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

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Unformatted text preview: 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 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 .- ( r ) = G Z V ( r ) | r- r | d 3 r = G M ( r ) r + 4 G Z r ( r ) r dr = Z r G M ( r ) r 2 dr F ( r ) m = d v dt =- ( r ) = G Z V ( r ) r- r | r- r | 3 d 3 r =- G M ( r ) r 2 =- V c ( r ) 2 r By convention, ( r ) 0 as r . Outside a spherically symmetric object, ( r ) =- G M / r . Inside a spherically symmetric uniform density shell, ( r ) = 0. The divergence of F gives Poissons equation:- 1 m F ( r ) = 2 ( r ) = 4 G ( r ) . Using Gauss Theorem, 4 G M = 4 G Z V ( 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 ( r )( r ) d 3 r =- 1 8 G Z V | | 2 d 3 r =- G Z M M ( r ) r d M . J.M. Lattimer AST 346, Galaxies, Part 4 Analytic Density-Potential Pairs in Spherical Symmetry I Homogenous sphere (radius R , ( r < R ) = C ) Inside: ( r ) =- 2 G ( R 2- r 2 / 3) , F ( r ) =- G M ( r ) / r 2 V 2 c = G M ( r ) / r = 4 G r 2 / 3 ( ( r ) = constant). I Singular isothermal sphere ( ( r ) = o r 2 o / r 2 ) ( r ) = 4 G o r 2 o ln r + C , M ( r ) = 4 o r 2 o r , V 2 c = 4 G r 2 o . I Power law ( ( r ) = o ( r / r o )- , 2 < < 3) ( r ) =- 4 G o r 2 o ( r / r o ) 2- / [(3- )( - 2)] , M ( r ) = 4 G o r 3 o ( r / r o ) 3- / (3- )....
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This note was uploaded on 01/25/2012 for the course AST 346 taught by Professor Lattimer during the Spring '11 term at SUNY Stony Brook.

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

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