CircularOrbitSelfForceData

Overview

CircularOrbitSelfForceData is a collection of black hole perturbation and self-force data for a point particle on a circular (equatorial) orbit about a Schwarzschild or Kerr black hole. It contains:

gravitational-wave energy fluxes to infinity and down the horizon, for Schwarzschild and for Kerr spins from $a = -0.99M$ to $a = 0.999M$, plus near-extremal data;
linear-in-spin corrections to the fluxes from a spinning secondary;
local gauge-invariant self-force quantities: Detweiler’s redshift invariant, the spin-precession invariant, tidal and octupolar invariants, and the second-order binding energy.

The data come from many groups. Each file or folder README names the paper the data were published in. If you use the data, please cite the relevant work (listed below) and acknowledge the Toolkit, for example: “This work made use of data hosted as part of the Black Hole Perturbation Toolkit at bhptoolkit.org.”

All data are plain-text, whitespace-separated columns in geometric units with $M = 1$. No code is needed to read them.

Installation

Clone the repository (about 20 MB):

git clone https://github.com/BlackHolePerturbationToolkit/CircularOrbitSelfForceData.git

You can also read individual files directly from GitHub without cloning, using their raw URLs:

https://raw.githubusercontent.com/BlackHolePerturbationToolkit/CircularOrbitSelfForceData/master/<path>

Data format

Schwarzschild

File	Columns	Reference
`Schwarzschild/Flux_Edot.dat`	$r_0/M$, $\dot E_\infty$, $\dot E_{\rm H}$ (10,000 radii from $r_0 = 6M$ to $10^4M$)	—
`Schwarzschild/Redshift_and_spin_invariants.dat`	$r_0/M$, $\Delta U\,(M/\mu)$, $\Delta\psi\,(M/\mu)$	Dolan et al., Phys. Rev. D 91, 023009 (2015), arXiv:1406.4890, Table III
`Schwarzschild/Tidal_invariants.dat`	$r_\Omega/M$, $\Delta\lambda^E_1$, $\Delta\lambda^E_2$, $\Delta\lambda^E_3$, $\Delta\chi$, … (see the file header)	Dolan et al., Phys. Rev. D 91, 023009 (2015), arXiv:1406.4890, Table I
`Schwarzschild/delta_K3plus.dat`	$r_0/M$, $\delta K_{3+}$ (the effective-one-body quantity of Eq. (2.47) of the reference)	Nolan et al., Phys. Rev. D 92, 123008 (2015), arXiv:1505.04447
`Schwarzschild/SecondOrderEbindingEnergy.dat`	$1/y$, $E_{\rm SF}$, estimated error in $E_{\rm SF}$, $E^{\rm cons}_{\rm SF}$, $E^{\rm 1st\,law}$, $\Delta E_{\rm SF}$, $\Delta E^{\rm cons}_{\rm SF}$	Pound, Wardell, Warburton and Miller, Phys. Rev. Lett. 124, 021101 (2020), arXiv:1908.07419
`Schwarzschild/Spin/SpinFluxFixedr0.dat`	$r_0$, $F_0$, $F^{\rm H}_\sigma$, $F^{\rm Inf}_\sigma$, $(dE/dt)_\sigma$, $\Delta_{\rm rel}$	Akcay et al., Phys. Rev. D 102, 064013 (2020), arXiv:1912.09461, Table I
`Schwarzschild/Spin/SpinFluxFixedy.dat`	$y$, $F_0$, $F^{\rm H}_\sigma$, $F^{\rm Inf}_\sigma$, $(dE/dt)_\sigma$, $\Delta_{\rm rel}$	Akcay et al., arXiv:1912.09461, Table II

Apart from the flux file, every Schwarzschild file starts with # comment lines describing the data, its column format and its source.

