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Astron. Astrophys. 344, 459-471 (1999)

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4. Constraining the scale height ratio and the total population from the normalized NH histogram

We now will derive from Eq. 11 the total population of supersoft sources in M31 and the scale height of this population with respect to the scale height of the M31 gas. In a first step we define the models used for the M31 gas, in a second step we derive the scaled [FORMULA] distribution and in a third step we apply a least-square fit to the scaled [FORMULA] distribution.

4.1. Possible [FORMULA]-models

From Eq. 8 follows that the scaled [FORMULA] distribution has to be considered in order to derive the relative scale height of the source distribution and the total number of the population. We now discuss two possible models for the [FORMULA]-distribution: a schematic one by Supper et al. (1997) and a detailled based on radio observations.

4.1.1. The Supper-[FORMULA] model

In a first approach we use the galaxy [FORMULA] model given in Fig. 12 of Supper et al. (1997). The galaxy is divided into 3 concentric ellipsoids covering the disk and one circle at the central bulge. The positions of the candidate supersoft sources have been projected onto the galaxy disk (cf. Fig. 4). The disk of M31 may be warped and flaring at the outer part (cf. Evans et al. 1998). Such a warping and flaring of the M31 disk may affect the scale height assumptions of those supersoft sources which are found in annulus III of the [FORMULA]-model of Supper et al. (1997). This point deserves further investigation (cf. the galactocentric dependence of a galaxy scale height given by Evans et al. 1998 for the M31 disk, cf. also Braun, 1991).

[FIGURE] Fig. 4. [FORMULA] model of M31 as taken from Supper et al. (1997), Fig. 12 for the SW-sample (upper panel ) and C-sample (middle panel ). Bottom: [FORMULA] image of M31 deduced from the radial profile of the HI intensity (Urwin 1980). The positions of the supersoft source candidates (the SW-sample and the C-sample) are given (lower panel ).

4.1.2. The Urwin-[FORMULA] model

As a more refined model for the [FORMULA] distribution in M31 the Urwin (1980) model is used. The radial distribution of the hydrogen column is calculated from the profile given in Fig. 8 of Urwin (1980) making use of the equation given in Dickey & Lockman (1990)

[EQUATION]

with the brightness temperature [FORMULA], the integral is over the velocity profile. With an inclination of [FORMULA] of the galaxy a maximum column density of [FORMULA] is derived for the NE HI profile and a maximum of [FORMULA] for the SW profile. The HI profile (not corrected for inclination) as determined from Eq. 12 with [FORMULA] taken from Fig. 8 of Urwin (1980) is shown in Fig. 5.

[FIGURE] Fig. 5. Radial hydrogen column density profile of M31 not corrected for the inclination of the galaxy. The solid histogram gives the north-eastern (NE) profile and dashed histogram gives the south-western (SW) histogram (calculated from the [FORMULA] given in Fig. 8 of Urwin (1980) and Eq. 12).

We do not take molecular hydrogen into account. We just mention that a value of [FORMULA] has been measured at the location RA (1950) = [FORMULA], Decl (1950) = [FORMULA] due to molecular hydrogen (cf. Urwin 1980, page 257).

4.2. Expected incompleteness of coverage as a function of hydrogen column density and white dwarf mass

In Kahabka (1998) the theoretically expected source count rate has been derived at the distance of M31 from non-LTE white dwarf atmosphere models (model M4) for white dwarf masses in the range [FORMULA] under the assumption the source is on Iben's stability line of surface hydrogen burning (cf. Iben 1982). We extended these calculations (model M5) to white dwarf masses as low as [FORMULA] (cf. Sect. 2.1). Taking the theoretical white dwarf mass distribution derived by Yungelson et al. (1996) into account a number/count rate diagram was calculated as a function of the hydrogen column. The result is given for models M4 and M5 in Fig. 6. From this diagram the completeness correction factor as a function of the hydrogen column has been derived making the following assumption. The X-ray survey of M31 by Supper et al. (1997) is according to Fig. 13 of Supper complete for ROSAT PSPC count rates [FORMULA]. From our Fig 6 the fractional number of sources seen for a specific hydrogen column assuming a cut-off count rate of [FORMULA] is derived. This fraction is equal to 1.0 for hydrogen columns [FORMULA]. The inverse of this fraction has been used as the correction factor to derive the completeness corrected normalized [FORMULA] histogram making use of the specific galaxy [FORMULA] model. Making e.g. use of the Supper galaxy [FORMULA] model it follows that for all annuli of the galaxy ellipse (including the bulge) completeness is not guaranteed and the lower hemisphere population is only partially seen. This agrees with the rather small fraction of 0.22 of the total galaxy population found for [FORMULA] i.e. at the other side of the midplane of the disk of M31. From Fig. 6 it becomes clear that for [FORMULA] only candidates with masses [FORMULA] are detectable. This means adding the mean foreground [FORMULA] of [FORMULA] in Supper's model annulus III is opaque for the lower hemisphere population with [FORMULA] but the bulge and annulus II are transparent for somewhat less massive white dwarfs ([FORMULA]).

