2: WEEK BINARY INSPIRAL AT LEADING ORDER AND COMPACT BINARIES IN THE UNIVERSE
(Dated: Jan 16, 2008)
THERES GOING TO BE NO LECTURE ON MONDAY, 21 JANUARY. But Yanbei is going to be at 124 Bridge Annex for oce hour from 5:15PM to 6:15PM.
I.
READINGS AND REFERENCE MATERIALS
1. Leading order inspiral of circular orbits: Sec. 6.1 of Alessandra Buonanno, Gravitational waves, arXiv:0709.4682. 2. Evolution of Keplerian orbits: P.C. Peters and J. Mathews, Phys. Rev. 131, 435 (1963); P.C. Peters, Phys. Rev. 136, 1224 (1964). 3. (Equatorial) Innermost Stable Circular Orbit in Kerr (a) For simple treatment, see Chapter 15 of Hartle, Gravity (see password protected section of the course website) (b) For more complete treatment with analytic formulas, see Section II of J.M. Bardeen, W.H. Press and S.A. Teukolsky, ApJ 178, 347 (1972) 4. Empirical determination of binary neutron-star rate in our galaxy: (a) One of the earliest papers: E.S. Phinney, ApJ 380, L17 (1991). (b) Detailed paper with more information on galaxy model, selection eects, and a Baysian analysis: C. Kim, V. Kalogera and D.R. Lorimer, ApJ 584, 985 (2003). (c) Inclusion of all 3 binary pulsars currently found in our Galaxy: C. Kim, V. Kalogera and D.R. Lorimer, astro-ph/0608280. 5. Basics of Stars and Galaxies: any standard astrophysics textbooks, in particular (a) For accretion in binaries, Chapter 4 of J. Frank, A. King and D. Raine, Accretion Power in Astrophysics, will be available at password protected section of the course website. 6. Population synthesis for stellar-mass compact binaries: V. Kalogera et al., Formation of double compact objects, Phys. Rep. 442, 75 (2007). 7. Supermassive Black Holes, Ferrarese and Ford, Space Science Reviews, Volume 116, Numbers 3-4, February 2005 , pp. 523-624. 8. Detailed process of of galaxy mergers, Begelman, Blandford and Rees, Nature, 287, 307 (1980). 9. Extreme-mass-ratio Inspirals, P. Amaro-Seoane, Class. Quantum Grav. 24 R113R169 (2007).
II.
PROBLEMS
This homework is due on Wednesday, January 23, before class. Although the maximum grade for this week will be 50, this problem set contains 9 problems with a total of 120 points. This is intentional, because these problems involve very dierent physics; the same problem may be interesting for some, but boring or totally trivial for others. Please choose problems that are most interesting to you (hopefully they are not all boring, in which case you should complain to Yanbei). Problem 1. Evolution of Keplerian Orbits under radiation reaction. [15 Points] Derive the evolution equations of da/dt and de/dt, and da/de of elliptic Keplerian orbits, under quadrupole radiation reaction, and compare with results from Peters and Mathews. It seems easiest to use the following parameterization of Keplerian motion (as Yanbei did in his lecture): x = a(cos u e), y = b sin u (1)
2 with u e sin u = 0 t , Problem 2. 0 = GM/a3 (2)
ISCO in Schwarzchild and Equatorial ISCO in Kerr [15 Points]
(a) Conrm that, for circular geodesics in Schwarzchild, we have d = dt GM/r3 (3)
(b) Argue that in general, for Schwarzchild and Kerr, for events that happen on circular geodesics once each period, the distance observer is going to see an angular frequency of = d/dt , and therefore the quadrupole gravitational-wave frequency would be 2. (c) For equatorial orbits in Kerr, using two conserved quantities ut = and u = , obtain the 2 eective potential Ve , and then obtain rISCO and ISCO . Your result can be numerical, e.g., presented in a plot which would be very easy to obtain using Mathematica. (d) In 2003, infrared ux uctuations with a period of 17 minute is observed from the galactic center [R. Genzel et al., Nature 425, 934 (2003)]. If we assume 17 minute to be due to events that happen once each orbital period, we would then be able to say that ISCO is larger than 2/(17min). Given a black-hole mass of 3.6 106 M [obtained from Keplerian motion, e.g., F. Eisenhauer et al., Astrophys. J. 628 246-259 (2005)], what do we infer about a? In addition, if we take 17 minute as really arising from an equatorial ISCO orbit, what is the actual (or proper) frequency at which this uctuation happens? Problem 3. Waveform in the frequency domain [10 Points] It is often convenient to have an analytic expression for the Fourier transform of h(t). One can obtain such h(f ) using the so-called stationaryphase approximation, which is explained by Alessandra in Sec. 6.4 of arXiv:0709.4682. Read through this section, and derive Eq. (6.39) yourself. In particular: (a) Pay attention to the phase factor i/4 where does it come from? why isnt it +i/4? (b) Now that h(f ) goes as f 7/6 which means signal is stronger in higher frequencies why is it so? Where in the derivation does this arise? Whats the physical reason? Problem 4. Characteristic age of pulsars assuming magnetic dipole damping [10 Points] In order to extract BNS merger rate from galactic BNS observations, we would like to know the total lifetime of a binary pulsar, i.g., time to merger plus time since birth. Show that with magnetic dipole radiation being the sole mechanism for damping, P/(2P ) gives a good estimate of the lifetime of the pulsar at very late times, when frequency is already much lower compared with birth frequency. [The only information needed would be that for magnetic dipole damping, = 3 . Please justify this result.] However, there are scenarios where characteristic age does not work! Problem 5. Baysian estimate for binary neutron star merger rates [15 Points] In addition to making an estimate like R 1/ , we can plot gures of rate probability distribution with the help of Baysian statistics, which helps people understand the status of their knowledge about certain things. In such studies, one have a set of assumptions (or theories), parameterized by R, and a set of observations (or experimental outcomes), parameterized by n. What can be obtained from rst principle, would be the probability of having n happen, if theory R turns out to be correct, i.e., p(n|R). This is called the likelihood function. Here we cite a related conclusion in conditional probability: p(n|R) = which can be converted into p(R|n) = p(n|R)p(R) p(n) (6) p(n, R) p(n, R) p(n) p(R|n)p(n) = = p(R) p(n) p(R) p(R) (5) (4)
