KerrGeodesics

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The KerrGeodesics package for Mathematica provides functions for computing bound timelike geodesics and their properties in Kerr spacetime.

Example usage

As a quick example, the figure at the top of this page is made using the simple commands:

orbit = KerrGeoOrbit[0.998, 3, 0.6, Cos[π/4]];
{t, r, θ, φ} = orbit["Trajectory"];

Followed by the plot command:

Show[
 ParametricPlot3D[{r[λ] Sin[θ[λ]] Cos[φ[λ]], r[λ] Sin[θ[λ]] Sin[φ[λ]], r[λ] Cos[θ[λ]]}, {λ, 0, 20}, 
  ImageSize -> 700, Boxed -> False, Axes -> False, PlotStyle -> Red, PlotRange -> All],
 Graphics3D[{Black, Sphere[{0, 0, 0}, 1 + Sqrt[1 - 0.998^2]]}]
 ]

Orbital parametrization

The orbits are parameterized by the following

$a$ - the black hole spin
$p$ - the semi-latus rectum
$e$ - the eccentricity
$x_\text{inc} = \cos\theta_\text{inc}$ - the orbital inclination.

The parametrization $\{a,p,e,\theta_\text{inc}\}$ is described in, e.g., Sec. II of arXiv:gr-qc/0509101

Orbital constants and frequencies

The constants of the motion can be computed using

KerrGeoEnergy[a,p,e,x]
KerrGeoAngularMomentum[a,p,e,x]
KerrGeoCarterConstant[a,p,e,x]

The above three can be computed together using KerrGeoConstantsOfMotion[a,p,e,x].

The orbital frequencies (w.r.t Boyer-Lindquist time $t$) are computed using KerrGeoFrequencies[a,p,e,x]. For this function you can pass the option Time->"Mino" to compute the frequencies w.r.t. Mino time.

Special orbits

The package allows you compute a variety of special orbits including the innermost stable circular/spherical orbit (ISCO/ISSO), innermost bound spherical orbit (IBSO), the photon orbit and the location of the separatrix between stable and plunging orbits. The relevant functions are:

KerrGeoISCO[a,x]
KerrGeoISSO[a,x]
KerrGeoPhotonSphereRadius[a,x]
KerrGeoIBSO[a,x]
KerrGeoSeparatrix[a,e,x]

Further examples

See the Documentation Centre for a tutorial and documentation on individual commands. Example notebooks can also be found in the Mathematica Toolkit Examples repository.