviour  during this period. This mixed spectral behaviour with time is in line with other studies.  The source was relatively in bright phase during 2013, in low-phase during 2014 and  entered into its brightest phase in January 2015. \n\n\n\\section{Conclusions}\nThe IBL blazar S5 716 was observed for 46 nights with high temporal resolution during a period of more than two years (2013--2015) in optical BVRI wavelength bands from Mt. Abu InfraRed Observatory (MIRO). It was monitored for more than two hours during 29 nights to address INV. The nightly averaged B, V, R \\& I band brightness magnitudes with 6256, 159, 214 \\& 177 data points,  were used to discuss long-term variability and color behaviour of the source. The source exhibited intra-night as well as inter-night variability at significant levels. From the present study, following conclusions are drawn: \n\\begin{itemize}\n\\item Source showed variability over diverse timescales i.e., a few tens of minutes to months and a duty cycle of variation of more than 31\\%. The DCV appears to be dependent upon monitoring time. Two major outbursts with $\\sim$ 370 and 500 days duration superimposed with several flares are noticed. \n\\item The structure function analysis leads to the  shortest variability timescale of $~$45.6 minutes, based on which  upper limit on the size of emission region  of the order of $10^{15}$ cm is estimated. There are several time scales longer than this indicating to multi-sized emission regions in the jet. Based on the longest time scale,  the size of emission region is estimated as  $4.8\\times10^{15}$ cm.\n\\item Assuming the  rapid variations to be originated in the vicinity of central engine, black hole mass is estimated to be   5.6$\\times10^{8} M_{\\odot}$  using shortest variability time scale.\n\\item The structure function analysis is used to infer a period of about 1.2 hr on the night of 2013 December 28. However, it could easily be red-noise signature as flux enhancements are within 3$\\sigma$.\n\\item The source exhibited a bluer when brighter (BWB) spectral behaviour in the long term $LC$ which supports shock-in-jet model.\n\\item The brightness of S5 0716+71 shows a mild increase with time during 2013 January--2015 June along with a mild bluer color.\n\\item A larger amplitude of variation when the source was in relatively brighter state is detected, indicating to synchrotron dominated jet emission. It, perhaps, indicates that long-term and intra-night variabilities are linked.\n\\end{itemize}\nIt should be noted that these inferences are drawn from the data with large gaps. However, the data presented here should be very useful for other related statistical and modeling  studies on this very interesting source.\n\n\\section{Acknowledgement}\nThis work is supported by the Department of Space, Govt. of India. We are grateful to the anonymous learned referee for constructive remarks which improved the quality of this work.  We express our thanks to  Mr. Kumar Venkatramani  and past observers as well as MIRO staff  for their help in  observations. We also acknowledge the use the data from the Steward Observatory spectropolarimetric monitoring project which is supported by Fermi Guest Investigator grants NNX08AW56G, NNX09AU10G, NNX12AO93G, and NNX15AU81G \\citep{smith2009}. \n\n{\\it Facility:- MIRO:1.2m(PRL-CCD), MIRO:ATVS}\n\n\\bibliographystyle{apj}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\nThe search for any unambiguous signal of quark-gluon plasma (QGP) in heavy-ion collisions is motivated by properly investigating the behaviour and properties of hot and dense HG \\cite{Singh:1993}. Here we formulate a new thermodynamically consistent excluded-volume model where we assign a finite hard-core volume to each baryon but mesons in the theory can easily overlap, fuse and interpenetrate into each other. Earlier, we have used this model successfully in obtaining the conjectured QCD phase boundary and thus determining precisely the location of QCD critical end point \\cite{Singh:2009,Srivastava:2010}. We calculate various thermodynamical quantities like number density etc. of HG and compare our model results with that of URASiMA event generator \\cite{Sasaki:2001}. We use our freeze-out picture \\cite{Tiwari:2012} for calculating various hadron ratios and compare our results with the experimental data and various excluded-volume models. In order to make the discussion complete, we further derive $\\eta\/s$ etc. from our model and compare them with other models. Further, we extend our model to deduce the rapidity as well as transverse mass spectra of hadrons and compare them with the experimental data available in order to illustrate the role of flow present in the fluid. Finally, we give summary of this work.           \n\n\n\n\\section{Model Descriptions}\n\nRecently, we have proposed a thermodynamically consistent excluded-volume model for a hot, dense HG \\cite{Singh:2009,Srivastava:2010,Tiwari:2012}. The attractive interaction between baryons and mesons is realized by including the baryon and meson resonances in our model calculation. The repulsive interaction between baryons is modelled via giving an equal and finite size to each baryon. Mesons can fuse and interpenetrate into each other so, they are treated as pointlike particles. Using quantum statistics, the grand canonical partition function for baryons can be written as follows :   \n\n\\begin{equation}\nln Z_i^{ex} = \\frac{g_i}{6 \\pi^2 T}\\int_{V_i^0}^{V-\\sum_{j} N_j V_j^0} dV\n\\int_0^\\infty \\frac{k^4 dk}{\\sqrt{k^2+m_i^2}} \\frac1{[exp\\left(\\frac{E_i - \\mu_i}{T}\\right)+1]}\n\\end{equation}\nwhere $g_i$ is the degeneracy factor of ith species of baryons,$E_{i}$ is the energy of the particle ($E_{i}=\\sqrt{k^2+m_i^2}$), $V_i^0$ is the eigen volume assigned to each baryon of ith species. Apparently, our approach is more simple in comparison to other thermodynamically consistent excluded-volume approach \\cite{Rischke:1991} which often involve transcendental expressions and are difficult to solve. We determine chemical freeze-out temperature and chemical potential by fitting our results with some experimental data and then use them in determining all other quantities and ratios. \n\n\n The rapidity distributions of baryons using thermal source can be written as follows \\cite{Tiwari1:2012} :\n\n\\begin{equation}\n\\Big(\\frac{dN}{dy}\\Big)_{th}=\\frac{g_iV\\lambda_i}{2\\pi^2}\\;\\Big[(1-R)-\\lambda_i\\frac{\\partial{R}}{\\partial{\\lambda_i}}\\Big]\nexp\\left(\\frac{-m_i\\;coshy}{T}\\right)\\Big[m_i^2T+\\frac{2m_iT^2}{coshy}+\\frac{2T^3}{cosh^2y}\\Big].\n\\end{equation}\n\nwhere $m_i$ is the mass of the ith species. $V$ is the freeze-out volume of the system. The rapidity spectra of hadrons with the effect of flow can be calculated by using following formula \\cite{Tiwari1:2012} :\n\n\\begin{eqnarray}\n\\frac{dN_i}{dy}=\\int_{-\\eta_{max}}^{\\eta_{max}} \\Big(\\frac{dN_i}{dy}\\Big)_{th}(y-\\eta)\\;d\\eta,\n\\end{eqnarray}\nwhere $\\eta_{max}$ is a free parameter related with the longitudinal flow velocity ($\\beta_L$) \\cite{Tiwari1:2012}.\n\nOur excluded-volume approach involves hard-core repulsion arising between two baryons but mesons do not possess any such repulsion. We have taken an equal volume $\\displaystyle V^{0}=4\\pi r^{3}\/3$ for each baryon with a hard-core radius $r=0.8\\; fm$. We have also taken all baryons, mesons and their resonances having masses upto $2\\;GeV\/c^{2}$ in our calculation of HG pressure. We have also imposed the condition of strangeness neutrality by considering $\\sum_{i}S_{i}(n_{i}^{s}-\\bar{n}_{i}^{s})=0$ where $S_{i}$ is the strangeness of ith hadron.\n\n\\section{Results and Discussions}\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.3]{totalno.density.eps}\n\\caption{Variation of total number density of hadrons with respect to temperature at constant net baryon density. lines show our model calculation and points are the data calculated by Sasaki using URASiMA event generator.}\n\\label{fig9}\n\\end{center}\n\\end{figure} \n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.3]{lambda_pimi.eps}\n\\caption{The energy dependence of $\\Lambda\/\\pi^{-}$ ratio. Dashed line is the result of Cleymans-Suhonen model \\cite{Cleymans:1987}. Symbols are the experimental data \\cite{Tiwari:2012}. RHIC data are at mid-rapidity.}\n\\label{fig9}\n\\end{center}\n\\end{figure} \n\n\n\nIn Fig. 1, we have plotted the variation of number density of hadrons with respect to temperature at fixed net baryon density. Our model results show a very good agreement with the results of Sasaki \\cite{Sasaki:2001} except at higher $T$. Figure 2 represents the variation of $\\Lambda\/\\pi^{-}$ ratio with $\\sqrt{s_{NN}}$. We find that our model calculation gives much better fit to the experimental data at all energies in comparison to Cleymans-Suhonen model \\cite{Cleymans:1987}. Figure 3 depicts the variation of $\\eta\/s$ with respect to temperature as obtained in our model for HG \\cite{Tiwari:2012} having a baryonic hard-core size $r=0.5$ fm, and compared the results with those of Gorenstein $et\\; al.$ \\cite{Gorenstein:2008}. We find that near the expected QCD phase transition temperature ($T_{c}=170-180$ MeV), $\\eta\/s$ shows a lower value in our HG model than the value in other model. In Fig. 4, we show the rapidity distributions of $\\pi^+$ for central $Au+Au$ collisions at $\\sqrt{s_{NN}}=200\\; GeV$. Dotted line shows the distribution of $\\pi^+$ due to stationary thermal source. Solid line shows the rapidity distributions of $\\pi^+$ after the incorporation of longitudinal flow in our thermal model and results give a good agreement with the experimental data \\cite{Tiwari1:2012}.\n\n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.3]{eta_s_t1.eps}\n\\caption{Variation of $\\eta\/s$ with temperature for $\\mu_{B}=0$ in our model and a comparison with the results obtained by Gorenstein $et\\; al.$ \\cite{Gorenstein:2008}.}\n\\label{fig9}\n\\end{center}\n\\end{figure} \n\n\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.3]{pi200.eps}\n\\caption{Rapidity distributions of $\\pi^+$ at $\\sqrt{s_{NN}}= 200 GeV $. Dotted line shows the rapidity distribution calculated in our thermal model. Solid line and dashed line show the results obtained after incorporating longitudinal flow in our thermal model. Symbols are the experimental data \\cite{Tiwari1:2012}.}\n\\label{fig9}\n\\end{center}\n\\end{figure} \n\n\n\n\n\n\\section{Summary}\nIn summary, we find that our model results mostly show very close agreement with those of Sasaki, although the two approaches are completely different in nature. Our model calculations for the particle ratios describe the experimental data very well. Transport quantities are also successfully described by our model. The rapidity distributions which essentially are dependent on thermal parameters, also show a systematic behaviour and their interpretations most clearly involve the presence of a collective flow involved in the final description of the fireball. \n\n\\section{Acknowledgment}\nSKT is grateful to Council of Scientific and Industrial Research (CSIR), New Delhi for providing a research grant. \n\n\n\n\n\n\n\\noindent\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nTopological quantum materials, such as topological insulators (TIs) and Dirac or Weyl semimetals, have recently been extensively investigated due to their highly nontrivial properties, e.g. topological surface states, linear dispersion relation, and protection from back scattering, which make them potentially applicable for new technologies. \\cite{Qi2010a,Hasan2010,Moore2010,Ando2013,RevModPhys.90.015001}.\nAt the same time, unconventional physical phenomena have been observed in light driven nonequilibrium states \\cite{Wang2013,Giorgianni2016,McIver2019,RMAD2020}, in which the dynamical properties have been relatively less investigated \\cite{Oka19}.\\\\\n\\indent A direct probe of the time-dependent evolution of the surface states in a nonequilibrium TI system can be provided by time- and angle-resolved electron emission spectroscopy (trARPES) \\cite{Sobota2012,Hajlaoui2012,Crepaldi2012,Wang2012,Hajlaoui2014,Neupane2015}. Apart from the bulk states as in a conventional insulator, the surface states act as an additional channel for the system to relax back to the equilibrium state. \nDepending on the specific band structures, the lifetime of the nonequilibrium states in a TI can vary from a few picoseconds (ps), e.g. in (Bi$_{0.2}$Sb$_{0.8}$)$_2$Te$_3$, to the order of microseconds ($\\mu$s) such as in Bi$_2$Te$_2$Se \\cite{Neupane2015}.\nNonequilibrium dynamics of charge carriers in TIs has also been studied using time-resolved optical spectroscopic techniques.\nOptical-pump THz-probe (OPTP) measurements in Bi$_{2}$Se$_{3}$ thin films\nrevealed a negative change of low-frequency THz optical conductance which was attributed to the metallic response of the surface states \\cite{Sim2014}.\nBy studying thickness dependence of the OPTP response in Bi$_{2}$Se$_{3}$ thin films, \na shorter time scale of $\\sim 5$~ps was observed for thicker films which was ascribed to the dynamics of bulk charge carriers, whereas a longer time scale of $\\sim 10$~ps in thinner films was assigned to surface states \\cite{Aguilar2015}. \nUsing mid-infrared-pump THz-probe and THz-pump THz-probe spectroscopy on Bi$_{2}$Se$_{3}$ thin films, a negative change of the THz optical conductivity was observed for both situations \\cite{Luo2019}.\nThese results were explained by the co-existence of two types of electron plasmas, one for bulk and the other for surface states \\cite{Luo2019}. \nA non-linear dependence of the pump-induced change of THz electric field transmission of pump power was observed in the bulk-insulating TI Bi$_{1.5}$Sb$_{0.5}$Te$_{1.7}$Se$_{1.3}$, which was related to the relaxation through the interaction between bulk and surface states, in particular, the injection of photo induced bulk carriers into the surface states\\cite{Choi2018}.\\\\\n\\indent Here, using time-resolved mid-infrared-pump THz-transmission-probe spectroscopy, we study the nonequilibrium electronic dynamics in the TI systems (Bi$_{1-x}$Sb$_{x}$)$_2$Te$_3$ (BST) as a function of temperature and pump-pulse fluence. By varying the content of Sb, the charge carriers of the system bulk can be tuned to be of electron-type (\\textit{n}-type) or of hole-type (\\textit{p}-type) \\cite{Zhang2011}.\nThe nonequilibrium dynamics is further investigated by changing the pump-pulse energies which allows a selective excitation below or above the bandgap. \nWhile for the below-bandgap excitation the relaxation dynamics exhibits an exponential decay as usually expected for a noninteracting system, a clear deviation from the exponential decay is revealed for the above-bandgap excitation.\nPhenomenologically, the observed relaxation behavior can be described by a compressed exponential decay, hinting at strong interaction of nonequilibrium states\\cite{Izrailev2006}.\nIn addition, our measurements reveal a non-linear dependence on pump-pulse fluence for the above-bandgap excitation, in contrast to the linear dependence observed in the below-bandgap excitation measurements.