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+{"text":"\\section{Introduction}\nThe large-scale halo of hot gas  provides\na unique way to measure the baryonic and gravitational\nmass of galaxy clusters.\nThe baryonic mass can be measured directly from\nthe observation of the hot X-ray emitting intra-cluster medium (ICM), and \nof the associated stellar component \\citep[e.g.][]{giodini2009,gonzales2007},\nwhile measurements of the gravitational mass require\nthe assumption of hydrostatic equilibrium between the gas and\ndark matter. \nCluster cores are subject to a variety of non-gravitational\nheating and cooling processes that may result in deviations\nfrom hydrostatic equilibrium, and in inner regions beyond the\ncore the ICM is expected to be in hydrostatic equilibrium\nwith the dark matter potential. At the\noutskirts, the low-density ICM and the proximity to the\nsources of accretion results in the onset\nof new physical processes\nsuch as  departure from hydrostatic equilibrium \\citep[e.g.,][]{lau2009},\nclumping of the gas \\citep{simionescu2011},\ndifferent temperature between electrons and ions \\citep[e.g.,][]{akamatsu2011},\nand flattening of the entropy profile \\citep{sato2012}, leading\nto possible sources of systematic uncertainties in the measurement of masses.\n\n\nThe detection of hot gas at large radii is limited\nprimarily by its intrinsic low surface brightness,  uncertainties\nassociated with the subtraction of background (and foreground) emission,\nand the ability to remove contamination from\ncompact sources unrelated to the cluster.\nThanks to its low detector background, \\it Suzaku\\rm\\ reported the\nmeasurement of ICM temperatures  to $r_{200}$\\ and beyond\nfor a few nearby clusters\n\\citep[e.g.][]{akamatsu2011,walker2012a,walker2012b,simionescu2011,burns2010,kawaharada2010,\nbautz2009,george2009}; to date \\emph{Abell~1835}\\\nhas not  been the target of a \\it Suzaku\\rm\\ observation.\n\nIn this paper we report the \\it Chandra\\rm\\ detection of X-ray emission\n  in \\emph{Abell~1835}\\ beyond $r_{200}$, using three observations \nfor a total of 193~ksec exposure time, extending the analysis\nof these \\it Chandra\\rm\\ data performed by \\cite{sanders2010}.\nThe radius $r_{\\Delta}$ is defined as the radius within which\nthe average mass density is $\\Delta$ times the critical density of\nthe universe at the cluster's redshift for our choice of \ncosmological parameters. The virial radius of a cluster\nis defined as the equilibrium radius of the collapsed\nhalo, approximately equivalent to one half of its turnaround radius \n\\cite[e.g.][]{lacey1993, eke1998}.\nFor an $\\Omega_{\\Lambda}$-dominated universe,\nthe virial radius is approximately $r_{100}$ \\citep[e.g.][]{eke1998}.\n\\emph{Abell~1835}\\ is the most luminous cluster in the \n\\cite{dahle2006} sample of clusters at $z=0.15-0.3$ selected\nfrom the \\emph{Bright Cluster Survey}.\nThe combination of high luminosity and availability of\n deep \\it Chandra\\rm\\ observations with local background make \\emph{Abell~1835}\\\nand ideal candidate to study its emission to the virial radius. \n\\emph{Abell~1835}\\ has a redshift of $z=0.253$,\nwhich for $H_{0}=70.2$~km~s$^{-1}$~Mpc$^{-1}$, $\\Omega_{\\Lambda}=0.73$,\n$\\Omega_M=0.27$ cosmology \\citep{komatsu2011} corresponds\nto an angular-size distance of $D_A=816.3$~Mpc, and a scale of \n237.48 kpc per arcmin.\n\n\n\\section{Chandra and ROSAT observations of Abell~1835 and the detection\nof cluster emission beyond $r_{200}$}\n\\label{sec:Sx}\n\\subsection{Chandra observations} \n\\it Chandra\\rm\\ observed \\emph{Abell~1835}\\ three times between December 2005 and August 2006\n(observations ID 6880, 6881 and 7370), with a combined clean exposure time\nof 193~ks. The three observations had similar aimpoint towards the\ncenter of the cluster (R.A. 14h01m02s, Dec. +02d51.5m J2000) and different\nroll angles. All observations were taken with the ACIS-I detector configuration,\nwhich consists of four ACIS front-illuminated chips in a two-by-two square,\nplus a fifth identical chip that may be used to measure \nthe \\emph{in situ} soft X-ray background.\nFigure~\\ref{fig:a1835} is an image from the longest observation (ID 6880, 118ks)\nin the soft X-ray band (0.7-2 keV). In addition to a large number of \ncompact X-ray sources that were excluded from further analysis, the data\nshow a clear detection of diffuse X-ray emission associated with two\nadditional low-mass clusters identified from the \\emph{Sloan Digital Sky Survey}, \nMAXBCG J210.31728+02.75364 and WHL J140031.8+025443.\nThe cluster MAXBCG J210.31728+02.75364 is the only cluster in the vicinity of \\emph{Abell~1835}\\\nreported in the MAXBCG catalog of \\cite{koester2007}, and it has\na measured  photo-$z$\nof 0.238, while the catalog of \\cite{wen2009} reports a photo-$z$\nof 0.269 for the same source; given the uncertainties associated\nwith photometric redshifts, it is likely that the cluster is\nin physical association with \\emph{Abell~1835}\\ ($z=0.253$).\nThe \\cite{wen2009} catalog also reports another optically-identified cluster \nin the area, WHL J140031.8+025443, with a spectroscopic redshift of $z=0.2505$.\nThe association of these two groups with \\emph{Abell~1835}\\ is confirmed by redshift\ndata provided by C. Haines (personal communication), who measures\na redshift of $z=0.250$ for WHL J140031.8+025443, and $z=0.245$ for MAXBCG J210.31728+02.75364.\n\nSince the goal of this paper is to study the\ndiffuse emission associated with \\emph{Abell~1835}, we excise a region of radius 90~arcsec\naround the position of the two clusters (black circles in \nFigure~\\ref{fig:a1835}), and study their emission separately from that of \\emph{Abell~1835}\\\n(see Section~\\ref{sec:low-mass-clusters}).\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=3.1in,angle=-90]{a1835Chandra.ps}\n\\caption{Image of \\emph{Abell~1835}\\ from observation 6880, in the 0.7-2 keV band.\nThe data were smoothed with a Gaussian kernel of $\\sim$6 arcsec standard error.\nThe dashed circles correspond to radial distances of approximately $r_{500}$\\ and $r_{200}$,\nand the full black circles mark the position of the two low-mass clusters associated with\n\\emph{Abell~1835}.}\n\\label{fig:a1835}\n\\end{figure}\n\n\n\\subsection{Chandra data analysis}\nThe reduction of the \\it Chandra\\rm\\ observations follow the procedure\ndescribed in \\cite{bonamente2006} and \\cite{bonamente2011}, which consists of filtering the\nobservations for possible periods of flaring background,\nand applying the latest calibration; no significant flares\nwere present in these observations. The reduction was\nperformed in CIAO~4.2, using CALDB 4.3; in Sec.~\\ref{sec:robustness} we discuss the impact of calibration\nchanges on our results. One of the calibration issues\nthat can affect the measurement of cluster emission is the uncertainty in\nthe contamination of the optical blocking filter, which causes\na reduction in the low energy quantum efficiency of the \\it Chandra\\rm\\ detectors.\nThe spatial and time dependence of this contaminant affects primarily\nthe effective area at $\\leq$0.7 keV~\\footnote{See \\cite{marshall2004} and \n\\it Chandra\\rm\\  calibration memos at cxc.harvard.edu.}, with an estimated residual\nerror of $\\leq$ 3\\% at higher energy. We therefore limit our spatial and spectral \nanalysis to the $\\geq$0.7~keV band.\nThe superior angular resolution of the \\it Chandra\\rm\\ mirrors \\citep{weisskopf2000}\nresults in a point-spread function with a 0.5~arcsec FWHM, and therefore \nthere is negligible contribution from the bright cluster core to \nthe emission in the outer annuli, and from secondary scatter (stray light) by sources outside\nthe field of view.\n\nThe subtraction of particle and sky background is one of the\nmost crucial aspects of the analysis of low surface brightness cluster regions.\nWe use \\it Chandra\\rm\\ blank-sky background observations,\nrescaled according to the high-\nenergy flux of the cluster, to ensure a correct subtraction\nof the particle background that is dominant at $E\\geq9.5$~keV, \nwhere the Chandra detectors have no effective area.\nThe temporal and spatial variability of the soft X-ray background at $E<2$~keV\nalso requires that a peripheral region free of cluster emission\nis used to measure any local enhancement (or deficit) of soft X-ray\nemission relative to that of the blank-sky fields, and account for this\ndifference in the analysis. This method \nis accurate for the determination of the temperature profile, but may result\nin small errors in the measurement of the surface brightness profile.\nIn fact, the blank-sky background is a combination of a particle component \nthat is not vignetted, and a sky component that is vignetted. To determine the\nsurface brightness of the cluster and of the local soft X-ray background,\na more accurate procedure consists of subtracting the non-vignetted\nparticle component as measured from \\it Chandra\\rm\\ observations in which\nthe ACIS detector was stowed \\cite[e.g.,][]{hickox2007}, after rescaling the\nstowed background to match the $E\\geq9.5$~keV cluster count rate, as in \nthe case of the blank-sky background.\n\nPoint sources are identified and removed using a wavelet detection method\nthat correlates the cluster observation with wavelet functions of \ndifferent scale sizes (\\emph{wavdetect} in CIAO). Subtraction\nof point soures from the blank-sky observations were \nperformed by eye, with results that closely match those\nof the wavelet method.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=2.5in, angle=-90]{Sx_stow_07-2.ps}\n\\caption{Exposure corrected surface brightness profile of \\emph{Abell~1835}\\ in the soft X-ray band (0.7-2 keV),\nobtained by subtraction of the particle background from the ACIS stowed\nobservations. The radii $r_{500}$, $r_{200}$\\ and the virial radius ($\\sim r_{100}$) are \nestimated from the data in Section~\\ref{sec:r500} (see Table~\\ref{tab:vikh-masses}).\nThe dashed red line is the average background level in the region $\\geq$ 700 arcsec.}\n\\label{fig:Sx}\n\\end{figure}\n\n\\subsection{Measurement of the surface brightness profile with Chandra}\nThe surface brightness profile obtained using this background subtraction\nis shown in Figure~\\ref{fig:Sx}, in which the red line represents\nthe average value of the background at radii $\\geq$~700 arcsec, where\nthe surface brightness profile is consistent with a constant level.\nTo determine the outer radius at which \\it Chandra\\rm\\ has a significant detection\nof the cluster, we also include sources of systematic errors in our analysis.\nOne source of uncertainty is the error in the measurement of the background level,\nshown in Figure~\\ref{fig:Sx-closeup} as the solid red lines.\nThe error is given by the standard deviation of the weighted mean of the datapoints\nat radii greater than 700~arcsec,\n to illustrate that\neach bin in the surface brightness profile beyond this radius\nis consistent with a constant level of the background. \n\nAnother source of uncertainty is the amount by which the stowed background is to\nbe rescaled to match the cluster count rate at high energy. The stowed background\ndataset applicable to the dates of observation of \\emph{Abell~1835}\\ has an exposure\ntime of 367~ksec, and the relative error in the rescaling of the background to match\nthe cluster count rate at high energy is 0.7\\%, as determined by the Poisson\nerror in the photon counts at high energy. \nMoreover, \\cite{hickox2006}\nhas shown that the spectral distribution of the particle background is remarkably stable,\neven in the presence of changes in the overall flux, and that \nthe ratio of soft-to-hard (2-7 keV to 9.5-12 keV) count rates remains constant to within $\\leq$2~\\%. We therefore\napply a systematic error of 2~\\% in the stowed background flux, to account for this\npossible source of uncertainty, in addition to the 0.7\\% error\ndue to the uncertainty in the rescaling of the background. \n\n\\begin{figure}\n\\centering\n\\includegraphics[width=2.5in, angle=-90]{Sx_stow_07-2_closeup.ps}\n\\caption{Close-up view of Figure~\\ref{fig:Sx}, in which the red lines represent the\n1-$\\sigma$ confidence in the background level as determined from the $\\geq$700~arcsec region,\nand the green error bars combine the statistical and systematic errors in the determination\nof the surface brightness.\n}\n\\label{fig:Sx-closeup}\n\\end{figure}\n\n\nIn Figure~\\ref{fig:Sx-closeup} the green error\nbars represent the cumulative effect of the statistical error due to the counting\nstatistic,  and the\nsources of errors associated with the use of the stowed background; the systematic\nerrors were added linearly to the statistical error as a conservative measure.\nThis error analysis shows that the emission from \\emph{Abell~1835}\\ remains significantly \nabove the background beyond $r_{200}$\\ and\nuntil approximately a radius of 600 arcsec, or approximately 2.4~Mpc. \nThe significance of the detection in the region 450-600\" (the\nfive datapoints in Figure~\\ref{fig:Sx-closeup} after the $r_{200}$ marker)\nis calculated as 5.5$\\sigma$, and is obtained by using the larger systematic\nerror bars for the surface brightness profile (in green in Figure~\\ref{fig:Sx-closeup}),\nadded in quadrature to the error in the determination of the background level\nfrom the $\\geq 700$\" region (red lines in Figure~\\ref{fig:Sx-closeup}).\n\nTo further test the effect of the background subtraction, we repeat our\nbackround subtraction process using the $\\geq 600$\" region \n(instead of the $\\geq 700$\" region) . The background level\nincreases by less than 1$\\sigma$ of the value previously determined (e.g., the\ntwo levels are statistically indistinguishable), and the significance of detection in\nthe region 450-600\" is 4.7$\\sigma$. Therefore we conclude that it is unlikely that\nthe excess of emission beyond $r_{200}$\\ and out to the virial radius is due to errors in the background\nsubtraction process.\nA similar result can be obtained including the 2-7 keV band,\nbut the signal-to-noise is reduced because at large radii this band is \ndominated by the background due to the softening of the cluster emission. \nWe estimate $r_{200}$\\ and the virial radius ($\\sim r_{100}$) from\nthe \\it Chandra\\rm\\ data in Section~\\ref{sec:r500}.\n\n\\subsection{Measurement of the\nsurface brightness profile with the ROSAT Position Sensitive Proportional Counter}\n\n\\it ROSAT\\rm\\ observed \\emph{Abell~1835}\\ on July 3--4 2003 for 6~ks with the Position\nSensitive Proportional Counter (PSPC), observation ID was 800569.\nThe PSPC has a 99.9\\% rejection of particle background in the 0.2-2~keV band\n\\citep{plucinsky1993} and an average angular resolution of $\\sim$30~arcsec that makes it \nvery suitable for observations of low surface brightness objects such as\nthe outskirts of galaxy clusters \\citep[e.g.][]{bonamente2001,bonamente2002,bonamente2003}.\nWe reduce the event file following the procedure described in  \\cite{snowden1994}\nand \\cite{bonamente2002}, which consists of corrections for detector gain fluctuations, and\nremoval of periods with a \\emph{master veto} rate of $\\leq$170 counts~s$^{-1}$ in order\nto discard periods of high background. These filters result in a clean exposure time of\n5.9~ks.\n\nSince the PSPC background is given only by the photon background, we generate an\nimage in the 0.2-2 keV band and use the exposure map to correct for the position--dependent \nvariations in the detector response and mirror vignetting.\nWe masked out the two low-mass cluster regions as we did for the \\it Chandra\\rm\\ data and \nall visible point sources, and obtained and exposure-corrected surface brightness profile\nout to a radial distance of $\\sim$20 arcmin, which corresponds to the location of the\ninner support structure of the PSPC detector. The \\it ROSAT\\rm\\ surface brightness profile therefore\ncovers the entire azimuthal range.\nIn Figure~\\ref{fig:rosat} we  show the radial profile of the surface brightness in the 0.2-2 keV\nband, showing a $\\sim$2~$\\sigma$ excess of emission in the 400-600\" region using the background\nlevel calculated from the region $\\geq$700\", as done for the \\it Chandra\\rm\\ data.\nThe \\it ROSAT\\rm\\ data therefore provide additional evidence of emission beyond $r_{500}$\\ and out to the\nvirial radius, although the short \\it ROSAT\\rm\\ exposure does not have sufficient number of counts\nto provide a detection with the same significance as in the \\it Chandra\\rm\\ data.\n\n\\begin{figure}\n\\includegraphics[width=2.4in,angle=-90]{.\/Sx-rosat.ps}\n\\caption{Surface brightness profile in 0.2-2 keV band from a 6~ks\nobservation with ROSAT PSPC. The background level is determined from \nthe data at radii $\\geq$~700\", as in the \\it Chandra\\rm\\ data.}\n\\label{fig:rosat}\n\\centering\n\\end{figure}\n\n\n\\section{Analysis of the Chandra spectra}\n\\label{sec:kT}\n\\subsection{Measurement of the temperature profile of Abell~1835}\n\\label{sec:spectral-fits}\nWe measure the temperature profile of \\emph{Abell~1835}\\ following\nthe background subtraction method described in Sec.~\\ref{sec:Sx},\nwhich makes use of the blank-sky background dataset and a\nmeasurement of\nthe local enhancement of the soft X-ray background, as\nis commonly done for \\it Chandra\\rm\\ data \\citep[e.g.][]{vikhlinin2006, maughan2008, bulbul2010}.\nIn Figure~\\ref{fig:soft-back} we show the spectral distribution\nof the local soft X-ray background enhancement, as determined from\na region beyond the virial radius ($\\geq$700~arcsec);\nthis emission was modelled with an APEC emission\nmodel of $kT\\sim 0.25$~KeV and of Solar abundance, consistent with\nGalactic emission, and then subtracted from all\nspectra. The spectra were fit in the 0.7-7 keV band using the \nminimum $\\chi^2$ statistic, after binning to ensure that there are\nat least 25 counts per bin. We use XSPEC version 12.6.0s\nfor the spectral analysys.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=2.3in,angle=-90]{backSpectrum.ps}\n\\caption{Spectrum of the local enhancement of the soft X-ray\nbackground from observation 6880. The other two exposures have\nsimilar levels of soft X-ray fluxes above the blank-sky emission,\nwhich is modeled as an unabsorbed $\\sim0.25$~keV thermal plasma at $z=0$.\nThe best-fit model has a $\\chi^2_{min}=73.9$ for 78 degrees of freedom,\nfor a null hypothesis probability of 61\\%.}\n\\label{fig:soft-back}\n\\end{figure}\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2.5in,angle=-90]{330-450.ps}\n\\includegraphics[width=2.5in,angle=-90]{450-600.ps}\n\\caption{Blank-sky background subtracted spectra of\nregions 330-450\" and 450-600\" from observation 6880.\nThe solid lines are the best-fit model of the local\nsoft X-ray enhancement of Figure~\\ref{fig:soft-back} (red),\nand its 90\\% upper limit (green).}\n\\label{fig:spectra}\n\\end{figure*}\n\nIn Figure~\\ref{fig:spectra} we show the spectra of\nthe outermost two regions, to show the impact of the \nsoft X-ray residuals in the background subtraction.\nThe importance of background systematics in the detection\nof emission and measurement of cluster temperatures for\nregions of low surface brightness was\nrecently addressed by \\cite{leccardi2008} using \\it XMM-Newton\\rm\\ data.\nFor our \\it Chandra\\rm\\ observations, the two main sources of uncertainty when determining the temperature\nof the outer regions are the subtraction of the blank-sky background, and\nthe subtraction of the locally-determined soft X-ray background.\nTable~\\ref{tab:background} reports the statistics of the background relevant\nto the outer regions of the cluster, with  both regions $\\sim$~10-20\\% above\nthe blank-sky background,  determined with a precision of 1-2\\%.\nThe additional soft X-ray background accounts for a significant portion\nof the remaining signal, as shown in Figure~\\ref{fig:spectra}; the 90\\% \nupper limit to the measurement  of this background is shown as the\ngreen lines, and emission from the cluster is still detected with\nhigh statistical significance. Both sources of error are \nincluded in the temperature measurements at large radii.\n\n\n\\begin{table*}\n\\centering\n\\caption{Background levels in outer regions}\n\\label{tab:background}\n\\begin{tabular}{lccc}\n\\hline\n \t\t& \\multicolumn{3}{c}{Observation ID}\\\\\n\t \t& 6880 \t& 6881\t&7370\\\\ \n\\hline\nExposure time (ks) & 114.1 & 36.0 & 39.5 \\\\\nCorrection to Blank-sky Subtraction$^{a}$ & -0.04$\\pm$0.01 & -0.125$\\pm$0.015 & -0.04$\\pm$0.015 \\\\\n\\hline\n\\multicolumn{4}{c}{Region 330-450\"}\\\\\nTotal Counts\t& 18,124 & 4,938 & 5,686 \\\\\nCount rate (c s$^{-1}$) & 0.158$\\pm$0.001 & 0.137$\\pm$0.002 & 0.144$\\pm$0.002\\\\\nNet count rate$^{b}$ ($10^{-2}$ c s$^{-1}$) \t& $2.75\\pm0.10$ & $2.54\\pm0.20$ & $2.36\\pm0.20$ \\\\\nPercent above back. & 17.4$\\pm$0.6 & 18.2$\\pm$1.4 & 16.4$\\pm$1.4 \\\\\nSXB count rate ($10^{-3}$ c s$^{-1}$) \t&  3.34$\\pm$0.77 & 7.20$\\pm$0.94 & 7.10$\\pm$0.71 \\\\\n\\hline\n\\multicolumn{4}{c}{Region 450-600\"} \\\\\nTotal Counts    & 15,811 & 4,901 & 5,483 \\\\\nCount rate (c s$^{-1}$) & 0.139$\\pm$0.001 & 0.136$\\pm$0.002 & 0.139$\\pm$0.002 \\\\\nNet count rate$^{b}$ ($10^{-2}$ c s$^{-1}$)     &1.02$\\pm$0.10 & 1.58$\\pm$0.20 & 1.23$\\pm$0.20 \\\\\nPercent above back. & 7.3$\\pm$0.7 & 11.6$\\pm$1.5 & 8.8$\\pm$1.4 \\\\\nSXB count rate ($10^{-3}$ c s$^{-1}$)   & 3.06$\\pm$0.70 & 7.50$\\pm$0.98 & 7.32$\\pm$0.73\\\\\n\\hline\n\\end{tabular}\n\n\\flushleft\n$a$: This is the fractional correction of the blank-sky data, to match the high-energy flux\nin the cluster observation.   \\\\\n$b$ This is the background-subtracted count rate, including cluster and soft X-ray background \n(SXB) signal.\n\\end{table*}\n\nWe use the APEC code \\citep[][code version 1.3.1]{smith2001} to model the \\it Chandra\\rm\\ spectra, with a\nfixed Galactic HI column density of $N_H=2.04\\times10^{20}$~cm$^{-2}$ \\citep{kalberla2005}.\nThe region at radii $\\leq 330$\" have a variable metal abundance, while the outer\nregion have a fixed abundance of $A=0.3$. In addition to the statistical errors\nobtained from the XSPEC fits, we add a systematic error of 10\\% in the\ntemperature measured in the core and a 5\\% error to the other region,\nto account for possible systematic uncertainties due to the \\it Chandra\\rm\\\ncalibration \\citep[see, e.g.,][]{bulbul2010}. \nOne possible source of systematic uncertainty in our results is indicated \nby the systematic difference between the \\it Chandra\\rm\/ACIS and\\it XMM-Newton\\rm\/EPIC \ntemperature measurements of galaxy clusters \\citep{nevalainen2010}\nThis amounts to a $\\pm$10\\% bias in the calibration of the effective area at 0.5 keV, \nwhich decreases roughly linearly towards 0\\% bias at 2 keV. \nAssuming that \\it XMM-Newton\\rm\/pn has a more accurately calibrated effective area, \nwe reduced the \\it Chandra\\rm\\ effective area by multiplying it with a linear function \nas indicated by the \\it Chandra\\rm\/\\it XMM-Newton\\rm\\ comparison. As a result, the temperature at the \noutermost radial bin decreases by $\\sim$ 5\\%. Thus, the cross-calibration \nuncertainties between \\it Chandra\\rm\\ and \\it XMM-Newton\\rm\\ do not explain the low temperature we measure in the outermost radial bin.\nUncertainties in the Galactic column density of HI do not impact significantly our results.\nChanging the value of  $N_H$ by $\\pm$10\\%, consistent with the variations between the \\cite{kalberla2005}\nand the \\cite{dickey1990} measurements, results in a change of best-fit temperature\nin each bin by less than 2\\%.\n\n\nGiven the emphasis of this paper on the detection of\nemission at large radii, we investigate\nthe sources of uncertainty caused by the background subtraction\nin the outer region at $\\geq$330\". We report the results of this \nerror analysis in Table~\\ref{tab:kT-err}, where \\emph{cornorm} refers\nto the normalization of the blank-sky background, and \\emph{soft residuals}\nrefers to the normalization of the soft X-ray residual model, as \nreported in Table~\\ref{tab:background}. In the analysis that follows,\nwe add the systematic errors caused by these sources linearly to the\nstatistical error.\nOur data do not constrain well the metal abundance of the plasma\nin the outer regions.\nUsing an abundance of $A=0.5$ instead of the nominal $A=0.3$\nleads to negligible changes in the best-fit temperature for both of\nthe outer annuli.\nIn the extreme case of an $A=0.0$ metal abundance, \nboth regions have an acceptable fit with the best-fit temperatures\nchange respectively by $+6$\\% for the 330-450\" region ($\\Delta \\chi^2=+1.3$), and\nby $-22$\\% for the 450-600\" region ($\\Delta \\chi^2=+9.2$,\nbest fit decreases from 1.26 to 0.98 keV). \nWe therefore find that, in the case of exceptionally low metalllicity, the\ntemperature profile we measure from these \\it Chandra\\rm\\ data would be even significantly steeper\nthan indicated by the result in Table~\\ref{tab:kT-err}.\nGiven that these data do not provide direct indication that the plasma in the outer regions \nmay have null metal content, we do not fold in this source of systematic error in the\nanalysis that follows.\n\n\n\n\\begin{table}\n\\caption{Temperature measurement and error analysis from\nthe \\it Chandra\\rm\\ data.}\n\\centering\n\\label{tab:kT-err}\n\\begin{tabular}{lcc}\n\\hline\nRegion & \\multicolumn{2}{c}{Projected Temperature (keV)}\\\\\n\\hline\n       & Measurement$^{a}$  & Calibration error$^{b}$  \\\\\n0-10\"  & 4.78$\\pm$0.06 & $\\pm$0.48 \\\\\n10-20\" & 7.09$\\pm$0.14 & $\\pm$0.71 \\\\\n20-30\" & 8.72$\\pm$0.27 & $\\pm$0.87 \\\\\n30-60\" & 9.47$\\pm$0.21 & $\\pm$0.47 \\\\ \n60-90\" & 10.57$\\pm$0.33 & $\\pm$0.53 \\\\\n90-120\" & 9.97$\\pm$0.44 & $\\pm$0.50 \\\\\n120-180\"& 9.68$\\pm$0.49 & $\\pm$0.48 \\\\\n180-240\" & 7.85$\\pm$0.65 &$\\pm$0.39 \\\\\n240-330\" & 6.02$\\pm$0.65 &$\\pm$0.30 \\\\\n330-450\" & 3.75$\\pm$0.72 & $\\pm$0.19\\\\\n450-600\" & 1.26$\\pm$0.16 & $\\pm$0.06 \\\\\n\\hline\n\t\\multicolumn{3}{c}{Measurement of Temperature Using} \\\\\n\t \\multicolumn{3}{c}{Background Systematic Errors (keV)}\\\\\n         & $+1\\sigma$ cornorm$^{c}$ & $-1\\sigma$ cornorm \\\\\n330-450\" & 3.02$\\pm$0.54 & 4.67$\\pm$1.00 \\\\\n450-600\" & 1.09$\\pm$0.10 & 1.31$\\pm$0.18 \\\\\n\t & $+1\\sigma$ soft res.$^{d}$ & $-1\\sigma$ soft res.\\\\\n450-600\" & 4.53$\\pm$1.03 & 3.05$\\pm$0.54\\\\\n450-600\" & 1.37$\\pm$0.25 & 1.16$\\pm$0.12\\\\\n \\multicolumn{3}{c}{Summary of Background Systematic Errors$^{e}$} \\\\\n330-450\" & \\multicolumn{2}{c}{$\\pm 0.83\\pm 0.74$ keV} \\\\ \n450-600\" & \\multicolumn{2}{c}{$\\pm 0.11\\pm 0.10$ keV} \\\\\n\\hline\n\\end{tabular}\n\n\\flushleft\n$a$: Uncertainty is 1$\\sigma$ statistical error from counting statistics only.\\\\\n$b$: Includes \\it XMM-Newton\\rm\/\\it Chandra\\rm\\ cross-calibration uncertainty of the effective area \\citep{nevalainen2010}.\\\\\n$c$: This is temperature obtained by varying by $\\pm 1\\sigma$ the fractional\ncorrection of the blank-sky data, to match the high-energy flux in the cluster observations.