A few more edits and tweaks. I am happy with the current version for …

…submission

paper/paper.md

@@ -23,26 +23,25 @@ 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.
-[comment: maybe we just say "...by a geodesic - the minimal path between two points in a curved space"] 
+In general relativity, the motion of a free-falling test particle in a curved spacetime is 
+described by a timelike geodesic - the minimal path between two points in 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 
+hole perturbation theory, because they describe the zeroth-order motion of a small object 
 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
+extreme-mass-ratio inspiral (EMRI) - an astrophysical binary in which a stellar mass
 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
-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 
+it has both time-translation symmetry and rotational symmetry, energy and (the $z$-component of) 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 motions are periodic.
 
-`KerrGeoPy` is a Python package which computes both stable and plunging geodesics in Kerr spacetime using the 
+`KerrGeoPy` is a Python package which computes both stable and plunging timelike geodesics in Kerr spacetime using the 
 analytic solutions to the geodesic equation derived in [@fujita] and 
 [@dyson]. It mirrors and builds upon much of the functionality of the `KerrGeodesics` [@kerrgeodesics] Mathematica library.
 Geodesic solutions are written in terms of Legendre elliptic integrals, which are 
@@ -70,33 +69,34 @@ Antenna (LISA), a future space-based gravitational wave observatory consisting o
 constellation of three satellites in orbit around the sun. LISA is an ESA-led mission 
 with significant contributions from NASA which is set to launch in the 2030s. It will
 complement existing ground-based detectors by opening up the millihertz band of the 
-gravitational wave spectrum [@lisa]. Because sources in this band evolve slowly over time and remain observable 
+gravitational wave spectrum [@lisa]. Because sources in this band evolve more slowly over time and remain observable 
 for a period of days to years, LISA is expected to detect many overlapping signals at all times. 
 Thus, accurate waveform models are needed in order to identify gravitational wave sources and 
 perform parameter estimation - the process of approximating characteristics of a source.
 
 For most LISA sources, well-developed waveform models based on either numerical relativity 
-or post-Newtonian theory already exist. However, EMRIs are instead best 
+or post-Newtonian theory already exist. However, EMRIs are instead more naturally 
 described by black hole perturbation theory, and the EMRI waveform models that currently exist 
-are comparatively underdeveloped. In a perturbation theory model, the orbital trajectory is assumed to be a geodesic at 
-leading order. Higher order corrections are then computed by introducing the gravitational 
-field of the inspiraling object as a perturbation to the background spacetime of the massive black hole, 
-expanded in powers of the mass ratio.
-
-In order to meet the accuracy requirements for LISA parameter estimation, EMRI waveform 
-models must include second order corrections to the orbital trajectory. However, to date, 
-most development has focused on first order corrections to only the simplest cases, 
-such as Schwarzschild orbits and equatorial Kerr orbits [@emri]. Additionally, most development 
-has taken place in Mathematica, which is an expensive and proprietary piece of software. `KerrGeoPy` is 
+are underdeveloped compared to other LISA sources. In a perturbation theory model, the orbital trajectory 
+is assumed to be a geodesic at leading order. Higher-order corrections are then computed by introducing 
+the gravitational field of the inspiraling object as a perturbation to the background spacetime of the 
+massive black hole, expanded in powers of the mass ratio.
+
+To meet the accuracy requirements for LISA parameter estimation, EMRI waveform 
+models must include both first- and second-order corrections to the orbital trajectory. However, to date, 
+second-order corrections are only available for the most simple systems, 
+quasi-circular inspirals in Schwarzschild [@emri]. Open-source tools can aid in rapidly expanding EMRI models
+to more complicated orbits in Kerr spacetime, but at the moment many tools for modeling EMRIs 
+are only available in Mathematica, which is an expensive and proprietary piece of software. `KerrGeoPy` is 
 intended to support future development of higher-order waveform models in preparation for
 LISA by providing a free alternative to the existing `KerrGeodesics` Mathematica library for other
 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 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,
+any point in time. This is essential for EMRI models, which typically require taking long time-averages over the geodesic motion. 
+Second, they produce several useful intermediate terms which are not calculated by other packages. 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.
 
@@ -114,6 +114,9 @@ perturbative models of binary black holes and their gravitational waves.
 # Acknowledgements
 
 We would like to thank Niels Warburton and Barry Wardell for their assistance in releasing 
-`KerrGeoPy` as part of the Black Hole Perturbation Toolkit.
+`KerrGeoPy` as part of the Black Hole Perturbation Toolkit. SP acknowledges support through
+NASA's Office of STEM Engagement, while ZN acknowledges support by an appointment 
+to the NASA Postdoctoral Program at the NASA Goddard Space Flight Center, administered by Oak Ridge 
+Associated Universities under contract with NASA.
 
 # References