\n \n\n\\begin{figure}[t]\n        \\center{\\includegraphics[width=0.45\\textwidth]\n        {Fig1.png}}\n        \\caption{\\label{fig1} (a) Schematic picture of band structure for the \\textit{n}- and \\textit{p}-type samples with CB, DP, VB and E$_F$, being the conduction band, Dirac point, valence band and Fermi energy, respectively. (b) Hall measurements of the \\textit{n}- and \\textit{p}-type BST samples at \\SI{2}{K} and \\SI{300}{K}. (c) Electric field of the THz probe pulse in the time domain, the inset illustrates the MIR pump THz probe experiment.}\n\\end{figure}\n\n\n\\section{Experimental details}\nThe BST thin films for our spectroscopic studies were grown on sapphire substrates by molecular beam epitaxy (MBE) and characterized by transport measurements \\cite{Yang2014}, to extract information about the charge carrier type and density.\nThe majority charge carriers are electrons or holes which are indicated by the negative or positive slope in the Hall measurements\\cite{Yang2014}, respectively, for the \\textit{n}- or \\textit{p}-type samples at \\SI{2}{K}, as shown in Fig. \\ref{fig1} (b).\nThese features are illustrated in the band structures in Fig.~\\ref{fig1} (a).\nThe thickness of the \\textit{n}-type and the \\textit{p}-type samples was \\SI{14}{nm} and \\SI{15}{nm}, respectively.\nThe electronic bangap in both samples was $\\approx$ \\SI{200}{meV} at \\SI{300}{K} \\cite{Ando2013,Yang2014, Zhang2011}.\nTo investigate the nonequilibrium dynamics, we performed mid-infrared pump THz-probe measurements in a transmission geometry as depicted in the inset of Fig. \\ref{fig1} (c).\nThe THz-pulses were generated by optical rectification\\cite{Ulbricht2011} of a \\SI{800}{nm} pulses at a repetition rate of \\SI{1}{kHz} from an amplified laser system in a \\SI{0.3}{mm} thick GaP crystal, and detected by electro-optic sampling \\cite{Ulbricht2011} in a GaP crystal with the same thickness. \nTime dependence of the electric field of the THz probe-pulse is shown in Fig.~\\ref{fig1} (c) with a maximum electric field strength of $\\approx$ \\SI{1.5}{kV\/cm}. \nThe complete THz beam path was kept inside vacuum to avoid water absorption.\nIn our experiment the pump pulses with a central energy of \\SI{150}{meV} and \\SI{500}{meV} were used for excitations below and above the bandgap, respectively. The \\SI{500}{meV} was generated from an optical parametric amplifier (OPA), while the \\SI{150}{meV} was generated by noncollinear difference frequency mixing using the signal and idler output of the OPA.\nMaximum powers of \\SI{0.4}{mW} and \\SI{16}{mW} with the temporal width of\n$\\approx$ \\SI{850}{fs} and $\\approx$ \\SI{120}{fs} were achieved for the \\SI{150}{meV} and \\SI{500}{meV} pump pulses, respectively.\nThe pump beam was modulated by a chopper for lock-in amplification, and the change of the peak electric field of the THz pulse was measured as a function of pump-probe time delay.\nThe samples were kept in a continuous flow cryostat for a temperature range from \\SI{4.2}{} to \\SI{300}{K}.\n\n\\section{Experimental results}\n\n\\subsection{\\textit{n}-type (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$}\n\n\\begin{figure}[t]\n        \\center{\\includegraphics[width=0.45\\textwidth]\n        {Fig2.png}}\n        \\caption{\\label{fig2} Pump-probe measurements on the \\textit{n}-type sample with \\textit{below-bandgap} excitation at \\SI{5}{K}. (a) Complete pump-probe trace at \\SI{27}{\\micro J\/cm^2} (symbols). The dashed line is a fit to a two-level system as described by Eq. \\ref{func1} and \\ref{func2}. (b) Fluence dependence at \\SI{5}{K} for four different fluences, the two-level-system fits (dashed lines) are shown as dashed lines with the same color as the corresponding data (symbols).}\n\\end{figure}\n\n\\begin{figure}[t]\n        \\center{\\includegraphics[width=0.45\\textwidth]\n        {Fig3.png}}\n        \\caption{\\label{fig3} \\textit{Above-bandgap} excitation measurements on the \\textit{n}-type sample. The dashed lines are fits with the compressed exponential function. (a) Fluence dependence measurements at \\SI{5}{K} with the corresponding fits as dashed lines. (b) Temperature dependence measurement at \\SI{1}{\\micro J\/cm^2} with the corresponding fits as dashed lines.}\n\\end{figure}\n\n\nFigure~\\ref{fig2} presents the results for the below-bandgap excitation in the \\textit{n}-type sample. Figure~\\ref{fig2} (a) shows $-\\Delta E\/E$, the pump-induced change of the peak THz electric field $\\Delta E$ relative to the unperturbed THz field $E$, as a function of the time delay $\\tau$ between the mid-IR pump pulse and THz probe pulse, measured at \\SI{5}{K} with a pump fluence of \\SI{27}{\\micro J\/cm^2}.\nUpon pump-probe overlap at $\\tau = 0$ the peak THz electric field is reduced up to a change of \\SI{-3}{\\percent} at $\\tau = \\SI{2}{ps}$.\nAfter this minimum, the signal relaxes towards its equilibrium value in about \\SI{20}{ps}, following an exponential decay.\nFigure~\\ref{fig2} (b) shows fluence dependence measurements at \\SI{5}{K}.\nWith decreasing pump fluence from \\SI{24}{} to \\SI{13.9}{\\micro J\/cm^2}, the maximum change of $-\\Delta E$ decreases by a factor of 2.\nThis dependence often is observed in pump-probe experiments because the change of the THz electric field typically depends on the number of excited charged carriers\\cite{Ulbricht2011}.\\\\\n\\indent Figure \\ref{fig3} shows the pump-probe measurements on the \\textit{n}-type sample for the \\textit{above-bandgap} excitation.\nThe maximum of $-\\Delta E\/E$ reaches up to \\SI{24}{\\percent} at a fluence of \\SI{15.9}{\\micro J\/cm^2}.\nEven at a much lower fluence of \\SI{1}{\\micro J\/cm^2} there is still a change of \\SI{4}{\\percent}, exceeding the \\SI{3}{\\percent} change of the below-bandgap measurement at the highest fluence of \\SI{27}{\\micro J\/cm^2} (see Fig.~\\ref{fig2}).\nIn contrast to the results with below-bandgap excitation, the signal exhibits a broader maximum and at high fluences (Fig.~\\ref{fig3} (a)) a change from an exponential decay to a slower decay. \nFurthermore, the maximum of $-\\Delta E\/E$ and the relaxation time both increase with increasing fluence, as observed in the below-bandgap measurements (Fig.~\\ref{fig2} (b)).\nFigure~\\ref{fig3} (b) shows the temperature dependence of the pump-probe signal at a fluence of \\SI{1}{\\micro J\/cm^2}.\nThe maximum of $-\\Delta E\/E$ decreases from \\SI{8}{\\percent} to \\SI{3}{\\percent} with increasing temperature from \\SI{5}{K} to \\SI{300}{K}.\nNoticeably, the relaxation in the above-bandgap measurement, in the lower fluence region, is almost twice faster, than in the below-bandgap measurement.\n\n\n\\subsection{\\textit{p}-type (Bi$_{1-x}$Sb$_x$)$_2$Te$_3$}\n\nFigure \\ref{fig4} (a) shows the pump-probe measurement on the \\textit{p}-type sample for the \\textit{below-bandgap} excitation at \\SI{5}{K} and \\SI{27}{\\micro J\/cm^2}.\nComparing this to the below-bandgap measurement of the \\textit{n}-type sample (Fig. \\ref{fig2}) shows differences in the relaxation behavior.\nFirstly, the \\textit{p}-type system relaxes within about \\SI{10}{ps}, while for the \\textit{n}-type sample it takes about \\SI{20}{ps}.\nSecondly, at the highest fluence of \\SI{27}{\\micro J\/cm^2}, the maximum induced change of the electric field is \\SI{1}{\\percent} in the \\textit{p}-type sample, hence evidently lower than the \\SI{3}{\\percent} change in the \\textit{n}-type sample.\nDespite these differences, the \\textit{p}-type sample (Fig.~\\ref{fig4} (b)) shows a similar dependence on the fluence as the \\textit{n}-type sample, i.e. $-\\Delta E$ and the relaxation time increase with increasing fluence.\n\n\\begin{figure}[t]\n        \\center{\\includegraphics[width=0.45\\textwidth]\n        {Fig4.png}}\n        \\caption{\\label{fig4} Pump-probe measurements on the \\textit{p}-type sample with \\textit{below-bandgap} excitation at \\SI{5}{K}. (a) Complete pump-probe trace at \\SI{27}{\\micro J\/cm^2}, black squares and red dashed line are the measured data and a two-level-system fit, described by Eq.~(\\ref{func1}) and (\\ref{func2}), respectively. (b) Fluence dependence \\SI{5}{K} for three different fluences, the two level fits are shown as dashed lines with the same color as the corresponding data.}\n\\end{figure}\n\n\\begin{figure}[t]\n        \\center{\\includegraphics[width=0.45\\textwidth]\n        {Fig5.png}}\n        \\caption{\\label{fig5} \\textit{Above-bandgap} excitation measurements on the \\textit{p}-type sample. The dashed lines are fits with the compressed exponential function. (a) Fluence dependence measurements at \\SI{5}{K} with the corresponding fits as dashed lines. (b) Temperature dependence measurement at \\SI{1}{\\micro J\/cm^2} with the corresponding fits as dashed lines.}\n\\end{figure}\n\n\nFigure \\ref{fig5} shows the pump-probe measurement on the \\textit{p}-type sample for \\textit{above-bandgap} excitation.\nThe fluence dependence measurements in Fig.~\\ref{fig5} (a) show a similar change in signal shape from exponential to a slower decay as the measurements on the \\textit{n}-type sample (Fig.~\\ref{fig3} (a)). \nWith increasing fluence from \\SI{1.6}{\\micro J\/cm^2} to \\SI{5.8}{\\micro J\/cm^2},\nthe signal amplitude increases from \\SI{4}{\\percent} to \\SI{18}{\\percent}, while the relaxation time increases from $\\approx \\SI{10}{ps}$ to $\\approx \\SI{15}{ps}$.\nFigure~\\ref{fig5} (b) shows that the pump-probe signal decreases from \\SI{4}{\\percent} to \\SI{1.5}{\\percent} with increasing temperature from 5 to 300~K.\nInterestingly, the \\textit{p}-type sample relaxes within $\\approx \\SI{10}{ps}$ in the above-bandgap excitation, which is a similar value as in the below-bandgap excitation.\nThis is in contrast to the \\textit{n}-type sample, where the above-bandgap excitation measurement relaxes within $\\approx \\SI{10}{ps}$ and the below-bandgap measurement within $\\approx \\SI{20}{ps}$ (see Fig. \\ref{fig2} and \\ref{fig3}).\\\\\n\n\\section{Discussion}\n\n\\begin{figure}[t]\n        \\center{\\includegraphics[width=0.45\\textwidth]\n        {Fig6.png}}\n        \\caption{\\label{fig6} Parameters of the fits on the fluence dependence measurements for the \\textit{below-bandgap} measurements at \\SI{5}{K} (see Figs. \\ref{fig2} and \\ref{fig4}). (a)  Max($-\\Delta E$) with the measured data in red squares and blue dots. The dashed lines are guide for the eye. (b) The relaxation time is linearly dependent on the fluence, the \\textit{p}-type sample relaxes twice as fast as the \\textit{n}-type sample.}\n\\end{figure}\n\n\\begin{figure}[t]\n        \\center{\\includegraphics[width=0.45\\textwidth]\n        {Fig7.png}}\n        \\caption{\\label{fig7} Parameter obtained from the compressed exponential fit (Eq. \\ref{func3}) of the \\textit{above-bandgap} excitation as a function of fluence for the \\textit{n}- and \\textit{p}-type sample (see Fig. \\ref{fig3} and \\ref{fig5}). (a) The \\textit{p}-type sample shows clear linear dependence on the fluence, while the \\textit{n}-type sample shows a nonlinear dependence. (b) The relaxation time shows a nonlinear dependence on the fluence and is similar in magnitude for both samples. (c) The compression parameter is constant within its uncertainty over the whole measurement at a value of $\\simeq \\SI{1.5}{}$ for both samples.}\n\\end{figure}\n\n\\begin{figure}[t]\n        \\center{\\includegraphics[width=0.45\\textwidth]\n        {Fig8.png}}\n        \\caption{\\label{fig8}  Parameter obtained from the compressed exponential fit (Eq. \\ref{func3}) of the \\textit{above-bandgap} excitation as a function of temperature for the \\textit{n}- and \\textit{p}-type sample (see Fig. \\ref{fig3} and \\ref{fig5}). (a) Max($-\\Delta E\/E)$ decreases linear with increasing temperature with a different slope for both samples, with the \\textit{n}-type sample showing the stronger decrease. (b) The relaxation time shows almost no temperature dependence in both samples. (c) The compression parameter shows a slight decrease with increasing temperature.}\n\\end{figure}\n\nTo quantify the differences for the different types of samples and the different excitation energies (see Fig.~\\ref{fig2}-\\ref{fig5}), we use phenomenological models to fit the results.\nIn the below-bandgap excitation measurements, one can clearly observe an exponential decay, while in the above-bandgap measurements the decay behavior deviates from an exponential one.\nAs shown in Fig.~\\ref{fig2} and \\ref{fig4}, the below-bandgap excitation measurements are well described by a system which relaxes exponentially. \nMax$(-\\Delta E)$ in Fig.~\\ref{fig6} (a) is the maximum value directly read from the experimental data as a function of pump fluence.\nThe fluence dependence exhibits a quasi-linear behavior, \nindicating that the excitation process is a linear one-photon process.\nBased on the relaxation, the linearity of the excitation, and the expected underlying band structure of BST we adopt a two-level model to describe the results for the below-bandgap excitations.\nThe relaxation dynamics of a two-level system is described by the following set of equations\n\\begin{align}\n\\label{func1}\n\\frac{\\text{d}C(t)}{\\text{d}t} &= +p(t)-\\frac{C(t)}{\\tau_1} \\\\\n\\label{func2}\n\\frac{\\text{d}G(t)}{\\text{d}t} &= -p(t)+\\frac{C(t)}{\\tau_1},\n\\end{align}\nwith $G(t)$, $C(t)$, and $p(t)$ being the population of the ground state, the excited state, and a Gaussian pulse, respectively, and $\\tau_1$ the time constant.\nThe change $\\Delta E$ is assumed to be proportional to the population of the excited state\\cite{Ulbricht2011}. \nAs one can see, the two-level-system description fits the relaxation of the signal well and the rise time of the signal is well described with an excitation with a Gaussian pulse (see dashed lines in Figs.~\\ref{fig2} and  \\ref{fig4}).\nIn Fig.~\\ref{fig6} (b) we show the relaxation time extracted from the \\textit{n}- and \\textit{p}-type samples. \nThe \\textit{p}-type sample with a relaxation time of \\SI{3}{ps} relaxes twice as fast as the \\textit{n}-type sample with a relaxation time of \\SI{6}{ps}, at the highest fluence of \\SI{27}{\\micro J\/cm^2}.\nThese relaxation times, in the order of a few ps, are often found for the more metallic TIs in literature\\cite{Choi2018,Cheng2014}, and this timescale is typically attributed to phonon mediated relaxation.\nThe difference in amplitude and relaxation time for the two samples could be due to the following difference of the underlying physical properties.\nOne possibility is that the effective mass of the charge carrier reported for the \\textit{n}- and \\textit{p}- type TIs are different\\cite{Witting2020, Taskin2011}, resulting in different scattering rates.\nAnother one is that the underlying excitation is a process involving states inside the bandgap, like the surface state, impurity bands or charge carrier puddles\\cite{Borgwardt2016,Tang2013,Zhang2011}.\\\\\n\\indent The above-bandgap excitation measurements show a clear deviation from an exponential decay (see Fig. \\ref{fig3} and \\ref{fig5}), therefore the corresponding relaxation dynamics needs a different description than the exponential decay.\nThe compressed exponential function was chosen because it fits well to the deviation from the exponential decay, that we observed in all the above-bandgap measurements (see dashed lines in Fig. \\ref{fig3} and \\ref{fig5}).\nThe compressed exponential function is defined as\n\\begin{align}\n\\label{func3}\n\\Delta E\/E = Ae^{-\\left(\\frac{\\tau-\\tau_0}{\\tau_1}\\right)^\\alpha},\n\\end{align}\nwith $A$, $\\tau_0$, $\\tau_1$, and $\\alpha$ being the amplitude, peak position, relaxation time, and compression parameter, respectively.\nThe amplitude $A$ is set to be the maximum change of $-\\Delta E\/E$, with $\\tau_0$ being the pump-probe delay corresponding to the maximum.\nThe fitting results are summarized in Fig.~\\ref{fig7} as a function of pump fluence.\nAs shown in Fig.~\\ref{fig7} (a), the maximum change of $-\\Delta E\/E$ in the \\textit{p}-type sample exhibits a linear dependence on the fluence at least up to \\SI{4}{\\micro J\/cm^2}, while for the \\textit{n}-type sample a saturation-like behavior is observed toward higher fluence above \\SI{4}{\\micro J\/cm^2}.\\\\\n\\indent In Fig.~\\ref{fig7} (b) one can see that the relaxation time increases from \\SI{1.5}{ps} at fluences of $\\approx \\SI{0.6}{\\micro J\/cm^2}$ to \\SI{6.2}{ps} at fluences of \\SI{5.8}{\\micro J\/cm^2} for the \\textit{p}-type BST.\nThe relaxation time of the \\textit{n}-type BST increases as well from \\SI{2.5}{ps} at fluences of $\\approx \\SI{0.6}{\\micro J\/cm^2}$ to \\SI{7}{ps} at fluences of \\SI{7.8}{\\micro J\/cm^2}.\nThe increase in relaxation time with increasing fluence is a dependence often observed in metals, which hints to a metallic nature of the electron dynamics\\citep{Choi2018}.\nThis metallic nature of the electron dynamics has been shown to exist in other TIs\\cite{Cheng2014}.\nIn combination with the \\si{ps} timescale of the relaxation time this indicates that the electron relaxation is predominantly through the scattering with phonons \\cite{Groeneveld1995}.\nFurthermore, the compression parameter with $\\alpha = 1.6$ is nearly constant over the whole measured range fluence range from \\SI{0.6}{} to \\SI{7.7}{\\micro J\/cm^2}.\nTypically, compressed exponential functions are used to describe a system that deviates from the normal exponential decay\\cite{Whitehead2009}.\nThis occurs when the underlying processes get more complex than a simple relaxation from one independent state to another\\cite{Whitehead2009} (see Appendix for an illustration). \nIn the case of BST this can be due to the interaction of electrons in the conduction band with states inside the bandgap.\nProbable candidates for these are the surface states,  impurities, or charge carrier puddles\\cite{Choi2018,Borgwardt2016}.\nThe fact that the compression parameter is independent of the fluence shows that the interaction between the excited states is independent of the number of excited electrons.\\\\\n\\indent For the temperature dependence measurement we use the compressed exponential function as well, because at a fluence of \\SI{1}{\\micro J\/cm^2} we still observed an deviation from normal exponential decay (see Fig.~\\ref{fig5}).\nFigure \\ref{fig8} shows all extracted fitting parameters from the \\textit{n}-type and \\textit{p}-type samples.\nIn Fig. \\ref{fig8}(a), Max($-\\Delta E\/E$) for both samples decreases monotonically with increasing temperature over the whole range from \\SI{5}{K} to \\SI{300}{K}. \nThe difference between the samples is, that the \\textit{p}-type sample has the smaller signal and slope.\nWe speculate that the difference in signal is due to higher number of charged carriers of the \\textit{p}-type sample in its equilibrium state shown by the Hall measurements in Fig. \\ref{fig1} (b).\nWith the charge carrier densities extracted from the Hall measurements and effective masses in Refs.~\\cite{Witting2020, Taskin2011}, we estimate a plasma frequency of $\\approx \\SI{50}{THz}$(\\SI{70}{THz}) and $\\approx \\SI{134}{THz}$(\\SI{72}{THz}) at \\SI{2}{K}(\\SI{300}{K}) for the \\textit{n}- and \\textit{p}-type BST, respectively.\nThe difference in signal at low temperatures therefore can be explained by a larger relative increase in plasma frequency upon pump excitation.\nAnother indicator for the influence of the charge carrier density is the smaller difference at higher temperature, here the plasma frequency is almost identical.\nFigure~\\ref{fig8}(b) shows that the relaxation time increases slightly with increasing temperature, which is similar to the behavior of metals in pump-probe experiments.\nThis dependence can be described with the two-temperature-model\\cite{Groeneveld1995}, in the same way as the fluence dependence of the relaxation time.\nThis further hints at the metallic nature of the relaxation in the above-bandgap excitation.\nA similar behavior was found in measurements in the TI compound Bi$_{1.5}$Sb$_{0.5}$Te$_{1.8}$Se$_{1.2}$\\cite{Cheng2014}.\n\n\\section{Summary}\n\nTo summarize, we performed time-resolved mid-infrared-pump terahertz-transmission-probe study of nonequilibrium charge-carrier dynamics in the \\textit{n}-type and the \\textit{p}-type topological insulators (Bi$_{1-\\text{x}}$Sb$_{\\text{x}}$)$_2$Te$_3$.\nThe relaxation dynamics of charge carriers was investigated as a function of temperature and pump fluence, for pump-pulse energies below and above the bandgap in the two types of samples.\nWhile for the below-bandgap excitation the relaxation dynamics exhibits an exponential decay signalling a noninteracting process, the above-bandgap excitation leads to a compressed exponential decay reflecting a more complex relaxation process involving interactions of different states in the out-of-equilibrium system.\nThese states are very likely the in-gap states, e.g. surface states, impurity states and charge puddles, as  \na saturation behaviour in the fluence dependence measurement was observed for both the \\textit{n}-type and the \\textit{p}-type samples.\nIn addition, the observed difference in temperature dependence for the two types of samples points to the  effects of equilibrium-state charge carrier density on the relaxation dynamics.\n\n\n\\begin{acknowledgments}\nWe acknowledge partial supported by\nthe DFG German Research Foundation) via Project No. 277146847 \u2014 Collaborative Research Center 1238: Control and Dynamics of Quantum Materials (Subprojects No. A04, B05). G.L. acknowledges the support by the Research Foundation \u2014 Flanders (FWO, Belgium), project Nr. 27531 and 52751.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{{\\rm REFERENCES}}\n\\sloppy \\hyphenpenalty10000\n\\begin{list}{}{\\leftmargin1cm\\listparindent-1cm\n\\itemindent\\listparindent\\parsep0pt\\itemsep0pt}\n{\\end{list}\\vspace{2mm}}\n\n\\def~{~}\n\\newlength{\\DW}\n\\settowidth{\\DW}{0}\n\\newcommand{\\hspace{\\DW}}{\\hspace{\\DW}}\n\n\\newcommand{\\refitem}[5]{\\item[]{#1} #\n\\def#5}\\ifx\\REFARG\\TYLDA\\else, {#5}\\fi.{#3}\\ifx#5}\\ifx\\REFARG\\TYLDA\\else, {#5}\\fi.~\\else, {\\it#3}\\fi\n\\def#5}\\ifx\\REFARG\\TYLDA\\else, {#5}\\fi.{#4}\\ifx#5}\\ifx\\REFARG\\TYLDA\\else, {#5}\\fi.~\\else, {\\bf#4}\\fi\n\\def#5}\\ifx\\REFARG\\TYLDA\\else, {#5}\\fi.{#5}\\ifx#5}\\ifx\\REFARG\\TYLDA\\else, {#5}\\fi.~\\else, {#5}\\fi.}\n\n\\newcommand{\\Section}[1]{\\section{#1}}\n\\newcommand{\\Subsection}[1]{\\subsection{#1}}\n\\newcommand{\\Acknow}[1]{\\par\\vspace{5mm}{\\bf Acknowledgments.} #1}\n\\pagestyle{myheadings}\n\n\\newfont{\\bb}{ptmbi8t at 12pt}\n\\newcommand{\\rule{0pt}{2.5ex}}{\\rule{0pt}{2.5ex}}\n\\newcommand{\\rule[-1.8ex]{0pt}{4.5ex}}{\\rule[-1.8ex]{0pt}{4.5ex}}\n\\def\\fnsymbol{footnote}{\\fnsymbol{footnote}}\n\n\\begin{center}\n{\\Large\\bf Searching for Potential Mergers among 22~500 Eclipsing Binary\nStars in the OGLE-III Galactic Bulge Fields\\footnote{Based on observations\nobtained with the 1.3-m Warsaw telescope at the Las Campanas Observatory of the\nCarnegie Institution for Science.}}\n\\vskip1cm\n{\\bf\nP.~~P~i~e~t~r~u~k~o~w~i~c~z$^1$,~~I.~~S~o~s~z~y~\\'n~s~k~i$^1$,~~A.~~U~d~a~l~s~k~i$^1$,\\\\\n~~M.~K.~~S~z~y~m~a~\\'n~s~k~i$^1$,~~{\\L}.~~W~y~r~z~y~k~o~w~s~k~i$^{1}$,~~R.~~P~o~l~e~s~k~i$^{1,2}$,\\\\\n~~S.~~K~o~z~{\\l}~o~w~s~k~i$^1$,~~J.~~S~k~o~w~r~o~n$^1$,~~P.~~M~r~\\'o~z$^1$,~~M.~P~a~w~l~a~k$^1$,\\\\\nand~~K.~~U~l~a~c~z~y~k$^{1,3}$\\\\}\n\\vskip3mm\n{\n$^1$ Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland\\\\\ne-mail: [email protected]\\\\\n$^2$ Department of Astronomy, Ohio State University, 140 W. 18th Ave.,\\\\\nColumbus, OH 43210, USA\\\\\n$^3$ Department of Physics, University of Warwick, Coventry CV4 7AL, UK\\\\\n}\n\\end{center}\n\n\\Abstract{Inspired by the discovery of the red nova V1309 Sco\n(Nova Scorpii 2008) and the fact that its progenitor was a binary system\nwith a rapidly decreasing orbital period, we have searched for\nperiod changes in OGLE binary stars. We have selected a sample of 22~462\nshort-period ($P_{\\rm orb}<4$~d) eclipsing binary stars observed\ntoward the Galactic bulge by the OGLE-III survey in years 2001--2009.\nThis dataset was extended with photometry from OGLE-II (1997--2000)\nand the first six years of OGLE-IV (2010--2015). For some stars,\nthe data were supplemented with OGLE-I photometry (1992--1995).\nAfter close inspection of the whole sample we have found 56 systems with\nrealistic period decrease and 52 systems with realistic period increase.\nWe have also recognized 35 systems with cyclic period variations.\nThe highest negative period change rate of $-1.943\\times 10^{-4}$~d\/y\nhas been detected in detached eclipsing binary OGLE-BLG-ECL-139622 with\n$P_{\\rm orb}=2.817$~d, while all other found systems are contact\nbinaries with orbital periods mostly shorter than 1.0~d. For 22 our systems\nwith decreasing orbital period the absolute rate is higher than the value\nreported recently for eclipsing binary KIC 9832227. Interestingly,\nthere is an excess of systems with high negative period change rate over\nsystems with positive rate. We cannot exclude the possibility that some\nof the contact binaries with relatively long orbital period and high\nnegative period change rate will merge in the future. However, our results\nrather point to the presence of tertiary companions in the observed\nsystems and\/or spot activity on the surface of the binary components.}\n\n{Galaxy: bulge -- Galaxy: disk -- binaries: eclipsing}\n\n\n\\Section{Introduction}\n\nRed novae belong to a very intriguing class of rarely observed luminous\ntransients. They are associated with a dynamical phase of common envelope\nevolution (Soker and Tylenda 2003), albeit alternative explanations have\nbeen also proposed, such as classical nova eruption with a slowly moving,\nmassive envelope (Shara {\\it et al.\\ } 2010), or explosion of the star in the\ndust-enshrouded phase of the evolution at the extremum of the asymptotic\ngiant branch (Thompson {\\it et al.\\ } 2009), or intensive mass loss from\nthe binary system through the outer Lagrangian point (Pejcha 2014).\nSo far, only a handful number of this kind of transients have been noted\nin the Milky Way: V4332 Sgr (Hayashi {\\it et al.\\ } 1994), V838 Mon (Brown {\\it et al.\\ } 2002),\nV1309 Sco (Nakano {\\it et al.\\ } 2008), and OGLE-2002-BLG-360 (Tylenda {\\it et al.\\ } 2013).\nThanks to precise high-cadence OGLE photometry covering seven years before\nthe eruption of the red nova V1309 Sco, it was realized that the progenitor\nwas a contact eclipsing binary with the orbital period shrinking\nfrom about 1.438~d in 2002 to 1.425~d in 2007 (Tylenda {\\it et al.\\ } 2011).\nBefore the eruption, the light curve shape of the object transformed\nfrom a double wave to single wave which indicated the immersion of two\nstellar bodies in an elongated common envelope. According to the\nestimation made by Kochanek {\\it et al.\\ } (2014), the number of Galactic events\nlike the V1309 Sco outburst is about once per decade.\n\nThe main goal of our work is searching for candidates for future\nstellar mergers or systems with high negative period change rate\namong eclipsing binaries observed by the OGLE survey toward\nthe Milky Way bulge. From the sample of eclipsing binaries,\nwe also select systems with positive period change rate and\nsystems with noticeable cyclic period variations.\n\n\n\\Section{Observations}\n\nThe Optical Gravitational Lensing Experiment (OGLE) is a long-term\nwide-field variability sky survey launched in 1992 with the\noriginal aim of searching for microlensing events (Udalski {\\it et al.\\ } 1992).\nThe survey is conducted at Las Campanas Observatory which is\noperated by the Carnegie Institution for Science. OGLE uses\nthe 1.3-m Warsaw Telescope. Since 2010 the project\nis in its fourth phase, OGLE-IV (Udalski {\\it et al.\\ } 2015), collecting\ndata with a 32-CCD mosaic camera of a field of view of 1.4 deg$^2$.\nCurrently, OGLE monitors over 3000~deg$^2$ of the sky.\nPrevious phases of the project were conducted in the following years:\n1992--1995 (OGLE-I), 1997--2000 (OGLE-II), and 2001--2009 (OGLE-III).\nIn 2016 the OGLE database exceeded $10^{12}$ single photometric measurements.\nThe survey has discovered and classified nearly one million genuine\nvariable stars toward the Galactic bulge, Galactic disk, and Magellanic\nClouds ({\\it e.g.},\\, Soszy\\'nski {\\it et al.\\ } 2013,2014,2015,2016, Pietrukowicz {\\it et al.\\ } 2013,\nMr\\'oz {\\it et al.\\ } 2015).\n\nEclipsing binary systems, among which we look for potential mergers,\nwere observed by OGLE-III in the direction of the Galactic bulge.\nIn the third phase of OGLE, an eight-CCD mosaic camera with a field\nof view of about 0.35 deg$^2$ was attached to the Warsaw Telescope\n(Udalski {\\it et al.\\ } 2003). Angular pixel size in OGLE-III and OGLE-IV\nis the same: 0\\makebox[0pt][l]{.}\\arcs26. During the OGLE-III phase 267 fields\ncovering about 92~deg$^2$ toward the Milky Way bulge were observed\n(Szyma\\'nski {\\it et al.\\ } 2011). In OGLE-II, a single CCD camera with a pixel\nsize of 0\\makebox[0pt][l]{.}\\arcs42 was used in driftscan mode (Udalski {\\it et al.\\ } 1997).\nIn that phase, 49 Galactic bulge fields of $14\\makebox[0pt][l]{.}\\arcm2\\times57\\ifmmode{'}\\else$'$\\fi$\neach covering a total area of around 11~deg$^2$ were monitored.\nThe bulge coverage of OGLE-I, conducted on the 1.0-m Swope telescope\nalso at Las Campanas Observatory, was even much smaller: 18 fields\nof $15\\ifmmode{'}\\else$'$\\fi\\times15\\ifmmode{'}\\else$'$\\fi$ each covered about 1.1~deg$^2$.\nOGLE monitors the sky mainly in the $I$-band, while $V$-band observations\nare collected to secure color information of the objects and for accurate\ntransformation to the standard Johnson-Cousins system. The number of\n$I$($V$)-band measurements in the most frequently observed Galactic\nbulge fields is the following: 261(44) in OGLE-I, 568(16) in OGLE-II,\n2540(34) in OGLE-III, and 12~889(144) in OGLE-IV (2010--2015).\nReduction of the OGLE data is performed with the difference image\nanalysis (DIA) technique (Alard and Lupton 1998, Wo\\'zniak 2000).\n\n\n\n\\Section{Binary Systems Selection}\n\nFor the purpose of our analysis we selected 53 OGLE-III Galactic\nbulge fields observed in the $I$-band for at least 6 seasons\nwith a minimum number of 40 epochs per season. Location of these\nfields in Galactic coordinates is presented in Fig.~1.\nOver 28 million detections in the brightness range $12.6<I<19.0$~mag\nwere a subject of the initial period search with the FNPEAKS\ncode\\footnote{http:\/\/helas.astro.uni.wroc.pl\/deliverables.php?lang=en\\&active=fnpeaks}\nfor each season separately. For further analysis we left about\n14 million detections with assessed periods in all well-covered seasons.