\\\\\n$d$: This is the temperature obtained by varying by $\\pm 1\\sigma$ the normalization\nof the best-fit model to the soft X-ray background residuals.\\\\\n$e$: Obtained from the average deviation of the $\\pm 1\\sigma $ `cornorm' and 'soft. res'\nmeasurements from the measurement with nominal values of these parameters.\n\\end{table}\n\n\\cite{sanders2010} measured temperature profiles for \\emph{Abell~1835}\\ out\nto approximately $r_{500}$\\ with \\it Chandra\\rm\\ and \\it XMM-Newton\\rm. Using the same \\it Chandra\\rm\\ \nobservations we analyze in this paper, their temperature profile\nhas a similar drop from the peak value to their outermost\nannulus ($322\\pm42$\"), where they measure a temperature of kT=$4.67\\pm^{0.82}_{0.52}$~keV\nthat is consistent with our measurements. Likewise, from the \\it XMM-Newton\\rm\\ data their outermost\nradial bin ($300\\pm10$\") has a temperature of kT=$5.2\\pm^{1.2}_{0.7}$~keV, also\nin agreement with our results. \nThe only measurement of the \\emph{Abell~1835}\\ temperature to the virial radius\navailable in the literature is that of \\cite{snowden2008}, who\ndoes report a temperature profile out to a distance of 12~arcmin\nfrom a long \\it XMM-Newton\\rm\\ observation (and out to 7' from a shorter observation).\nIn particular,\nthey report a temperature of $kT=3.14\\pm0.93$ for the region\n420-540\", which straddles our measurements at 330-450\" ($3.75\\pm0.72$~keV)\nand at 450-600\" ($1.26\\pm0.16$, statistical errors only). The same paper\nalso reports a measurement of $kT=3.33\\pm1.75$~keV for the region 540-720\",\ni.e., beyond our outer annulus. Their temperature is somewhat higher that\nours, although the large error bars cannot exclude that the \\it Chandra\\rm\\\nand \\it XMM-Newton\\rm\\ measurements\nare consistent. Therefore our results confirm and extend the\nearlier \\it XMM-Newton\\rm\\ analysis of \\cite{snowden2008}.\n\n\\subsection{Measurement of the average temperature of \nMAXBCG J210.31728+02.75364\nand WHL J140031.8+025443}\n\\label{sec:low-mass-clusters}\nWe also measure the temperature of the two SDSS clusters detected\nin our \\it Chandra\\rm\\ images, \nMAXBCG J210.31728+02.75364 \nand WHL J140031.8+025443. The two clusters  are located between a distance of $\\sim$380-650\"\nfrom the cluster center, and therefore we start by extracting a spectrum for this annulus\nexcluding two regions of 1.5' radius centered at the two clusters.\nThis radius was determined by visual inspection, after smoothing of the \\it Chandra\\rm\\ image with a\nGaussian kernel of $\\sigma=6$ arcsec. \nFor this annulus, we measure a temperature of $kT=1.85\\pm0.36$~keV for a fixed\nabundance of $A=0.3$ Solar. We then use this spectrum as the local background for the\ntwo cluster regions, and measure a temperature $kT=2.73\\pm^{0.93}_{0.54}$~keV i\nfor MAXBCG J210.31728+02.75364\n(357 source photons, 19\\% above the average emission of the annulus),\nand $kT=2.09\\pm^{4.6}_{0.55}$~keV for WHL J140031.8+025443 (538 photons, 27\\% above background). \nFor both clusters, we assumed the same Galactic $HI$ column density as for \\emph{Abell~1835}, and\na fixed metal abundance of $A=0.3$ Solar. For both clusters we also extract spectra in\nregions larger than 1.5', and determine that no additional source photons are present\nfrom these two clusters beyond this radius.\n\n\n\\subsection{Tests of robustness of the temperature measurement at large radii}\n\\label{sec:robustness}\nTo further test the measurement of temperatures especially at large radii,\nwhere\nthe background subtraction is especially important,\nwe also measure the temperature profile using the same stowed\nbackground data that was used for the surface brightness measurement\nof Figures~\\ref{fig:Sx} and \\ref{fig:Sx-closeup}. As in the case\nof the blank-sky background, we first rescale the stowed data to match\nthe high-energy count rate of the cluster observation, and\nuse a region at large radii ($\\geq$ 700 arcsec) to measure\nthe local X-ray background. We model the background using\nan APEC plus a power-law model, the latter component necessary\nto model the harder emission due to unresolved AGNs that is typically removed when\nthe blank-sky background is used instead, and apply this\nmodel to all cluster regions. We find that the temperature profile\nis consistent within the 1~$\\sigma$ statistical errors \nof the values provided in Table~\\ref{tab:kT-err} for each region,\nand therefore conclude that the temperature drop at large radii, and\nespecially in the outermost region, is not sensitive\nto the background subtraction method.\n\nThe temperature measurement is also dependent on an accurate subtraction\nof background (and foreground) sources of emission. \nPoint sources in the field of view are detetected using the CIAO\ntool  \\emph{wavdetect},  which correlates the image with wavelets\nat small angular scales (2 and 4 pixels, one pixel is 1.96\"), searches the results for 3-$\\sigma$ correlations, and\nreturns a list of elliptical regions to be excluded from the analysis. \nWe study in particular the\neffect of background sources on the measurement of the temperature in the outermost\nannulus (450-600\"). In this region,  \\emph{wavdetect} finds\n24 point sources, plus portions of the two low-mass galaxy clusters\ndescribed in Section~\\ref{sec:low-mass-clusters}.\nWe extract a spectrum for this region from the longest observation (ID 6880),\nand now include in the spectrum all\npoint sources excluded in the previous analysis.\nWe find a count rate of $3.20\\pm0.13\\times 10^{-2}$ counts~s$^{-1}$, compared\nto the point source-subtracted rate of $1.02\\pm0.10\\times 10^{-2}$ counts~s$^{-1}$,\ncorresponding to an increase in background-subtracted flux by a factor of three.\nWe then fit the spectrum with the same APEC model as described in Section~\\ref{sec:spectral-fits},\nand find a best-fit temperature of $kT=1.96\\pm0.17$~keV for a best-fit goodness statistic\nof $\\chi^2=537$ for 429 degrees of freedom (or $\\chi^2_{red}$=1.25), compared\nto the temperature of $1.22\\pm0.19$~keV for a $\\chi^2=415$ for 389 degrees of freedom (or $\\chi^2_{red}$=1.08).\nWe therefore conclude that an accurate subtraction of point sources and unrelated\nsources of diffuse emission is crucial to obtain an accurate measurement of the\ntemperature profile, especially in regions of low-surface brightness such as those\nnear the virial radius.\n\nChanges in the instrument calibration affect the measurement of temperatures.\nWe therefore repeat the same data reduction and spectral analysis  using the latest \nsoftware and calibration database available at time of writing (CIAO 4.4 and CALDB 4.5.1) for\nthe longest observation (ID 6880),\nand obtain a new temperature profile for the same regions as reported in Table~\\ref{tab:kT-err}.\nIn the outermost two regions, we measure a temperature of 3.04$\\pm0.69$~keV (330-450\") and \n1.23$\\pm$0.21~keV (450-600\"), well within the 1-$\\sigma$ confidence intervals of the measurements\nusing the older calibration (3.40$\\pm$0.76 and 1.22$\\pm$0.19 respectively, also in agreement with the\nvalues of Table ~\\ref{tab:kT-err} obtained from the combination of all exposures). \nThe temperature of the inner regions are also always within 1-$\\sigma$\nof the results obtained with the earlier calibration, and we therefore conclude that changes in the instrument\ncalibration do not affect significantly our results.\n\n\n\n\\section{Measurement of masses and gas mass fraction}\n\nWe fit the surface brightness and the temperature profiles with\nthe \\cite{vikhlinin2006} model. The electron density is modelled\nwith a double-$\\beta$ profile modified by a cuspy core component and an exponential cutoff at large radii,\nfor a total of eleven model parameters;\n the temperature has both a cool-core component to follow\nthe cooler gas in the core, and a decreasing profile at large radii, for an additional\nnine parameters. For our analysis, we follow  \\cite{vikhlinin2006} and fix the\n$\\gamma$=3.0 parameter, and do not use the cuspy-core component ($\\alpha=0$)\nor the second $\\beta$-model component, so that the density is modelled by just \none $\\beta$-model with an exponential cutoff, for just four free parameters\n(core radius $r_c$, exponent $\\beta$, scale radius $r_s$ and exponential cutoff\nexponent $\\epsilon$, see Table~\\ref{tab:vikh-fit}). \nFor the temperature profile, we fix the parameter $a=0$, and the\nremaining eight parameters are reported in Table~\\ref{tab:vikh-fit}.\n\n\n\nWe use a Monte Carlo Markov chain (MCMC) method that we used in\nprevious papers \\citep[e.g.,][]{bonamente2004,bonamente2006}.\nThe MCMC analysis consists of a projection of the three-dimensional\nmodels and a comparison \nof the projected surface brightness and temperature profiles,\nand results in simultaneous estimation\nof the posterior distributions of all model paramters. Uncertainties in the  parameters are obtained\nfrom the posterior distributions, with 1-$\\sigma$ errors  assigned using the 68.3\\%\nconfidence interval around the median of the distribution. \n\nThe gas mass is directly calculated from the electron density\nmodel parameters via\n\\begin{equation}\nM_{gas}(r) = m_p \\mu_e \\int_0^r n_e(r) 4 \\pi r^2 dr\n\\end{equation}\nand the total gravitational mass via the equation of\nhydrostatic equilibrium,\n\\begin{equation}\nM(r) = - \\frac{kT(r) r}{\\mu_e m_p G} \\left(\\frac{d \\ln n_e}{d \\ln r} + \\frac{d \\ln kT}{d \\ln r} \\right),\n\\label{eq:hse}\n\\end{equation}\nwhere $m_p$ is the proton mass, $\\mu_e \\simeq 1.17$ the mean electron molecular weight, and \n$G$ the gravitational constant. The total density of matter is simply obtained via\n\\begin{equation*}\n\\rho(r) = \\frac{1}{4 \\pi r^2} \\frac{d M(r)}{dr}\n\\end{equation*}\nand therefore can be obtained via a derivative of the mass profile.\nIn Equation~\\ref{eq:hse}, the term $A = d \\ln n_e\/d \\ln r + d \\ln kT\/d \\ln r$\nand its first derivative are always negative, as is $d kT(r)\/d r$ at large radii.\nTherefore, the density can be rewritten as\n\\begin{equation}\n\\rho(r) = - \\frac{1}{4 \\pi r^2 \\mu_e m_p G} \\left[ kT \\left( A +r \\frac{dA}{dr} \\right)  + r A \\frac{d kT}{dr} \\right] \n\\label{eq:density}\n\\end{equation}\nin which the only negative term  is the one containing $A \\cdot d kT(r)\/d r$, while\nthe other two terms remain positive out to large radii.\n\n\\subsection{Modelling of the Chandra data out to the virial radius}\n\\label{sec:hse}\nThe \\cite{vikhlinin2006} model provides a satisfactory fit\nout to the outermost radius of 600\";\nFigure~\\ref{fig:kt-0-600-fit} shows the best-fit models to the temperature \nand surface brightness profiles,\nbest-fit parameters\nof the model are reported in Table~\\ref{tab:vikh-fit}.\nThe temperature profile measured by \\it Chandra\\rm\\ in Figure~\\ref{fig:kt-0-600-fit}\nis so steep that it causes the total matter density $\\rho(r)$ to become \\emph{negative}\nat approximately 400\", indicating that the temperature profile \ncannot originate from gas in hydrostatic equilibrium. \nThe situation is illustrated in Figure~\\ref{fig:mass-0-600}, where the relevant terms\nof Equation~\\ref{eq:density} are plotted individually; \n the density inferred from hydrostatic equilibrium becomes negative \nwhere the\nnegative term crosses the positive ones,\n and the mass profile has a negative slope beyond that point. These fit parameters\ntherefore lead to an unacceptable situation, and responsibility for this inconsistency\ncan be attributed to an overly steep temperature profile, with a drop by a factor of\nten between approximately 1.5' to 10'. \n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2.25in, angle=-90]{vikh_temp_0-600chain_90CI.ps}\n\\includegraphics[width=2.25in, angle=-90]{SB_vikh_0-600chain_0-600_600-1100.ps}\n\\caption{Left: Best-fit Vikhlinin model for the projected temperature\nprofile out to 600\", with 90\\% confidence intervals. Right:  Best-fit\nVikhlinin model to the 0.7-2 keV surface brightness (model+background) profile.\nEmission beyond 600\" is statistically\nconsistent with the background, in  blue is the extrapolation out to 1100\". \nPrior removal of the stowed background\ncaused the lower backgroud  level in Figure~\\ref{fig:Sx}.\n}\n\\label{fig:kt-0-600-fit}\n\\end{figure*}\n\n\n\n\\begin{table*}\n\\centering\n\\caption{Best-fit parameters for the Vikhlinin model using \\it Chandra\\rm\\ data out to 330\"}\n\\label{tab:vikh-fit}\n\\begin{tabular}{cccccccccc}\n\\hline\n$n_{e0}$ & $r_{c}$ & $\\beta$ & $r_{s}$ & $\\epsilon$ & $n_{e02}$ & $\\gamma$ & $\\alpha$ & $\\chi^{2}_{tot} \\textrm{(d.o.f.)}$\\\\\n(10$^{-2} $cm$^{-3}$) & (arcsec) & & (arcsec) & & & & & &\\\\\n\\hline\n\\multicolumn{10}{c}{Using \\it Chandra\\rm\\ data out to 330\"}\\\\\n$9.602\\pm^{0.488}_{0.415}$ & $6.743\\pm^{0.373}_{0.403}$ & $0.498\\pm^{0.009}_{0.009}$ & $119.8\\pm^{13.3}_{13.4}$ & $1.226\\pm^{0.098}_{0.097}$ \n& 0.0 & 3.0 & 0.0 & \\nodata\\\\ \n\\hline\n\\multicolumn{10}{c}{Using \\it Chandra\\rm\\ data out to 600\"}\\\\\n$9.763\\pm^{0.447}_{0.450}$ & $6.346\\pm^{0.385}_{0.343}$ & $0.488\\pm^{0.009}_{0.009}$ & $96.44\\pm^{9.55}_{8.67}$ & $1.067\\pm^{0.075}_{0.079}$\n& 0.0 & 3.0 & 0.0 & \\nodata\\\\\n\\hline\n\\hline\n$T_{0}$ & $T_{min}$ & $r_{cool}$ & $a_{cool}$ & $r_{t}$ & $a_{t}$ & $b_{t}$ & $c_{t}$ &\\\\\n(keV) & (keV) & (arcsec) & & (arcsec) & & & & &\\\\\n\\hline\n\\multicolumn{10}{c}{Using \\it Chandra\\rm\\ data out to 330\"}\\\\\n$38.25\\pm^{19.63}_{17.23}$ & 3.0 & $92.48\\pm^{52.63}_{40.52}$ & 1.0 & $257.5\\pm^{143.0}_{66.72}$ & 0.0 & $1.024\\pm^{0.426}_{0.283}$ & 2.0 &  39.0 (83)\\\\\n\\hline\n\\multicolumn{10}{c}{Using \\it Chandra\\rm\\ data out to 600\"}\\\\\n$10.17\\pm^{0.85}_{0.60}$ & 3.0 & $11.82\\pm^{3.61}_{2.29}$ & $1.924\\pm^{0.802}_{0.568}$ & 600.0 & 0.0 & $2.800\\pm^{0.224}_{0.210}$ & 10.0 & 106.4 (154)\\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\n\\begin{table*}\n\\centering\n\\caption{Masses Calculated using \\it Chandra\\rm\\ data out to 330\", and Extrapolated out to $r_{100}$}\n\\label{tab:vikh-masses}\n\\begin{tabular}{ccccc}\n\\hline\n$\\Delta$ & $r_{\\Delta}$ & $M_{gas}$ & $M_{total}$ & $f_{gas}$\\\\\n         & (arcsec)     & $\\times 10^{13}~M_{\\odot}$ & $\\times 10^{14}~M_{\\odot}$ & \\\\\n\\hline\n2500 & $164.9\\pm^{4.1}_{3.9}$ & $4.70\\pm^{0.15}_{0.14}$ & $5.03\\pm^{0.38}_{0.35}$ & $0.093\\pm^{0.004}_{0.004}$\\\\\n500  & $326.6\\pm^{7.1}_{6.9}$ & $10.75\\pm^{0.23}_{0.23}$ & $7.80\\pm^{0.52}_{0.49}$ & $0.138\\pm^{0.006}_{0.006}$\\\\\n200  & $453.3\\pm^{15.2}_{15.1}$ & $15.36\\pm^{0.48}_{0.48}$ & $8.35\\pm^{0.86}_{0.81}$ & $0.184\\pm^{0.014}_{0.012}$\\\\\n100  & $570.9\\pm^{26.6}_{25.3}$ & $19.53\\pm^{0.84}_{0.82}$ & $8.34\\pm^{1.22}_{1.06}$ & $0.234\\pm^{0.024}_{0.022}$\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{table*}\n\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2.3in, angle=-90]{mass-0-600.ps}\n\\includegraphics[width=2.3in, angle=-90]{terms-0-600.ps}\n\\caption{Mass profile using data out to 600\" and the temperature fit of Figure~\\ref{fig:kt-0-600-fit},\nand the radial distribution of the positive and negative terms in the density\nequation (Equation~\\ref{eq:density}).}\n\\label{fig:mass-0-600}\n\\end{figure*}\n\n\nThe results presented in this section provide\nevidence  that the gas detected by \\it Chandra\\rm\\ near the virial radius is\n\\emph{not} in hydrostatic equilibrium, and a number of theoretical\nstudies do in fact suggest that  beyond $r_{500}$\\ the intergalactic\nplasma is not supported solely by thermal pressure \\citep[e.g.][]{lau2009}.\n\\it Suzaku\\rm\\ has reported the measurement of emission near the\nvirial radius for several clusters, including \\emph{Abell~1413},\n\\emph{Hydra~A}, \\emph{Perseus}, \\emph{PKS0745-191}, \\emph{Abell~1795}, \\emph{Abell~1689}\nand \\emph{Abell~2029}\n\\citep{hoshino2010,sato2012,simionescu2011,george2009, bautz2009,kawaharada2010,\nwalker2012a,walker2012b}. Some of these results do in fact report an apparent\ndecrease in total mass with radius \\citep[e.g.]{george2009,kawaharada2010} \nand lack of hydrostatic equilibrium at large radii \\citep[e.g.][]{bautz2009},  similar\nto the results presented in this paper.\nTemperature profiles measured by \\it Suzaku\\rm\\ typically do not feature as extreme a temperature drop\nas the one reported in Figure~\\ref{fig:kt-0-600-fit}, i.e., a factor of nearly 10\nfrom peak to outer radius, although in some cases the drop of temperature from the peak\nvalue to that at $r_{200}$\\ is consistent with the one reported in this paper.\n\n\n\\subsection{Modelling of the Chandra data out to $r_{500}$}\n\\label{sec:r500}\nThe steepening of the radial profile beyond 400\" is driven by the\ntemperature of the last datapoint beyond $r_{200}$. \nWe also model the surface brightness and temperature profiles of the\n\\it Chandra\\rm\\ data out to only 330\", or approximately $r_{500}$, and find the\nbest-fit \\cite{vikhlinin2006} model for the temperature profile\nreported  in Figure~\\ref{fig:kT-0-330} and Table~\\ref{tab:vikh-fit}.\nWe measure a gas mass fraction of $f_{gas}(r_{500})=0.138\\pm0.006$;\nif we add the mean stellar fraction as measured by either \\cite{giodini2009}\n($f_{\\star}=0.019\\pm0.002$) or by \\cite{gonzales2007} ($f_{\\star}\\simeq0.012$)\nassuming $M(r_{500})=7.1\\times 10^{14}$ $M_{\\odot}$,\nwe find that \\emph{Abell~1835}\\ has an average baryon content within  $r_{500}$\\\nthat is consistent with the cosmic abundance of $\\Omega_b\/\\Omega_M=0.167\\pm0.007$ \\citep{komatsu2011}\nat the 2-$\\sigma$ level. As is the case in most clusters, especially relaxed\nones, the radial distribution of the gas mass fraction increases with radius\n\\citep[e.g.,][]{vikhlinin2006}.\n\nWe use this modelling of the data to measure $r_{500}$, and to provide estimates\nfor $r_{200}$\\ and the virial radius.\nThe extrapolation of this model to 600\" now falls above the measured temperature profile,\nand the mass profile using hydrostatic equilibrium is monotonic.\nThis best-fit model is marginally compatible with the assumption of hydrostatic equilibrium.\nIn fact, Table~\\ref{tab:vikh-masses} shows that the extrapolated  mass profile\nflattens around $r_{200}$, with virtually no additional mass being necessary\nbeyond this radius to sustain the hot gas in hydrostatic equilibrium.\nMoreover, between $r_{500}$\\ and $r_{200}$, all of the gravitational mass is accounted\nby the hot gas mass, i.e., \\emph{no} dark matter is required beyond $r_{500}$.\nThis extrapolation of the $\\leq$~$r_{500}$\\ data to the virial radius therefore\nleads to a dark matter halo that is much more concentrated than\nthe hot gas.\n\n\n\\section{Entropy profile and convective instability at\nlarge radii}\n\\label{sec:entropy}\nThe Schwarzschild\ncriterion for the onset of convective instability is given by the\ncondition of buoyancy of an infinitesimal blob of gas that is displaced by an\namount $dr$, $d \\rho_{blob} < d \\rho$,\nwhere $\\rho_{blob}$ is the density of the displaced blob, assumed to attain pressure\nequilibrium with the surrounding, and $\\rho$ is the density of ambient medium.\nIf the blob is displaced adiabatically, using pressure $P$ and entropy $s$ as the\nindependent thermodynamic variables in the derivatives of $\\rho_{blob}$ and $\\rho$, \nthe buoyancy condition gives \n\\begin{equation}\n\\left. \\frac{\\partial \\rho}{\\partial s} \\right|_{p} ds > 0\n\\label{eq:buoyancy}\n\\end{equation}\nas condition for convective instability, i.e., a blob that is displaced radially outward will\nfind itself in a medium of higher density and continue to rise to larger radii. Since\n$({\\partial \\rho}\/{\\partial s})_P=-\\rho^2 (\\partial T \/ \\partial P)_s<0$ (material\nis heated upon adiabatic compression),  Equation~\\ref{eq:buoyancy} simply\nreads that \\emph{a radially decreasing entropy profile is convective unstable}.\n\nAn ideal gas has an entropy of\n\\begin{equation}\nS = \\nu R \\left( \\frac{3}{2} \\ln T - \\ln \\rho + C\\right)\n\\end{equation}\nwhere $\\nu$ is the number of moles, $R$ is the gas constant, and $C$ is a constant.\nIn astrophysical applications, it is customary \\citep[e.g.][]{cavagnolo2009} to use a definition\nof entropy that is related to the thermodynamic entropy by an operation of\nexponential and a constant offset,\n\\begin{equation}\nS = \\frac{kT}{n_e^{2\/3}},\n\\label{eq:entropy}\n\\end{equation}\nThe entropy $S$ defined by Equation~\\ref{eq:entropy} has\nunits\nof keV cm$^{2}$, and it is required to be radially increasing to maintain convective equilibrium.\nNumerical simulations\nindicate that entropy outside the core is predicted to increase with radius approximately \nas $r^{1.1}$ or $r^{1.2}$  \\citep{voit2005,tozzi2001}.\nIn Figure~\\ref{fig:entropy-profile} we show the radial profile of the entropy out to\nthe outer radius of 10 arcmin, with a significant decrease at large radii that indicates\nan incompatibility of the best-fit model with convective equilibrium. For comparison,\nwe also show the entropy profile measured using the modelling of the data\nout to only $r_{500}$, as described in Sec.~\\ref{sec:r500}. This entropy profile\nuses the shallower temperature profile of Figure~\\ref{fig:kT-0-330}, and its\nextrapolation to larger radii remains non-decreasing, i.e., marginally consistent\nwith convective equilibrium.\n\nThe Schwarzschild criterion \ndoes not apply in the presence of a magnetic field. For typical\nvalues of the thermodynamic quantities of the ICM, the electron and ion gyroradii are\nseveral orders of magnitude smaller than the mean free path for Coulomb collisions\n\\citep[e.g.][]{sarazin1988}, even for a magnetic field of order 1 $\\mu G$, and therefore\ndiffusion takes place primarily along field lines \\citep[e.g.][]{chandran2007}.\nThere is strong evidence of magnetic\nfields in the central regions of clusters \\citep[e.g., radio halos, ][]{venturi2008,cassano2006},\nthough it is not clear whether magnetic fields are ubiquitous\nnear the virial radius, as in the case of Abell~3376 \\citep{bagchi2006}.\nIn the presence of magnetic fields, \\cite{chandran2007} has shown that the\ncondition for convective instability is simply $dT\/dR<0$.\n\n  \nThe \\it Chandra\\rm\\ data out to the virial radius therefore indicate\nthat the ICM is convectively unstable, regardless of the\npresence of a magnetic field. In fact, in the absence of magnetic\nfields near the virial radius, Figure~\\ref{fig:entropy-profile} shows that \\emph{Abell~1835}\\\nfails the standard Schwarzschild criterion, i.e., the entropy decreases with radius;\nin the presence of magnetic fields, the negative gradient in the temperature profile alone\nis sufficient for the onset of convective instability \n\\citep[e.g., as discussed by ][]{chandran2007}.\n  Convective instabilities would carry hotter\ngas from the inner regions towards the outer region within a few sound crossing\ntimes. As shown by \\cite{sarazin1988}, the sound crossing time for a 10~keV\ngas is $\\sim 0.7$~Gyr for a 1~Mpc distance,\nand an unstable temperature gradient such as that of Figure~\\ref{fig:kt-0-600-fit}\nwould be flattened by convection within a few Gyrs.\nConvection could in principle also result in an additional pressure gradient\ndue to the flow of hot plasma to large radii, which can in turn help support the gas\nagainst gravitational forces. \n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2.3in, angle=-90]{vikh_temp_0-330chain_extrapolated_to_600_90CI.ps}\n\\includegraphics[width=2.3in, angle=-90]{fgas_profile.ps}\n\\caption{\nTemperature and gas mass fraction profiles measured from a fit to the \\it Chandra\\rm\\ data out to 330\", and extrapolation of the\nbest-fit model out to 600\".}\n\\label{fig:kT-0-330}\n\\end{figure*}\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=2.3in, angle=-90]{entropy_0-600.ps}\n\\includegraphics[width=2.3in, angle=-90]{entropy_0-330.ps}\n\\caption{\nDeprojected entropy profiles using the full \\it Chandra\\rm\\ data out to 600\" (left, see Section~\\ref{sec:hse}),\nand using only data out to $r_{500}$\\ (right, see Section~\\ref{sec:r500}).}\n\\label{fig:entropy-profile}\n\\end{figure*}\n\n\n\\section{Discussion and interpretation}\nIn this paper we have reported the detection of X-ray emission in \\emph{Abell~1835}\\ with \\it Chandra\\rm\\ that extends out to approximately\nthe cluster's virial radius. The emission can be explained by the presence of a cooler\nphase of the plasma that is dominant at large radii, possibly linked to the infall\nof gas from large-scale filamentary structures. We also investigate the effects of clumping of the gas \nat large radii, and conclude that in principle a radial gradient in the clumping factor of the hot ICM\ncan explain the apparent flattening of the entropy profile and the turn-over of the mass profile.\n\\subsection{Detection of X-ray emission out to the virial radius}\nThe detection of X-ray emission out to a radial distance of 10 arcmin, or approximately 2.4~Mpc,\nindicates the presence of diffuse gas out to the cluster's virial radius.\nThis is the first detection of gas out to the virial radius with \\it Chandra\\rm, matching\nother detections obtained with \\it Suzaku\\rm\\ for nearby clusters \n\\citep[e.g.][]{akamatsu2011,walker2012a,walker2012b,simionescu2011,burns2010,kawaharada2010,\nbautz2009,george2009}.\nDespite its higher background, \\it Chandra\\rm\\ provides a superior angular resolution to image and remove emission from unrelated sources.\nAs can be seen from Figure~\\ref{fig:a1835}, there are approximately 100 point-like sources that were automatically\ndetected and removed, and we were also able to identify two low-mass clusters that are likely associated with \\emph{Abell~1835}.\n\\it Chandra\\rm\\ therefore has the ability to constrain the emission of clusters to the virial radius, especially for higher-redshift\ncool-core clusters for which the \\it Suzaku\\rm\\ point-spread function would cause significant contamination from the \ncentral signal to large radii.\n\n\nIt is not easy to interpret the emission at the outskirts as an extension \nof the hot gas detected at radii $\\leq$~$r_{500}$. In fact, as shown in Section~\\ref{sec:hse}, the steepening of the\ntemperature profile is incompatible with the assumption of\nhydrostatic equilibrium at large radii. \nWe also showed in Section~\\ref{sec:entropy} that\nthe gas has a negative entropy gradient beyond this radius, rendering it convectively unstable. \nTherefore, if the temperature profile of Figure~\\ref{fig:kt-0-600-fit} originates from\na single phase of the ICM, convection would transport hotter gas towards the outskirts, flattening\nthe temperature profile within a few Gyrs. Cooling of the gas by thermal radiation cannot be\nresponsible for off-setting the heating by convection, since the cooling time\n ($t_{cool} \\sim kT^{1\/2} n_e^{-1}$) is longer at the outskirts than in the peak-temperature regions\ndue to the higher density.\n\n\\subsection{Warm-hot gas towards the cluster outskirts}\nA possible interpretation for the detection of emission near the virial radius and its\nsteep temperature profile is the presence of a separate phase at the cluster outskirts\nthat is not in hydrostatic equilibrum with the cluster's potential.\nIn this case, cooler gas may be the result of infall from filamentary structures that\nfeed gas into the cluster, and the temperature of this \\emph{warm-hot} gas may in fact be\nlower than that shown in Figure~\\ref{fig:kt-0-600-fit} \n(i.e., $kT \\sim 1.25$~keV for the region $\\geq 450$\") if\nthis gas lies in projection against the tail end of the hotter ICM.\n\nWe estimate the mass of this putative warm-hot gas assuming that all of the\nemission from the outermost region is from a uniform density gas\nseen in projection. This assumption may result in an overestimate\nof the emission measure; in fact,  the extrapolation of the gas density profile \nin the hydrostatic or convective scenarios may yield a significant amount of\nemission in the last radial bin. \nWe were unable to perform a self-consistent modelling\nof the emission  in the full radial range, since the low signal-to-noise\nratio does not allow a two-phase modelling in the last radial bin.\nIn this simple uniform density warm-hot gas scenario, \nthe gas is in a filamentary structure\nof length $L$ and area $A=\\pi(R_{out}^2-R_{in}^2)$, where\n$R_{out}=600$\" and $R_{in}=450\"$; this is the same model\nalso considered in \\cite{bonamente2005} for the cluster \\emph{Abell~S1101}.\nSince the length $L$ of the filament along the sightline is unknown,\nwe must either assume $L$ or the electron density $n_e$, and \nestimate the mass implied by the detected emission.\nThe emission integral for this region is proportional to\n\\begin{equation}\nK = \\frac{10^{-14}}{4 \\pi D_A^2 (1+z)^2} n_e^2 V,\n\\end{equation}\nwhere $K$ is measured in XSPEC from a fit to the spectrum, $D_A$ is\nthe angular distance in cm, $z$ is the cluster redshift, and \nthe volume is $V=A \\times L$. For this estimate we assume \nfor simplicity that\nthe mean atomic weights of hydrogen and of the electrons are\nthe same, $\\mu_e=\\mu_H$.\nUsing the best-fit spectral model with $kT=1.26\\pm0.16$ keV,\nwe measure $K=1.05\\pm 0.13 \\times 10^{-4}$. If we assume a filament of\nlength $L=10$~Mpc, then the average density is $n_e=2.4\\pm0.3$~cm$^{-3}$,\nand the filament mass is $4.6\\pm0.6 \\times 10^{13}$~$M_{\\odot}$.\nAlternatively, a more diffuse filament gas of $n_e=10^{-5}$~cm$^{-3}$\nwould require a filament of length $L=58\\pm8$~Mpc, with\na mass of $1.1\\pm0.2\\times 10^{14}$~$M_{\\odot}$, comparable to the\nentire hot gas mass within $r_{200}$. The fact that a lower density gas\nyields a higher mass is given by the fact that, for a measured value of $K$\nwe obtain $n_e \\propto L^{-1\/2}$, and therefore the mass is proportional to $L^{1\/2}$.\nFor comparison, the gas mass for this shell inferred from the standard\nanalysis, i.e.,  assuming that the gas is in the shell itself,\nis $\\sim 3\\times 10^{13}$~$M_{\\odot}$, as can be also seen from Table~\\ref{tab:vikh-masses}.\n\nIf the gas is cooler, then the mass budget would increase further.\nIn fact, the bulk of the emission from cooler gas falls outside of the \\it Chandra\\rm\\ bandpass,\nand for a fixed number of detected counts the required emission integral increases.\nWe illustrate this situation by fitting the annulus to an emission\nmodel with a fixed value of $KT=0.5$~keV, which result in a value\nof $K=1.88\\pm 0.24 \\times 10^{-4}$ (the fit is significantly poorer, with\n$\\Delta \\chi^2=+10$ for one fewer degree of freedom). \nAccordingly, the filament mass estimates would be increased \napproximately by a factor of two. \n\n\nA warm-hot phase at $T\\leq 10^7$~K is expected to be a significant reservoir of baryons\nin the universe \\citep[e.g.][]{cen1999,dave2001}. Using the \\it ROSAT\\rm\\ soft X-ray \nPosition Sensitive Proportional Counter (PSPC)  detector --  better suited to\ndetect the emission from sub-keV plasma --  we\nhave already reported \nthe detection of a large-scale halo of emission around the \\emph{Coma} cluster out to $\\sim$~5 Mpc, well beyond the\ncluster's virial radius \\citep{bonamente2003,bonamente2009}. \nIt is possible to speculate that the high mass of \\emph{Abell~1835}, one\nof the most luminous and massive clusters on the \\emph{Bright Cluster Survey} sample \\citep{ebeling1998},\nis responsible for the heating of the infalling gas to temperatures that makes it\ndetectable by \\it Chandra\\rm, and that other massive clusters may therefore provide\nevidence of emission to the virial radius with the \\it Chandra\\rm\\ ACIS detectors.  \nThe infall scenario is supported by the \\emph{Herschel} observations\nof \\cite{pereira2010}, who measure a galaxy velocity distribution for \\emph{Abell~1835}\\\nthat does not appear to decline at large radii as in most of the other clusters\nin their sample. A possible interpretation for their data is the presence of a \nsurrounding filamentary structure  that is infalling into the cluster.\n\n\\subsection{Effects of gas clumping at large radii}\nMasses and entropy measured in this paper assume that the gas has a uniform density\nat each radius.  To quantify the effect of departures from uniform density, we\n   define the clumping factor $C$ \nas the ratio of density averages over a large region,\n\\begin{align}\nC & = \\frac{\\langle n_e^2 \\rangle}{\\langle n_e \\rangle^2}\n\\end{align}\nwith $C \\geqslant 1$.\nClumped gas emits more efficiently than gas of uniform\ndensity, \nand the same surface brightness $I$ results in a lower estimate for the gas density and mass,\n\\begin{equation}\nI \\propto \\int <n_e^2> dl = \\int <n_e>^2 C dl,\n\\end{equation}\nwhere $l$ is a distance along the sightline.\nFrom Figure~\\ref{fig:entropy-profile} we see that the entropy drop from\napproximately 400\" to 600\" would be offset by a decrease in $n_e^{2\/3}$ by a factor\nof 3, or a decrease in $n_e$ by a factor of 5. We therefore suggest that a clumping\nfactor of $C \\simeq 25$ at 600\" would in principle be able to provide\na flat entropy profile, and even higher clumping factors would provide\nan increasing entropy profile in better agreement with theory \\citep[e.g.][]{voit2005,tozzi2001}.\nNumerical simulations by \\cite{nagai2011} suggest  values of the clumping factor\n$C \\leq 3$ near $r_{200}$, with significantly higher clumping possible at larger radii. \nUse of the \\cite{nagai2011}  model in the analysis of a large sample of galaxy clusters by \\cite{eckert2012}\nresults in better agreement of observations with numerical simulations. \n\nClumping can also affect the measurement of hydrostatic masses.\nIn particular, gas with an increasing radial profile of the clumping factor\ncould result in a steeper gradient of the density profile, when compared with what is measured assuming\na uniform density. According to Equation~\\ref{eq:hse}, this \nwould result in larger estimates of the hydrostatic mass, in principle able to reduce or entirely\noffset the apparent decreas of $M(r)$ reported in Figure~\\ref{fig:mass-0-600}.\nWe therefore conclude that a radial increase in the clumping of the gas can in principle\naccount for the apparent decrease of the mass profile and of the entropy profile\nreported in this paper (Figures~\\ref{fig:mass-0-600} and \\ref{fig:entropy-profile}), and therefore\nit is a viable scenario to interpret our \\it Chandra\\rm\\ observations.\nClumping of the gas at large radii has also been suggested based on \\it Suzaku\\rm\\ observations\n\\citep[e.g.,][]{simionescu2011}.\n\n\\section{Conclusions}\nIn this paper we have reported the detection of emission from \\emph{Abell~1835}\\ with \\it Chandra\\rm\\\nout to the cluster's virial radius. The cluster's surface brightness\nis significantly above the background level out to a radius of\napproximately 10 arcminutes, which correspond to $\\sim$2.4 Mpc at the\ncluster's redshift. We have investigated several sources of systematic\nerrors in the background subtraction process, and determined that the\nsignificance of the detection in the outer region (450-600\") is\n$\\geq 4.7$~$\\sigma$, and the emission cannot be explained\nby fluctuations in the background. Detection out to the virial\nradius is also implied by the \\it XMM-Newton\\rm\\ temperature profile\nreported by \\cite{snowden2008}.\n\nThe \\it Chandra\\rm\\ superior angular resolution made it straightforward to\nidentify and subtract sources of X-ray emission that are unrelated to the cluster. \nIn addition to a large number of point sources, we have identified X-ray emission\nfrom two low-mass clusters that were selected from the SDSS data,\nMAXBCG J210.31728+02.75364 \\citep{koester2007}\nand WHL J140031.8+025443 \\citep{wen2009}.\nThe two clusters have photometric and spectroscopic redshifts that make them\nlikely associated with \\emph{Abell~1835}. These are the only two\nSDSS-selected clusters that are in the vicinity of \\emph{Abell~1835}.\n\n\nThe outer regions of the \\emph{Abell~1835}\\ cluster have a sharp drop in the temperature\nprofile, a factor of about ten from the peak temperature. The sharp drop\nin temperature implies that the hot gas cannot be in hydrostatic equilibrium, and\nthat the hot gas would be convectively unstable. A possible scenario to\nexplain the observations is the presence of \\emph{warm-hot} gas\nnear the virial radius that is not in hydrostatic equilibrium with\nthe cluster's potential, and with a mass budget comparable to that\nof the entire ICM. The data  are also consistent with an alternative scenario\nin which a significant clumping of the gas at large radii is responsible\nfor the apparent negative gradients of the mass and entropy profiles\nat large radii.\n\n\n\n\n\\bibliographystyle{mn2e}\n\\bibliographystyle{apj}\n\\input{ms.bbl}\n\n\n\n\\label{lastpage}\n\\end{document}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe advent of the {\\it Einstein} observatory changed the belief that\nearly-type galaxies contain little interstellar gas by revealing hot\nX-ray emitting halos associated with many of them (e.g.  Forman et\nal. 1979).  Subsequent X-ray observations led to the conclusion that\nthese galaxies can retain large amounts (up to $\\sim 10 ^{11}\nM_{\\odot}$) of hot ($T \\sim 10^{7}$ K) interstellar medium\n(Forman, Jones \\& Tucker 1985; Trinchieri \\& Fabbiano 1985; Canizares,\nFabbiano \\& Trinchieri 1987).\n\n\\begin{figure*}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.6\\hsize\n \\epsffile{xray_fig.eps}\n \\caption{Grey-scale {\\it ROSAT} HRI image of the core of the cluster\nAbell~2634. The image has been smoothed using a Gaussian kernel with a\ndispersion of 8 arcsec. The positions of the cluster galaxies with\nmeasured redshift are marked with crosses.} \n\\end{center}\n\\label{grayscale}\n\\end{figure*}\n\n\nHowever, this picture might be expected to be different for galaxies\nthat reside near the centres of rich clusters of galaxies, since their\nproperties must be affected by their dense environment.  For example,\nwe might expect interstellar medium (ISM) gas to be stripped from the\ngalaxy by the ram pressure resulting from the passage of the galaxy\nthrough the intracluster medium (ICM) (Gunn \\& Gott 1972).  Stripping\nof the ISM can also result from tidal interactions with other nearby\ngalaxies (Richstone 1975; Merritt 1983, 1984).  The most dramatic and\nwell-studied example of a galaxy which appears to be in the process of\nbeing stripped of its ISM is the elliptical galaxy M86 in the Virgo\ncluster, which shows a `plume' of X-ray emission emanating from it\n(Forman et al. 1979; White et al. 1991; Rangarajan et al. 1995).\n\nIn addition to the mechanisms which remove the ISM of a galaxy, gas\ncan also be replenished.  The gravitational pull of a galaxy attracts\nthe surrounding ICM.  This gas ends up being concentrated in or behind\nthe galaxy, depending on the velocity of the galaxy relative to the\nICM (see, for example, Sakelliou, Merrifield \\& McHardy 1996). Stellar\nwinds can also replenish the hot gas in a galaxy's ISM.\n\nAll the processes mentioned above take place simultaneously. The\nrelative importance of each process depends on: the galaxies'\nvelocities; the local density of the ICM; the number density of\ngalaxies; their orbits in the cluster; and the gravitational potential\nof each galaxy.  It is therefore {\\it a priori} difficult to say which\nmechanism dominates in the cores of rich clusters of galaxies, and\nhence whether cluster galaxies are surrounded by the extensive X-ray\nemitting halos that we see associated with galaxies in the field.\n\nUnfortunately, X-ray observations of rich clusters have generally not\nbeen of high enough quality to answer this question, since any\nemission from the galaxies is hard to detect against the high X-ray\nbackground produced by the cluster's ICM (Canizares \\& Blizzard 1991;\nVikhlinin, Forman \\& Jones 1994; Bechtold et al. 1983; Grebenev et\nal. 1995; Soltan \\& Fabricant 1990; Mahdavi et al. 1996). In the cases\nwhere galaxy X-ray emission has been reported, the studies have been\nrestricted to a few bright cluster galaxies, and it has not proved\npossible to investigate the general galaxy population in a\nstatistically complete manner.\n\nIn order to search for X-ray emission from galaxies in a moderately\nrich environment, we obtained a deep {\\it ROSAT} HRI observation of\nthe core of the rich cluster Abell~2634, which is a nearby (z=0.0312)\ncentrally-concentrated cluster of richness class I.  In \\S2.1 we\ndescribe the analysis by which the X-ray emission from the galaxies in\nthis cluster was detected.  In \\S2.2 we explore the properties of this\nX-ray emission, and show that the galaxies in this cluster lack the\nextensive gaseous halos of similar galaxies in poorer environments.\nIn \\S3, we show how this difference can be attributed to ram\npressure stripping.\n\n\n\\section{X-ray observations and analysis}\n\nThe core of Abell~2634 was observed with the {\\it ROSAT} HRI in two\npointings, in January and June 1995, for a total of $62.5\\,{\\rm\nksec}$.  The analysis of these data was performed with the IRAF\/PROS\nsoftware.\n\nInspection of the emission from the cD galaxy and other bright X-ray\nsources in the images from the two separate observations indicates\nthat the two sets of observations do not register exactly and that a\ncorrection to the nominal {\\it ROSAT} pointing position is\nrequired. Therefore, the second set of observations was shifted by\n$\\sim$ 2.0 arcsec to the east and $\\sim$ 0.8 arcsec to the south; such\na displacement is consistent with typical {\\it ROSAT} pointing\nuncertainties (Briel et al. 1996).  Both images were then registered\nwith the optical reference frame to better than an arcsecond.  A\ngrey-scale image of the total exposure is shown in Fig. 1.  The image\nhas been smoothed with a Gaussian kernel of 8 arcseconds dispersion.\nAt the distance of Abell~2634, $1\\,{\\rm arcsec}$ corresponds to\n$900\\,{\\rm pc}$.\\footnote{Here, as throughout this paper, we have\nadopted a Hubble constant of $H_{0}=50\\; {\\rm km\\:s^{-1}\\:Mpc^{-1}}$.}\n\n\\begin{table}\n \\caption{Bright Sources}\n \\begin{tabular}{cccc}\n\\hline \\hline\nSource & $\\alpha$(J2000) & $\\delta$(J2000) & ID\/notes \\\\\n     & $^{\\rm h} \\; ^{\\rm m} \\; ^{\\rm s} $ & $\\degr \\; \\arcmin \\;\n\\arcsec$ & \\\\\n\\hline\n1 & 23 38 29.1 & 27 01 53.5 & cD galaxy \\\\\n2 & 23 37 56.1 & 27 11 31.3 & cluster \\\\\n3 & 23 39 01.6 & 27 05 35.9 & star \\\\\n4 & 23 39 00.5 & 27 00 27.9 & star\\\\\n5 & 23 38 31.7 & 27 00 30.5 & nothing \\\\\n6 & 23 38 41.5 & 26 48 04.1 & star \\\\\n7 & 23 38 19.8 & 26 56 41.5 & ? \\\\\n8 & 23 38 07.4 & 26 55 52.8 & star \\\\\n9 & 23 37 57.5 & 26 57 30.1 & galaxy ?\\\\\n10 & 23 37 45.3 & 26 57 53.1 & two objects \\\\\n11 & 23 37 26.2 & 27 08 14.6 & ? \\\\\n\n\\hline\n \\end{tabular}\n\\end{table}\n\nThis deep image of Abell~2634 reveals the largescale X-ray emission\nfrom the hot ICM of the cluster and a few bright sources, which are\nnumbered on Fig.~1.  Source 1 is the cD galaxy NGC~7720, located near\nthe centre of Abell~2634. It hosts the prototype wide-angle tailed\nradio source 3C~465 (e.g.~Eilek et al.  1984).  Source 2 is a\nbackground cluster at a redshift of $cz \\simeq 37,000 \\ {\\rm km \\\ns^{-1}}$ (Pinkney et al.  1993; Scodeggio et al.  1995).  For the rest\nof the X-ray bright sources, the Automatic Plate Measuring \nmachine, run by the Royal Greenwich Observatory in Cambridge, was used\nto obtain optical identifications.  Table 1. gives the positions of\nthese sources as determined from the X-ray image, and the class of\ntheir optical counterparts.  The position of source 7 coincides with a\nfaint object in the Palomar sky survey, but there is also a nearby\nstar, and source 11 does not seem to have a discernible optical\ncounterpart.  All these sources were masked out in the subsequent\nanalysis.  \n\nThe positions of galaxies that are members of Abell~2634 are also\nindicated on Fig.~1.  Pinkney et al.\\ (1993) collected the redshifts\nof $\\sim$150 galaxies that are probable members Abell~2634 (on the\nbasis that their redshifts lie in the range $6,000 < cz < 14,000\\,\n{\\rm km\\,s^{-1}}$), and Scodeggio et al. (1995) have increased the\nnumber of galaxies whose redshifts confirm that they are cluster\nmembers up to $\\sim 200$.  The sample of redshifts is complete to a\nmagnitude limit of 16.5, and from this magnitude-limited sample we\nhave selected those galaxies that appear projected within a circle of\n15 arcmin radius, centered on the cD galaxy.  This selection yields 62\ngalaxies, of which the vast majority are of type E and S0 -- only 10\nare classified as spirals or irregular.  The positions of these\ngalaxies are taken from the CCD photometry of Pinkney (1995) and\nScodeggio et al. (1995), and are accurate to $\\sim 1\\,{\\rm arcsec}$.\nThey are marked as crosses on Fig.~1.\n\nInspection of Fig.~1 reveals several cases where the location of a\ngalaxy seems to coincide with an enhancement in the cluster's X-ray\nemission, and it is tempting to interpret such enhancements as the\nemission from the galaxy's ISM.  However, it is also clear from Fig.~1\nthat the X-ray emission in this cluster contains significant\nsmall-scale fluctuations and non-uniformities.  We must therefore\nconsider the possibility that the apparent associations between galaxy\nlocations and local excesses in the X-ray emission may be\nchance superpositions.  We therefore now present a more objective\napproach to searching for the X-ray emission from cluster galaxies.\n\n\n\n\\subsection{Detection of the cluster galaxies}\n\nBefore adopting an approach to detecting the emission from cluster\ngalaxies, we must first have some notion as to how bright we might\nexpect the emission to appear in this deep HRI image.  Previous X-ray\nobservations have shown that the X-ray luminosities of E and S0\ngalaxies in the 0.2-3.5 keV energy band range from $\\sim10^{39}$ to\n$\\sim 10^{42} \\ {\\rm erg \\ s^{-1}}$ (Kim, Fabbiano \\& Trinchieri\n1992a, b; Forman et al. 1985).  These limits at the distance of\nAbell~2634 correspond to fluxes of $5 \\times 10^{-16}$ to $5 \\times\n10^{-13} \\ {\\rm erg \\ s^{-1} \\ cm^{-2}}$.  We have used the PIMMS\nsoftware to convert these limits to count rates for the {\\it ROSAT}\nHRI detector.  The emission from the galaxies was modeled by a\nRaymond-Smith plasma (Raymond \\& Smith 1977) with a temperature\n$kT=0.862$ keV and a metal abundance of 25\\% solar; these quantities\nare consistent with the values previously found from observations of\nearly-type galaxies (Kim et al.  1992a; Matsushita et al. 1994; Awaki\net al. 1994). The absorption by the galactic hydrogen was also taken\ninto account by using the column density given by Stark et al. (1992)\nfor the direction of Abell~2634 ($N_{\\rm H}= 4.94 \\times 10^{20} \\\n{\\rm cm^{-2}}$).  These calculations predict that the $62.5\\,{\\rm\nksec}$ HRI observation of this cluster should yield somewhere between\n$\\sim1$ and $\\sim1200$ counts from each galaxy.  Motivated by this\nprediction of a respectable, but not huge, number of counts per\ngalaxy, we set out to detect emission associated with cluster\ngalaxies.\n\nWe are trying to detect this fairly modest amount of emission against\nthe bright background of the ICM emission.  We therefore seek to\nimprove the statistics by stacking together the X-ray images in the\nvicinity of the 40 E and S0 galaxies marked in Fig.~1.\nFig. 2. presents a contour plot of the combined image, which covers a\nregion of 1 arcmin radius around the stacked galaxies.  The centre of\nthe plot coincides with the optical centres of the individual\ngalaxies. Clearly, there appears to be X-ray emission associated with\nthe cluster galaxies, and it is centered at their optical positions.\nThis coincidence provides us with some confidence that the X-ray and\noptical frames are correctly registered.  We have also constructed a\ncomposite brightness profile for the 40 galaxies by adding the\nunsmoothed counts detected in concentric annuli centered on each\ngalaxy.  The width of each annulus in this profile was set at 6 arcsec\nand the local background, as measured in an annulus between 1.0 and\n2.0 arcmin around each galaxy, was subtracted.  The resulting profile\nis presented in Fig. 3.  Once again, the excess of emission in the\nvicinity of the cluster galaxies is apparent.\n\nIn order to assess the significance of this detection, we generated\n100 sets of simulated data from randomly selected points on the image.\nThe diffuse emission from the ICM varies systematically with radius,\nand so we might expect the probability that a galaxy is coincidentally\naligned with a clump in the ICM emission to vary systematically with\nradius.  Further, the sensitivity of the HRI varies with radius, and\nso the detectability of the emission from a single galaxy will vary\nwith radius as well.  We therefore constructed the simulated data sets\nby extracting counts from the HRI image at the same radii as the true\ngalaxy locations, but at randomized azimuthal angles.  The mean\nprofile and the RMS fluctuations amongst the simulated data sets are\nshown in Fig.~3.  As might be expected, the average number of counts\nin these random data sets is zero; the larger RMS error bars at small\nradii reflect the smaller sizes of these annuli.  From a $\\chi^{2}$\ncomparison between the observed galaxy profile and the simulated\nprofile, we can conclude that there is less than 0.1\\% probability\nthat the apparent peak in the galaxy emission is produced by chance.\nThus, the detection of emission from the galaxies in Abell~2634 is\nhighly statistically significant.\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.9\\hsize\n \\epsffile{contour.ps}\n \\caption{Contour plot of the combined image of all the early-type\ngalaxies that belong to Abell~2634. The pixel size of the image is 2\narcsec and it has been smoothed with a Gaussian kernel of 2\npixels. The center of the plot coincides with the optical centres of\nthe galaxies. The contour lines are from 20 to 100 per cent the peak\nvalue and are spaced linearly in intervals of 5 per cent.}\n\\end{center}\n\\label{contour}\n\\end{figure}\n\n\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.9\\hsize\n \\epsffile{40gal_simul.ps}\n \\caption{The combined surface brightness profile of all the\nearly-type galaxies (filled squares) normalized to one galaxy. Open squares\nrepresent the average profile from  the simulations.} \n\\end{center}\n\\label{earlysimul}\n\\end{figure}\n\n\\subsection{Origin of the X-ray emission}\n\nAs mentioned in the introduction, early-type galaxies have been found\nto retain large amounts of hot gas, which extends far beyond the\noptical limits of the galaxies. X-ray binaries also contribute to the\ntotal emission, and they become more dominant in X-ray faint galaxies.\nWe might also expect some of the emission to originate from faint\nactive galactic nuclei (AGNs) in the cores of these galaxies.\nAlthough none of the galaxies in our sample has been reported as an\nactive galaxy, there is increasing dynamical evidence that the vast\nmajority of elliptical galaxies contain central massive black holes\n(van der Marel et al. 1997; Kormendy et al.  1996a, 1996b; for a\nreview Kormendy \\& Richstone 1995), and so we might expect some\ncontribution from low-level activity in such systems.  We therefore\nnow see what constraints the observed X-ray properties of the galaxies\nin Abell~2634 can place on the origins of the emission.\n\n\n\\subsubsection{The extent of the X-ray emission}\n\nOne diagnostic of the origins of the X-ray emission is the measurement\nof its spatial extent.  AGN emission should be unresolved by the HRI,\nwhile emission from X-ray binaries should be spread over a similar\nspatial scale as the optical emission, and halos of hot gas should be\nstill more extended.\n\n\\begin{figure*}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.45\\hsize\n \\epsffile{cDplotno2.ps}\n \\epsfxsize 0.45\\hsize\n \\epsffile{s3plotno2.ps}\n\\end{center}\n\\begin{center}\n \\leavevmode \n \\epsfxsize 0.45\\hsize\n \\epsffile{s4plotno2.ps}\n \\epsfxsize 0.45\\hsize\n \\epsffile{s8plotno2.ps}\n\\end{center}\n\\caption{Surface brightness distribution of the bright sources in the\nHRI image. (s1, s3, s4, and s8). Source 1 is the central cD\ngalaxy. The profile is fitted by the appropriate HRI PSF for the\ndistance of the point source from the centre of the image (dash line)\nand a Gaussian (solid line). The calculated width ($\\sigma$) of the\nbest fit Gaussian is also given.}\n\\label{sources}\n\\end{figure*}\n \nIn order to assess the spatial extent of the X-ray emission, we need\nto characterize the PSF in this HRI observation.  The point sources\ndetected in these data are more extended than the model PSF for the\nHRI detector given by Briel et al. (1996) as is seen in Fig.~4, where\nfour of the sources are fitted by this model PSF (dash line).  This\ndiscrepancy can be attributed to residual errors in the reconstruction\nof {\\it ROSAT}'s attitude, which broaden the PSF in long integrations.\nWe have therefore empirically determined the PSF that is appropriate\nfor this observation by fitting the profiles of the point sources with\na Gaussian PSF model (Fig.~4, solid line). Only sources 1, 3, 4, 8 are\nused for the determination of the width of the Gaussian.  Source 6 is\nvery elongated and can not be represented by a symmetrical\nfunction. The mean dispersion of the best-fit model was found to be $(4.1\n\\pm 0.1)$ arcsec.  All of the point sources detected in this image\nhave widths consistent with this value, and so there is no evidence\nthat the PSF varies with radius.  We therefore adopt this PSF for the\nemission from all the galaxies in the observation.\n\n\nFigure~5 shows the comparison between the adopted PSF and the emission\nfrom the cluster galaxies.  The emission appears to be more extended\nthan the PSF; fitting the data to the PSF yields a $\\chi^{2}$ value of\n14.2 with 9 degrees of freedom, which is marginally consistent with\nthe emission being unresolved.  