[FIGURE] Fig. 6. Theoretical number/count rate diagram of supersoft sources for a galaxy of the size of the Milky Way and for a distance of 700 kpc (M31). Note that a galaxy like M31 has a mass twice as large as the Milky Way. These numbers follow from the population synthesis calculations of Yungelson et al. (1996) for the Milky Way galaxy. Labels mark hydrogen column densities ([FORMULA]). Dots mark white dwarf masses (counted from bottom to top) 1.34,1.29,1.25,1.20,1.15,1.10,1.00(twice),0.95,0.90,0.85 [FORMULA]. This figure shows the results for model M5 ([FORMULA]) and for model M4 ([FORMULA]).

Making use of the number/count rate distribution derived from model M4 and M5 the fraction of objects seen for different hydrogen columns has been calculated and the result is given in Fig. 7. This fraction is equal to 1.0 for columns below [FORMULA], which means completeness is fulfilled, and decreases for increasing columns.

[FIGURE] Fig. 7. Fraction of sources with white dwarf masses above [FORMULA] seen in the Andromeda galaxy (M31) for different hydrogen absorbing columns. This fraction is equal to 1.0 for columns below [FORMULA], which means completeness is fulfilled and decreases for increasing columns.

4.3. Number of supersoft sources expected for the Supper-[FORMULA] model

The scaled [FORMULA] values have been derived making use of the [FORMULA] ranges given in Table 1, deriving the local [FORMULA] values from the Supper [FORMULA] model and by applying the completeness correction. The scaled [FORMULA] histogram is plotted in Fig. 8. This distribution extends from n=0 to n=2 and comprises both hemispheres of the galaxy. It turns out that of the 18 sources in the distribution 22% (4 of 18) fall below the galaxy disk. This may not be unexpected as the masses and hence temperatures of the white dwarfs involved are substantial and the total [FORMULA] of the galaxy disk is only in one [FORMULA] ellipse large enough (i.e. [FORMULA]) to hide the lower hemisphere population nearly completely.

[FIGURE] Fig. 8. Normalized [FORMULA] distribution corrected for completeness with 80 (corrected) candidates with well determined [FORMULA] values in the n=0.0-1.0 interval. The best-fit for a population of 1000, 2500, and 5000 sources is given as dashed histograms.

The total population can be inferred with Eq. 11. Using the number of the corrected population above and below the galaxy disk of 80 (which is uncertain in the range 30-130 considering the errors, cf. Fig. 8) and assuming that only a fraction of 6.0% of the whole population in the [FORMULA] and of 6.9% in the hydrogen-burning shell approximation is covered as only objects with white dwarf masses in excess of 0.90[FORMULA] are detected a total population of 1300 (500-2200) and 1200 (430-1900) respectively is derived for the Andromeda galaxy. These numbers are consistent with the range of [FORMULA]800-5000 supersoft sources predicted from the population synthesis calculations of DiStefano & Rappaport (1994). The distribution would be consistent to be centered at [FORMULA]. This fits with a disk population of a scale height significantly smaller than the gas scale height. A scale height ratio can be constrained from this histogram. This means the scale height of the source distribution [FORMULA] can be determined with Eq. 9 if the scale height of the gas distribution [FORMULA] is known. As a function of galactocentric radius [FORMULA] varies from 150 pc to 600 pc (Braun 1991). A chi-squared fit has been applied to the normalized [FORMULA] distribution. The result of a chi-squared fit of Eq. 11 to the distribution given in Fig. 8 is given in Fig. 9. The range of the population follows from the chi-squared fit to the measured distribution taking the errors into account. A total population of 1,800-5,800 sources is obtained for h-values 1[FORMULA]h[FORMULA]6, which means for a source population which is more confined to the galaxy disk than the gas distribution. If there is a large population of supersoft sources in M31 then the sources are very confined to the galaxy plane. There may exist a number of the order 200 hot ([FORMULA]) and X-ray luminous planetary nebula nuclei in a spiral galaxy of the size of M31 according to the estimates of Iben & Tutukov (1985). They can be a minor sub-population of a larger population of luminous supersoft sources but with a larger scale height (h[FORMULA]1). From the formal fit of Eq. 11 to the normalized [FORMULA] distribution (making use of the Supper-[FORMULA] model) we would exclude that a population of hot and luminous planetary nebula nuclei (of order of 200 objects) alone account for the observed sample.

[FIGURE] Fig. 9. Result of chi-squared fit (of Eq. 11) to the distribution of Fig. 8. The scale height ratio [FORMULA]-population plane is shown. The fit has been applied to the [FORMULA] distribution (upper hemisphere). The 90% and 95% confidence bounds are shown as dashed and solid lines. The bar gives the estimate derived from the galactic population, cf. Sect. 4.5.

4.4. Number of sources expected with the Urwin-[FORMULA] model

The normalized [FORMULA] histogram is also calculated by making use of the Urwin [FORMULA] model for the north-eastern (NE) part of the galaxy. This [FORMULA] model consists of a radial distribution with 57 rings. This distribution has been converted into a galaxy [FORMULA] model of M31 by assuming an inclination of the galaxy of [FORMULA] (cf. Fig. 4). This is a more refined model than the Supper-[FORMULA] model. The normalized distribution is given in Fig. 10. This distribution appears to cover only parts of the normalized [FORMULA]-bins. The main histogram extends over the range [FORMULA]=0.0-0.6. This fact can be explained if one considers the galactocentric distribution (12-16 kpc) of the sources which fall into this interval (cf. Sect. 5) and the projected hydrogen columns of the M31 galaxy [FORMULA] for these radii. The hydrogen columns are that large that indeed only part of the upper hemisphere population is detectable in agreement with the histogram extending to values well below n=1.0. The entries in the histogram for n[FORMULA] are from the population found at radii 18-23 kpc. Here the projected hydrogen columns of [FORMULA] are lower and the lower hemisphere population is detectable. But this part of the histogram is not very significant. We constrain our fit of Eq. 11 only to the n=0.0-0.6 regime. This allows to constrain the size of the population and the scale height ratio. As we do not cover the top of the distribution we are not able to determine an upper bound for the population. Only by constraining the scale height ratio to realistic values for stellar populations we can determine an upper bound for the population.

[FIGURE] Fig. 10. Normalized [FORMULA] distribution corrected for completeness with 24 (corrected) candidates with well determined [FORMULA] values in the n=0.0-0.6 interval. The best-fit for a population of 1000, 2500, and 5000 sources is given as drawn and dashed histograms.

The size of the population as derived with the Urwin-[FORMULA] model is consistent to be in the range [FORMULA]1000-10,000 sources for a scale height ratio h=1-5 (cf. Fig. 11).

[FIGURE] Fig. 11. Result of chi-squared fit (of Eq. 11) to the distribution of Fig. 10. The confidence plane for the scale height ratio [FORMULA]-population plane shown. The 95% confidence bound is given as solid line and the 90% confidence bound as dashed line. The 10% confidence is given by a dotted line. The fit has been applied to the [FORMULA] distribution (part of upper hemisphere).

As a refined model the radial profiles of the north-eastern (NE) and the south-western (SW) galaxy as given in Urwin (1980) have been used to calculate a [FORMULA]-map of the galaxy and to deduce the hydrogen-column at the location of each supersoft source. The normalized [FORMULA] distribution has been calculated which is given in Fig. 12. This distribution extends over a similar [FORMULA] range as for the Urwin model. A fit of Eq. 11 to this distribution for a population is given in Fig. 13 as a function of the scale height ratio [FORMULA]. This distribution again extends mainly over the n=0.0-0.6 interval (see discussion above). The size of the population is [FORMULA]1000-10,000 for a scale height ratio h=1-5. There are sources from Table 1 which fall beyond the n=2 limit and are rejected (in the specific [FORMULA]-model). For the NE-SW Urwin model these sources are found either at radii [FORMULA]15 kpc or at radii [FORMULA]5 kpc. The nature of these sources is unclear or the [FORMULA]-model is still too crude (but see discussion in Sect. 5). Some sources correlate with a foreground star or a M31 supernova remnant. Another possibility is that these sources are located at a large distance from the galaxy plane ([FORMULA]500 pc) and are projected due to the considerable inclination of the galaxy towards the wrong reference hydrogen column. But this appears to be quite unlikely.

[FIGURE] Fig. 12. Normalized [FORMULA] distribution corrected for completeness with 23 (corrected) candidates with well determined [FORMULA] values deduced with the Urwin [FORMULA] model (NE and SW galaxy) in the [FORMULA] interval n=0.0-0.6. The dashed histograms give the best-fit for a population of 1000, 2500, and 5000 sources.

[FIGURE] Fig. 13. Result of chi-squared fit (of Eq. 11) to the distribution of Fig. 12. The scale height ratio [FORMULA]-population plane is shown. The 90% and 90% confidence bounds are shown as dashed and solid lines. The 10% confidence bound is shown as dotted line. The fit has been applied to [FORMULA] distribution (part of upper hemisphere). The bar gives the estimate derived from the galactic population, cf. Sect. 4.4.

