Celestial mechanics cheat sheet

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\title{Celestial mechanics cheatsheet}
\author{Riku Salminen}
\date{\today}

\begin{document}

\begin{multicols}{4}

\section{Laws of planetary motion}

\begin{itemize}
    \item Orbit path is a conic section
    \item Mechanical energy is conserved
    \item Angular momentum is conserved
    \begin{itemize}
        \item Orbital motion in a plane fixed in inertial space
        \item Constant area velocity
    \end{itemize}
\end{itemize}

\section{Law of universal gravitation}

\[
    \vec{F} = - \frac{G m_0 m_1}{r^3} \vec{r}
\]

% $\mu = G \left( m_0 + m_1 \right)$

\section{Symbols and quantities}

\begin{description}
    \item[$\mu$] Gravity parameter
    \item[$\epsilon$] Specific orbital energy
    \item[$h$] Specific relative angular momentum

    \item[$e$] Eccentricity
    \item[$p$] Semi-latus rectum
    \item[$a$] Semi-major axis
    \item[$b$] Semi-minor axis
    \item[$c$] Focal distance
    \item[$q$] Pericenter distance

    \item[$f$] True anomaly
    \item[$E$] Eccentric anomaly
    \item[$F$] Hyperbolic anomaly
    \item[$D$] Parabolic anomaly
    \item[$M$] Mean anomaly
    \item[$n$] Mean motion

    \item[$s$] Universal variable
    \item[$\alpha$] Inverse semi-major axis

    \item[$r$] Radius
    \item[$v$] Velocity
    \item[$\phi$] Flight path angle
\end{description}

\section{Conic sections}

radius and true anomaly

\[
    r = \frac{p}{1 + e \cos f}
\]

focus-directrix property

focal-radii property

\begin{align*}
    M &= E - e \sin E \\
      &= e \sinh F - F
\end{align*}


\subsection{Eccentric anomaly}

\[
    M =
        \left\{
        \begin{array}{ll}
            E - e \sin E
                & \quad e < 1 \\
            e \sinh F - F
                & \quad e > 1 \\
            \frac{1}{6} D^3 + \frac{1}{2} D
                & \quad e = 1 > 0
        \end{array}
        \right.
\]

eccentric anomaly

hyperbolic anomaly

parabolic anomaly

\subsection{Subsection}

And a few subsections

lorem ipsum


\columnbreak

\section{f \& g functions}

\begin{align*}
    \vec{r} &= f\vec{r_0} + g\vec{v_0} \\
    \vec{v} &= \dot{f}\vec{r_0} + \dot{g}\vec{v_0}
\end{align*}

\begin{equation*}
    f \dot{g} - \dot{f} g = 1
\end{equation*}

\begin{align*}
    h &= x_0 \dot{y_0} - y_0 \dot{x_0} \\
    f &= \frac{x_1 \dot{y_0} - \dot{x_0} y_1}{h} \\
    g &= \frac{x_0 y_1 - x_1 y_0}{h} \\
    \dot{f} &= \frac{\dot{x_1} \dot{y_0} - \dot{x_0} \dot{y_1}}{h} \\
    \dot{g} &= \frac{x_0 \dot{y_1} - \dot{x_1} y_0}{h}
\end{align*}

\section{f \& g power series}

\section{Universal variables}

\[
    \alpha = \frac{\mu}{a} = \frac{2 \mu}{r} - v^2 = -2 \epsilon
\]

    $ dt = r ds $

\begin{align*}
    t &= r_0 s c_1 + r_0 \dot{r_0} s^2 c_2 + \mu s^3 c_3 \\
    r &= r_0 c_0 + r_0 \dot{r_0} c_1 + \mu s^2 c_2
\end{align*}

\columnbreak


\section{Stumpff functions}

\[
    c_0 =
        \left\{
        \begin{array}{ll}
            1
                & \quad z = 0 \\
            \cosh \sqrt{-z}
                & \quad z < 0 \\
            \cos \sqrt{z}
                & \quad z > 0
        \end{array}
        \right.
\]

\[
    c_1 =
        \left\{
        \begin{array}{ll}
            1
                & \quad z = 0 \\
            \frac{\sinh \sqrt{-z}}{\sqrt{-z}}
                & \quad z < 0 \\
            \frac{\sin \sqrt{z}}{\sqrt{z}}
                & \quad z > 0
        \end{array}
        \right.
\]

\[
    c_2 =
        \left\{
        \begin{array}{ll}
            \frac{1}{2}
                & \quad z = 0 \\
            \frac{\cosh \sqrt{-z} - 1}{-z}
                & \quad z < 0 \\
            \frac{1 - \cos \sqrt{z}}{z}
                & \quad z > 0
        \end{array}
        \right.
\]

\[
    c_3 =
        \left\{
        \begin{array}{ll}
            \frac{1}{6}
                & \quad z = 0 \\
            \frac{\sinh \sqrt{-z} - \sqrt{-z}}{-z \sqrt{-z}}
                & \quad z < 0 \\
            \frac{\sqrt{z} - \sin \sqrt{z}}{z \sqrt{z}}
                & \quad z > 0
        \end{array}
        \right.
\]

\[
    c_k = { \sum \limits_{i = 0}^{\infty} }
        \frac{ \left( -z \right) ^i }{ \left( k + 2i \right) ! }
\]

\begin{align*}
    z c_{k+2} \left( z \right) &=
        \frac{1}{k!} - c_k \left( z \right) \\
    c_0 \left( 4 z \right) &=
        2 c_0 \left( z \right) ^2 - 1 \\
    c_1 \left( 4 z \right) &=
        c_0 \left( z \right) c_1 \left( z \right) \\
    c_2 \left( 4 z \right) &= 
        \frac{1}{2} c_1 \left( z \right) ^2 \\
    c_3 \left( 4 z \right) &=
        \frac{
        \left( c_2 \left( z \right) +
            c_0 \left( z \right) c_3 \left( z \right)
        \right)
        }{4}
\end{align*}

\[
    \frac{d}{ds} c_0 \left( \alpha s^2 \right) =
        - \alpha s c_1 \left( \alpha s^2 \right)
\]

\[
    \frac{d}{ds} s^{k+1} c_{k+1} \left( \alpha s^2 \right) =
        s^k c_k \left( \alpha s^2 \right)
\]

\end{multicols}

\end{document}