View source for List of equations in gravitation

{{short description|None}}

This article summarizes [[equation]]s in the theory of [[gravitation]].<!--for now just Newtonian gravitation and the GEM weak gravitational field and slow-motion limit of general relativity-->

== Definitions==

===Gravitational mass and inertia===

A common misconception occurs between [[Center of mass|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.

{| class="wikitable"
|-
! scope="col" width="200" | Quantity (common name/s)
! scope="col" width="125" | (Common) symbol/s
! scope="col" width="200" | Defining equation
! scope="col" width="125" | SI units
! scope="col" width="100" | Dimension
|-
! Centre of gravity 
| '''r'''<sub>cog</sub> (symbols vary)
| ''i''<sup>th</sup> moment of mass <math> \mathbf{m}_i = \mathbf{r}_i m_i \,\!</math>

Centre of gravity for a set of discrete masses:<br />
<math> \begin{align} \mathbf{r}_\mathrm{cog} & = \frac{1}{M \left | \mathbf{g} \left ( \mathbf{r}_i \right ) \right |}\sum_i \mathbf{m}_i \left | \mathbf{g} \left ( \mathbf{r}_i \right ) \right | \\
 & = \frac{1}{M \left | \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right |}\sum_i \mathbf{r}_i m_i \left | \mathbf{g} \left ( \mathbf{r}_i \right ) \right | \end{align}\,\!</math>

Centre of gravity for a continuum of mass: <br />
<math> \begin{align} \mathbf{r}_\mathrm{cog} & = \frac{1}{M \left | \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right |}\int \left | \mathbf{g} \left ( \mathbf{r} \right ) \right |\mathrm{d}\mathbf{m} \\
 & = \frac{1}{M \left | \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right |}\int \mathbf{r} \left | \mathbf{g} \left ( \mathbf{r} \right ) \right | \mathrm{d}^n m \\
 & = \frac{1}{M \left | \mathbf{g} \left ( \mathbf{r}_\mathrm{cog} \right ) \right |}\int \mathbf{r} \rho_n \left | \mathbf{g} \left ( \mathbf{r} \right ) \right | \mathrm{d}^n x \end{align} \,\!</math>
| m
| [L]
|-
![[Standard gravitational parameter]] of a mass 
| ''μ''
|<math> \mu = Gm \,\!</math>
| N m<sup>2</sup> kg<sup>−1</sup>
| [L]<sup>3</sup> [T]<sup>−2</sup>
|-
|}

===Newtonian gravitation===

{{main|Newtonian gravitation}}
{{multiple image
   | align = right
   | direction = vertical
   | footer = Interpretations of the gravitational field.
   | image1    = Newtonian gravity field (physics).svg
   | caption1  = In [[classical gravitation]], mass is the source of an attractive [[gravitational field]] '''g'''.
   | width1     = 190
   | image2    = Gravitomagnetic field due to angular momentum.svg
   | caption2  = [[Gravitomagnetic field]] '''H''' due to (total) [[angular momentum]] '''J'''.<ref name="Gravitation and Inertia">Gravitation and Inertia, I. Ciufolini and J.A. Wheeler, Princeton Physics Series, 1995, {{ISBN|0-691-03323-4}}</ref>
   | width2     = 190
  }}
{| class="wikitable"
|-
! scope="col" width="200" | Quantity (common name/s)
! scope="col" width="125" | (Common) symbol/s
! scope="col" width="200" | Defining equation
! scope="col" width="125" | SI units
! scope="col" width="100" | Dimension
|-
![[Gauss' law for gravity|Gravitational field]], field strength, potential gradient, acceleration
| '''g'''
|<math>\mathbf{g} = \mathbf{F}/m \,\!</math>
| N kg<sup>−1</sup> = m s<sup>−2</sup>
| [L][T]<sup>−2</sup>
|-
![[Gauss' law for gravity|Gravitational flux]]
| ''Φ<sub>G</sub>''
|<math>\Phi_G = \int_S \mathbf{g} \cdot \mathrm{d}\mathbf{A} \,\!</math>
| m<sup>3</sup> s<sup>−2</sup>
| [L]<sup>3</sup>[T]<sup>−2</sup>
|-
!Absolute [[gravitational potential]]
| ''Φ'', ''φ'', ''U'', ''V''
|<math> U = - \frac{W_{\infty r}}{m} = - \frac{1}{m} \int_\infty^{r} \mathbf{F} \cdot \mathrm{d}\mathbf{r} = - \int_\infty^{r} \mathbf{g} \cdot \mathrm{d}\mathbf{r} \,\!</math>
| J kg<sup>−1</sup>
| [L]<sup>2</sup>[T]<sup>−2</sup>
|-
!Gravitational potential difference
| Δ''Φ'', Δ''φ'', Δ''U'', Δ''V''
|<math> \Delta U = - \frac{W}{m} = - \frac{1}{m} \int_{r_1}^{r_2} \mathbf{F} \cdot \mathrm{d}\mathbf{r} = - \int_{r_1}^{r_2} \mathbf{g} \cdot \mathrm{d}\mathbf{r} \,\!</math>
| J kg<sup>−1</sup>
| [L]<sup>2</sup>[T]<sup>−2</sup>
|-
![[Gravitational energy|Gravitational potential energy]]
| ''E<sub>p</sub>''
|<math> E_p = - W_{\infty r} \,\!</math>
| J
| [M][L]<sup>2</sup>[T]<sup>−2</sup>
|-
!Gravitational torsion field
| '''Ω'''
|<math> \boldsymbol{\Omega} = 2 \boldsymbol{\xi} \,\!</math>
| Hz = s<sup>−1</sup>
| [T]<sup>−1</sup>
|-
|}

===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.

{| class="wikitable"
|-
! scope="col" width="200" | Quantity (common name/s)
! scope="col" width="125" | (Common) symbol/s
! scope="col" width="200" | Defining equation
! scope="col" width="125" | SI units
! scope="col" width="100" | Dimension
|-
!Gravitational torsion flux
| ''Φ<sub>Ω</sub>''
|<math>\Phi_\Omega = \int_S \boldsymbol{\Omega} \cdot \mathrm{d}\mathbf{A} \,\!</math>
| N m s kg<sup>−1</sup> = m<sup>2</sup> s<sup>−1</sup>
| [M]<sup>2</sup> [T]<sup>−1</sup>
|-
![[gravitoelectromagnetism|Gravitomagnetic field]]
| '''H''', '''B'''<sub>g</sub>, '''B''', '''ξ'''
|<math> \mathbf{F} = m \left ( \mathbf{v} \times 2 \boldsymbol{\xi} \right ) \,\!</math>
| Hz = s<sup>−1</sup>
| [T]<sup>−1</sup>
|-
!Gravitomagnetic flux
| ''Φ<sub>ξ</sub>''
|<math>\Phi_\xi = \int_S \boldsymbol{\xi} \cdot \mathrm{d}\mathbf{A} \,\!</math>
| N m s kg<sup>−1</sup> = m<sup>2</sup> s<sup>−1</sup>
| [M]<sup>2</sup> [T]<sup>−1</sup>
|-
!Gravitomagnetic vector potential <ref name="Gravitation and Inertia" />
| '''h'''
|<math> \mathbf{\xi} = \nabla \times \mathbf{h} \,\!</math>
| m s<sup>−1</sup>
| [M] [T]<sup>−1</sup>
|-
|}

==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.

{| class="wikitable"
|-
! scope="col" width="200" | Physical situation 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
!Gravitational potential gradient and field
| {{plainlist}}
*''U'' = gravitational potential
*''C'' = curved path traversed by a mass in the field
{{endplainlist}}
| <math> \mathbf{g} = - \nabla U </math>

<math> \Delta U = -\int_C \mathbf{g} \cdot d\mathbf{r}\,\!</math>
|-
!Point mass
|
| <math> \mathbf{g} = \frac{Gm}{\left | \mathbf{r} \right |^2 }\mathbf{\hat{r}} \,\!</math>
|-
!At a point in a local array of point masses
|
| <math>\mathbf{g} = \sum_i \mathbf{g}_i = G \sum_i \frac{m_i}{\left | \mathbf{r}_i - \mathbf{r} \right |^2}\mathbf{\hat{r}}_i \,\!</math>
|-
!Gravitational torque and potential energy due to non-uniform fields and mass moments
|{{plainlist}}
*''V'' = volume of space occupied by the mass distribution
*'''m''' = ''m'''''r''' is the mass moment of a massive particle
{{endplainlist}}
| <math> \boldsymbol{\tau} = \int_{V_n}  \mathrm{d} \mathbf{m} \times \mathbf{g} \,\!</math>

<math> U = \int_{V_n} \mathrm{d} \mathbf{m} \cdot \mathbf{g} \,\!</math>
|-
! Gravitational field for a rotating body
| {{plainlist}}
*<math>\phi</math> = zenith angle relative to rotation axis
*<math>\mathbf{\hat{a}} \,\!</math> = unit vector perpendicular to  rotation (zenith) axis, radial from it
{{endplainlist}}
| <math>\mathbf{g} = - \frac{GM}{\left | \mathbf{r} \right |^2} \mathbf{\hat{r}} - (\left | \boldsymbol{\omega} \right |^2\left | \mathbf{r} \right | \sin \phi )\mathbf{\hat{a}} \,\!</math>
|-
|}

===Gravitational potentials===

General classical equations.