We can obtain a better fit by modeling\nthe radial profile of the emission using a Gaussian, which we convolve\nwith the PSF to model the observed profile.  Fitting this model to the\nobservations, we find that the intrinsic width of the X-ray emission\nis $4.3^{+2.2}_{-2.8}$ arcsec. The best-fit model is also shown in\nFig.~5.\n\n\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.9\\hsize\n \\epsffile{40gal_gauss_2psfs.ps}\n\\caption{The combined surface brightness distribution of the\n40 early-type galaxies, normalized to one galaxy. The profile is fitted\nby the measured HRI PSF (dashed line) and the spatially-extended model \n(solid line).}\n\\end{center}\n\\label{earlyprof}\n\\end{figure}\n\nThe radius of the X-ray halos of early-type galaxies with optical\nluminosities comparable to those in this cluster has been shown to be\n$\\sim 20$ -- 60 kpc (e.g.~Fabbiano et al. 1992), with the lower values\ncharacterizing optically fainter galaxies. At the distance of\nAbell~2634 these values correspond to $\\sim 20$ -- 60 arcsec, much\nlarger than the upper limit of $\\sim$ 7 arcsec we found for the extent\nof the galactic X-ray emission.  Thus, the X-ray emission from the\ncluster galaxies, although apparently extended, clearly does not\noriginate from the large halos of hot gas found around comparable\ngalaxies in poorer environments.\n\nOne possible explanation for the spatial extent of the emission from\nthese galaxies is that it could arise from errors in the adopted\npositions for the galaxies.  Such errors would broaden the\ndistribution of X-rays when the data from different galaxies are\nco-added even if the individual sources are unresolved.  However, the\nzero-point of the X-ray reference frame is well tied-down by\nthe detected point sources in the field.  Further, the optical\nlocations of the galaxies come from CCD photometry with positional\nerrors of less than an arcsecond.  It therefore cannot explain the\n$\\sim 4$ arcsec extent of the observed X-ray emission.\n\nWe therefore now turn to the extent of the X-ray emission that we\nmight expect from X-ray binaries.  Nearly half of the early-type\ngalaxies that we use for our analysis have been imaged in the I-band\nby Scodeggio, Giovanelli \\& Haynes (1997).  They have fitted the\noptical galaxy profile with a de Vaucouleur law, and found a mean\nvalue for their effective radii of $\\sim$8 arcsec, with only 4\ngalaxies smaller than 3 arcsec and another 3 larger than 13 arcsec.\nThese values are directly comparable to the spatial extent of the\nX-ray emission derived above.  Thus, it would appear that the\nobservations are consistent with what we would expect if the X-ray\nemission from the galaxies in Abell~2634 originates from X-ray\nbinaries in these systems, although we have not ruled out the\npossibility that some fraction of the emission comes from AGN.\n\n\\subsubsection{The luminosity of the X-ray emission}\n\nA further test of the origins of the X-ray emission in the cluster\ngalaxies comes from its luminosity.  It has been found that the blue\nluminosities of galaxies correlates with their X-ray luminosities,\nwith the optically brighter galaxies being more luminous in X-rays\n(e.g.~Forman et al. 1985; Fabbiano et al.  1992).  This correlation\nfor the early-type galaxies in the Virgo cluster is presented in\nFig. 6. The optical and X-ray luminosities of these galaxies are taken\nfrom Fabbiano et al. (1992).  The line in this plot divides the\n$L_{\\rm B} - L_{\\rm X}$ plane into two distinct galaxy types (Fabbiano\n\\& Schweizer 1995).  In addition to the differences in the ratio of\nX-ray-to-optical luminosities, galaxies in these two regions have been\nshown to possess different spectral properties.  The spectra of the\nX-ray bright galaxies [group (I)] are well fitted by Raymond-Smith\nmodels of 1 keV temperature, and it is believed that these galaxies\nretain large amounts of hot ISM.  In the spectra of the X-ray faint\ngalaxies of group (II), on the other hand, a hard component is\npresent; X-ray binaries are believed to be the major source of the\nX-rays in these galaxies.\n\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.9\\hsize\n \\epsffile{fabb.ps}\n\\caption{X-ray luminosity versus blue luminosity for the early-type\ngalaxies. The line delineates the boundary between the locations of\ngalaxies where the hot ISM makes a significant contribution to the\ntotal emission [region (I)], and the locations of galaxies where the\nentire emission can be ascribed to X-ray binaries [region (II)].  The\nlocations in this plane of galaxies that belong to the Virgo cluster\nare marked by filled circles.  The average properties of galaxies in\nAbell~2634 lying in\ndifferent optical luminosity ranges are indicated by crosses.}\n\\end{center} \n\\label{LxLB}\n\\end{figure}\n\n\nIn order to see where the galaxies of Abell~2634 lie in this plot, we\nmust calculate their optical and X-ray luminosities.  Butcher \\&\nOemler (1985) have measured {\\it J} and {\\it F} optical magnitudes for\na large number of galaxies in Abell~2634.  We have converted these\nmagnitudes to the blue band by applying the colour relations provided\nby Oemler (1974) and Butcher \\& Oemler (1985), correcting for galactic\nextinction, and using the appropriate K-correction.  We find that the\nabsolute blue magnitude of the galaxies in the HRI image lie in the\nrange from $-18.5$ to $-21.4$.  We have divided these galaxies into\nthree groups according to their optical luminosities: group A\n$(0.4-2.0) \\times 10^{10} \\ {\\rm L_{\\sun}}$ with 17 galaxies; group B\n$(2.1-3.7) \\times 10^{10} \\ {\\rm L_{\\sun}}$ with 8 galaxies; and group\nC $(3.8-5.5) \\times 10^{10} \\ {\\rm L_{\\sun}}$ with only 2 galaxies.\n\nThe X-ray luminosity of each group was obtained by repeating the\nanalysis of \\S2.1 using just the galaxies in each sub-sample.  Using\nthe PIMMS software we converted the observed count rate from the HRI\nimage to X-ray luminosity in the energy range 0.2-3.5 keV.  The thermal\nmodel used for the conversion is the same that was used by Fabbiano et\nal. (1992) to derive the plot shown in Fig. 6, and is discussed in\n\\S2.1.   \n\nThe resulting values for optical and X-ray luminosities in each\nsub-sample are shown in Fig.~6.  The horizontal error bars represent\nthe width of each optical luminosity bin and the vertical ones show\nthe errors in the measured X-ray luminosities.  This plot shows that\nthe galaxies in our sample follow the established correlation: the\noptically-brighter galaxies are also more luminous in the X-rays.  The\nexistence of this correlation also implies that the detected X-ray\nflux from the galaxies in Abell~2634 is not dominated by a few bright\ngalaxies, but that the optically-fainter galaxies also contribute to\nthe detected X-ray emission.\n\nThe galaxies in Abell~2634 probe the fainter end of the $L_{\\rm B}$ --\n$L_{\\rm X}$ relation as covered by Virgo galaxies.  It should be borne\nin mind that there is a bias in the Virgo data which means that the\ntwo data sets in Fig.~6 are not strictly comparable.  At the lower\nflux levels, a large number of Virgo galaxies have not been detected\nin X-rays, and so this plot preferentially picks out any X-ray-bright\nVirgo galaxies.  For the Abell~2634 data, on the other hand, the\nco-addition of data from all the galaxies in a complete sample means\nthat the data points represent a true average flux.  However, it is\nclear that the X-ray fluxes from galaxies in these two clusters are\ncomparable.\n\nThe similarity between the X-ray properties of galaxies in these two\nclusters is of particular interest because their environments differ\nsignificantly.  The galaxies from the Virgo cluster shown in Fig.~6\nlie in a region between 360 kpc and 2 Mpc from the centre of the\ncluster.  Recent {\\it ROSAT} PSPC observations have shown that the\nnumber density of the hot ICM of this cluster drops from $3 \\times\n10^{-4}$ to $3 \\times 10^{-5} \\ {\\rm cm^{-3}}$ in this region\n(Nulsen \\& B\\\"{o}hringer 1995).  The galaxies from Abell~2634 that\nhave gone into this plot lie in the inner 0.8 Mpc of Abell~2634, and\nin this region the number density of the ICM varies between $1\n\\times 10^{-3}$ and $2 \\times 10^{-4} \\ {\\rm cm^{-3}}$ (Sakelliou \\&\nMerrifield 1997). Thus, the galaxies in the current analysis come from\na region in which the intracluster gas density is, on average, an order of\nmagnitude higher that surrounding the Virgo cluster galaxies.\n\nThe location of the galaxies in region (II) of Fig.~6 adds weight to\nthe tentative conclusion of the previous section that the X-ray\nemission from these galaxies can be explained by their X-ray binary\npopulations, since any significant ISM contribution would place them in\nregion (I).  Similarly, the low X-ray fluxes of these galaxies leaves\nlittle room for a significant contribution from weak AGN.  If the\nX-ray binary populations are comparable to those assumed by Fabbiano\n\\& Schweizer (1995) in calculating the dividing line in Fig.~6, then\nessentially all the X-ray emission from these galaxies can be\nattributed to the X-ray binaries.  Thus, any average AGN emission\nbrighter than a few times $10^{40} \\ {\\rm erg \\ s^{-1}}$ can be\nexcluded, as such emission would also move the galaxies into region\n(I) of the $L_{\\rm B}$ -- $L_{\\rm X}$ plane.\n\n\n\\subsection{Spiral galaxies}\n\n\\begin{figure}\n\\begin{center}\n \\leavevmode\n \\epsfxsize 0.9\\hsize\n \\epsffile{spiral_sim.ps}\n \\caption{The combined surface brightness profile of the spiral \ngalaxies that lie in the field of view of the HRI (filled squares)\nnormalized to one galaxy. Open squares \nrepresent the average profile from the simulations.} \n\\end{center}\n\\label{spiral}\n\\end{figure}\n\nHaving discussed the X-ray properties of the early-type galaxies in\nAbell~2634 at some length, we now turn briefly to the properties of\nthe spiral galaxies in the cluster.  Abell~2634 is a reasonably rich\nsystem, and we therefore do not expect to find many spiral galaxies\nwithin it.  Indeed, only 7 of the 62 galaxies whose redshifts place\nthem at the distance of Abell~2634, and which lie within the field of\nthe HRI, have been classified as spirals.  The statistics are\ncorrespondingly poor when the X-ray emission around these galaxies is\nco-added: the combined profile is shown in Fig.~7, together with the\nresults from the control simulations (see \\S 2.1 for details).  It is\nclear from this figure that the spirals have not been detected in this\nobservation, and a $\\chi^2$ fit confirms this impression.\n\nThe failure to detect these galaxies is not surprising.  Not only are\nthere relatively few of them, but their X-ray luminosities are lower\nthan those of early-type galaxies.  In the {\\it Einstein} energy band\n(0.2 - 3.5 keV), their luminosities have been found to lie in the\nrange $\\sim 10^{38}$ to $\\sim 10^{41} \\ {\\rm erg \\ s^{-1}}$ (Fabbiano\n1989).  Modeling this emission using a Raymond-Smith model with a\nhigher temperature than for the early-type galaxies, as appropriate\nfor spiral galaxies (Kim et al. 1992a), we find that the expected\ncount rate for these galaxies is a factor of $\\sim 40$ lower than for\nthe ellipticals in the cluster.  It is therefore unsurprising that we\nfail to detect the small number of spiral galaxies present in the\ncluster.  \n\n\n\n\\section{Discussion}\n\nIn this paper, we have detected the X-ray emission from the normal\nelliptical galaxies in Abell~2634.  The limited spatial extent of this\nemission coupled with its low luminosity is consistent with it\noriginating from normal X-ray binaries in the galaxies' stellar\npopulations.   These galaxies do not seem to have the extended\nhot ISM found around galaxies that reside in poorer cluster\nenvironments.  We therefore now discuss whether this difference can be\nunderstood in terms of the physical processes outlined in the\nintroduction.\n\nIntuitively, the simplest explanation for the absence of an extensive\nhalo around a cluster galaxy is that it has been removed by ram\npressure stripping as the galaxy travels through the ICM.  A simple\ncriterion for the efficiency of this process can be obtained by\ncomparing the gravitational force that holds the gas within the\ngalaxy to the force due to the ram pressure, which tries to remove it\n(Gunn \\& Gott 1972).\n\nThe gravitational force is given by:\n\\begin{equation}\nF_{\\rm GR} \\sim G \\ \\frac{M_{\\rm gal} \\ M_{\\rm gas}}{R_{\\rm gal}^{2}}\n\\end{equation}\nwhere $M_{\\rm gal}$ is the total mass of the galaxy, $M_{\\rm gas}$ is\nthe mass of the X-ray emitting gas, and $R_{\\rm gal}$ is the radius of\nthe galaxy's X-ray halo.  For typical values for the masses of the\ngalaxy and the gas of $10^{12} \\ M_{\\sun}$ (Forman et al.\\ 1985) and\n$5 \\times 10^{9} \\ M_{\\sun}$ (e.g.~Canizares et al.\\ 1987)\nrespectively, and a mean value for $R_{\\rm gal}$ of $40 \\ {\\rm kpc}$\n(Canizares et al. 1986), which is a representative value for galaxies\nof the same optical luminosity as the galaxies in Abell~2634,\nequation~(1) implies that $F_{\\rm GR} \\sim 1 \\times 10^{30} \\ {\\rm\nN}$.\n \nThe force due to ram pressure is described by:\n\\begin{equation}\nF_{\\rm RP}=\\rho_{\\rm ICM} \\ v_{\\rm gal}^{2} \\ \\pi R_{\\rm gal}^{2} = \\mu\n\\ m_{\\rm p} \\ n\\ v_{\\rm gal}^{2} \\ \\pi R_{\\rm gal}^{2}\n\\end{equation}\nwhere $\\rho_{\\rm ICM}$ is the density of the ICM, $\\mu$ is the mean\nmolecular weight, $m_{\\rm p}$ is the proton mass, $n$ is the number\ndensity of the ICM, and $v_{\\rm gal}$ is the galaxy velocity.  From\nthe velocity dispersion profile of Abell~2634 presented by den Hartog\n\\& Katgert (1996) we find that the velocity dispersion, $\\sigma_{\\rm\nv}$, in the inner 15 arcmin of this system is $ \\approx 710 \\ {\\rm km\n\\ s^{-1}}$. Assuming an isotropic velocity field, the characteristic\nthree-dimensional velocity of each galaxy is hence $v_{\\rm gal}=\\surd\n3 \\ \\sigma_{\\rm v} \\approx 1230 \\ {\\rm km \\ s^{-1}}$.  The number\ndensity of the ICM in the same inner region has been derived from\nrecent {\\it ROSAT} PSPC data (Schindler \\& Prieto 1997) and this HRI\nobservation (Sakelliou \\& Merrifield 1997), and is found to vary from\n$1 \\times 10^{-3} \\ {\\rm cm^{-3}}$ down to $2 \\times 10^{-4} \\ {\\rm\ncm^{-3}}$, consistent with previous {\\it Einstein} observations (Jones\n\\& Forman 1984; Eilek et al. 1984).  Inserting these values into\nequation~(2), we find $F_{\\rm RP} \\sim (1 - 10) \\times 10^{30} \\ {\\rm\nN}$.\n\nThus, the ram pressure force exerted on the galaxies in Abell~2634 is\nfound to be larger than the force of gravity, and so we might expect\nram pressure stripping to be an effective mechanism for removing the\nISM from these galaxies.  In poorer environments, the density of the\nICM is likely to be at least a factor of ten lower, and the velocities\nof galaxies will be a factor of $\\sim 3$ smaller.  We might therefore\nexpect $F_{\\rm RP}$ to be a factor of $\\sim 100$ lower in poor\nenvironments.  Since such a change would make $F_{\\rm RP} < F_{\\rm\nGR}$, it is not surprising that galaxies in poor environments manage\nto retain their extensive halos.\n\nThe absence of extensive X-ray halos around the galaxies in Abell~2634\nimplies that ram pressure stripping dominates the processes of\naccretion and stellar mass loss which can replenish the ISM.  By\ncarrying out similar deep X-ray observations of clusters spanning a\nwide range of ICM properties, it will be interesting to discover more\nprecisely what sets of physical conditions can lead to the efficient\nISM stripping that we have witnessed in Abell~2634.\n\n \n\n\\section*{ACKNOWLEDGEMENTS} \n \nWe are indebted to the referee, Alastair Edge, for a very insightful\nreport on the first incarnation of this paper.  We thank Jason Pinkney\nfor providing us with the positions and redshifts of the galaxies and\nRob Olling for helpful discussions.  This research has made use of the\nNASA\/IPAC Extragalactic Database (NED) which is operated by the Jet\nPropulsion Laboratory, California Institute of Technology, under\ncontract with the National Aeronautics and Space Administration. Much\nof the analysis was performed using {\\sc iraf}, which is distributed\nby NOAO, using computing resources provided by STARLINK.  MRM is\nsupported by a PPARC Advanced Fellowship (B\/94\/AF\/1840).\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\noindent     \n\n\n In Si based semiconductor technology Sb is considered an important \ndopant for its role in the development of field effect transistors \nand infrared detectors \\cite{temp}. Ion implantation is a useful \ntechnique for fabricating such devices as it produces buried layers \nwith well defined interfaces, expanding possibility of designing \nnovel structures. The increased density in VLSI circuits also makes \nthe technological applications of the ion implantation , especially \nin MeV energy range, increasingly important. MeV implantation however \ncan also produce severe modifications in the material depending on \nthe nature and the energy of the impinging ion, and the implantation \ndose \\cite{tam1}.\nExtensive usage of ion implantation in device fabrication and the \ncontinued miniaturization of device structures has brought the \nissue of surface modifications, via ion implantations, to the \nforefront. However, the factors responsible for such modifications \nand the surface morphology after ion implantation, have received \nlittle attention \\cite{car}.\n\n Atomic force microscope (AFM) is a very effective tool for \nexamining surface modifications and surface structures. However,\nthere are very few studies in literature that have investigated \nthe morphological changes of the ion implanted surfaces by AFM. \nFurthermore, most of these surface studies are performed after keV \nimplantations \\cite{cou,pia,wan}, or at low fluences for individual \ncascade studies \\cite{wil}. Surface modifications after high energy,\n100 MeV, ion irradiation have also been investigated \\cite{sin}. \nHowever, the role of MeV ion implantation \non the surface topography remains poorly understood. In the present \nstudy we have made detailed investigation on Si(100) surfaces after \n1.5~MeV Sb implantation. The technique of AFM has been applied to \nunderstand the modification in roughness and morphology of silicon \nsurfaces upon ion implantation. We also investigate here the formation\n of nano-sized defects zones on Si(100) surfaces after MeV implantations.\n The results of shape transition in these nanostructures, from being\n elliptical at low fluences to becoming deformed circular at high fluences,\n will also be discussed here. Experimental procedure are mentioned in section~2\n and the results will be discussed in section~3. Conclusion are presented in section~4.\n \n\\section {Experimental}\n\\noindent     \n\nA mirror polished (100)-oriented Si single crystal (p-type) \nwafer was used in the\npresent study. The samples were implanted at room temperature\nwith a scanned beam of 1.5 MeV Sb$^{2+}$ ions at various fluences\nranging from  1$\\times 10^{11}$ to 5$\\times10^{14} ions\/cm^2$.\n The implantations were performed with\nthe samples oriented 7$^o$ off-normal to the incident beam to\navoid channeling effects. \n\nAFM Nanoscope E and Nanoscope III from Veeco were used to image\nthe implanted silicon sample surfaces. Images have been acquired\nin contact and tapping modes. Images ranging from 0.2 to 10 $\\mu{m}$ square \nwere obtained.\n\n\n\\section {Results and Discussion}\n\nFigure~1 shows \nthe $1 \\times 1\\mu{m^2}$ and $200 \\times 200 {nm^2}$ \n3D-images of the virgin silicon surface. It is observed that the\n virgin Si surface is smooth. 1.5 MeV implantation was carried out at \n various fluences and several $1 \\times 1\\mu{m^2}$ and $200 \\times 200 {nm^2}$ \nimages were taken at all the fluences. The  $1 \\times 1\\mu{m^2}$ images \nwere utilized for measuring the rms surface roughness of Si(100) surfaces\n after implantation.\nThe average roughness at each fluence is plotted \nin Figure~2. The rms roughness for a virgin Si(100) surface, \nmeasured to be 0.234~nm, is also marked. It can be clearly \nseen from Fig.~2 that the rms surface roughness exhibits \nthree prominent behaviors as a function of fluence. For \nlow fluences, up to $1\\times10^{13} ions\/cm^{2}$, the \nroughness is small and does not increase much compared to \nthe virgin surface roughness. Beyond this fluence an enhanced \nsurface roughness, increasing at a much steeper rate is observed. \nThis trend continues up to the fluence of \n$1\\times10^{14} ions\/cm^{2}$ where a high roughness of 0.296~nm \nis measured. A saturation in surface roughness with a slight \ndecrease in the roughness is observed beyond this fluence. \nThe decrease in surface roughness, at \n$5\\times10^{14} ions\/cm^{2}$, seems reasonable in view of high \nlevel of amorphicity at this dose \\cite{sdey2}, as beyond a certain \nhigh amorphicity, further higher levels of amorphization should \ntend to make the surface more homogeneous. A similar decrease \nin surface roughness with increasing fluence, beyond a critical \nfluence, has been observed for keV implantations of P and As \nin amorphous films \\cite{edr}.\n\n\n Our earlier RBS\/C and Raman scattering results \n\\cite{sdey2,sdey3} show that Si lattice disorder also displays\n3 similar behaviours as a function of ion fluence. Initially a low lattice \ndamage, due to simple defects, is seen upto the fluence of \n$1\\times10^{13}ions\/cm^{2}$. The disorder becomes larger with the\nonset of crystalline\/amorphous (c\/a) transition in Si-bulk \nat $1\\times10^{13}ions\/cm^{2}$. Finally the disorder \nsaturates with the Si-lattice as well as Si-surface becoming \namorphised at $5\\times10^{15}ions\/cm^{2}$.\nThe roughness on the Si\nsurface will be determined by several roughening and smoothening\nprocess that undergo on an ion implanted surface. Nuclear Energy loss\neffects are also crucial. In addition lattice disorder and the\nassociated stress will also be important in the evolution of the\nion implanted surface.\n\n\n High resolution $200\\times200nm^{2}$ \nimages of the Si-surfaces were acquired for all the fluences\nand are shown in Figure~3 for two representative Sb fluences\nof 1$\\times10^{13}$ and 5$\\times10^{14}ions\/cm^2$. \nThe images of Si surface acquired upto the  fluence of \n1$\\times10^{12}ions\/cm^2$ (not shown) are similar to the \nvirgin surface (of Fig.~1b) and their surface roughness \nis also similar (Fig.~2). However, after a \nfluence of 1$\\times10^{13}ions\/cm^2$, several nanostructures\ncan be seen on the Si-surface (see Fig.~3a). Fig.~4a is same \nas Fig.~3a and shows the approximate outlines of some of the \nnanostructures. The nanostructures represent the damage due to \nion implantation and are roughly of elliptical \nshape. For a quantitative analysis of \nthese nanostructures, the two axial lengths \nwere measured and the mean lengths of the minor and the major \naxes are found to be 11.6 and 23.0~nm respectively. The mean lengths \nof the two axes and the mean areas of the surface features \nare tabulated in Table~1 for various incident ion fluences. \n\nFor a Sb fluence of 5$\\times10^{13}ions\/cm^2$, although the \nsilicon surface is again found to be decorated with the \nelliptical nanostructures, the features have expanded\nalong both the axial directions (with the mean lengths \nof axes being 14.5 and 26.1~nm respectively). The average \narea of the nanostructures at this stage is calculated to be \n$297~nm^2$, which is about 41\\% higher than that at \n1$\\times10^{13}ions\/cm^2$. For a fluence of \n1$\\times10^{14}ions\/cm^2$, the area of these  nanostructures\nfurther inflates to $325\\pm31~nm^2$. Although the length of \nthe minor axis has not changed much at this fluence compared \nto that at 5$\\times10^{13}ions\/cm^2$, the major axis is elongated \nand has an average value of 31.6~nm.  Upto this stage the \neccentricity of the elliptical structures, for all fluences \nis found to be $\\sim$ 0.85$\\pm$ 0.4. The eccentricity of the \nelliptical structures, at each fluence is listed in Table~ 1.  \nInterestingly, after a fluence 5$\\times10^{14}ions\/cm^2$, the \nsurface structures undergo a shape transition with the \nnanostructures having axial lengths of $30.1\\pm4.4~nm$ and \n$30.7\\pm2.4~nm$, respectively (see Fig.~3b). Fig.~4b is same \nas Fig.~3b and shows the approximate outlines of some of the \nnanostructures. The nanostructures have become much bigger in \nsize and appear somewhat of circular shape. The nanostructures \nare not fully circular and have eccentricity $\\sim$ 0.19$\\pm$ 0.05 \n(the eccentricity for circle=0). However, the eccentricity of these \nnanostructures is much reduced compared to those at lower fluences \n(where eccentricity $\\sim$ 0.85$\\pm$ 0.4). Hence, we refer to these \nnanostructures as {\\it approximately circular}. An explosion in \nsize  ($\\sim$ 120\\%) of these features compared to that at \n1$\\times10^{14}ions\/cm^2$ suggests a tremendous modification in \nsurface morphology at this stage. Our results are in contrast to \nthe keV implantation study of Sb in Si where, for doses lower \nthan 1$\\times10^{14}ions\/cm^2$, no change in the surface topography \nwas observed \\cite{cou}.\n\nThe random arrival of ions on the surface constitute the stochastic\nsurface roughening. Surface diffusion, viscous flow, and surface sputtering\netc. contribute towards the smoothening of the surface \\cite{eklund}.\nThe mechanism for the formation of surface damage is also postulated as a\nresults of cascade collision due to nuclear energy loss ($S_n$). In the \npresent study also, $S_n$ seems to be the dominating factor in the creation of\nthe nanostructures after Sb implantation. In addition several factor like\nc\/a transition in Si lattice, the strain in the surface and in bulk, defect \nand disorder in the medium etc. may also responsible for the \nstructure formation at the surface. For Sb implantation in Si we observe \nthe formation of nanostructure at Si(100) surface only after the \nfluence of 1$\\times10^{13}ions\/cm^2$. These nanostructures inflate in size \nwith increasing fluence. The size inflation may also be related to the increased \ndisorder \\cite{sdey2,sdey3} in the Si lattice. The shape transition of \nnanostructures, from being elliptical at lower fluence to deformed  circular at \n5$\\times10^{14} ions\/cm^2$ , may be caused by the increase in the density\nof the electronic excitations. Our earlier studies \\cite{sdey2,sdey3} \nshow that the amorphization of Si-surface at this stage also leads to stress\nrelaxations on the ion implanted surface.\n\n\n\\section{Summary and conclusions}\n\\noindent\n\nIn the present study we have investigated  the modifications in \nthe morphology of the Si(100) surfaces after 1.5~MeV Sb implantation. \nWe observe the presence of nano-sized defect zones on the Si surfaces \nfor the Sb fluences of 1$\\times10^{13} ions\/cm^2$ and higher.