\nFrom this sample we removed detections with signals around 1\/3~d,\n1\/2~d, and 1~d being very likely daily aliases. Since we concentrate\non short-period systems only, we removed objects with the initially\ndetected periods longer than 2~d or orbital periods $P_{\\rm orb}>4$~d.\nAfter some verification tests, we decided to work further on 1\\% of stars\nwith the highest variability signal. The initial periods for about 137~500\nstars were corrected with the TATRY code (Schwarzenberg-Czerny 1996).\nCross-matching of this sample with the list of OGLE-III Galactic bulge\nRR Lyr-type stars (Soszy\\'nski {\\it et al.\\ } 2011) led to the rejection\nof about 2\\% of stars. RR Lyr stars, particularly of RRc type showing\nclose-to-sinusoidal light curve shapes and with periods of a few tens\nof day, could contaminate our sample. We made a visual inspection\nof $I$-band light curves of around 134~700 detections. This time-consuming\noperation allowed us to reject other contaminants from the sample, such as\nspotted variables, rotating variables, and $\\delta$~Sct-type pulsators.\nImportantly, the inspection helped us to verify the orbital periods\nof candidates for eclipsing binaries. We corrected our list of candidate\nOGLE-III eclipsing variables for artifacts and duplicates detected\nin adjacent fields.\n\nIn the next step, we searched for OGLE-II and OGLE-IV counterparts.\nOGLE-IV data used cover six seasons, from 2010 to 2015.\nAbout 43\\% and 95\\% of OGLE-III bulge binaries are present in the\nOGLE-II and OGLE-IV images, respectively. As in the case of the\nOGLE-III data, we took into account only binaries with at least 40\n$I$-band measurements per season. Several times more frequent OGLE-IV\nobservations allowed us to verify the classification of the variables\nand to correct the orbital periods determined from the OGLE-III data.\n\nThe final number of detected eclipsing binaries is 22~462.\nFor 19~885 binaries, the OGLE-III data are extended with photometry from\nOGLE-IV. For 8788 binaries, the data are also extended with photometry from\nOGLE-II. For 7825 eclipsing binaries our dataset covers 20 years\nof OGLE-II, OGLE-III, and OGLE-IV. In the case of the most interesting\nsystems, this dataset was supplemented with OGLE-I photometry increasing\nthe time coverage up to 24 years. In our final sample, there are 1657\nbinaries with solely OGLE-III photometry. All light curves were\ncleaned from outlying points by phasing and binning the data. After some\ntests we set the cleaning limit at a mild level of $5\\sigma$ to avoid\nrejection of good data points in the case of systems with possibly\nhigh period change rates. For objects that are present in the OGLE\ncollection of eclipsing binary stars toward the Galactic bulge published\nby Soszy\\'nski {\\it et al.\\ } (2016), we use their format: OGLE-BLG-ECL-NNNNNN.\nFor objects that are absent in that collection, we left the standard\nformat used in the OGLE database: FIELD.CHIP.ID.\n\n\\begin{figure}[htb]\n\\centerline{\\includegraphics[angle=0,width=130mm]{fig1.ps}}\n\\FigCap{Location of 53 OGLE-III Galactic bulge fields selected\nfor eclipsing binary stars searches.}\n\\end{figure}\n\n\n\n\\Section{Period Changes}\n\nA simple method was applied to find systems with reliable period\nchange rates and candidates for possible mergers in the sample\nof 22~462 eclipsing binaries. We searched for negative as well as\npositive linear period changes using $I$-band photometry from\nOGLE-II, OGLE-III, and OGLE-IV, depending on the coverage.\nOGLE-III and OGLE-IV data were divided into seasons.\nAfter some tests we decided not to divide the less rich OGLE-II data.\nFor each OGLE-III\/IV season and the whole four-year OGLE-II dataset\naccurate orbital periods were determined using the TATRY code\n(Schwarzenberg-Czerny 1996). Moments corresponding to each\nperiod of time were calculated as average from the epochs.\nOur approach allowed us to avoid possible problems with variations\nin mean brightness, amplitude, and light curve shapes (due to star\nspots), also problems with the presence of some remnant outlying\npoints near eclipses and small magnitude offsets between the photometry\nfrom different OGLE phases to all of which the classical $O-C$ method\nis sensitive ({\\it e.g.},\\, Conroy {\\it et al.\\ } 2014, Gies {\\it et al.\\ } 2015 in the application\nto data from the Kepler satellite). After fitting a linear regression\nto all obtained ``period change curves'', we made a close inspection\nof 2403 systems for which $|\\dot P_{\\rm orb}|>5\\sigma_{\\dot P{\\rm orb}}$.\nIn many period change curves with particularly high rate derived from the\nautomated fit, we noticed outlying points corresponding to less frequently\nobserved seasons. After the inspection we found only 59 systems with\nrealistic period decrease and 53 systems with realistic period increase.\nWe identified 35 systems with evident cyclic period variations.\n\nIn the last stage, we verified whether some of the found interesting\nbinaries had been observed by OGLE-I in years 1992--1995. Photometry\nwas collected for twelve of these binaries. It turned out that three\ncandidate systems with decreasing period and one candidate system\nwith increasing period seem to show rather non-monotonic variations.\nThe final number of detected systems with the increasing, decreasing,\nand cyclic period changes is 56, 52, and 35 objects, respectively.\n\nWe recognized four of our systems in the list of 569 contact binaries with\nreliable period change rates determined by Kubiak {\\it et al.\\ } (2006) based on\nOGLE data from years 1992--2005. Systems BW1.125206 = OGLE-BLG-ECL-265310\nand BWC.169286 = OGLE-BLG-ECL-276943 were found, at that time, to have\nthe decreasing period, while systems BW7.159932 = OGLE-BLG-ECL-288099\nand BW4.5243 = OGLE-BLG-ECL-279991 to have the increasing period.\nHowever, observations spanning 24 years indicate non-monotonic\nvariations of the period in these four binaries.\n\nPeriod change curves together with phased $I$-band light curves for seven\nof our binary stars with the highest negative orbital period change rate are\npresented in Fig.~2. In Table~1, we list basic parameters of all 56 systems\nwith reliable negative period changes sorted from the highest absolute rate.\nThe given orbital periods correspond roughly to the middle of the OGLE-III\nphase (year 2005). System OGLE-BLG-ECL-139622 with the highest derived negative\nperiod change rate of $\\dot P_{\\rm orb}=-1.943\\times10^{-4}$~d\/y is a detached\neclipsing binary of Algol (EA) type. Unfortunately, this object was monitored\nonly in the OGLE-III phase. New time-series data would help in verification\nof the observed trend. Other systems with the decreasing orbital period are\ncontact binaries. All of them have absolute rates at least one order\nof magnitude lower than OGLE-BLG-ECL-139622. Interestingly, high negative\nperiod change rates are observed in contact systems with relatively\nlong orbital periods: all four contact binaries with $P_{\\rm orb}>1.0$~d\nhave $|\\dot P_{\\rm orb}|>10^{-5}$~d\/y; all seven contact binaries with\n$P_{\\rm orb}>0.8$~d have $|\\dot P_{\\rm orb}|>6\\times10^{-6}$~d\/y.\n\nIn Fig.~3, we present seven binaries with the highest derived positive\nperiod change rate. Table~2 provides information on all 52 systems with the\nincreasing orbital period. All such systems have the orbital period\n$<0.8$~d and the period change rate $<10^{-5}$~d\/y or lower\nthan the absolute value in eight systems with the most rapid negative\nperiod changes. Distributions of systems with negative and positive period\nchange rates are compared in Fig.~4. All detected systems with the increasing\norbital period are contact binaries.\n\nFinally, in Fig.~5 and Table~3, we present systems with cyclic period\nvariations. We assess the cycle period for each of the system.\nAll these systems are contact binaries with $P_{\\rm orb}<0.9$~d.\n\n\\begin{figure}[htb]\n\\centerline{\\includegraphics[angle=0,width=130mm]{fig2.ps}}\n\\FigCap{Binary systems with the highest negative period change rates.\nPresented OGLE-III light curves are phased with constant period\ndetermined for this dataset.}\n\\end{figure}\n\n\\begin{table}[h!]\n\\centering\n\\caption{\\small Parameters of the systems with derived negative period change rate}\n\\medskip\n{\\footnotesize\n\\begin{tabular}{ccccccc}\n\\hline\nOGLE-BLG-ECL- &   RA    &    Dec  & $I_{\\rm max}$ & $V-I$ & $P_{\\rm orb}$ & $\\dot P_{\\rm orb}$ \\\\\n              & J2000.0 & J2000.0 &     [mag]     & [mag] &      [d]      &         [d\/y] \\\\\n\\hline\n139622 & $17\\uph51\\upm33\\makebox[0pt][l]{.}\\ups33$ & $-29\\arcd54\\arcm38\\makebox[0pt][l]{.}\\arcs7$ & 15.28 & 1.41 & 2.817436 & $-1.94\\pm0.28$~E$-4$ \\\\\n299145 & $18\\uph05\\upm34\\makebox[0pt][l]{.}\\ups20$ & $-29\\arcd40\\arcm21\\makebox[0pt][l]{.}\\arcs5$ & 16.42 & 1.26 & 1.252603 & $-1.72\\pm0.24$~E$-5$ \\\\\n344477 & $18\\uph10\\upm24\\makebox[0pt][l]{.}\\ups89$ & $-28\\arcd34\\arcm26\\makebox[0pt][l]{.}\\arcs5$ & 16.08 & 1.23 & 1.032195 & $-1.50\\pm0.18$~E$-5$ \\\\\n176377 & $17\\uph54\\upm41\\makebox[0pt][l]{.}\\ups59$ & $-29\\arcd29\\arcm18\\makebox[0pt][l]{.}\\arcs1$ & 16.80 & 1.74 & 1.245946 & $-1.42\\pm0.05$~E$-5$ \\\\\n339765 & $18\\uph09\\upm52\\makebox[0pt][l]{.}\\ups64$ & $-27\\arcd25\\arcm53\\makebox[0pt][l]{.}\\arcs1$ & 18.62 & 1.47 & 0.377840 & $-1.18\\pm0.08$~E$-5$ \\\\\n170070 & $17\\uph54\\upm09\\makebox[0pt][l]{.}\\ups38$ & $-29\\arcd42\\arcm53\\makebox[0pt][l]{.}\\arcs7$ & 16.43 & 1.85 & 1.401907 & $-1.15\\pm0.03$~E$-5$ \\\\\n175354 & $17\\uph54\\upm36\\makebox[0pt][l]{.}\\ups55$ & $-33\\arcd44\\arcm51\\makebox[0pt][l]{.}\\arcs9$ & 17.31 & 1.58 & 0.840163 & $-1.06\\pm0.16$~E$-5$ \\\\\n154749 & $17\\uph52\\upm54\\makebox[0pt][l]{.}\\ups10$ & $-29\\arcd08\\arcm29\\makebox[0pt][l]{.}\\arcs7$ & 17.55 & 2.45 & 0.932247 & $-1.04\\pm0.18$~E$-5$ \\\\\n220028 & $17\\uph58\\upm24\\makebox[0pt][l]{.}\\ups67$ & $-29\\arcd57\\arcm42\\makebox[0pt][l]{.}\\arcs0$ & 16.68 & 1.53 & 0.920792 & $-6.02\\pm1.10$~E$-6$ \\\\\n172659 & $17\\uph54\\upm22\\makebox[0pt][l]{.}\\ups13$ & $-30\\arcd01\\arcm41\\makebox[0pt][l]{.}\\arcs9$ & 18.10 & 1.77 & 0.626226 & $-5.85\\pm0.21$~E$-6$ \\\\\n195664 & $17\\uph56\\upm19\\makebox[0pt][l]{.}\\ups97$ & $-30\\arcd55\\arcm40\\makebox[0pt][l]{.}\\arcs9$ & 18.55 & 1.71 & 0.436900 & $-4.56\\pm0.19$~E$-6$ \\\\\n192939 & $17\\uph56\\upm06\\makebox[0pt][l]{.}\\ups40$ & $-29\\arcd29\\arcm21\\makebox[0pt][l]{.}\\arcs4$ & 18.23 & 2.09 & 0.531934 & $-4.40\\pm0.20$~E$-6$ \\\\\n243151 & $18\\uph00\\upm27\\makebox[0pt][l]{.}\\ups01$ & $-29\\arcd27\\arcm41\\makebox[0pt][l]{.}\\arcs1$ & 18.41 & 1.62 & 0.403628 & $-4.35\\pm0.50$~E$-6$ \\\\\n347999 & $18\\uph10\\upm49\\makebox[0pt][l]{.}\\ups69$ & $-29\\arcd25\\arcm35\\makebox[0pt][l]{.}\\arcs8$ & 18.39 & 1.31 & 0.365383 & $-4.30\\pm0.66$~E$-6$ \\\\\n212252 & $17\\uph57\\upm44\\makebox[0pt][l]{.}\\ups48$ & $-31\\arcd03\\arcm40\\makebox[0pt][l]{.}\\arcs8$ & 18.71 & 1.95 & 0.413250 & $-3.88\\pm0.17$~E$-6$ \\\\\n168049 & $17\\uph53\\upm59\\makebox[0pt][l]{.}\\ups85$ & $-30\\arcd07\\arcm37\\makebox[0pt][l]{.}\\arcs4$ & 17.66 & 1.64 & 0.506522 & $-2.71\\pm0.18$~E$-6$ \\\\\n320382 & $18\\uph07\\upm43\\makebox[0pt][l]{.}\\ups53$ & $-25\\arcd42\\arcm42\\makebox[0pt][l]{.}\\arcs8$ & 17.31 & 1.68 & 0.388373 & $-2.63\\pm0.40$~E$-6$ \\\\\n218785 & $17\\uph58\\upm18\\makebox[0pt][l]{.}\\ups73$ & $-32\\arcd12\\arcm24\\makebox[0pt][l]{.}\\arcs8$ & 18.04 & 1.82 & 0.353455 & $-2.53\\pm0.36$~E$-6$ \\\\\n119460 & $17\\uph49\\upm11\\makebox[0pt][l]{.}\\ups03$ & $-35\\arcd01\\arcm02\\makebox[0pt][l]{.}\\arcs8$ & 17.12 & 1.41 & 0.394387 & $-2.49\\pm0.36$~E$-6$ \\\\\n351804 & $18\\uph11\\upm16\\makebox[0pt][l]{.}\\ups73$ & $-25\\arcd42\\arcm55\\makebox[0pt][l]{.}\\arcs4$ & 16.71 & 1.46 & 0.385420 & $-2.12\\pm0.26$~E$-6$ \\\\\n130708 & $17\\uph50\\upm35\\makebox[0pt][l]{.}\\ups67$ & $-29\\arcd19\\arcm13\\makebox[0pt][l]{.}\\arcs6$ & 17.49 & 2.10 & 0.366826 & $-2.05\\pm0.39$~E$-6$ \\\\\n149279 & $17\\uph52\\upm26\\makebox[0pt][l]{.}\\ups28$ & $-32\\arcd13\\arcm12\\makebox[0pt][l]{.}\\arcs7$ & 17.62 & 2.08 & 0.314728 & $-2.03\\pm0.32$~E$-6$ \\\\\n183004 & $17\\uph55\\upm16\\makebox[0pt][l]{.}\\ups99$ & $-31\\arcd03\\arcm52\\makebox[0pt][l]{.}\\arcs4$ & 15.73 & 1.37 & 0.467429 & $-2.03\\pm0.10$~E$-6$ \\\\\n190636 & $17\\uph55\\upm55\\makebox[0pt][l]{.}\\ups09$ & $-31\\arcd00\\arcm26\\makebox[0pt][l]{.}\\arcs6$ & 17.63 & 1.85 & 0.305221 & $-1.96\\pm0.16$~E$-6$ \\\\\n287671 & $18\\uph04\\upm31\\makebox[0pt][l]{.}\\ups54$ & $-28\\arcd38\\arcm06\\makebox[0pt][l]{.}\\arcs5$ & 17.08 & 1.19 & 0.532926 & $-1.94\\pm0.07$~E$-6$ \\\\\n201682 & $17\\uph56\\upm50\\makebox[0pt][l]{.}\\ups80$ & $-29\\arcd44\\arcm49\\makebox[0pt][l]{.}\\arcs5$ & 17.32 & 1.90 & 0.497314 & $-1.89\\pm0.29$~E$-6$ \\\\\n213011 & $17\\uph57\\upm48\\makebox[0pt][l]{.}\\ups50$ & $-29\\arcd21\\arcm59\\makebox[0pt][l]{.}\\arcs0$ & 17.65 & 1.94 & 0.573979 & $-1.78\\pm0.11$~E$-6$ \\\\\n159116 & $17\\uph53\\upm16\\makebox[0pt][l]{.}\\ups11$ & $-29\\arcd48\\arcm59\\makebox[0pt][l]{.}\\arcs7$ & 15.83 & 1.26 & 0.587393 & $-1.72\\pm0.06$~E$-6$ \\\\\n229500 & $17\\uph59\\upm13\\makebox[0pt][l]{.}\\ups78$ & $-30\\arcd49\\arcm06\\makebox[0pt][l]{.}\\arcs9$ & 16.01 & 1.56 & 0.398365 & $-1.71\\pm0.19$~E$-6$ \\\\\n163655 & $17\\uph53\\upm38\\makebox[0pt][l]{.}\\ups86$ & $-32\\arcd55\\arcm59\\makebox[0pt][l]{.}\\arcs1$ & 17.00 & 1.70 & 0.492482 & $-1.62\\pm0.09$~E$-6$ \\\\\n192437 & $17\\uph56\\upm03\\makebox[0pt][l]{.}\\ups96$ & $-29\\arcd59\\arcm12\\makebox[0pt][l]{.}\\arcs6$ & 16.71 & 1.67 & 0.402313 & $-1.60\\pm0.07$~E$-6$ \\\\\n233821 & $17\\uph59\\upm37\\makebox[0pt][l]{.}\\ups95$ & $-28\\arcd50\\arcm22\\makebox[0pt][l]{.}\\arcs4$ & 15.29 & 1.23 & 0.359920 & $-1.59\\pm0.04$~E$-6$ \\\\\n238356 & $18\\uph00\\upm01\\makebox[0pt][l]{.}\\ups26$ & $-29\\arcd09\\arcm07\\makebox[0pt][l]{.}\\arcs4$ & 17.23 & 1.52 & 0.292658 & $-1.47\\pm0.05$~E$-6$ \\\\\n176119 & $17\\uph54\\upm40\\makebox[0pt][l]{.}\\ups35$ & $-29\\arcd56\\arcm26\\makebox[0pt][l]{.}\\arcs1$ & 16.37 & 1.68 & 0.594569 & $-1.44\\pm0.06$~E$-6$ \\\\\n181311 & $17\\uph55\\upm08\\makebox[0pt][l]{.}\\ups47$ & $-29\\arcd33\\arcm46\\makebox[0pt][l]{.}\\arcs8$ & 16.51 & 1.37 & 0.422617 & $-1.43\\pm0.05$~E$-6$ \\\\\n217990 & $17\\uph58\\upm14\\makebox[0pt][l]{.}\\ups55$ & $-31\\arcd30\\arcm12\\makebox[0pt][l]{.}\\arcs7$ & 18.39 & 1.83 & 0.412545 & $-1.14\\pm0.17$~E$-6$ \\\\\n182438 & $17\\uph55\\upm14\\makebox[0pt][l]{.}\\ups05$ & $-30\\arcd10\\arcm56\\makebox[0pt][l]{.}\\arcs0$ & 17.54 & 1.75 & 0.549392 & $-1.11\\pm0.09$~E$-6$ \\\\\n280295 & $18\\uph03\\upm51\\makebox[0pt][l]{.}\\ups13$ & $-27\\arcd54\\arcm12\\makebox[0pt][l]{.}\\arcs3$ & 16.19 & 1.35 & 0.484237 & $-1.10\\pm0.10$~E$-6$ \\\\\n181083 & $17\\uph55\\upm07\\makebox[0pt][l]{.}\\ups21$ & $-29\\arcd58\\arcm17\\makebox[0pt][l]{.}\\arcs8$ & 16.20 & 1.45 & 0.349391 & $-1.02\\pm0.04$~E$-6$ \\\\\n114280 & $17\\uph48\\upm28\\makebox[0pt][l]{.}\\ups67$ & $-35\\arcd05\\arcm09\\makebox[0pt][l]{.}\\arcs9$ & 16.30 & 1.28 & 0.453031 & $-9.99\\pm1.64$~E$-7$ \\\\\n106230 & $17\\uph47\\upm21\\makebox[0pt][l]{.}\\ups97$ & $-34\\arcd40\\arcm49\\makebox[0pt][l]{.}\\arcs3$ & 16.06 & 1.32 & 0.489019 & $-8.88\\pm0.45$~E$-7$ \\\\\n184767 & $17\\uph55\\upm25\\makebox[0pt][l]{.}\\ups22$ & $-29\\arcd56\\arcm15\\makebox[0pt][l]{.}\\arcs9$ & 17.38 & 2.42 & 0.473305 & $-8.57\\pm1.18$~E$-7$ \\\\\n225278 & $17\\uph58\\upm52\\makebox[0pt][l]{.}\\ups64$ & $-28\\arcd42\\arcm13\\makebox[0pt][l]{.