choclo.point.gravity_uu

choclo.point.gravity_uu(easting_p, northing_p, upward_p, easting_q, northing_q, upward_q, mass)

Upward-upward component of the gravitational tensor due to a point source.

Returns the upward-upward component of the gravitational tensor produced by a single point source on a single computation point

Parameters:

easting_p, northing_p, upward_pfloat

Easting, northing and upward coordinates of the observation point in meters.

easting_q, northing_q, upward_qfloat

Easting, northing and upward coordinates of the point source in meters.

massfloat

Mass of the point source in kilograms.

Returns:

g_uufloat

Upward-upward component of the gravitational tensor generated by the point source on the observation point in \(\text{s}^{-2}\).

Notes

Returns the upward-upward component \(g_{zz}(\mathbf{p})\) of the gravitational tensor \(\mathbf{T}\) on the observation point \(\mathbf{p} = (x_p, y_p, z_p)\) generated by a single point source located in \(\mathbf{q} = (x_q, y_q, z_q)\) and mass \(m\).

\[g_{zz}(\mathbf{p}) = G m \left[ \frac{ 3 (z_p - z_q)^2 }{ \lVert \mathbf{p} - \mathbf{q} \rVert_2^5 } - \frac{ 1 }{ \lVert \mathbf{p} - \mathbf{q} \rVert_2^3 } \right]\]

where \(\lVert \cdot \rVert_2\) refer to the \(L_2\) norm (the Euclidean distance between \(\mathbf{p}\) and \(\mathbf{q}\)) and \(G\) is the Universal Gravitational Constant.