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	${\vec {h}}={\vec {r}}\times {\vec {v}}$ $h$ $={\sqrt {\mu p}}$ $=r^{2}{\dot {\theta }}$ $=r_{a}v_{a}$ $=r_{p}v_{p}$
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	${\vec {e}}={\frac {\left(v^{2}-{\frac {\mu }{r}}\right){\vec {r}}-({\vec {r}}\cdot {\vec {v}}){\vec {v}}}{\mu }}$
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 $\phi$	$\phi =0$	$\sin \phi ={\frac {e\sin E}{\sqrt {1-e^{2}\cos ^{2}E}}}$ $\cos \phi ={\sqrt {\frac {1-e^{2}}{1-e^{2}\cos ^{2}E}}}$	$\phi ={\frac {\theta }{2}}$	$\sin \phi ={\frac {-e\sinh E}{\sqrt {e^{2}\cosh ^{2}E-1}}}$ $\cos \phi ={\sqrt {\frac {e^{2}-1}{e^{2}\cosh ^{2}E-1}}}$
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 $r_{a}$	$r_{a}=a$	$r_{a}=a(1+e)$	$r_{a}=\infty$	$r_{a}=a(1+e)$ (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 $T={\frac {2\pi }{n}}$	$T=2\pi {\sqrt {\frac {a^{3}}{\mu }}}$	$T=2\pi {\sqrt {\frac {a^{3}}{\mu }}}$	$T=\infty$	$T=\infty$