\nThese nanostructures are elliptical in shape and their size \nincrease with fluence. We observe an abrupt\nincrease in size of nanostructures accompanied by a shape\ntransition after the fluence of  5$\\times10^{14}ions\/cm^2$.  \nThe nanostructures become approximately circular at this stage. \nWe have also investigated the modifications in the surface\nroughness of the ion implanted Si surfaces and find that\nsurface roughness demonstrates 3 different stages as a function \nof fluence. \n\n\\section{Acknowledgments}\n\\noindent\n\nThis work is partly supported by ONR grant no. N00014-97-1-0991.\nWe would like to thank A.M. Srivastava for very useful comments \nand suggestions. We would also like to thank Puhup Manas\nfor his help with the figures.\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nVarious types of oscillating stars reside in binary systems and   therefore a precise determination of their system and stellar parameters can be performed. In particular the combined photometric-spectroscopic analysis of eclipsing binaries (EBs hereafter) leads to the determination of absolute masses and radii of the components in a direct way \\citep[e.g.][]{2013A&A...557A..79L, 2013A&A...556A.138F}. Thanks to spectral disentangling techniques \\citep[e.g.][]{1994A&A...281..286S, 1995A&AS..114..393H, 2001LNP...573..269I, 2019A&A...623A..31S}, faint components giving rise to a contribution of a few per cent can be detected in high-resolution spectra \\citep[e.g.][]{2013A&A...557A..79L, 2014MNRAS.438.3093T, 2014MNRAS.443.3068B}. Such techniques ensure the determination of precise (up to one per cent) model-independent dynamical masses of stars, which can further be confronted with asteroseismic values provided at least one of the binary components pulsates. This also means that the results can be used to calibrate the asteroseismic mass determination that becomes more and more important, in particular for planet hosting stars \\citep[e.g.][]{2007ApJ...670L..37H, 2008JPhCS.118a2016H, 2012A&A...543A..98H}.\n\nAlgol-type systems (Algols, hereafter) are semi-detached, interacting EBs consisting of a main-sequence star of spectral type B-A (primary, hereafter) and an evolved F-K type companion (secondary, hereafter). As a consequence of the above-mentioned facts, studying pulsating Algols from complementary spectroscopic and photometric data provides a valuable test of stellar evolutionary models. The group of oscillating eclipsing Algol stars (oEAs, hereafter) \\citep{2002ASPC..259...96M} consists of eclipsing Algols with mass transfer where the mass-accreting primary shows $\\delta$\\,Sct-like oscillations. These stars are extraordinary objects for asteroseismic studies because they allow us to investigate short-term dynamical stellar evolution during mass-transfer episodes, most probably caused by the magnetic activity cycle of the less massive secondary. Basic principles of the interaction between the magnetic cycle of the cool secondary, the occurrence of rapid mass-transfer episodes, the dynamical behaviour of the system, and the excitation of different non-radial pulsation (NRP hereafter) modes of the mass-gaining primary can be studied in great detail.  \\citet{2018MNRAS.475.4745M} showed for the first time that mass transfer and accretion influences amplitudes and frequencies of NRP modes of the oEA star RZ\\,Cas, where amplitude changes are caused by the sensitivity of the mode selection mechanism to conditions in the outer envelope \\citep[e.g.][]{1998A&A...333..141P}  and frequency variations by the acceleration of the outer layers by mass and angular momentum transfer.\n\nThe details of angular momentum exchange in Algols determining their evolution are still not completely understood. This is in particular valid for systems like the so-called R\\,CMa stars  \\citep[e.g.][]{2011MNRAS.418.1764B, 2013A&A...557A..79L, 2018A&A...615A.131L} that include companions of extremely low masses. But also for the oEA star RZ\\,Cas, calculations showed that its actual configuration cannot be explained when assuming a purely conservative mass-transfer scenario in the past \\citep{2008A&A...486..919M}. In several Algols, such as RZ\\,Cas \\citep{2008A&A...480..247L} or TW Dra \\citep{2008A&A...489..321Z, 2010AJ....139.1327T}, orbital period variations were observed that can be attributed to periods of rapid mass exchange. The gas stream hits the equatorial zones of the atmosphere of the pulsating star, transfers an essential amount of angular momentum, and forces the acceleration of its outermost surface layers, thus causing strong differential rotation. While rotation alters the frequencies of the individual axisymmetric modes \\citep[e.g.][]{1981ApJ...244..299S, 2008ApJ...679.1499L}, it also lifts the degeneracy in frequency for the non-axisymmetric modes as observed for some $\\beta$\\,Cep variables \\citep[e.g.][]{2004A&A...415..241A, 2005MNRAS.360..619J}. This means that NRP modes are sensitive to the acceleration of the surface layers and that it is possible to probe the acceleration via the rotational splitting effect, as suggested by \\citet{2018MNRAS.475.4745M} for RZ\\,Cas. The study of the corresponding frequency shifts, together with the direct measurement of changes in the projected equatorial velocity ($v\\sin{i}$, hereafter) of the primary from its spectral line profiles leads to an estimation of the amount of matter transferred to the primary. The results can be compared to those obtained from 3D hydrodynamical calculations, assuming different rates of mass transfer \\citep[e.g.][]{2007ASPC..370..194M}, and finally explain the kind of mass and angular momentum transfer of the Algols.\n\nOne further advantage of oEAs is that the so-called spatial filtration or eclipse mapping effect, occurring as a result of the obscuration of parts of the stellar disc of the oscillating primary by the secondary during eclipses, simplifies the identification of the pulsation modes. The effect was predicted and found in photometric \\citep[e.g.][]{2003ASPC..292..369G, 2004A&A...419.1015M, 2005ApJ...634..602R} and spectroscopic \\citep{2018A&A...615A.131L} observations. The dynamic eclipse mapping method was introduced by \\citet{2011MNRAS.416.1601B} with the aim of NRP mode identification by reconstructing the surface intensity patterns on EBs. With modern methods such as the least-squares deconvolution (LSD) technique \\citep{1997MNRAS.291..658D}  and the pixel-by-pixel method of the FAMIAS program \\citep{2008CoAst.157..387Z}, high-$l$-degree NRP modes can also be detected and identified from the signal-to-noise ratio (S\/N hereafter) enhanced line profiles; a unique identification however is difficult \\citep[e.g.][]{2009CoAst.159...45L}. \n\nRZ\\,Cas (spectral type A3\\,V\\,+\\,K0\\,IV) is a short-period ($P$\\,=\\,1.1953\\,d) Algol and one of the best studied oEA stars.  A partial eclipse is observed during primary minimum \\citep{1994AJ....107.1141N}. The primary was found by \\citet{1998IBVS.4581....1O, 2001AJ....122..418O} to exhibit short-period light variability with a dominant oscillation mode of a frequency of 64.2\\,c\\,d$^{-1}$. This finding was later confirmed from dedicated multi-site photometric campaigns by both \\citet{2003ASPC..292..113M} and \\citet{2004MNRAS.347.1317R}. The latter authors obtained simultaneous Str\\\"omgren uvby light curves of RZ\\,Cas. These authors present a detailed photometric analysis for both binarity and pulsation, deriving WD \\citep{1971ApJ...166..605W} solutions in the four mentioned passbands as well as absolute parameters of both components, and confirming the pulsational behaviour of the primary component as found by \\citet{1998IBVS.4581....1O}. We use the radii and flux ratios between the components derived by  \\citet{2004MNRAS.347.1317R} as a reference in our spectroscopic investigation. A comprehensive overview on the observations and analysis of RZ\\,Cas can be found in \\citet{2018MNRAS.475.4745M}. \n\nStarting our  spectroscopic investigation of RZ\\,Cas in 2001, we were the first to detect rapid oscillations in its spectra \\citep{2004A&A...413..293L}, and later on in spectra taken in 2006 \\citep{2008A&A...480..247L, 2009A&A...504..991T}. From the different amplitudes of the Rossiter-McLaughlin effect \\citep[][RME hereafter]{1924ApJ....60...15R, 1924ApJ....60...22M} observed during the primary eclipses in different seasons and the modelling of line profiles over the full orbital cycle using the Shellspec07\\_inverse program, we deduced that RZ\\,Cas was in an active phase of mass transfer in 2001, whereas in 2006 it was in a quiet state. To model the surface intensity distribution of the secondary of RZ\\,Cas, we had to include a large cool spot facing the primary, presumably originating from a cooling mechanism by the enthalpy transport via the inner Lagrangian point as suggested by \\citet{1994PASJ...46..613U}. Comparing the rapid oscillations found in the radial velocities (RVs hereafter) from 2001 and 2006 also with those derived from the light curves of RZ\\,Cas taken over many years \\citep[see][]{2018MNRAS.475.4745M}, we found that the NRP pattern of RZ\\,Cas changed from season to season. Different NRP modes have been excited with different amplitudes in different years and also frequency variations of single modes were observed.  \\citet{2018MNRAS.475.4745M} suggested that these frequency variations could be caused by a temporary acceleration of the outer layers of the primary owing to angular momentum exchange by mass-transfer effects.  \n\nAfter observing RZ\\,Cas in 2001 and 2006, we took new time series of high-resolution spectra in 2008 and 2009. The fact that we found a typical timescale of about nine years from the behaviour of the pulsation amplitudes but also from light-curve analysis \\citep[see][]{2018MNRAS.475.4745M} forced us to start a spectroscopic monitoring of the star covering the years 2013 to 2017. We now investigate the complete data set with the aim to use the spectra and  observed variations in RVs and line profile variations (LPV hereafter) to deduce stellar and system parameters, check for timely variable NRP pulsation patterns, and try to correlate all observed variations with the occurrence of quiet and active phases of the Algol system.  Observations are described in Sect.\\,\\ref{Sect2}. The spectra taken with the HERMES spectrograph are used for a detailed analysis with the aim to derive precise atmospheric parameters for both components of RZ\\,Cas (Sect.\\,\\ref{Sect3}). The extraction of RVs and calculation of mean, high-S\/N line profiles with the newly developed LSDbinary program is described in Sect.\\,\\ref{Sect4}. Using different methods, we measure the projected equatorial rotation velocity of the primary (Sect.\\,\\ref{Sect5}). In Sect.\\,\\ref{Sect6} we use the O-C values collected from literature to compute the orbital period variations of RZ\\,Cas over the last decades. We check for non-Keplerian effects in the orbital RV curves in Sect.\\,\\ref{Sect7.2} and try to model them using the PHOEBE \\citep{2005ApJ...628..426P} program. The results are discussed in Sect.\\,\\ref{Sect8} followed by concluding remarks in Sect.\\,\\ref{Sect9}.\n\nOur investigation of NRP is based on high-frequency oscillations in the RVs and in LPV. Applied methods and results will be described in a forthcoming article (Paper\\,II) and discussed together with the results presented in this work.\n\n\\begin{table} \\centering\n \\tabcolsep 1.8mm\n \\caption{Journal of observations listing the instrument, its spectral resolving power, year of observation, and mean Julian date. The last four columns give the number of spectra, total time span of observations in days, number of observed nights, and number of groups of observations.}\\label{Tab01}\n \\begin{tabular}{lclcrrrr}\n \\toprule\n Source & $R$     & Year & mean JD     & $s$ & $t$ & $n$ & $g$\\\\\n \\midrule\n TCES   & 32\\,000 & 2001     & 2\\,452\\,190 &  962 &   15 &13&1\\\\\n TCES   & 32\\,000 & 2006     & 2\\,453\\,800 &  517 &  150 &7 &3\\\\\n TCES   & 32\\,000 & 2008     & 2\\,454\\,717 &   94 &   27 &5 &2\\\\\n HERMES & 85\\,000 & 2009     & 2\\,455\\,156 &  228 &    5 &5 &1\\\\\n TCES   & 32\\,000 & 2013     & 2\\,456\\,600 &  835 &  157 &21&3\\\\\n TCES   & 32\\,000 & 2014     & 2\\,456\\,938 &  696 &   13 &8 &2\\\\ \n TCES   & 58\\,000 & 2015     & 2\\,457\\,300 &  998 &  152 &31&4\\\\\n TCES   & 58\\,000 & 2016     & 2\\,457\\,647 &  586 &   14 &10&1\\\\\n TCES   & 58\\,000 & 2017     & 2\\,458\\,097 & ~~43 &    4 &3 &1\\\\\n \\bottomrule\n \\end{tabular}\n\\end{table}\n\n\\section{Observations}\\label{Sect2}\n\nSpectra were taken over a total time span of 16 years with the TCES spectrograph at the 2 m Alfred Jensch Telescope of the Th\\\"uringer Landessternwarte (TLS) Tautenburg and over one year with the HERMES spectrograph \\citep{2011A&A...526A..69R} at the 1.25 m Mercator Telescope on La Palma. The TCES instrument is an echelle spectrograph in Coude focus and covers the wavelength range 4450-7550\\,\\AA. A 2015 upgrade resulted in improved efficiency so that this spectrograph could be used with higher spectral resolution (narrower entrance slit of one instead of two arcsec width). Table\\,\\ref{Tab01} gives the journal of observations. \n\nThe TCES spectra were reduced using standard MIDAS packages for Echelle spectrum reduction and HERMES spectra were reduced using the HERMES spectrum reduction pipeline. We used our own routines for the normalisation of the spectra to the local continuum. Instrumental shifts  were corrected using an additional calibration based on a large number of telluric O$_2$ absorption lines.\n\n\\section{Spectrum analysis}\\label{Sect3}\n\n\\subsection{Spectra}\n\nThe HERMES spectra taken in  2009 comprise the  H$_\\epsilon$ to H$_\\alpha$ range, whereas the TLS spectra only include H$_\\beta$ and H$_\\alpha$. Moreover, the HERMES spectra provide the higher spectral resolution. That is why we decided to use them for spectrum analysis. We were able to decompose the spectra into the spectra of the components using the KOREL program \\citep{1995A&AS..114..393H}. Analysing the decomposed spectra, we faced problems with the continuum normalisation. As a result of the very faint contribution of the secondary, small deviations in the continuum of the composite spectrum from the true continuum leads to large deviations in the continuum of the decomposed spectrum of the secondary. A first normalisation was applied during spectrum reduction by fitting higher polynomials to nodes assumed to represent the local continuum in each of the extracted echelle orders. This approach is not accurate enough when the contribution of the secondary is as faint as in the present case. Corrections to the local continuum have to be applied during spectrum analysis by a comparison with the continua of synthetic spectra. The spectra are then renormalised by multiplying them with the local ratio of the continua of the synthetic spectrum to that of the reduced spectrum. The spectrum decomposition with KOREL, on the other hand, is based on Fourier transform and introduces undulations in the continuum that are additive and have to be subtracted. This introduces an ambiguity with the multiplicative corrections described before that we could not solve to reach the required accuracy. Instead, we used the average of nine composite HERMES spectra of RZ\\,Cas taken at maximum separation of the components and having about the same RV. In this case, we have to apply multiplicative continuum corrections only, for which we used spline functions. Because the signal of the secondary gets fainter for smaller wavelengths and because of the occurrence of telluric lines in the red part of the spectra we limited the spectral range to 4000-5550\\,\\AA, including the three Balmer lines H$_\\beta$, H$_\\gamma$, and H$_\\delta$. \n\n\\subsection{Methods}\n\n\\begin{figure}\n \\includegraphics[width=\\linewidth]{Fig01.png}\n \\caption{Continuum flux ratios between the components vs. wavelength. The dotted lines are calculated from synthetic continuum spectra with $T_{\\rm eff}$$_1$ of 8700\\,K and $T_{\\rm eff}$$_2$ of 3800\\,K to 5000\\,K. The best agreement with the flux ratio from photometry (red line) is obtained for $T_{\\rm eff}$$_2$\\,=\\,4400\\,K (black solid line). The upper green line represents $T_{\\rm eff}$$_1$\\,=\\,9000\\,K, $T_{\\rm eff}$$_2$\\,=\\,4500\\,K, the lower green line for $T_{\\rm eff}$$_1$\\,=\\,8400\\,K, $T_{\\rm eff}$$_2$\\,=\\,4300\\,K.}\n \\label{Fig01}\n\\end{figure}\n\n\\begin{table*}\n \\tabcolsep 4mm\n \\caption{Atmospheric parameters of primary and secondary of RZ\\,Cas derived with multiple methods.}\\label{Tab02}\n \\begin{tabular}{lllllll}\n \\toprule\n           & \\multicolumn{2}{c}{Method a)} & \\multicolumn{2}{c}{Method b)} & \\multicolumn{2}{c}{Method c)}\\\\\n\\midrule\n$T_{\\rm eff}$\\ (K)   &  ~8650$\\pm$60                 & ~4860$_{-150}^{+190}$    & ~8643$\\pm$57              & ~4474$\\pm$83              & ~8635$\\pm$49     & ~4800$_{-120}^{+130}$         \\vspace{1mm}\\\\\n$\\log{g}$\\ (dex) & ~~4.41$\\pm$0.09           & ~~3.7 fix                        & ~~4.42$\\pm$0.07                & ~~3.7 fix                    & ~~4.41$\\pm$0.06  & ~~3.7 fix                        \\vspace{1mm}\\\\\n$v_{\\rm turb}$\\ (km\\,s$^{-1}$)& ~~3.60$_{-0.15}^{+0.32}$  & ~~1.42$\\pm$0.80              & ~~~3.61$_{-0.13}^{+0.17}$ & ~~1.03$_{-0.96}^{+0.63}$ & ~~3.59$\\pm$0.13  & ~~1.83$\\pm$0.75                 \\vspace{1mm}\\\\  \n$[$Fe\/H$]$ & $-$0.43$_{-0.06}^{+0.01}$ & $-$0.50$\\pm$0.20             & $-$0.42$\\pm$0.03              & $-$0.49$\\pm$0.20      & $-$0.43$\\pm$0.02 & $-$0.38$\\pm$0.18              \\vspace{1mm}\\\\\n$[$C\/H$]$  & $-$0.82$_{-0.20}^{+0.14}$ & $-$0.53$_{-0.36}^{+0.27}$& $-$0.80$_{-0.18}^{+0.13}$ & $-$0.63$_{-0.42}^{+0.29}$& $-$0.82$_{-0.18}^{+0.13}$ & $-$0.24$_{-0.46}^{+0.18}$\\vspace{1mm}\\\\\n$v\\sin{i}$\\ (km\\,s$^{-1}$)& 65.40$\\pm$0.95                & ~~~84.5$_{-7.9}^{+9.3}$  & 65.65$\\pm$0.93            & ~~~92.3$_{-9}^{+12}$  & 65.74$\\pm$0.88   & ~~~81.5$_{-7.6}^{+8.7}$   \\vspace{1mm}\\\\\n$F_2\/F_1$              & \\multicolumn{2}{c}{free}                             & \\multicolumn{2}{c}{free}                          & \\multicolumn{2}{c}{taken from photometry}                \\\\\n$(R_2\/R_1)^{\\rm calc}$ & \\multicolumn{2}{c}{0.926$\\pm$0.008}      & \\multicolumn{2}{c}{~~~~~~~1.17 fixed}    & \\multicolumn{2}{c}{0.870}                           \\\\\n$(R_2\/R_1)^{\\rm adj}$  & \\multicolumn{2}{c}{0.780}                & \\multicolumn{2}{c}{1.065}                            & \\multicolumn{2}{c}{0.870}                             \\\\\nrms (line depth)       & \\multicolumn{2}{c}{0.004499}             & \\multicolumn{2}{c}{0.004696}                         & \\multicolumn{2}{c}{0.004525}                  \\\\ \n \\bottomrule\n \\end{tabular}\n\\end{table*}\n\n\\begin{figure*}[hbt!]\n \\includegraphics[width=\\linewidth]{Fig02.png}\n \\caption{Results of spectrum analysis using method a) (cf. Sect.\\,\\ref{Sect3.3}, Table\\,\\ref{Tab02}), shown for the H$_{\\delta}$-H$_{\\gamma}$ region (left) and the region around the Mg\\,I\\,b triplet (right). First row: Observed, continuum adjusted composite spectrum (black), best-fitting synthetic spectrum (red), and shifted difference spectrum (green). The inverse of the applied continuum correction is shown in blue. Second row: The same for the secondary. Third row: Flux ratio of the secondary to primary from photometry (black) from spectrum analysis based on $R_2\/R_1$=0.926 (red) and assuming the adjusted ratio of 0.78 (green).}\n \\label{Fig02}\n \\vspace{4mm}\n \\includegraphics[width=\\linewidth]{Fig03.png}\n \\caption{As Fig.\\,\\ref{Fig02}, but using method b). Third row: Flux ratio from photometry (black) and from spectrum analysis based on $R_2\/R_1$=1.17 (red) and on the calculated ratio of 1.065 (green).}\n \\label{Fig03}\n \\vspace{4mm}\n \\includegraphics[width=\\linewidth]{Fig04.png}\n \\caption{As Fig.\\,\\ref{Fig02} but using method c). Third row: Flux ratio from photometry (black) from spectrum analysis based on the photometric flux ratio assuming $R_2\/R_1=1.17$ (red) and on the adjusted ratio of $R_2\/R_1=0.87$ (green). Bottom row: Ratio of the photometric flux ratio to the flux ratio obtained from spectrum analysis based on $R_2\/R_1=1.17$.}\n \\label{Fig04}\n\\end{figure*}\n\nWe used the GSSP program \\citep{2015A&A...581A.129T} to derive the atmospheric parameters of the components of RZ\\,Cas. It is based on the spectrum synthesis method and performs a grid search in stellar parameters. Synthetic spectra were computed with SynthV \\citep{1996ASPC..108..198T}, based on a library of atmosphere models computed with LLmodels \\citep{2004A&A...428..993S} for the hot primary and on MARCS models \\citep{2008A&A...486..951G} for the cool secondary. Atomic data were taken from the VALD database \\citep{2000BaltA...9..590K}.\n\nBesides the atmospheric parameters, GSSP also has to solve for the a priori unknown flux ratio of the components. In the following, we use the continuum flux ratio of the secondary (component\\,2) to  primary (component\\,1) for comparison.  It is interpolated from the $uvby$ luminosities provided by \\citet{2004MNRAS.347.1317R} and shown by the red line in Fig.\\,\\ref{Fig01}. To get an impression of the influence of the $T_{\\rm eff}$\\ of primary and secondary on the resulting flux ratios with wavelength, we computed synthetic continuum spectra for different $T_{\\rm eff}$\\ of the components, assuming $\\log{g}$$_2$\\,=\\,3.7  \\citep{2004MNRAS.347.1317R}, and $\\log{g}$$_1$\\,=\\,4.4 and [Fe\/H]=$-$0.42 as derived in Sect.\\,\\ref{Sect3.3}. Synthetic flux ratios were computed from the ratio of the continuum spectra assuming a radii ratio of $R_2\/R_1$\\,=\\,1.17, also taken from \\cite{2004MNRAS.347.1317R}. The best representative theoretical curve is shown by the black line in Fig.\\,\\ref{Fig01} and corresponds to $T_{\\rm eff1}$\\,=\\,8700\\,K, $T_{\\rm eff2}$\\,=\\,4400\\,K. From the green lines, we see that a variation of $T_{\\rm eff}$$_2$ by 100\\,K requires a variation of $T_{\\rm eff}$$_1$ by 300\\,K to approximately fit the photometric flux ratio.\n\nWe applied three different methods (labelled a, b, and c in what follows) to determine the atmospheric parameters together with the radii ratio and continuum flux ratio of the secondary to primary.\n\na) We use the standard method of the GSSP program that takes the wavelength dependence of the flux ratio from the ratio of the synthetic continuum spectra calculated with SynthV and scales it with the square of the radii ratio. The latter is obtained from comparing the observed with the synthetic normalised spectra on a grid of atmospheric parameters. To obtain the atmospheric parameters together with the radii ratio, we minimise\n\\begin{equation}\n \\chi^2 = \\sum_\\lambda\\left(\\frac{\\displaystyle o(\\lambda)-s(\\lambda)}{\\displaystyle\\sigma}\\right)^2,\n \\label{GSSP1}\n\\end{equation}\n where $o(\\lambda)$ and $s(\\lambda)$ are the observed and synthetic composite spectra, respectively, and $\\sigma$ is the estimated mean error of $o(\\lambda)$. The synthetic spectrum is computed from\n\\begin{equation}\n s(\\lambda) = \\frac{\\displaystyle s_1(\\lambda,v_1)+V_F(\\lambda)\\,s_2(\\lambda,v_2)}{\\displaystyle 1+V_F(\\lambda)},\n \\label{GSSP2}\n\\end{equation}\nwhere $s_1$ and $s_2$ are the synthetic spectra of the components shifted for their RVs $v_1, v_2$ (all spectra in line depths), and the flux ratio\n\\begin{equation}\n V_F(\\lambda) = \\left(\\frac{\\displaystyle R_2}{\\displaystyle R_1}\\right)^2\\frac{\\displaystyle C_2(\\lambda)}{\\displaystyle C_1(\\lambda)}\\label{GSSP3}    \n\\end{equation}\n is the product of the squared radii ratio and the ratio of the continuum fluxes $C_2$ and $C_1$ per unit surface, determined with SynthV.\n\nb) We use Equations\\,\\ref{GSSP1} to \\ref{GSSP3} as before, but fix the radii ratio to 1.17 as obtained from the uvby photometry.\n\nc) We replace the flux ratio $V_F$ computed so far from Eq.\\,\\ref{GSSP3} by the flux ratio obtained from the $uvby$ photometry. No synthetic spectra are needed in this case. The best-fitting radii ratio can then be determined from Eq.\\,\\ref{GSSP3} using a simple least-squares fit.\n\nBecause of the known degeneracy between various free parameters, we reduced their number. The spectrum of the secondary with its small contribution to total light suffers from low S\/N and we fixed its $\\log{g}$\\ to 3.7, as derived from light curve analysis \\citep{2004MNRAS.347.1317R}. Furthermore, we used the same values for its elemental abundances as derived for the primary, except for the iron and carbon abundances, which we determined separately. For the primary, we basically know the photometric value of $\\log{g}$\\,=\\,4.33(3) from \\citet{2004MNRAS.347.1317R}. Because of the good S\/N of the spectra and the fact that the Balmer lines shapes are very sensitive to $\\log{g}$\\ in this temperature range, we decided to use $\\log{g}$\\ of the primary as a free parameter to compare the results with the photometric results. \n\nThe GSSP allows us to iterate atmospheric parameters such as $T_{\\rm eff}$, $\\log{g}$, and microturbulent velocity $v_{\\rm turb}$\\ together with $v\\sin{i}$\\ and the surface abundance of one chemical element at the same time. We started, in this way, to optimise [Fe\/H] together with the other parameters, first for the primary, then for the secondary, and repeated until the change in parameters and $\\chi^2$ became marginal. In a next step, we fixed all parameters to the values obtained so far and optimised the elemental abundances of all other chemical elements for which significant contributions in the spectra could be found.\n\n\\subsection{Results}\\label{Sect3.3}\n\nTables\\,\\ref{Tab02} and \\ref{Tab03} list the results. $(R_2\/R_1)^{\\rm calc}$ in Table\\,\\ref{Tab02} is the radii ratio obtained from the analysis using methods a, b), or c). $(R_2\/R_1)^{\\rm adj}$ is the adjusted radii ratio that follows when we apply the least-squares fit based on Eq.\\,\\ref{GSSP3}, as used in method c) for the two other methods as well. This means that when keeping all other parameters obtained with the various methods, the radii ratio has to be changed from $(R_2\/R_1)^{\\rm calc}$ to $(R_2\/R_1)^{\\rm adj}$  to give the best agreement with the photometric flux ratio. For method c), both radii ratios are identical. The quantity $(R_2\/R_1)^{\\rm adj}$ is used in the following as a measure for the agreement of the spectroscopic with the photometric results. It should be equal to 1.17 in the optimum case.