# 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 earthâ€™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 earthâ€™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}