Gravitational Dynamics Formulae

Link phase space quantities r J(r,v) K(v)  (r) VtVt E(r,v) dθ/dt vrvr

Link quantities in spheres g(r)  (r)  (r) v esc 2 (r) M(r) Vcir 2 (r) σ r 2 (r) σ t 2 (r) f(E,L)

Motions in spherical potential

PHASE SPACE DENSITY:Number of stars per unit volume per unit velocity volume f(x,v). dN=f(x,v)d 3 xd 3 v TOTAL # OF PARTICLES PER UNIT VOLUME: MASS DISTRIBUTION FUNCTION:

TOTAL MASS : TOTAL MOMENTUM: MEAN VELOCITY: = =0 (isotropic) & = =σ 2 (x)

NOTE: d 3 v=4πv 2 dv (if isotropic) d 3 x=4πr 2 dr (if spherical) GAMMA FUNCTIONS:

GRAVITATIONAL POTENTIAL DUE TO A MASS dM: RELATION BETWEEN GRAVITATIONAL FORCE AND POTENTIAL: FOR AN N BODY CASE:

LIOUVILLES THEOREM: (volume in phase space occupied by a swarm of particles is a constant for collisionless systems) IN A STATIC POTENTIAL ENERGY IS CONSERVED: Note:E=energy per unit mass

POISSON’S EQUATION : INTEGRATED FORM:

EDDINGTON FORMULAE:

RELATING PRESSURE GRADIENT TO GRAVITATIONAL FORCE: GOING FROM DENSITY TO MASS:

GOING FROM GRAVITATIONAL FORCE TO POTENTIAL :

SINGULAR ISOTHERMAL SPHERE MOD

Conservation of momentum:

PLUMMER MODEL: GAUSS’ THEOREM:

ISOTROPIC SELF GRAVITATING EQUILIBRIUM SYSTEMS

Cont:

CIRCULAR SPEED: ESCAPE SPEED: ISOCHRONE POTENTIAL:

JEANS EQUATION (steady state axisymmetric system in which σ 2 is isotropic and the only streaming motion is in the azimuthal direction)

VELOCITY DISPERSIONS (steady state axisymmetric and isotropic σ) OBTAINING σ USING JEANS EQUATION:

ORBITS IN AXISYMMETRIC POTENTIALS Φ eff

EQUATIONS OF MOTION IN THE MERIDIONAL PLANE:

CONDITION FOR A PARTICLE TO BE BOUND TO THE SATELLITE RATHER THAN THE HOST SYSTEM: TIDAL RADIUS:

LAGRANGE POINTS: Gravitational pull of the two large masses precisely cancels the centripetal acceleration required to rotate with them. EFFECTIVE FORCE OF GRAVITY: JAKOBI’S ENERGY:

DYNAMICAL FRICTION:

Cont: Only stars with v  v M contribute to dynamical friction. For small v M : For sufficiently large v M :

FOR A MAXWELLIAN VELOCITY DISTRIBUTION:

ORBITS IN SPHERICAL POTENTIALS

RADIAL PERIOD:Time required for the star to travel from apocentre to pericentre and back. AZIMUTHAL PERIOD: Where: In general  θ will not be a rational number  orbits will not be closed.

STELLAR INTERACTIONS

FOR THE SYSTEM TO NO LONGER BE COLLISIONLESS: RELAXATION TIME: CONTINUITY EQUATION:

Helpful Math/Approximations (To be shown at AS4021 exam) Convenient Units Gravitational Constant Laplacian operator in various coordinates Phase Space Density f(x,v) relation with the mass in a small position cube and velocity cube