geoana.gravity.Sphere.gravitational_potential

Sphere.gravitational_potential(xyz)

Gravitational potential due to a sphere.

\[ \begin{align}\begin{aligned}r > R\\\phi (\mathbf{r}) = \gamma \frac{m}{r}\\r < R\\\phi (\mathbf{r}) = \gamma \frac{2}{3} \pi \rho (3R^2 - r^2)\end{aligned}\end{align} \]

Parameters:

xyz(…, 3) numpy.ndarray

Locations to evaluate at in units m.

Returns:

(…, ) numpy.ndarray

Gravitational potential at sphere location xyz in units \(\frac{m^2}{s^2}\).

Examples

Here, we define a sphere with mass m and plot the gravitational potential as a function of distance.

>>> import numpy as np
>>> import matplotlib.pyplot as plt
>>> from geoana.gravity import Sphere
>>> from geoana.utils import ndgrid

Define the sphere.

>>> location = np.r_[0., 0., 0.]
>>> rho = 1.0
>>> radius = 1.0
>>> simulation = Sphere(
>>>     location=location, rho=rho, radius=radius
>>> )

Now we create a set of gridded locations, take the distances and compute the gravitational potential.

>>> X, Y = np.meshgrid(np.linspace(-1, 1, 20), np.linspace(-1, 1, 20))
>>> Z = np.zeros_like(X) + 0.25
>>> xyz = np.stack((X, Y, Z), axis=-1)
>>> r = np.linalg.norm(xyz, axis=-1)
>>> u = simulation.gravitational_potential(xyz)

Finally, we plot the gravitational potential as a function of distance.

>>> plt.plot(r, u)
>>> plt.xlabel('Distance from sphere')
>>> plt.ylabel('Gravitational potential')
>>> plt.title('Gravitational Potential as a function of distance from sphere')
>>> plt.show()

(Source code, png, pdf)