Astrophysically relevant bound trajectories around a Kerr black hole

11 Feb 2019
•
Rana Prerna Indian Institute of Astrophysics
•
Mangalam A. Indian Institute of Astrophysics

We derive alternate and new closed-form analytic solutions for the
non-equatorial eccentric bound trajectories, $\{ \phi \left( r, \theta
\right)$, $\ t \left( r, \theta \right),\ r \left( \theta \right) \}$, around a
Kerr black hole by using the transformation $1/r=\mu \left(1+ e \cos \chi
\right)$. The application of the solutions is straightforward and numerically
fast...We obtain and implement translation relations between energy and angular
momentum of the particle, ($E$, $L$), and eccentricity and inverse-latus
rectum, ($e$, $\mu$), for a given spin, $a$, and Carter's constant, $Q$, to
write the trajectory completely in the ($e$, $\mu$, $a$, $Q$) parameter space. The bound orbit conditions are obtained and implemented to select the allowed
combination of parameters ($e$, $\mu$, $a$, $Q$). We also derive specialized
formulae for spherical and separatrix orbits. A study of the non-equatorial
analog of the previously studied equatorial separatrix orbits is carried out
where a homoclinic orbit asymptotes to an energetically bound spherical orbit. Such orbits simultaneously represent an eccentric orbit and an unstable
spherical orbit, both of which share the same $E$ and $L$ values. We present
exact expressions for $e$ and $\mu$ as functions of the radius of the
corresponding unstable spherical orbit, $r_s$, $a$, and $Q$, and their
trajectories, for ($Q\neq0$) separatrix orbits; they are shown to reduce to the
equatorial case. These formulae have applications to study the gravitational
waveforms from EMRIs besides relativistic precession and phase space
explorations. We obtain closed-form expressions of the fundamental frequencies
of non-equatorial eccentric trajectories that are equivalent to the previously
obtained quadrature forms and also numerically match with the equivalent
formulae previously derived. We sketch several orbits and discuss their
astrophysical applications.(read more)