}\\arcs3$ & 16.96 & 1.36 & 0.417923 & $-7.21\\pm0.55$~E$-7$ \\\\\n241306 & $18\\uph00\\upm17\\makebox[0pt][l]{.}\\ups74$ & $-29\\arcd16\\arcm25\\makebox[0pt][l]{.}\\arcs1$ & 14.38 & 1.09 & 0.371268 & $-7.11\\pm0.22$~E$-7$ \\\\\n246693 & $18\\uph00\\upm46\\makebox[0pt][l]{.}\\ups32$ & $-28\\arcd52\\arcm27\\makebox[0pt][l]{.}\\arcs3$ & 15.69 & 0.84 & 0.363524 & $-7.01\\pm0.32$~E$-7$ \\\\\n122558 & $17\\uph49\\upm35\\makebox[0pt][l]{.}\\ups14$ & $-30\\arcd48\\arcm09\\makebox[0pt][l]{.}\\arcs5$ & 16.79 & 1.92 & 0.410946 & $-6.79\\pm1.20$~E$-7$ \\\\\n162594 & $17\\uph53\\upm33\\makebox[0pt][l]{.}\\ups56$ & $-30\\arcd03\\arcm08\\makebox[0pt][l]{.}\\arcs9$ & 16.79 & 1.70 & 0.356191 & $-6.24\\pm0.26$~E$-7$ \\\\\n207879 & $17\\uph57\\upm22\\makebox[0pt][l]{.}\\ups78$ & $-28\\arcd56\\arcm59\\makebox[0pt][l]{.}\\arcs1$ & 15.10 & 1.48 & 0.393082 & $-5.51\\pm0.33$~E$-7$ \\\\\n215121 & $17\\uph58\\upm00\\makebox[0pt][l]{.}\\ups28$ & $-29\\arcd57\\arcm49\\makebox[0pt][l]{.}\\arcs2$ & 17.46 & 1.59 & 0.307842 & $-5.51\\pm1.13$~E$-7$ \\\\\n250731 & $18\\uph01\\upm07\\makebox[0pt][l]{.}\\ups03$ & $-30\\arcd42\\arcm13\\makebox[0pt][l]{.}\\arcs7$ & 16.63 & 1.69 & 0.260272 & $-4.91\\pm0.57$~E$-7$ \\\\\n294795 & $18\\uph05\\upm10\\makebox[0pt][l]{.}\\ups60$ & $-29\\arcd21\\arcm03\\makebox[0pt][l]{.}\\arcs9$ & 13.04 & 1.02 & 0.293658 & $-4.84\\pm0.41$~E$-7$ \\\\\n198303 & $17\\uph56\\upm33\\makebox[0pt][l]{.}\\ups75$ & $-30\\arcd14\\arcm33\\makebox[0pt][l]{.}\\arcs8$ & 14.81 & 1.20 & 0.438734 & $-3.81\\pm0.72$~E$-7$ \\\\\n168012 & $17\\uph53\\upm59\\makebox[0pt][l]{.}\\ups73$ & $-30\\arcd04\\arcm48\\makebox[0pt][l]{.}\\arcs6$ & 16.40 & 1.58 & 0.371039 & $-3.81\\pm0.68$~E$-7$ \\\\\n279326 & $18\\uph03\\upm46\\makebox[0pt][l]{.}\\ups01$ & $-29\\arcd47\\arcm40\\makebox[0pt][l]{.}\\arcs6$ & 15.77 & 1.52 & 0.227733 & $-3.29\\pm0.24$~E$-7$ \\\\\n187430 & $17\\uph55\\upm38\\makebox[0pt][l]{.}\\ups54$ & $-29\\arcd17\\arcm24\\makebox[0pt][l]{.}\\arcs1$ & 17.47 & 2.01 & 0.275312 & $-3.18\\pm0.54$~E$-7$ \\\\\n217596 & $17\\uph58\\upm12\\makebox[0pt][l]{.}\\ups70$ & $-28\\arcd48\\arcm16\\makebox[0pt][l]{.}\\arcs6$ & 15.91 & 1.35 & 0.354705 & $-3.06\\pm0.13$~E$-7$ \\\\\n\\hline\n\\noalign{\\vskip3pt}\n\\end{tabular}}\n\\end{table}\n\n\n\\begin{figure}[htb]\n\\centerline{\\includegraphics[angle=0,width=130mm]{fig3.ps}}\n\\FigCap{Binary systems with the highest positive period change rates.\nOGLE-III light curves are presented.}\n\\end{figure}\n\n\\begin{table}[h!]\n\\centering\n\\caption{\\small Parameters of the systems with derived positive period change rate}\n\\medskip\n{\\footnotesize\n\\begin{tabular}{ccccccc}\n\\hline\nOGLE-BLG-ECL- &   RA    &    Dec  & $I_{\\rm max}$ & $V-I$ & $P_{\\rm orb}$ & $\\dot P_{\\rm orb}$ \\\\\n              & J2000.0 & J2000.0 &     [mag]     & [mag] &      [d]      &         [d\/y] \\\\\n\\hline\n145302 & $17\\uph52\\upm04\\makebox[0pt][l]{.}\\ups81$ & $-30\\arcd35\\arcm24\\makebox[0pt][l]{.}\\arcs0$ & 18.74 & 2.89 & 0.518252 & $7.91\\pm1.32$~E$-6$ \\\\\n154461 & $17\\uph52\\upm52\\makebox[0pt][l]{.}\\ups57$ & $-30\\arcd57\\arcm27\\makebox[0pt][l]{.}\\arcs1$ & 18.80 & 1.84 & 0.457895 & $7.48\\pm0.82$~E$-6$ \\\\\n291221 & $18\\uph04\\upm50\\makebox[0pt][l]{.}\\ups83$ & $-29\\arcd33\\arcm26\\makebox[0pt][l]{.}\\arcs4$ & 18.16 & 1.92 & 0.389955 & $4.95\\pm0.85$~E$-6$ \\\\\n311367 & $18\\uph06\\upm47\\makebox[0pt][l]{.}\\ups01$ & $-30\\arcd47\\arcm49\\makebox[0pt][l]{.}\\arcs2$ & 17.94 & 1.44 & 0.420201 & $4.44\\pm0.80$~E$-6$ \\\\\n325941 & $18\\uph08\\upm19\\makebox[0pt][l]{.}\\ups42$ & $-26\\arcd00\\arcm05\\makebox[0pt][l]{.}\\arcs2$ & 16.84 & 1.62 & 0.326240 & $4.39\\pm0.21$~E$-6$ \\\\\nBLG157.1.71541  & $17\\uph58\\upm30\\makebox[0pt][l]{.}\\ups84$ & $-32\\arcd39\\arcm53\\makebox[0pt][l]{.}\\arcs8$ & 18.68 & 1.91 & 0.268803 & $4.02\\pm0.59$~E$-6$ \\\\\n317088 & $18\\uph07\\upm22\\makebox[0pt][l]{.}\\ups46$ & $-29\\arcd18\\arcm39\\makebox[0pt][l]{.}\\arcs2$ & 17.68 & 1.33 & 0.487443 & $3.47\\pm0.69$~E$-6$ \\\\\n183497 & $17\\uph55\\upm19\\makebox[0pt][l]{.}\\ups37$ & $-32\\arcd55\\arcm41\\makebox[0pt][l]{.}\\arcs0$ & 17.77 & 1.18 & 0.523197 & $3.36\\pm0.79$~E$-6$ \\\\\n126797 & $17\\uph50\\upm07\\makebox[0pt][l]{.}\\ups17$ & $-30\\arcd09\\arcm38\\makebox[0pt][l]{.}\\arcs1$ & 17.75 & 2.56 & 0.367062 & $3.36\\pm0.57$~E$-6$ \\\\\n306303 & $18\\uph06\\upm16\\makebox[0pt][l]{.}\\ups05$ & $-29\\arcd00\\arcm46\\makebox[0pt][l]{.}\\arcs8$ & 17.88 & 1.15 & 0.415518 & $3.31\\pm0.59$~E$-6$ \\\\\n243435 & $18\\uph00\\upm28\\makebox[0pt][l]{.}\\ups54$ & $-30\\arcd27\\arcm58\\makebox[0pt][l]{.}\\arcs7$ & 18.31 & 1.56 & 0.419818 & $3.25\\pm0.58$~E$-6$ \\\\\n201168 & $17\\uph56\\upm48\\makebox[0pt][l]{.}\\ups09$ & $-29\\arcd30\\arcm10\\makebox[0pt][l]{.}\\arcs7$ & 18.16 & 1.92 & 0.465644 & $2.90\\pm0.56$~E$-6$ \\\\\n200942 & $17\\uph56\\upm46\\makebox[0pt][l]{.}\\ups93$ & $-30\\arcd00\\arcm57\\makebox[0pt][l]{.}\\arcs6$ & 18.30 & 2.08 & 0.358376 & $2.86\\pm0.52$~E$-6$ \\\\\n122590 & $17\\uph49\\upm35\\makebox[0pt][l]{.}\\ups42$ & $-30\\arcd32\\arcm44\\makebox[0pt][l]{.}\\arcs1$ & 16.33 & 1.48 & 0.306970 & $2.36\\pm0.10$~E$-6$ \\\\\n169859 & $17\\uph54\\upm08\\makebox[0pt][l]{.}\\ups29$ & $-29\\arcd44\\arcm38\\makebox[0pt][l]{.}\\arcs1$ & 18.13 & 1.50 & 0.393950 & $2.34\\pm0.13$~E$-6$ \\\\\n245597 & $18\\uph00\\upm40\\makebox[0pt][l]{.}\\ups49$ & $-28\\arcd39\\arcm52\\makebox[0pt][l]{.}\\arcs1$ & 17.99 & 1.50 & 0.402376 & $2.29\\pm0.11$~E$-6$ \\\\\n197927 & $17\\uph56\\upm32\\makebox[0pt][l]{.}\\ups09$ & $-29\\arcd15\\arcm43\\makebox[0pt][l]{.}\\arcs0$ & 17.22 & 1.76 & 0.545565 & $2.26\\pm0.27$~E$-6$ \\\\\n203756 & $17\\uph57\\upm00\\makebox[0pt][l]{.}\\ups79$ & $-29\\arcd39\\arcm38\\makebox[0pt][l]{.}\\arcs4$ & 17.95 & 1.71 & 0.418336 & $2.14\\pm0.41$~E$-6$ \\\\\n192324 & $17\\uph56\\upm03\\makebox[0pt][l]{.}\\ups39$ & $-29\\arcd25\\arcm10\\makebox[0pt][l]{.}\\arcs6$ & 17.81 & 2.02 & 0.436902 & $2.05\\pm0.24$~E$-6$ \\\\\n282055 & $18\\uph04\\upm00\\makebox[0pt][l]{.}\\ups73$ & $-28\\arcd39\\arcm33\\makebox[0pt][l]{.}\\arcs7$ & 17.61 & 1.39 & 0.482814 & $2.04\\pm0.14$~E$-6$ \\\\\nBLG206.4.258074 & $18\\uph00\\upm56\\makebox[0pt][l]{.}\\ups13$ & $-28\\arcd35\\arcm56\\makebox[0pt][l]{.}\\arcs0$ & 18.07 & 1.66 & 0.405867 & $2.00\\pm0.22$~E$-6$ \\\\\n216089 & $17\\uph58\\upm05\\makebox[0pt][l]{.}\\ups00$ & $-28\\arcd38\\arcm34\\makebox[0pt][l]{.}\\arcs9$ & 15.77 & 1.48 & 0.490068 & $1.92\\pm0.12$~E$-6$ \\\\\n196868 & $17\\uph56\\upm26\\makebox[0pt][l]{.}\\ups44$ & $-29\\arcd18\\arcm34\\makebox[0pt][l]{.}\\arcs7$ & 16.12 & 1.70 & 0.400507 & $1.86\\pm0.09$~E$-6$ \\\\\n233754 & $17\\uph59\\upm37\\makebox[0pt][l]{.}\\ups55$ & $-29\\arcd12\\arcm27\\makebox[0pt][l]{.}\\arcs8$ & 16.86 & 1.35 & 0.570240 & $1.84\\pm0.09$~E$-6$ \\\\\n157410 & $17\\uph53\\upm07\\makebox[0pt][l]{.}\\ups74$ & $-30\\arcd27\\arcm52\\makebox[0pt][l]{.}\\arcs4$ & 15.39 & 1.35 & 0.363414 & $1.69\\pm0.08$~E$-6$ \\\\\n124567 & $17\\uph49\\upm50\\makebox[0pt][l]{.}\\ups90$ & $-29\\arcd38\\arcm17\\makebox[0pt][l]{.}\\arcs9$ & 16.20 & 1.78 & 0.400343 & $1.61\\pm0.03$~E$-6$ \\\\\n233878 & $17\\uph59\\upm38\\makebox[0pt][l]{.}\\ups27$ & $-28\\arcd45\\arcm47\\makebox[0pt][l]{.}\\arcs8$ & 13.80 & 0.92 & 0.741434 & $1.60\\pm0.04$~E$-6$ \\\\\n194517 & $17\\uph56\\upm14\\makebox[0pt][l]{.}\\ups16$ & $-29\\arcd43\\arcm37\\makebox[0pt][l]{.}\\arcs7$ & 17.20 & 1.57 & 0.421592 & $1.58\\pm0.20$~E$-6$ \\\\\n340259 & $18\\uph09\\upm55\\makebox[0pt][l]{.}\\ups74$ & $-29\\arcd33\\arcm39\\makebox[0pt][l]{.}\\arcs6$ & 14.77 & 0.79 & 0.453261 & $1.38\\pm0.19$~E$-6$ \\\\\n199259 & $17\\uph56\\upm38\\makebox[0pt][l]{.}\\ups43$ & $-29\\arcd48\\arcm45\\makebox[0pt][l]{.}\\arcs8$ & 13.74 & 0.97 & 0.389284 & $1.37\\pm0.09$~E$-6$ \\\\\n270325 & $18\\uph02\\upm56\\makebox[0pt][l]{.}\\ups52$ & $-30\\arcd17\\arcm39\\makebox[0pt][l]{.}\\arcs6$ & 16.59 & 1.38 & 0.381891 & $1.32\\pm0.11$~E$-6$ \\\\\n230641 & $17\\uph59\\upm20\\makebox[0pt][l]{.}\\ups86$ & $-28\\arcd46\\arcm18\\makebox[0pt][l]{.}\\arcs7$ & 15.56 & 1.24 & 0.382904 & $1.29\\pm0.03$~E$-6$ \\\\\n195044 & $17\\uph56\\upm16\\makebox[0pt][l]{.}\\ups92$ & $-30\\arcd44\\arcm39\\makebox[0pt][l]{.}\\arcs6$ & 16.39 & 1.47 & 0.409737 & $1.16\\pm0.03$~E$-6$ \\\\\n135055 & $17\\uph51\\upm04\\makebox[0pt][l]{.}\\ups99$ & $-29\\arcd40\\arcm10\\makebox[0pt][l]{.}\\arcs7$ & 15.75 & 1.69 & 0.522241 & $1.10\\pm0.13$~E$-6$ \\\\\n153698 & $17\\uph52\\upm48\\makebox[0pt][l]{.}\\ups61$ & $-29\\arcd06\\arcm42\\makebox[0pt][l]{.}\\arcs9$ & 16.72 & 1.69 & 0.386626 & $1.08\\pm0.14$~E$-6$ \\\\\n234160 & $17\\uph59\\upm39\\makebox[0pt][l]{.}\\ups66$ & $-29\\arcd05\\arcm13\\makebox[0pt][l]{.}\\arcs9$ & 16.94 & 1.35 & 0.528657 & $1.06\\pm0.06$~E$-6$ \\\\\n200313 & $17\\uph56\\upm43\\makebox[0pt][l]{.}\\ups77$ & $-30\\arcd49\\arcm34\\makebox[0pt][l]{.}\\arcs2$ & 17.24 & 1.54 & 0.479131 & $1.03\\pm0.05$~E$-6$ \\\\\n174839 & $17\\uph54\\upm34\\makebox[0pt][l]{.}\\ups06$ & $-29\\arcd24\\arcm38\\makebox[0pt][l]{.}\\arcs3$ & 16.42 & 1.46 & 0.461225 & $9.62\\pm0.24$~E$-7$ \\\\\n219977 & $17\\uph58\\upm24\\makebox[0pt][l]{.}\\ups47$ & $-28\\arcd41\\arcm05\\makebox[0pt][l]{.}\\arcs4$ & 15.62 & 1.65 & 0.486807 & $8.94\\pm0.31$~E$-7$ \\\\\n127233 & $17\\uph50\\upm10\\makebox[0pt][l]{.}\\ups38$ & $-35\\arcd02\\arcm26\\makebox[0pt][l]{.}\\arcs6$ & 16.73 & 1.44 & 0.350237 & $8.74\\pm1.59$~E$-7$ \\\\\n205743 & $17\\uph57\\upm12\\makebox[0pt][l]{.}\\ups10$ & $-29\\arcd56\\arcm21\\makebox[0pt][l]{.}\\arcs1$ & 15.75 & 1.40 & 0.300489 & $8.24\\pm0.97$~E$-7$ \\\\\n212539 & $17\\uph57\\upm45\\makebox[0pt][l]{.}\\ups97$ & $-29\\arcd12\\arcm04\\makebox[0pt][l]{.}\\arcs9$ & 17.61 & 1.79 & 0.286102 & $8.15\\pm1.49$~E$-7$ \\\\\n317072 & $18\\uph07\\upm22\\makebox[0pt][l]{.}\\ups35$ & $-29\\arcd42\\arcm13\\makebox[0pt][l]{.}\\arcs1$ & 15.53 & 1.25 & 0.353712 & $7.28\\pm0.82$~E$-7$ \\\\\n162146 & $17\\uph53\\upm31\\makebox[0pt][l]{.}\\ups37$ & $-29\\arcd53\\arcm13\\makebox[0pt][l]{.}\\arcs0$ & 17.13 & 1.87 & 0.410687 & $5.97\\pm0.62$~E$-7$ \\\\\n125481 & $17\\uph49\\upm57\\makebox[0pt][l]{.}\\ups46$ & $-29\\arcd16\\arcm03\\makebox[0pt][l]{.}\\arcs0$ & 15.08 & 1.60 & 0.358558 & $5.88\\pm0.27$~E$-7$ \\\\\n207420 & $17\\uph57\\upm20\\makebox[0pt][l]{.}\\ups44$ & $-30\\arcd36\\arcm58\\makebox[0pt][l]{.}\\arcs6$ & 14.27 & 1.02 & 0.254388 & $5.63\\pm0.11$~E$-7$ \\\\\n158555 & $17\\uph53\\upm13\\makebox[0pt][l]{.}\\ups46$ & $-31\\arcd13\\arcm56\\makebox[0pt][l]{.}\\arcs5$ & 17.60 & 1.96 & 0.231003 & $5.44\\pm0.48$~E$-7$ \\\\\n284531 & $18\\uph04\\upm14\\makebox[0pt][l]{.}\\ups75$ & $-29\\arcd52\\arcm19\\makebox[0pt][l]{.}\\arcs8$ & 13.56 & 0.93 & 0.435555 & $5.17\\pm0.21$~E$-7$ \\\\\n159557 & $17\\uph53\\upm18\\makebox[0pt][l]{.}\\ups25$ & $-29\\arcd53\\arcm13\\makebox[0pt][l]{.}\\arcs3$ & 14.37 & 0.98 & 0.522287 & $5.01\\pm0.27$~E$-7$ \\\\\n202706 & $17\\uph56\\upm55\\makebox[0pt][l]{.}\\ups80$ & $-28\\arcd39\\arcm16\\makebox[0pt][l]{.}\\arcs6$ & 15.88 & 1.68 & 0.449735 & $4.08\\pm0.71$~E$-7$ \\\\\n182350 & $17\\uph55\\upm13\\makebox[0pt][l]{.}\\ups65$ & $-29\\arcd26\\arcm44\\makebox[0pt][l]{.}\\arcs8$ & 15.55 & 1.29 & 0.282379 & $3.46\\pm0.16$~E$-7$ \\\\\n181955 & $17\\uph55\\upm11\\makebox[0pt][l]{.}\\ups74$ & $-30\\arcd02\\arcm12\\makebox[0pt][l]{.}\\arcs1$ & 16.14 & 1.41 & 0.414302 & $2.62\\pm0.14$~E$-7$ \\\\\n\\hline\n\\noalign{\\vskip3pt}\n\\end{tabular}}\n\\end{table}\n\n\\begin{figure}[htb]\n\\centerline{\\includegraphics[angle=0,width=130mm]{fig4.ps}}\n\\FigCap{Comparison of the distributions of systems with negative and positive\nperiod change rates. Note the excess of systems with high negative rate.}\n\\end{figure}\n\n\\begin{figure}[htb]\n\\centerline{\\includegraphics[angle=0,width=130mm]{fig5.ps}}\n\\FigCap{Example binary systems with cyclic orbital period variations.\nOGLE-III light curves are phased with constant period determined\nfor this dataset.}\n\\end{figure}\n\n\\begin{table}[h!]\n\\centering\n\\caption{\\small Parameters of candidate systems with cyclic period variations}\n\\medskip\n{\\footnotesize\n\\begin{tabular}{ccccccc}\n\\hline\nOGLE-BLG-ECL- &   RA    &    Dec  & $I_{\\rm max}$ & $V-I$ & $P_{\\rm orb}$ & $P_{\\rm cyc}$ \\\\\n              & J2000.0 & J2000.0 &     [mag]     & [mag] &      [d]      &     [y] \\\\\n\\hline\n124213 & $17\\uph49\\upm48\\makebox[0pt][l]{.}\\ups43$ & $-33\\arcd44\\arcm49\\makebox[0pt][l]{.}\\arcs2$ & 18.34 & 1.96 & 0.83464370 &  6   \\\\\n124728 & $17\\uph49\\upm51\\makebox[0pt][l]{.}\\ups95$ & $-29\\arcd41\\arcm48\\makebox[0pt][l]{.}\\arcs8$ & 17.35 & 2.10 & 0.35675024 &  6.5 \\\\\n125525 & $17\\uph49\\upm57\\makebox[0pt][l]{.}\\ups66$ & $-34\\arcd39\\arcm54\\makebox[0pt][l]{.}\\arcs5$ & 14.76 & 1.20 & 0.41200122 & 15   \\\\\n127744 & $17\\uph50\\upm14\\makebox[0pt][l]{.}\\ups07$ & $-30\\arcd06\\arcm22\\makebox[0pt][l]{.}\\arcs3$ & 14.98 & 1.77 & 0.89206984 &  6.5 \\\\\n129787 & $17\\uph50\\upm29\\makebox[0pt][l]{.}\\ups23$ & $-29\\arcd42\\arcm24\\makebox[0pt][l]{.}\\arcs8$ & 16.43 & 2.11 & 0.60583946 &  7   \\\\\n144092 & $17\\uph51\\upm58\\makebox[0pt][l]{.}\\ups17$ & $-29\\arcd46\\arcm41\\makebox[0pt][l]{.}\\arcs5$ & 15.54 & 1.58 & 0.37023316 &  5   \\\\\n148291 & $17\\uph52\\upm21\\makebox[0pt][l]{.}\\ups01$ & $-29\\arcd46\\arcm59\\makebox[0pt][l]{.}\\arcs3$ & 16.94 & 1.59 & 0.33885024 & 21   \\\\\n153387 & $17\\uph52\\upm47\\makebox[0pt][l]{.}\\ups02$ & $-33\\arcd14\\arcm16\\makebox[0pt][l]{.}\\arcs0$ & 18.09 & 1.78 & 0.41679258 & 14   \\\\\n154199 & $17\\uph52\\upm51\\makebox[0pt][l]{.}\\ups23$ & $-29\\arcd46\\arcm23\\makebox[0pt][l]{.}\\arcs6$ & 16.29 & 1.50 & 0.35765448 &  9   \\\\\n157195 & $17\\uph53\\upm06\\makebox[0pt][l]{.}\\ups58$ & $-30\\arcd06\\arcm54\\makebox[0pt][l]{.}\\arcs9$ & 16.25 & 1.59 & 0.30197516 &  3.0 \\\\\n163206 & $17\\uph53\\upm36\\makebox[0pt][l]{.}\\ups89$ & $-32\\arcd43\\arcm33\\makebox[0pt][l]{.}\\arcs6$ & 17.65 &  $-$ & 0.32505982 & 16   \\\\\n163661 & $17\\uph53\\upm38\\makebox[0pt][l]{.}\\ups87$ & $-29\\arcd39\\arcm03\\makebox[0pt][l]{.}\\arcs8$ & 18.56 & 1.81 & 0.30801484 &  6.5 \\\\\n163907 & $17\\uph53\\upm40\\makebox[0pt][l]{.}\\ups05$ & $-30\\arcd07\\arcm16\\makebox[0pt][l]{.}\\arcs9$ & 15.25 & 1.37 & 0.28606972 &  7   \\\\\n168647 & $17\\uph54\\upm02\\makebox[0pt][l]{.}\\ups50$ & $-33\\arcd00\\arcm18\\makebox[0pt][l]{.}\\arcs2$ & 16.10 & 1.65 & 0.31771956 & 10   \\\\\n171861 & $17\\uph54\\upm18\\makebox[0pt][l]{.}\\ups15$ & $-29\\arcd33\\arcm30\\makebox[0pt][l]{.}\\arcs8$ & 17.62 & 2.03 & 0.33601454 &  2.0 \\\\\n174546 & $17\\uph54\\upm32\\makebox[0pt][l]{.}\\ups42$ & $-29\\arcd34\\arcm58\\makebox[0pt][l]{.}\\arcs1$ & 15.66 & 1.31 & 0.45498264 & 20   \\\\\n177307 & $17\\uph54\\upm46\\makebox[0pt][l]{.}\\ups24$ & $-29\\arcd44\\arcm55\\makebox[0pt][l]{.}\\arcs2$ & 17.08 & 1.64 & 0.29253374 &  9   \\\\\n184596 & $17\\uph55\\upm24\\makebox[0pt][l]{.}\\ups33$ & $-29\\arcd33\\arcm41\\makebox[0pt][l]{.}\\arcs8$ & 13.57 & 0.98 & 0.52178366 &  1.5 \\\\\n192607 & $17\\uph56\\upm04\\makebox[0pt][l]{.}\\ups85$ & $-30\\arcd20\\arcm50\\makebox[0pt][l]{.}\\arcs3$ & 18.06 & 2.46 & 0.36283864 &  4   \\\\\n194120 & $17\\uph56\\upm12\\makebox[0pt][l]{.}\\ups29$ & $-30\\arcd46\\arcm02\\makebox[0pt][l]{.}\\arcs8$ & 15.93 & 1.47 & 0.31553406 &  5.5 \\\\\n207113 & $17\\uph57\\upm19\\makebox[0pt][l]{.}\\ups01$ & $-33\\arcd53\\arcm47\\makebox[0pt][l]{.}\\arcs1$ & 17.23 & 1.13 & 0.44961886 &  2.7 \\\\\n208059 & $17\\uph57\\upm23\\makebox[0pt][l]{.}\\ups64$ & $-30\\arcd49\\arcm15\\makebox[0pt][l]{.}\\arcs3$ & 16.52 & 1.53 & 0.41324802 &  1.5 \\\\\n209299 & $17\\uph57\\upm29\\makebox[0pt][l]{.