CHANDRA MONITORING OBSERVATIONS OF THE ANTENNAE GALAXIES. II. X-RAY LUMINOSITY FUNCTIONS

The Astrophysical Journal, 661:135 148, 2007 May 20 # 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A. CHANDRA MONITORING OBSERVATIONS OF THE ANTENNAE GALAXIES. II. X-RAY LUMINOSITY FUNCTIONS A. Zezas, G. Fabbiano, and A. Baldi Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 François Schweizer Carnegie Observatories, Pasadena, CA 91101-1292 A. R. King Theoretical Astrophysics Group, University of Leicester, Leicester LE1 7RH, UK A. H. Rots Harvard-Smithsonian Center for Astrophysics, Cambridge, MA 02138 and T. J. Ponman School of Physics and Astronomy, University of Birmingham, Birmingham B15 2TT, UK Received 2006 September 5; accepted 2007 January 18 ABSTRACT We present the X-ray luminosity functions ( XLFs) of the X-ray source population detected in the Chandra monitoring observations of NGC 4038/4039 (the Antennae). The seven individual XLFs are well described by a flat power law with a cumulative slope 0:5 0:8. A similar slope ( ¼ 0:480:08 þ0:09 ) is measured for the sources detected in the co-added observation, which reaches a limiting luminosity of 10 37 erg s 1. In our analysis we account for observational biases by deriving incompleteness functions and including them in the fitting process. We do not detect significant variations between the shape of the XLF of the seven observations. The two shorter exposures appear to have steeper XLFs, but these are still consistent with the other observations. These results indicate that the XLFs of starforming galaxies are indeed flatter than those of more evolved stellar populations, even down to the typical luminosities of X-ray binaries. Based on this, as well as the X-ray variability and spectral properties of the X-ray sources, we suggest that the observed population down to our detection limit consists predominantly of X-ray binaries accreting close to their Eddington limit, similar to the high or very high states of Galactic X-ray binaries. In the case of ultraluminous X-ray sources (L X > 10 39 erg s 1 ), we cannot rule out the contribution of a beamed component (because of either mechanical focusing or Doppler boosting) in their observed emission. However, even without beaming, we estimate that the maximum observed luminosity (L X 10 40 erg s 1 ) could be produced by a 80 M black hole accreting at its Eddington limit; such black holes can be the result of regular stellar evolution of double stellar systems. Subject headinggs: galaxies: interactions galaxies: peculiar X-rays: galaxies 1. INTRODUCTION With the launch of the Chandra X-ray Observatory, X-ray luminosity functions ( XLFs) have become a standard tool for the characterization of the populations of discrete X-ray sources detected in nearby galaxies. These observations agree in the general picture that starbursting galaxies have flatter XLFs than spiral or early-type galaxies (e.g., Kilgard et al. 2002; Zezas & Fabbiano 2002; Colbert et al. 2004). However, for very few starforming galaxies has it been possible to obtain high-quality XLFs covering luminosities from the most luminous sources observed (L X k 10 39 erg s 1 ) down to the typical luminosities of active X-ray binaries (10 37 erg s 1 ). Broad luminosity coverage is particularly important in order to probe a representative sample of the X-ray binary populations and to study the link between ultraluminous X-ray sources ( ULXs) and lower luminosity sources. Moreover, although it is well known that X-ray variability is a defining property of X-ray binaries, very little is known about its effect on the shape of their XLF. This is particularly important for comparisons between galaxies, as well as for the use of the XLFs as a descriptor of the source populations (e.g., for comparisons with simulations from X-ray binary population synthesis models, see Belczynski et al. [2004a]). 135 The Antennae galaxies ( NGC 4038/4039) provide a unique laboratory to study the XLFs of star-forming galaxies; they are nearby (19 Mpc; Whitmore et al. 1999), allowing the detection of low-luminosity sources without serious confusion problems, and they feature a large population of X-ray sources (60 sources down to luminosities of 10 38 erg s 1 ), which allows us to study, in detail, the connection of the sources with their local star formation environments. The Antennae are also exceptional in that, in a single system, they host the largest population of X-ray sources with luminosities in excess of those expected based on Galactic neutron star or black hole X-ray binaries. This gives us the opportunity to investigate how this population of ultraluminous X-ray sources ( ULXs) is related to the lower luminosity sources, as well as to quantify the effect of their significant variability on the shape of the overall XLF. In particular, the Chandra monitoring observations of the Antennae (Fabbiano et al. 2003; Zezas et al. 2006) offer an opportunity to investigate the effect of source variability on their XLF and to probe the X-ray source populations down to luminosities of 2:5 ; 10 37 erg s 1, well within the range of typical luminosities of active high-mass and low-mass X-ray binaries ( HMXBs, LMXBs; McClintock & Remillard 2006). The monitoring campaign consisted of seven observations performed between 2000 December and 2002 November and probed source

136 ZEZAS ET AL. Vol. 661 Fig. 1. A 0.3 7.0 kev band background image from observation 2 (left) and the co-added, total exposure (right) of the Antennae (produced by wavdetect). The contours are in levels of 0.5 and 1.0 counts pixel 1 for observation 2, and 0.5, 1.0, 1.25, and 1.75 counts pixel 1 for the co-added image. On this image we mark the sources used for the XLF fit with circles and the sources which were excluded from the fit with crosses (see x 2.1). variability on timescales of weeks to years. Five of the individual exposures have detection limits of 10 38 erg s 1, while two shorter exposures have detection limits of 5 ; 10 38 erg s 1.A detailed description of the observations and the data analysis is presented in Zezas et al. (2006, hereafter Paper I). In the present paper we focus on the XLFs of the discrete sources detected in the individual as well as the co-added exposures. In x 2 we describe how we construct the XLFs, the correction of observational biases, and the different methods used for the XLF fits. In x 3, we describe the results from the XLF fits and compare the individual XLFs. Finally, in x 4 we discuss the implications of these results for the nature of the observed X-ray sources. Throughout the paper we assume the generally accepted distance to the Antennae of 19.0 Mpc (Whitmore et al. [1999]; however, a distance of 14 Mpc has been proposed by Saviane et al. [2004]. Adoption of this distance would result in downshifting the luminosity scale by a factor of 2). All errors are at the 68% confidence level unless otherwise stated. 2. X-RAY LUMINOSITY FUNCTION OF POINT SOURCES 2.1. Derivation of the XLF In order to derive the XLF of the point sources in the Antennae we used the photometric data presented in Paper I. As described in detail in that paper, the photometric data have been corrected for variations of the effective area over the detector and between observations, which allows us to directly compare the XLFs of the different observations even in terms of source counts. For our analysis we exclude any sources with a strong extended component (x 2.4 of Paper I), as well as sources 51 and 61, which are associated with the two nuclei. We also exclude source 90 (source X-37 in the notation of Zezas et al. [2002]), which was identified with a background AGN (Clark et al. 2005). The luminosity of each source is calculated assuming a powerlaw spectrum ( ¼ 1:7) with Galactic foreground absorption (N H ¼ 3:24 ; 10 20 cm 2 ; Stark et al. 1992). Such a spectrum is representative of the typical source spectra. A more accurate approach would be to use the individual spectra of each source or the average spectrum of sources of different luminosities. However, its implementation in the fitting process used in this study (x 3) would be complicated. We used this method in the XLF fits for the first data set, which was analyzed in a slightly different way (Zezas & Fabbiano 2002), but we do not find any significant difference between these earlier results and our present analysis. 2.2. Incompleteness Correction The calculation of an unbiased XLF in the Antennae is not straightforward, because of the spatial variations of the diffuse emission intensity over the system (Fig. 1; see, e.g., Zezas & Fabbiano 2002; Baldi et al. 2006), which result in varying detection thresholds over the galaxy. To account for this effect without narrowing the luminosity coverage by setting an overly conservative completeness limit, we calculated the source detection probability as a function of source and background intensity (in counts). Since the source detection only depends on the number of source and background counts within the detection cell, these detection probabilities are independent of the specifics of the detector (e.g., effective area) and the source and background spectra. The detection probabilities were calculated by simulating data sets with sources of intensities between 2.5 and 50.0 counts, and background surface brightness levels between 0.025 and 7.8 counts pixel 1 (based on the typical background levels in the different observations of the Antennae). The simulations were performed with MARX version 3.0 (Wise et al. 2004), following a similar procedure as in Zezas & Fabbiano (2002). For each source andbackgroundintensity,wesimulated3datasetsof25sources each. The simulated sources were located within 1.5 0 from the optical axis, in order to minimize the effects of PSF degradation (almost all sources in the Antennae are located within 1.5 0 from the center of the galaxy). Each data set was processed in the same way as the actual data; first, we extracted a broadband (0.3 7.0 kev) image, which we searched for sources in scales of 1.0, 2.0, 4.0,

No. 1, 2007 X-RAY LUMINOSITY FUNCTION OF ANTENNAE GALAXIES 137 TABLE 1 Fits to the Source Detection Probability Functions Background (counts pixel 1 ) (1) k 0 (2) k 1 (3) k 2 (4) Fig. 2. Plot of the source detection probability curves for different background levels. From left to right: 0.025, 0.05, 0.10, 0.25, 0.35, 0.50, 0.85, 1.00, 1.25, 1.75, 2.35, and 3.0 counts pixel 1. 8.0, and 16.0 pixels using the wavdetect tool with a false source detection probability of 10 6 (for more details see Zezas et al. 2006). The detection probability for each source and background intensity is then the fraction of sources recovered at a significance greater than 3 above their local background (assuming the Gehrels approximation; Gehrels 1986). This significance criterion was applied to minimize effects due to Poisson noise (see below) and to match the criteria used to derive the observed source list. Since, as pointed out in the previous paragraph, the detection efficiency is only a function of the number of source and background counts in the detection cell, the completeness curves calculated this way can also be used for sources detected in other bands. Moreover, they can be used for observations taken with different detectors or at different times and regardless of the spectrum of each source, as long as the source detection process is the same. However, since the size of the detection cell depends on the size of the point-spread function ( PSF), these detection probabilities do depend on the off-axis angle (e.g., Kim & Fabbiano 2004). The effective area, exposure time and source spectrum become a factor when we calculate the detection efficiency as a function of the source luminosity. In Figure 2 we present the detection probability as a function of source intensity (in counts) and background (in counts pixel 1 ). We also parameterize the detection probability curves at each background level by a function of the form A(C ) ¼ 1:0 k 0 C k 1 e k 2C ; where C is the source intensity (in counts) used for the simulation. The best-fit parameters for these analytic functions are given in Table 1. The parameterization of the source detection probability functions with analytic curves has the advantage of smoothing the statistical noise due to the finite number of simulated sources (which is particularly important for sources with low detection probabilities) and allowing their implementation in analytic fitting schemes. We note that because the number of detected counts for each source follows a Poisson distribution, it will be biased toward a higher number with respect to the true intensity, especially in the case of weak sources close to the detection limit. In order to overcome this bias, we include in our study only sources with significance greater than 3 above their local background and brighter ð1þ 0.025... 11:12 þ12:6 5:64 0:83 þ0:47 0:50 0:43 þ0:07 0:08 0.05... 11:25 þ3:62 2:65 1:24 þ0:18 0:19 0:3 þ0:02 0:02 0.10... 38:69 þ10:4 8:00 2:52 þ0:19 0:19 0:40 þ0:02 0:02 0.25... 15:08 þ2:51 2:10 1:91 þ0:13 0:13 0:29 þ0:01 0:01 0.35... 18:76 þ3:26 2:72 2:41 þ0:16 0:16 0:30 þ0:01 0:01 0.50... 11:89 þ2:20 1:80 2:22 þ0:21 0:21 0:20 þ0:01 0:01 0.78... 9:38 þ1:39 1:18 2:11 þ0:17 0:18 0:22 þ0:01 0:01 0.85... 19:98 þ4:21 3:36 3:69 þ0:30 0:31 0:30 þ0:02 0:02 1.00... 11:63 þ1:90 1:59 3:21 þ0:26 0:26 0:26 þ0:01 0:01 1.25... 7:23 þ0:81 0:70 2:50 þ0:19 0:19 0:20 þ0:01 0:01 2.35... 4:65 þ0:29 0:27 2:42 þ0:12 0:12 0:16 þ0:00 0:00 3.00... 3:73 þ0:20 0:18 5:00 þ0:30 0:30 0:24 þ0:01 0:01 7.8... 1:27 þ0:05 0:05 2:45 þ0:17 0:17 0:10 þ0:01 0:01 Notes. Col. (1): Background level in counts pixel 1. Cols. (2), (3), and (4): Best-fit coefficients for the fitting function (see eq. [1]). than the 50% completeness limit for the average background (shown by a horizontal dashed line in Fig. 2). To ensure that down to our limiting source intensity all sources have a finite detection probability, we exclude regions of the galaxy with very high background (>1.0 counts pixel 1 for the individual exposures and >1.7 counts pixel 1 for the co-added data set). These selection criteria are shown in Figure 3, where we plot the local background surface brightness against the intensity of each source and the 50% and 30% completeness limits (solid and dashed lines, respectively) based on the detection probability curves. The hatched area shows the part of the source-background intensity space which we exclude from our analysis. In Figure 1 we show a full-band (0.3 7.0 kev) background map for observation 2 and the co-added observation, produced by wavdetect, with the sources used for the XLF fit marked by circles. In the same figure we show the sources excluded from the fit, either because of their low intensity or because of their high background, marked by crosses. Although the same sources were not always excluded in different observations, their spatial distribution is very similar. The total number of sources detected in each observation, the number of them used to derive the XLF, and their corresponding limiting luminosities and number of counts are listed in Table 2. 2.3. Fitting Method Because of the relatively small number of sources, and in order to obtain the maximum information from our data, we fitted their unbinned luminosity function. We performed the fit by two methods: (1) a maximum likelihood fit of the distribution of the source intensities, and (2) a fit of a histogram of the source intensities using the Sherpa fitting package. Next we discuss these two fitting procedures in detail. 2.3.1. Maximum Likelihood Fit Our maximum likelihood ( ML) fit is based on the method of Schmitt & Maccacaro (1986; hereafter SM86), which assumes Poisson uncertainties on the intensity of each source and on the total number of sources. In order to account for the local background of each source, we treat the overall XLF as a combination of XLFs from multiple samples observed at different backgrounds. However, we generalize the multisample method of SM86 to

138 ZEZAS ET AL. Vol. 661 Fig. 3. Plot of the local source background surface brightness vs. net source number of counts for sources detected in observation 2 and the co-added exposure. The solid and dashed lines show the 50% and 30% completeness limits for a given background level and source intensity. Sources to the left and top of these curves have lower detection probability. The hatched area shows the sources excluded for the XLF fits because of low detection probability at these background levels. include the incompleteness function and a broken power-law fitting function. In the Appendix, we derive the likelihood function taking into account the incompleteness and describe its numerical implementation. We then test this method by fitting several simulated data sets of different slopes, and we find very good agreement between the input parameters and those we recover from the fit. Note that the commonly used analytic estimator for the slope of a log N-log S distribution (or a luminosity function) derived by Crawford et al. (1970) is biased because it does not include any uncertainties on the source intensities, and the method of Murdoch et al. (1973) is not appropriate for faint sources because it assumes that the source intensities follow a Gaussian distribution (e.g., SM86). We fit the unbinned distribution of the source intensities to a simple power law of the form N(>L) ¼ KL ; where L is the luminosity (above a cutoff L 0 ), is the cumulative slope, and K is the normalization of the XLF. Note that although this functional form refers to the cumulative distribution, the fit is performed on the differential number of sources. We estimate ð2þ the 90% confidence interval of the slope from the range in which the logarithm of the likelihood changes by 1.3 from the best-fit value (e.g., Bevington & Robinson 1992). We confirm the confidence intervals with Monte Carlo simulations. For each source we draw 1000 samples of its intensity, given its observed number of counts and background. As a sampling distribution we use the posterior predictive distribution ( PPD) of the source counts (van Dyk et al. 2001; see also Park et al. 2006). Then we perform the fit for each of these 1000 simulated XLFs, and from the histogram of the slope we estimate the 90% quantile. These confidence intervals are in good agreement with those estimated from the distribution of the log likelihood as a function of the fitted parameter. 2.3.2. Sherpa Fit We also fit the XLF with the Sherpa fitting package. Since Sherpa handles only histogrammed data, we bin the list of source intensities in counts space, from the faintest to the brightest sources and with a bin size of one count. This natural binning scheme gives the maximum resolution without artificially increasing the number of bins. We take into account the incompleteness by calculating an ancillary response function (ARF) on the same TABLE 2 Summary of Data Used for the Luminosity Function Fits Observation (1) Total (2) Sample Size Complete (3) Observation Limit [counts a (L X b )] (4) 50% Completeness Limit [counts a (L X b )] (5) 1... 60 52 8.24 (0.59) 12.79 (0.92) 2... 72 61 8.24 (0.62) 12.47 (0.94) 3... 79 64 7.79 (0.60) 12.47 (0.96) 4... 82 70 7.79 (0.60) 12.05 (0.93) 5... 84 71 7.60 (1.08) 12.05 (1.71) 6... 84 69 6.56 (0.98) 12.05 (1.80) 7... 87 73 6.56 (0.47) 12.05 (0.86) 8... 102 61 12.77 (0.19) 15.41 (0.23) Notes. Col. (1): Observation number. Col. (2): Total number of sources detected with S/N > 3:0 above the local background. Col. (3): Number of sources used in the XLF fit. Col. (4): Minimum number of counts (and luminosity) of the detected sources. Col. (5): Number of counts (and luminosity) at the 50% completeness limit for the typical background of the sources used in the fit. a The number of counts for each observation are normalized to the aim point of observation 2. b The X-ray luminosity (in units of 10 38 erg s 1 ) is in the 0.1 10.0 kev band and is calculated assuming a power-law spectrum ( ¼ 1:7, with Galactic line-of-sight absorption) and the effective area of observation 2 (see Table 2 of Zezas et al. 2006).

No. 1, 2007 X-RAY LUMINOSITY FUNCTION OF ANTENNAE GALAXIES 139 TABLE 3 Power-Law Fits to the XLF Observation (1) ML (2) (3) Sherpa Norm (4) L ref (5) 1... 0:43 þ0:09 0:15 0:39 þ0:1 0:1 2:5 þ0:6 2... 0:52 þ0:10 0:13 0:53 þ0:10 0:10 3:9 þ0:8 3... 0:59 þ0:10 0:13 0:59 þ0:10 0:10 4:5 þ0:9 4... 0:60 þ0:14 0:11 0:67 þ0:10 0:09 5:5 þ1:1 5... 0:80 þ0:17 0:13 0:86 þ0:13 0:12 7:2 þ1:4 6... 0:85 þ0:17 0:17 0:87 þ0:13 0:13 6:9 þ1:3 7... 0:63 þ0:11 0:15 0:65 þ0:10 0:09 5:5 þ1:1 Co-added... 0:42 þ0:08 0:13 0:53 þ0:1 0:1 5:0 þ1:5 0:5 4.9 0:7 5.8 0:8 6.1 0:9 5.5 1:2 10.2 1:2 10.7 0:9 5.1 1:1 5.1 Notes. Col. (1): Observation number. Col. (2): Best-fit cumulative slope from maximum likelihood fit. Col. (3): Best-fit cumulative slope from the Sherpa fit. Col. (4): Differential number of counts at the reference luminosity. Col. (5): Luminosity of the reference point (fixed at 10 counts) in units of 10 37 erg s 1. grid as the XLF (see also Kenter & Murray 2003; Wang 2004). In the case of a sample with a uniform background, this ARF is effectively the detection probability as a function of source intensity. However, in our case, where the sources have very different backgrounds, the ARF consists of the detection probability of each source in each bin based on its observed background, number of counts, and the incompleteness function discussed in x 2.2. For bins which include more than one source, the value of the ARF is their average detection probability, while for bins with no source we estimate the value of the ARF by interpolation. Because of the small number of counts the fit was performed using the Cash statistic (Cash 1979). Fig. 4. Plot of the log likelihood (normalized to the maximum likelihood of each fit) vs. the power-law cumulative slope. Orange, red, green, blue, light blue, magenta, and yellow curves correspond to data sets 1 7, respectively, while the black curve corresponds to the co-added data set. The horizontal dashed line shows the 90% [log (L)] confidence interval. 3. RESULTS 3.1. Fitting Results Based on the above methods, we find that the XLFs of the individual and the co-added data are fitted with a single power law of similar slope (0.5 0.6) and normalization. The reference point for the power-law model is fixed at 10 counts for the seven individual exposures and 50 counts for the co-added exposure. Only the XLFs of observations 5 and 6 show somewhat steeper slopes (0.8), but this is partly because their exposures are shorter than those of the other observations. To assess the importance of this effect, we performed the same analysis on 30 ks segments of the other exposures, in order to match the exposure times of observations 5 and 6. We found that these XLFs had slightly steeper slopes than those derived from the full exposures, but consistent within the uncertainties. Moreover, coincidentally, the brightest sources in the Antennae were observed at lower intensities during these observations, which also contributes in the slightly steeper slopes. The best-fit slopes for the power-law fits calculated using the ML and the Sherpa methods are presented in Table 3. In Figure 4 we plot the difference of the log likelihood with respect to the best-fit value [log (L)] as a function of the cumulative slope (). A dashed line shows the log (L) ¼ 0:5 range which corresponds to the 68% confidence interval for one interesting parameter. These results show that the two methods give very consistent results. Also, the XLF parameters for observation 1 are in good agreement with those estimated in Zezas & Fabbiano (2002), following a different fitting method. In Figure 5 we plot the XLFs from the individual as well as the co-added observations. The solid and dashed lines show the incompleteness corrected and the observed XLFs, respectively. The hatched area shows the 90% confidence bounds based on Monte Carlo simulations of the unbinned XLF (x 2.3.1). Each of the individual XLFs are binned to the natural binning grid of the observed XLF. The standard deviation in each bin is a measure of the uncertainty of the observed XLF due to the uncertainty of the intensity of each source. To this uncertainty we add in quadrature the uncertainty of the number of sources in each bin of the observed XLF, following the Gehrels approximation (Gehrels 1986). Finally we plot these errors on the cumulative XLF. Note that we use this method for illustrative purposes only; all fits are performed on the differential XLFs as described in x 2.3. From Figure 5 there appears to be a bump on the XLF of the co-added observation at a luminosity of 1:4 ; 10 38 erg s 1. Therefore, we fit the XLF of the co-added observations with a broken power law of the form 8 L 1 >< A for L L b ; L ref N(L) ¼ A 0 L 2 >: for L L b ; L ref where A 0 ¼ AL ð b /L ref Þ 2 1 ; 1 and 2 are the slopes below and above the break point, respectively; A is the number of sources at the reference luminosity L ref ; and L b is the luminosity of the break point. We initially performed the fit using Sherpa, which gives 1 ¼ 1:721:36 þ0:31, 2 ¼ 1:350:14 þ0:14,andl b ¼ 83:0(>39:4) counts [or L b ¼ 7:5 ; 10 37 erg s 1 (>3:5 ; 10 37 erg s 1 )]. As is clear from these results, the slope of the faint end of the XLF and the break point are poorly constrained, and the fit statistic is only slightly improved. Because of the very small number of sources in each bin (less than 5, and typically 0), as well as because the position of the break point has a fixed low bound (zero), it is not legitimate to use the F-test to compare the broken and single power-law fits (e.g., Protassov et al. 2002). Instead, we use a likelihood ratio test (LRT; e.g., Cowan, 1998). Based on the simpler model (power law), we simulate 1000 XLFs which we then fit with both models. From the best-fit statistic for each fit we calculate the likelihood ratio LR ¼ S p /S bp, where S p and

140 ZEZAS ET AL. Vol. 661 Fig. 5. Plot of the completeness-corrected (solid line) and uncorrected (dashed line) cumulative XLFs from the individual and the co-added observations of the Antennae. The hatched areas show the 1 uncertainty of the XLF, taking into account uncertainties of the number of sources as well as of their luminosity (including Poisson noise on the number of observed source counts and spectral uncertainties). S bp are the fit statistic values (Cash statistic in this case) for the single and broken power-law models, respectively. The probability of obtaining, by chance, an improved fit with the more complicated model is given by the percentage of the likelihood ratios which are higher than the ratio for the observed data sets. We find that the broken power law provides a marginally improved fit over the simple power law at the 90% confidence level for the combined data set. We also performed a fit with a power law and an exponential cutoff at the high-luminosity end in order to test the suggestion that there is an upper limit in the XLF of starburst galaxies (Gilfanov et al. 2004; Colbert et al. 2004). We find that (1) this model does not provide a statistically significant improvement in the fit and (2) its parameters are poorly constrained. 3.2. XLF Variability One of the defining properties of X-ray binaries is their intensity and/or spectral variability. From the photometric analysis of the seven observations, we find that the majority of the sources detected in each observation do indeed exhibit spectral variability (e.g., Fabbiano et al. 2003; Zezas et al. 2006). However, the results from the XLF fits indicate that the slopes of the individual observations are consistent, with the only exceptions being observations 5 and 6, which appear to have steeper slopes. We also tested the hypothesis that the XLFs are consistent with the same parent population using the Kruskal-Wallis nonparametric test (e.g., Conover 1980). This test applies in the case of multiple samples and tests the hypothesis that at least one data set has a different mean from the others. Because observations 5 and 6 were taken with shorter exposure times (30.0 ks instead of 70.0 ks for the other exposures), we compared the XLFs in two groups: data sets 1, 2, 3, 4, and 7, and data sets 5 and 6. We also compared all data sets truncated at the 50% completeness limit of the shortest exposure. The probabilities that each of these three sets of XLFs are consistent with the same parent population are 68%, 89%, and 86%, respectively (see Table 4). Therefore, we conclude that the seven XLFs are consistent with each other. 4. DISCUSSION In x 3 we calculated the parameters of the XLF of the discrete sources in the Antennae galaxies. The XLFs for the seven

No. 1, 2007 X-RAY LUMINOSITY FUNCTION OF ANTENNAE GALAXIES 141 Fig. 5 Continued individual exposures extend to luminosities of 8 ; 10 37 erg s 1, while the XLF for the co-added exposure extends down to 10 37 erg s 1, well within the luminosity range of active X-ray binaries. We now use these results, together with the spectral and multiwavelength properties of the sources presented in Paper I, to set constraints on the nature of these X-ray sources. TABLE 4 Results of Kruskal-Wallis Nonparametric Test Datasets (1) L X Cutoff (10 37 erg s 1 ) (2) Probability (3) 1, 2, 3, 4, 7... No 0.68 2, 3, 4, 7... No 0.97 1, 2, 3, 4, 7... 1.8 0.99 5, 6... No 0.89 All... 2.0 0.86 Notes. Col. (1): Data sets used in the comparison. Col. (2): Luminosity cutofffor all data sets (in units of 10 37 erg s 1 ). Col. (3): Probability that the data sets are not drawn from the same parent population. 4.1. Luminosity Function The results from the fits of the co-added XLF show that it is well represented by a relatively flat, straight power law with a cumulative slope of 0.5, down to a luminosity of 10 37 erg s 1. This slope is similar to the slopes measured for other star-forming galaxies (e.g., Hartwell et al. 2004; Ott et al. 2005; Kilgard et al. 2002; Colbert et al. 2004). The fact that it extends down to luminosities typical of X-ray binaries provides further support for the notion that the XLFs of X-ray sources in star-forming galaxies are generally flatter than those of X-ray sources in galaxies with older stellar populations, or in other words, star-forming galaxies have a larger population of ULXs. These results, which are derived from a large sample of X-ray sources over 3 orders of magnitude in luminosity, indicate that the flatter slopes of XLFs in star-forming galaxies are not only a manifestation of the relative preponderance of ULXs in young stellar systems (e.g., Ptak & Colbert 2004; Swartz et al. 2004), but an intrinsic property of the luminosity distribution of even their regular X-ray binary population. Although the data are well represented by a single power law, there is a suggestion of a knee, or a relative lack of sources at a

142 ZEZAS ET AL. Vol. 661 Fig. 6. Plot of the source luminosity as measured from the co-added exposure against the average luminosity of the sources from the individual exposures they were detected in (only sources detected in at least two exposures are included in this plot). The diagonal line indicates the sources for which the two luminosities are the same. Sources which fall below the one-to-one line have underestimated luminosity in the co-added exposure. luminosity of 10 38 erg s 1. However, a broken power law does not provide an improved fit at a confidence level greater than 90%, and the slopes of the two components are consistent within the errors. Moreover, this knee appears to be near the detection limit of the individual exposures, which suggests that it may be due to the fact that only persistent sources will be detected in the co-added observation at luminosities below the detection limit of the individual exposures. Variable and transient sources, on the other hand, may not be detected in some observations, resulting in two competing effects, depending on their detection in the co-added exposure; if they are detected, their luminosity inferred from the co-added exposure will be lower than their luminosity in the individual exposures, resulting in a flattening of the bright end of the XLF of the co-added data set and a steepening in fluxes close to the detection limit. This effect is demonstrated in Figure 6, which shows the average luminosity of sources in the individual exposures they were detected and their luminosity estimated from the co-added data set (only sources detected it at least two exposures are included in this plot); a significant number of sources (many of which have variability above the 3 level), fall below the line of equality, indicating that their luminosity as measured from the co-added exposure is underestimated. On the other hand, sources which are not detected (or not included in the fit because of their high local background and/or low significance; see x 2.2), will not be present in the XLF of the co-added data set, resulting in a flatter slope, since these sources tend to be in relatively low luminosities. For example, in the case of the Antennae, we find that 42 of the 102 pointlike sources are not included in the fit of the XLF from the co-added exposure, either because they are not detected or because their local background exceeds the threshold required for a uniform incompleteness correction. This is demonstrated in Figure 7, where we plot the cumulative function of the average luminosity of sources detected in the individual exposures (black line), the XLF from the co-added exposure (red line), the XLFs of transient sources ( green line), and sources detected only in the co-added exposure (blue line). In order to show the contribution of the different source types in the XLF from the co-added data set, all XLFs apart from that of the average luminosities are based on the co-added data, and the selection of the sources is based on the criteria presented Fig. 7. Comparison of the XLFs of the average luminosity of sources from the individual exposures (black line), the XLF from the co-added exposure (red line), the XLFs of transient sources ( green line), and sources detected only in the co-added exposure (blue line). The XLFs are based on the co-added data, except for the XLF of the average source luminosity, which is based on the average source luminosity in the individual exposures. The XLFs have not been corrected for incompleteness. in x 2. We did not apply any completeness correction in these XLFs. From this figure it is clear that the slight flattening of XLF from the co-added data set is due to the exclusion of a large number of sources (because of the requirement for a finite completeness correction) and sources detected only in individual exposures. The additional number of faint sources detected in the co-added image contribute only in the very low-luminosity end of the XLF. Although the actual difference in the shape of the XLF is not significant in our case, it indicates that special care needs to be exercised when interpreting results from co-added exposures. The lack of a strong break in the XLF suggests that the majority of luminous sources (L X > 10 38 erg s 1 ) are the highluminosity tail of the fainter sources, rather than an additional population. If so, the high-luminosity tail of the XLF in the Antennae could be associated with a population of sources with accretion rates close to their Eddington limits, similar to the high or very high state of Galactic X-ray binaries. This is in agreement with their spectra, which have power-law photon indices in the range 1:7 2.5 (Zezas et al. 2006; A. Zezas et al. 2007, in preparation), as well as with their spectral variability, which for many of the luminous sources resembles that of black hole binaries in a very high state (high/hard low/soft transitions; e.g., McClintock & Remillard 2006). Also, several sources show spectral variations without significant luminosity variations that are similar to the high state very high state transitions (spectral changes not accompanied by large luminosity changes). It is also supported by the high luminosities of these sources; the transition from the high to the low state in Galactic black hole binaries occurs on average at 10 2 of their bolometric Eddington luminosity (e.g., Maccarone 2005), which corresponds to less than 10 37 erg s 1 for typical black holes in the 2.0 20.0 M mass range and assuming a bolometric correction of 20% ( Portegies-Zwart et al. 2005). Therefore, it seems unlikely that there is a significant population of low-state X-ray binaries in the luminosity range that we are probing with our observations. A population of young X-ray binaries with high accretion rates is consistent with recent population synthesis models that show that systems accreting through Roche lobe overflow (RLOF) are the dominant population at ages between 50 and 250 Myr

No. 1, 2007 X-RAY LUMINOSITY FUNCTION OF ANTENNAE GALAXIES 143 (e.g., Belczynski et al. 2007). However, even in younger populations, wind-fed black hole binaries may reach Eddington luminosities if the mass-loss rate of their donors is high enough (e.g., Belczynski et al. 2004a). Both types of systems can sustain high luminosities over long periods of time and do not show the transient behavior associated with wind-fed systems or typical black hole low-mass X-ray binaries. This is also consistent with the variability results from our monitoring campaign, which show that only one of the sources with luminosities above 5 ; 10 38 erg s 1 exhibits transient behavior (Zezas et al. 2006). The lack of high-low transitions also indicates that the state-transition timescales of these X-ray sources are much longer than the duration of our campaign (1 yr). This provides support for the scenario that the majority of the X-ray sources in the Antennae are in a long-term high state (e.g., King et al. 2001; Rappaport et al. 2005), resulting in flatter XLFs than those observed in normal spiral galaxies. This is also consistent with the lack of large-scale variability, which indicates that the majority of the sources accrete at a relatively constant (within a factor of a few) fraction of their Eddington accretion rates, and it may explain the lack of significant variations between the XLFs of the different observations. Although the highest luminosity of the XLF, which in different observations does not correspond to the same sources, changes by up to 1 order of magnitude, this does not affect the shape of the XLF, which is driven by the larger number of faint sources. For example even in the cases of observations 4, 5, and 6, where the maximum luminosity of the ULXs is reduced by 50%, we do not detect any statistically significant difference in the slope of the XLFs. Finally, the lack of XLF variability gives us confidence that single observations of a galaxy can provide a representative picture of the XLF of its X-ray source populations. This allows for meaningful comparisons between the XLFs of different galaxies and also for meaningful comparisons with theoretical XLFs calculated from X-ray binary population synthesis models (e.g., Belczynski et al. 2004a, 2007). 4.2. Nature of the ULXs As discussed in x 4.1, the luminosity range of the observed sources, as well as their long-term spectral variability, indicate that they are related to compact objects accreting close to their Eddington limits. The fact that the XLF is well represented by a power law over 3 orders of magnitude suggests that even the most luminous sources (i.e., those with luminosities above 10 39 erg s 1 ; ULXs) are simply the high end of the X-ray binary distribution, rather than an additional source population. A similar conclusion was reached by Grimm et al. (2003) and Liu et al. (2006), which, however, are based on comparisons between different galaxies and much narrower luminosity ranges. If the ULXs are X-ray binaries accreting at Eddington or mildly super-eddington rates, their maximum observed luminosity of 10 40 erg s 1 corresponds to a compact-object mass of 50 M,assumingnobeaming.However, even in the case of the standard Shakura-Sunyaev accretion disk (Shakura & Sunyaev 1973; see also Begelman et al. 2006), if the accretion rate Ṁ becomes highly super-eddington, the Eddington luminosity limit can be exceeded by ln (Ṁ /Ṁ Edd ). In the case of HMXBs accreting through Roche lobe overflow, the accretion rate can even reach 10 4 Ṁ Edd, resulting in a compact object mass within the range of black holes than can be produced via normal stellar evolution. In fact, stellar populations of ages between 10 120 Myr (which are representative of the most recent starburst events in the Antennae) can produce a significant number of binary systems with black hole masses 25 M, and with significantly higher masses for lower metallicities (Belczynski et al. 2004b). Although the majority of the X-ray binaries in the Antennae could be explained in terms of disk emission from critically accreting objects, a beamed component cannot be ruled out. In fact, critical accretion rates can lead to the formation of physically thick accretion disks, which result in mechanical focusing of the emitted radiation (e.g., Abramowicz et al. 1988, 1988; Urry et al. 1991; King et al. 2001). The luminosity enhancement factors depend on the structure of the accretion disk and on the viewing angle, but are expected to be quite low (10% 30%; Madau 1988). In this case the mass of the most luminous source would be even lower. In the case of thick accretion disks, and under the assumption of a power-law distribution of compact objects, one would expect their luminosity distribution to follow a broken power law ( Urry et al. 1991; see also Zezas & Fabbiano 2002), with the position of the break with respect to the maximum luminosity depending on the opening of the collimating structure. The fact that we do not observe a pronounced break indicates that this accretion mode is observed only in a fraction of the luminous sources, and the break is masked by the statistical noise and the unbeamed sources. Another possibility to consider is the contribution by a jet component. Jet emission is found to contribute significantly to the bolometric luminosity of X-ray binaries in the very high state (e.g., Fender et al. 2004). For example, Koerding et al. (2002) proposed a simple model which can reproduce the general shape of the XLFs in spiral and starburst galaxies by a mixture of disk and jet components. Given the small number of sources in the ULX regime, it is not possible to distinguish between different types of beaming or even quantify beaming s importance for the shape of the XLF. The only constraints on the importance of beaming can be derived from the X-ray spectral and timing properties of the ULXs and their radio counterparts. These results show that the majority of the ULXs in the Antennae are most likely the high-luminosity tail of the normal X-ray binary population. Therefore, if by the term ULX we wish to describe an unusual source population, we suggest that a more appropriate luminosity limit would be closer to 10 40 erg s 1, rather than 10 39 erg s 1, which has been used traditionally. An intrinsic luminosity of 10 40 erg s 1 would require isotropic emission from black holes in excess of 80 M, which is at the limit of masses produced by regular stellar evolution of binary stellar systems (Belczynski et al. 2004b). We also note that we cannot rule out the possibility that some of the observed ULXs in the Antennae are associated with black holes with masses in the range 100 1000 M (intermediate mass black holes; IMBHs). However, our results indicate that they are not the dominant population. 5. CONCLUSIONS We have presented an analysis of the XLFs of the discrete sources detected in the seven Chandra monitoring observations of the Antennae galaxies, as well as their combined data set. The XLFs for the seven individual exposures extend to luminosities of 8 ;10 37 erg s 1, while the XLF for the co-added exposure extends down to 10 37 erg s 1, well within the luminosity range of active X-ray binaries. From the analysis of these XLFs we find: 1. The shape of the XLF does dot change significantly between the different exposures, despite the variability of the individual sources.

144 ZEZAS ET AL. Vol. 661 2. Luminosities of variable sources measured from the coadded data set are systematically underestimated. Moreover, the requirement to have a finite completeness correction leads to the rejection of several sources from the co-added data set, which in turn results in a somewhat flatter ( but not statistically significant) slope. 3. The XLF from the co-added data set is described by a single power law with a cumulative slope ¼ 0:53 þ0:1 0:1. Neither a broken power law nor a power law with an exponential cutoff provide a statistically significant improvement in the fit. 4. The lack of a strong break in the XLF down to luminosities of 10 37 erg s 1 indicates that the majority of the luminous sources (L X > 10 38 erg s 1 ), are the high-luminosity tail of the fainter sources, rather than an additional source population. This is consistent with a population of objects in a long-term high or very high state, for example, young X-ray binaries accreting through Roche lobe overflow, which are expected to dominate at ages between 50 and 250 Myr (e.g., Belczynski et al. 2007). 5. The continuous shape of the XLF over 3 orders of magnitude in luminosity and the long-term spectral variability of the ULXs indicate that the majority belong to the same population as the lower luminosity sources. Sources in the 10 39 10 40 erg s 1 luminosity range can be explained in the framework of regular X-ray binary formation and evolution. However, the more luminous sources may be associated with a more exotic population of compact objects (e.g., IMBHs). This work was supported by NASA contract NAS 8 39073 (CXC) and NASA Grant NAG 5 9983. A. Z. acknowledges support from NASA LTSA grant NAG5-13056. A. R. K. gratefully acknowledges a Royal Society Wolfson Research Merit Award. We also thank the California-Harvard Astrostatistics collaboration and in particular Vinay Kashyap for many useful discussions. We would like to thank the anonymous referee for useful suggestions. APPENDIX DERIVATION OF THE LIKELIHOOD FUNCTION In this section we describe the derivation of the likelihood function and its implementation in the maximum likelihood fitting procedure. We follow the methodology and notation of Schmitt & Maccacaro (1986, hereafter SM86). First, we derive the likelihood function for a single power law and then the likelihood function for the broken power-law fit, and in the end we describe their numerical implementation. A1. SINGLE POWER LAW Assuming a luminosity function of the form N(>L) ¼ KL ða1þ above a cutoff luminosity L 0,whereL is the luminosity, is the cumulative slope, and K is the normalization of the XLF. Following the nomenclature of SM86, within the source aperture we detect C þ B counts, where C is the number of counts due to the source and B is the number of counts due to background. Then the relation between count rate and luminosity is L(C; i ) ¼ S i C; ða2þ where S i is the conversion factor for a spectrum with parameters i, which also depends on the detector effective area and the energy band over which we estimate the luminosity. Then ignoring for the moment any spectral uncertainties, we can write equation (A1) in differential form and in terms of the source intensity C in counts, N(C ) ¼ K 0 (S i C ) 1 S i ; ða3þ where K 0 ¼ K. The probability of detecting m counts from a source with intensity C and background B is P(m) ¼ Poi(C þ B) ¼ e (CþB) (C þ B) m ; ða4þ m! where Poi(C þ B) is the Poisson probability density. In the following discussion, we assume for simplicity the same background and counts-to-luminosity conversion factor for all sources, and we perform the fit in counts space. However, in the end of the section we discuss two generalizations which include the background for each source and their individual count rate to luminosity conversion factors based on their spectra and corresponding uncertainties. The detection probability A(C; B) for a source of intensity C observed over background B can be estimated from simulations analyzed in the same way as the data (see x 2.2 and, e.g., Zezas & Fabbiano 2002; Kim & Fabbiano 2004). Here, in order to include the incompleteness in the analytic calculation of the likelihood function we parameterize it as (see x 2.2). A(C ) ¼ 1:0 k 0 C k 1 e k 2C : ða5þ

No. 1, 2007 X-RAY LUMINOSITY FUNCTION OF ANTENNAE GALAXIES 145 Normalization of the source detection probability requires that NC ð ÞAC; ð BÞdC ¼ N 0 ð Þ 1:0 k 0 C k 1 e k 2 C dc ¼ 1:0; KC 1 ða6þ which gives a normalizing constant of h i N 0 ¼ K 1 C 0 k 0 k ð k 1Þ 1; 2 ðk 1 ; k 2 Þ ða7þ where ðk 1 ; k 2 Þ is the incomplete gamma function (e.g., Abramowicz & Stegun 1964). Then given the luminosity function (A1) and probability distributions (A4) and (A5), the number of sources detected with m counts at a given background B is Nm ð Þ ¼ N 0 Pm ð ÞNC ð ÞdC ¼ KC ð Þ 1 CþB e ð Þ ðc þ B m! Þ m 1:0 k 0 C k 1 e k 2 C dc: ða8þ After a binomial expansion, this equation becomes N(m) ¼ N 0 K Xm i¼0 B mi e C 1:0 k 0 C k 1 e k C 2C i1 dc; ðm iþ! i! ða9þ which has the same form as equation (8) of SM86, with the only difference being the inclusion of the incompleteness term A(C; B). Rearranging the above equation gives where Nm ð Þ ¼ KC 0 ½ Qm; ð B; ; aþrm; ð B; ; k 0 ; k 1 ; k 2 ;ÞŠ; ða10þ Qm; ð B; ; aþ ¼ Xm B mi (m i)! i¼0 which is identical to the definition of Q in SM86 (their eq. [10]), and i¼0 i1 C C e dc ¼ Xm B mi i! i!(m i)! ð i ; Þ; ða11þ i¼0 Rm; ð B; ; k 0 ; k 1 ; k 2 ;Þ ¼ Xm B mi k 0 e (k 2þ1)C C i1þk1 dc ða12þ (m i)! i¼0 i! X m B mi ¼ k 0 i!(m i)! (k 2 þ 1) (iþk1) ði þ k 1 ; ðk 2 þ 1Þ Þ: ða13þ Since we also apply a signal-to-noise ratio (S/N) criterion, following SM86, the likelihood function for this fit will have the form where L ¼ exp N 0 C 0 P Nth N 0 KP Nth M T! Y M T i¼1 Q(i; B; ; a) R(i; B; ; k 0 ; k 1 ; k 2 ;) P Nth ; ða14þ P Nth ¼ X1 i¼nth Q(i; B; ; a) R(i; B; ; k 0 ; k 1 ; k 2 ;): ða15þ Taking the logarithm of the likelihood, minimizing with respect to the normalization, and substituting back the best-fit normalization, we obtain the log likelihood as a function of the slope : log (L) ¼ XM T i¼1 log Q(i; B; ; a) R(i; B; ; k 0 ; k 1 ; k 2 ;) P Nth : ða16þ