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! $\nu vv\nu v\nu \nu v$. 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	$\overrightarrow{h}=\overrightarrow{r}\times \overrightarrow{v}$ $h$ $=\sqrt{\mu p}$ $=r^2\dot{\theta }$ $=r_a{v}_a$ $=r_p{v}_p$
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	$\overrightarrow{e}=\frac{(v^2-\frac{\mu }{r})\overrightarrow{r}-(\overrightarrow{r}\cdot \overrightarrow{v})\overrightarrow{v}}{\mu }$
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	$\dot{\theta }=\frac{n{a}^2}{r^2}\sqrt{1-e^2}$
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 $E$	$E=\theta$	$E=2{\tan }^{-1}(\sqrt{\frac{1-e}{1+e}}\tan \left(\frac{\theta }{2}\right))$ $\sin E=\frac{sin\theta \sqrt{1-e^2}}{1+e\cos \theta }$ $\cos E=\frac{e+cos\theta }{1+e\cos \theta }$ $E_{n+1}=E_n+\frac{M-E_n+e\sin E}{1-e\cos E}$	$E=\tan \frac{\theta }{2}$	$\sinh E=\frac{-\sin \theta \sqrt{e^2-1}}{1+e\cos \theta }$ $\cosh E=\frac{\cos \theta +e}{1+e\cos \theta }$ $E_{n+1}=E_n+\frac{M+E_n-e\sinh E}{e\cosh E-1}$
True Anomaly $\theta$	$\theta =E$	$\theta =2{\tan }^{-1}(\sqrt{\frac{1+e}{1-e}}\tan \left(\frac{E}{2}\right))$ $\theta ={\cos }^{-1}(\frac{p}{re}-\frac{1}{e})$	$\sin \theta =\frac{pE}{r}$ $\cos \theta =\frac{p-r}{r}$ $\theta ={\cos }^{-1}(\frac{p}{re}-\frac{1}{e})$	$\sin \theta =\frac{-\sinh E\sqrt{e^2-1}}{1-e\cosh E}$ $\cos \theta =\frac{\cosh E-e}{1-e\cosh E}$ $\theta ={\cos }^{-1}(\frac{p}{re}-\frac{1}{e})$
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 $\varphi$	$\varphi =0$	$\sin \varphi =\frac{e\sin E}{\sqrt{1-e^2{\cos }^{2}E}}$ $\cos \varphi =\sqrt{\frac{1-e^2}{1-e^2{\cos }^{2}E}}$	$\varphi =\frac{\theta }{2}$	$\sin \varphi =\frac{-e\sinh E}{\sqrt{e^2{\cosh }^{2}E-1}}$ $\cos \varphi =\sqrt{\frac{e^2-1}{e^2{\cosh }^{2}E-1}}$
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 $r_a$	$r_a=a$	$r_a=a(1+e)$	$r_a=\mathrm{\infty }$	$r_a=a(1+e)$ (negative)
Semi-parameter $p$	$p=a$	$p=a(1-e^2)$	$p=2r_p$	$p=a(1-e^2)$ (positive)
Semi-major axis $a$	$a=r$	$a=-\frac{\mu }{2\xi }$ $a=\frac{r_a+r_p}{2}$	$a=\mathrm{\infty }$	$a=-\frac{\mu }{2\xi }$ (negative)
Velocity $v$	$v=\sqrt{\frac{\mu }{r}}$	$v$ $=\sqrt{2(\frac{\mu }{r}+\xi )}$ $=\sqrt{\frac{2\mu }{r}-\frac{\mu }{a}}$ $=\sqrt{\frac{\mu }{r}(2-\frac{1-e^2}{1+e\cos \theta })}$	$v=v_{esc}=\sqrt{\frac{2\mu }{r}}$	$v=\sqrt{\frac{2\mu }{r}-\frac{\mu }{a}}$
Hyperbolic excess velocity $v_{\mathrm{\infty }}$			$v_{\mathrm{\infty }}=0$	$v_{\mathrm{\infty }}=\sqrt{-\frac{\mu }{a}}$
Asymptotic True Anomaly ${\theta }_{\mathrm{\infty }}$			${\theta }_{\mathrm{\infty }}=\pi$	${\theta }_{\mathrm{\infty }}={\cos }^{-1}(-\frac{1}{e})$
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 $T=\frac{2\pi }{n}$	$T=2\pi \sqrt{\frac{a^3}{\mu }}$	$T=2\pi \sqrt{\frac{a^3}{\mu }}$	$T=\mathrm{\infty }$	$T=\mathrm{\infty }$