# KerrGeodesics

Install this package!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.