\n\nFigures\\,\\ref{Fig02} to  \\ref{Fig04} compare the results from the three methods. These figures show in the first row the observed spectrum together with the best-fitting synthetic composite spectrum. The second row compares the ``observed'' spectrum of the secondary with the best-fitting synthetic spectrum found for the secondary. The observed spectrum of the secondary was therefore computed by subtracting the best-fitting spectrum of the primary from the observed composite spectrum. These spectra are rescaled according to the obtained flux ratio and the observed spectrum of the secondary is correspondingly noisy. The bottom row compares the obtained flux ratio as a function of wavelength with that obtained from photometry. The differences seen by eye are marginal. \n\nFrom Table\\,\\ref{Tab02} and Figures\\,\\ref{Fig02} to \\ref{Fig04} we conclude as follows:\nFirst, there are only small differences in the atmospheric parameters and elemental abundances derived with the different methods for the primary.\nSecond, methods a) and c) give $T_{\\rm eff}$\\ of the secondary that agree with each other within the 1\\,$\\sigma$ error bars, but both are distinctly higher than the value expected from photometry.  The $T_{\\rm eff}$\\ derived from method b), on the other hand, is in agreement with that value.\nThird, the iron and carbon abundances of the secondary cannot be distinguished from those of the primary within the 1$\\sigma$ errors (Table\\,\\ref{Tab02}), but all three methods yield a distinctly lower [C\/H] of the primary component compared to the solar value. \nFourth, all three methods deliver flux ratios higher than the photometric ratio, as can be seen from the bottom panels in Figures\\,\\ref{Fig02} to \\ref{Fig04}.  Based on the atmospheric parameters derived with methods a) and c), RZ\\,Cas should have much smaller radii ratios (the $(R_2\/R_1)^{\\rm adj}$ in Table\\,\\ref{Tab02}) than computed. From the results of light curve analysis by \\citet{2004MNRAS.347.1317R}, however, we expect a radii ratio of 1.17 and can exclude that the radius of the secondary is larger than that of the primary. From method b), on the other hand, we obtain a value of $(R_2\/R_1)^{\\rm adj}$ that is larger than unity and not so far from 1.17.\n\n\\begin{table}\n \\tabcolsep 3.3mm\n \\caption{Elemental abundances. We list the solar values based on  \\citet{2009ARA&A..47..481A}, given as 12\\,+\\,log{(E\/H)}, and the abundances of the primary of RZ\\,Cas relative to the solar values, measured with the multiple methods.}\\label{Tab03}\n\\begin{tabular}{lcccccc}\n\\toprule\n & Solar & Method a) & Method b) & Method c) \\\\\n\\midrule\nC   &8.43 & $-0.82_{-0.20}^{+0.14}$ & $-0.80_{-0.18}^{+0.13}$ & $-0.82_{-0.18}^{+0.13}$\\vspace{1mm}\\\\\nO   &8.69 & $-0.11_{-0.34}^{+0.21}$ & $-0.22_{-0.48}^{+0.24}$ & $-0.17_{-0.39}^{+0.22}$\\vspace{1mm}\\\\\nMg  &7.60 & $-0.14_{-0.05}^{+0.05}$ & $-0.17_{-0.05}^{+0.05}$ & $-0.17_{-0.05}^{+0.05}$\\vspace{1mm}\\\\\nSi  &7.51 & $-0.20_{-0.14}^{+0.12}$ & $-0.18_{-0.14}^{+0.13}$ & $-0.19_{-0.14}^{+0.13}$\\vspace{1mm}\\\\\nCa  &6.34 & $-0.39_{-0.08}^{+0.07}$ & $-0.38_{-0.08}^{+0.08}$ & $-0.39_{-0.07}^{+0.07}$\\vspace{1mm}\\\\\nSc  &3.15 & $-0.23_{-0.09}^{+0.09}$ & $-0.21_{-0.09}^{+0.09}$ & $-0.24_{-0.09}^{+0.09}$\\vspace{1mm}\\\\\nTi  &4.95 & $-0.18_{-0.04}^{+0.04}$ & $-0.20_{-0.04}^{+0.04}$ & $-0.21_{-0.04}^{+0.04}$\\vspace{1mm}\\\\\nV   &3.93 & $-0.06_{-0.20}^{+0.15}$ & $-0.11_{-0.22}^{+0.16}$ & $-0.07_{-0.19}^{+0.15}$\\vspace{1mm}\\\\\nCr  &5.64 & $-0.35_{-0.07}^{+0.06}$ & $-0.36_{-0.06}^{+0.06}$ & $-0.36_{-0.06}^{+0.06}$\\vspace{1mm}\\\\\nMn  &5.43 & $-0.45_{-0.13}^{+0.11}$ & $-0.47_{-0.14}^{+0.12}$ & $-0.46_{-0.13}^{+0.11}$\\vspace{1mm}\\\\\nFe  &7.50 & $-0.43_{-0.06}^{+0.01}$ & $-0.42_{-0.04}^{+0.02}$ & $-0.43_{-0.02}^{+0.03}$\\vspace{1mm}\\\\\nNi  &6.22 & $-0.41_{-0.15}^{+0.12}$ & $-0.41_{-0.15}^{+0.12}$ & $-0.43_{-0.15}^{+0.12}$\\vspace{1mm}\\\\\nY   &2.21 & $-0.32_{-0.26}^{+0.19}$ & $-0.32_{-0.26}^{+0.20}$ & $-0.32_{-0.26}^{+0.19}$\\vspace{1mm}\\\\\nBa  &2.18 & $-0.45_{-0.18}^{+0.17}$ & $-0.42_{-0.18}^{+0.17}$ & \n$-0.45_{-0.17}^{+0.17}$\\\\            \n \\bottomrule                    \n \\end{tabular}                  \n\\end{table}\n\nThere is a large degeneracy between various parameters such as $T_{\\rm eff}$, $\\log{g}$, $v_{\\rm turb}$, [Fe\/H] (both components), [Mg\/H] (cool component), and radii ratio. This explains why we end up in both methods a) and c) with extraordinary small radii ratios on costs of higher $T_{\\rm eff}$\\  of the secondary (much too high when comparing with Fig.\\,\\ref{Fig01}). We assume that the number of degrees of freedom is too high when trying to optimise the radii ratio together with the other parameters; the signal from the faint companion in our composite spectra is simply too low for that. Thus we observe, although it does not give the smallest root mean square (rms) of the O-C residuals, that the most reliable results are obtained when fixing the flux ratio to the photometric ratio as we did in method b). \n\n\\section{Radial velocities and LSD profiles with LSDbinary}\\label{Sect4}\n\n\\begin{figure}\\centering\n \\includegraphics[width=\\linewidth]{Fig05.png}\n \\caption{LSD profiles computed with LSDbinary from spectra taken during primary eclipse for the primary (left) and secondary (right) of RZ\\,Cas. Phase zero corresponds to Min\\,I.}\n \\label{Fig05}\n\\end{figure}\n\nThe classical method of LSD \\citep[][]{1997MNRAS.291..658D} is based on using one line mask as a template. In the result,  a strong deconvolved line profile is obtained for the star that best matches this template, whereas the contribution of the other component is more or less suppressed. \\citet{2013A&A...560A..37T} generalised the method so that it  can simultaneously compute an arbitrary number of LSD profiles from an arbitrary number of line masks. In this work, we used the LSDbinary program written by V. Tsymbal, which computes separated LSD profiles for the two stellar components using as templates two synthetic spectra that are based on two different atmosphere models. The program also delivers optimised values of the RVs of the components and their radii ratio. A first successful application of the program to the short-period Algol \\object{R CMa} is presented in \\cite{2018A&A...615A.131L}; this work also includes a short description of the program algorithms and shows the advantages of LSDbinary against TODCOR \\citep{1994ApJ...420..806Z} in the case of very small flux ratios between the components of binary stars.\n\nWe applied LSDbinary to the spectra of RZ\\,Cas to obtain the separated LSD profiles and RVs of its components. Figure\\,\\ref{Fig05} shows LSD profiles computed from RZ\\,Cas spectra taken during the primary eclipse in 2016 as an example. The RME can clearly be seen in the profiles of the primary, as well as the RV variation due to orbital motion in the profiles of both components. We will use the obtained RVs and LSD profiles for a detailed investigation of the stellar and system parameters of RZ\\,Cas in this first part and of the pulsations of its primary component in Paper\\,II.\n\n\\section{Rotation velocity of the primary}\\label{Sect5}\n\nWe applied three different methods to determine  $v\\sin{i}$\\ of the primary of RZ\\,Cas. In all cases, we only use spectra taken in out-of-eclipse phases.\n\n\\subsection{Fourier method}\\label{Sect5.1}\nFirst, we applied the Fourier method \\citep{1933MNRAS..93..478C, 1976PASP...88..809S, 2005oasp.book.....G} to the calculated LSD profiles. This method (DFT hereafter) is based on the determination of the rotation-broadening related zero points in the Fourier power spectra of the profiles. We assumed a linear limb darkening law with a limb darkening coefficient $\\beta$\\,=\\,1.5 (or $b$\\,=\\,0.6 with $b=\\beta\/(1+\\beta)$). Varying the value of $\\beta$ leads to slight systematic effects in $v\\sin{i}$, but changes were minor compared to the differences to the results from the two other methods described below. The selection of zero points was based on a 3$\\sigma$ clipping, comparing the $v\\sin{i}$\\ from single zero points with the mean values from all zero points.\n\n\\noindent In the result, we obtained $v\\sin{i}$\\ values strongly varying with orbital phase. Figure\\,\\ref{Fig06} shows an example.  We assume that this variation comes from Algol-typical effects such as an inhomogeneous circum-primary gas-density distribution and\/or surface structures caused by the influence of mass transfer and gas stream from the secondary. In a next step, we removed all the variations found in the line profiles with the pixel-by-pixel method of FAMIAS \\citep{2008CoAst.157..387Z} by subtracting all frequency contributions computed with the mentioned program from the profiles. This method will be explained in detail in Paper\\,II. For each season, we considered a certain pixel of all LSD profiles as part of one time series from which we subtracted the found contributions and did this pixel-by-pixel to build the undistorted profiles. We performed that for all seasons but 2008 for which we did not have enough data to apply the pixel-by-pixel method. The subtraction of the high-amplitude, low-frequency contributions found with FAMIAS (most of them are harmonics of the orbital frequency) are responsible for the cleaning, not the faint high-frequency oscillations due to pulsation.  The resulting $v\\sin{i}$\\ are shown in Fig.\\,\\ref{Fig06} in red. The distribution is now much flatter and we used it to calculate the $v\\sin{i}$\\ of the corresponding season as its arithmetic mean. Results are listed in the third column of Table\\,\\ref{Tab04}.\n\n\\subsection{Single spectral line} \nNext, we looked for stronger spectral lines of the primary in the composite spectra that are free of blends from the cool companion. We found only one line consisting of the Fe\\,II doublet 5316.61\/5316.78\\,\\AA\\ that fulfils that condition. For each season, we fitted the line profiles by synthetic spectra computed with the parameters listed in Table\\,\\ref{Tab02}, method b) to determine the best-fitting $v\\sin{i}$\\ and its error. This means that we were fixing all atmospheric parameters except for  $v\\sin{i}$\\ to the solution determined from the spectra observed in 2009 when the star was in a relatively quiet phase. However, this approach does not account for the effects in active phases of RZ\\,Cas such as the attenuation of light by circumstellar material as found for the year 2001 for example \\citep{2009A&A...504..991T}. Thus we used two more free parameters in our fit, correction factor $a$ counting for different line depths caused by the presumed effects, and factor $b$ to correct the continuum of the observed spectrum in the vicinity of the Fe\\,II line.  Both were determined from a least-squares fit\n\\begin{equation}\n    \\{1-[1-a\\,P_s(v\\sin{i})]-b\\,P_o\\}^2~\\longrightarrow~min.\n\\end{equation}\nwhere $P_s$ is the synthetic and $P_o$ the observed line profile. Finally, we built the mean $v\\sin{i}$\\ per season from the weighted mean of well-selected data points using  3$\\sigma$ clipping. Figure\\,\\ref{Fig07} shows two examples: one for 2001 in which the star was in an active phase, and one for 2014 in which the $v\\sin{i}$\\ shows a much smoother behaviour with orbital phase. The results are listed in the fourth column of Table\\,\\ref{Tab04}.\n\n\\begin{figure}\\centering\n \\includegraphics[angle=-90, width=\\linewidth]{Fig06.png}\n \\caption{Values of $v\\sin{i}$\\ obtained from DFT vs. out-of-eclipse phases measured from the LSD profiles observed in 2016 (black) and from the same profiles corrected for the LPV found with FAMIAS (red).}\n \\label{Fig06}\n\\end{figure}\n\n\\begin{figure}\\centering\n \\includegraphics[width=.82\\linewidth]{Fig07.png}\n \\caption{Values of $v\\sin{i}$\\ determined from the Fe\\,II 5317\\,\\AA\\ line vs. orbital phase, shown for 2001 and 2014. The mean values were built from the values indicated in green.}\n \\label{Fig07}\n\\end{figure}\n\n\\subsection{Using FAMIAS}\nIn Paper\\,II, we will \nuse the moment methods \\citep{1992A&A...266..294A} of the FAMIAS program for a mode identification of low-$l$ degree modes. For that, we cleaned the observed LSD profiles for all low-frequency distributions in the same way as described in Sect.\\,\\ref{Sect5.1}. One free parameter in applying the moment method is the $v\\sin{i}$\\ of the primary, which optimum value and 1$\\sigma$ error we obtained from the resulting $\\chi^2$-distribution. The values are listed in the last column of Table\\,\\ref{Tab04}. The number of observations in 2008 were not sufficient to apply this method.\n\n\\subsection{Comparison}\\label{Sect5.4}\nThe $v\\sin{i}$\\ determined with the three different methods are shown in Fig.\\,\\ref{Fig08}. The values obtained from DFT and FAMIAS agree in most cases within the 1$\\sigma$ error bars; there is a larger difference for 2006. A systematic offset can be observed for the $v\\sin{i}$\\ values obtained from the Fe\\,II line. This is in particular the case for 2001 when RZ\\,Cas was in an active phase. We assume that the offset comes from the cleaning procedures that we applied when using the other two methods. In these cases we removed the low-frequency contributions that we found with the pixel-by-pixel method of FAMIAS from the LSD profiles so that we removed the line broadening effects due to orbital-phase dependent dilution effects by circumstellar material in this way. \n\n\\begin{table}\n\\tabcolsep 1.4mm\n\\caption{Values of $v\\sin{i}$\\  obtained from the three methods in multiple years. JD is JD\\,2\\,450\\,000+. The last column gives the rotation-to-orbit synchronisation factor. Here and in the following, values in parentheses are the errors in units of the last digits.}\\label{Tab04}\n\\begin{tabular}{cccccc}\n\\toprule\nYear &  JD & $v\\sin{i}$$_{\\rm DFT}$ &$v\\sin{i}$$_{\\rm FeII}$ & $v\\sin{i}$$_{\\rm FAMIAS}$ & $F_1$ \\\\\n     &   &  (km\\,s$^{-1}$)        & (km\\,s$^{-1}$)          & (km\\,s$^{-1}$) & \\\\\n\\midrule\n2001 & 2192 & 68.9(1.2) & 74.1(2.4) & 69.47(39) & 0.978(15)\\\\\n2006 & 3768 & 64.33(74) & 67.0(1.6) & 65.95(28) & 0.921(12)\\\\\n2008 & 4717 & 63.39(27) & 63.6(1.0) &  --       & 0.896(11)\\\\\n2009 & 5156 & 63.91(45) & 64.49(59) & 62.99(25) & 0.897(11)\\\\\n2013 & 6635 & 65.16(87) & 66.9(1.3) & 64.43(35) & 0.916(13)\\\\\n2014 & 6936 & 64.33(61) & 65.3(1.1) & 64.94(26) & 0.914(12)\\\\\n2015 & 7278 & 64.31(67) & 66.11(91) & 64.08(23) & 0.908(12)\\\\\n2016 & 7646 & 63.62(65) & 64.6(1.0) & 64.87(29) & 0.909(12)\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\\centering\n\\includegraphics[angle=-90, width=\\linewidth]{Fig08.png}\n\\caption{Values of $v\\sin{i}$\\  measured from a) DFT (black crosses), b) the Fe\\,II line (red squares), and c) FAMIAS (green plus signs). The solid and dotted lines show possible sinusoidal variations (see text).}\n\\label{Fig08}\n\\end{figure}\n\nFinally, we built a spline fit through the averaged values obtained from DFT and  FAMIAS and interpret this as the typical variation of $v\\sin{i}$\\ over the epoch of our observations.\nWe see that $v\\sin{i}$\\ was distinctly larger in 2001 compared to the other seasons. We assume that its value increased during the active phase of RZ\\,Cas near to 2001 because of the acceleration of the outer layers of  the primary by angular momentum transport via mass transfer from the cool component (see Sect.\\,\\ref{Sect8} for a more detailed discussion). A period search in the values averaged from DFT and FAMIAS gives a period of 9.6\\,yr, overlaid on a long-term trend (the solid line in Fig.\\,\\ref{Fig08}). A fit based on the 9.0\\,yr period as discussed in Sect.\\,\\ref{Sect7.2.3} is shown by the dotted line. The difference is marginal. The last column in Table\\,\\ref{Tab04} lists the rotation-to-orbit synchronisation factor of the primary, $F_1$, calculated from its radius, taken from \\citet{2004MNRAS.347.1317R} as 1.67\\,$\\pm$\\,0.02\\,R$_\\odot$, and the arithmetic mean of the $v\\sin{i}$\\ measured with DFT and FAMIAS. The result shows that the primary rotates sub-synchronously, only in 2001 it reaches almost synchronous rotation velocity.\n\n\\section{Orbital period changes}\\label{Sect6}\n\nIt is hard to search for orbital period changes in the range of a few seconds when the orbital RV curves are heavily distorted by non-Keplerian effects such as in the case of RZ\\,Cas  (see Fig.\\,\\ref{Fig13}). Table\\,\\ref{Tab05} lists the orbital periods and times of minimum ($T_{\\rm Min}$ hereafter) derived from the RVs in different seasons using the method of differential corrections \\citep{1910PAllO...1...33S, 1941PNAS...27..175S}. The 1$\\sigma$ errors of the period are of the order of two seconds or more, which is larger than the expected period changes.  Also the inclusion of effects such as Roche geometry of the components and spots on the stellar surfaces into the PHOEBE calculations (Sect.\\,\\ref{Sect7.2}) did not help us reach the desired accuracy in orbital period. The $T_{\\rm Min}$, on the other hand, could be very precisely determined.\n\n\\begin{table}\\centering\n\\tabcolsep 1.3mm\n\\caption{Orbital periods $P_{RV}$ and times of minimum $T_{\\rm RV}$ derived from the RVs from single seasons, and period changes $\\triangle P_{phot}$ and periods $P_{phot}$ derived from the photometric $T_{\\rm Min}$.}\n\\label{Tab05}\n\\begin{tabular}{cllrc}\n\\toprule\nYear  & \\multicolumn{1}{c}{$P_{RV}$}                  & \\multicolumn{1}{c}{$T_{\\rm RV}$} \n      & \\multicolumn{1}{c}{$\\triangle P_{phot}$} & \\multicolumn{1}{c}{$P_{phot}$}\\\\\n      & \\multicolumn{1}{c}{(d)}                    & \\multicolumn{1}{c}{BJD\\,2450000+}\n      & (sec)                                      & (d) \\\\\n\\midrule\n2001 &  1.19572(30)  & 2190.9954063(32) &    1.07 & 1.1952626\\\\\n2006 &  1.195248(20) & 3775.9076591(23) & $-$0.15 & 1.1952486\\\\\n2008 &  1.19525(26)  & 4717.7661978(51) & $-$0.03 & 1.1952500\\\\\n2009 &  1.1954(13)   & 5156.4217537(36) &    0.40 & 1.1952549\\\\\n2013 &  1.195307(65) & 6663.6333727(22) &    0.31 & 1.1952539\\\\\n2014 &  1.19533(28)  & 6936.1537115(17) &    0.45 & 1.1952554\\\\\n2015 &  1.195278(17) & 7279.1944163(12) &    0.70 & 1.1952583\\\\\n2016 &  1.19497(26)  & 7644.9432256(20) &    0.67 & 1.1952580\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\n\\includegraphics[angle=-90, width=.9\\linewidth]{Fig09.png} \n\\caption{Radial velocities of the primary in 2006 (black) and 2001 (red) vs. orbital phase. Phase is calculated from $P$ and $T_{\\rm min}$ as derived from the RVs in 2006.}\n\\label{Fig09}\n\\end{figure}\n\nWhen we plot the RVs versus orbital phase where the latter is calculated from $T_{\\rm min}$ and $P$ of one single season, we see a clear phase shift in RV compared to other seasons (Fig.\\,\\ref{Fig09}). Since we do not have any information about the behaviour of the system in between, we cannot deduce period shifts from our RVs alone, however. To fix the problem, we used the photometric $T_{\\rm Min}$ from literature. We collected data from the O-C Gateway\\footnote{http:\/\/var2.astro.cz\/ocgate\/index.php?lang=en}, Bob Nelson's Data Base of O-C Values\\footnote{w.aavso.org\/bob-nelsons-o-c-files}, $T_{\\rm Min}$ kindly provided by J.\\,Kreiner\\footnote{https:\/\/www.as.up.krakow.pl\/o-c\/} \\citep[also see][]{2004AcA....54..207K}, and unpublished data from D.\\,Mkrtichian. Cross-checking for duplicates and rejecting all visual observations, we ended up with 605 $T_{\\rm Min}$ covering the time span from 1896 to 2019, to which we added the eight $T_{\\rm Min}$ derived from our spectra. We converted all dates given as HJD to BJD based on terrestial time TT. Then we computed the overall best fitting period from a least-squares fit $T_{\\rm Min}$ versus season number $E$, yielding $P_0$\\,=\\,1.195250392(61)\\,d and $T_0$\\,=\\,2\\,453\\,775.89453(79). The resulting values\n \\begin{equation}\n     O-C = T_E-T_0-P_0E\n \\end{equation}\nare shown in Fig.\\,\\ref{Fig10}. \n\nThere exist different approaches to determine the local period from an O-C diagram. The classic method is to fit segments by linear or parabolic functions to calculate constant periods or linearly changing periods per segment, respectively. But there are also approaches that assume a continuous change of the orbital period like that of \\citet{1994A&A...282..775K}, see \\citet{2001OAP....14...91R} for an overview on other methods. We applied a similar procedure as used in \\citet{2018MNRAS.475.4745M}, interpolating the O-C data to a grid of step width one in season $E$, using spline fits together with 3$\\sigma$-clipping and computing the period change from the local slope of the resulting fit. It is\n \\begin{equation}\\label{deltaP}\n     (O-C)_E-(O-C)_{E-1} = T_E-T_{E-1}-P_0\n \\end{equation}\nwhere $T_E$\\,$-$\\,$T_{E-1}$ is the local period and thus Eq.\\,\\ref{deltaP} describes the local difference $\\triangle P$\\,=\\,$P$\\,$-$\\,$P_0$. A necessary precondition for our method is that the observed data points are sufficiently dense so that the spline fit gives a reliable prediction of the behaviour between these points. For that reason, we only considered all $T_{\\rm Min}$ observed after BJD 2\\,437\\,000.\n\n\\begin{figure}\n\\includegraphics[angle=-90, width=\\linewidth]{Fig10.png}\n\\caption{O-C values calculated from the corrected $T_{\\rm Min}$, full range in JD.}\n\\label{Fig10}\n\\end{figure}\n \n\\begin{figure}\n\\includegraphics[width=\\linewidth]{Fig11.png}\n\\caption{O-C values and derived period changes. Top: O-C values (filled black circles) fitted by splines (red line). Values considered as outliers are shown by open circles and those derived from our spectra in green. Bottom: Period changes calculated from the local slope of the spline fit (see text). Open circles indicate the seasons of our spectroscopic observations.}\n\\label{Fig11}\n\\end{figure}\n\nFigure\\,\\ref{Fig11} shows in its top panel the O-C values together with the spline fit. The $T_{\\rm Min}$ obtained from our RVs ($T_{\\rm RV}$ in Table\\,\\ref{Tab05}) are shown in green and fit very well. The bottom panel shows the resulting period changes by the solid line, where we assumed a ``best-selected'' smoothness parameter for the underlying spline fit. To give an impression of the influence of the smoothness parameter, we also show (dotted line) the period changes resulting from a much larger parameter, leading to a  more relaxed spline fit. The positions of our spectroscopic observations are indicated. Obtained period changes and periods are listed in Table\\,\\ref{Tab05} as $\\triangle P_{LC}$ and $P_{LC}$. \n\n\\citet{2018MNRAS.475.4745M} derived typical timescales of 4.8, 6.1, and 9.2\\,yr from the O-C variations. We did a frequency search in the obtained period changes using the PERIOD04 program \\citep{2005CoAst.146...53L} and found the six periods listed in Table\\,\\ref{Tab06}. We do not assume that the observed variation can be described as strictly cyclic but consider the found periods as typical timescales that describe the behaviour in certain seasons. Figure\\,\\ref{Fig12} illustrates this. It shows the calculated period changes together with the best fit of the six periods and also single contributions from four of the six periods. The longest period of 52.7\\,yr describes a long-term trend in the data. The 6.3\\,yr period describes the variations before BJD 2\\,442\\,560 (segment A), and the 8.6\\,yr period the behaviour between 2\\,445\\,240 and 2\\,456\\,070 (segment B). The 14.5\\,yr period is responsible for an amplitude variation over a longer timescale. The remaining two periods of 6.9 yr and 10.8\\,yr cannot directly be linked to the variations in this way, but improve the fit by counting for their non-cyclicity. When doing a separate frequency search in segments A and B, we find the dominant periods as 5.9\\,yr and 8.4\\,yr, respectively. This shows that we can only estimate the order of the time scales underlying the orbital period variations but cannot use the errors obtained from frequency search as a measure for the accuracy of the obtained values.\n\n\\begin{figure}\\centering\n\\includegraphics[angle=-90, width=\\linewidth]{Fig12.png}\n\\caption{Calculated period changes (solid line) fitted by six frequencies (dashed line). The dotted lines show (from top to bottom) the contributions of the 52.7, 6.3, 8.6, and 14.5\\,yr periods.}\n\\label{Fig12}\n\\end{figure}\n\n\\begin{table}\n\\tabcolsep 2.65mm\n\\caption{Timescales and amplitudes of orbital period variation.}\n\\label{Tab06}\n\\begin{tabular}{lrrrrrr}\n\\toprule\n$P$ (y)  & 6.32 & 6.94 & 8.60 & 10.78 & 14.55 & 52.70\\\\\n$A$ (sec)& 0.34 & 0.37 & 0.37 &  0.24 &  0.26 &  0.45\\\\\n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{figure*}\\centering\n\\includegraphics[width=0.76\\textwidth]{Fig13.png}\n\\caption{Radial velocity residuals. Left: the Rossiter-McLaughlin effect in the RVs of the primary in different years (red) after subtracting the best-fitting Keplerian orbits. For comparison, the RVs from 2006 are shown in black. In 2013, the RVs are divided into BJD\\,<\\,2\\,456\\,600 (2013a, red) and BJD\\,>\\,2\\,456\\,600 (2013b, green), and in 2015 into BJD\\,<\\,2\\,457\\,250 (2015a, red) and BJD\\,>\\,2\\,457\\,250 (2015b, green). Right: The same for the RVs of the secondary, shown over a larger range in orbital phase. Phase zero corresponds to Min\\,I in each case.}\n\\label{Fig13}\n\\end{figure*}\n\n\\section{Radial velocity variations with orbital phase}\n\\subsection{Keplerian approach}\\label{Sect7.1}\n\nFigure\\,\\ref{Fig13} shows the RVs folded with the orbital period after subtracting the best-fitting Keplerian orbits computed with the method of differential corrections as mentioned in Sect.\\,\\ref{Sect6}. We use the observations from 2006 when RZ\\,Cas was in a quiet state for comparison, shown by black dots. The deviations from a straight line (pure Keplerian motion) of the RVs of the secondary seen in 2006 are due to its non-spherical shape and inhomogeneous surface intensity distribution as discussed in \\citet{2009A&A...504..991T}, which we investigate in detail in Sect.\\,\\ref{Sect7.2}. More or less strong deviations from the behaviour in 2006 can be seen in different years. \n\nLooking at the behaviour of the RVs of the secondary, we find the strongest deviations in 2001 where we see large variations with orbital phase. Almost no differences to 2006 are observed in 2009, 2015, and 2016, whereas moderate differences occur in 2013 and 2014. Interpreting the strength of the deviations as activity indicator, we conclude that we observed RZ\\,Cas in or just after a mass-transfer episode in 2001 and in a quiet state in 2006 (as already stated in \\citet{2008A&A...480..247L}, and \\citet{2009A&A...504..991T}), and that this quiet phase continued until 2009, followed by a slightly more active phase around 2013 and 2014, and falling back into a quiet state in 2015 and 2016.