Unbound motion of massive particles in the Schwarzschild metric: Analytical description in case of strong deflection

Abstract

Deflection angles of massive test particles moving along an unbound trajectory in the Schwarzschild metric are considered for the case of large deflection. We analytically consider the strong deflection limit, which is opposite to the commonly applied small deflection approximation and corresponds to the situation when a massive particle moves from infinity, makes several revolutions around a central object and goes to infinity. For this purpose we rewrite an integral expression for the deflection angle as an explicit function of the parameters determining the trajectory and expand it. Remarkably, in the limiting case of strong deflection, we succeed in deriving for the first time the analytical formulas for deflection angles as explicit functions of parameters at infinity. In particular, we show that in this case the deflection angle can be calculated as an explicit function of the impact parameter and velocity at infinity beyond the usual assumption of small deflection.