Gravitational capture cross-section in Zipoy-Voorhees spacetimes

TL;DR

This paper investigates how the quadrupole moment of a static, axisymmetric spacetime affects the capture cross-section of test particles and photons, providing insights into realistic compact objects like neutron stars.

Contribution

It derives explicit formulas for capture cross-sections and escape angles in the quadrupolar ($q$-metric), extending Schwarzschild results to more realistic astrophysical objects.

Findings

Capture cross-sections depend on the quadrupole parameter $q$.

Explicit expression for photon escape angle as a function of $q$.

Results reduce to Schwarzschild case when $q$ approaches zero.

Abstract

We consider geodesics of massive and massless test particles in the gravitational field of a static and axisymmetric compact object described by the quadrupolar metric ($q$-metric), which is the simplest generalization of the Schwarzschild metric, containing an independent quadrupole parameter $q$. We analyze the effective potential profile and calculate the orbital parameters and capture cross-sections of test particles in this spacetime. Moreover, we derive the explicit expression for the escape angle of photons as a function of the quadrupole parameter. All the results reduce in the corresponding limit of vanishing quadrupole to the well-known case of the Schwarzschild spacetime. We argue that our results could be used to investigate realistic compact objects such as white dwarfs and neutron stars.