International Gravity Formula

ahrs.utils.wgs84.international_gravity(lat: float, epoch: str = '1980') → float

Compute the normal gravity, \(\mathrm{g}\), using the International Gravity Formula [Lam45], adapted from Stokes’ formula, and adopted by the International Association of Geodesy in 1930.

The expression for gravity on a spheroid, which combines gravitational attraction and centrifugal acceleration, at a certain latitude, \(\phi\), can be written in the form of a series:

\[\mathrm{g} = \mathrm{g}_e\big(1 + \beta\sin^2(\phi) - \beta_1\sin^2(2\phi) - \beta_2\sin^2(\phi)\sin^2(2\phi) - \beta_3\sin^4(\phi)\sin^2(2\phi) - \dots\big)\]

where the values of the \(\beta\)’s are:

\[\begin{split}\begin{array}{ll} \beta &= \frac{5}{2}m\Big(1-\frac{17}{35}f - \frac{1}{245}f^2 - \frac{13}{18865}f^3 - \dots\Big) - f \\ \beta_1 &= \frac{1}{8}f(f+2\beta) \\ \beta_2 &= \frac{1}{8}f^2(2f+3\beta) - \frac{1}{32}f^3(3f+4\beta) \\ & \vdots \\ & \mathrm{etc.} \end{array}\end{split}\]

and \(\mathrm{g}_e\) is the measured normal gravity on the Equator. For the case of the International Ellipsoid, the third-order terms are negligible. So, in practice, the term \(\beta_2\) and all following terms are dropped to yield the form:

\[\mathrm{g} = \mathrm{g}_e \big(1 + \beta \sin^2\phi - \beta_1 \sin^2(2\phi)\big)\]

In the original definition the values of \(\beta\) and \(\beta_1\) are rounded off to seven decimal places to simply get the working formula:

\[\mathrm{g} = 9.78049 \big(1 + 0.0052884 \sin^2\phi - 0.0000059 \sin^2(2\phi)\big)\]

Back in 1930 the definitions of the elementary properties (\(a\), \(\mathrm{g}_e\), etc.) weren’t as accurate as now. At different moments in history, the values were updated to improve the accuracy of the formula. Those different moments are named epochs and are labeled according to the year they were updated:

epoch	\(\mathrm{g}_e\)	\(\beta\)	\(\beta_1\)
1930	9.78049	5.2884 x 10^-3	5.9 x 10^-6
1948	9.780373	5.2891 x 10^-3	5.9 x 10^-6
1967	9.780318	5.3024 x 10^-3	5.9 x 10^-6
1980	9.780367715	5.3024 x 10^-3	5.8 x 10^-6
1984	9.7803253359	5.3024 x 10^-3	5.8 x 10^-6

The epoch 1980, which is the most commonly used, is implemented here by default.

The epoch 1984 was updated with the WGS84 ellipsoid model, but it does not offer a significant improvement in accuracy [Mor80] [HvS13].

Parameters:

• lat (float) – Geographical Latitude, in decimal degrees.

• epoch (str, default: ‘1980’) – Epoch of the Geodetic Reference System. Options are '1930', '1948', '1967', '1980', and '1984'.

Returns:

g – Normal gravity, in m/s^2, at given latitude.

Return type:

float

Examples

>>> ahrs.utils.international_gravity(10.0)
9.781884110728155
>>> ahrs.utils.international_gravity(10.0, epoch='1930')
9.7820428934191