------------------------------------------------------------------------ From: Ashley J. Ruiter aruiter@nmsu.edu To: gcnews@aoc.nrao.edu Subject: submit ruiter_GCnews.tex ApJL submitted %astro-ph/0511813 %http://arxiv.org/abs/astro-ph/0511813 \documentclass[]{article} \usepackage{emulateapj} \usepackage{graphicx} \def\msun{{\rm ~M}_{\odot}} \def\rsun{{\rm ~R}_{\odot}} \def\myr{{\rm ~Myr}} \def\mdot{\dot M} \def\mpy{{\rm ~M}_{\odot} {\rm ~yr}^{-1}} \begin{document} \title{The Nature of the Faint {\em Chandra} X-ray Sources in the Galactic Centre} \author{Ashley J. Ruiter\altaffilmark{1}, Krzysztof Belczynski\altaffilmark{1,2}, and Thomas E. Harrison\altaffilmark{1}} $^{1}$ New Mexico State University, Dept. of Astronomy, 1320 Frenger Mall, Las Cruces, NM 88003\\ $^{2}$ Tombaugh Fellow\\ aruiter@nmsu.edu,kbelczyn@nmsu.edu,tharriso@nmsu.edu} Recent {\em Chandra} observations have revealed a large population of faint X-ray point sources in the Galactic Centre. The observed population consists of $\gtrsim 2000$ faint sources in the luminosity range $\sim 10^{31}-10^{33}$ erg s$^{-1}$. The majority of these sources (70\%) are described by hard spectra, while the rest are rather soft. The nature of these sources still remains unknown. Belczynski \& Taam (2004) demonstrated that X-ray binaries with neutron star or black hole accretors may account for most of the soft sources, but are not numerous enough in order to account for the observed number and X-ray properties of faint hard sources. Both wind-fed systems and quiescent Roche lobe overflow transients were tested as potential source candidates. Muno et al. (2004) proposed that intermediate polars (subclass of magnetic cataclysmic variables) may be able to explain the faint hard population. Since an observational test of this hypothesis is not currently feasible due to {\em (i)} the large extinction toward the Galactic Centre, and {\em (ii)} low luminosity of intermediate polar donors (K-M dwarfs), we propose a theoretical test. A full population synthesis calculation of the Galactic Centre region has been carried out. Our results indicate that the numbers and X-ray luminosities of intermediate polars are consistent with the observed faint hard Galactic Centre population. We discuss the properties of the intermediate polar population, and suggest future tests for the hypothesis. For example, the derived slope of the X-ray luminosity function from our synthetic population of $\sim 0.8$ could be compared with the observed slope, once one has been obtained from observations. \end{abstract} \keywords{Galaxy: center --- X-rays: binaries --- stars: white dwarfs} \section{Introduction} An X-ray survey of the Galactic Centre (GC) with the ACIS-I on {\em Chandra} (Wang, Gotthelf \& Lang 2002) first revealed the presence of $\sim$ 1000 spectrally hard X-ray sources ($2-10$ keV) with luminosities $L_{\rm x} \lesssim 10^{35}$ erg s$^{-1}$. Pfahl et al. (2002) have claimed that wind-fed neutron star accretors with intermediate- and high-mass companions are responsible for a significant fraction of the hard sources in the Wang et al. (2002) survey. A deeper {\em Chandra} survey of the nuclear bulge region (Muno et al. 2003; hereafter MM03) revealed over 2000 X-ray point sources with X-ray luminosities $\sim 10^{31} - 10^{33}$ erg s$^{-1}$. The majority of these sources (1427) are described by hard spectra (with a photon index of an absorbed power law $\Gamma < 1$), while the rest (652) are characterized by softer spectra ($\Gamma > 1$).\footnote{Foreground sources excluded; M.Muno, private communication.