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Class 18. N -body Techniques, Part 1The N -body Problem• Study of the dynamics of interacting particles, usually involving mutual forces. E.g.,Mutual Force Applicationgravity stellar dynamics, planetesimalsQM molecular dynamics, solid-state physicsEM plasma physicsetc. etc.• Stick with gravitation for now.• Only a few literature references available, e.g., Aarseth, Danby (Ch. 9), etc.Generalized Newton’s Laws¨ri=Xj6=iFij= −Xj6=iGmj(ri− rj)|ri− rj|3.• These are 3N coupled 2nd-order ODEs.• As usual, reduce to 1st-order:˙ri= vi(velocity),˙vi= −Xj6=iGmj(ri− rj)|ri − rj|3(acceleration).– This makes 6N coupled 1st-order ODEs.– We know how to solve these!• Key is to solve the equations efficiently:1. Solve Newton’s 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 ' 2: Jupiter and Sun, extrasolar planets.N ' 9: Solar system.1N ' 10–100: Small stellar system.N ' 100–1000: Open cluster, rubble pile!N ' 105–106: Globular cluster, planetesimals.N ' 107–108: Cosmological volume (DM halos).N ' 109: Planetary rings.N ' 1011: Galaxy.Also have “restricted” problems where one or more “test” particles exert no gravita-tional 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 time–evolution time (1–1 09yrs).Stellar system: Relaxation time (∼ 100’s of crossing times).Globular cluster: Core collapse (∼ 10’s of relaxation times).Galaxy: 1010yrs (many steps).Universe: 1010yrs (fewer steps).• Often to achieve steady state over many dynamical times it seems Nτ/δt ∼ constant.=⇒ timescale and lengthscale closely coupled.– E.g., crossing time for closed dynamical system.Virial theorem: 2K + W = 0 , K =12Mhv2i, W = −GM2/rg.Crossing time = [L]/[V ] ' rg/hv2i1/2' r3/2g/√GM.∗ Typically want δt ' τD/30 = τcross/30.– Another handy formula:τD'3√Gρ.E.g., for typical asteroid, ρ ' 2 g/cc so τD' 2.3 h. For Earth, spread out massof Sun to 1 AU: ρ = M/43πr3⊕=⇒ τD' 1 yr. Why? ω2r = GM/r2⇒ 4π2/τ2=GM/r3=43πGρ. ∴ τ ∼ 3/√Gρ.Units• In MKS, G = 6.7 × 10−11, 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/2π andspeeds in v⊕= 30 km s−1.– For galaxies, could use masses in 109M, distances in kpc. Then times would bein ∼ 15 Myr and speeds in kpc/15 Myr.2Constants of motion• If there are no outside forces/torques, Newton’s 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 withconstant velocity.– Can therefore set rg= vg≡ 0.N = 2 problem• Solved by Kepler, explained by Newton.• General solution (ellipse):r = a(1 − e cos ψ)cos θ =cos ψ − e1 − e cos ψwhere a = semi-major axis, e = eccentricity, ψ = eccentric anomaly, and mean anomalyωt = ψ − e sin ψ (Kepler’s equation).• Useful facts: if r and v are relative coordinates of two bodies, t henE =12m1v21+12m2v22−Gm1m2|r1− r2|=12m1m2Mv2+12Mv2g−Gm1m2r,where M ≡ m1+ m2. Hence, since we can always set vg≡ 0,Eµ=v22−GMr,where µ ≡ m1m2/M = reduced mass. Also haveE = −Gm1m22a,(Cf. Goldstein), so:1a=2r−v2GM.In addition, if h = r × v = L/µ = angular momentum per unit reduced mass, thene =r1 −h2aGM.Note h = rpvp= rava, where p and a denote periapse and apoapse, respectively, andrp= (1 − e)a, ra= (1 + e)a, rp+ ra= 2a.Finally,cos i =hzh,where i = orbital inclination wrt z = 0 plane.3N > 2 problemThe orbit of any one planet depends on the combined motion of all the planets,not to men tion the actions of all these o n each other. To consider simultaneouslyall these causes of motion and to define these motions by exact laws allowi ngof convenient calculation exceeds, unless I am mistaken, the force s of the entirehuman intellect.—Isaac Newton 1687.• One of the earliest N-body simulations (collision of two galaxies) used lightbulbs tocompute the forces! (Cf. Holmberg 1941, ApJ 94, 385


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UMD ASTR 415 - N -body Techniques Part 1

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