Gravity anomaly due to a buried sphere

Navigating Gravity Anomaly of a Buried Sphere

Buried Sphere Model

Model Assumptions

Open Processing function

PERL code

P5.js implementation

Draw the problem geometry with latex

References and Citation

Introduction

It is possible to calculate the exact gravity anomaly due to a wide variety of simple shapes. The simplified geometry of these shapes (spheres, rods, plates) simplifies the analytical solution to find their gravity anomalies (e.g., application of Gauss's law). A surprisingly large number of natural gravity anomalies can be assessed by comparing them to the gravity anomalies of simple shapes. A general form from the vertical component of the gravity anomaly due to a buried mass is: $$ g_z = G \int \frac{dm}{r^2} \cos \theta $$ where $g_z$ is the gravity anomaly, $G$ is the gravitational constant, $m$ is the anomalous mass, $r$ is distance to the center of the mass, and $\theta$ is the angle between the center of the mass, the point where gravity is measured, and vertical. Vertical is defined by the Earth's gravity field. It is assumed that the deflection of the equipotential surface (and deflection of the vertical) can be neglected. That is, the anomalous mass is very small compared to the magnitude of the Earth's field.

Here, code is presented to calculate the gravity anomaly due to a buried sphere.The buried sphere model is useful for approximating the geometries of magma chambers, ore bodies and some structural domes.

Calculation of the gravity anomaly due to a sphere implemented in p5 javascript: