Astron. Astrophys. 319, 525-534 (1997)

2. Theoretical expectations

2.1. Assumptions

For a star moving supersonically with velocity v relative to the interstellar gas, the amount of mass which is accreted per unit time is , where is the gravitational capture radius, is the mass of the neutron star, n the number density of the ISM (, where is the hydrogen number density, for standard chemical composition) and is a parameter of order unity which encompasses the theoretical uncertainties on the value of . In the following we take = 1 (see e.g. Novikov & Thorne 1973). At the typical values of the ISM density (outside molecular clouds), it is not clear that the fluid approximation applies at the gravitational capture radius (see Koester 1976) and the accretion rate could be greatly reduced. However, low magnetic fields, frozen in the interstellar medium, may effectively couple the gas particles. In this case, if the Larmor radius is smaller than the gravitational capture radius, the fluid approximation can be still considered valid and the accretion rate is reasonably expressed through the previous formula. The total luminosity L emitted by an ONS is then

where is the relativistic efficiency ( for a neutron star radius of km and a mass of ) and c is the light velocity. If the star moves subsonically relative to the ISM, Eq. (1) remains valid provided that the ISM sound speed is used in place of v ().

As shown by Eq. (1), in order to investigate the detectability of ONSs, it is crucial to have knowledge of the density distribution of the ISM and the velocity distribution of ONSs in the Galaxy. Calculations of the temporal evolution of the ONS distribution function have been carried out by many authors (Paczyski 1990; Hartmann, Epstein & Woosley 1990; Blaes & Rajagopal 1991; Blaes & Madau 1993; Zane et al. 1995a). Since we are interested to investigate the detectability of ONSs in the solar neighbourhood and close to the Galactic plane, in the following we will use the analytic approximation to the ONS distribution computed by Zane et al. (1995a).

The structure of the Local Interstellar Medium has been widely investigated: within 50-100 pc from the Sun, the gas is tenuous ( cm^-3) and warm ( K), although the presence of a number of relatively high density regions ( cm^-3) can rise the observed column density up to cm^-2 along certain line of sights, such as those in the direction of Cygnus Rift and Cygnus OB7 (Frisch and York 1983; Paresce 1984; Welsh et al. 1994; Diamond et al. 1995). On much larger scales, according to Dickey & Lockman (1990), the average density in the Galactic plane is cm^-3 with a scale height variable with the distance from the Sun, although there are a number of tenuous ( cm^-3) hot ( - K) bubbles. Since Cygnus Rift and Cygnus OB7 are very close to the Galactic plane (), in the following the ISM will be approximately described as a piecewise constant density medium with cm^-3 for pc and cm^-3 elsewhere. For the sources embedded in the clouds and the background sources, the contribution of the cloud material to the total absorption will be added.

A thorough investigation of the emitted spectrum turns out to be a further key ingredient in order to correctly address the issue of detecting ONSs, since the choice of the energy bands where to look for and the absorption of the interstellar medium are strongly related to the ONSs emission properties. As shown by Alme & Wilson (1973), if binary Coulomb collisions between the infalling ions and the atmospheric electrons dominate, as it is expected at very low accretion rates, the accretion flow can be stopped at several Thomson depths and the resulting spectrum can be thermalized at a temperature approximately equal to the star effective temperature

where is the fraction of the surface area which undergoes accretion (see below). Then, ONSs accreting directly from the ISM emit typically in the ultraviolet and soft X-ray bands. The low bolometric luminosity ( erg s^-1) and the softness of the spectrum ( eV) explain the difficulty in observing an isolated ONS. Previous investigations were restricted to the calculation of the emitted spectrum at high luminosities. A detailed numerical analysis of the spectral properties of unmagnetized neutron stars accreting well below the Eddington limit has been recently presented by Zampieri et al. (1995). The emergent spectrum turns out to be significantly hardened with respect to a black body at the star effective temperature, with a broad maximum shifted toward higher frequencies by a factor 3 at erg s^-1. This fact is due entirely to the frequency dependence of the free-free opacity, for which higher energy photons decouple at larger depths and temperatures in the neutron star atmosphere.

