Gravitational capture

Let us consider now the motion of a test particle in which its trajectory terminates in the black hole. Two types of such a motion are possible. First, the trajectory of the particle starts at infinity and ends in the black hole. Second, the trajectory starts and ends in the black hole. Of course, a particle cannot be ejected from the black hole. Hence, the motion on the second-type trajectory becomes possible either if the particle was placed on this trajectory via a non-geodesic curve or if the particle was created close to the black hole.

The gravitational capture of a particle coming from infinity is of special interest. Let us have a better look at this case. It is clear from the analysis of motion given in the preceding section that a particle coming from infinity can be captured if its specific energy Ẽ is greater, for a given ˜L, than the maximum (Ẽ_max) of the curve V (r ). Let us consider the gravitational capture in two limiting cases, one for a particle whose velocity at infinity is much lower than the speed of light (v_∞/c << 1) and another for a particle which is ultrarelativistic at infinity.

In the former case, Ẽ ≈ 1. The curve V (r ), which has Ẽ_max = 1, corresponds to ˜L_cr = 2 (line c in figure 5.2). The maximum of this curve lies at r = 2r_S. This radius is minimal for the periastra of the orbits of the particles with v_∞ = 0 which approach the black hole and again recede to infinity. If ˜L ≤ 2, gravitational capture takes place. The angular momentum of a particle moving with the velocity v_∞ at infinity is L = mv_∞b, where b is the impact parameter. The condition ˜L ≡ L/mcr_S = 2 defines the critical value b_cr,nonrel = 2r_S (c/v_∞) of the impact parameter for which the capture takes place. The capture cross section for a non-relativistic particle is

 (1.1)

For an ultrarelativistic particle, b_cr = 3√3r_S/2, and the capture cross section is

 (1.2)

Owing to a possible gravitational capture, not every particle whose velocity exceeds the escape limit flies away to infinity. In addition, it is necessary that the angle ψ between the direction to the black hole center and the trajectory be greater than a certain critical value ψ_cr. For the velocity equal to the escape threshold this critical angle is given by the expression

 (1.3)

The plus sign is chosen for r > 2r_S (ψ_cr < 90^o), and the minus sign for r < 2r_S (ψ_cr > 90^o).

For an ultrarelativistic particle, the critical angle is given by the formula

 (1.4)

The plus sign is taken for r > 1.5r_S and the minus for r < 1.5r_S.