There are several important timescales and length scales that govern the dynamics of the stellar system and MBH. They are listed here with estimates of their value in the Galactic Center. A solar-type star and M_● = 3 × 10⁶ M_ּ (Genzel et al 2000) are assumed throughout. Physical lengths are also expressed as angular sizes assuming that the distance to the Galactic Center is R₀ = 8 kpc (Reid 1993).

1.1 Timescales

The dynamical time, or orbital time, t_d, is the time it takes a star to cross the system

 (1.1)

where r is the typical size of the system and M_tot is the total mass enclosed in radius r .

The two body relaxation time, t_r , is related to the 1D velocity dispersion σ, the mean stellar mass  and the stellar number density n_* by

 (1.2)

where log Λ is the Coulomb logarithm, the logarithm of the ratio between the largest and smallest impact parameters possible in the system for elastic collisions. Because the relaxation timescale in the Galactic Center is shorter than the age of the Galaxy (∼10 Gyr), the old stars are expected to be well relaxed by now.

The mass segregation timescale is of the same order as the relaxation timescale,

 (1.3)

The rate (per star) of grazing collisions between two stars of mass and radius M^a_* , R^a_* and M^b_* , R^b_* , each, is

 (1.4)

where it is assumed that the stars follow a mass-independent Maxwell-Boltzmann velocity distribution with velocity dispersion σ (this is a good approximation near the MBH). There are two contributions to the total rate, one due to the geometric cross section (first term in the square brackets) and one due to ‘gravitational focusing’ (second term in the square brackets). Gravitational focusing expresses the fact that the two stars do not move on straight lines, but are attracted to each other. This effect is important when the typical stellar velocities are much smaller than the escape velocity, ve, from the stars, σ² < GM_*/2R_* = v²_e /4.

1.2 Length scales

The size of the event horizon of a non-rotating black hole, the Schwarzschild radius, is

 (1.5)

The tidal radius, r_t, is the minimal distance from the MBH where the stellar self-gravity can still resist the tidal forces of the MBH. If the star's orbit takes it inside the tidal radius, it will be disrupted, and roughly half of its mass will fall into the MBH, while the other half will be ejected (e.g. Ayal et al 2000). The exact value of the tidal radius depends on the stellar structure and the nature of the orbit, and up to a factor of order unity is given by

 (1.6)

Tidal disruption is relevant as long as the tidal radius lies outside the event horizon. Since r_t ∝ M_●^1/3, while r_S ∝ M_●, there exists a maximal MBH mass for tidal disruption, which for a solar-type star is ∼10⁸ M_ּ.

The radius of influence, r_h, is the region where the MBH potential dominates the dynamics. If the MBH is embedded in an isothermal stellar system (i.e. σ is constant), then the radius of influence can be defined as

 (1.7)

In practice, the distribution is not isothermal and σ is not constant, and so r_h is evaluated loosely by choosing a representative value of σ far enough from the MBH. The stellar mass enclosed within r_h is of the same order as the mass of the MBH.