In the following, we will consider two possibilities for the emitted flux: first, black body emission at the neutron star effective temperature (see Eq. 2) which has been frequently used in the previous investigations and can be regarded as a useful approximation; second, the spectra computed by Zampieri et al. (1995). In addition, we assume that a relic magnetic field is present and that the accretion flow is channeled into the polar caps. If we neglect all the radiative effects produced by the magnetic field (on this regard see e.g. Miller 1992; Shibanov et al. 1992; Nelson et al. 1995), the main consequence is to limit the size of the emitting region by a factor , where is the area of the polar cap and is the Alfvén radius. For G, as we consider here, . This fact will produce a hardening of the spectrum with respect to the unmagnetized case with the same luminosity, since the flux emitted per unit surface is .

2.2. Expected number of ONSs

The technique which has been used to calculate the expected number of ONSs in the molecular clouds is the generalization of a similar procedure introduced by Zane et al. (1995b) and is described in detail in the Appendix. With this technique it is possible to calculate also the expected number of foreground and background ONSs accreting from the average ISM, which are seen in the direction of the clouds but are not embedded within them. This aspect deserves particular notice since, as shall see, they are expected to be quite numerous at low sensitivity limits and could be distinguished from the ONSs embedded in the molecular clouds only by means of their observed spectral properties.

The characteristic parameters of the two clouds are quoted in Table 1. Results of the expected number of sources observable in Cygnus Rift, Cygnus OB7 and also in other spatial regions in the cloud directions are presented in Tables 2 and 3 (fifth column) for different values of the sensitivity limit S (, G). These numbers are calculated assuming the spectrum of Zampieri et al. (1995) and a black body spectrum (values in brackets). The fraction of clouds which has been covered by the ROSAT pointings at a given sensitivity depends mainly on the exposure time. In the second column of Tables 2 and 3 we quote the fractional coverage of the cloud areas f as a function of threshold. As can be seen, the covering increases from 1% at c s^-1 to 5% at c s^-1. To compare the theoretical estimates with the actual number of unidentified X-ray sources detected in the ROSAT pointings of our sample, for each interval of distance we have computed the expected number of detectable sources according to the following formula

where is the expected number of ONSs above threshold detectable in the direction of the whole cloud areas and is the fractional coverage of the pointings at the same threshold (see Tables 2 and 3). Here c s^-1, c s^-1, c s^-1, c s^-1, c s^-1 and . The first term on the right hand side of Eq. (3) gives the number of sources observable in the pointings with limiting sensitivity , while the other terms account for the contributions from pointings at lower sensitivity. Finally, the total expected number of sources is obtained integrating over distance and is shown in Fig. 2 (summing the contributions from the two clouds).

Table 1. Characteristic parameters for Cygnus Rift and Cygnus OB7

Table 2. Expected number of ONSs detectable in the direction of Cygnus Rift as a function of threshold (, G). Values in the third column (hardness ratio HR) and in the last two columns refer to the Zampieri's spectrum and to a black body (values in brackets). is the interval of distance considered, the expected number of ONSs above threshold S detectable in the direction of the whole cloud area (see the Appendix) and the expected number of sources in the fraction of the cloud f covered by ROSAT pointings (see Eq. (3)).

Table 3. Expected number of ONSs detectable in the direction of Cygnus OB7 as a function of threshold (, G). Values in the third column (hardness ratio HR) and in the last two columns refer to the Zampieri's spectrum and to a black body (values in brackets). is the interval of distance considered, the expected number of ONSs above threshold S detectable in the direction of the whole cloud area (see the Appendix) and the expected number of sources in the fraction of the cloud f covered by ROSAT pointings (see Eq. (3)).

Fig. 1. Map of the analyzed ROSAT PSPC pointings. Each circle is radius. The black circles correspond to pointings containing ONS candidates (see text). The contours of the clouds are from Dame & Thaddeus (1985).

Fig. 2. The - distribution for all the sources detected by ROSAT (continuous line) and all the optically unidentified sources detected by ROSAT (dashed line). Also shown is the expected number of ONSs, , in the fraction of the clouds covered by ROSAT pointings (dash-dotted line; Zampieri's spectrum).

In the fraction of the clouds actually observed by ROSAT, we expect the detection of 10-24 sources (the exact value depends on the assumption on the emitted spectrum) at the sensitivity limit of c s^-1 in the direction of Cygnus Rift and of 5-13 sources in the direction of Cygnus OB7. Among these sources, 6-11 sources are expected to be really embedded in Cygnus Rift and 3-6 in Cygnus OB7. The other objects are foreground or background sources; in particular, we estimate that 3-8 and 2-6 foreground ONSs should be detectable in the direction of Cygnus Rift and Cygnus OB7, respectively. We note that at high thresholds c s^-1 the detection of a significant number of sources would be expected.

Then, in agreement with previous investigations (Blaes & Madau 1993; Colpi, Campana & Treves 1993; Zane et al. 1995a) Cygnus Rift and Cygnus OB7 are expected to be particularly favourable sites for the observability of ONSs with ROSAT PSPC. In addition, we have found that in their direction a significant number of foreground and background ONSs should be observable.

2.3. Expected emission properties

In principle we could compare directly the theoretical spectral distribution with the observed one but, as we shall see, the sources selected in our sample have too few photons to extract a spectrum. Then, in order to compare our theoretical expectations with the observations, in each range of distances considered we have simulated PSPC spectra by folding the theoretical model with interstellar absorption, detector effective area and response matrix, and calculated the PSPC hardness ratio, defined by

where and are the count rates in the PSPC channel ranges 11-40 and 41-240, corresponding roughly to the energy bands 0.1-0.4 keV and 0.4-2.4 keV. We have repeated the calculation assuming either the spectrum computed by Zampieri et al. (1995) or a black body at the neutron star effective temperature. Results are presented in Tables 2 and 3, where the lowest and highest values of the hardness ratio is reported for each interval of distance (the highest value refers to the model with maximum luminosity). It is interesting to note that, for distances above pc for the black body and pc for the synthetic spectra, the absorption of the interstellar medium and the cloud material causes the hardness ratio to be very close to unity. Then, although ONSs are relatively soft sources, above these distances they should appear significantly hardened.

We have estimated also the apparent visual magnitude of ONSs below 100 pc from the Sun and find that . Then, these sources should lack of optical counterparts in the digitized red plates of the Palomar All Sky Survey (POSS, limiting magnitude 20). According to recent findings by Blaes, Warren & Madau (1995), for ONSs embedded within the clouds and considering polar cap emission the reprocessing of the UV-soft X-ray radiation by the surrounding gas might increase the emitted flux at optical wavelengths by 1-2 magnitudes. However, even in this case the very low visual magnitude prevents any possibility of optical identification for sources beyond 100 pc. Then, we will use the lack of optical counterparts in the digitized POSS plates as a distinguishing criterion for selecting ONSs candidates among ROSAT sources.

Inserting in Eq. (A2) the appropriate effective areas and frequencies (taken from the world wide web site of the Center for Extreme Ultraviolet Astrophysics), we have calculated also the expected count rates in the Lex band (67-178 Å) of the Extreme Ultraviolet Explorer Deep Survey (EUVE DS). Below 250-300 pc, the count rate turns out to be smaller than c s^-1 even for the more luminous sources and no detectable UV flux can be observed at Earth.

Finally, comparing the observed hardness ratios with the values quoted in Tables 2 and 3 could be used to discriminate ONSs among other optically unidentified X-ray ROSAT sources.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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