Last active
August 29, 2015 14:10
-
-
Save rikusalminen/5dc81a18999b256c6e55 to your computer and use it in GitHub Desktop.
Celestial mechanics cheat sheet
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
| % vim:nolist lbr tw=78 expandtab autoindent nocindent | |
| \documentclass[a4paper]{article} | |
| \usepackage[utf8]{inputenc} | |
| \usepackage[landscape,top=1cm,bottom=1cm,left=1cm,right=1cm]{geometry} | |
| %\usepackage{amsmath} | |
| \pagestyle{empty} | |
| \setcounter{secnumdepth}{0} | |
| \title{Celestial mechanics cheatsheet} | |
| \author{Riku Salminen} | |
| \date{\today} | |
| \begin{document} | |
| \begin{center} | |
| \begin{tabular}{ l | c | c | c | c | c | c} | |
| & | |
| & | |
| Conic section ($f$) & | |
| Ellipse ($E$) & | |
| Hyperbola ($F$) & | |
| Parabola ($D$) & | |
| Universal ($s$) \\ | |
| \hline | |
| & | |
| & | |
| $ \frac{p}{e} = x + \frac{r}{e} $ & | |
| $ \frac{\left( x + c \right) ^2}{a^2} + \frac{y^2}{b^2} = 1 $ & | |
| $ \frac{\left( x - c \right) ^2}{a^2} - \frac{y^2}{b^2} = 1 $ & | |
| $ 2p \left( x - q \right) = - y^2 $ & | |
| $ $ \\ | |
| eccentricity & | |
| $ e $ & | |
| $ e = \sqrt{1 + \frac{2 \epsilon h^2}{\mu ^2}} $ & | |
| $ 0 < e < 1 $ & | |
| $ e > 1 $ & | |
| $ e = 1 $ & | |
| $ $ \\ | |
| semi-major axis & | |
| $ a $ & | |
| $ \frac{p}{1 - e^2} $ & | |
| $ a > 0 $ & | |
| $ a < 0 $ & | |
| $ \infty $ & | |
| $ \alpha = \frac{\mu}{a} = \frac{2 \mu}{r} - v^2 $ \\ | |
| semi-minor axis & | |
| $ b $ & | |
| $ $ & | |
| $ \frac{p}{\sqrt{1 - e^2}} = a \sqrt{1 - e^2} $ & | |
| $ \frac{p}{\sqrt{e^2 - 1}} = -a \sqrt{e^2 - 1} $ & | |
| $ \infty $ & | |
| $ $ \\ | |
| focal distance & | |
| $ c $ & | |
| $ $ & | |
| $ c = \sqrt{a^2 - b^2} = a e $ & | |
| $ c = \sqrt{a^2 + b^2} = -a e $ & | |
| $ \infty $ & | |
| $ $ \\ | |
| %pericenter distance & | |
| %$ q $ & | |
| %$ \frac{p}{1 + e} $ & | |
| %$ $ & | |
| %$ $ & | |
| %$ \frac{p}{2} $ & | |
| %$ $ \\ | |
| \hline | |
| % angular momentum & | |
| angul. momentum & % XXX: doesn't fit in one page otherwise :( | |
| $ h $ & | |
| $ \vec{h} = \vec{r} \times \vec{v} = \sqrt{\mu p} = r^2 \dot{f} $ & | |
| $ \sqrt{\mu a \left( 1 - e^2 \right)} $ & | |
| $ \sqrt{-\mu a \left( e^2 - 1 \right)} $ & | |
| $ \sqrt{\mu p} $ & | |
| $ $ \\ | |
| orbital energy & | |
| $ \epsilon $ & | |
| $ \epsilon = - \frac{\mu}{2 a} = \frac{v^2}{2} - \frac{\mu}{r} $ & | |
| $ \epsilon < 0$ & | |
| $ \epsilon > 0$ & | |
| $ \epsilon = 0$ & | |
| $ \epsilon = - \frac{\alpha}{2} $ \\ | |
| \hline | |
| true anomaly & | |
| $ f $ & | |
| $ $ & | |
| $ | |
| \left\{ | |
| \begin{array}{ll} | |
