3. Estimating the total population
In the work of DiStefano & Rappaport (1994) a population of supersoft sources has been derived e.g. for the M31 galaxy by making certain assumptions about their spatial distribution, temperature and luminosity distribution. Here the work of DiStefano & Rappaport (1994) can be significantly extended as the sample derived from the observations has been enlarged significantly and a white dwarf mass distribution and a hydrogen column density distribution is derived for this sample.
3.1. The observation derived distribution
In Fig. 1 the distribution of supersoft sources from Table 1 with well determined values is shown as a histogram. Each source is distributed with fractional numbers into a number of bins determined by the uncertainty of the value of given in Table 1. Errors for these fractional numbers have been calculated in the following way. Distribution of the same fractional numbers in a range of bins which is twice as large (twice the error) reduces the fractional number per bin by a factor of 2. Therefore we used 0.5 the fractional value per bin as the error per bin. Two histograms are given. The white histogram comprises all sources (i.e. 32) for which the could be constrained reasonably well, the black histogram comprises objects (i.e. 26) not coinciding with foreground stars or M31 supernova remnants. The fact that for a comparatively large sample of 26 objects hydrogen column densities can be inferred allows to probe their spatial distribution in the galaxy disk assuming a simple scale height law like an exponential law. This method allows not only to probe their distribution but also to derive a mean scale height of the detectable population. It might well be required to extend this analysis to much larger values in excess of in order to cover the more deeply embedded objects. We extended our calculations to absorbing column densities as high as and applied them e.g. to the source with the catalog index 156 (cf. Table 1). We find that this source is consistent to be highly absorbed (), the total column density of the M31 disk at that location is .
3.2. The derived white dwarf mass distribution
Deriving a mass distribution of a galaxy population is by far not trivial. In case of white dwarfs in supersoft sources it appears to be possible to derive reliable estimates of the masses using certain assumptions which have to be shown in later work to be correct or at least not completely unreasonable (cf. discussion in Kahabka 1998). From the numbers in Table 1 a mass distribution has been set up (cf. Fig. 2). Each source is distributed with fractional numbers into a number of bins determined by the uncertainty of the value of given in Table 1. The histogram comprises all objects (i.e. 26) not coinciding with foreground stars or M31 supernova remnants for which the mass could be constrained reasonably well. The figure shows that only objects with masses in excess of can be detected which is in agreement with the expected detection limit in ROSAT PSPC count rates for the used exposure time of the observations (cf. Kahabka 1998). The reason is simply that for lower white dwarf masses the X-ray luminosities ate too low to be detectable.
In Fig. 3 the cumulative number distribution of white dwarfs in supersoft sources with masses is given as deduced for the Milky Way in the calculations of Yungelson et al. (1996) for the - and the hydrogen-burning shell approximation. is the time it takes the white dwarf to decline by 3 magnitudes in its bolometric luminosity. Four distributions of supersoft sources are given for each approximation, the distribution of the CV class, the subgiant, symbiotic class and the total distribution. The total distributions have been used in our further discussion, e.g. to derive from the observed white dwarf distribution the predicted total distribution. In the approximation 113 (out of 1895) objects are expected to be seen in the Milky Way with white dwarf masses in excess of after correction for the limited visibility due to the constraint and about 226 and objects in the twice as large M31 galaxy (see Sect. 3.3). In the hydrogen-burning shell approximation 107 (out of 1553) objects are expected to be seen in the Milky Way with white dwarf masses in excess of after correction for the limited visibility due to the constraint and about 214 objects in the twice as large M31 galaxy. If one assumes that one is complete for masses in excess of then the observed number of 4 systems (cf. Fig. 2) would give a total population of 500 in the approximation and a total population of 2000 in the hydrogen-burning shell approximation (as 16 for a population of 1895 are predicted in the and 3.3 for a population of 1553 in the hydrogen-burning shell approximation). The number of the total population deduced from the first approximation appears somewhat small for the M31 galaxy.
3.3. Correcting for the and distribution
Using the observationally derived and distribution one can, by comparing with the predicted distributions infer a total number of the population.
In a simple approach a double exponential description as e.g. introduced in DiStefano & Rappaport (1994) can be used to describe the source and the distribution. The scale height of the source distribution is assumed to be different from the scale height of the distribution . Expressing the density distribution of the gas and hence the distribution as
where z is the distance from the galaxy plane, the gas density, and the gas (roughly the hydrogen) column density at the base of the galaxy disk and expressing the source distribution as
with the integrated (total) number of sources in the upper hemisphere of the galaxy disk (half of the total population) and the exponential scale height of the source population, then one gets from Eq. 5
Then Eq. 6 can be reduced to
then Eq. 8 reduces to
Eqs. 8 and 11 give the expected number of sources (above the galaxy disk) and within the normalized interval . From the distribution of observed numbers per interval the scale height ratio can be derived (as well as the total number of the population ). This distribution has a powerlaw behavior with slope (h-1). If the slope of the source distribution equals the slope of the gas distribution then the scale height ratio is . The scale height for the gas may be in the range 150-600 pc for the M31 galaxy (cf. Braun 1991).
© European Southern Observatory (ESO) 1999
Online publication: March 18, 1999