choclo.prism.gravity_e#

choclo.prism.gravity_e(easting, northing, upward, prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_top, density)[source]#

Easting component of the gravitational acceleration due to a prism.

Returns the easting component of the gravitational acceleration produced by a single rectangular prism on a single computation point.

Parameters:

easting, northing, upwardfloat

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

prism_west, prism_east, prism_south, prism_north, prism_bottom, prism_topfloat

The boundaries of the prism. Must be in meters.

densityfloat

Density of the rectangular prism in kilograms per cubic meter.

Returns:

g_efloat

Easting component of the gravitational acceleration generated by the rectangular prism on the observation point in $\text{m}/{\text{s}}^2$.

Notes

Returns the easting component $g_x(\mathbf{p})$ of the gravitational acceleration $\mathbf{g}$ on the observation point $\mathbf{p}=(x_p,y_p,z_p)$ generated by a single rectangular prism defined by its boundaries $x_1,x_2,y_1,y_2,z_1,z_2$ and with a density $\rho$:

$$g_x(\mathbf{p})=G\rho |||k_x(x,y,z){|}_{X_1}^{X_2}{|}_{Y_1}^{Y_2}{|}_{Z_1}^{Z_2}$$

where

$$k_x(x,y,z)=[y\mathrm{safe-ln}(z,r)+z\mathrm{safe-ln}(y,r)-x\mathrm{safe-arctan}(yz,xr)]$$

$$r=\sqrt{x^2+y^2+z^2},$$

and

$$\begin{array}{r}{X}_1=x_1-x_p\\ X_2=x_2-x_p\\ Y_1=y_1-y_p\\ Y_2=y_2-y_p\\ Z_1=z_1-z_p\\ Z_2=z_2-z_p\end{array}$$

are the shifted coordinates of the prism boundaries and $G$ is the Universal Gravitational Constant.

The $\mathrm{safe-ln}$ and $\mathrm{safe-arctan}$ functions are defined as follows:

$$\begin{array}{r}\mathrm{safe-ln}(x,r)=\{\begin{array}{ll}0& x=0,r=0\\ \ln (x+r)& x\ge 0\\ \ln ((r^2-x^2)/(r-x))& x<0,r\ne -x\\ -\ln (-2x)& x<0,r=-x\end{array}\end{array}$$

$$\begin{array}{r}\mathrm{safe-arctan}(y,x)=\{\begin{array}{ll}\text{arctan}\left(\frac{y}{x}\right)& x\ne 0\\ \frac{\pi }{2}& x=0\text{and}y>0\\ -\frac{\pi }{2}& x=0\text{and}y<0\\ 0& x=0\text{and}y=0\end{array}\end{array}$$

These were defined after [Fukushima2020] and guarantee a good accuracy on any observation point.

References

• [Nagy2000]

• [Nagy2002]

• [Fukushima2020]