Some 3 body problems Kozai resonance 2 planets in mean motion resonance Lee, M. H. 2004

3 body perturbing function Assume one body is pretty far away and define R 3 to be distance to this body and center of mass of the inner binary To second order in R 3 m3m3 m1m1 m2m2 R3R3 COM ψ just a 2 body problem Like a disturbing function R

Orbit averaging First average over one complete outer orbit then over one cycle of inner orbit First averaging over outer orbit (with elements subscripted _e Want to consider all possible mean anomalies for outer orbit and integrate up V e Integrate from 0,2π

Averaging over outer orbit Using manipulation of Keplerian orbits Now we have to expand cos 2 ψ Depends on relative inclination and angles of perihelion of inner and outer orbits

Averaging over inner orbit Valtonen and Karttunen expand in terms of Eccentric anomaly and then take expectation values over orbit, finding Does not depend on angle of perihelion of outer body, so its eccentricity, e e, is conserved Since we averaged over mean motions, the Hamiltonian does not depend on mean anomali, that means that the semi-major axes are conserved

Elimination of nodes Total angular momentum is conserved Choose a coordinate system such that z is aligned with total angular momentum L Angular momentum vectors of 2 bodies set their longitude of ascending nodes. They lie along a line in the planet perpendicular to the total angular momentum vector. Therefore Ω 1 =Ω 2 +π The Hamiltonian cannot depend on the longitude of ascending node of the outer body because this is just a choice of rotation in the coordinate system. Because Ω 1 =Ω 2 +π the coordinate system also cannot depend on Ω 2. Since H is independent of both, associated momenta are conserved

Orbit averaged Because e e and a e conserved after orbit averaging, i e does not change either For inner body, a i and i i are related by above conserved quantity. Changes in e related to changes in I Valtonen & Karttunen then work with relative inclination only --- I find this confusing since conserved quantities seemed to require that we work in coordinate system with i defined with respect to total angular momentum. Though if m 3 sufficiently far away and massive, then it doesn’t matter

Kozai resonance Important coefficient Convert this into a timescale for inner system with n i =(G(m 1 +m 2 )/a i 3 ) 1/2 Giving timescale for Kozai oscillations of order τ Rapidly decreases with increasing distance Kozai oscillations include variations in angle of perihelion which are either circulating or librating

Min Max i,e At low eccentricity and high inclination large oscillations can take place

Kozai cycles High inclination asteroid belt objects would experience large eccentricities However Morbidelli mentions that secular perturbations can cause sufficient precession that outer tidal perturbation is averaged and the Kozai cycle erased Relevant to binary star/planetary systems

Galactic tides The ecliptic is not aligned with the Galactic plane ( COBE images showing both zodiacal cloud and galactic plane)

Galactic Tides Similar type of perturbation caused by Galactic tide With high inclination Similar conservation law setting relation between e,i Oort cloud bodies can have large swings in e,i due to Galactic tide

2 planets in resonance Write the Hamiltonian as a sum of Keplerian parts and a Disturbing function Cannonical variables just mass weighted versions of Delauny or Poincare variables Perform a cannonical transformation to new coordinates the following angles, λ 1,λ 2 σ 1 =iλ 1 - jλ 2 - ϖ 1, σ 2 = iλ 1 - jλ 2 - ϖ 2 …. Resulting Hamiltonian does not depend on λ 1, or λ 2 so there is a conserved quantity from each.

2 planets in resonances averaged theory Following Michtchenko + Beuge + Ferraz-Mello Conserved quantities coupled oscillations between e,a

Apsidal resonance Libration of Δ ϖ = ϖ 1 - ϖ 2 about 0 or π. Search for stable resonance solutions to right m 2 >m 1 Different values of relative angles

Trajectories in resonance capture Depends on mass ratio and time allowed to drift Angle between planets can be a simple 0,π or “assymetric” from Beuge et al. 06, similar plot with less information shown by Lee, MH 04

ACR (Apsidal corotation resonance) Idea is to look for particularly stable solutions Maxima of averaged Hamiltonian Quantified numerically by a stability index

2 planets in resonance After elimination of short period motion (orbit averaging) the Hamiltonian only has 2 degrees of freedom so can plot level curves However real system may exhibit chaos Michtchenko plots spectral number (number of peaks in spectrum) and averages over fast frequency oscillations to compare to averaged resonant Hamiltonian Which angles librate and circulate is set by mass ratio and eccentricity At higher eccentricity orbits are more chaotic See recent papers by Michtchenko, Beuge and Ferraz-Mello

Resonance capture 2 planets Eccentricity damping rate compared to migration rate can allow a steady state with limiting final eccentricity Or it can be a problem, leading to a fine tuning problem as high eccentricities could be reached leading to instability I did not find interesting limits on migration rates for capture in to the 2:1 resonance but did for the 3:1 resonance using simplistic versions of the non-adiabatic limit

Reading Valtonen & Karttunen Chap 9 on the Kozai resonance Recent papers by Michtchenko, Ferraz Mello, Beuge and collaborators

Problems Consider the possibility that the ~0.8 Solar mass star TW PSA which is 60,000 AU (on sky) away from Fomalhaut is bound to the Fomalhaut system. How long would a Kozai cycle induced by this binary star in planet Fom B take?