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# part4 - Stellar Dynamics Stellar systems vs gases...

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Stellar Dynamics I Stellar systems vs. gases I Gravitational potential I Spherical and disk potentials I Orbits in the stellar neighborhood I Orbits of single stars I Orbits of stars in clusters I The virial theorem I Measuring masses from motions I Effective potentials and epicycles I Relaxation of orbits and encounters I The Boltzmann equation J.M. Lattimer AST 346, Galaxies, Part 4

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Stellar Systems vs. Gases Similarities I Comprise many interacting point-like objects I Can be described by distribution functions of position and velocity I Obey continuity equations (are not created or destroyed) I Interactions and the systems as a whole obey conservation laws of energy and momentum I Concepts like pressure and temperature apply Differences I Relative importance of short (gas) and long-range (stellar systems) forces I Stars interact continuously with entire ensemble via long-range force of gravity I Gases interct continuously via frequent, short-range, strong, elastic, repulsive collisions I Stellar pairwise encounters are very rare I Pressures in stellar systems can be anisotropic I Stellar systems have negative specific heat and evolve away from uniform temperature I Gases evolve toward uniform temperature and have positive specific heats J.M. Lattimer AST 346, Galaxies, Part 4
Potential Theory Gravitational potential is a scalar field whose gradient gives the net gravitational force (per unit mass), a vector field . - Φ( r ) = G Z V ρ ( r 0 ) | r 0 - r | d 3 r 0 = G M ( r ) r + 4 π G Z r ρ ( r 0 ) r 0 dr 0 = Z r G M ( r 0 ) r 0 2 dr 0 F ( r ) m = d v dt = -∇ Φ( r ) = G Z V ρ ( r 0 ) r 0 - r | r 0 - r | 3 d 3 r 0 = - G M ( r ) r 2 = - V c ( r ) 2 r By convention, Φ( r ) 0 as r → ∞ . Outside a spherically symmetric object, Φ( r ) = - G M / r . Inside a spherically symmetric uniform density shell, Φ( r ) = 0. The divergence of F gives Poisson’s equation: - 1 m ∇ · F ( r ) = 2 Φ( r ) = 4 π G ρ ( r ) . Using Gauss’ Theorem, 4 π G M = 4 π G Z V ρ ( r ) d 3 r = - 1 m Z V ∇ · F ( r ) d 3 r = - 1 m Z A F ( r ) · d 2 S Gravitational potential energy (last equality for spherical symmetry) W = 1 2 Z V ρ ( r )Φ( r ) d 3 r = - 1 8 π G Z V |∇ Φ | 2 d 3 r = - G Z M M ( r ) r d M . J.M. Lattimer AST 346, Galaxies, Part 4

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Analytic Density-Potential Pairs in Spherical Symmetry I Homogenous sphere (radius R , ρ ( r < R ) = C ) Inside: Φ( r ) = - 2 π G ( R 2 - r 2 / 3) , F ( r ) = - G M ( r ) / r 2 V 2 c = G M ( r ) / r = 4 π G ρ r 2 / 3 ( ω ( r ) = constant). I Singular isothermal sphere ( ρ ( r ) = ρ o r 2 o / r 2 ) Φ( r ) = 4 π G ρ o r 2 o ln r + C , M ( r ) = 4 πρ o r 2 o r , V 2 c = 4 π G ρ 0 r 2 o . I Power law ( ρ ( r ) = ρ o ( r / r o ) - α , 2 < α < 3) Φ( r ) = - 4 π G ρ o r 2 o ( r / r o ) 2 - α / [(3 - α )( α - 2)] , M ( r ) = 4 π G ρ o r 3 o ( r / r o ) 3 - α / (3 - α ). I Hernquist ( ρ ( r ) = M a / [2 π r ( r + a ) 3 ]) Φ( r ) = - G M / ( r + a ) , M ( r ) = M r 2 / ( r + a ) 2 . I Jaffe ( ρ ( r ) = M a / [4 π r 2 ( r + a ) 2 ]) Φ( r ) = - ( G M / a ) ln(1 + a / r ) , M ( r ) = M r / ( r + a ) . I Plummer ( ρ ( r ) = 3 a 2 M / [4 π ( r 2 + a 2 ) 5 / 2 ]) Φ( r ) = - G M / r 2 + a 2 , M ( r ) = M r 3 / ( r 2 + a 2 ) 3 / 2 .
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part4 - Stellar Dynamics Stellar systems vs gases...

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