}\\ups74$ & $-30\\arcd53\\arcm29\\makebox[0pt][l]{.}\\arcs3$ & 16.76 & 1.72 & 0.29563266 & 10   \\\\\n219570 & $17\\uph58\\upm22\\makebox[0pt][l]{.}\\ups46$ & $-29\\arcd45\\arcm49\\makebox[0pt][l]{.}\\arcs1$ & 16.48 & 1.39 & 0.31779704 &  5   \\\\\n222038 & $17\\uph58\\upm35\\makebox[0pt][l]{.}\\ups09$ & $-29\\arcd53\\arcm02\\makebox[0pt][l]{.}\\arcs3$ & 18.38 & 1.77 & 0.34785152 &  4.5 \\\\\n224344 & $17\\uph58\\upm47\\makebox[0pt][l]{.}\\ups91$ & $-26\\arcd50\\arcm46\\makebox[0pt][l]{.}\\arcs5$ & 15.38 & 1.52 & 0.39265194 &  9.5 \\\\\n235487 & $17\\uph59\\upm46\\makebox[0pt][l]{.}\\ups18$ & $-29\\arcd16\\arcm00\\makebox[0pt][l]{.}\\arcs4$ & 15.90 & 1.35 & 0.46023396 &  6.5 \\\\\n240451 & $18\\uph00\\upm13\\makebox[0pt][l]{.}\\ups09$ & $-29\\arcd14\\arcm25\\makebox[0pt][l]{.}\\arcs9$ & 16.70 & 1.39 & 0.36804430 & 22   \\\\\n246852 & $18\\uph00\\upm47\\makebox[0pt][l]{.}\\ups15$ & $-28\\arcd36\\arcm58\\makebox[0pt][l]{.}\\arcs2$ & 17.57 & 1.55 & 0.42095148 &  8.5 \\\\\n250500 & $18\\uph01\\upm05\\makebox[0pt][l]{.}\\ups72$ & $-30\\arcd13\\arcm22\\makebox[0pt][l]{.}\\arcs7$ & 18.27 & 1.71 & 0.47156330 &  7   \\\\\n289401 & $18\\uph04\\upm41\\makebox[0pt][l]{.}\\ups02$ & $-28\\arcd46\\arcm03\\makebox[0pt][l]{.}\\arcs7$ & 16.91 & 1.22 & 0.48576687 &  2.8 \\\\\n290247 & $18\\uph04\\upm45\\makebox[0pt][l]{.}\\ups58$ & $-29\\arcd31\\arcm37\\makebox[0pt][l]{.}\\arcs7$ & 16.36 & 1.47 & 0.40756464 &  5   \\\\\n292769 & $18\\uph04\\upm59\\makebox[0pt][l]{.}\\ups46$ & $-30\\arcd36\\arcm47\\makebox[0pt][l]{.}\\arcs3$ & 16.92 & 1.86 & 0.24031322 &  5   \\\\\n313336 & $18\\uph06\\upm58\\makebox[0pt][l]{.}\\ups97$ & $-27\\arcd41\\arcm34\\makebox[0pt][l]{.}\\arcs2$ & 18.70 & 1.60 & 0.32255660 &  3.5 \\\\\n314765 & $18\\uph07\\upm07\\makebox[0pt][l]{.}\\ups94$ & $-29\\arcd39\\arcm00\\makebox[0pt][l]{.}\\arcs9$ & 14.63 & 1.02 & 0.30319922 &  6.5 \\\\\n\\hline\n\\noalign{\\vskip3pt}\n\\end{tabular}}\n\\end{table}\n\n\n\\Section{Summary and Conclusions}\n\nIn the huge sample of 22~462 eclipsing binaries with $P_{\\rm orb}<4$~d\ndetected in the OGLE-III Galactic bulge fields, we found 108 systems\nwith reliable monotonic period changes: 56 systems with negative\nrate and 52 systems with positive one. We also indicated 35 systems with\nevident cyclic period variations. All reported systems but object\nOGLE-BLG-ECL-139622 with the highest derived negative period change rate\nare contact binaries. We did not find any binary with rapid\norbital period decrease as expected for a system heading toward merger\nwithin a few years. The period change rate in V1309 Sco five years\nbefore the merger event was about $-8.3\\times10^{-4}$~d\/y,\nwhile two years before the event it reached $-3.8\\times10^{-3}$~d\/y.\nFor comparison, contact binary OGLE-BLG-ECL-299145 with the second\nfastest measured period decrease in the whole our sample has the\nrate of merely $-1.7\\times10^{-5}$~d\/y. Twenty-two our systems\nwith negative period changes have the absolute rate higher then\n$-2\\times10^{-6}$~d\/y, that is the value estimated for contact system\nKIC 9832227 by Molnar {\\it et al.\\ } (2017). We cannot exclude the possibility\nthat this particular binary system and also our contact binaries with\nrelatively long orbital period ($P_{\\rm orb}>1.0$~d) and relatively\nhigh negative period change rate ($|\\dot P_{\\rm orb}|>10^{-5}$~d\/y), such\nas OGLE-BLG-ECL-344477, OGLE-BLG-ECL-176377, and OGLE-BLG-ECL-170070,\nwill merge in near future (in tens or hundreds of years).\nHowever, the fact that eclipsing binary OGLE-BLG-ECL-139622 with the\nhighest derived period change rate in our sample is a detached\nsystem and all the remaining binaries with reliable period changes\nare short-period contact binaries, some of which show cyclic variations,\nstrongly indicate for the presence of third bodies in the investigated\nsystems. Results from various observations of nearby close\nbinaries ($P_{\\rm orb}<1.0$~d) support the idea that tertiary companions\nto such binaries are very common (D'Angelo {\\it et al.\\ } 2006,\nPribulla and Rucinski 2006, Tokovinin {\\it et al.\\ } 2006, Rucinski {\\it et al.\\ } 2007).\n\nAnother possible explanation of the observed period changes could\nbe slow movement of starspots on the surface of the binary components.\nClose binary systems are often chromospherically active and thus they\ncan be strong X-ray emitters. We looked for X-ray counterparts to our 143\neclipsing binaries with the detected period changes and we found that\nall 32 binaries located within the Chandra Galactic Bulge Survey area\n($-3\\ifmmode^{\\circ}\\else$^{\\circ}$\\fi \\lesssim l \\lesssim 3\\ifmmode^{\\circ}\\else$^{\\circ}$\\fi$, $1\\ifmmode^{\\circ}\\else$^{\\circ}$\\fi \\lesssim |b| \\lesssim 2\\ifmmode^{\\circ}\\else$^{\\circ}$\\fi$,\nJonker {\\it et al.\\ } 2011, Wevers {\\it et al.\\ } 2016) have such counterparts. Four\nother binaries have counterparts in the XMM-Newton data (Page {\\it et al.\\ } 2012).\nSlowly drifting starspots would result in long-term brightness variations\nin the light curves of binaries. Some of our objects exhibit\nsuch mean brightness variations (see examples in Fig.~6).\n\n\\begin{figure}[htb]\n\\centerline{\\includegraphics[angle=0,width=130mm]{fig6.ps}}\n\\FigCap{Long-term brightness variations in selected binary systems with\nnegative (upper panels), positive (middle panels), and cyclic period changes\n(lower panels). In some cases, the brightness variations seem to correlate\nwith the period changes. All stars are isolated objects in the OGLE images.}\n\\end{figure}\n\nThe negative result of our search for contact systems with rapid period\ndecrease in the OGLE-III data is in agreement with the lack of observed\nGalactic red nova outbursts during the current fourth phase of OGLE.\nDespite our unsuccessful search for future mergers, binaries with relatively\nlong orbital period and high negative period change rate are worth further\nmonitoring.\n\n\n\\Acknow{\nWe would like to thank Profs. M. Kubiak and G. Pietrzy\\'nski,\nformer members of the OGLE team, for their contribution to the\ncollection of the OGLE photometric data over the past years.\nThe OGLE project has received funding from the National Science\nCentre, Poland, grant MAESTRO 2014\/14\/A\/ST9\/00121 to A.U. This work\nhas been also supported by the Polish Ministry of Sciences and Higher\nEducation grants No. IP2012 005672 under the Iuventus Plus program\nto P.P. and No. IdP2012 000162 under the Ideas Plus program to I.S.}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
{"text":"\\section{Introduction}\n\nA new perspective on the equilibrium morphology of \nthin accretion discs has been introduced \nin \\cite{C05,CR06}, where the effect of the plasma \nback-reaction on the magnetic field of the central object is outlined:\nit can induce a radially oscillating `crystal profile', meaning that the inner field acquires a well-defined periodic behaviour. This could be described via the formation of small-scale structures in the magnetic surfaces \n(like the one shown in \\manualref{fig:crystal}{2}; see also \\cite{LM10,MB11}), and it is inferred that such microstructures distort the background morphology, eventually breaking up the disc into a ring series, in the limit of a strong non-linear back-reaction.\n\nThis issue is different -- from the very beginning -- from the standard model of disc dynamics and its well-known open questions, such as the problem of turbulence-enhanced viscosity (for review purposes, you can see \\cite{BKL01,B03REVIEW}). The main purpose of this work is to generalize this approach to the time-dependent case, showing that this kind of structures holds locally in space and time: it could be therefore used to address the explanation of local processes and transient phenomena.\n\nThe brand new point of view stems from the implementation of an ideal \nMHD scheme where the Ferraro \nCorotation Theorem \\cite{F37} holds,\nallowing the angular velocity of the disc to be expressed in terms of magnetic flux surfaces only (for a stationary visco-resistive extension of this model see \\cite{BM10,BMP11}). \nIn fact, the background centrifugal force is balanced by the central \nobject gravity -- resulting in the Keplerian rotation of the disc --\nas well as by the background magnetic field, since the initial magnetosphere has a\ncurrent-free morphology.\nWhen the back-reaction corrections to the centrifugal \nforce and the Lorentz force are expressed in terms \nof the magnetic flux function, their balance results in the formation of toroidal current channels embedded in a radial oscillating magnetic structure.\nA remarkable assumption in constructing \nthis scenario is the pure rotation of the disc, which is not \nendowed with any poloidal component of the velocity field.\n\nIndeed, in many real astrophysical systems, the disc plasma is not very far from the quasi-ideal behaviour, \nbut the following three main points need to be discussed.\n(i) Given the mean thermodynamic parameters of the \nplasma in an accreting disc (temperature and number density), \nthe kinetic theory uniquely determines finite non-zero \nvalues of the viscosity and resistivity coefficients. \nEven when the value of such coefficients is rather small, \nthe impact of their existence -- i.e. the damping they can induce on the magnetic structure -- \ncould not be negligible, due to the \nlong lifetime of the accreting systems. \\\\\n(ii) The Shakura model for accretion within a thin disc \nconfiguration \\cite{S73,SS73} relies on very large values \nof the viscosity coefficient, able to balance the angular \nmomentum transport responsible for a significant non-zero \naccretion rate. \nWe will show how the microstructures framework and \nthe Shakura model of accretion are not \nstraightly comparable scenarios (e.g. the \npurely rotating disc is unable to directly accrete mass); none the less, \nit is of significant interest to \nunderstand if the periodic structures\npredicted in \\cite{C05,CR06} \nstill survive in the presence of the visco-resistive effects \nrequired to account for a turbulent accreting \nplasma. \\\\\n(iii) A non-zero resistivity coefficient is always present in every numerical simulation, at least to reproduce the effect of numerical dissipation due to machine's finite precision \\cite{BCFMR11}.\nFurthermore, works like \\cite{ZFRBM07} include non-ideal terms to explore the coupling of disc and jet physics, aiming at the problem of angular momentum transport.\n\nIn this work we include visco-resistive effects to the \nmicrostructures paradigm and grant the time dependence \nof the magnetic surfaces, needed to preserve the \nconsistence of the Generalized Ohm Law.\nIn fact, as far as we include a resistive contribution, the azimuthal \ncomponent of the Ohm Law acquires a term \nproportional to the toroidal current: this would turn out to be \nunbalanced in a purely rotating configuration, \nunless the non-stationarity of the model ensures \nthe presence of a non-zero azimuthal electric field.\nIn such a non-stationary case, we are able to \nrecover a solution consistent with the Corotation Theorem and to provide \nthe full consistence of the equilibrium configurations, using rather natural assumptions in the thin disc \nlimit and requiring the toroidal component \nof the magnetic field to be negligible\n(we adopt a dipole configuration to characterize the \nmagnetic properties of the central object).\n\nFocusing on the magnetic back-reaction, but neglecting its \neffect on the mass density distribution, \nwe obtain a periodic profile for the disc magnetic field, \nisomorphic to the `crystal profile' studied in \\cite{C05}\nbut for the presence of an exponential damping in time.\nNote that this is the most general regime for a non-stationary purely rotating disc, which is not requested to steadily accrete matter; we are dealing with a model where accretion could only exist by means of some intermittent instability (e.g., 3D modes excited along a separatrix \\cite{C09}).\nThe characteristic lifetime of this structure \ndepends on the resistivity \n(or equivalently on the viscosity, \nsince the Prandtl number is constrained to the unity \nby the consistence of the configuration equations) \nand crucially \non the spatial scale of the radial \noscillations. \nThe obtained time-scales could aim at accounting for transient \nphenomena: lifetimes can be easily \nfitted in the range of density and temperature typical for stellar accretion discs, \ncorresponding to radial scales that preserve the request to deal with \na local model around a fixed value of the radius (i.e. the length scale of the oscillations \nis much below the disc radial size).\nThe present analysis is therefore crucial in focusing \nthe correct phenomenological scenario to which this new paradigm can be referred to, ruling out this morphology of the magnetic field from the steady-state configuration of a stellar accretion disc.\n\nThe paper is organized as follows.\nIn \\Sref{sec:eqns}, we review the fundamental equations of axially symmetric two-dimensional MHD describing an accretion disc embedded in the gravitational and magnetic fields of a central object, taking into account the ones which notably differ from the equilibrium equations.\nIn \\Sref{sec:perturbation}, we develop the perturbation scheme for the considered problem and fix a fiducial radius for the local study, in order to expand the relevant equations up to the first order in the length scale of magnetic perturbation.\nIn \\Sref{sec:solution}, we derive the time-dependent form of separable solutions and recover the compatibility of this form with previously obtained equilibrium configurations.\nIn \\Sref{sec:time}, we give estimations of the lifetime of microstructures, checking the consistence with dynamical requests and establishing that this paradigm can address local transient events rather than steady-state configurations.\nConcluding remarks will follow in \\Sref{sec:final}.\n\n\\section{Relevant equations} \\label{sec:eqns}\n\nNeglecting the electron pressure gradient,\nthe Generalized Ohm Law retains its validity in the same form of stationary MHD:\n\\begin{equation} \\label{GOL}\n\\vec{E} + \\df{1}{c} \\<{ \\vec{v} \\mathbf{\\times} \\vec{B} } = \\eta_{\\scriptscriptstyle \\mathrm{B}} \\vec{J} \\, ,\n\\end{equation}\nobviously meaningful at every different time $t$.\nThis equation links the main physical quantities (electric and magnetic field $\\vec{E}$ and $\\vec{B}$, velocity field $\\vec{v}$, current density field $\\vec{J}$) by means of a microscopical transport coefficient -- the resistivity $\\eta_{\\scriptscriptstyle \\mathrm{B}}$.\nIntroducing a scalar electric potential $\\Phi$ and a vector magnetic potential $\\vec{A}$, we get\n\\SUBEQ{\n\\begin{equation}\n\\vec{E} = - \\vec{\\nabla} \\! \\Phi - \\df{1}{c} \\partial_{t} \\vec{A} \\, ,\n\\end{equation}\n\\begin{equation}\n\\vec{B} = \\vec{\\nabla} \\! \\times \\vec{A} \\, ,\n\\end{equation}\n}\nwhere it is crucial to note that a rotational electric field is generated when the magnetic potential depends on time.\nWe adopt the magnetic flux surfaces formalism and claim that $\\vec{A} = \\vec{A} \\<{\\psi}$, with $\\psi\\<{\\vec{r};t}$ the magnetic flux function.\nSince the system is axially symmetric, we can choose a cylindrical coordinate system $\\<{r,\\phi,z}$, where the $\\hat{\\mathrm{e}}_z$ axis is the symmetry axis; the magnetic potential and field assume then the following form:\n\\begin{equation} \\label{B_form}\n\\vec{A} = \\df{\\psi}{r} \\hat{\\mathrm{e}}_\\phi \\; \\Longrightarrow \\; \\vec{B} = \\df{\\vec{\\nabla} \\! \\psi}{r} \\mathbf{\\times} \\hat{\\mathrm{e}}_\\phi \\, ,\n\\end{equation}\nand we specify that $\\psi\\<{\\vec{r};t} = \\psi\\<{r,z^2;t}$ is symmetric under reflection over the equatorial plane $z=0$ -- in agreement with the symmetry of the background hydrostatic equilibrium quantities.\nThis form leads to strictly poloidal components of the magnetic field $\\vec{B}$, for it is possible to show, under the hypotheses of the present work, that if the azimuthal magnetic field is set to zero at the initial time, it will remain zero at every following time.\nThis could be read \\virg{on average}, constraining the turbulence-generated field to preserve a zero mean value if no external component is available; we are decoupling the study from the phenomenon of magnetic field generation due to shearing, assuming that our disc is not able to generate and sustain a coherent and significant azimuthal field.\n\nSince we are dealing with a purely rotating configuration in local approximation (in a narrow annulus around a fixed radius), the azimuthal component of Generalized Ohm Law \\eqref{GOL} becomes:\n\\begin{equation}\\label{psi_t}\n\\DERO{\\psi}{t} - \\df{c^2 \\eta_{\\scriptscriptstyle \\mathrm{B}}}{4 \\pi} \\DELTA{ \\psi } = 0 \\, ,\n\\end{equation}\nhaving used the Amp\\`{e}re Law to remove the current:\n\\begin{equation}\n\\vec{J} = \\df{c}{4 \\pi} \\vec{\\nabla} \\! \\times \\<{ \\vec{\\nabla} \\! \\times \\vec{A} } \\, .\n\\end{equation}\nThe same expression must be obtained from the Induction Equation with the usual derivation,\nand this can be done only if the resistivity is nearly a constant, precisely if\n\\begin{equation} \\label{grad_eta}\n\\abs{ \\df{\\vec{\\nabla} \\! \\eta_{\\scriptscriptstyle \\mathrm{B}}}{\\eta_{\\scriptscriptstyle \\mathrm{B}}} } \\ll \\abs{ \\df{ \\vec{\\nabla} \\! \\<{\\Delta \\psi}}{\\Delta \\psi} } \\, ,\n\\end{equation}\nwhere $\\Delta \\<{\\cdot}$ is the Laplace operator.\nIt is worth noting that this work is focused on the inner bulk region of the disc where this condition can hold (i.e. far from the interface between the disc and the surrounding magnetosphere, where the resistivity falls abruptly to zero).\nMoreover, the azimuthal component of the Induction Equation:\n\\begin{equation}\n\\DERO{B_\\phi}{t} = \\left[ \\vec{\\nabla} \\! \\times \\<{ \\vec{v} \\times \\vec{B} } \\right]_\\phi - \\df{c}{4 \\pi} \\vec{\\nabla} \\! \\times \\<{ \\eta_{\\scriptscriptstyle \\mathrm{B}} \\vec{\\nabla} \\! \\times \\<{B_\\phi \\vec{E}rsphi} } \\, ,\n\\end{equation}\nin the case of zero azimuthal field, yields to the following constraint:\n\\begin{equation} \\label{I_t}\n\\vec{\\nabla} \\! \\omega \\times \\vec{\\nabla} \\! \\psi = 0\\, ,\n\\end{equation}\nwhere we have introduced the angular velocity $\\omega\\<{\\vec{r};t}$ such that $\\vec{v} = \\omega r \\vec{E}rsphi$, which turns out to be a flux function $\\omega\\<{r,z;t} = \\omega\\<{\\psi}$. This result generalizes the Corotation Theorem \\cite{F37}, which holds in the visco-resistive framework provided there is no azimuthal component of the central magnetic field.\n\nFinally, we deal with the MHD momentum conservation equation:\n\\begin{equation} \\label{momentum}\n\\begin{split}\n\\rho \\<{ \\partial_{t} \\vec{v} + \\<{ \\vec{v} \\mathbf{\\cdot} \\vec{\\nabla} \\! } \\vec{v} } = & - \\vec{\\nabla} \\! p - \\rho \\vec{\\nabla} \\! \\chi + \\vec{F}_L + \\\\ & + \\eta_{\\scriptscriptstyle \\mathrm{V}} \\left[ \\nabla^2 \\vec{v} + \\df{1}{3} \\vec{\\nabla} \\! \\<{ \\vec{\\nabla} \\! \\mathbf{\\cdot} \\vec{v} } \\right] \\, ,\n\\end{split}\n\\end{equation}\nwhere $p$ is the thermodynamic pressure, $\\chi$ is the gravitational potential of the central object, $\\vec{F}_L$ is the Lorentz force, and the viscosity $\\eta_{\\scriptscriptstyle \\mathrm{V}}$ is assumed to be a constant.\nIts azimuthal component gives us the evolution law for the angular velocity:\n\\begin{equation} \\label{omega_t}\n\\DERO{\\omega}{t} - \\df{\\eta_{\\scriptscriptstyle \\mathrm{V}}}{\\rho} \\DELTA{ \\omega } = 0\\, ,\n\\end{equation}\nwhile the other components retain their stationary forms because of the pure rotation assumption.\n\n\\section{Perturbative Scheme} \\label{sec:perturbation}\n\nWe can now separate the contribution of the background dipole-like magnetic field\nfrom the back-reaction induced by the plasma current, i.e.\n\\begin{equation}\n\\psi \\<{r,z^2;t} = \\psi_0 \\<{R_0,z^2} + \\psi_1 \\<{ r-R_0,z^2;t} ,\n\\end{equation}\nwhere $ R_0 $ is the fiducial value for the local approximation, centred on $ \\abs{r-R_0} \\ll R_0 $.\nWe highlight that the background surface $\\psi_0$ is a stationary vacuum solution of Laplace's equation \\mbox{(i.e.  $\\Delta \\psi_0 = 0$)}, so \\eref{psi_t} can be written by means of the back-reaction $\\psi_1$ only.\n\nCorrespondingly, since \\eref{I_t} constrains the angular velocity to depend upon the magnetic surfaces (and mainly on the background one), it gains the following local expression:\n\\begin{equation} \\label{ferraro}\n\\omega \\<{ R_0 , r-R_0 , z ; t } \\equiv \\omega \\<{\\psi} \\simeq \\omega\\<{\\psi_0} + \\omega'_0 \\psi_1 \\, ,\n\\end{equation}\nwith\n\\SUBEQ{\n\\begin{equation}\n\\omega \\<{ \\psi_0 } = \\Omega_{\\mathrm{K}} \\<{R_0} = \\sqrt{ \\df{G M_{\\mathrm{*}}}{R_0^3} } \\, ,\n\\end{equation}\n\\begin{equation}\n\\omega'_0 = \\left. \\DERT{\\omega}{\\psi}{} \\right|_{\\psi_0} = \\mathrm{const.} \\, ,\n\\end{equation}\n}\nand we recover the Keplerian profile $\\Omega_{\\mathrm{K}}\\<{r}$ for the background contribution.\nNote now that the evolution of the angular velocity is determined by \\eref{omega_t}: using the expansion \\eqref{ferraro}, it reduces itself exactly to \\eref{psi_t} if\n\\begin{equation} \\label{prandtl}\n\\df{\\eta_{\\scriptscriptstyle \\mathrm{V}}}{\\rho} = \\df{ c^2 \\eta_{\\scriptscriptstyle \\mathrm{B}} }{ 4 \\pi } \\, ,\n\\end{equation}\nwhich states that the Magnetic Prandtl Number $\\mathbb{PR}$ is set to one, a condition also adopted in \\cite{BM10} with the purpose of recovering the existence of a solution consistent with the Corotation Theorem.\n\nFurthermore, \\eref{prandtl} shows that the height-dependence of $\\eta_{\\scriptscriptstyle \\mathrm{B}} = \\eta_{\\scriptscriptstyle \\mathrm{B}} \\<{\\rho}$ is determined only by the density profile, giving a deeper meaning to \\eref{grad_eta}. This can be explained after the introduction of three dimensionless variables defined as follows:\n\\SUBEQ{\n\\begin{equation} \\label{adim_space}\n\\bar{r} = k\\<{r - R_0}, \\quad \\bar{z} = \\sqrt[4]{3 \\beta_0} \\df{z}{H_0},\n\\end{equation}\n\\begin{equation} \\label{adim_time}\n\\bar{t} = \\df{t}{\\tau} = \\df{k^2 \\eta_{\\scriptscriptstyle \\mathrm{V}}}{\\rho_0} t \\, ,\n\\end{equation}\n} \nand two dimensionless functions written as\n\\begin{equation}\nY \\<{\\bar{r},\\bar{z}^2;\\bar{t}} = \\df{k \\psi_1}{\\left. {\\partial \\psi_0}\/{\\partial r} \\right|_0}, \\qquad  D\\<{\\bar{z}^2} = \\df{\\rho}{\\rho_0} \\, ,\n\\end{equation}\nwhere $\\rho_0 = \\rho\\<{\\bar{r}=0,\\bar{z}=0}$ is the density value on the equatorial plane at the fixed radius, $H_0$ is the half-thickness of the disc, and $k^{-1}$ is the radial scale of the back-reaction flux function $\\psi_1$.\n\nThe parameter $\\beta_0$ used in \\eref{adim_space} to scale the height-dependence of magnetic surfaces is the usual plasma $\\beta$-parameter, but it takes into account the magnitude of the background field only:\n\\begin{equation} \\label{beta}\n\\beta_0 = 8 \\pi \\df{p_0}{B_{z0}^2} = \\<{k H_0}^2 \\, ,\n\\end{equation}\nwhere $p_0$ is the background thermodynamic pressure.\nWe can then write down the components of \\eref{grad_eta} to obtain:\n\\SUBEQ{\n\\begin{equation}\n\\df{\\partial_{z} \\eta_{\\scriptscriptstyle \\mathrm{B}}}{\\eta_{\\scriptscriptstyle \\mathrm{B}}} \\simeq \\df{1}{H_0} \\ll \\df{\\sqrt[4]{\\beta_0}}{H_0} \\simeq \\df{\\partial_{z} \\psi_1}{\\psi_1}\n\\end{equation}\n\\begin{equation} \\label{kR0}\n\\df{\\partial_{r} \\eta_{\\scriptscriptstyle \\mathrm{B}}}{\\eta_{\\scriptscriptstyle \\mathrm{B}}} \\simeq \\df{1}{R_0} \\ll k \\simeq \\df{\\partial_{r} \\psi_1}{\\psi_1} \\, ,\n\\end{equation}\n}\nwhich are identically satisfied if\n\\begin{equation} \\label{regime}\n\\beta_0 \\gg 1 \\, ,\n\\end{equation}\nsince \\eref{kR0} -- via \\eref{beta} -- can be restated as\n\\begin{equation}\nk R_0 \\simeq \\sqrt{\\beta_0} \\df{R_0}{H_0} \\gg 1 \\, ,\n\\end{equation}\nand this is implied by \\eref{regime} and the thin disc assumption $H_0 \\ll R_0$.\nThen it follows that \\eref{regime} specifies the regime of validity for our treatment, for it assures the slowly varying behaviour of the resistivity.\n\nIn terms of these variables and parameters, \\eref{psi_t} becomes\n\\begin{equation} \\label{heateq}\nD\\<{\\bar{z}^2} \\DERO{Y}{\\bar{t}} - \\DELTAE{Y} = 0 \\, ,\n\\end{equation}\nwhere \n\\begin{equation}\n\\DELTAE{\\<{\\cdot}} \\doteq \\DER{\\<{\\cdot}}{\\bar{r}}{2} + \\dfrac{1}{\\sqrt{3\\beta_0}} \\DER{\\<{\\cdot}}{\\bar{z}}{2}\n\\end{equation}\nis the dimensionless Laplacian.\n\n\\section{Damped Solutions} \\label{sec:solution}\n\n\\eref{heateq} is a linear parabolic PDE with an infinite number of solutions: in particular, it admits a separable solution analogue to those found in \\cite{CR06}, currently endowed with an explicit time-dependence:\n\\begin{equation} \\label{solution}\nY\\<{\\bar{r},\\bar{z}^2;\\bar{t}} = \\mathcal{Y} \\, F\\<{\\bar{z}^2} \\sin\\<{a \\, \\bar{r}} e^{-\\bar{t}} \\, ,\n\\end{equation}\n$\\mathcal{Y}$ and $a$ being real constants. The height-dependence is fixed by\n\\begin{equation} \\label{F_u2}\n\\DERT{F\\<{\\bar{z}^2}}{\\bar{z}}{2} - \\sqrt{3 \\beta_0} \\<{a^2 - D\\<{\\bar{z}^2}} F\\<{\\bar{z}^2}=0 \\, ,\n\\end{equation}\nafter the choice of an equation of state, needed -- together with the vertical hydrostatic background equilibrium -- to set an expression for $D\\<{\\bar{z}^2}$.\nIf we choose a polytropic form for the gravothermal background, such that:\n\\SUBEQ{\n\\begin{equation}\np\\<{\\bar{z}^2} = p_0 \\, D\\<{\\bar{z}^2}^{1 + {1}\/{\\Gamma}}\n\\end{equation}\n\\begin{equation} \\label{D_polytropic}\nD\\<{\\bar{z}^2} = \\left[ 1- \\<{\\df{z}{H_0}}^2 \\right]^{\\Gamma} \\simeq 1 - \t\\df{\\Gamma}{\\sqrt{3\\beta_0}} \\bar{z}^2 \\, ,\n\\end{equation}\n}\nwhere $\\Gamma$ is the polytropic index,\nthen we can find an analytic solution of \\eref{F_u2} by means of Generalized Hermite Polynomials\n$\\mathcal{H}\\!\\left[ \\cdot , \\cdot \\right]$:\n\\begin{equation} \\label{hermite}\n\\begin{split}\nF\\<{\\bar{z}^2} = & e^{-\\sqrt{\\Gamma} \\bar{z}^2 \/ 2} \\cdot \\\\\n& \\cdot \\mathcal{H}\\!\\!\\left[- \\df{\\sqrt{\\Gamma \/ 3 \\beta_0} + a^2 - 1}{2 \\sqrt{\\Gamma \/ 3 \\beta_0}} , \\sqrt{\\sqrt{\\Gamma} \\bar{z}^2} \\right] ,\n\\end{split}\n\\end{equation}\nhaving omitted a constant factor and adopting the notation $\\mathcal{H}\\!\\left[ \\xi , x \\right]$ for the Hermite Polynomial of order $\\xi$ in the variable $x$.\n\\begin{figure} \\label{fig:profile}\n\\centering\n\\includegraphics[scale=.5]{image1.pdf}\n\\caption{Behaviour of the back-reaction magnetic surfaces in \\eref{hermite} versus the dimensionless vertical coordinate $\\bar{z}$, normalized to the gaussian equatorial value with $a = a_{\\mathrm{min}}$.\nWe fix $\\beta_0 = 400\/3$ and $\\Gamma=1$, and we change $a$ from $ a_{\\mathrm{min}} $.\nAs $a$ increases, the plots are drawn as follows: thick, thin, dashed, dotted, dot-dashed.\nIt is worth noting that the gaussian profile has the greatest full width at half maximum, while it has not the greatest initial amplitude.}\n\\end{figure}\nThese Generalized Polynomials are monotonically decreasing provided their order is negative.\nChoosing the radial wave-number as:\n\\begin{equation}\na^2 = a^2_{\\mathrm{min}} = {1-\\sqrt{\\Gamma \/ 3 \\beta_0}} \\, ,\n\\end{equation}\nthe solution \\eqref{hermite} reproduces the gaussian profile already obtained in \\cite{C05}.\nThis is the lower limit for the range of the parameter values, since the solution becomes non-physical when $a < a_{\\mathrm{min}}$; on the contrary, an upper limit doesn't exist, although the amplitude of the structure becomes smaller \nas $a$ increases, and it is eventually negligible slightly over the unity (see \\manualref{fig:profile}{1}).\nOnly a narrow range of values around the unity is therefore suitable to tune the radial wavenumber.\nAfter these considerations, the solution shown in \\eref{solution} is plotted in \\manualref{fig:crystal}{2} for the isothermal case, as a prototype of a magnetic microstructure.\n\\begin{figure} \\label{fig:crystal}\n\\centering\n\\includegraphics[scale=1.3]{image2.pdf}\n\\caption{Behaviour of the initial ($\\bar{t}=0$) back-reaction magnetic surfaces in \\eref{solution} versus the dimensionless radial and vertical coordinates $\\bar{r}$ and $\\bar{z}$, normalized to the maximum equatorial value obtained at the fixed radius. Parameters are chosen to recover the profile of \\cite{CR06}, so $\\beta_0 = 300\/4$, $\\Gamma=1$ and $a = a_{\\mathrm{min}}$. The rigid oscillatory profile shown is what we referred to as a `magnetic microstructure'.}\n\\end{figure}\n\\subsection*{Matching With Previous Models} \\label{sec:matching}\nWe underline that, under the assumption of pure rotation $\\vec{v} = \\omega R_0 \\vec{E}rsphi$, the only equations explicitly depending on time are those shown in \\Sref{sec:eqns}, though there are other interesting equations.\n\nRadial and vertical components of momentum conservation \\eqref{momentum} retain their stationary formulation, returning the exact same dimensionless equations offered in \\cite{C05,CR06}, now coupled to \\eref{heateq}.\nThis equation introduces the new variable $\\bar{t}$, but acts as a closure condition for the system which determines the equilibrium of perturbative pressure $\\hat{P}$ and density $\\hat{D}$, namely:\n\\begin{subequations} \\label{coppi_eqns}\n\\begin{equation}\n\\partial_{\\bar{z}^2} \\hat{P} + \\df{1}{\\sqrt{3 \\beta_0}} \\hat{D} + 2 \\DELTAE{Y} \\partial_{\\bar{z}^2} Y = 0\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n\\df{1}{2} \\partial_{\\bar{r}} \\hat{P} + \\left( D\\<{\\bar{z}^2} + \\df{1}{\\beta_0} \\hat{D} \\right) & Y + \\\\ + \\DELTAE{Y} & \\<{ 1+\\partial_{\\bar{r}} Y} = 0 \\, .\n\\end{split}\n\\end{equation}\n\\end{subequations}\nIn the linear regime $Y \\ll 1$, when pressure and density perturbations are assumed to be negligible, system \\eqref{coppi_eqns} is fully satisfied solving the equation\n\\begin{equation}\n\\DELTAE{Y} = - D\\<{\\bar{z}^2} Y \\, ,\n\\end{equation}\nwhich is consistent with the separable solution \\eqref{solution} and owns the same magnetic structure (although damped in time) developed in \\cite{LM10} and \\cite{BMP11}, according to the additional approximations considered in those papers.\nIt is interesting to note that \\eref{heateq} could be a closure condition also in the general non-linear regime, but this is forbidden by the Continuity equation:\n\\begin{equation} \\label{continuity}\n\\DERO{\\rho}{t} + \\vec{\\nabla} \\! \\mathbf{\\cdot} \\<{ \\rho \\vec{v} } = 0 \\, ,\n\\end{equation}\nwhich shows that the density has to keep its steady-state profile. \nThis suggests that a complete model with the disc decomposition in a ring-like structure resembling the one in \\cite{CR06} (which needs the density to be time-dependent) should exhibit non-vanishing poloidal velocities.