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Unformatted text preview: Assignment #10 PHYSICS 8.284 Due 11:04 am Wednesday 3 May 2006 • Reading: Binney and Tremaine, Chapter 4 through equation (4-45). (1) In casting the Boltzmann equation in spherical coordinates we found we needed the quantities ˙ v r , v ˙ , and ˙ v . There are several ways to obtain these. One is to use La- grange’s equations. The other is to assume spherical symmetry, write down expres- sions for E , L 2 , and L z , all of which are conserved, and differentiate these. The sec- ond approach has the shortcoming that it looses generality, dropping terms ρ /ρν and ρ /ρπ . Using one method or the other, derive the equations 2 v 2 + v ρ v ˙ r = , r − ρr v r v cot νv 2 1 ρ v ˙ = and − r + r − r ρν v r v cot νv v 1 ρ v ˙ = . − r − r − r sin ν ρπ (2) The Cartesian Jeans equation yields a particularly simple equation for the special case of a plane parallel stellar system with θ 2 independent of distance z from the zz plane, d 2 ln β d 2 θ 2 dz 2 = = −...
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- Spring '06
- Physics, vr v� cot