diff --git a/README.md b/README.md index 4846e6f..44575a2 100644 --- a/README.md +++ b/README.md @@ -19,13 +19,13 @@ $e$ - orbital eccentricity
$x$ - cosine of the orbital inclination -\begin{equation} +$$ a = \frac{J}{M^2}, \quad\quad p = \frac{2r_{\text{min}}r_{\text{max}}}{M(r_{\text{min}}+r_{\text{max}})}, \quad\quad e = \frac{r_{\text{max}}-r_{\text{min}}}{r_{\text{max}}+r_{\text{min}}}, \quad\quad x = \cos{\theta_{\text{inc}}} -\end{equation} +$$ Note that $a$ and $x$ are restricted to values between -1 and 1, while $e$ is restricted to values between 0 and 1. Retrograde orbits are represented using a negative value for $a$ or for $x$. Polar orbits, marginally bound orbits, and orbits around an extreme Kerr black hole are not supported. -First, construct a [`StableOrbit`](stable_orbit.StableOrbit) using the four parameters described above. +First, construct a `StableOrbit` using the four parameters described above. ```python @@ -35,7 +35,7 @@ from math import cos, pi orbit = kg.StableOrbit(0.999,3,0.4,cos(pi/6)) ``` -Plot the orbit from $\lambda = 0$ to $\lambda = 10$ using the [`plot()`](stable_orbit.StableOrbit.plot) method +Plot the orbit from $\lambda = 0$ to $\lambda = 10$ using the `plot()` method ```python @@ -48,7 +48,7 @@ fig, ax = orbit.plot(0,10) -Next, compute the time, radial, polar and azimuthal components of the trajectory as a function of Mino time using the [`trajectory()`](stable_orbit.StableOrbit.trajectory) method. By default, the time and radial components of the trajectory are given in geometrized units and are normalized using $M$ so that they are dimensionless. +Next, compute the time, radial, polar and azimuthal components of the trajectory as a function of Mino time using the `trajectory()` method. By default, the time and radial components of the trajectory are given in geometrized units and are normalized using $M$ so that they are dimensionless. ```python @@ -88,11 +88,6 @@ plt.ylabel(r"$\phi(\lambda)$") - Text(0, 0.5, '$\\phi(\\lambda)$') - - - - ![png](README_files/Getting%20Started_6_1.png) @@ -100,7 +95,7 @@ plt.ylabel(r"$\phi(\lambda)$") ## Orbital Properties -Use the [`constants_of_motion()`](stable_orbit.StableOrbit.constants_of_motion) method to compute the dimensionless energy, angular momentum and Carter constant. By default, constants of motion are given in geometrized units where $G=c=1$ and are scale-invariant, meaning that they are normalized according to the masses of the two bodies as follows: +Use the `constants_of_motion()` method to compute the dimensionless energy, angular momentum and Carter constant. By default, constants of motion are given in geometrized units where $G=c=1$ and are scale-invariant, meaning that they are normalized according to the masses of the two bodies as follows: \begin{equation} \mathcal{E} = \frac{E}{\mu}, \quad \mathcal{L} = \frac{L}{\mu M}, \quad \mathcal{Q} = \frac{Q}{\mu^2 M^2} @@ -108,7 +103,7 @@ Use the [`constants_of_motion()`](stable_orbit.StableOrbit.constants_of_motion) Here, $M$ is the mass of the primary body and $\mu$ is the mass of the secondary body. -Frequencies of motion can be computed in Mino time using the [`mino_frequencies()`](stable_orbit.StableOrbit.mino_frequencies) method and in Boyer-Lindquist time using the [`fundamental_frequencies()`](stable_orbit.StableOrbit.fundamental_frequencies) method. As with constants of motion, the frequencies returned by both methods are given in geometrized units and are normalized by $M$ so that they are dimensionless. +Frequencies of motion can be computed in Mino time using the `mino_frequencies()` method and in Boyer-Lindquist time using the `fundamental_frequencies()` method. As with constants of motion, the frequencies returned by both methods are given in geometrized units and are normalized by $M$ so that they are dimensionless. ```python @@ -171,18 +166,18 @@ $\mathcal{Q}$ - Carter constant It is assumed that all orbital parameters are given in geometrized units where $G=c=1$ and are normalized according to the masses of the two bodies as follows: -\begin{equation} +$$ a = \frac{J}{M^2}, \quad \mathcal{E} = \frac{E}{\mu}, \quad \mathcal{L} = \frac{L}{\mu M}, \quad \mathcal{Q} = \frac{Q}{\mu^2 M^2} -\end{equation} +$$ -Construct a [`PlungingOrbit`](plunging_orbit.PlungingOrbit) by passing in these four parameters. +Construct a `PlungingOrbit` by passing in these four parameters. ```python orbit = kg.PlungingOrbit(0.9, 0.94, 0.1, 12) ``` -As with stable orbits, the components of the trajectory can be computed using the [`trajectory()`](plunging_orbit.PlungingOrbit.trajectory) method +As with stable orbits, the components of the trajectory can be computed using the `trajectory()` method ```python @@ -220,13 +215,6 @@ plt.ylabel(r"$\phi(\lambda)$") ``` - - - Text(0, 0.5, '$\\phi(\\lambda)$') - - - - ![png](README_files/Getting%20Started_15_1.png)