Front Cover Equations

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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 θ(theta) instead of ν(nu) for true anomaly, as it looks too much like v velocity. Go ahead and try it out! νvvνvννv. 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.

[edit] Common two-body equations

Specific angular momentum \vec{h}=\vec{r}\times\vec{v}
h=\sqrt{\mu p}
=r^2\dot{\theta}
= rava
= rpvp
Specific Mechanical Energy
ξ=\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}}

[edit] Anomalies

Circle Ellipse Parabola Hyperbola
Eccentric Anomaly E E = θ 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 θ θ = 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 = θ M = E − esinE M=\frac{E^3}{3}+E M = esinh(E) − E

[edit] Other Parameters

Circle Ellipse Parabola Hyperbola
Flight Path Angle φ φ = 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 rp rp = a rp = a(1 − e) r_p=\frac{p}{2} rp = a(1 − e)
Apoapsis ra ra = a ra = a(1 + e) r_a=\infty ra = a(1 + e) (negative)
Semi-parameter p p = a p = a(1 − e2) p = 2rp p = a(1 − e2) (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
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