} Belczynski \& Taam (2004) have studied the entire population of X-ray binaries with neutron star (NS) and black hole (BH) accretors in the context of the MM03 survey. It was demonstrated that neither wind-fed (low-, intermediate-, high-mass) systems nor Roche Lobe Overflow (RLOF) transients in quiescence can explain the entire faint population. However, the quiescent transients may be responsible for most of the faint soft GC sources. Muno et al. (2004) suggested that the observed faint hard sources are most likely intermediate polars (IPs); a subclass of magnetic cataclysmic variables (CVs). Among observations of magnetic CVs, IPs are asynchronous rotators, with the white dwarf (WD) spin period being shorter than the orbital period. Currently there are $\sim$ 31 IP binaries known (Gansicke et al. 2005). They consist of a magnetized WD and a low-mass companion. The companion transfers mass via RLOF to the WD, with typical mass transfer rates of $\sim 10^{-11}$ M$_{\rm \odot}$/yr. Matter spirals in toward the WD and forms an accretion disc. It is thought that the white dwarf magnetosphere truncates the accretion disc where the magnetic pressure exceeds the gas pressure. Matter is channeled into accretion columns over both magnetic poles of the WD (e.g., Belle et al. 2005). Typical IP magnetic fields are $\leq 10$ MG (de Martino et al. 2004). It is from below the accretion shock, above the WD surface, where the hard X-rays originate (see Patterson 1994; Warner 1995 for a review). IPs are known to exhibit both soft and hard X-ray emission and are thought to be the most luminous subclass of CVs in X-rays, owing to their typically larger orbital periods ($\sim 3-10$ hr, though some IPs have periods outside this range; de Martino 2005), fairly high mass transfer rates, and large emission regions as compared to other CVs, e.g., polars (Norton \& Watson 1989a; Patterson 1994). We address the issue of the nature of the faint hard source population observed in the deep GC exposure of MM03, and test the validity of the Muno et al. (2004) hypothesis that these sources are IPs. We construct a simple phenomenological model of an IP and using population synthesis (\S\,2) we calculate the number of IPs and their X-ray luminosities in the GC (\S\,3). In \S\,4 we discuss our results in context of available observations. \section{Model Description} Our study has been carried out using the updated population synthesis code {\tt StarTrack} (Belczynski, Kalogera \& Bulik 2002; Belczynski et al. 2005a). The single star evolution is followed with modified analytic formulae presented by Hurley, Pols \& Tout (2000). All stars (singles and binaries) are evolved with solar metallicity ($Z=0.02$, appropriate for the GC) and with standard wind mass loss rates. We assume a continuous star formation rate in the GC over the last 10 Gyrs, and a binary fraction of 50\%. A number of physical processes important for binary evolution are accounted for: tidal interactions, detailed mass transfer (MT) calculations, common envelope (CE) events, supernovae explosions, and different mechanisms for angular momentum losses such as magnetic braking (MB) and gravitational radiation (GR), among others. The primary goal of this study is to test the hypothesis that intermediate polars can account for the observed population of the faint hard X-ray point source population in the GC. We use our standard model to check whether within the general framework of binary evolution the number of predicted IPs coincides with that of the GC faint hard sources. The standard model is described in detail in Belczynski et al. (2005a), and here we reiterate only the description of the most important input physics for IP formation. It has been realized that CVs are the likely outcome of the common envelope phase (Paczynski, 1976). Despite years of investigation, no consistent physical model for this phase of binary evolution exists. Therefore, we use two current alternative pictures of CE evolution to provide an estimate of associated model uncertainties. {\em Standard CE model.} Our standard model calculations assume energy balance (Webbink 1984); the donor envelope is ejected from the system at the expense of the binary orbital energy. The binary separation following the CE phase depends on parameters $\alpha_{\rm ce}$ and $\lambda$, relating to the efficiency with which the CE is expelled from the system, and donor internal structure, respectively. We adopt $\alpha_{\rm ce} \times \lambda = 1.0$. {\em Alternative CE model.} In our alternative CE model, we have adopted the prescription described in Nelemans \& Tout (2005) which employs angular momentum balance, and assumes that the angular momentum is lost from the binary in a linear fashion as a function of mass loss. Following Nelemans \& Tout (2005), we adopt a scaling factor $\gamma = 1.5$ for this model. See Belczynski, Bulik \& Ruiter (2005b) for the full CE equations. {\em X-ray Luminosity Calculation.