# gravity derivations

1. normal gravity at sea level from latitude

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(g\) | normal gravity at sea level | \(\frac{m}{s^2}\) | gravity {:}
   \(\phi\) | latitude | \(degN\) | latitude {:}

   The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

   \[\begin{eqnarray} g & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)} {\sqrt{1 - 0.00669437999013{\sin}^2(\frac{\pi}{180}\phi)}} \end{eqnarray}\]

2. gravity at specific altitude

   symbol | name | unit | variable name
   --- | --- | --- | ---
   \(a\) | WGS84 semi-major axis | \(m\) |
   \(b\) | WGS84 semi-minor axis | \(m\) |
   \(f\) | WGS84 flattening | \(m\) |
   \(g_{h}\) | gravity at specific height | \(\frac{m}{s^2}\) | gravity {:,vertical}
   \(g\) | normal gravity at sea level | \(\frac{m}{s^2}\) | gravity {:}
   \(GM\) | WGS84 earth’s gravitational constant | \(\frac{m^3}{s^2}\) |
   \(z\) | altitude | \(m\) | altitude {:,vertical}
   \(\phi\) | latitude | \(degN\) | latitude {:}
   \(\omega\) | WGS84 earth angular velocity | \(rad/s\) |

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \begin{eqnarray} m & = & \frac{\omega^2a^2b}{GM} \\ g_{h} & = & g \left[ 1 - \frac{2}{a}\left(1+f+m-2f{\sin}^2(\frac{\pi}{180}\phi)\right)z + \frac{3}{a^2}z^2 \right] \\ \end{eqnarray}

3. gravity at earth surface

   symbol | name | unit | variable name
   --- | --- | --- | ---
   \(a\) | WGS84 semi-major axis | \(m\) |
   \(b\) | WGS84 semi-minor axis | \(m\) |
   \(f\) | WGS84 flattening | \(m\) |
   \(g_{surf}\) | gravity at surface altitude | \(\frac{m}{s^2}\) | surface_gravity {:}
   \(g\) | normal gravity at sea level | \(\frac{m}{s^2}\) | gravity {:}
   \(GM\) | WGS84 earth’s gravitational constant | \(\frac{m^3}{s^2}\) |
   \(z_{surf}\) | surface altitude | \(m\) | surface_altitude {:}
   \(\phi\) | latitude | \(degN\) | latitude {:}
   \(\omega\) | WGS84 earth angular velocity | \(rad/s\) |

   The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

   \begin{eqnarray} m & = & \frac{\omega^2a^2b}{GM} \\ g_{surf} & = & g \left[ 1 - \frac{2}{a}\left(1+f+m-2f{\sin}^2(\frac{\pi}{180}\phi)\right)z + \frac{3}{a^2}z^2 \right] \\ \end{eqnarray}