Astro 250 Planetary Dynamics Problem Set 3 Do at least 1 problem.

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1 Astro 250 Planetary Dynamics Problem Set 3 Do at least 1 problem. Readings: Murray & Dermott (don t let disturbing expansions disturb you), , ; ALSO check out lowest order forms of Lagrange s equations in equations (8.12) (8.17). Problem 1. The Titan ringlet and 1:0 Apsidal Resonance The Colombo ringlet, also known informally as Titan ringlet, is a narrow planetary ring around Saturn that sits within 1:0 apsidal resonance established by largest of Saturnian moons, Titan. This means that precession rate of apsidal line of ringlet matches mean motion of Titan; Titan appears to pull ring along. Denote Titan s mass over Saturn s mass by M T /M = , its semi-major axis by a T = km, its eccentricity e T = 0, its mean motion by Ω T, and its mean longitude by λ T. Denote a single ring particle s semi-major axis by a = km, its eccentricity by e = , and its mean motion by Ω = rad/ s. These numbers are given just for reference; problem below does not require any numerical evaluation. a) Write down, to leading order in e, SINGLE term of disturbing function due to Titan ( perturber) that represents 1:0 apsidal resonance. Leave all variables in symbolic form (do not plug in numbers). b) Use Lagrange s equations to compute ȧ, ė, and ω for ring particle. Express in terms of constant η = (M T /M)ΩαH 10 /2, where α = a/a T and H 10 = 2b (1) 1/2 (α) + α(d/dα)b (1) 1/2 (α) 3α. c) The above expression for ω is incomplete because it only accounts for mean-motion resonant forcing by Titan. What is missing is forcing by secular potential. Let s just say complete answer is ω = answer in part b + ω sec (1) where ω sec = ω Saturn + ω stuff is total additional precession rate induced by oblateness of Saturn and secular potential of everything else nearby rings (remember Titan ringlet is just 1 narrow ring embedded in Saturn s gigantic ring complex), Titan, or satellites, Sun, lost pens, etc. In fact, Saturnian oblateness completely overwhelms or contributions. There is no need to write out explicitly what all se terms are; we will 1

2 just work with ω sec (a). (Those of you who did problem 1 of PS 1 know what this function is, but explicit form is not needed for this problem!) Furr define ɛ(a) ω sec (a) Ω T (2) φ ω λ T (3) where φ is resonant libration angle. Also define a = a 0 such that ɛ(a 0 ) = 0. Your equations for ė and ω are coupled ordinary differential equations. Replace e and φ in favor of variables, h e cos φ (4) k e sin φ (5) Write down ḣ and k in terms of η, ɛ, h, and k. d) Solve your equations for ḣ and k. Your solution should contain two arbitrary constants: an amplitude and a phase associated with a sinusoidal oscillation. e) Plot possible trajectories in h and k space. Identify conditions under which φ is circulating (running gamut from 0 to 2π) or librating (oscillating about a fixed value). If particle is librating, what are libration centers, φ? What libration centers are associated with a > a 0? What centers are associated with a < a 0? If you have correct solution, you should notice that something terrible happens at a = a 0. This is simply a deficiency of our low-order ory. Problem 2. Tilted Rings Consider two planets on circular orbits around a star. The inner planet has mass m 1 and semimajor axis a 1, and outer planet has mass m 2 and semimajor axis a 2. At t = 0, orbit plane of m 2 coincides with x-y plane; orbit plane of m 1 is inclined by i 1 and has longitude of ascending node Ω 1 = 0 (on x-axis); and two planets happen to be passing conjunction at mean longitude λ 1 = λ 2 = 0 (on x-axis). a) The period ratio between two planets (a 2 /a 1 ) 3/2 is found to be very well approximated by integer ratio k : j, where k and j are relatively prime (only common factor 2

3 of two numbers is 1). It is proposed that two planets occupy a k : j mean-motion resonance. How many conjunctions occur before planets pass conjunction again at λ 1 = λ 2 = 0? Argue from this result that when k j, mean-motion resonance is weak. b) Secular approximation: Smear m 2 along its orbit so that it becomes a circular wire of uniform linear mass density. Derive to leading order in i 1 time-averaged disturbing potential felt by m 1 due to m 2. You will need to perform two integrals: one over wire that is m 2 (integral over θ), and a second over orbit of m 1 (integral over ψ). See Figure 1. Use Laplace coefficients (see integral definition on page 237 of MD). Hint: Write down z in terms of a 1, i 1, and ψ. Also write down d in terms of a 1, a 2, θ, and z. This d is denominator of disturbing potential. Write down integral over θ for potential of outer wire as evaluated at a single point on inner orbit. Taylor expand integrand in small parameter z BEFORE trying to perform integral over θ. Your θ-integral should produce Laplace coefficients (following physicist s maxim that any integral we can t do is given an honorary name). Finally time-average (integrate) over ψ. You can check your answer by looking up appropriate secular term ( term that does not depend on any mean longitudes) in Appendix B of MD. Note that s in notation of MD is actually sin i/2. 2. h h z r d a 2 a 1 ψ θ m 1 m 2 line of nodes a) We ll calculate torque by writing potential and taking its derivative with respect to angle 1.Let sfirstfindpotentialduetoouterwireatasinglepoint Figure 1: Schematic on of tilted inner rings wire problem. located The a height angle z θ is above in x-y plane plane of (in outer orbital wire. From plane of m 2 ). The diagram angle ψ we is see orbital that plane squared of mdistance 1. Notebetween that as our shown point here on in this inner wire and diagram, two planets agivenpointonouterwireis are not actually passing conjunction. d 2 =(a 2 1 z2 )+a 2 2 2(a2 1 z2 ) 1/2 a 2 cos θ + z 2 3 a a 2 2 2a 1 a 2 cos θ + z 2 a 2 cos θ. a 1 Let mass of our point on inner wire be dm 1. We know mass per unit angle of outer wire is m 2 /(2π), so potential is ( ) 1/2

4 Problem 3. Principal Lindblad Resonances This problem is derived from Goldreich & Tremaine s (1980, ApJ, 241, 425, hereafter GT) paper on disk-satellite interactions. In a coordinate system that attaches origin to (primary) star of mass M, perturbation potential due to a (secondary) planet of mass M p reads φ p (r, θ, t) = GM p r r p + GM p r p 3 r p r where r is vector position (measured from origin) where potential is to be evaluated, and r p is vector displacement from origin to planet. Note that in equation (4) of GT, re is an error; ir (M s /M p )Ω 2 (r) should be replaced by GM s /r 3 s. (This error does not propagate to rest of ir paper.) It is useful to expand φ p in a Fourier series: φ p (r, θ, t) = l= m=0 φ p l,m (r) cos{mθ [mω p + (l m)κ p ]t} where Ω p is mean angular frequency of planet ( rotational frequency of guiding center of planet s orbit), and κ p is planet s epicyclic frequency ( frequency of radial oscillations due to non-zero eccentricity of planet). In a frame that rotates at angular frequency Ω p + (l m)κ p /m, perturbation potential is time-independent and has an m-fold azimuthal symmetry. Assume that planet s eccentricity is zero so that r p is a constant. strength of principal m th component of potential, φ p m,m(r). Evaluate This expression is sufficient to describe perturbation potential of a planet on a perfectly circular orbit, and it is component that establishes principal Lindblad resonances (Galacto-speak) or first-order mean-motion resonances (planeto-speak) in disk. Principal Lindblad resonances excited in a disk dominate evolution of semi-major axis of planet; y are responsible for planet migration. Express your answer in terms of Laplace coefficients (see integral definition on page 237 of Murray and Dermott). Watch out for cases m = 0 and m = 1. Compare your answer to equation (7) of GT. 4

5 Problem 4. Poincare, Lagrange, and Hamilton Hamilton s equations read: ṗ i = H/ q i (6) q i = H/ p i (7) Any set of variables {q i, p i } that satisfy Hamilton s equations are called canonical. Unfortunately, Keplerian osculating elements are not canonical variables. appropriately constructed combinations of Kepler elements are canonical. combination is Poincare s set: However, One such q 1 = λ p 1 = µa (8) q 2 = ω p 2 = µa (1 ) 1 e 2 (9) q 3 = Ω p 3 = µa(1 e 2 ) (1 cos i) (10) (See section 2.10 of MD but note that y add an extra µ into ir equations for which we have no use.) Insert Poincare s canonical variables as written above into Hamilton s equations to derive Lagrange s equations (6.145), (6.146), and (6.148) (6.150). Ignore (6.147) for which we will have no use in this course. Everywhere you see ɛ in (6.145) (6.150) replace it with λ (see discussion on page 252). This is more-or-less a plug-and-chug problem. Historically this is not way Lagrange actually derived his equations. But problem does highlight fact that Lagrange s equations are really just Hamilton s equations, re-written in a nice practical way for celestial mechanicians. Hint: R/ e = 3 i=1 ( R/ p i)( p i / e), and similarly for or Kepler elements. 5