} We assume that the IP X-ray luminosity is a function of the accretion rate, accretor physical properties, and the efficiency with which the accretion luminosity is converted to hard X-ray luminosity in the {\em Chandra} band. For degenerate donors (WDs) of mass $M_{\rm don}$, we assume that the mass transfer is driven by GR and we calculate it from \begin{equation} \mdot_{\rm don} = M_{\rm don} D^{-1} {\,d J_{\rm gr}/\,d t \over J_{\rm orb}} \label{mt13} \end{equation} where \begin{equation} D={5 \over 6}+ {1 \over 2} \zeta_{\rm don}-{1-f_{\rm a} \over 3 (1+q)}- { (1-f_{\rm a}) (1+q) \beta_{\rm mt}+f_{\rm a} \over q}. \label{mt17} \end{equation} $J_{\rm orb}$ is the orbital angular momentum, $dJ_{\rm gr}/dt$ is the angular momentum loss due to GR, $q$ ($\le 1.0$) is the binary mass ratio, $f_{\rm a}$ is the fraction of transferred mass accreted by the accretor of mass $M_{\rm acc}$ (we assume Eddington limited accretion; see below), and $\beta_{\rm mt} = M_{\rm acc} M_{\rm don}^{2}/(M_{\rm don} + M_{\rm acc})^{2}$. The radius mass exponent for the donor $\zeta_{\rm don}$ is obtained from stellar models (see Belczynski et al. 2005a). For non-degenerate donors we estimate the mass transfer rate as follows: \begin{equation} \mdot_{\rm don} = - {\zeta_{\rm evl}+{2 \over \tau_{\rm mb}} + {2 \over \tau_{\rm tid}} + {2 \over \tau_{\rm gr}} \over \zeta_{\rm don} - \zeta_{\rm lob}} M_{\rm don} \label{mt7} \end{equation} where $\zeta_{\rm evl}$ is the change of the donor radius due to its nuclear evolution, $\zeta_{\rm lob}$ is the radius mass exponent for the donor Roche lobe, and $\tau_{\rm mb}$, $\tau_{\rm tid}$ and $\tau_{\rm gr}$ are the timescales associated with MB, tidal interactions and GR, respectively. In some cases mass transfer proceeds on a thermal timescale, and thus we use \begin{equation} \mdot_{\rm th} = - {M_{\rm don} \over \tau_{\rm th}} \label{mt10} \end{equation} where the thermal timescale may be obtained from $\tau_{\rm th} = ({30 \times {M_{\rm don}^{2}}) / (R_{\rm don} L_{\rm don}})$ (e.g., Kalogera \& Webbink 1996). Then, we can calculate mass accretion rate as \begin{equation} \mdot_{\rm acc}= {\rm min}(0.95 \times \mdot_{\rm don}, \mdot_{\rm edd}). \label{mt10b} \end{equation} In the above we limit the accretion rate to the critical Eddington rate ($\mdot_{\rm edd}$), such that if the transfer rate is super-Eddington, the excess material leaves the binary system with the specific angular momentum of the accreting star. Additionally, we impose a mass-loss rate of 5\% in the sub-Eddington regime, since some IPs are observed to experience a mass loss rate of few percent ($\sim 10$\% in the extreme case of AE Aqr; Wynn, King \& Horne 1997). The hard {\em Chandra} band X-ray luminosity of an IP system is calculated from \begin{equation} L_{\rm x} = \eta_{\rm bol}\, \eta_{\rm geo} \, L_{\rm bol} = \eta_{\rm bol} \, \eta_{\rm geo}\, \epsilon \, {G M_{\rm acc} \mdot_{\rm acc} \over R_{\rm acc}} \label{Lx} \end{equation} where $G$ is the gravitational constant, $\epsilon$ is the conversion efficiency of gravitational binding energy to radiation (1 for surface accretion onto the WD), and $M_{\rm acc}$ and $R_{\rm acc}$ are the accreting WD mass and radius, respectively. We use $\eta_{\rm bol}=0.09$ for the bolometric correction to $2-8$ keV X-ray luminosity, although it is noted that this value is quite uncertain and may span a wide range ($\sim 0.01-0.2$; Norton \& Watson 1989b). The IP X-ray emission is likely anisotropic, at least to some extent, since the accretion proceeds through the WD magnetic poles. However, it has been demonstrated that the emission region in IPs may be quite extensive (i.e., may encompass more than a quarter of the WD surface, Norton \& Watson 1989a) and thus the anisotropy is not expected to be large. Here we assume isotropic emission and so $\eta_{\rm geo}=1$, although we note that a more sophisticated model including polar cap accretion should be invoked once more observations are available. In general $\eta_{\rm geo}$ depends on the size of the emitting region/regions and their relative orientation to the observer. {\em Simulations.} The initial mass of single stars and the primary (more massive) components in binaries are drawn within the range $0.8-150 \msun$ from a broken three-component power law initial mass function, with a slope of $-1.3/-2.2/-2.7$ in mass ranges $0.08-0.5/0.5-1/1-150 \msun$ (Kroupa, Tout \& Gilmore 1993). The secondary mass is obtained through a flat mass ratio distribution. Initial binary orbits are specified by semi-major axis (distribution flat in the logarithm $\sim 1/a$) and eccentricity (thermal distribution $\sim 2e$). The stellar population of the GC is evolved through 10 Gyr (age of the Milky Way) with a constant star formation rate and all IP systems are extracted and their luminosities and numbers are compared to those of the MM03 survey. We call {\em any} binary system experiencing RLOF in which the accretor is a white dwarf and the donor is any type of star a CV (i.e. including AM CVn systems, see Warner 1995). We consider various types of WDs: helium (He WD), carbon-oxygen (CO WD), oxygen-neon (ONe WD), hydrogen (H WD; formed by stripping the envelope of a low-mass main sequence star in RLOF), and hybrid (Hyb WD; containing a helium envelope and a helium-carbon-oxygen mantle, formed by stripping of a low-mass helium star in RLOF). From the CV population, we then assign a fraction of CV systems that are intermediate polars. In our model we adopt an IP fraction (IP$_{\rm frac}$) of 5\% following MM03. We have calibrated our results to pertain to the surveyed GC region by scaling by stellar mass. Following MM03 (see their \S\,3), we have assumed that the 17' $\times$ 17' {\em Chandra} field of view of the GC corresponds to a cylinder 440 pc deep with a radius of 20 pc, encompassing $1.3 \times 10^{8}$ M$_{\odot}$ in stars\footnote{There may be 50\% uncertainty associated with this estimate (Launhardt et al. 2002), which will propagate linearly in our results.}. \section{Results} It is found that the synthetic GC population of IP systems span a wide range of X-ray luminosities $\sim 3 \times 10^{29} - 5 \times 10^{33}$ erg s$^{-1}$. In the following we discuss only the systems above the MM03 survey detection limit, i.e., with X-ray luminosities $\geq 10^{31}$ erg s$^{-1}$ unless otherwise noted. The main results of our calculations for the standard and alternative CE prescriptions are presented in Table~\ref{tab01} and Figure~\ref{fig01}. {\em Standard CE model.} The number of IPs depends strongly on the adopted IP fraction. We find $\sim 800-8000$ IPs in the GC for IP fractions of $1-10\%$, respectively. To match the observed number of faint hard X-ray sources in the GC (1427) with IPs we would require an IP fraction of $\sim 2\%$. The distribution of orbital periods of our IPs peaks between $\sim 1.5 - 3.5$ hr and extends toward longer periods ($\geq 5$ hr), which is in agreement with the observed orbital periods of magnetic CVs (see i.e., Tovmassian et al. 2004). We find that the most frequent IP type (89\%) is a magnetic WD accreting from a main sequence (MS) star. The majority of MS donors (87\%) are low mass stars $\le 0.7$ M$_{\odot}$, i.e. K-M dwarfs. Accretors in the WD-MS subclass are CO WDs (66\%) and He WDs (23\%). The second most frequent (10.5\%) type of IP is a double degenerate system (i.e., close WD-WD binary with stable RLOF). The most common configurations found within this subclass are: CO WD-He WD (6.6\%), CO WD-Hyb WD (1.7\%) and CO WD-H WD (1.7\%). Finally, we find very few (0.5\%) IPs with giant-like donors. In this subclass almost all systems consist of a low-mass ($\sim 0.3-0.4 \msun$) sub-giant or a red giant transferring mass to a CO WD. The relative occurrence frequencies of different IP types reflect various evolutionary timescales for donors of varying types and masses. The WD-MS IPs are abundant since low-mass MS stars have very long evolutionary timescales ($\sim 10^{11} -10^{12}$ yr) and RLOF at the IP phase is also driven on long timescales; by GR ($\tau_{\rm gr} \sim 10^{10}$ yr) with the addition of MB ($\tau_{\rm mb} \gtrsim 10^9-10^{10}$ yr) for some systems. The evolution of double degenerates is driven by GR on shorter timescales on the order $\tau_{\rm gr} \sim 10^9$ yr (since these binaries are tighter); moreover it is difficult to form these systems (survival of two CE phases rather than one in the case of WD-MS formation). For WD-giant donor systems, the evolutionary and MB timescales are shorter than in the above cases, making them the least represented subclass of IPs. In Figure~\ref{fig01} we show the overall luminosity distribution and the corresponding X-ray luminosity function (XLF) for the model IPs. The IPs above the MM03 detection limit are marked separately for easy comparison. It is noted that only $\sim 55$\% of IPs are bright enough to make the X-ray luminosity threshold of the survey. The brightest IPs in our simulations are those with giant donors, with average X-ray luminosities $L_{\rm x} \sim 5 \times 10^{32}$ erg s$^{-1}$. The power-law slope of the cumulative distribution ($N(>L_{\rm x}) \sim L_{\rm x}^{-\beta}$) of the XLF for the synthetic IPs (above the survey detection limit) is $\beta \sim 0.8$. This corresponds to a slope of $\alpha \sim 1.8$ in the $N \sim L_{\rm x}^{-\alpha}$ distribution, which is comparable to the slope found for the observed systems $\alpha \sim 1.3-1.7$ (see MM03; eq.~5). {\em Alternative CE model.} It is found that $\sim 170-1700$ IPs may be present in the GC for IP fractions of $1-10\%$, and one would require an IP fraction of $\sim 8\%$ to match the number of GC faint hard sources. Orbital periods are found in the same range as for the standard CE model. Once again, the most frequent IP type is a magnetic WD accreting from a low-mass MS star (90.8\%) with the majority of them being CO WD-MS star binaries (75\%). Double degenerates are the second most populated subclass (8.6\%), first with CO WD-H WD (4\%), followed by CO WD-He WD and then CO WD-Hyb WD (2.1\% and 1.9\%, respectively) with other types constituting the rest. Again only a small fraction (0.6\%) of the IPs involve a WD-giant star binary. In Figure~\ref{fig01} we show the luminosity distribution of synthetic IPs and the corresponding XLF for the alternative CE model. The slope of the XLF for IPs above the survey detection limit is found again to be $\beta \sim 0.8$. {\em Typical evolution.} Two MS stars (2 and $0.6 \msun$) start out on a highly eccentric orbit ($e = 0.9$) with an orbital period of 1460 days when the Galaxy is 2.1 Gyr old. At 3.6 Gyr the system has circularized (period of 109 days), a CE phase is initiated by the primary (now an asymptotic giant), and upon ejection of the envelope the orbital period shrinks by $\sim$ two orders of magnitude (now 16 hours). The primary shortly thereafter becomes a CO WD. At 8.4 Gyr the MS secondary initiates RLOF and the system becomes an IP. At 10 Gyr we find an IP system with a period of 2.9 hours. We note that for this same system, if the alternative CE prescription model is used an IP is never formed. The reason for this is that upon ejection of the CE, the binary does not lose enough orbital angular momentum in order to end up in a tight enough orbit such that RLOF, and an IP phase, can ensue. Hence a smaller number of IPs are found in our alternative CE model. \section{Discussion} We have explored the possibility that the faint hard sources in the GC are intermediate polars. It is found that for both current common envelope models, the IP population is ample enough to explain the GC faint hard sources. The required IP fractions are then $\sim 2\%$ and $\sim 8\%$ for the standard and alternative CE models, respectively. The fraction is rather uncertain and currently estimated to be $\sim 5\%$ by MM03 based on the Kube et al. (2003) catalog of CVs. Once improved observational constraints on the IP fraction are obtained, we will be able to test the validity of different CE models. At this point a full picture of the GC X-ray point sources begins to emerge. Most of these sources are faint and are spectrally hard (1427) and they can be explained by a population of IPs as suggested by Muno et al.(2004) and confirmed in this study. Additionally, there may be a small contribution, at the level of few percent, of wind-fed sources with NS and BH accretors to the faint hard population, as proposed by Pfahl et al. (2002) and later revised by Belczynski \& Taam (2004). The faint soft sources (652) are likely RLOF transients with NS and BH accretors in quiescence as proposed by Belczynski \& Taam (2004). The bright GC sources ($\lesssim 20$; Wang et al. 2002 and MM03 can be explained by a population of NS/BH persistent sources and transients in outburst (e.g., Belczynski \& Taam 2004). We have found that most of the Galactic Centre IPs are either magnetic white dwarfs feeding from low-mass MS late-type companions, or double degenerate systems (e.g., AM CVn binaries). These systems are very unlikely to be detected at wavelengths other than X-rays, due to the high extinction toward the GC, and the intrinsically low brightness of the IP systems (K-M dwarfs). Recently, Laycock et al. (2005) carried out a search for infrared counterparts of the GC X-ray sources. It was found that high mass X-ray binaries (with donors brighter than B2V) are unlikely candidates for the majority of the GC faint sources, in agreement with our findings and those of Belczynski \& Taam (2004). Bandyopadhyay et al. (2005) are conducting a deeper near-infrared survey, which will detect all giant type donors and MS donors with spectral types earlier than G. However, if the typical donors in IPs are indeed K-M MS stars, as found in our study, and also observed for some Galactic IPs (e.g., M3 V for EX Hya, Dhillon et al. 1997; K5 V for AE Aqr, Tanzi et al. 1981), these systems will go undetected in this survey. The only test we are able to propose at this time is to compare the observed X-ray luminosity distribution, or alternatively the XLF slope, with the predictions found for IPs in our simulations (see Fig.~\ref{fig01}). We would like to acknowledge the support of KBN grant 1P03D02228, and to thank M.Muno, J.Grindlay and K.Belle for informative and helpful discussions. \begin{references} \reference{} Bandyopadhyay, R.M., et al.\ 2005, \mnras, in press (astro-ph/0509346) \reference{} Belczynski, K., Bulik, T., \& Ruiter, A.J.\ 2005b, \apj, 629, 915 \reference{} Belczynski, K., Kalogera, V., \& Bulik, T.\ 2002, \apj, 572, 407 \reference{} Belczynski, K., et al.\ 2005a, \apj, submitted \reference{} Belczynski, K., \& Taam, R., 2003 \apj, 616, 1159 \reference{} Belle, K. E., et al.\ 2005, AJ, 129, 1985 \reference{} de Martino, D., et al.\ 2004, Nuclear Physics B, 132, 693 \reference{} de Martino, D., et al.\ 2005, \aap, 437, 935 \reference{} Dhillon, V. S., Marsh, T. R., Duck, S. R., \& Rosen, S. R., 1997, \mnras, 285, 95 \reference{} Gansicke, B.T., et al.\ 2005, \mnras, 361, 141 \reference{} Hurley, J. R., Pols, O. R., \& Tout, C. A., 2000, \mnras, 315, 543 \reference{} Kalogera, V., Webbink, R. 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R., \& Horne, K., 1997, \mnras, 286, 436 \end{references} \clearpage \begin{deluxetable}{lcccc} \tablewidth{300pt} \tablecaption{Galactic Centre Population of Intermediate Polars} \tablehead{Model & WD-MS & WD-giant & WD-WD & Total} \startdata Standard CE & & & &\\ \hspace*{0.5cm}IP$_{\rm frac}$ 1\% & 694 & 5 & 82 & 781\\ \hspace*{0.5cm}IP$_{\rm frac}$ 5\% & 3474 & 25 & 410 & 3909\\ \hspace*{0.5cm}IP$_{\rm frac}$ 10\% & 6948 & 50 & 820 & 7818\\ Alternative CE & & & & \\ \hspace*{0.5cm}IP$_{\rm frac}$ 1\% & 157 & 1 & 14 & 172\\ \hspace*{0.5cm}IP$_{\rm frac}$ 5\% & 784 & 5 & 74 & 863\\ \hspace*{0.5cm}IP$_{\rm frac}$ 10\% & 1568 & 9 & 149 & 1726\\ \enddata \label{tab01} \end{deluxetable} \begin{figure} \includegraphics[width=0.6\columnwidth,angle=0]{f1.ps} \caption{Left: X-ray luminosity distribution and corresponding XLF for Galactic Centre IPs for the standard CE model and an IP fraction of 5\%. The entire population is shown with a dotted line while IPs brighter than the Muno et al. (2003) X-ray luminosity threshold are shown with a solid line. The XLF slope of cumulative distribution ($N(>L_{\rm x}) \sim L_{\rm x}^{-\beta}$) is marked. Right: same as the left panels but for the alternative CE model. } \label{fig01} \end{figure} \end{document}