18. Class N -body Techniques, Part 1
The N -body Problem
Study of the dynamics of interacting particles, usually involving mutual forces. E.g., Application Mutual Force gravity stellar dynamics, planetesimals QM molecular dynamics, solid-state physics EM plasma physics etc. etc. Stick with gravitation for now. Only a few literature references available, e.g., Aarseth, Danby (Ch. 9), etc.
Generalized Newtons Laws
i = r
j=i
F ij =
j=i
Gmj (ri rj ) . |ri rj |3
These are 3N coupled 2nd -order ODEs. As usual, reduce to 1st -order: ri = vi (velocity), Gmj (ri rj ) vi = (acceleration). |ri rj |3 j=i This makes 6N coupled 1st -order ODEs. We know how to solve these! Key is to solve the equations eciently: 1. Solve Newtons Laws using ODE integrator. 2. Evaluate interparticle forces Fij several techniques.
Typical Parameters
First, need to get a feeling for the problem... What are typical problem sizes? N N 2: Jupiter and Sun, extrasolar planets. 9: Solar system. 1
N N N N N N
10100: Small stellar system. 1001000: Open cluster, rubble pile! 105 106 : Globular cluster, planetesimals. 107 108 : Cosmological volume (DM halos). 109 : Planetary rings. 1011 : Galaxy.
Also have restricted problems where one or more test particles exert no gravitational forces but still feel forces due to more massive particles, e.g., Lagrange problem, comets in the Oort cloud, etc. What are typical timescales? ([T ] = [L]/[V ]) Solar system: Orbital timeevolution time (1109 yrs). Stellar system: Relaxation time ( 100s of crossing times). Globular cluster: Core collapse ( 10s of relaxation times). Galaxy: 1010 yrs (many steps). Universe: 1010 yrs (fewer steps). Often to achieve steady state over many dynamical times it seems N /t constant. = timescale and lengthscale closely coupled. E.g., crossing time for closed dynamical system.
1 Virial theorem: 2K + W = 0, K = 2 M v 2 , W = GM 2 /rg . 3/2 Crossing time = [L]/[V ] rg / v 2 1/2 rg / GM . Typically want t D /30 = cross /30.
Another handy formula:
3 . G E.g., for typical asteroid, 2 g/cc so D 2.3 h. For Earth, spread out mass 4 3 r of Sun to 1 AU: = M 3 / = D 1 yr. Why? 2 r = GM/r 2 4 2 / 2 = 4 GM/r 3 = 3 G. 3/ G. D
Units In MKS, G = 6.7 1011 , M = 2 1030 , r = 1.5 1011 . Often want to work in scaled units to keep values close to unity. Typically set G 1. For solar system, use masses in M , distances in AU. Then times in yr/2 and speeds in v = 30 km s1 . For galaxies, could use masses in 109 M , distances in kpc. Then times would be in 15 Myr and speeds in kpc/15 Myr. 2
Constants of motion If there are no outside forces/torques, Newtons Laws for a gravitating system imply: 1. Total energy is conserved. 2. Total angular momentum is conserved. 3. System center of mass is either stationary in the inertial frame, or moves with constant velocity. Can therefore set rg = vg 0. N = 2 problem Solved by Kepler, explained by Newton. General solution (ellipse): r = a(1 e cos ) cos e cos = 1 e cos where a = semi-major axis, e = eccentricity, = eccentric anomaly, and mean anomaly t = e sin (Keplers equation).
Useful facts: if r and v are relative coordinates of two bodies, then 1 1 Gm1 m2 1 m1 m2 2 1 Gm1 m2 2 2 2 E = m1 v1 + m2 v2 = v + Mvg , 2 2 |r1 r2 | 2M 2 r where M m1 + m2 . Hence, since we can always set vg 0, v 2 GM E = , 2 r
where m1 m2 /M = reduced mass. Also have Gm1 m2 E...