OOM Info

Gravitational Force

F = GMm/R²

G in Msol

4.3E-3 pc (km/s)² Msol^-1

Gravitational PE

U = GM²/R

characteristic velocity of a small obj in circular orbit

v=(GM/R)^1/2 w/ central object is mass M, distance from the obj is R

characteristic gravitational collapse time & where it comes from

t_ff = (Grho)^-1/2 or (D³/GM)^1/2

derived by saying t_dyn=D/v (b/c dist = vel*time), uses the characteristic velocity of v=(GM/D)^1/2

mean free path

1/nσ, σ = cross section of the object passing through the field (for the oort cloud question, pi R_oort²)

Local star velocity (relative & plain)

Most going ~200-250 km/s, so have a relative velocity of ~30km/s

Encounter rates

Γ = nσv

Average local stellar density

0.1 Msol/pc³

circular velocity

v_c = ωr

Formula for orbital periods (used to get mass)

P² = (4π²)/(GM_tot) a³

Mass version: (4π²/G) x (a³/P²)

centripetal acceleration

a_c = v²/r = GM_(<r) / r²

Work

W = F*d

Characteristic rotation frequency

Ω_grav = (GM/D³)^1/2 (inverse of characteristic time, freq is 1/period

Gives the max spin rate of an object held together only with gravity, or the average rotation rate of something orbiting mass M at distance D

Charcteristic tidal force

F_tidal=(GM₁M₂)/D² * R/D ; same as gravitational force but is reduced by R/D

(useful if within the virial radius of something, like tidal stripping from a galaxy within the virial radius)

Self gravitational force

F_grav = GM₁²/R²

circular velocity in a gravitational potential

v_c²( r ) = GM_(<r) / r