8.4 Log N-log S relations
8.4.3 Results
In the analysis of the log N-log S relations, the bulge region was omitted. A comparison ofChandraand
XMM-Newtonobservations of that region shows that, due to the larger point spread function,XMM-Newton
cannot resolve all individual sources. Thus the number of sources and the derived source fluxes are biased. Figure 8.5 shows the derived CNC (blue), background CNC (red (observed) and green (extrapolated)) and the background corrected CNC (black) for the whole galaxy. From the intersection of the derived CNC with the background CNC we can estimate the flux below which the derived CNC becomes incomplete. This
limit is∼8×10−15erg cm−2s−1or∼6×1035erg s−1(2.0–10.0 keV). Assuming the source spectrum used
to derive the ECF values (cf.Table 8.2), this limiting luminosity corresponds to∼1×1036 erg s−1 in the
also included source fluxes below 2 keV, are∼1036erg s−1 (Trudolyubov et al. 2002b; Kong et al. 2003a,
SBK2009). The energy range used for source detection was 0.3–7 keV in Kong et al. (2003a), 0.3–10 keV in SBK2009, and not given in Trudolyubov et al. (2002b), who stated that they followed the analysis method described in Shirey et al. (2001), where the 0.3–12 keV band was used. This means that the completeness limits of my study is comparable with the limits of these previous studies, which is not surprising as the
previous studies are also based onChandraandXMM-Newtonobservations. The studies by SBK2009 and
Trudolyubov et al. (2002b), who used XMM-Newton observations that cover (part of) the major axis of
M 31, had similar integration times as used in my study. Each of the fields studied in Kong et al. (2003a)
was observed three times withChandraACIS-S, where one observation had an exposure of∼15 ks. The
difference in my study compared to previous studies, is the large spatial coverage of M 31. The background
corrected CNC flattens in the range of 3–4×10−14erg cm−2s−1or 2–3×1036erg s−1. At even lower fluxes
the cumulative number of sources decreases with decreasing flux. This behaviour is totally unexpected for a cumulative log N-log S relation. Assuming that the number of background sources as well as the number of sources of M 31 increase with decreasing flux the cumulative number of sources should keep raising with fainter and fainter limiting flux. A flattening could imply that most of the sources of M 31 in our sample emit
radiation at fluxes above 3–4×10−14erg cm−2s−1, which means that the examined source sample starts to
be dominated by background sources below that flux. The decreasing number of sources clearly indicates that the contribution of background sources is overestimated at these fluxes. There are several possible explanations for the discrepancy between the assumed and the actual number of background sources. First, the normalisation of the background CNC from the COSMOS field might differ from that of the local background CNC in the field of M 31. Unfortunately, it is not possible to determine the actual value for the field of M 31, as such a determination will always be affected by the presence of the galaxy. The estimation of the background log N-log S relation might be improved by using deep observations of fields close to M 31 that only contain background sources and foreground stars to adjust the normalisation from the COSMOS field. However it is a demanding task to select these fields, as the globular cluster system of M 31 is very
extended. It is also questionable whether>∼100 ks of observation time would be approved to observe those
fields, as the background contribution derived from these fields would still be only an approximation of the real background behind M 31. Second, the effects of absorption of X-rays from background objects by the interstellar medium of M 31, which were neglected so far as they reduce the flux in the 2.0–10.0 keV band
by a few percent only (cf.Table 8.1), might become important for fluxes near the detection limit. There,
a small reduction of the flux can bring the source flux below the detection threshold and hence the source is not detected. Third, faint background sources located close to bright sources of M 31 or behind regions of X-ray emitting gas might not be detected, due to the limiting conditions imposed by the detectors. To minimise the effects of the uncertainty in the number of background sources the following considerations are limited to fluxes above the flattening.
Going from faint to brighter luminosities, the background corrected CNC flattens at∼2.3×1037erg s−1
(∼3.2×10−13 erg cm−2s−1) and steepens again at ∼5.8×1037 erg s−1 (∼8×10−13 erg cm−2 s−1). This
feature was also reported in SBK2009 (cf.Fig. 3 of their paper). Given that SBK2009 determined the source
fluxes from spectral fits to the individual sources, whereas I used fluxes derived with emldetect and
assuming the same spectrum for all sources, the observed “bump” in the CNC seems not to be an artefact resulting from the adopted method, but an inherent property of the source population of M 31. Possible explanations for this “bump” are: (1) an independent source population, which only contains sources in the
flux range∼3.2–8×10−13erg cm−2 s−1, or (2) an excess of sources in the XRB population of M 31 in that
flux range.