| \sin f &= \frac{\sqrt{1 - e^2} \sin E}{1 - e \cos E} \\ | |
| \cos f &= \frac{\cos E - e}{1 - e \cos E} \\ | |
| \tan \frac{f}{2} &= | |
| \sqrt{\frac{1+e}{1-e}} \tan \frac{E}{2} \\ | |
| \end{array} | |
| \right. | |
| $ & | |
| $ | |
| \left\{ | |
| \begin{array}{ll} | |
| \sin f &= \frac{\sqrt{e^2 - 1} \sinh F}{e \cosh F - 1} \\ | |
| \cos f &= \frac{e - \cosh F}{e \cosh F - 1} \\ | |
| \tan \frac{f}{2} &= | |
| \sqrt{\frac{e+1}{e-1}} \tanh \frac{F}{2} \\ | |
| \end{array} | |
| \right. | |
| $ & | |
| $ | |
| \left\{ | |
| \begin{array}{ll} | |
| \sin f &= \frac{2 D}{1 + D^2} \\ | |
| \cos f &= \frac{1 - D^2}{1 + D^2} \\ | |
| \tan \frac{f}{2} &= D \\ | |
| \end{array} | |
| \right. | |
| $ & | |
| $ $ \\ | |
| eccentric anomaly & | |
| $ E $ & | |
| $ $ & | |
| $ | |
| \left\{ | |
| \begin{array}{ll} | |
| \sin E &= \frac{\sqrt{1 - e^2} \sin f}{1 + e \cos f} \\ | |
| \cos E &= \frac{e + \cos f}{1 + e \cos f} \\ | |
| \end{array} | |
| \right. | |
| $ & | |
| $ | |
| \left\{ | |
| \begin{array}{ll} | |
| \sinh F &= \frac{\sqrt{e^2 - 1} \sin f}{1 + e \cos f} \\ | |
| \cosh F &= \frac{e + \cos f}{1 + e \cos f} \\ | |
| \end{array} | |
| \right. | |
| $ & | |
| $ D = \tan \frac{1}{2} f $ & | |
| $ dt = r ds $ \\ | |
| & | |
| $ $ & | |
| $ $ & | |
| $ | |
| \left\{ | |
| \begin{array}{ll} | |
| e \sin E &= \frac{\vec{r} \cdot \vec{v}}{\sqrt{\mu a}} \\ | |
| e \cos E &= 1 - \frac{r}{a} \\ | |
| \end{array} | |
| \right. | |
| $ & | |
| $ | |
| \left\{ | |
| \begin{array}{ll} | |
| e \sinh F &= \frac{\vec{r} \cdot \vec{v}}{\sqrt{- \mu a}} \\ | |
| e \cosh F &= 1 - \frac{r}{a} \\ | |
| \end{array} | |
| \right. | |
| $ & | |
| $ | |
| \left\{ | |
| \begin{array}{ll} | |
| D &= \frac{\vec{r} \cdot \vec{v}}{\sqrt{\mu p}} \\ | |
| D^2 &= \frac{r}{q} - 1 \\ | |
| \end{array} | |
| \right. | |
| $ & | |
| $ $ \\ | |
| partial derivatives & | |
| $ \frac{d}{dt} $ & | |
| $ \dot{f} = \sqrt{\frac{\mu}{p^3}} \left( 1 + e \cos f \right) ^2 $ & | |
| $ \dot{E} = \frac{1}{r} \sqrt{\frac{\mu}{a}} $ & | |
| $ \dot{F} = \frac{1}{r} \sqrt{\frac{\mu}{- a}} $ & | |
| $ \dot{D} = \frac{1}{r} \sqrt{\frac{\mu}{p}} $ & | |
| $ \dot{s} = \frac{ds}{dt} = \frac{1}{r} $ \\ | |
| & | |
| $ \frac{d}{dM} $ & | |
| $ \frac{df}{dM} = \frac{df}{dE} \frac{dE}{dM} $ & | |
| $ \frac{dE}{dM} = \frac{1}{1 - e \cos E} $ & | |
| $ \frac{dF}{dM} = \frac{1}{e \cosh F - 1} $ & | |
| $ \frac{dD}{dM} = \frac{2}{D^2 + 1} $ & | |
| $ \frac{d}{dt} = \frac{ds}{dt} \frac{d}{ds} = \frac{1}{r} \frac{d}{ds} $ \\ | |
| & | |
| $ \frac{df}{dE} $ & | |
| $ $ & | |
| $ \frac{df}{dE} = \frac{\sqrt{1-e^2}}{1 - e \cos E} $ & | |
| $ \frac{df}{dF} = \frac{\sqrt{e^2 - 1}}{e \cosh F - 1} $ & | |
