From e05c0d97fae7c04dce6d8c92947ad07ad6c887be Mon Sep 17 00:00:00 2001 From: Zachary Nasipak <> Date: Tue, 19 Dec 2023 12:45:49 -0500 Subject: [PATCH] Some small edits/tweaks plus a suggested edit at the beginning. Let me know if there are any issues/concerns/questions. --- paper/paper.md | 36 ++++++++++++++++++++---------------- 1 file changed, 20 insertions(+), 16 deletions(-) diff --git a/paper/paper.md b/paper/paper.md index 5cafd1a..8cd66aa 100644 --- a/paper/paper.md +++ b/paper/paper.md @@ -24,32 +24,34 @@ bibliography: paper.bib # Summary In general relativity, the motion of a free falling test particle in a curved spacetime is -described by a geodesic - the generalization of a "straight line" path to a curved space. +described by a geodesic - the generalization of a "straight line" path to a curved space. +[comment: maybe we just say "...by a geodesic - the minimal path between two points in a curved space"] The geodesics of Kerr spacetime are of particular interest in the field of black hole perturbation theory because they describe the zeroth order motion of a small object -moving through the background spacetime of a large spinning black hole. For this reason, computing -geodesics is an important step in modelling the gravitational radiation emitted by an +moving through the background spacetime of a much more massive spinning black hole. For this reason, computing +geodesics is an important step in modeling the gravitational radiation emitted by an extreme mass ratio inspiral (EMRI) - an astrophysical binary in which a stellar mass -compact object, such as a neutron star or a small ($10^1 - 10^2 M_\odot$) black hole, -spirals into a massive ($10^4 - 10^7 M_\odot$) black hole. +compact object, such as a neutron star or black hole (with mass $10^1 - 10^2 M_\odot$), +spirals into a massive black hole (with mass $10^4 - 10^7 M_\odot$). Kerr spacetime has several nice properties which simplify the problem of computing geodesics. Since -it has both time translation symmetry and rotational symmetry, energy and angular momentum are conserved quantities. It is also +it has both time-translation symmetry and rotational symmetry, energy and angular momentum are conserved quantities. It is also equipped with a higher order symmetry which gives rise to a third constant of motion called the Carter constant. These three constants of motion, along with the spin of the black hole, uniquely define a geodesic up to -initial conditions [@schmidt].Alternatively, geodesics can be identified using a suitably generalized -version of the parameters used to define a Keplerian orbit (eccentricity, semi-latus rectum and inclination angle). -Bound geodesics also possess fundamental frequencies since their radial, azimuthal and polar motion is periodic. +initial conditions [@schmidt]. Alternatively, geodesics can be identified using a suitably generalized +version of the parameters used to define a Keplerian orbit (eccentricity, semi-latus rectum, and inclination angle). +Bound geodesics also possess fundamental frequencies since their radial, azimuthal, and polar motions are periodic. -`KerrGeoPy` is a Python implementation of the `KerrGeodesics` [@kerrgeodesics] Mathematica library -which computes both stable and plunging geodesics in Kerr spacetime using the +`KerrGeoPy` is a Python package which computes both stable and plunging geodesics in Kerr spacetime using the analytic solutions to the geodesic equation derived in [@fujita] and -[@dyson]. These solutions are written in terms of elliptic integrals, which are +[@dyson]. It mirrors and builds upon much of the functionality of the `KerrGeodesics` [@kerrgeodesics] Mathematica library. +Geodesic solutions are written in terms of elliptic integrals, which are evaluated using `SciPy`. Users can construct a geodesic by providing the initial position and four-velocity, or by providing either the constants of motion or the Keplerian parameters described above. `KerrGeoPy` also provides methods for computing the fundamental frequencies and constants of motion associated with a given geodesic and implements the algorithm described -in [@stein] for finding the location of the last stable orbit orbit, known as the separatrix. +in [@stein] for finding the location of the last stable orbit, known as the separatrix. The package also +includes several methods for visualizing and animating geodesics. ![Example of an equatorial (left), spherical (center) and generic (right) orbit computed by `KerrGeoPy`](orbits.png) @@ -85,9 +87,11 @@ researchers to build on in their own projects. Although other Python packages [@kerrgeodesicgw] with similar functionality do exist, they mostly rely on numerical integration to compute geodesics. The analytic solutions used by `KerrGeoPy` have two main advantages -over this approach. First, they are guaranteed to be numerically stable and can be quickly evaluated at -any point in time. Second, they produce several useful intermediate terms which cannot easily be -computed using numerical integration. +over this approach. First, they can be much more numerically stable over long time periods and can be quickly evaluated at +any point in time. Second, they produce several useful intermediate terms which are not calculated by other packages that rely on +numerical integration. Modeling EMRIs typically requires long time-averages over the geodesic motion. Therefore, +`KerrGeoPy`, with its analytic solutions and various orbital parametrizations, is specifically tuned to support +perturbative models of binary black holes and their gravitational waves. `KerrGeoPy` is a part of the [Black Hole Perturbation Toolkit](https://bhptoolkit.org). The source code is hosted on [Github](https://github.com/BlackHolePerturbationToolkit/KerrGeoPy) and the package is