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