{| class="wikitable"
! scope="col" width="200" | Physical situation 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
!Potential energy from gravity, integral from Newton's law
|
|<math> U = -\frac{G m_1 m_2}{\left | \mathbf{r} \right |} \approx m \left | \mathbf{g} \right | y\,\!</math>
|-
!Escape speed
| {{plainlist}}
*''M'' = Mass of body (e.g. planet) to escape from
*''r'' = radius of body
 {{endplainlist}}
|<math> v = \sqrt{\frac{2GM}{r}}\,\!</math>
|-
!Orbital energy
|  {{plainlist}}
*''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)
 {{endplainlist}}
|<math> \begin{align} E & = T + U \\ 
& = -\frac{G m M}{\left | \mathbf{r} \right |} + \frac{1}{2} m \left | \mathbf{v} \right |^2 \\
& = m \left ( - \frac{GM}{\left | \mathbf{r} \right |} + \frac{\left | \boldsymbol{\omega} \times \mathbf{r} \right |^2}{2} \right ) \\
& = - \frac{GmM}{2 \left | \mathbf{r} \right |} 
\end{align} \,\!</math>
|-
|}

===Weak-field relativistic equations===

{| class="wikitable"
|-
! scope="col" width="100" | Physical situation 
! scope="col" width="250" | Nomenclature
! scope="col" width="10" | Equations
|-
![[Gravitomagnetic field]] for a rotating body
| '''ξ''' = gravitomagnetic field
|<math>\boldsymbol{\xi} = \frac{G }{2 c^2} \frac{\mathbf{L}  3(\mathbf{L} \cdot \mathbf{\hat{r}} ) \mathbf{\hat{r}}}{\left | \mathbf{r} \right |^3}</math>
|-
|}

==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]]

==Footnotes==

{{Reflist}}

==Sources==

* {{cite book| author=P.M. Whelan, M.J. Hodgeson| title=Essential Principles of Physics| publisher=John Murray|edition=2nd| date=1978 | isbn=0-7195-3382-1}}
* {{cite book| author=G. Woan| title=The Cambridge Handbook of Physics Formulas| url=https://archive.org/details/cambridgehandboo0000woan| url-access=registration| publisher=[[Cambridge University Press]]| date=2010| isbn=978-0-521-57507-2}}
* {{cite book| author=A. Halpern| title=3000 Solved Problems in Physics, Schaum Series| publisher=Mc Graw Hill| date=1988| isbn=978-0-07-025734-4}}
* {{cite book|pages=12–13| author=[[Rita G. Lerner|R.G. Lerner]], G.L. Trigg| title=Encyclopaedia of Physics| publisher=VHC Publishers, Hans Warlimont, Springer|edition=2nd| date=2005| isbn=978-0-07-025734-4}}
* {{cite book| author=C.B. Parker| title=McGraw Hill Encyclopaedia of Physics| publisher=McGraw Hill| edition=2nd| date=1994| isbn=0-07-051400-3| url-access=registration| url=https://archive.org/details/mcgrawhillencycl1993park}}
* {{cite book| author=P.A. Tipler, G. Mosca| title=Physics for Scientists and Engineers: With Modern Physics| publisher=W.H. Freeman and Co|edition=6th| date=2008| isbn=978-1-4292-0265-7}}
* {{cite book|title=Analytical Mechanics|author=L.N. Hand, J.D. Finch|publisher=Cambridge University Press |date=2008|isbn=978-0-521-57572-0}}
* {{cite book|title=Mechanics, Vibrations and Waves|author=T.B. Arkill, C.J. Millar|publisher=John Murray |date=1974|isbn=0-7195-2882-8}}
* {{cite book|title=Dynamics and Relativity|author=J.R. Forshaw, A.G. Smith|publisher=Wiley |date=2009|isbn=978-0-470-01460-8}}

==Further reading==

* {{cite book|title=Physics with Modern Applications|author=L.H. Greenberg|publisher=Holt-Saunders International W.B. Saunders and Co|date=1978|isbn=0-7216-4247-0|url-access=registration|url=https://archive.org/details/physicswithmoder0000gree}}
* {{cite book|title=Principles of Physics|author=J.B. Marion, W.F. Hornyak|publisher=Holt-Saunders International Saunders College|date=1984|isbn=4-8337-0195-2}}
* {{cite book|title=Concepts of Modern Physics|edition=4th|author=A. Beiser|publisher=McGraw-Hill (International)|date=1987|isbn=0-07-100144-1}}
* {{cite book|title=University Physics – With Modern Physics|edition=12th|author=H.D. Young, R.A. Freedman|publisher=Addison-Wesley (Pearson International)|date=2008|isbn=978-0-321-50130-1}}

{{SI units navbox}}

[[Category:Physical quantities]]
[[Category:SI units]]
[[Category:Lists of physics equations|gravitation]]
[[Category:Gravity]]