\n\nLooking at the behaviour of the RVs of the primary, in particular at the amplitude and shape of the RME, we see a different picture. The amplitude of the RME in 2001 is much larger and its shape is strongly asymmetric. In all the other years, however, we see almost no difference to 2006, except for three nights of observation covering the ingress of the eclipses: one in 2009, one in 2013, and one in 2015. \n\n\\subsection{Analysis with PHOEBE}\\label{Sect7.2}\n\nRZ Cas is a semi-detached Algol-type binary system and its cool component fills its critical Roche lobe. Tidal distortions occur and lead to non-spherical shapes. According to \\citet{2013MNRAS.431.2024S}, non-negligible effects on RVs occur for $a<20\\,(R_1+R_2)$, that is when the separation of the components is smaller than 20 times the sum of their radii. Thus, the a priori assumption of Keplerian orbit, which considers the mass centre of the stellar disc coinciding with its intensity centre, is no longer valid. As already mentioned in previous sections, RZ Cas is undergoing stages of active mass transfer. A hot region around the equatorial belt has long been known \\citep{1982ApJ...259..702O}. Moreover, the recent comprehensive tomographic study of \\citet{2014ApJ...795..160R} revealed clear indicators of mass stream activity in several short period Algols, including RZ Cas. \\citet{1994PASJ...46..613U} found that the secondary of RZ\\,Cas can be modelled only when assuming an unusually large value of the gravity darkening exponent of 0.53 and inferred that dark spots are present on the front and back sides of the secondary with respect to the primary. These authors supposed that quasi-radial flow in the sub-adiabatic stellar envelope from the deep interior is the cause of darkening. \\citet{2008ysc..conf...33T, 2009A&A...504..991T} confirmed the finding of the two spots, resuming the interpretation. \\citet{2018MNRAS.481.5660D} and \\citet{2018A&A...611A..69B}, on the other hand, give an alternative explanation, showing for the short-period Algols \\object{$\\delta$~Lib} and \\object{$\\lambda$~Tau} that the mass stream can produce a light scattering cloud in front of the surface of the Roche lobe filling secondary facing the primary.\n\n\\subsubsection{Method}\n\nTo account for all these effects, we need to use sophisticated models to simulate RV changes during the whole orbital motion via integrating both components surface intensities. For this purpose, we used the well-known Wilson-Devinney code \\citep{1971ApJ...166..605W, 2005Ap&SS.296..121V} through the PHOEBE interface \\citep{2005ApJ...628..426P}. The WD code consists of light and\/or RV curve synthesiser (LC) and parameter optimiser (DC) for fitting purpose. As  we describe later, our attempts to optimise the multi-parameter fit failed with DC, which is expected for such complex configurations. Thus we used LC via our Markov Chain Monte Carlo (MCMC hereafter) optimiser written in Phoebe-scripter extension, to both optimise the parameters and explore the parameter space \\citep[see][]{2018MNRAS.481.5660D}.\n\nIn MCMC runs, we start from a random initial parameter set and add accepted parameters to the Markov chain. We need to run a lot of chains (or \"walkers\") to avoid the issue of one Markov chain sticking in a local $\\chi^2$ valley. That is why it is advisable to start the simulation with at least two times the number of walkers of the number of parameters to be optimised \\citep[see][]{2013PASP..125..306F}. For our last sets of simulation, we set the number of walkers > 20 and the lowest iteration numbers are dynamically increased to fulfil the condition by \\citet{2013PASP..125..306F} that the iteration number should be at least ten times the auto-correlation time (i.e. typically 500,000 iterations per season). \n\n\\subsubsection{Application}\n\nSpots are described in WD by two coordinates, co-latitude $\\delta$ and longitude $\\lambda$, and two physical parameters, temperature ratio compared to the normal surface $T$ and radius $R$. In our initial run, we started to model the RVs of the cool component by taking one spot (Spot1) on the cool secondary facing the primary into account to mimic a scattering cloud or diffusive material between the components. All of our models converged to spot position $\\delta\\approx 90^\\circ$, $\\lambda\\approx 0^\\circ$, that is the region that faces the primary. However, we found a strong correlation between spot size and temperature ratio of the form of $R^2T^4$\\,=\\,constant, balancing a lower temperature ratio by a more extended spot. Thus, we could only fit the \"strength\" or \"contrast\" of the spot with respect to the stellar surface, fixing the temperature ratio to a reasonable lower limit of 0.76. As mentioned at the beginning of Sect.\\,\\ref{Sect7.2}, \\citet{1994PASJ...46..613U} suggested dark spots at the front and back sides of the cool secondary towards the primary caused by mass transfer. We therefore implemented a second spot (Spot2) on the opposite side of the surface of the secondary to check if this offers further improvement.\n\nTo check if we can detect a variation of the filling factor between active and inactive phases of RZ\\,Cas, we used the \"unconstrained mode \" of WD and the surface potential of the secondary as a free parameter. In all of our runs, however, the filling factor of the cool component converged to unity. So for the final analysis, we fixed it to unity and used the \"semi-detached mode\". Eccentricity was set to zero and the orbital inclination to 82$^\\circ$. The temperature ratio was fixed to 0.76 for both spots on the secondary and, as already mentioned, the position of Spot1 to $\\delta_1$\\,=\\,90$^\\circ$, $\\lambda_1$\\,=\\,0$^\\circ$. The location of Spot2 converged to $\\delta_2$\\,=\\,90$^\\circ$ without showing larger scatter, so we fixed $\\delta_2$ as well.\n\n\\subsubsection{Results}\\label{Sect7.2.3}\n\nFree parameters were the synchronisation parameter for the primary, $F_1$, the radii of Spot1 and Spot2 on the secondary, $R_1^{sec}$ and $R_2^{sec}$, the longitude of Spot2, $\\lambda_2^{sec}$, and the systemic velocity $V_\\gamma$. The RV semi-amplitudes were allowed to vary within the error bars derived in previous attempts and led to the values as listed in Table\\,\\ref{Tab07}. \n\n\\begin{table}\\centering\n\\tabcolsep 4.1mm\n\\caption{Basic absolute values derived with PHOEBE.} \n    \\label{Tab07}\n    \\begin{tabular}{llll}\n    \\toprule  \n    $q$            & 0.350782(47) & $M_1$ (M$_\\sun$) & 1.9507(54)\\\\\n    $a$ (R$_\\sun$) & 6.5464(54)   & $M_2$ (M$_\\sun$) & 0.6843(13) \\\\\n    \\bottomrule\n    \\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\tabcolsep 1.45mm\n\\caption{Synchronisation factor of the primary, radii of both spots on the secondary, longitude of Spot2 on the secondary, and mean scatter of the RV residuals.}\\label{Tab08}\n\\begin{tabular}{lccccc}\n\\toprule\nSeason & $F_1$ & $R_1^{sec}$ & $R_2^{sec}$ & $\\lambda_2^{sec}$ & rms\\\\\n       &       & \\small (deg)& \\small (deg)& \\small (deg)      & \\small (km\\,s$^{-1}$)\\\\\n\\midrule\n2001  & 1.217(20) & $23.09_{-1.40}^{+0.80}$  & $22.8_{-2.2}^{+2.1}$ & $136.7_{-4.1}^{+3.9}$& 5.31\\vspace{1mm}\\\\\n2006  & 0.924(13) & $41.04_{-0.92}^{+0.92}$  & $10.6_{-2.5}^{+1.9}$ & $150  _{-20 }^{+26 }$& 2.47\\vspace{1mm}\\\\\n2009  & 1.006(11) & $36.55_{-0.74}^{+0.74}$  & $10.8_{-1.7}^{+1.4}$ & $244.4_{-6.1}^{+6.6}$& 1.62\\vspace{1mm}\\\\ \n2013a & 0.897(08) & $28.69_{-1.08}^{+0.64}$  & $24.0_{-1.0}^{+0.9}$ & $180.0_{-2.4}^{+2.4}$& 3.46\\vspace{1mm}\\\\\n2013b & 0.991(14) & $28.01_{-1.09}^{+0.99}$  & $23.7_{-1.2}^{+1.2}$ & $218.4_{-2.9}^{+3.1}$& 3.60\\vspace{1mm}\\\\  \n2014  & 0.812(12) & $33.16_{-0.84}^{+0.96}$  & $21.3_{-1.2}^{+1.2}$ & $209.4_{-3.1}^{+3.3}$& 2.92\\vspace{1mm}\\\\  \n2015a & 1.057(13) & $42.51_{-0.72}^{+0.66}$  & $19.2_{-1.9}^{+1.8}$ & $194.4_{-3.2}^{+3.6}$& 2.71\\vspace{1mm}\\\\  \n2015b & 0.782(11) & $41.53_{-0.54}^{+0.69}$  & $14.3_{-1.0}^{+0.9}$ & $180.0_{-3.0}^{+3.2}$& 2.46\\vspace{1mm}\\\\\n2016  & 0.826(10) & $42.71_{-0.73}^{+0.63}$  & $16.2_{-1.1}^{+1.1}$ & $124.5_{-3.3}^{+3.0}$& 2.26\\\\ \n\\bottomrule\n\\end{tabular}\n\\end{table}\n\n\\begin{figure}\\centering\n\\includegraphics[width=\\linewidth]{Fig14.png}\n\\caption{Residuals of the RVs of the secondary after subtracting the best-fitting PHOEBE solutions without including spots (black), including one spot (red), and including two spots (green) on the secondary, obtained (from top to bottom) for seasons 2013a, 2015b, and 2016.}\\label{Fig14}\n\\end{figure}\n\n\\begin{table}\\centering\n\\caption{Absolute values of positive and negative RV deviations due to RME.}\n\\begin{tabular}{lccc}\n    \\toprule\nSeason& JD & $K_p$ (km\\,s$^{-1}$)& $K_n$ (km\\,s$^{-1}$)\\\\\n    \\midrule\n2001  & 2\\,452\\,192 & 26.5 & 32.4\\\\\n2006  & 2\\,453\\,768 & 19.9 & 22.4\\\\\n2009  & 2\\,455\\,156 & 24.7 & 21.0\\\\\n2013a & 2\\,456\\,584 & 22.6 & 21.7\\\\\n2013b & 2\\,456\\,672 & 26.4 & 20.0\\\\\n2014  & 2\\,456\\,936 & 19.3 & 19.7\\\\ \n2015a & 2\\,457\\,238 & 24.6 & 19.8\\\\\n2015b & 2\\,457\\,293 & 17.5 & 19.8\\\\\n2016  & 2\\,457\\,646 & 19.1 & 19.0\\\\\n\\bottomrule\n    \\end{tabular}\n    \\label{Tab09}\n\\end{table}\n\nIn Fig.\\,\\ref{FigA1}, we show the corner plot for season 2009. This shows that MCMC not only gives the most optimal parameter set, it also provides an error estimation on each parameter and possible correlations between the parameters. Finally obtained values are listed in Table\\,\\ref{Tab08}, in which we also included the mean scatter calculated from the RV residuals after subtracting the PHOEBE solutions obtained for each season. These residuals are shown in Fig.\\,\\ref{FigA2}. \n\nFigure\\,\\ref{Fig14} shows the influence of the inclusion of spots on the secondary into the model. We selected three seasons as examples. Including spots on the secondary did not effect the modelled RVs of the primary and vice versa. Thus, we only show the residuals of the RVs of the secondary. The black dots corresponds to the best-fitting solutions without spots, that is when only the non-spherical shapes of the components (in particular of the secondary) are taken into account. The RVs show a systematic deviation around Min\\,II that is distinctly reduced when including Spot1 that faces the primary, resulting in the red dots. The RVs show smaller, non-systematic deviations around Min\\,I, which is clearly present in the data from 2013a, small in 2016, and almost not visible in 2015b. Adding the second spot on the opposite side reduces this scatter, as shown by the green dots, but in some cases the reduction is marginal.\n\nOne explanation for the asymmetry observed in the RME in the years 2009, 2013b, and 2015a (see Fig.\\,\\ref{Fig13}) could be the impact of a hot spot on the primary on its RVs. For comparison, we therefore additionally included a hot spot on the primary for\nthese seasons, taking three more free parameters (temperature ratio, radius, and longitude) into account. The fit did not improve the solutions, however. \n\nTable\\,\\ref{Tab09} lists absolute values of positive and negative amplitudes of the RME (maximum deviations in RV during ingress and egress, respectively) obtained after subtracting the best-fitting PHOEBE solution for $F_1$\\,=\\,0, that is for a non-rotating primary, from the observed RVs. Panels h) and i) in Fig.\\,\\ref{Fig15} show that the positive amplitudes reveal a similar behaviour as $F_1$ or $R_2$, whereas the negative amplitudes show a systematically decreasing trend with time (see next Section for a discussion). \n\n\\begin{figure}\n\\includegraphics[width=\\linewidth]{Fig15.png}\n\\caption{Time variations of different parameters (see text). The solid lines are calculated from the best-fitting periods and long-term trends. The dotted lines show possible sinusoidal variations with a period of 9\\,yr plus a long-term trend, except for panel f), which shows an 18 y variation. Values from 2013b and 2015a were considered as outliers.}\\label{Fig15}\n\\end{figure}\n\n\\begin{table}\\centering\n\\caption{Timescales in years determined from the seasonal variations of different parameters.}\n\\begin{tabular}{ccccccccc}\n\\toprule\nd$P$ & $v\\sin{i}$ & $F_1$ & $R_1^{sec}$ & $R_2^{sec}$ & $\\lambda_2^{sec}$ & rms & $K_p$ & $K_n$\\\\\n\\midrule\n8.6 & 9.6 & 8.4 & 9.4 & 8.8 & 16.7 & 9.3 & 8.5 & 9.6\\\\\n\\bottomrule\n\\end{tabular}\\label{Tab10}\n\\end{table}\n\nFigure\\,\\ref{Fig15} shows the seasonal variations of all investigated parameters. In panel a) we added the period change derived from the O-C values (cf. Sect.\\,\\ref{Sect6}), in panel b) the $v\\sin{i}$\\ values from the fit determined in  Sect.\\,\\ref{Sect5.4}, and in panel g) the mean scatter of the RV residuals after subtracting the best-fitting PHOEBE solutions from the input data (see Fig.\\,\\ref{FigA2}). All variations can be described by a sinusoid plus a long-term trend. Because the seasonal sampling was not sufficient to determine a second frequency describing the long-term trend, we fixed it to $10^{-6}$\\,c\\,d$^{-1}$\\ and determined only amplitudes and phases. Table\\,\\ref{Tab10} lists the timescales derived from the best-fitting sinusoids.\n\nThe best-fitting curves are shown in Fig.\\,\\ref{Fig15} by solid lines. We see that $F_1$, $R_2$, rms, and $K_p$ vary almost in phase, whereas $R_1$ varies in anti-phase. Moreover, $v\\sin{i}$\\ and $K_p$ show almost identical shapes of variation. Searching for a possible common period that explains the variations of all parameters, we found a period of 9 years that best fits, together with the mentioned long-term trend, the variations of all parameters except for $\\lambda_2^{sec}$ that can be fitted by a period of 18\\,years, twice the period of 9\\,years. For d$P$ we had to add a third (optimised) period of 6.3\\,yr. The resulting fits are shown in Fig.\\,\\ref{Fig15} by  dotted lines. In all cases, the quality is comparable to that obtained from the optimised values listed in Table\\,\\ref{Tab10}.\n\n\\section{Discussion}\\label{Sect8}\n\nOur spectral analysis yields atmospheric parameters of the components that are in agreement with the results of light curve analysis by \\citet{2004MNRAS.347.1317R}. For the first time, we determine the elemental surface abundances. For both components, we find [Fe\/H] close to $-0.4$, as for Ca, Cr, Mn, and Ni of the primary, whereas O, Mg, Si, Sc, Ti, and V show relative abundances between $-0.2$ and $-0.1$. For the carbon abundance of the primary, we find [C\/H]\\,=\\,$-0.80^{+0.13}_{-0.18}$, which is remarkably different from the other elements. We think that the difference is significant. First, the considered spectral range shows a sufficiently large number of C\\,I lines of the primary (SynthV computes line depths >5\\% for 32 rotationally unbroadened C\\,I lines). Second, differences between the fits with [C\/H] of $-0.4$ and $-0.8$ can clearly be seen by eye. Figure\\,\\ref{Fig16} shows this for the strongest carbon lines in the H$\\beta$ region of the spectrum of the primary (observed composite spectrum after subtracting the synthetic spectrum of the secondary). In the spectrum of the cool secondary, carbon is mainly present in form of the CH molecule bands. Compared to the primary, the signal is very weak and we can only say that [C\/H] is below solar abundance, between $-0.3$ and $-1.0$ within the 1$\\sigma$ error bars. \n\n\\citet{2008A&A...486..919M} tried to model the binary evolution of RZ\\,Cas and found consistent solutions only for an initial mass ratio of $q$\\,$\\approx$\\,3, which is about the inverse of the actual ratio. From the large initial mass of the donor in that case we can assume that the primary has switched during its evolution from pp-chain to CNO-cycle hydrogen burning, resulting in depleted carbon and enhanced nitrogen abundances in its core \\citep[e.g.][]{2010A&A...517A..38P}. From the reversal of mass ratio we conclude that the donor was stripped by mass loss down to its core so that the gainer accreted  CNO-cycled, carbon-deficient material in the late phase of its evolution. This material has then mixed with the surface layers of the gainer, leading to the observed carbon abundance. In that case, the surface carbon abundance of the gainer cannot be lower than that of the donor star, however. Multiple authors have used the sketched scenario to explain the surface abundance anomalies observed in several other Algol-type stars; these include \\citet{2012MNRAS.419.1472I} for \\object{GT~Cep}, \\object{AU~Mon}, and \\object{TU~Mon}, \\citet{2014MNRAS.444.3118K} for \\object{u~Her}, and \\citet{2018MNRAS.481.5660D} for $\\delta$\\,Lib.\n\n\\begin{figure}\n\\includegraphics[angle=-90, width=\\linewidth]{Fig16.png}\n\\caption{Fit of the observed spectrum of the primary (black) in the H$_\\beta$\\,region by synthetic spectra with [C\/H] of 0.0 (blue), $-0.4$ (green), and $-0.8$ (red).}\n\\label{Fig16}\n\\end{figure}\n\n\\begin{figure*}\n\\includegraphics[width=.5\\linewidth]{Fig17a.png}\n\\includegraphics[width=.5\\linewidth]{Fig17b.png}\n\\caption{Logarithmic plots of gas density distributions (in $10^{11}$\\,cm$^{-3}$) obtained from 3D hydrodynamic simulations  of the RZ\\,Cas system (Nazarenko \\& Mkrtichian, priv. comm.)  for  mass-transfer rates of 1\\,$\\times$\\,$10^{-9}$\\,M$_\\odot$y$^{-1}$ (left) and 6\\,$\\times$\\,$10^{-8}$\\,M$_\\odot$y$^{-1}$ (right). The viewing angles at first ($\\phi$\\,=\\,$-0.1$) and last ($\\phi$\\,=\\,$+0.1$) contact are indicated.}\n\\label{Fig17}\n\\end{figure*}\n\nOur further investigation was based on the separated LSD profiles and RVs of the two components of RZ\\,Cas computed with LSDbinary, and on the times of minimum (O-C values) taken from literature. The change in orbital period computed from the slope of the O-C diagram cannot be characterised by one single timescale, as already found by \\citet{2018MNRAS.475.4745M}. These authors derived timescales of 4.8, 6.1, and 9.2 years. We observe from our slightly extended data set different timescales of variation in different seasons, such as 6.3\\,yr around JD\\,2\\,440\\,000, and 8.6\\,yr from 2\\,450\\,000 to 2\\,455\\,000 (which is about the time span of our spectroscopic observations), overlaid by longer periods. From our previous analysis using the Shellspec07\\_inverse code \\citep{2009A&A...504..991T}, we know that RZ\\,Cas was in an active state of mass transfer in 2001 and in a quiet state in 2006. Comparison of the RV residuals after subtracting a pure Keplerian orbit with those from 2006 gave us initial hints pointing to further activity periods (Fig.\\,\\ref{Fig13}). From the RVs of the primary we see that the RME was distinctly enhanced only in 2001. A comparison of the RV residuals of the secondary with those from 2006 shows the largest deviation in 2001, weaker deviations in 2013 and 2014, and almost no deviations in 2009, 2015, and 2016. The amount of scatter found in the RV residuals after subtracting the best-fitting PHOEBE solutions is strongly correlated with this finding. Thus, we conclude that no further mass-transfer episode as strong as in or close to 2001 occurred in RZ\\,Cas.\n\nThe modelling of the observed RVs of both components with MCMC-PHOEBE gave the best results when adding two dark spots on the surface of the cool companion: Spot1 facing the primary, Spot2 on the opposite side.  Spot1 was already found from the RVs from 2001 and 2006 by \\citet{2009A&A...504..991T}, whereas \\citet{1994PASJ...46..613U} included two spots into their model, explaining the observed  anomalous gravity darkening by a cooling mechanism by enthalpy transport due to mass outflow that leads to a reverse process of gravitational contraction. \n\nWe monitored both spots for the first time over decades.\nWe find that Spot1 always exactly points towards the primary, with radii (a synonym for strength or contrast as mentioned in before) between 23$^\\circ$ and 43$^\\circ$. Spot2, on the other hand, is much weaker, shows a variation in its position, and induces only a second order improvement in some of the observed seasons (cf. Fig.\\,\\ref{Fig14}). The main findings are that the strengths of the two spots vary in anti-phase and that Spot2 shows different positions in longitude, varying around the longitude of the L2 point of 180$^\\circ$.\n\nThe fact that the strength of Spot1 is largest when the star is between 2006 and 2009 in a quiet phase (Fig.\\,\\ref{Fig15}) speaks against the cooling by mass-outflow mechanism. Instead, we argue that the variations of the strengths of both spots can be explained by magnetic activity of the cool companion, assuming an activity cycle of 9 years, based on an 18-year cycle of magnetic field change, including a reversal of the magnetic poles. The 9-year cycle can be found in the variations of all investigated parameters except for the longitude of Spot2, as we showed in the last section, and was also found by \\citet{2018MNRAS.475.4745M}. Spot2 shows a migration in longitude, returning after an 18-year cycle to its position from 2001 (cf. Fig.\\,\\ref{Fig15}). We assume that we observe similar surface structures on the cool secondary of RZ\\,Cas such as for cool, rapidly rotating RS\\,CVn binaries or single rapidly rotating variables of FK Com and BY Dra type. Long-living active regions were observed on opposite sides of these stars. The spots are of different intensity and also show switching of activity from one spot to the other on timescales of years or decades, which is known as the flip-flop effect \\citep[see][and references herein]{2018A&A...613A...7Y}.\n\nWe believe that there is a  direct analogy with sunspot activity. Sunspots show a saucer-shaped depression in the photosphere caused by the Lorentz force of the strong magnetic field within the spot, the so-called Wilson depression \\citep{1774RSPT...64....1W}. This means that inside the spot the level of $\\tau$\\,=\\,1 is located below the level of the photosphere outside the spot. For the Sun, the geometric depth of the depression is of the order of 600\\,km \\citep[e.g.][]{1972SoPh...26...52G, 2018A&A...619A..42L}. For the Roche-lobe filling donor, the existence of such atmospheric depression close to the L1 point means that L1 is \"fed\" by atmospheric layers of lower density and thus the mass transfer is the more suppressed the deeper the depression is. The local magnetic field strength controls the strength (depression) of Spot1 and the height of atmospheric layers feeding the L1 point and in this way the mass-transfer rate. RZ\\,Cas showed, in perfect agreement with the drafted scenario, high mass-transfer rate in 2001 (and possibly in 2011, when observations are missing) when the Spot1 size was around 20 degrees and low activity state in 2006-2007 and in 2015-2016 when the spot size was about 40 degrees.\n\nThe explanation for the asymmetry and different amplitudes observed in the RME in the years 2001, 2009, 2013b, and 2015a (see Fig.\\,\\ref{Fig13}) could be the impact of the combined effect of acceleration of the photospheric layers in 2001 caused by the high mass-transfer rate and by screening the surface of the primary by the dense gas stream \\citep{2008A&A...480..247L}. The amplitudes of the RME in RZ\\,Cas in general show very different behaviour during ingress ($K_p$) and egress ($K_n$) of primary eclipse. The variation of $K_p$ is similar to those of $R_2$ and rms, whereas the shape of the $K_n$ variation perfectly matches that of $v\\sin{i}$\\ (see Fig.\\,\\ref{Fig15}). We assume that $K_n$ is related to the true $v\\sin{i}$\\ seen outside the eclipses and $K_p$ is strongly influenced by mass-transfer effects. This can be explained from Fig.\\,\\ref{Fig17}, showing the gas density distribution around RZ\\,Cas for two different mass-transfer rates, calculated from 3D-hydrodynamic simulations by Nazarenko \\& Mkrtichian (priv. comm.). It can be seen that the equatorial zone of the surface of the primary is masked during ingress (at orbital phases $\\phi$\\,=\\,$-0.1..0$) by the optical thick gas stream from the secondary, whereas we directly see the complete disc of the primary during egress ($\\phi$\\,=\\,$0..0.1$), only very slightly hampered by optical thin circumbinary matter. In consequence, $K_p$ is affected by the seasonal varying density of the gas stream (or mass-transfer rate) and $K_n$ is correlated with the surface rotation velocity of the primary. All the variability seen in RME in the years from 2006 on is mainly related to the variations of $K_p$, forced by changes in gas-stream density and variable screening and attenuation effects, while the rotation speed was nearly constant. \n\nThe synchronisation factor derived with PHOEBE is mainly based on the shape of the RME and this shape is strongly influenced by timely varying Algol-typical effects (see paragraph below). Our MCMC simulation did not count for the observed asymmetry in the RME either (i.e. the fact that the amplitudes $K_p$ and $K_n$ are different from each other) and so the synchronisation factor $F_1$ has to be considered as some kind of mean value. The shape of its time variation resembles the mean of the shapes of the $K_p$ and $K_n$ variations (cf. Fig.\\,\\ref{Fig15}). The results obtained for the synchronisation factor in Sect.\\,\\ref{Sect5.4} from radius and $v\\sin{i}$\\ of the primary, on the other hand, give sub-synchronous rotation of the primary. Its outer layers are only accelerated to almost synchronous rotation during the active phase\nin 2001. The finding of sub-synchronous rotation is surprising. It was suggested by \\citet{2014A&A...570A..25D} to occur during the rapid phase of mass exchange in Algols, when the donor star is spun down on a timescale shorter than the tidal synchronisation timescale and material leaving the inner-Lagrangian point is accreted back onto the donor, enhancing orbital shrinkage. But these authors state that once the mass-transfer rate decelerates and convection develops in the surface layers, tides should be effective enough to re-synchronise the primary. To our knowledge, sub-synchronous rotation was found so far in only one short-period Algol, namely TV\\,Cas, by \\cite{1992A&A...257..199K}, in which the authors found synchronisation factors of 0.85 for the primary and 0.65 for the secondary component.\n\nThe RV residuals after subtracting the solutions obtained with our MCMC simulations (Fig.\\,\\ref{FigA2}) finally show that our model distinctly reduces the scatter compared to the residuals obtained from pure Keplerian motion. On the other hand, we can still recognise many of the signs of activity that we discussed in connection with Fig.\\,\\ref{Fig13}. We assume that  these still unexplained features are due to the  distribution and density of circumbinary matter along the line of sight in different orbital phases, varying between different seasons according to the varying activity of the star. Finally, we can conclude from Fig.\\,\\ref{FigA2} in the same way as from Fig.\\,\\ref{Fig13} that RZ\\,Cas showed an extraordinary phase of activity around 2001 followed by a quiet phase in 2006 to 2009, slightly enhanced activity in 2013\/2014, followed by a quiet phase again. The calculated fits on the variation of the radii of the two spots and the RME amplitude $K_p$ based on the nine-year cycle, on the other hand, show extrema of the same amplitude such as in 2001 for the time around 2010-2011. Therefore, because of missing data in this period, we cannot exclude the possibility that a second mass-transfer episode of comparable strength like in 2001 occurred close to 2010-2011.