| $ \frac{df}{dD} = \frac{2}{D^2 + 1} $ & | |
| $ \frac{d}{ds} = r \frac{d}{dt} $ \\ | |
| universal variable & | |
| $ s $ & | |
| $ $ & | |
| $ \sqrt{a} \left( E - E_0 \right) $ & | |
| $ \sqrt{-a} \left( F - F_0 \right) $ & | |
| $ \sqrt{p} \left( D - D_0 \right) $ & | |
| $ $ \\ | |
| \hline | |
| time of flight & | |
| $ t $ & | |
| $ \frac{1}{n} M $ & | |
| $ \sqrt{\frac{a^3}{\mu}} \left( E - e \sin E \right) $ & | |
| $ \sqrt{\frac{-a^3}{\mu}} \left( e \sinh F - F \right) $ & | |
| $ \sqrt{\frac{p^3}{\mu}} \left( \frac{1}{6} D^3 + \frac{1}{2} D \right) $ & | |
| $ r_0 s c_1 + r_0 \dot{r_0} s^2 c_2 + \mu s^3 c_3 $ \\ | |
| radius & | |
| $ r $ & | |
| $ \frac{p}{1 + e \cos f} $ & | |
| $ a \left( 1 - e \cos E \right) $ & | |
| $ a \left( 1 - e \cosh F \right) $ & | |
| $ q \left( D^2 + 1 \right) $ & | |
| $ r_0 c_0 + r_0 \dot{r_0} s c_1 + \mu s^2 c_2 $ \\ | |
| radial velocity & | |
| $ \dot{r} $ & | |
| $ \sqrt{\frac{\mu}{p}} e \sin f $ & | |
| $ \sqrt{\frac{\mu}{a}} \frac{e \sin E}{1 - e \cos E} $ & | |
| $ \sqrt{\frac{\mu}{-a}} \frac{e \sinh F}{e \cosh F - 1} $ & | |
| $ \sqrt{\frac{\mu}{p}} \frac{2 D}{D^2 + 1} $ & | |
| $ $ \\ | |
| horizontal velocity & | |
| $ r \dot{f} $ & | |
| $ \sqrt{\frac{\mu}{p}} \left( 1 + e \cos f \right) $ & | |
| $ \sqrt{\frac{\mu}{a}} \frac{\sqrt{1 - e^2}}{1 - e \cos E} $ & | |
| $ \sqrt{\frac{\mu}{-a}} \frac{\sqrt{e^2 - 1}}{e \cosh F - 1} $ & | |
| $ \sqrt{\frac{\mu}{p}} \frac{2}{D^2 + 1} $ & | |
| $ $ \\ | |
| orbital speed & | |
| $ v $ & | |
| $ \sqrt{\frac{\mu}{p} \left( e^2 + 2 e \cos f + 1 \right) } $ & | |
| $ \sqrt{\frac{\mu}{a} \frac{1 + e \cos E}{1 - e \cos E}}$ & | |
| $ \sqrt{\frac{\mu}{-a} \frac{e \cosh F + 1}{e \cosh F - 1}} $ & | |
| $ \sqrt{\frac{\mu}{p} \frac{4}{D^2 + 1}} $ & | |
| $ $ \\ | |
| flight path angle & | |
| $ \phi $ & | |
| $ \tan \phi = \frac{\dot{r}}{r \dot{f}} = \frac{e \sin f}{1 + e \cos f} $ & | |
| $ \tan \phi = \frac{e \sin E}{\sqrt{1 - e^2}} $ & | |
| $ \tan \phi = \frac{e \sinh F}{\sqrt{e^2 - 1}} $ & | |
| $ \tan \phi = D $ & | |
| $ $ \\ | |
| \hline | |
| position & | |
| $x$ & | |
| $ \frac{p \cos f}{1 - e \cos f} $ & | |
| $ a \left( \cos E - e \right) $ & | |
| $ a \left( \cosh F - e \right) $ & | |
| $ q \left( 1 - D^2 \right) $ & | |
| $ $ \\ | |
| & | |
| $y$ & | |
| $ \frac{p \sin f}{1 - e \cos f} $ & | |
| $ b \sin E $ & | |
| $ b \sinh F $ & | |
| $ p D $ & | |
| $ $ \\ | |
| velocity & | |
| $ \dot{x} $ & | |
| $ - \sqrt{\frac{p}{\mu}} \sin f $ & | |
| $ \sqrt{\frac{\mu}{a^3}} \frac{-a \sin E}{1 - e \cos E} $ & | |
| $ \sqrt{\frac{\mu}{- a^3}} \frac{a \sinh F}{e \cosh F - 1} $ & | |
