Physics Formulae/Gravitation Formulae

Lead Article: Tables of Physics Formulae

This article is a summary of the laws, principles, defining quantities, and useful formulae in the analysis of Gravitation.

Gravitational Field Definitions
A common misconseption occurs between centre of mass and centre of gravity. They are defined in simalar ways but are not exactly the same quantity. Centre of mass is the mathematical descrition of placing all the mass in the region considered to one position, centre of gravity is a real physical quantity, the point of a body where the gravitational force acts. They are only equal if and only if the external gravitational field is uniform.

Contrary to the strong analogy between (classical) gravitation and electrostatics, there are no "centre of charge" or "centre of electrostatic attraction" analogues.

Gravitational Potential Gradient and Field


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 * $$ \mathbf{g} = - \nabla U $$
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Modern Laws
Gravitomagnetism (GEM) Equations:

In an relativley flat spacetime due to weak gravitational fields (by General Relativity), the following gravitational analogues of Maxwell's equations can be found, to describe an analogous Gravitomagnetic Field. They are well established by the theory, but have yet to be verified by experiment.

Classical Laws
It can be found that Kepler's Laws, though originally discovered from planetary observations (also due to Tycho Brahe), are true for any central forces.

For Kepler's 1st law, the equation is nothing physically fundamental; simply the polar equation of an ellipse where the pole (origin of polar coordinate system) is positioned at a focus of the ellipse, centred on the central star.

e = elliptic eccentricity

a = elliptic semi-major axes = planet aphelion

b = elliptic semi-minor axes = planet perihelion

$$ e = \sqrt{1-\left ( \frac{b}{a} \right )^2} \,\!$$

Gravitational Fields
The general formula for calculating classical gravitational fields, due to any mass distribution, is found by using Newtons Law, definition of g, and application of calculus:


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 * $$ \mathbf{g} = G \int_{V_n} \frac{\mathbf{r} \rho_n \mathrm{d}{V_n}}{\left | \mathbf{r} \right |^3}\,\!$$
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Uniform Mass Corolaries
For uniform mass distributions the table below summarizes common cases.

For a massive rotating body (i.e. a planet/star etc), the equation is only true for much less massive bodies (i.e. objects at the surface) in physical contact with the rotating body. Since this is a classical equation, it is only approximatley true at any rate.

For non-uniform fields and mass-moments, applying differentials of the scalar and vector products then integrating gives the general gravitational torque and potential energy as:


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 * $$U=\int \left ( \mathbf{g}\cdot\mathrm{d}\mathbf{m}+\mathbf{m}\cdot\mathrm{d}\mathbf{g} \right ) \,\!$$

$$\boldsymbol{\tau}=\int \left ( \mathrm{d}\mathbf{m}\times\mathbf{g}+\mathbf{m}\times\mathrm{d}\mathbf{g} \right )\,\!$$
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