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2012-12-12T15:57:52Z

92.200.75.117: fixed math formula

The '''Jeans equations''' describe the motion of a collection of stars in a gravitational field.

If ''n'' = ''n''(''x'',''t'') is the density of stars in space, as a function of position ''x'' = (''x''1, ''x''2, ''x''3) and time ''t'', ''v'' = (''v''1, ''v''2, ''v''3) is the velocity, and Φ = Φ(''x'',''t'') is the gravitational potential, the Jeans equations may be written as<ref>pp. 195-197, §4.2, ''Galactic dynamics'', James Binney, Scott Tremaine, Princeton University Press, 1988, ISBN 0-691-08445-9.</ref><ref name=DEGN>{{cite book|last=Merritt|first=David|author-link=David Merritt|title=Dynamics and Evolution of Galactic Nuclei|year=2013|publisher=Princeton University Press|location=Princeton, NJ|url=http://openlibrary.org/works/OL16802359W/Dynamics_and_Evolution_of_Galactic_Nuclei}}</ref>

<math>
\frac{\partial n }{\partial t} + \sum_i \frac{\partial(n \langle{v_i}\rangle)}{\partial x_i}=0,</math>

<math>
\frac{\partial(n \langle{v_j}\rangle)}{\partial t} + n \frac{\partial \Phi}{\partial x_j}
+ \sum_i \frac{\partial(n \langle{v_i v_j}\rangle)}{\partial x_i}= 0 \qquad (j=1, 2, 3.)
</math>

Here, the <…> notation means an average at a given point and time (x,t), so that, for example, <math>\langle{v_1}\rangle</math> is the average of component 1 of the velocity of the stars at a given point and time.
The second set of equations may alternately be written as

<math>
n \frac{\partial \langle{v_j}\rangle}{\partial t}
+ \sum_i n \langle{v_i}\rangle \frac{\partial{\langle{v_j}\rangle}}{\partial x_i}
= -n \frac{\partial \Phi}{\partial x_j} - \sum_i \frac{\partial (n \sigma_{ij}^2)}{\partial x_i} \qquad (j=1, 2, 3.)
</math>

where <math>\sigma_{ij}^2=\langle{v_i v_j}\rangle-\langle{v_i}\rangle \langle{v_j}\rangle</math> measures the velocity dispersion in components ''i'' and ''j'' at a given point.

The Jeans equations are analogous to the [[Euler equations (fluid dynamics)|Euler equations]] for fluid flow and may be derived from the [[collisionless Boltzmann equation]]. They were originally derived by [[James Clerk Maxwell]] but were first applied to stellar dynamics by [[James Jeans]].<ref>p. 82, "On the theory of star-streaming and the structure of the universe", J. H. Jeans, ''Monthly Notices of the Royal Astronomical Society'' '''76''' (December 1915), pp. 70-84, {{bibcode|1915MNRAS..76...70J}}.</ref>

==References==
{{reflist}}

[[Category:Stellar dynamics]]
[[Category:Statistical mechanics]]
[[Category:Equations]]

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