# Front Cover Equations

Adapted From Fundamentals of Astrodynamics and Applications, Second Edition by David A. Vallado (Vallado Orange Book)

Note! This is not an exact copy of the front cover. I have made changes to symbols where appropriate for legibility and such. For instance, I use $\theta$(theta) instead of $\nu$(nu) for true anomaly, as it looks too much like $v$ velocity. Go ahead and try it out! $\nu v v \nu v \nu \nu v$. Also, I use the same symbol $E$ for elliptic, parabolic, and hyperbolic eccentric anomaly. They are all used the same way, and it is always clear from context which is which.

I also group things differently, and I add in a couple of equations which are in pencil in my book.

These are provided pretty much without comment, as a reference for someone who knows what to do with the equation, but just needs to see it written out.

## Common two-body equations

Specific angular momentum $\vec{h}=\vec{r}\times\vec{v}$
 $h$ $=\sqrt{\mu p}$ $=r^2\dot{\theta}$ $=r_av_a$ $=r_pv_p$
Specific Mechanical Energy
 $\xi$ $=\frac{v^2}{2}-\frac{\mu}{r}$ $=-\frac{\mu}{2a}$
Orbital period $T=2\pi\sqrt{\frac{a^3}{\mu}}$
Eccentricity Vector $\vec{e}=\frac{\left(v^2-\frac{\mu}{r}\right)\vec{r}-(\vec{r}\cdot\vec{v})\vec{v}}{\mu}$
Radius $r=\frac{p}{1+e\cos\theta}$
Radial Rate $\dot{r}=\frac{r\dot{\theta}e\sin\theta}{1+e\cos\theta}$
Angular Rate $\dot{\theta}=\frac{na^2}{r^2}\sqrt{1-e^2}$
Semi-parameter $p=\frac{h^2}{\mu}$
Semi-major axis
 $a$ $=\sqrt[3]{\mu\left(\frac{T}{2\pi}\right)^2}$ $=\sqrt[3]{\frac{\mu}{n}}$

## Anomalies

 Circle Ellipse Parabola Hyperbola Eccentric Anomaly $E$ $E=\theta$ $E=2\tan^{-1}\left(\sqrt{\frac{1-e}{1+e}}\tan\left(\frac{\theta}{2}\right)\right)$ $\sin{E}=\frac{sin{\theta}\sqrt{1-e^2}}{1+e\cos{\theta}}$ $\cos{E}=\frac{e+cos{\theta}}{1+e\cos{\theta}}$ $E_{n+1}=E_n+\frac{M-E_n+e\sin E}{1-e\cos E}$ $E=\tan{\frac{\theta}{2}}$ $\sinh E=\frac{-\sin \theta\sqrt{e^2-1}}{1+e\cos \theta}$$\cosh E=\frac{\cos{\theta}+e}{1+e\cos{\theta}}$ $E_{n+1}=E_n+\frac{M+E_n-e\sinh E}{e\cosh E-1}$ True Anomaly $\theta$ $\theta=E$ $\theta=2\tan^{-1}\left(\sqrt{\frac{1+e}{1-e}}\tan\left(\frac{E}{2}\right)\right)$ $\theta=\cos^{-1}\left(\frac{p}{re}-\frac{1}{e}\right)$ $\sin\theta=\frac{pE}{r}$$\cos\theta=\frac{p-r}{r}$ $\theta=\cos^{-1}\left(\frac{p}{re}-\frac{1}{e}\right)$ $\sin\theta=\frac{-\sinh E\sqrt{e^2-1}}{1-e\cosh E}$$\cos\theta=\frac{\cosh{E}-e}{1-e\cosh{E}}$ $\theta=\cos^{-1}\left(\frac{p}{re}-\frac{1}{e}\right)$ Mean Anomaly $M=nt$ $M=E=\theta$ $M=E-e\sin E$ $M=\frac{E^3}{3}+E$ $M=e\sinh(E)-E$

## Other Parameters

Circle Ellipse Parabola Hyperbola
Flight Path Angle $\phi$ $\phi=0$ $\sin\phi=\frac{e\sin E}{\sqrt{1-e^2\cos^2E}}$

$\cos\phi=\sqrt{\frac{1-e^2}{1-e^2\cos^2E}}$

$\phi=\frac{\theta}{2}$ $\sin\phi=\frac{-e\sinh E}{\sqrt{e^2\cosh^2E-1}}$

$\cos\phi=\sqrt{\frac{e^2-1}{e^2\cosh^2E-1}}$

Polar Form Equation $r=a$ $r=\frac{p}{1+e\cos\theta}$
 $r$ $=\frac{p}{1+\cos\theta}$ $=\frac{p(1-E^2)}{2}$
$r=\frac{p}{1+e\cos\theta}$
Periapsis $r_p$ $r_p=a$ $r_p=a(1-e)$ $r_p=\frac{p}{2}$ $r_p=a(1-e)$
Apoapsis $r_a$ $r_a=a$ $r_a=a(1+e)$ $r_a=\infty$ $r_a=a(1+e)$ (negative)
Semi-parameter $p$ $p=a$ $p=a(1-e^2)$ $p=2r_p$ $p=a(1-e^2)$ (positive)
Semi-major axis $a$ $a=r$ $a=-\frac{\mu}{2\xi}$
$a=\frac{r_a+r_p}{2}$
$a=\infty$ $a=-\frac{\mu}{2\xi}$ (negative)
Velocity $v$ $v=\sqrt{\frac{\mu}{r}}$
 $v$ $=\sqrt{2\left(\frac{\mu}{r}+\xi\right)}$ $=\sqrt{\frac{2\mu}{r}-\frac{\mu}{a}}$ $=\sqrt{\frac{\mu}{r}\left(2-\frac{1-e^2}{1+e\cos\theta}\right)}$
$v=v_{esc}=\sqrt{\frac{2\mu}{r}}$ $v=\sqrt{\frac{2\mu}{r}-\frac{\mu}{a}}$
Hyperbolic excess velocity $v_\infty$ $v_\infty=0$ $v_\infty=\sqrt{-\frac{\mu}{a}}$
Asymptotic True Anomaly $\theta_\infty$ $\theta_\infty=\pi$ $\theta_\infty=\cos^{-1}\left(-\frac{1}{e}\right)$
Eccentricity $e$ $e=0$
 $e$ $=\frac{r_a-r_p}{r_a+r_p}$ $=\sqrt{1+\frac{2\xi h^2}{\mu^2}}$
$e=1$ $e=\sqrt{1+\frac{2\xi h^2}{\mu^2}}$
Mean Motion $n=\frac{2\pi}{T}$ $n=\sqrt{\frac{\mu}{a^3}}$ $n=\sqrt{\frac{\mu}{a^3}}$ $n=2\sqrt{\frac{\mu}{p^3}}$ $n=\sqrt{-\frac{\mu}{a^3}}$
Period $T=\frac{2\pi}{n}$ $T=2\pi\sqrt{\frac{a^3}{\mu}}$ $T=2\pi\sqrt{\frac{a^3}{\mu}}$ $T=\infty$ $T=\infty$