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Astron. Astrophys. 358, 57-64 (2000)

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2. Model properties

In this paper the model presented in Paper I has been improved in different respects. A mass distribution for the stars has been introduced as well as a star radius distribution. A more realistic description of star collisions has been also developed by introducing: i) the angle [FORMULA] (between the direction of motion of one star and the trajectory of the other one, [FORMULA] with [FORMULA] corresponding to the case of equal star velocities); ii) the impact parameter b (defined as the smallest distance between the two star centers). The influence of these new parameters is described hereafter.

As a first step, mass segregation has not been taken into account in the expression for the star density distribution (stars/pc3/dM) which reads:

[EQUATION]

In this expression x is the Salpeter parameter, R is the distance from the central black hole and M is the star mass in units of the solar mass [FORMULA], [FORMULA]. The spatial distribution of stars [FORMULA] is the same as used in Paper I, namely

[EQUATION]

The presence of stars of different masses has important consequences on the energetics of stellar collisions since the energy released in each collision is strictly linked to the amount of the involved mass. In fact, the energy available for emission in the collision between stars [FORMULA] and [FORMULA] (star masses in terms of [FORMULA]) is the star kinetic energy relative to the center of mass:

[EQUATION]

In this expression [FORMULA] is the mass center velocity,

[EQUATION]

and [FORMULA] is the star keplerian velocity

[EQUATION]

around the central mass, [FORMULA]. [FORMULA] is composed of the black hole mass [FORMULA] and the sum of all stellar masses inside the sphere of radius R:

[EQUATION]

[EQUATION]

where [FORMULA] cm is the Schwarschild radius.

The introduction of a more realistic geometry for star collisions takes into account that the relative velocity of the two stars, [FORMULA], is related to the [FORMULA] angle:

[EQUATION]

and that the cross section for a collision is [FORMULA]. Since a collision can occur only if the distance between the two star centers is less or equal to the sum of the two star radii, the impact parameter is limited by the condition

[EQUATION]

where b is in units of the sun radius, [FORMULA].

Star radii, [FORMULA], are defined in terms of star masses by the following the relationship (Foellmi 1998)

[EQUATION]

obtained from a fit over the results of stellar evolutionary models by Schaller et al. 1992.

Assuming an isotropic distribution of stars, the collision rate within a shell [FORMULA] at a distance R from the black hole is

[EQUATION]

[EQUATION]

Some of the parameters entering the collision description can not be fixed from outside, these are: the distance R and the time t at which the collision takes place, the masses [FORMULA] and [FORMULA] of the two stars involved, the angle [FORMULA] between their orbits and the impact parameter b. Even a complete solution of the dynamic of a star cluster around a black hole can not completely define these parameters since initial conditions in the cluster are unknown. In our analysis these parameters have been randomly chosen for each collision taking into account the constraint that the final distribution of each series of parameters must fulfill specific requirements. These requirements are: final mass distribution must follow expression (1); [FORMULA] and b must have a distributions like [FORMULA] and [FORMULA]; R distribution must take into account that the collision probability at distance R is given by the collision rate [FORMULA]; the time at which collisions occur must follow a Poisson distribution of mean rate [FORMULA].

The other parameters present in the above description have been chosen following the physical considerations described hereafter.

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© European Southern Observatory (ESO) 2000

Online publication: June 26, 2000
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