List of equations in gravitation explained

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 (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Centre of gravity	rcog (symbols vary)	i^th moment of mass m _{i} = r _{i} m _{i} Centre of gravity for a set of discrete masses: \begin{align} r _{cog} & = (1)/(M \left | g \left ( r _{i} \right ) \right |) \sum _{i} m _{i} \left	\mathbf \left (\mathbf_i \right) \right	\\ & = \frac\sum_i \mathbf_i m_i \left	\mathbf \left (\mathbf_i \right) \right	\end\,\! Centre of gravity for a continuum of mass: \begin{align} r _{cog} & = (1)/(M \left | g \left ( r _{cog} \right ) \right |) \int \left	\mathbf \left (\mathbf \right) \right	\mathrm\mathbf \\ & = \frac\int \mathbf \left	\mathbf \left (\mathbf \right) \right	\mathrm^n m \\ & = \frac\int \mathbf \rho_n \left	\mathbf \left (\mathbf \right) \right	\mathrm^n x \end \,\!	m	[L]
Standard gravitational parameter of a mass	μ	\mu = Gm	N m^2 kg^−1	[L]^3 [T]^−2

Newtonian gravitation

See main article: Newtonian gravitation.

Quantity (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Gravitational field, field strength, potential gradient, acceleration	g	g = F /m	N kg^−1 = m s^−2	[L][T]^−2
Gravitational flux	ΦG	\Phi _{G} = \int _{S} g ⋅ d A	m^3 s^−2	[L]^3[T]^−2
Absolute gravitational potential	Φ, φ, U, V	U = - (W _{infty})/(m) = - (1)/(m) \int ^{r} _{inf ty} F ⋅ d r = - \int ^{r} _{inf ty} g ⋅ d r	J kg^−1	[L]^2[T]^−2
Gravitational potential difference	ΔΦ, Δφ, ΔU, ΔV	\Delta U = - (W)/(m) = - (1)/(m) \int ^{r _{2}} _{r _{1}} F ⋅ d r = - \int ^{r _{2}} _{r _{1}} g ⋅ d r	J kg^−1	[L]^2[T]^−2
Gravitational potential energy	Ep	E _{p} = - W _{infty}	J	[M][L]^2[T]^−2
Gravitational torsion field	Ω	\boldsymbol{\Omega} = 2 \boldsymbol{\xi}	Hz = s^−1	[T]^−1

Gravitoelectromagnetism

In the weak-field and slow motion limit of general relativity, the phenomenon of gravitoelectromagnetism (in short "GEM") 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 (common name/s)	(Common) symbol/s	Defining equation	SI units	Dimension
Gravitational torsion flux	ΦΩ	\Phi _{\Omega} = \int _{S} \boldsymbol{\Omega} ⋅ d A	N m s kg^−1 = m^2 s^−1	[M]^2 [T]^−1
Gravitomagnetic field	H, Bg, B, ξ	F = m \left ( v x 2 \boldsymbol{\xi} \right )	Hz = s^−1	[T]^−1
Gravitomagnetic flux	Φξ	\Phi _{\xi} = \int _{S} \boldsymbol{\xi} ⋅ d A	N m s kg^−1 = m^2 s^−1	[M]^2 [T]^−1
Gravitomagnetic vector potential	h	\xi = \nabla x h	m s^−1	[M] [T]^−1

Equations

Newtonian gravitational fields

See also: Spherical polar coordinates.

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	g = - \nabla U \Delta U = - \int _{C} g ⋅ d r
Point mass		g = (Gm)/(\left | r \right | ^{2}) \hat{r } \,\!
At a point in a local array of point masses		g = \sum _{i} g _{i} = G \sum _{i} (m _{i})/(\left | r _{i} - r \right | ^{2}) \hat{r }_i \,\!
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	\boldsymbol{\tau} = \int _{V _{n}} d m x g U = \int _{V _{n}} d m ⋅ g \
Gravitational field for a rotating body	\phi = zenith angle relative to rotation axis\hat{a } \,\! = unit vector perpendicular to rotation (zenith) axis, radial from it	g = - (GM)/(\left | r \right | ^{2}) \hat{r } - (\left	\boldsymbol \right	^2\left	\mathbf \right	\sin \phi)\mathbf \,\!

Gravitational potentials

General classical equations.

Physical situation	Nomenclature	Equations
Potential energy from gravity, integral from Newton's law		U =- (G m _{1} m _{2})/(\left | r \right |) ≈ m \left	\mathbf \right	y\,\!
Escape speed	M = Mass of body (e.g. planet) to escape from r = radius of body	v = \sqrt{ (2 GM)/(r) }\,\!
Orbital energy	m = mass of orbiting body (e.g. planet) M = mass of central body (e.g. star) ω = angular velocity of orbiting mass r = separation between centres of mass T = kinetic energy U = gravitational potential energy (sometimes called "gravitational binding energy" for this instance)	\begin{align} E & = T + U \ & =- (G m M)/(\left | r \right |) + (1)/(2) m \left	\mathbf \right	^2 \\& = m \left (- \frac + \frac \right) \\& = - \frac \end \,\!

Weak-field relativistic equations

See also

• Defining equation (physical chemistry)
• List of electromagnetism equations
• List of equations in classical mechanics
• List of equations in nuclear and particle physics
• List of equations in quantum mechanics
• List of equations in wave theory
• List of photonics equations
• List of relativistic equations
• Table of thermodynamic equations

Sources

• Book: P.M. Whelan, M.J. Hodgeson. Essential Principles of Physics. John Murray. 2nd. 1978 . 0-7195-3382-1.
• Book: G. Woan. The Cambridge Handbook of Physics Formulas. registration. Cambridge University Press. 2010. 978-0-521-57507-2.
• Book: A. Halpern. 3000 Solved Problems in Physics, Schaum Series. Mc Graw Hill. 1988. 978-0-07-025734-4.
• Book: 12–13. R.G. Lerner, G.L. Trigg. Encyclopaedia of Physics. VHC Publishers, Hans Warlimont, Springer. 2nd. 2005. 978-0-07-025734-4.
• Book: C.B. Parker. McGraw Hill Encyclopaedia of Physics. McGraw Hill. 2nd. 1994. 0-07-051400-3. registration.
• Book: P.A. Tipler, G. Mosca. Physics for Scientists and Engineers: With Modern Physics. W.H. Freeman and Co. 6th. 2008. 978-1-4292-0265-7.
• Book: Analytical Mechanics. L.N. Hand, J.D. Finch. Cambridge University Press . 2008. 978-0-521-57572-0.
• Book: Mechanics, Vibrations and Waves. T.B. Arkill, C.J. Millar. John Murray . 1974. 0-7195-2882-8.
• Book: Dynamics and Relativity. J.R. Forshaw, A.G. Smith. Wiley . 2009. 978-0-470-01460-8.

Further reading

• Book: Physics with Modern Applications. L.H. Greenberg. Holt-Saunders International W.B. Saunders and Co. 1978. 0-7216-4247-0. registration.
• Book: Principles of Physics. J.B. Marion, W.F. Hornyak. Holt-Saunders International Saunders College. 1984. 4-8337-0195-2.
• Book: Concepts of Modern Physics. 4th. A. Beiser. McGraw-Hill (International). 1987. 0-07-100144-1.
• Book: University Physics – With Modern Physics. 12th. H.D. Young, R.A. Freedman. Addison-Wesley (Pearson International). 2008. 978-0-321-50130-1.