The Schwarzschild/Spin README also points to additional spinning-secondary data hosted on Zenodo:

Description	Authors	Reference	Data
Spin-aligned flux for four different spin supplementary conditions	E. Harms, G. Lukes-Gerakopoulos, S. Bernuzzi, A. Nagar	Phys. Rev. D 94, 104010 (2016), arXiv:1609.00356	Zenodo
Spin-aligned flux, including linearized (in spin) flux	S. Bernuzzi, E. Harms, G. Lukes-Gerakopoulos	Phys. Rev. D 100, 104056 (2019), arXiv:1907.12233	Zenodo

Kerr

Fluxes (Kerr/Fluxes/)

File	Columns	Reference
`Flux_Edot_a<a>.dat` for $a/M \in \{\pm0.1, \pm0.2, \pm0.3, \pm0.4, \pm0.5, \pm0.6, -0.7, \pm0.8, \pm0.9, 0.95, \pm0.99, 0.995, 0.999\}$	$r_0/M$, $\dot E_\infty$, $\dot E_{\rm H}$ (10,000 radii from $10^4M$ down to the innermost stable circular orbit)	For $\lvert a\rvert \le 0.9M$: computed by S. A. Hughes for Taracchini et al., Phys. Rev. D 90, 084025 (2014), arXiv:1404.1819
`Flux_Edot_eps_1e-9.dat`	$r_0/M$, $\dot E_\infty$, $\dot E_{\rm H}$ for the near-extremal spin $a = (1-\epsilon)M$, $\epsilon = 10^{-9}$	Gralla, Hughes and Warburton, Class. Quant. Grav. 33, 155002 (2016), arXiv:1603.01221

Negative values of $a$ correspond to retrograde orbits. For prograde orbits the horizon flux $\dot E_{\rm H}$ can be negative, wherever superradiance extracts energy from the black hole.

Spinning-secondary flux corrections (Kerr/Fluxes/Spin/)

File	Columns	Reference
`Fluxes_spin_corr_a<a>.dat` for $a/M \in \{0.1, \dots, 0.9, 0.95, 0.97, 0.99, 0.995\}$	$r$, `flux_corr`: the absolute value of the linear-in-spin correction to the flux, normalized to $q^2$ with $q$ the mass ratio (one header line)	Piovano, Maselli and Pani, Phys. Rev. D 102, 024041 (2020), arXiv:2004.02654

All of these flux corrections are negative over the range of $r$ covered. The files store their absolute values.

Local invariants (Kerr/Local_Invariants/)

File	Columns	Reference
`redshift_invariant.dat`	$r_0/M$, then $\Delta U$ for $a/M = -0.9, -0.7, -0.5, 0, 0.5, 0.7, 0.9$ (`-` where no value is available)	Shah, Friedman and Keidl, Phys. Rev. D 86, 084059 (2012), arXiv:1207.5595, Table III
`spin_invariant.dat`	$a/M$, $r/M$, $\delta\psi$, estimated absolute numerical error (no header; $a/M = -0.9, -0.8, \dots, 0.9$)	Bini, Damour, Geralico, Kavanagh and van de Meent, Phys. Rev. D 98, 104062 (2018), arXiv:1809.02516

Usage

Mathematica

Files without a header can be imported directly as a table. For files with # comment lines, keep only the numeric rows:

base = "https://raw.githubusercontent.com/BlackHolePerturbationToolkit/CircularOrbitSelfForceData/master/";

flux = Import[base <> "Schwarzschild/Flux_Edot.dat", "Table"];
redshift = Select[Import[base <> "Schwarzschild/Redshift_and_spin_invariants.dat", "Table"], VectorQ[#, NumericQ] &];
spinCorr = Import[base <> "Kerr/Fluxes/Spin/Fluxes_spin_corr_a0.9.dat", "Table", "HeaderLines" -> 1];

To read a local clone, replace base with the path to the cloned repository.