| $ \sqrt{\frac{\mu}{p}} \frac{-2 D}{D^2 + 1} $ & | |
| $ $ \\ | |
| %& | |
| %$ $ & | |
| %$ $ & | |
| %$ - \frac{1}{r} \sqrt{\mu a} \sin E $ & | |
| %$ - \frac{1}{r} \sqrt{- \mu a} \sinh F $ & | |
| %$ - \frac{1}{r} \sqrt{\mu p} D $ & | |
| %$ $ \\ | |
| & | |
| $ \dot{y} $ & | |
| $ \sqrt{\frac{p}{\mu}} \left( e + \cos f \right) $ & | |
| $ \sqrt{\frac{\mu}{a^3}} \frac{b \cos E}{1 - e \cos E} $ & | |
| $ \sqrt{\frac{\mu}{- a^3}} \frac{b \cosh F}{e \cosh F - 1} $ & | |
| $ \sqrt{\frac{\mu}{p}} \frac{2}{D^2 + 1} $ & | |
| $ $ \\ | |
| %& | |
| %$ $ & | |
| %$ $ & | |
| %$ \frac{1}{r} \sqrt{\mu a \left( 1 - e^2 \right) } \cos E $ & | |
| %$ \frac{1}{r} \sqrt {- \mu a \left( e^2 - 1 \right) } \cosh F $ & | |
| %$ \frac{1}{r} \sqrt {\mu p} $ & | |
| %$ $ \\ | |
| \hline | |
| f \& g functions & | |
| $ f $ & | |
| $ 1 - \frac{r}{p} \left( 1 - \cos \hat{f} \right) $ & | |
| $ 1 - \frac{a}{r_0} \left( 1 - \cos \hat{E} \right) $ & | |
| $ 1 - \frac{a}{r_0} \left( 1 - \cosh \hat{F} \right) $ & | |
| $ 1 - \frac{q}{r_0} \hat{D}^2 $ & | |
| $ 1 - \frac{\mu}{r_0} s^2 c_2 $ \\ | |
| & | |
| $ g $ & | |
| $ \frac{r r_0}{\sqrt{\mu p}} \sin \hat{f}$ & | |
| $ | |
| \sqrt{\frac{a}{\mu}} r_0 \sin \hat{E} + | |
| \frac{a r_0 \dot{r_0} \left(1 - \cos \hat{E} \right)}{\mu} | |
| $ & | |
| $ | |
| \sqrt{\frac{-a}{\mu}} r_0 \sinh \hat{F} + | |
| \frac{a r_0 \dot{r_0} \left(1 - \cosh \hat{F} \right)}{\mu} | |
| $ & | |
| $ | |
| \sqrt{\frac{p}{\mu}} r_0 \hat{D} + | |
| \frac{q r_0 \dot{r_0} \hat{D}^2}{\mu} | |
| $ & | |
| $ r_0 s c_1 + r_0 \dot{r_0} s^2 c_2 $ \\ | |
| & | |
| & | |
| & | |
| $ \hat{t} - \sqrt{\frac{a^3}{\mu}} \left( \hat{E} - \sin \hat{E} \right) $ & | |
| $ \hat{t} - \sqrt{\frac{-a^3}{\mu}} \left( \sinh \hat{F} - \hat{F} \right) $ & | |
| $ \hat{t} - \sqrt{\frac{p^3}{\mu}} \frac{1}{6} \hat{D}^3 $ & | |
| $ \hat{t} - \mu s^3 c_3 $ \\ | |
| & | |
| $ \dot{f} $ & | |
| $ | |
| \sqrt{\frac{\mu}{p}} \tan \frac{\hat{f}}{2} | |
| \left( \frac{1 - \cos \hat{f}}{p} - \frac{1}{r} - \frac{1}{r_0} \right) | |
| $ & | |
| $ - \frac{\sqrt{\mu a}}{r r_0} \sin \hat{E} $ & | |
| $ - \frac{\sqrt{- \mu a}}{r r_0} \sinh \hat{F} $ & | |
| $ - \frac{\sqrt{\mu p}}{r r_0} \hat{D} $ & | |
| $ - \frac{\mu}{r r_0} s c_1 $ \\ | |
| & | |
| $ \dot{g} $ & | |
| $ 1 - \frac{r_0}{p} \left( 1 - \cos \hat{f} \right) $ & | |
| $ 1 - \frac{a}{r} \left( 1 - \cos \hat{E} \right) $ & | |
| $ 1 - \frac{a}{r} \left( 1 - \cosh \hat{F} \right) $ & | |
| $ 1 - \frac{q}{r} \hat{D}^2 $ & | |
| $ 1 - \frac{\mu}{r} s^2 c_2 $ \\ | |
| \end{tabular} | |
| \end{center} | |
| \end{document} |
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment