14.7 Gravitational Radiation from Point Masses in a Keplerian Orbit

14 General Relativity14.7 Gravitational Radiation from Point Massesin a Keplerian Orbit(8 units)This project does not require prior knowledge of either of the Part II courses General Relativityor Cosmology, but does require Part IB Methods and Part IA Dynamics. (You may, however,find it useful to review some of your answers after taking appropriate Part II courses; but thecomputation may be attempted immediately.)For nearly Newtonian, slow-motion sources in General Relativity, the following are the formulaeforthetime-averagedlossesduetogravitationalradiationfortheenergyandangularmomentumof a gravitating source:dEdt = − G ...5c 5〈... I jk I jk 〉 (1)anddJ jdt = −2G...5c 5ε jkl〈ÏkiI il 〉. (2)Here i, j and k are the three space indices, the summation convention is used and ε is thealternating tensor. (Note that the coordinate system used is Euclidean, so there is no distinctionbetween upstairs and downstairs indices.) The dot represents a derivative with respect to timet (note the asymmetric time derivatives in (2)). I jk is the trace-free part of the second momentof the mass distribution,I jk = Q jk − 1 3 δ jkQ ii ,where Q jk is the second moment of the mass distribution (the moment of inertia tensor):Q jk = ∑ am (a) x (a)jx (a)kwherethesumisoverthe(point)masses. (Forcontinuousmassdistributionsthesumisreplacedby an integral.)The brackets 〈•〉 in (1) and (2) denote an average over a suitably large spacetime 4-volume (amore precise definition is given below).Consider a Keplerian orbit of two point masses m 1 and m 2 in the x–y plane, with semi-majoraxis a and eccentricity e. Assume the origin is the centre of mass. It is convenient to introducethe total mass M = m 1 +m 2 and the reduced mass µ = m 1 m 2 /M.Let d be the distance between the two masses and let ψ be the angle of one of the massesrelative to the x-axis (so that the other mass has angle ψ+π) at a particular point of the orbit;so both d and ψ are functions of time. The first point mass is then at d 1 (cosψ,sinψ,0) and thesecond point mass is at d 2 (−cosψ,−sinψ,0), where d 1 = dµ/m 1 and d 2 = dµ/m 2 .Question 1 Calculate the Q jk and hence the I jk as functions of d, µ and ψ. Simplifyyour answer and write it in matrix form.For Keplerian motion the orbit equation isd = a(1−e2 )1+ecosψJuly 2013/Part II/14.7 Page 1 of 3 ○c University of Cambridge

and the angular velocity is given by√GMa(1−e˙ψ =2 )d 2 .In equations (1) and (2), assume 〈•〉 denotes an average in time over one orbit. For a functionw of ψ this means thatwhereis the period of the orbit.〈w〉 = 1 T∫ T0w ( ψ(t) ) dt = 1 TT = 2πa3/2√GM∫ 2π0w(ψ)˙ψ dψAssume that a and e do not change appreciably in one orbit, so that their time derivatives canbe ignored in the averages.Question 2thatwhereand thatwhereNote that J 1 = J 2 = 0 (i.e., the x and y components of J vanish). Show〈 〉 dE= − 32G4 µ 2 M 3dt 5c 5 a 5 f(e) (3)f(e) =1+7324 e2 + 3796 e4(1−e 2 ) 7/2〈 〉 dJ3= − 32G7/2 µ 2 M 5/2dt 5c 5 a 7/2 g(e) (4)g(e) = 1+ 7 8 e2(1−e 2 ) 2.For Keplerian orbits we haveand(J 3 is usually denoted L).E = −G µM2aJ 3 = √ Gµ 2 Ma(1−e 2 )Question 3and thatwhereFor Keplerian orbits prove that〈 〉 da= − 64G3 µM 2dt 5c 5 a 3 f(e),〈 〉 de= − 304G3 µM 2dt 15c 5 a 4 h(e)h(e) =(1+121304 e2 )e(1−e 2 ) 5/2 .Under what conditions is it true that the time derivatives of a and e (as just calculated)can be ignored in the averages in (1) and (2) (as was assumed above in deriving (3) and(4))? (This is a consistency check.)July 2013/Part II/14.7 Page 2 of 3 ○c University of Cambridge

Question 4Find a as a function of time in the case when e ≡ 0 (i.e., a circular orbit).Question 5 The WUMa eclipsing binary star system has m 1 = 0.77M ⊙ and m 2 =0.56M ⊙ (the Sun has mass M ⊙ ≈ 2×10 30 kg), with period T = 0.33days. This determinesthe initial value of a but not of e, so we consider the situation as a function of e.Using a computer, calculate how much power is being radiated (now) gravitationally fore = 0, 0.5 and 0.95. Compare the results with the electromagnetic output of the Sun(about 4×10 26 Js −1 ).Write a program to determine how long it will take, as a function of initial eccentricitye 0 , before a decays to zero for WUMa. Use this program to calculate the decay time forinitial values e 0 = 0.00, 0.05, 0.10, ..., 0.95. Compare the results with the present age ofthe universe (13.7×10 9 years).Question 6 Write a program to calculate the initial semi-major axis a 0 that a binarymust have in order for the inspiral time to be equal to the age of the universe. Use thisto calculate a 0 (in units of the solar radius R ⊙ ≈ 7 × 10 8 m) for a binary consisting ofa solar mass star and giant gas planet of mass 0.001M ⊙ (about a Jupiter mass) withe 0 = 0.00, 0.05, 0.10, ..., 0.95.Hot Jupiters are a class of exoplanets with masses similar to Jupiter, but orbiting at< 0.5AU (where 1AU ≈ 1.5×10 11 m). Since Jupiter-like planets are expected to form ataround 5AU, using your above results comment on whether gravitational radiation is apossible mechanism for the Hot Jupiters to have subsequently migrated to their currentlocation.July 2013/Part II/14.7 Page 3 of 3 ○c University of Cambridge