\n\nAnother solution can be found in the special `crystal regime' specified by\n\\begin{equation} \\label{crystalregime}\n\\mathcal{Y} \\gtrsim 1 \\, , \\quad \\hat{D} \\ll D \\, ,\n\\end{equation}\noccurring when the back-reaction field is strong -- it is indeed determined by the constant $\\mathcal{Y}$ in \\eref{solution} -- whereas density perturbations are negligible. It is worth noting here that background and back-reaction field magnitudes are completely independent, so that \\eref{crystalregime} is still consistent with \\eref{regime}.\nIn this case, the system \\eqref{coppi_eqns} is reduced to: \n\\SUBEQ{\n\\begin{equation}\nD\\<{\\bar{z}^2} \\partial_{\\bar{t}} Y - \\DELTAE{Y} = 0\n\\end{equation}\n \n\\begin{equation}\n\\partial_{\\bar{z}^2} \\hat{P} + 2 \\DELTAE{Y} \\partial_{\\bar{z}^2} Y = 0\n\\end{equation}\n \n\\begin{equation}\n\\df{1}{2} \\partial_{\\bar{r}} \\hat{P} + D\\<{\\bar{z}^2} Y + \\DELTAE{Y} \\<{ 1+\\partial_{\\bar{r}} Y} = 0 \\, ,\n\\end{equation}\n}\nwhich admits the solution \\eqref{solution} and gives a separable form of the pressure too:\n\\begin{equation} \\label{PY^2}\n\\hat{P} \\<{\\bar{r},\\bar{z}^2;\\bar{t}}= \\mathcal{Y}^2 \\, D\\<{\\bar{z}^2} F^2\\<{\\bar{z}^2} \\sin^2\\<{a \\bar{r}} e^{-2\\bar{t}} \\, ,\n\\end{equation}\nwhere $D\\<{\\bar{z}^2}$ and $F\\<{\\bar{z}^2}$ can be expressed via \\eref{D_polytropic} and \\eref{hermite}, respectively.\nThis perturbative pressure increase corresponds to a temperature (or internal energy) increase, eventually modelled by a perturbative $\\hat{T} \\<{\\bar{r},\\bar{z}^2;\\bar{t}}$, for in this peculiar regime there is no density perturbation.\n\n\\section{Estimations of Damping Time} \\label{sec:time}\n\nIt turns out that every magnetic structure obtained in literature as an equilibrium configuration becomes a dynamical solution after the compatibility with \\eref{heateq} has been checked, admitting in this perspective also solutions slightly different from the proposed \\eref{solution}. \nAt the same time, the damping effect cannot be removed: the unperturbed magnetic configuration dominated by the central field is restored and the back-reaction microstructured field becomes negligible after the time\n\\begin{equation} \\label{life}\n\\tau = \\df{ \\rho_0 }{k^2 \\eta_{\\scriptscriptstyle \\mathrm{V}}} \\, ,\n\\end{equation}\ndefined in \\eref{adim_time}.\nA direct consequence is that such structures need the plasma to be quasi-ideal; in particular, this shows how this perspective cannot be consistent with the Standard Model \\cite{S73,SS73} and its huge effective viscosity, which damps the structures after very little time.\n\nThis quasi-ideal lifetime depends mainly on number density $n_\\mathrm{e}$ and temperature $T$, and their possible values are interrelated because of \\eref{prandtl}.\nIn what follows we make use of microscopical resistivity:\n\\begin{equation} \\label{microeta}\n\\eta_{\\scriptscriptstyle \\mathrm{B}} = \\df{m_{\\mathrm{e}} \\nu_{\\mathrm{ie}}}{n_\\mathrm{e}\\mathrm{e}^2} \\simeq 4 \\pi \\mathrm{e}^2 \\sqrt{\\df{m_{\\mathrm{e}}}{K_{\\scriptscriptstyle \\mathrm{B}}^3}} \\, \\df{\\mathrm{Log} \\Lambda \\<{n_\\mathrm{e}, T}}{\\left( T \\left[\\!\\, \\mathrm{K}\\right] \\right)^{3\/2}} \\; \\left[\\!\\, \\mathrm{s}\\right] \\, ,\n\\end{equation}\nand microscopical viscosity:\n\\begin{equation} \\label{microvisco}\n\\eta_{\\scriptscriptstyle \\mathrm{V}} = \\df{m_{\\mathrm{i}} n_\\mathrm{e} c_\\mathrm{S}^2}{\\nu_{\\mathrm{ii}}} \\simeq \\df{\\sqrt{m_{\\mathrm{i}} K_{\\scriptscriptstyle \\mathrm{B}}^5}}{4 \\pi \\mathrm{e}^4} \\, \\df{\\,\\<{T \\left[\\!\\, \\mathrm{K}\\right]}^{5\/2}}{\\mathrm{Log} \\Lambda \\<{n_\\mathrm{e}, T}} \\; \\left[\\! \\df{\\, \\mathrm{g}}{\\, \\mathrm{cm} \\cdot \\, \\mathrm{s}} \\right] \\, ,\n\\end{equation}\nwhere $m_{\\mathrm{e}}$ and $\\mathrm{e}$ are electronic mass and charge, $m_{\\mathrm{i}}$ is the ionic mass (here protons are considered), $\\nu_\\mathrm{ie}$ and $\\nu_{\\mathrm{ii}}$ are the collision frequencies of ions with electrons and ions with ions respectively, $K_{\\scriptscriptstyle \\mathrm{B}}$ is the Boltzmann constant, and $\\mathrm{Log} \\Lambda \\<{n_\\mathrm{e}, T}$ is the Coulomb Logarithm.\nBy means of these expressions we obtain the quasi-ideal Magnetic Prandtl Number:\n\\begin{equation} \\label{PRlong}\n\\begin{split}\n\\mathbb{PR} \\doteq & \\; \\df{4 \\pi \\eta_{\\scriptscriptstyle \\mathrm{V}}}{c^2 \\rho \\eta_{\\scriptscriptstyle \\mathrm{B}}} = \\\\\n& \\df{K_{\\scriptscriptstyle \\mathrm{B}}^4}{4 \\pi \\mathrm{e}^6 c^2 \\sqrt{m_{\\mathrm{e}} m_{\\mathrm{i}}}} \\, \\df{\\left( T \\left[\\!\\, \\mathrm{K}\\right] \\right)^4}{\\mathrm{Log} \\Lambda \\<{n_\\mathrm{e}, T} \\, n_\\mathrm{e}\\left[\\! \\, \\mathrm{cm}^{-3}\\right]} \\, ,\n\\end{split}\n\\end{equation}\nso that the request $\\mathbb{PR}=1$ sets a transcendental relation $n_\\mathrm{e} = n_\\mathrm{e} \\<{T}$, represented as a straight line in \\manualref{fig:PR}{3}.\nThe same Figure shows how the range of possible temperatures and densities shrinks when a real accretion disc is considered.\nWe deduce that the microstructures -- as described here -- could only be found in correspondence of specified values of the involved physical parameters, i.e.  with densities in $\\<{10^8, 10^{13}} \\, \\mathrm{cm}^{-3}$ and temperatures in $\\<{5\\cdot10^3, 10^5} \\, \\mathrm{K}$.\n\n\\begin{figure} \\label{fig:PR}\n\\centering\n\\includegraphics[scale=.45]{image3}\n\\caption{\nContour plot of Magnetic Prandtl Number in the log-log plane of temperature and number density via \\eref{PRlong}, restricted from values $\\mathbb{PR}=0.1$ (darkest stripe) to ${\\mathbb{PR}=10}$ (lightest stripe).\nThe white dashed line is $\\mathbb{PR}=1$ and points out the behaviour of the implied relation $n_\\mathrm{e}\\<{T}$.\nThe black dashed lines mark the ranges consistent with known accretion discs.\n}\n\\end{figure}\n\nHaving assigned the $n_\\mathrm{e}\\<{T}$ prescription, we are able to check if the microstructures exist beyond the dynamical time-scale $\\Omega_{\\mathrm{K}}^{-1}$, which is the time needed for the vertical hydrostatic equilibrium to be established: since we assumed it is preserved (vertical background equilibrium equation), the microstructures have to exceed it for the model to be consistent. \nThe \\eref{life} is recast in this terms as:\n\\begin{equation} \\label{tauomega}\n\\df{\\tau}{\\Omega_{\\mathrm{K}}^{-1}} = \\<{\\df{\\lambda}{2 \\pi c_\\mathrm{S} \\<{T}}}^2 \\nu_{\\mathrm{ii}}\\<{T} \\Omega_{\\mathrm{K}}\\<{M_\\mathrm{*},R_0} \\, ,\n\\end{equation}\nwhere $\\lambda \\doteq 2 \\pi k^{-1}$ is the back-reaction length scale.\n\nSeeking for the consistence check, we make use of an estimation provided for an analogous global structure \\cite{MB11}, which is of the order of $\\<{10^3,10^4} \\, \\mathrm{cm}$.\nIn \\manualref{fig:tauomega}{4} we then represent the lifetime at a fixed $\\lambda$ as a function of the radial parameter $R_0$, for different values of the temperature $T$; we deduce that this model has not to be applied to cold discs with a mean temperature below $10^4 \\, \\mathrm{K}$.\nMagnetic microstructures can appear in the inner regions of discs with $T$ in $\\<{10^4,10^5} \\, \\mathrm{K}$.\nLooking at the details of \\manualref{fig:tauomega}{4}, we note that at the lowest temperatures the lifetime lies in the shaded region, where microstructures vanish before vertical equilibrium is established or their rise is competitive with Magneto-Rotational Instability ($\\tau \\lesssim \\Omega_{\\mathrm{K}}^{-1}$) \\cite{BH91}, so there is no formation of periodic magnetic flux surfaces.\nRaising the temperature, the lifetime eventually crosses the threshold such that $\\tau \\gg \\Omega_{\\mathrm{K}}^{-1}$, starting from the lower values of radius: at $T=10^5 \\, \\mathrm{K}$ the structure is however confined within radii less than $10^9 \\, \\mathrm{cm}$, and the highest temperatures are forbidden by the constraint $\\mathbb{PR}=1$.\n\nWe have also to remember the request of \\eref{regime} on the $\\beta_0$ parameter, which in this narrow range of temperatures would imply a limitation on the background magnetic field magnitude.\nThis turns out to be easily feasible since it only asks the vertical magnetic field to not exceed $10^{12} \\, \\mathrm{G}$, ruling out only the strongest of the magnetars; even highly magnetized neutron stars reach this value only at their surface, with little or no influence on the disc.\n\n\\begin{figure} \\label{fig:tauomega}\n\\centering\n\\includegraphics[scale=.5]{image4}\n\\caption{\nPlot of microstructures lifetime versus fiducial radius $R_0$ via \\eref{tauomega}, with central mass $M_\\mathrm{*} = 1 M_\\mathrm{\\odot}$.\nMicrostructures length scale adopted here is $\\lambda = 5 \\cdot 10^3 \\, \\mathrm{cm}$, following the estimation in \\cite{MB11}.\nTime is in units of dynamical time, so the thin horizontal line marking the shaded region acts as the threshold for $\\tau \\gg \\Omega_{\\mathrm{K}}^{-1}$.\nThe $\\tau\\<{R_0}$ lines are drawn for $T=10^4 \\, \\mathrm{K}$ (dotted line), $5\\cdot10^4 \\, \\mathrm{K}$ (dashed) and $10^5 \\, \\mathrm{K}$ (solid).\n}\n\\end{figure}\n\nMicrostructures are then characterized locally in space (inner disc region) and time (finite lifetime).\nIn a more quantitative way, fixing the fiducial radius and cutting the lowest values of temperature, lifetime can be estimated in seconds as a function of the microstructures length scale (note that a fixed radius implies a fixed dynamical time-scale).\nIt simply stems from \\eref{life} again by means of \\eref{microvisco}:\n\\begin{equation} \\label{real_lifetime}\n\\tau\n= \\df{\\mathrm{e}^4}{\\pi} \\sqrt{\\df{m_{\\mathrm{i}}}{K_{\\scriptscriptstyle \\mathrm{B}}^5}} \\left( \\lambda \\left[\\!\\, \\mathrm{cm} \\right] \\right)^2  \\df{\\mathrm{Log} \\Lambda \\<{n_\\mathrm{e}, T} \\, n_\\mathrm{e}\\left[\\! \\, \\mathrm{cm}^{-3}\\right]}{\\left( T \\left[\\!\\, \\mathrm{K}\\right] \\right)^{5\/2}} \\, ,\n\\end{equation}\ngiving at the same time a lower bound for the scale of the spatial fluctuations of back-reaction field, as it is represented in \\manualref{fig:tau}{5}.\nExisting structures have size of at least tens of meters, outlive the dynamical time and vanish over minutes.\nRaising the temperature, smaller structures emerge with the same lifetime, while the biggest scales reach lifetimes of the order of hours.\nIt is worth noting that such scales are tunable, in the sense that the model cannot fix neither $\\lambda$ nor $\\tau$ because of the assumption of locality.\nTransient events whose duration lies in the range of minutes-hours has been observed in accretion discs, so we infer they can be related to the formation of microstructures.\n\n\\begin{figure} \\label{fig:tau}\n\\centering\n\\includegraphics[scale=.5]{image5}\n\\caption{Plot of microstructures lifetime versus their length scale via \\eref{real_lifetime}.\nThe horizontal thin line marking the value of dynamical time $\\Omega_{\\mathrm{K}}^{-1}\\<{M_\\mathrm{*},R_0}$ is drawn after fixing the values of central mass and disc scale radius to $M_\\mathrm{*} = 1 M_\\mathrm{\\odot}$ and $R_0 = 5\\cdot10^8 \\, \\mathrm{cm}$.\nThe $\\tau\\<{\\lambda}$ lines are drawn for $T=5\\cdot10^4 \\, \\mathrm{K}$ (dashed) and $10^5 \\, \\mathrm{K}$ (solid).\nThe dot-dashed horizontal lines mark the value of one hour and one day respectively, in order to show the lifetimes order of magnitude, which is mainly from minutes to hours and never exceeds the day.\n}\n\\end{figure}\n\n\\subsection*{Relaxing The Prandtl Constraint}\nIt is possible to preserve the $\\mathbb{PR}=1$ condition and avoid the ranges-shrinking at the same time.\nLet us adopt an effective resistivity, enhanced because of the far bigger value of viscosity, and parametrized in the form:\n\\begin{equation} \\label{effective_eta}\n\\eta_{\\scriptscriptstyle \\mathrm{B}}^{\\mathrm{eff}} = \\df{4 \\pi}{c^2 \\rho} \\eta_{\\scriptscriptstyle \\mathrm{V}} \\, ,\n\\end{equation}\npreserving in this way the validity of \\eref{prandtl}, regardless of the density-temperature relation determined in the quasi-ideal case.\nWith this assumption, the accessible values of the physical parameters are broadened up to the full ranges consistent with observations.\nIt is then possible to get the same lifetimes in a way more extended range of physical states, and even reach time-scales of the order of days for peculiar dense and cold discs.\n\nIt is useful to say that the Prandtl condition could also be avoided if some other model-feature is included: e.g. a radial component of the velocity field would keep the same perspective on finite lifetime, but allowing $\\mathbb{PR} \\neq 1$ \\cite{MC12}.\n\n\\section{Final Remarks} \\label{sec:final}\n\nWe discussed how the equilibrium configurations proposed by \\cite{C05} and the following works have to be generalized by means of non-ideal terms (an arbitrarily small but non-zero resistivity is required by kinetic theory, and its effects are not negligible), and how these equilibria evolve in time.\nThe peculiar morphology of magnetic microstructures is preserved, but it has been shown how the dissipative coefficients forbid its maintaining: in particular, the macroscopical viscosity of Shakura Standard Disc destroys it on a time-scale so much short that the structure formation itself is forbidden.\nThe microscopical plasma viscosity gives instead the chance to outlive the dynamical threshold: quasi-ideal microstructures vanish after minutes and eventually hours.\nThis time-scale is consistent with the description of transient events, once the model is refined to gain a perspective in the emissions possibly related to localized structures of magnetic field and induced current.\n\nThis result rules out the stationary morphology in stellar accretion disc (with a central neutron star or a solar mass black hole) adopted until now, because of the finite lifetime found also in the quasi-ideal case.\nRealistic \\virg{crystal structures} can exhibit a length scale of at least $10^3 \\, \\mathrm{cm}$, confined by the request to exceed the dynamical time (lower bound).\nIt is clear at this point that the aimed transient events have a local nature in both space and time -- spatially confined to microscales and temporally limited by the structures mean lifetime.\n\nIt has also been shown that only temperatures between $10^4$ and $10^5 \\, \\mathrm{K}$ allow the formation of quasi-ideal structures, and this formation is easier in the inner region of discs, up to a radial extent of the order of $10^9 \\, \\mathrm{cm}$.\nFor central magnetic fields stronger than $10^{12} \\, \\mathrm{G}$ structures are forbidden, but only magnetars could reach such an order of magnitude, while the range of temperatures and sizes is consistent with observed or estimated parameters in binary accretion discs with a magnetized central object (white dwarf, neutron star or low mass black hole), also with a meaningful magnitude of the magnetic field far from the object surface.\n\n\\paragraph*{Acknoledgments.}\nThis work has been partially developed in the framework\nof the CGW Collaboration (www.cgwcollaboration.it).\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}