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}