To estimate the slope of the background corrected CNC I fitted the differential log N-log S relation. The
best fit slope (of the CNC above 3.2×10−14 erg cm−2s−1) is∼0.7, which lies in the range expected for
(a) Eastern part (b) Western part
(c) Northern part (d) Southern part
Figure 8.6:Cumulative X-ray log N-log S relation (blue), background CNC (red (observed) and green (extrapolated)) and the background corrected CNC (black) for the (a) eastern, (b) western, (c) northern and (d) southern part of the galaxy. The solid lines display the fits to the background corrected CNCs. The slopes are given in Table 8.5.
Figure 8.7: A comparison of the cumulative luminosity functions for the northern (blue) and southern (green) part and the eastern (black) and western (red) part of the disc.
Figure 8.6 shows the CNCs for the southern and northern part and for the eastern and western part of M 31. The solid lines display the fits to the background corrected CNCs of these four regions. Only data
points with fluxes above 3.2×10−14 erg cm−2s−1 were used in the fits. The best fit slopes are given in
Table 8.5. For the western and northern part, the slopes seem to be flatter than for the southern and eastern part. However the slopes of all four parts are consistent (within the errors) among each other and with the
slope derived for the whole galaxy. Interestingly, the number of sources per deg2detected at the same flux
limit (for fluxes above 3.2×10−14erg cm−2s−1) is higher in the southern and eastern part compared with
the northern and western part, although at higher fluxes the differences in the CNCs are within the errors (see Table 8.5). The “bump” that was detected in the CNC of the whole galaxy is also present in the CNCs of the eastern and northern part, while it is not visible in the western and southern parts. A comparison of the background corrected CNCs or cumulative luminosity functions (CLFs) of the four regions is shown in Fig. 8.7.
The cumulative log N-log S relations for the regions with increasing distance to the centre of M 31 are shown in Fig. 8.8. The best fit slopes are given in Table 8.6. Again, only data points with fluxes above
3.2×10−14erg cm−2s−1 were used, where for the outer disc region and the region beyond the D
25ellipse
the data points for fluxes larger than 10−13 erg s−1 were ignored. Grimm et al. (2003) discovered that
(a) Inner disc (b) Dust ring
(c) Outer disc (d) Beyond D25
Figure 8.8:Cumulative X-ray log N-log S relation (blue), background CNC (red (observed) and green (extrapolated)) and the background corrected CNC (black) for the (a) inner disc, (b) dust ring, and (c) outer disc regions and (d) for the region beyond the D25ellipse. The solid lines display the fits to the
Table 8.5:Number counts in the northern, southern, eastern and western part of M 31 for two different limiting fluxes
flux>3.2×10−14erg cm−2s−1 flux>10−13erg cm−2s−1
Region slope err srcs err srcs err srcs err srcs err deg−2 area−1 deg−2 area−1
Northern disc 0.65 0.07 29.11 2.03 37.99 3.82 13.54 2.11 17.75 2.77 Southern disc 0.73 0.06 37.08 3.22 49.43 4.29 15.22 2.21 20.41 2.96 Eastern disc 0.75 0.07 38.32 3.24 51.62 4.36 15.00 2.17 20.34 2.94 Western disc 0.61 0.07 27.72 2.91 35.73 3.75 13.76 2.15 17.83 2.79 sum (N+S) 87.42 4.06 38.16 2.87 sum (E+W) 87.35 4.06 38.17 2.87 all 0.70 0.03 33.14 0.76 87.5 2 14.39 0.95 38.1 2.5
later studies the range of slopes for HMXB populations was extended to 0.5–0.7, as long as the slope and its error was consistent with 0.6 (Grimm et al. 2005; Shtykovskiy & Gilfanov 2005). From the study of the log N-log S relations of the different radial regions it follows that the slopes of the inner disc and dust ring regions are consistent with the slope expected for an HMXB population. This is an interesting finding as up-to-now, not a single HMXB in M 31 is known for sure (for a discussion of HMXB candidates see Sect. 9.4.3.2). However, the slope, by itself, may not be a good indication of the underlying X-ray source population (Kong et al. 2003a).
Prestwich (2002) showed that flatter slopes indicate higher star formation rates. Applying this finding to the slopes of the different radial regions, we should expect the highest star formation rate (SFR) for the
dust ring region, followed by the inner disc region. In the outer disc region and the region beyond the D25
ellipse the star formation rate should decrease. These results are in agreement with the SFRs derived from GALEX observations (Boissier et al. 2007). The quantitative analysis of the correlation between HMXBs and SFRs, and a detailed discussion about HMXBs and log N-log S relations is presented in Sect. 8.4.3.1.
A comparison of the different background corrected CNCs or CLFs can be found in Fig. 8.9. Going from
the inner disc region to the region beyond theD25ellipse, the CNC keeps raising till FX∼1.6×10−14erg
cm−2s−1, ∼4×10−14 erg cm−2s−1, ∼2×10−14 erg cm−2s−1, ∼3×10−14erg cm−2 s−1, respectively. In
the outer disc region the number of background sources is larger than the number of M 31 sources for
fluxes smaller∼2×10−13erg cm−2s−1. In the outermost region the source population is clearly dominated
by the background sources, but we still expect M 31 to have about 13 sources/deg2 with fluxes above
∼3×10−14erg cm−2s−1.
In the range between∼1.6×10−13erg cm−2s−1 and∼8×10−13erg cm−2s−1, where the CNC of the
whole observed field showed the “bump”, we find evidence of such a bump in the CNC of the dust ring region, while the CNC of the inner disc region is rather flat. For the outer disc region and the region beyond
the D25 ellipse, the statistics are not good enough above∼3.2×10−13erg cm−2s−1to decide whether the
“bump” is visible or not.
The number of “hard” X-ray sources of M 31 at limiting fluxes of 3.2×10−14erg cm−2s−1and10−13erg
cm−2s−1are given in Table 8.6. The sum of all sources obtained from the individual regions (row “sum” of
Table 8.6) is in agreement with the values derived from the number counts of the whole field (row “all” of Ta- ble 8.6). A graphical illustration of Table 8.6 is given in Fig. 8.10, where the number of sources depending on
the distance to the centre is shown. For limiting fluxes of 3.2×10−14erg cm−2s−1and10−13erg cm−2s−1,
Figure 8.9: A comparison of the CLFs for the inner disc (black), dust ring (red), and outer disc (blue) regions and the region beyond the D25ellipse (green).
by an exponential profile:
I =I0×e−
a
a0 (8.8)
whereais the major axis of the elliptical regions anda0is the scale length. The best fit values are given in
Table 8.7. Exponential profiles are known to describe the (optical) luminosity distribution of spiral galaxy discs. Comparing the best-fit values for the scale length shows that the number of brighter sources decreases faster with increasing distance than the number of faint sources.
I also compared the number of M 31 sources to the number of background sources for the total field
(without bulge) at a limiting flux of 3.2×10−14 erg cm−2 s−1. The result is that about 60% of all sources
brighter than 3.2×10−14 erg cm−2s−1 are background sources. Determining the number of background
sources for different radial distances results in a contribution of 20% of background sources for the inner disc region, 57% in the dust ring region and about 80% each in the outer disc region and the region beyond
the D25ellipse. For a limiting flux of10−13erg cm−2s−1 these values change to 7%, 34%, 65% and 95%
going from the inner disc region to the region located beyond the D25 ellipse. These numbers, once more,
illustrate that the fraction of sources belonging to M 31 decreases with increasing distance to the centre of the galaxy.
Table 8.6: Number counts in the inner disc, dust ring, outer disc and beyond D25 regions for two
different limiting fluxes
flux>3.2×10−14erg cm−2s−1 flux>10−13erg cm−2s−1
Region slope err srcs err srcs err srcs err srcs err deg−2 area−1 deg−2 area−1
inner disc 0.68 0.09 202 30 42 6 115.4 22.3 24.3 4.7 dust ring 0.56 0.11 37 7 22 4 16.7 4.5 9.9 2.7 outer disc 0.82 0.36 12 4 10 3 4.6 2.0 3.8 1.7 beyond D25 1.69 0.65 13 3 13 3 0.28 0.95 0.3 1.0 sum 87 4 38.3 2.5 all 0.70 0.03 33.14 0.76 87.5 2 14.39 0.95 38.1 2.5
Figure 8.10: Number of sources detected in the 2.0–10 keV band depending on the distance to the centre of M 31. The colours illustrate different flux limits: black for a limit of 3.2×10−14erg cm−2s−1 and red for a limit of10−13erg cm−2s−1.
Table 8.7:Fit parameters for the distance distribution shown in Fig. 8.10
flux limit I0 a0(deg) a0(kpc) χ2 d.o.f.
3.2×10−14erg cm−2s−1 58.37+21.7
−17.9 0.74+0−0..2036 10.1+4−2..97 3.47 2.0
10−13erg cm−2s−1 53.74+28.2
−20.2 0.38+0−0..1012 5.2+1−1..64 1.51 2.0
8.4.3.1 High mass X-ray binaries and the luminosity function
Grimm et al. (2003) give the following relation between the number of sources (HMXBs) which are brighter
than 2×1038erg s−1in the 2.0–10.0 keV band and the SFR:
N(L >2×1038erg s−1) = (2.9±0.23) SFR(M¯yr−1) (8.9)
(Eq. 5 of Grimm et al. 2003). Taking the SFRs from Boissier et al. (2007) and the HIsurface density of
Chemin et al. (2009), we can compare the SFRs to the CLFs for the inner disc and dust ring region. In
addition, we can compare those quantities for the whole disc of M 31, by applying an SFR of∼1.0 M¯/yr
(Williams 2003).
Figure 8.11 shows the number of sources brighter than 2×1038erg s−1and the SFR for different galaxies
(taken from Grimm et al. 2003) and in addition the values for the inner disc, the dust ring region, and for the whole disc of M 31. Shown are the number of sources derived from the CLFs (red, green) as well as the values reduced by the amount of known LMXBs (blue). We assumed a poissonian distribution to determine the errors in the numbers of sources. All data points agree within their errors with the values expected from
Eq. 8.9. For the outer disc region and the region beyond the D25ellipse the low statistics do not allow to
derive a meaningful number of sources with luminosities brighter than 2×1038erg s−1. As M 31 contains
only a small amount of sources that are brighter than 2×1038 erg s−1 (2.0–10.0 keV band; ∼4 sources),
the above results does not allow to decide whether the inner disc or dust ring region contain an unknown HMXB population.
Grimm et al. (2003) also provides an expression for a universal cumulative luminosity function for HMXB populations, namely:
N(> L38) =
¡
5.4+2−1..17¢SFR¡L38−0.61±0.12−210−0.61±0.12¢ (8.10)
(Eq. 19 of Grimm et al. 2003), whereL38=L/1038erg s−1(2.0–10.0 keV),i. e.the luminosity is given in
units of1038erg s−1, and SFR denotes the star formation rate. Comparing the number of sources derived
from Eq. 8.10 with the CLFs of the inner disc and dust ring region, where I used the same values for the star formation rate as in Fig. 8.11, shows that in the inner disc region (Fig. 8.12(a)) HMXBs make up only for a fraction of the observed CLF. This implies that the inner disc region contains a large population of LMXBs. The CLF of the dust ring region is in agreement (within the errors) with the number of sources expected for an HMXB population (Fig. 8.12), although it looks like for most of the data points the expected CLF is higher than the measured CLF. This could imply that the number of background sources was overestimated for the dust ring region, which seems plausible since the dust ring region is the part of the disc of M 31 were
the highestNHvalues have been measured (cf.Fig. 8.1). Taking the values listed in Table 8.1 we should
expect that the flux from the background sources is reduced by∼20%–30% in the dust ring region. This
absorption effect was included in the determination of the number of background sources in the following
way. For a given fluxFg the “absorption-corrected” fluxFac=Fg/(1−a)was determined, whereais the
Figure 8.11: Star formation rate versus number of sources with luminosities above 2×1038 erg s−1
for different star forming galaxies from Grimm et al. (2003) and for the whole disc (green) and inner disc and dust ring region (red) of M 31. For the whole disc and inner disc region additional number of sources that are reduced by the amount of known LMXBs (blue) are given. Also shown is the relation between the SFR and the CLFs (lines) derived in Grimm et al. (2003).
background sources were derived, using:
N(> Fac) = ½ A F−α1 ac , Fac> Sb, B F−α2 ac , Fac≤Sb, (8.11)
where α1 = 1.43±0.10, α2 = 0.59 ±0.33, Sb= 1.02+0−0..2519×10−14 erg cm−2s−1, A= 266 ±11, and
A=B Sα1−α2
b (Cappelluti et al. 2007).
Figure 8.13 shows the CLFs of the dust ring region for an absorption of 0%, 20%, and 30%. In addition, the expected number of HMXBs for the SFR measured in the dust ring region (Eq. 8.10) is plotted. We see that the number of sources detected in the dust ring region is in agreement with the values predicted by Eq. 8.10, and that this finding is independent from the absorption used. With the absorption included there are less data points where the expected CLF is higher than the measured CLF.
In summary, the comparison of the measured CLF with the number of sources predicted from theoretical considerations suggests that the dust ring region contains an up to now unknown population of HMXBs.
(a) Inner disc (b) Dust ring
Figure 8.12:Cumulative X-ray luminosity function (black) and expected CLF of an HMXB population (red), derived from Eq. 8.10, for the (a) inner disc and (b) dust ring regions. The errors of the expected CLF increases with decreasing luminosity, as they also contain the uncertainties in the normalisation and slope of the luminosity function given in Eq. 8.10.
sources located in the dust ring region and detected in the 2.0–10.0 keV band, five source are identified as SNRs or SNR candidates, and a further 14 sources are identified as XRBs or GlCs or as candidates for these two classes. From the five sources identified as SNRs or SNR candidates four sources have 2.0–
10.0 keV fluxes below 2.6×10−14erg cm−2s−1. This means that from the∼100 sources/deg2 (assuming
an absorption of 20%) with a flux larger than10−14erg cm−2 s−1 we expect∼24 sources to be LMXBs
and about seven to be SNRs. Taking into account the corrections for the absorption, and for the numbers of