Python

numpy.loadtxt skips # comment lines by default:

import numpy as np

base = "https://raw.githubusercontent.com/BlackHolePerturbationToolkit/CircularOrbitSelfForceData/master/"

r0, Edot_inf, Edot_H = np.loadtxt(base + "Kerr/Fluxes/Flux_Edot_a0.9.dat", unpack=True)
r0, dU, dpsi = np.loadtxt(base + "Schwarzschild/Redshift_and_spin_invariants.dat", unpack=True)
r, flux_corr = np.loadtxt(base + "Kerr/Fluxes/Spin/Fluxes_spin_corr_a0.9.dat", skiprows=1, unpack=True)

# missing entries ("-") in the Kerr redshift table become NaN
dU_kerr = np.genfromtxt(base + "Kerr/Local_Invariants/redshift_invariant.dat")

Examples

Plot the total gravitational-wave energy flux against orbital radius for a prograde orbit about a black hole with $a = 0.9M$:

import numpy as np
import matplotlib.pyplot as plt

base = "https://raw.githubusercontent.com/BlackHolePerturbationToolkit/CircularOrbitSelfForceData/master/"
r0, Edot_inf, Edot_H = np.loadtxt(base + "Kerr/Fluxes/Flux_Edot_a0.9.dat", unpack=True)

plt.loglog(r0, Edot_inf + Edot_H)
plt.xlabel(r"$r_0/M$")
plt.ylabel(r"$\dot E_\infty + \dot E_H$")
plt.show()

For computing fluxes and self-force quantities yourself, see the Teukolsky package and the other self-force modules in the Toolkit.

Authors and contributors

The repository is maintained by Niels Warburton. Data contributed by Scott Hughes; Samuel Gralla, Scott Hughes and Niels Warburton; Gabriel Andres Piovano, Andrea Maselli and Paolo Pani; Abhay Shah, John Friedman and Tobias Keidl; Donato Bini, Thibault Damour, Andrea Geralico, Chris Kavanagh and Maarten van de Meent; Sam Dolan, Patrick Nolan, Adrian Ottewill, Niels Warburton and Barry Wardell; Adam Pound, Barry Wardell, Niels Warburton and Jeremy Miller; Sarp Akcay, Sam Dolan, Chris Kavanagh, Jordan Moxon, Niels Warburton and Barry Wardell.

Citing CircularOrbitSelfForceData

If you use CircularOrbitSelfForceData in your research, please acknowledge the Toolkit:

This work makes use of the Black Hole Perturbation Toolkit.

CircularOrbitSelfForceData also requests the following citations:

For the Kerr fluxes with |a| ≤ 0.9M:
Taracchini, Buonanno, Khanna and Hughes, Small mass plunging into a Kerr black hole: Anatomy of the inspiral-merger-ringdown waveforms, Phys. Rev. D 90, 084025 (2014) arXiv Inspire

BibTeX

@article{Taracchini:2014zpa,
    author = "Taracchini, Andrea and Buonanno, Alessandra and Khanna, Gaurav and Hughes, Scott A.",
    title = "{Small mass plunging into a Kerr black hole: Anatomy of the inspiral-merger-ringdown waveforms}",
    eprint = "1404.1819",
    archivePrefix = "arXiv",
    primaryClass = "gr-qc",
    doi = "10.1103/PhysRevD.90.084025",
    journal = "Phys. Rev. D",
    volume = "90",
    number = "8",
    pages = "084025",
    year = "2014"
}

For the near-extremal Kerr fluxes:
Gralla, Hughes and Warburton, Inspiral into Gargantua, Class. Quant. Grav. 33, 155002 (2016) arXiv Inspire

BibTeX

@article{Gralla:2016qfw,
    author = "Gralla, Samuel E. and Hughes, Scott A. and Warburton, Niels",
    title = "{Inspiral into Gargantua}",
    eprint = "1603.01221",
    archivePrefix = "arXiv",
    primaryClass = "gr-qc",
    doi = "10.1088/0264-9381/33/15/155002",
    journal = "Class. Quant. Grav.",
    volume = "33",
    number = "15",
    pages = "155002",
    year = "2016",
    note = "[Erratum: Class.Quant.Grav. 37, 109501 (2020)]"
}

See how to cite for further guidance.

Overview

Installation

Data format

Schwarzschild

Kerr

Usage

Mathematica

Python

Examples

Authors and contributors

Citing CircularOrbitSelfForceData

Related tools

PostNewtonian-SelfForce