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

Appendix A: expected number of ONSs

In the calculation of the expected number of ONSs we have used the analytic approximation to the ONS distribution computed by Zane et al. (1995a), which is based on the evolution of the F population (characterized by the lowest value of the mean velocity) of model b of Narayan & Ostriker (1990) and on the assumption of a uniform spatial distribution of neutron stars:

where dV is the spatial volume integration element, pc^-3 is the number density of ONSs averaged in the local region around the Sun ( kpc, pc), is the present total number of ONSs in the Galaxy and is an analytic fit to the cumulative velocity distribution derived by Zane et al. (1995a).

The count rate (measured at Earth) in a certain spectral interval (, ) of a star which emits a monochromatic flux of radiation is

where d is the distance of the source, is the absorption cross-section of the ISM (Morrison & McCammon 1983) and the hydrogen column density. is the effective area and the bandpass of the detector ( keV and keV for ROSAT PSPC).

Clearly, at a certain distance an ONS will be detectable if its count rate is above the sensitivity limit S of the detector. Then, at there exists a limiting value of the star luminosity, which depends on the emission properties and the absorption of the ISM, below which an accreting neutron star does not give rise to a count rate above the threshold of the detector and hence is not observable. From Eq. (1) this translates directly into an upper limit for the star velocity, .

Alternatively, a star with a given luminosity L, or velocity v, will be observable up to a maximum distance at which goes below S. Then, the total number of ONSs which can be observed in a certain interval of distance [ , ] and within a solid angle can be calculated from Eq. (A1) summing up all the neutron stars that are contained within the volume and integrating over v:

In Eq. (A3) the integral goes from the maximum velocity of detectability at distance , , to the maximum velocity at , , whereas the second term on the right hand side accounts for all the stars with luminosity above threshold throughout all of the spatial volume considered. The integral has been evaluated numerically using a Lobatto quadrature and has been interpolated at the appropriate value of v from a table of entries previously calculated (an extensive use has been done of a local cubic interpolation procedure). We note that for each threshold there exists an absolute upper limit to the distance of detectability which corresponds to the star with the minimum accretion velocity and maximum emitted luminosity. We note also that in each interval of distance [ , ] the ISM density must be constant in order to ensure the regularity of the function and hence the correct computation of the integral.

We have estimated the spatial boundaries and of Cygnus Rift and Cygnus OB7 ( is the position of the cloud center) by inverting for the cloud width the expression , where is the cloud volume and its apparent angular surface (taken from Dame et al. 1987; see also Table 1). Using these boundaries, the expected number of ONSs in each cloud has been evaluated from Eq. (A3). Moreover, with the same 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.

© European Southern Observatory (ESO) 1997

Online publication: July 3, 1998
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