Astro 250: Solutions to Problem Set 1. by Eugene Chiang

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1 Asto 250: Solutions to Poblem Set 1 by Eugene Chiang Poblem 1. Apsidal Line Pecession A satellite moves on an elliptical obit in its planet s equatoial plane. The planet s gavitational potential has the fom U GM p 1 J 2 ( Rp P 2 (cos θ)], (1) whee is the distance fom the planet to the satellite, θ is the pola angle measued fom the planet s spin axis, P 2 (cos θ) = 1 2 (3cos2 θ 1) is the Legende polynomial of degee 2, M p and R p ae the planet s mass and adius, espectively, G is the gavitational constant, and J 2 is a constant that chaacteizes the dimensionless stength of the quadupole field of the planet (the degee of planetay oblateness). Celestial mechanicians define thei potentials U to be positive, by contast with the usual convention in physics. a) Use the appopiate petubation equation due to Gauss (equation of MD) to calculate ω, the time-aveaged pecession ate of the satellite s apsidal line. b) Show that a and e do not suffe any time-aveaged vaiations using the appopiate equations of Gauss (also found in section 2.9 of MD). a) Fist, some elations fo Kepleian ellipses to lowest ode in e that ae convenient to have at one s fingetips: = a(1 ecos f) (2) f = n(1 + 2ecos f) (3) GM p = n 2 a 3 (4) ṙ = naesin f (5) The petubation equations fo ω, e, and a in the plane ead, again to lowest ode in e: d ω dt = 1 ( R cos f + 2S sin f) (6) nae 1

2 de dt = 1 (R sin f + 2S cos f) na (7) da dt = 2 Resin f + S(1 + ecos f)] n (8) One fact woth emembeing is that when you pull inwads (R < 0) on a paticle nea its peiapse (f 0), its apsidal line advances ( d ω dt > 0; ω advances in the diection of inceasing tue anomaly). The planet s axisymmetic bulge causes a puely inwad adial petubation foce in its equatoial plane. Using P 2 (0) = 1/2, and emembeing that celestial mechanicians define the potential to be positive, we have: R = d d JGM p 2 ( Rp = 3JGM p 2 2 ( Rp (9) We inset this into petubation equation (6) and aveage ove 1 obit to find the timeaveaged pecession ate. < d ω dt >= 3JR2 p GM 2π p 0 2nae cos f 4 dt df df 2π/n 0 dt (10) Note that we must aveage ove time, not tue anomaly; the paticle spends moe time nea apoapse than peiapse, and the petubation foce must be weighted to account fo this. Inset equations (2 4) into (9) to find < d ω dt >= 3 GM pjr 2 p 2 a 7/2 (11) Physically, the 1/ 4 petubation foce is much stonge at peiapse than apoapse, so that despite the exta time spent at apoapse, the attactive petubation foce is felt pincipally at peiapse. The esult of the net exta inwad tug at peiapse is that the apsidal line advances. b) The adial foce R in the petubation equations fo a and e ae weighted by sinf, which aveages to zeo ove 1 obit. Poblem 2. The Pevesity of Osculating Elements Deduce the values and time dependences of the osculating elements that chaacteize a pefectly cicula equatoial obit of adius aound an oblate planet. Employ the potential given by equation (1) above. To get stated, compute the elation between the 2

3 angula velocity and the obital adius. Remembe that the osculating elements ae those elements of a Kepleian ellipse that just kisses (is instantaneously tangential to) the actual position and velocity of the paticle. We fit the Keple obit to the motion by assuming that the paticle moves in a point-mass potential. Hee the potential is not that of a point-mass, but we wish to descibe the motion of the paticle as if it wee. The osculating elements of a paticle at a paticula instant in time ae the a, e, and ω appopiate to a Kepleian (ead: point-mass potential) ellipse fitted to the paticle s motion at that instant. Hee the paticle is executing a pefect cicle about a nonpoint-mass potential, and we (pevesely) wish to descibe its cicula obit in tems of constantly changing Kepleian ellipses. It is impotant to emembe that the n, a and e appopiate to the fitted ellipse at any instant ae abstactions which fall out of the osculating element fomalism and do not coespond to anything teibly physical. Let s fist solve fo a and e as a function of (and quadupole coefficient J). Two unknowns call fo two equations. The fist equation is a geometical constaint: since the paticle is moving puely azimuthally in its pefect cicle, the paticle must be eithe at the peiapse o the apoapse of the fitted ellipse. Let s guess that the paticle is at the peiapse (if we choose incoectly, the fitted e will tun out to be negative): = a(1 e) (12) The second equation fits the cicula velocity; equate the centipetal acceleation to the total adial acceleation due to the planet: λ 2 = GM p J ( Rp ] = n 2 (1 + 4ecos 0) (13) whee fo the last expession we have used (3). Use (12 13) to solve fo a and e in tems of, emembeing that fo osculating elements, n 2 a 3 = GM p always (the fitted ellipse is a Kepleian ellipse): ( ) e = 3 2 J Rp 2 (14) ( ) a = ( J Rp 2) (15) 3

4 Now fo ω we ague as follows: the only way an ellipse can tace out a cicle is if its longitude of peiapse advances as quickly as its mean longitude, ω = λ. Then ω = λt + ω 0 (16) whee λ can be expessed in tems of using equation (13). Since ȧ and ė ae both popotional to R sin f, and since the paticle is always at f = 0, ȧ = ė = 0. Accodingly, the values fo a and e that we deduced have no time dependences. We can use Gauss s equations to pove that ω = λ. Using (6), d ω dt = 1 3GM p JRp 2 nae 2 4 = GM p na 2 = na 2 / 2 = n(1 + 2e) = λ (17) Poblem 3. Velocity Ellipsoid in Collisionless Kepleian Disks Conside a cicumstella disk composed of massless test paticles which move without colliding on obits of eccenticity e 1. What is the atio of the velocity dispesions in the adial and azimuthal diections? Mateial in the eading fom Binney & Temaine (1987) is elevant to this poblem. By velocity dispesion in an axisymmetic disk we mean the following. Imagine ouselves co-otating with the disk on a cicula obit. At a given instant in time, we measue the appaent velocities of all paticles whizzing by ou position. We then (1) squae and (2) aveage the appaent velocities in a paticula diection to obtain the squaed velocity dispesion, σ 2, in that diection. The poblem asks you to obtain the atio of squaed dispesions in the adial and azimuthal diections, σ/σ 2 φ 2. Povided the disk is collisionless (paticle pass though each othe), this atio is magically independent of the actual distibution of andom velocities; i.e., this atio is independent of the actual distibution of eccenticities, povided they ae small. To begin, let us fix ou obital adius at R 0 and ou angula velocity at Ω 0. The paticles which manage to coss ou position (so that we can measue thei velocities) oiginate fom a ange of paent guiding centes centeed about ou own obit. Those paticles which come fom elatively distant guiding centes will have elatively high eccenticities in ode to each us. Conside a paticle which cosses ou position fom a 4

5 (geneic) guiding cente at R g which is a adial distance x g > 0 inside ou fixed, cicula obit: R g + x g = R 0. The obit of the paticle is expessed in tems of its adial and azimuthal excusions, x and y, fom its cicula guiding cente obit, viz. x = X cos(κt + ψ) = X cos(nt + ψ) (18) y = Y sin(nt + ψ) = 2X sin(nt + ψ) (19) whee the ightmost equalities ae appopiate fo Kepleian ellipses of small eccenticity. In this egime, the adial epicyclic fequency, κ, equals the azimuthal fequency, n, so that obits ae closed. Moeove, the amplitude of excusions in the azimuthal diection, Y, is twice that in the adial diection, X. Calculate fist the adial velocity dispesion, σr 2. In the following, the ovehead ba denotes an aveage ove all paticles cossing ou position at a given instant in time, t 0. It is an aveage ove all paticles oiginating fom all contibuting guiding centes. σ 2 R = ẋ 2 = X 2 n 2 sin 2 (nt 0 + ψ) = n 2 X 2 sin 2 (nt 0 + ψ) = n2 2 X2 (20) The last equality follows fom ou assumption that amplitudes and phases of measued paticles ae completely uncoelated. (Fo two uncoelated vaiables, AB = A B.) We can go no futhe without knowing the explicit distibution of eccenticities and semi-majo axes of the paticles. Howeve, this infomation is not equied because the question asks only fo the atio of σ 2 R to σ2 θ, and the quantity X2 will divide out in that atio, as we show below. Calculate now the azimuthal velocity dispesion, emembeing that the appaent azimuthal velocity of a paticle (the one that you measue) is eally the diffeence between its inetial space azimuthal velocity and you own inetial space cicula velocity. Realize below that in cetain instances the diffeence between R 0 and R g is negligibly small. σ 2 θ = ( θr 0 Ω 0 R 0 = R0( 2 θ Ω 0 = R0 2 ( θ Ω g ) + (Ω g Ω 0 ) ] 2 5

6 = R 2 0( ẏ R g dω dr x g Now ealize that x g, the adial distance fom the guiding cente to you position, is meely x, the instantaneous adial excusion of the paticle away fom its guiding cente (fo some eason, this seemingly obvious fact took me 3 hous yesteday to ealize): 2Xn = R0 2 cos(nt0 + ψ) dω ] 2 R g dr X cos(nt 0 + ψ) 2Xn cos(nt 0 + ψ) + 3n ] 2 2 X cos(nt 0 + ψ) n2 2 X2 ( )2 = n2 8 X2 (21) Then dividing (20) by (21) and taking the squae oot, we obtain ou final answe: σ R σθ = 2 (22) which is the invese of the atio of Kepleian epicyclic motions about a given guiding cente. It is diffeent fom that atio because of the undelying mean shea of the disk. Of couse, the squaed atio is 4. Note that the velocity dispesion in the z diection is de-coupled fom the plana components (inclinations have nothing to do with eccenticities, povided both ae small). The squaed atio of 4-to-1 (adial component being lage) fo Kepleian disks figues pominently in studies of disks; Chandasekha was the fist to deive this atio, I believe; diffeent values will obtain fo diffeent otation (sheaing) pofiles in disks, be they Galactic o planetay; in disks composed of pefectly spheical, inelastic, collisional paticles, the velocity ellipsoid tends to ound itself into a sphee (i.e., fo optical depth τ 1, collision ates become so high that the paticles act like a gas of isotopic pessue, σ 2 = σ 2 φ = σ2 z). The coss component σ 2 φ v v φ detemines the ate of angula momentum tanspot acoss annuli i.e., adial mass tanspot in accetion disks (it is zeo in ou collisionless poblem). 6