List of equations in gravitation

This article summarizes equations in the theory of gravitation.

Definitions

Gravitational mass and inertia

A common misconception occurs between centre of mass and centre of gravity. They are defined in similar ways but are not exactly the same quantity. Centre of mass is the mathematical description 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 equal if and only if the external gravitational field is uniform.

Quantity	symbol/s	Defining equation	SI units	Dimension
Centre of gravity	r_cog	i^th moment of mass Centre of gravity for a set of discrete masses: Centre of gravity for a continuum of mass:	m
Standard gravitational parameter of a mass	μ		N m² kg⁻¹	³ ⁻²

Newtonian gravitation

Quantity	symbol/s	Defining equation	SI units	Dimension
Gravitational field, field strength, potential gradient, acceleration	g		N kg⁻¹ = m s⁻²	⁻²
Gravitational flux	Φ_G		m³ s⁻²	³⁻²
Absolute gravitational potential	Φ, φ, U, V		J kg⁻¹	²⁻²
Gravitational potential difference	ΔΦ, Δφ, ΔU, ΔV		J kg⁻¹	²⁻²
Gravitational potential energy	E_p		J	²⁻²
Gravitational torsion field	Ω		Hz = s⁻¹	⁻¹

Gravitoelectromagnetism

In the weak-field and slow motion limit of general relativity, the phenomenon of gravitoelectromagnetism occurs, creating a parallel between gravitation and electromagnetism. The gravitational field is the analogue of the electric field, while the gravitomagnetic field, which results from circulations of masses due to their angular momentum, is the analogue of the magnetic field.

Quantity	symbol/s	Defining equation	SI units	Dimension
Gravitational torsion flux	Φ_Ω		N m s kg⁻¹ = m² s⁻¹	² ⁻¹
Gravitomagnetic field	H, B_g, B, ξ		Hz = s⁻¹	⁻¹
Gravitomagnetic flux	Φ_ξ		N m s kg⁻¹ = m² s⁻¹	² ⁻¹
Gravitomagnetic vector potential	h		m s⁻¹	⁻¹

Equations

Newtonian gravitational fields

It can be shown that a uniform spherically symmetric mass distribution generates an equivalent gravitational field to a point mass, so all formulae for point masses apply to bodies which can be modelled in this way.

Physical situation	Nomenclature	Equations
Gravitational potential gradient and field	U = gravitational potential C = curved path traversed by a mass in the field
Point mass
At a point in a local array of point masses
Gravitational torque and potential energy due to non-uniform fields and mass moments	V = volume of space occupied by the mass distribution m = mr is the mass moment of a massive particle
Gravitational field for a rotating body	= zenith angle relative to rotation axis = unit vector perpendicular to rotation axis, radial from it

Gravitational potentials

General classical equations.

Physical situation	Nomenclature	Equations
Potential energy from gravity, integral from Newton's law
Escape speed	M = Mass of body to escape from r = radius of body
Orbital energy	m = mass of orbiting body M = mass of central body ω = angular velocity of orbiting mass r = separation between centres of mass T = kinetic energy U = gravitational potential energy

Weak-field relativistic equations

Physical situation	Nomenclature	Equations
Gravitomagnetic field for a rotating body	ξ = gravitomagnetic field