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Astron. Astrophys. 331, 535-540 (1998)

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2. The model

We solved numerically the system of differential equations of motions in the Galactic potential, taken in the form (Paczynski 1990):

[EQUATION]

with a quasi-spherical halo with a density distribution:

[EQUATION]

Here R and Z are the cylindrical coordinates, d the radius in the quasi-spherical halo. The parameters of the potential are given in the following table, [FORMULA] being determined from the halo mass, [FORMULA].


[TABLE]

The density in our model is constant in time. The local density is calculated using data and formulae from Bochkarev (1992) and Zane et al. (1995). n is total gas density, [FORMULA] and [FORMULA] are the densities of the neutral and molecular hydrogen, [FORMULA], [FORMULA] and [FORMULA] are the values of the densities for [FORMULA].

[EQUATION]

[EQUATION]

For [FORMULA] we assumed:

[EQUATION]

For [FORMULA] was assumed to be uniform:

[EQUATION]

Of course, this is not accurate for small R, so for the very central part of the Galaxy our results are only a rough estimate (see Zane et al. (1996) for detailed calculation of the NS emission from the Galactic center region). For [FORMULA] we assumed

[EQUATION]

[EQUATION]

[EQUATION]

For [FORMULA] we assumed

[EQUATION]

[FORMULA], [FORMULA] and [FORMULA] being taken from Bochkarev (1992).

The total gas density distribution in the R -Z plane used in our computations is shown in Fig. 1.

[FIGURE] Fig. 1. The density distribution in particle per cubic centimeter in the R -Z plane.

In our model we assumed that the birthrate of NS and BH is proportional to the square of the local density. Stars were assumed to be born in the Galactic plane (Z=0) with circular velocities plus additional isotropic kick velocities.

For the kick velocity distribution we used the formula from Lipunov et al. (1996). This formula was constructed as an analytical approximation of the three-dimensional velocity distribution of radio pulsars from Lyne & Lorimer (1994).

[EQUATION]

V being the space velocity of the compact object, [FORMULA] a characteristic velocity, [FORMULA] and [FORMULA] the probability (see the detailed description of the analytical approximation in Lipunov et al. (1996)). This formula reproduces the observed distribution with a mean velocity of 350 km/s for [FORMULA] =400 km/s. This velocity distribution seems more likely than a [FORMULA] - function and a Maxwellian distribution, which we used in Paper I. Kick velocities were taken equal for the NS and the BH. It is possible however that BH have lower kick velocities because of their higher masses (see White and van Paradijs, 1996). One of the reasons to make computations for [FORMULA] =200 km/s was to explore this situation.

For each star we computed the exact trajectory and the accretion luminosity. The accretion luminosity was calculated using Bondi's formula:

[EQUATION]

[EQUATION]

.

The sound velocity, [FORMULA], was taken to be 10 km/s everywhere. We used a mass [FORMULA] for NS and [FORMULA] for BH. [FORMULA] is the density, [FORMULA] being the mass of the hydrogen atom. The radii, [FORMULA], where the energy is liberated, were assumed to be equal to 10 km for NS and 90 km (i.e. [FORMULA], [FORMULA]) for BH. Calculations used a grid with a cell size 100 pc in the R-direction and 10 pc in the Z-direction. The luminosity is given on the figures in ergs per second per cubic parsec.

For the normalization of our results we assumed that [FORMULA] and [FORMULA] in the considered volume of the Galaxy. For a Salpeter mass function with [FORMULA] =2.35 the ratio of NS to BH is about 10 if NS are formed from stars with masses between [FORMULA] and [FORMULA], and BH from stars with masses higher than [FORMULA]. Motch et al. (1997) argued that [FORMULA] can be ruled out, [FORMULA] being a more probable value, but for the calculations of the distribution the total number is not so important, and for other numbers of compact objects the results (i.e. the value of the luminosity) can be easily scaled. It should be mentioned, as suggested by the unknown referee, that [FORMULA] is required to explain that the present heavy element abundance in the Galaxy is about Z=0.02.

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© European Southern Observatory (ESO) 1998

Online publication: February 16, 1998
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