\n\nThe orbital period was at maximum shortly after 2001, dropped down steeply until 2006, and was then more or less continuously rising. From Eq.\\,8 in \\citet{1973A&A....27..249B}, assuming conservative mass transfer and conservation of orbital angular momentum, we obtain a mass-transfer rate of 1.5\\,$\\times\\,10^{-6}$\\,M$_\\odot$\\,yr$^{-1}$. This is a typical value observed for Algol-type stars \\citep[e.g.][]{1976IAUS...73..283H} and also agrees \\citet{1976AcA....26..239H}, who calculate, from an O-C analysis, a mean mass-transfer rate of RZ\\,Cas of 1.0\\,$\\times\\,10^{-6}$\\,M$_\\odot$\\,yr$^{-1}$, averaged over several episodes of period change. As mentioned in the introduction, evolutionary scenarios for RZ\\,Cas have to count for mass loss from the system \\citep{2008A&A...486..919M}, which may be the case for Algols in earlier stages of their evolution in general \\citep[e.g.][]{2006MNRAS.373..435I, 2013A&A...557A..40D, 2015A&A...577A..55D}. Thus, conservatism is not necessarily justified and the assumption of mass conservation can only lead to a raw estimation of the amount of mass transferred during the active phase of RZ\\,Cas in 2001.\n\nWe assume that the enhanced value of $v\\sin{i}$\\ in 2001 points to an acceleration of the outer layers of the primary of RZ\\,Cas due to the mass-transfer episode occurring close to that year. This then dropped down by about 5\\,km\\,s$^{-1}$\\ and it could be that its slight increase after 2009 is correlated with a slightly increased activity as indicated by the variation of $F_1$ and rms.\n\n\\section{Conclusions}\\label{Sect9}\n\nIn this first of three articles related to spectroscopic long-term monitoring of RZ\\,Cas, we investigated high-resolution spectra with respect to stellar and system parameters based on the RVs and LSD profiles of its components calculated with the LSDbinary program. The main goal was to search for further episodes of enhanced mass transfer occurring after 2001 and for a general timescale of variations possibly caused by the magnetic cycle of the cool companion.\n\nFrom spectrum analysis we determined precise atmospheric parameters, among them low metal surface abundances, in particular [Fe\/H] of  $-0.42$ and [C\/H] of $-0.80$ for the primary and [Fe\/H] of the same order for the secondary. The carbon deficiency observed for the primary gives evidence that the outer layers of the cool secondary have been stripped in the fast mass-transfer phase down to its core so that CNO-cycled material was transferred to the outer layers of the primary in later stages of evolution. The derived $T_{\\rm eff}$\\ of the components and the $\\log{g}$\\ of the primary agree within the 1$\\sigma$ error bars with the results from LC analysis by \\citet{2004MNRAS.347.1317R}. From the RV analysis with PHOEBE, we derived very precise masses and separation of the components of $M_1$\\,=\\,1.951(5)\\,M$_\\odot$, $M_2$\\,=\\,0.684(1)\\,M$_\\odot$, and $a$\\,=\\,6.546(5)\\,R$_\\odot$.\n\nFrom several of the investigated parameters that show seasonal variations, such as orbital period, $v\\sin{i}$, strength of the spots on the secondary, synchronisation factor calculated with PHOEBE, and rms of the RV residuals after subtracting the PHOEBE solutions, we deduce a common time scale of the order of 9 years. The variation of the orbital period is complex and can be described in detail only when adding further periods. We conclude that we see the effects of a 9-year magnetic activity cycle of the cool companion of RZ\\,Cas, caused by an 18-year dynamo cycle that includes a reversal of polarity. This conclusion is strongly supported by the behaviour of the two dark spots on the surface of the secondary that show the flip-flop effect in their strengths and one  spot that shows an 18-year periodicity in longitudinal migration.\n\nFrom the variations of orbital period and $v\\sin{i}$\\ around 2001, we conclude that the determined $v\\sin{i}$\\ is the projected equatorial rotation velocity of the outer layers of the primary accelerated by transferred matter and does not stand for the rotation velocity of the star as a whole. In all other seasons, the measured $v\\sin{i}$\\ point to a sub-synchronous rotation of the gainer. At the present stage, we cannot give a physical explanation for this new and interesting finding, however.\n\nBased on our available data, we conclude that RZ\\,Cas was undergoing an episode of high mass transfer in 2001, in a quiet phase in 2006 and 2009, followed by a slightly more active phase in 2013 and 2014, and again in a quiet phase in 2015 and 2016. Because we did not observe the star in 2010 and 2011, we cannot exclude that a second episode of high mass transfer occurred in these years, which would agree with the derived magnetic activity cycle of nine years.\n\nThe results of our investigation of high-frequency oscillations of the primary of RZ\\,Cas will be presented in Part II of this article. In a third article, we will investigate the accretion-induced variability of He\\,I lines detected in the spectra.\n\n\\begin{acknowledgements}\nHL and FP acknowledge support from DFG grant LE 1102\/3-1. VT acknowledges support from RFBR grant No. 15-52-12371. AD is financially supported by the Croatian Science Foundation through grant IP 2014-09-8656 and Erciyes University Scientific Research Projects Coordination Unit under grant number MAP-2020-9749. The research leading to these results has (partially) received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement N$^\\circ$670519: MAMSIE), from the KU~Leuven Research Council (grant C16\/18\/005: PARADISE), from the Research Foundation Flanders (FWO) under grant agreement G0H5416N (ERC Runner Up Project), as well as from the BELgian federal Science Policy Office (BELSPO) through PRODEX grant PLATO. Results are partly based on observations obtained with the HERMES spectrograph, which is supported by the Research Foundation - Flanders (FWO), Belgium, the Research Council of KU Leuven, Belgium, the Fonds National de la Recherche Scientifique (F.R.S.-FNRS), Belgium, the Royal Observatory of Belgium, the Observatoire de Gen\\`eve, Switzerland and the Th\\\"uringer Landessternwarte Tautenburg, Germany.\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nRecently much attention has been given to connections between\ncriticality and self-organized criticality (SOC) and evolutionary\nphenomenon, particularly punctuated equilibrium, and to\nconnections between SOC and synchronization. \nWe describe a model \nwhich we hope draws some connection between these 2 ideas.\nSOC has been proposed to describe out of equilibrium systems\nthat are critical, that self-organize into a scale invariant\ncritical state without tuning of a control\nparameter and show fractal time series.\\cite{rf:52} \nEvolution SOC type models \\cite{rf:60,rf:68,rf:45}\nhave been proposed to explain \npunctuated equilibrium. \\cite{rf:66,rf:67}\nPunctuated equilibrium is the phenomenon observed\nin the fossil record where long periods of stasis are interrupted \nby sudden bursts of evolutionary change. \nKaufman and Johnsen \\cite{rf:64} have modeled co-evolution,\nwhere agents live on \na coupled fitness landscape and walk around by\nrandom mutation, only \nmoves to higher fitnesses are allowed.\nOnce at a local maximum the walk stops until moves by another\nagent deform the lanscape so the agent is no longer at a maximum.\nKauffman \\cite{rf:64} has linked this to SOC. Bak,\nSneppen and Flyvbjerg \\cite{rf:60,rf:68} have taken a similar\napproach. They define a species as \na barrier to increasing fitness and choose the\nleast fit, then randomly change its barrier and the barriers of\nother agents. The\nsystem evolves to a critical state with a self-organized\nfitness threshold.\nSOC has also been linked to periodic\nbehaviour.\\cite{rf:50,rf:56,rf:58,rf:70} \nA.Corral et al and Bottani \\cite{rf:56,rf:50}, say there is a close relationship between\nSOC and synchronization. SOC\nappears when a system is perturbed which\notherwise should synchronise totally or partially. \nThe perturbation may be open boundary conditions rather than\nperiodic\\cite{rf:50,rf:70} or it may be\nrandomness present in initial conditions which is preserved by\nthe dynamic, or it may be the addition of noise.\nThe model we study appears to link these 2 ideas with the\nemergence of avalanches of partial synchronization on all\nscales. However all these models are real space models whereas\nours is a mean-field model with no spatial dimension. Our model \nis entirely determistic, the critical state is produced by\ncertain initial conditions, indeed other initial conditions\nproduce completely periodic states. \nOur model also is not strictly speaking an SOC model since\ncritical behaviour only occurs for a certain range of the\nparameter and then only for certain initial conditions. A\ncomplete analysis of the \nintial conditions is outside the scope of this paper.\n\nOur model was originally\nmotivated as a model of the behaviour of speculating traders in a \nfinancial market in the spirit of co-evolution.\nRecent results have shown stock price time\nseries to be fractal with Hurst exponent different from 0.5,\n\\cite{rf:72} \nand with positive lyapunov exponent.\n\\cite{rf:33,rf:34,rf:39}\nScrambling daily returns changes the Hurst exponent back to 0.5.  \nLarge crashes have been supposed\nto be due to exogenous shocks, where information enters the\nmarket randomly.\nHowever large crashes interspersed with periods of slow\ngrowth are \nstrongly reminiscent of punctuated equilibrium. Indeed\nMandlebrot has noted large changes of cotton prices occur in\noscillatory groups \nand the movement in tranquil periods is smoother than\npredicted.\\cite{rf:77} \nScaling behavior has been noted in a financial \nindex and in the size of companies.\\cite{rf:100,rf:101}  Stanley et\nal have noted `scaling laws used to describe complex systems\ncomprised of many interacting inanimate particles (as in many\nphysical systems) may be usefully extended to describe complex\nsystems comprised of many interacting animate subsystems (as in\neconomics).'\n\nVarious models have been proposed to explain\nmarket movements.\\cite{rf:36,rf:38,rf:6}\nSato and Takayasu have proposed a threshold type\nmodel.\\cite{rf:73,rf:73a}\nSince critical states can produce avalanches on all scales,\nwithout the need for \nexogenous shocks, we believe critical type dynamics\nare present in financial market dynamics. \n\n\\section{Model}\n\\noindent\nWe hope to model co-evolutionary phenomenon where the micro-level\nitself defines the macro-level but is also slaved to the\nmacro-level. This is very evident in speculative financial market dynamics\nwhere a collection of individuals (micro) trade therby creating a price time\nseries (macro), but determine their trading behaviour by reference to\nthis same price series and other macro variables.  \nWe desired to make a model in analogy to this phenomenon. \n\nThis is a highly stylized toy model of a stock-market.There are \n$N$ agents which are represented by spins $s_{i}(t)$, where\n$s_{i}(t)=1$, means the agent $i$ owns the stock and\n$s_{i}(t)=-1$ means doesn't own the stock at time t. Each agent also has an \nabsolute fitness $F_{i}(t)$ and a relative fitnes\n$f_{i}(t)=F_{i}(t)-F(t)$ where the mean-fitness\n$F(t)=\\frac{1}{N}\\sum_{i=1}^{N}F_{i}(t)$. \nWe\nbelieve speculative traders are part of 2 crowds, bulls and bears, and our\nmacrovariable is `groupthink' $G(t)$, \ndefined by, \n\\begin{equation}\nG(t)=\\Delta P(t)=\\frac{1}{N}\\sum_{i=1}^{N}s_{i}(t)\n\\end{equation}\nThe dynamic is:\n\\begin{equation}\n\\Delta s_{i}(t)=s_{i}(t+1)-s_{i}(t)=\\left\\{\\begin{array}{ll}\n-2s_{i}(t) & f_{i}(t)\\leq 0\\\\\n0 & f_{i}(t)>0\n\\end{array}\n\\right.\n\\end{equation}\n\\begin{equation}\n\\Delta F_{i}(t)=F_{i}(t+1)-F_{i}(t)=-\\frac{1}{2}\\Delta s_{i}(t)G(t)+\\frac{1}{2}|\\Delta s_{i}(t)|c\n\\end{equation}\n\nThe dynamic is synchronous and deterministic. First $G(t)$ and\n$F(t)$ are calculated then all agents are updated according to\n(2) and (3). The price $P(t)$ is defined by (1) and\n$P(0)=0$. Initially $s_{i}(0)$ are chosen randomly with\nprobablity 1\/2 and $F_{i}(0)$ are chosen randomly from the\ninterval [-1,1].\n\n$G(t)$ measures the bullishness or bearishness of the crowd.\nAlthough different to ours Callan and Shapiro mention groupthink\nin Theory of Social \nImitation\\cite{rf:75} and Vaga's\\cite{rf:76,rf:72} Coherent\nMarket Hypothesis explicitly includes a variable called\n{\\it groupthink}. \n\nWe believe speculative agents determine their spin state\ndependent on whether they \nbelieve the market will move in their favour in the future. \nTherefore \nour agents have an absolute fitness $F_{i}(t)$ which measures their\nperception of whether they are in a good position with respect\nto the future. If $F_{i}(t)$ is relatively high their state will\nbe stable and if $F_{i}(t)$ is relatively low they will want to\nchange their current state. Many ways to define $F_{i}(t)$ are\npossible. In this model we define it by \nanalogy to Plummer. Agents\nconsider the market to be `overbought' or `oversold'. \nIn our simplistic model this is measured by $G(t)$. An agent is fit if it is in the minority\ngroup.  According to Plummer when most agents \nare in one position then there must be less buying into this\nposition (because there are only a finite amount of agents) and\ntherefore the market will eventually correct itself (change\ndirection) because \nits growth will not be sustainable. It is always profitable then \nto be in the minority group before a correction. At a correction \nthe dominant crowd breaks, the macro position dissolves, the market\nmay crash, and \nsubsequently bull and bear crowds will begin to reform. \nIn fact at these times the agents may trade in\n2 macro-clusters or chaotically with the market attaining high\nvolatility which persists for some time.  \nThis type of\ntrader has \nbeen called a {\\it sheep trader} \\cite{rf:36,rf:38}, in\ncontrast to fundamentalists speculators and\nnon-speculators. \nTherefore in this model an agents fitness is increased if it\nchanges from the \nmajority group to the minority group, with the increase\nproportional to the size of the majority. The opposite is\napplied if it changes the other way. \nIf an agent doesn't change its state then it's absolute fitness\n$F_{i}(t)$ is not\nchanged, regardless of whether $G(t)$ changes.\nAn agent also has a relative fitness $f_{i}(t)$. The $f_{i}(t)$ are the\nbehaviour controlling variables in this model. They may change\nin 2 ways. Firstly an agent $\\alpha$ may change its state $s_{\\alpha}(t)$ thereby\ndirectly changing $F_{\\alpha}(t)$ and $f_{\\alpha}(t)$. This is similar to\na single adaptive\nmove on a fitness landscape by an individual optimizing\nagent. Secondly co-evolution may occur. Here an agent's\nrelative fitness $f_{\\beta}(t)$ may change due to changes in the\nother agents fitnesses $F_{i}(t)$ changing $F(t)$ while\n$F_{\\beta}(t)$ remains constant.\n\nTo model evolution then we follow natural selection by analogy\nand mutate unfit agents and leave fit agents unchanged (although \ntheir relative fitnesses may change). \nas in Kaufman,\\cite{rf:64} and Bak et al\\cite{rf:60,rf:68}.\nMutation is considered to\nbe a state change and this changes an agent's fitness according\nto (3). In this model since\ntheir are only 2 possible states this means we simply flip\nstate. (In a more extensive model this would correspond to\nchanges to an\nownership portfolio vector). To decide which spins flip we could\ncompare pairs of \nfitnesses and change the least fit. \nThat is we could choose 2 agents $\\alpha$ and $\\beta$ and let\nthem compete so that\n$F_{\\alpha}>F_{\\beta}$ then we say\n$s_{\\alpha}(t+1)=s_{\\alpha}(t)$ and\n$s_{\\beta}(t+1)=-s_{\\beta}(t)$. \nHowever in this paper we simply\ntake a mean-fitness approach. That is all agents $i$ whose\nfitnesses $F_{i}(t)$ fulfill $F_{i}(t)\\leq F(t)$ ie\n$f_{i}(t)\\leq 0$\nflip there spins and their fitnesses change according to (3).\nAll other agent's states and absolute fitnesses $F_{i}(t)$ do\nnot change although their relative fitnesses $f_{i}(t)$ of\ncourse do.  \nTherefore fit agents which could be considered to be at a local\nmaximum do not change their states until the mean-fitness $F(t)$ has\nbecome equal to their fitnesses $F_{i}(t)$.\n\nThis means the fitnesses are all internally defined emergent\nproperties as in co-evolution. Of course if there is no overall crowd\npolarisation then changing state does not change fitness.\n\nTherefore the fitness update rule (3) can be seen as the\nadaptive walk part and this is the reason why we do not simply\nset  $F_{i}(t)=-s_{i}(t)G(t)$ or $\\Delta F_{i}(t)=-s_{i}(t)G(t)$\ncontinuously for all agents. We \nhope agents will take time to walk out of unfit states and that\nfit maxima will be created which persist for some time. Our\nabsolute fitness is therefore cumulative and is only changed for unfit\nagents. More realistically we could\nthink of agents imperfectly sampling the market ie $G(t)$ at a series of\ntimes to determine their current absolute fitness. In\nfact we see that the concept of relative fitness and absolute\nfitness are very similar to the concept of `bounded\nrationality.' An agents rationality is bounded because he only\nmakes local adaptive moves and can percieve only his absolute\nfitness but not the overall mean-fitness or his relative fitness\n\n\nThis fitness of the position is natural in the sense that it can be seen as \na kind of potential for future profit, usually termed `utility'\nin economics. The fitter an agent is\nthe less likely it will want to change its state, the more\nstable it is, because it believes the market to be oversold in\nits favour.\n\nSince $G(t)$ will on average be $0$,\nin addition in equation (3) we include a very small control\nparameter $c$ which controls the driving rate. We add this to \nall fitnesses below the mean so that unfit agents on\naverage will be come fitter and interact with the fit\nagents. This is a general characteristic of evolutionary systems \nthat single entity moves on fitness landscape should be on\naverage uphill. \n\nOur market price $P(t)$ is defined by $\\Delta P(t)=G(t)$, ie\nprice increases while more people own the share than don't own\nit, and the price is theoretically unbounded as it should\nbe. Positive groupthink means positive increase and\nvice-versa. This is similar to the way prices are usually\ndefined by $\\Delta p(t)\\propto Z(t)$, where $Z(t)$ \nis the excess demand for something. \nThis model does not included a fixed amount of\nshares. Indeed any trader may independently buy or sell a share\nwithout the notion of swapping. This reflects the fact that this \nis a model of only speculative behaviour, and part of a much\nlarger of pool of shares. However a more realistic model should\ninclude a fixed amount of shares.\n\nThis model is intended to be a suggestive\nillustration rather than a realistic stock-market. \n\n\\section{Results}\n\nShown in Fig.1a is a time series for the fitnesses $F_{i}(t)$\nfor an $N=80$ system for $c=0.01$. Punctuated equilibrium behaviour is\nclearly visible, with periods of relative stasis interspersed\nwith sudden jumps.\nAlthough not shown the mean-fitness time series $F(t)$\nshows changes on all scales similar to a devils staircase. Also\nshown in Fig.1b is the corresponding daily returns time-series\n$\\Delta P(t)=G(t)$,\nthis also shows calm periods and sudden bursts of high\nvolatility. Infact this behaviour is a kind of intermittent\npartial synchronization. Shown in Fig.2 is the same time\nseries but with a small portion magnified. Macroscopic \nsynchronization can be seen. Partial synchronizations show various\ndifferent periods and complexities, and persist for various\nlengths of time.\nClustering allows\nsynchronized spins to trade in phase with groupthink $G(t)$ thereby\nrapidly increasing there fitnesses, or out of phase thereby\nbecoming less fit, this is the origin of the sudden large\nchanges in fitness. \nAlso at these times the fitness deviation suddenly\nincreases,(not shown). Between periods of large-scale partial\nsynchronization with high volatility, periods \nof calm are characterized by a small even number of spins\nflipping in anti-phase, they therefore increase their $F_{i}$\nonly slowly due to the driving parameter c, the returns $G(t)$\nremaining roughly \nconstant during these periods with the fitness deviation\ndecreasing. (Of course anti-phase flipping with no average\nincrease in fitness is prevented in a real market by a\nfixed transaction cost. A more realistic model must include this.) \nWhen the mean-fitness $F(t)$ which is usually increasing\ncrosses some non-flipping $F_{i}$, this spin flips and may cause the\n$F(t)$ to cross some more $F_{i}$, possibly starting an \navalanche. This only happens when the total fitness deviation is \nsmall.\n\nShown in Fig.3 are two price $P(t)$ time series for\n$c=0.013$. Their fractal slightly repetetive pattern is highly\nreminiscent of real financial time series.\n\nSince this model is deterministic, completely periodic\nstates are also possible. Shown in Fig.4 is the average of the\nquantity $<R(t)>$\nwhere $R(t)=\\sum_{i=1}^{N}|\\Delta s_{i}(t)|$\nis the amount of spins which flip at any time and $<\\ldots >$\ndenotes time averaging. In Fig.4a are time series for $c=0.0113$ \nwhile Fig.4b is $c=0.01$. \n\nFig.4a shows one time series finds\nthe 2 cluster periodic state, where 2 groups alternately topple. \nHere 59 time series are included in the non-periodic state. If\nthis is a\ntransient it is super long even for the moderate size \n$N=200$. Fig.4b shows at more regular $c$ values the system\ncan find periodic states with larger amounts of\nclusters. Roughly half of the 60 series investigated become\nperiodic by $t=2.5\\times 10^{7}$.\nShown in Fig.5a is $<R(t)>$ plotted against $c$. \nInfact to construct this plot we\ndiscarded $3\\times 10^{7}$ time steps and then averaged over the next\n$20,000$, each point represents a different initial condition\nand there are $8$ for each cost $c=0.001x+0.000138$, $x$ is an integer.\nFor small cost\ncritical type behaviour is evident \nwith a sudden phase transition at $c=0$. This is of course\nbecause at negative $c$ less fit spins continuously flip and\nnever interact with spins at greater than mean fitness. In fact\nthe system divides into a frozen solid fit component and an\nunfit gaseous type component for negative $c$. The size of the\nfrozen component \ndepends on the initial conditions as can be seen from the points \nat negative cost. For larger positive cost an upper branch of\nperiodic attractors at  \n$<R(t)>=100$, half the system size, is evident, the\nsystem has settled into 2 alternately toppling clusters which\ninterleave the mean-fitness $F(t)$. The lower branch is\ncharacterised by the punctuated equilibrium \nstate shown in Fig.1.\nFig.5b shows the time average $<S(t)>$ of an entropy type\nquantity of the fitness\ndistribution $S(t)$ given by,\n$S(t)=-\\sum_{i=1}^{N}\\frac{|f_{i}(t)|}{f(t)}ln\\frac{|f_{i}(t)|}{f(t)}$\nwhere $f_{i}(t)$ is the fitness deviation and \n$f(t)=\\sum_{i=1}^{N}|f_{i}(t)|$. The averaging is the same as\nfor $<R(t)>$. \nFor this $N=200$ system the maximum $S=ln200\\approx\n5.3$ and the periodic points at positive and negative cost are\nvery near this. \nThe punctuated equilibrium state which exists near\nthe transition is more ordered at lower entropy. \n\nThis is our first evidence of critical behaviour for small $c$. \nSecond evidence is obtained by \nlooking at the distribution of avalanches. In the punctuated\nequilibrium state the system finds a state characterised by\nfluctuations on all scales. Shown in Fig.6 is the\ndistribution of $P(R)$ against $R$ where $P(R)$ is the\nprobability of an avalanche of size $R$. These are distributions \nof avalanches for 1 time series for 3 different system\nsizes. They are not ensembles of time series, this distribution\nis independent of the initial conditions and any non-periodic\ntime series contains all avalanche sizes. \n\nThe time series\nwere of length $T=16,000,000$, near the transition point at\n$c=0.0113$. \nThe distribution shows scale invariance,$P(R)\\sim R^{\\alpha }$ up\nto about \nhalf the system size. At half the system size there is a peak\nwhere the system almost finds the periodic attractor and spends\nmore time in these states.\nAfter this \nthe distribution continues  \nto the cutoff near the system size. The scaling exponent $\\alpha \n$ taken\nfrom the $N=3000$ distribution is $\\alpha =-1.085\\pm 0.002$.\nAlso shown in Fig.7 is the distribution of magnitude of\nchanges in mean \nfitness $\\Delta F(t)=|F(t+1)-F(t)|$, the steps in the devils\nstaircase. \nThe time series are\nthe same as in Fig.6 for 2 different system sizes, there is\nno ensemble averaging. In fact the two distributions for\n$N=1500,923$ are almost identical, if we were to superpose them,\nonly one could be seen. This is also true for other system\nsizes. Peaks appear at $\\Delta F\n\\approx 0.12,0.5,0.75$. Between the peaks we see scaling\nregimes. Here we see at least 2\nscaling regimes, $P(\\Delta F)\\sim \\Delta F^{\\beta}$ where for\n$\\Delta F \\le 0.1$,  $\\beta =-1.25 \\pm 0.003$ and for $0.1\\le\n\\Delta F \\le 0.4$, $\\beta =-1.39 \\pm 0.02$. Possibly there is\nanother scaling regime for $0.55\\le \\Delta F \\le 0.7$.\n\\section{Conclusion}\n\\noindent\nThis model illustrates an interesting relation between critical\nphenomenon and punctuated equilibrium on the one hand, and\nbetween partial synchronization and punctuated equilibrium on\nthe other hand. The system synchronizes for certain cost $c$ and\ncertain intial conditions, otherwise it shows critical\nbehaviour, similar to the SOC models cited in the introduction. We\nbelieve this deserves further  \ninvestigation. We also find an interesting phase transition. \n\nSome typical behaviour of \nmoney markets is present here, especially the periods of low\nvolatility, where the price is relatively stable and the fitness \ngrows slowly while the fitness deviation decreases slowly, \ninterrupted by shorter periods of persistent high volatility and macroscopic\noscilations which are observed in real time series. We wonder if \nlike in earthquake dynamics, which are often modelled by SOC\ndynamics, a large crash in a real financial market is preceded\nby some smaller self-reinforcing oscilatory pre-shock, as is\nseen in our dynamics here. Also the time price time series is\nhighly suggestive of real time series, with \nformations similar to `double tops' and `rebounds' described in\nquantitative analysis, produced by the \nnear periodic macro behaviour which can appear. The\nslightly repetetive self-similarity reminds us of financial time series.    \n\nMany possible models of financial market dynamics can be\nplausibly suggested, included many exhibiting threshold\ndynamics, since data concerning the micro behaviour of\nindividual traders is not available.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}