The NE-SW model may describe the distribution of the hydrogen column in M31 in a good approximation. It becomes evident that in the range of galactocentric radii 12-16 kpc where most supersoft sources are found the hydrogen column is that large that only part of the upper hemisphere population is visible. The total population can be constrained dependent on the scale height ratio.

In Table 2 the size of the population of supersoft sources in M31 as derived from different galaxy [FORMULA] models is summarized. In the Supper [FORMULA] model numbers have been derived from the n=0-1 histogram (the complete upper galaxy hemisphere) and in the Urwin [FORMULA] model from the n=0-0.6 histogram (60% of the upper galaxy hemisphere). Interestingly the range of the population derived from different [FORMULA] models does not differ much. This may be due to the fact that the errors associated with the (corrected) numbers are substantial due to the small number of selected sources. In order to better confine the range of the population detections of supersoft sources in the 12-16 kpc ring for values n[FORMULA]0.6 are required. Such sources are heavily absorbed, they have hydrogen columns [FORMULA] according to Fig. 6 and they are only detected in the ROSAT 1991 survey of Supper if the white dwarf mass is in excess of [FORMULA] (see possible candidates in the C-sample, cf. Table 1).


[TABLE]

Table 2. Population of supersoft sources in M31 derived from a chi-square fit of Eq. 11 to the normalized [FORMULA] histogram for different galaxy [FORMULA] models and different scale height ratios [FORMULA].


4.5. Comparison with the galactic population

There is evidence that the group of observed galactic supersoft sources is larger than assumed. Patterson et al. (1998) proposes three sources to belong to this family, e.g. V Sge, T Pyx and (possibly) WX Cen. These blue and optically bright binary systems have orbital periods of 12, 2, and 10 hours. Assuming distances of 1.3, 2.5, and 1.4 kpc the objects are found 200, 430, and 16 pc above the galactic plane. The two "standard" galactic supersoft sources RX J0925.7-4758 and RX J0019.8+2156 are 33 and 840 pc above the galactic plane assuming distances of 1 kpc. Assuming an exponential z-distribution (cf. Eq. 6) and assuming a scale height for the source population [FORMULA] the total population can be constrained in order to be consistent with this sample. This is an independent consistency check for the distribution and size of the galactic sample. Assuming a scale height of 150 pc the population has to be greater than 270 in order to explain the discovery of one source such as RX J0019.8+2156 at such a large scale height. Assuming a much smaller scale height of 30 pc the probability of observing one RX J0019.8+2156 is negligible. The scale height of a population of supersoft sources which can explain RX J0019.8+2156 has to be larger than 105 pc if the total population is [FORMULA]3000. This is not a problem as 105 pc is still a small scale height for stellar populations. T Pyx is a recurrent supersoft source which is found 200 pc above the galactic plane. T Pyx may harbor a massive white dwarf as it has a recurrence period of 20 years. According to the population synthesis calculations of Yungelson et al. (1996) there may be 113 galactic supersoft sources which are more massive than [FORMULA]. These sources can become recurrent supersoft sources with such a recurrence period (cf. Kahabka 1995). Assuming a scale height of 105 pc for the source distribution 2 sources are expected to be found at a distance from the galactic plane as large as in T Pyx (430 pc). To observe one T Pyx is therefore in full agreement with this number. The distance from the galactic plane of all other supersoft sources is considerably smaller and is in agreement with such a population. The conclusion is that the prediction of a population of [FORMULA]1900 supersoft sources in the Milky Way by the population synthesis calculations of Yungelson et al. (1996) is in agreement with the so far discovered galactic population if the scale height is [FORMULA]100 pc. One expects then 0.6 systems to be observed at the distance from the galactic plane of 840 pc, the distance of RX J0019.8+2156.

Assuming a scale height for the galactic supersoft sources of 100 pc and a scale height for the gas of 200-600 pc a value [FORMULA] (Eq. 9) is derived. Scaling with the mass ratio of the Andromeda galaxy and the Milky Way galaxy, which is about 2, one expects from the population synthesis calculations of Yungelson et al. (1996) that there exists a population of [FORMULA]3800 supersoft sources in M31. Such a population having [FORMULA] is consistent with the chi-squared fit to the normalized [FORMULA] distribution given in Figs. 9, 11 and 13. There is still the possibility of a bi-modal population consisting of a more extended population (e.g. the CV-type supersoft sources) and a more to the galaxy plane confined population (e.g. the subgiant class). We fitted such a bi-modal population to the normalized [FORMULA] histogram of Fig. 12 and find that an extended ([FORMULA]) population of 500 sources and a confined (h[FORMULA]10) population of [FORMULA] sources is possible.

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© European Southern Observatory (ESO) 1999

Online publication: March 18, 1999
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