3 Now coming back to Baysian statistics, before performing any experiments, one rst assigns a prior distribution, p(R), which reects our initial knowledge about R. After performing one test, suppose a result of n0 is obtained, then one writes p(R|n0 ) p(n0 |R)p(R) (7)
and obtains a new distribution by normalizing, and interprets p(R|n0 ) as our updated knowledge, which is corrected from p(R) using the likelihood function p(n0 |R). In an idealized situation, one would perform many tests, and update ones p(R) many times, which will then reect the reality more and more accurately. In this idealized situation, once we perform more and more useful experiments, the posterior distribution p(R) will not depend very much on the initial prior we use. In less idealized situations, when we only have a small number of samples, results from Baysian statistics depends a lot on the prior we put in, and are then not as easy to interpret. Let us now try to understand how Baysian statistics can be used to understand galactic BNS rate, in the following steps: (a) Suppose for a type of source with lifetime , we have a birth rate R, modied by a reduction factor N (due to our nite reach into the galaxy), and pulsar beaming factor fb , then show that p(n|R) = n e , n! = R /(N fb ) . (8)
The fact that we see one source would then mean p(R|1) R R /(N fb ) e p(R) . N fb (9)
Now, what do we put as p(R)? In studies done by Vicky Kalogera et al. (reading item 4), p(R) = 1 is assumed. However, it might be more fair to assume p(R) = 1/R, with an upper cut o frequency set by supernova rate (the reason being, since each BNS form through some sort of supernova event, the total BNS birth rate should not exceed the supernova rate), while without a lower cuto. The rationale behind such 1/R scaling is that this gives a at distribution of log R, which means it is equally possible to have a rate of 1/Myr, 0.1/Myr, 0.01/Myr, etc., which seems a more reasonable way of representing ignorance in astronomy. However, for the moment, let us try to recover their result using p(R) = 1. (b) For each of the three galactic BNS, nd the appropriate N and fb in the reading, you will then be able to obtain p(R(k) |1(k) ), where k = 1, 2, 3 representing population associated with each pulsar. (For the double pulsar, even though we dont know the beaming ratio, let us take the more optimistic point of view and assume its beaming ratio to be roughly equal to that of the rst two binary pulsars.) Note how , N and fb aects the distribution of R, and explain in simple terms, then you understand why the newest discovery of the double pulsar is so dramatic. (c) Let us pause here to examine another issue: theres suppose an unknown channel that forms BNS binaries with lifetime very small. Then naturally we would observe no such BNS in our galaxy. How do we understand this using Baysian statistics? What would p(R|0) look like? What does it mean? (d) Now returning to the 3 observed binary pulsars, what would be ppost (Rtot ) p[Rtot = R(1) +R(2) + R(3) |1(1) , 1(2) , 1(3) ]? Apparently, if we assume the three dierent observations are independent (can you give a reason why? or why not?), we simply need to calculate the distribution of the independent sum of R(k) , which is easy (one line in Mathematica) if we use the fact that the characteristic function of Rtot is the product of the characteristic functions of p(R(k) |1(k) ). (e) What if we use a prior of p(R) 1/R instead of p(R) 1? The immediate consequence is that, if we only observe one BNS, the posterior distribution does not have a peak at some R = 0 value. Of course, situation changes when we have more than one observation. Now with three BNS, how does our result compare with the previous result? Apparently we should have a lower peak rate than previously. With which condence level would this result have been ruled out by the previous study using at prior? Problem 6. Roche Lobe and tidal disruption of stars. [10 points] Even though the shape of Roche Lobe can only be calculated numerically, one can obtain simple expressions when one of the component is much less massive than the other, e.g., during the capture of a main sequence star m by a SMBH
4 M . For simplicity, let us assume a circular orbit. In this case, at leading order in m/M , show that (by Taylor-expanding gravitational force from M and centrifugal force, in the vicinity of m) that
m
(m/M )1/3 a
(10)
where m is the distance between L1 and m. Imposing m R , obtain the a which corresponds to tidal disruption of the main-sequence star, and then estimate the corresponding Keplerian frequency, and explain its signicant toward LISAs ability of observing inspirals of main-sequence stars into SMBHs. Problem 7. Common-Envelope Binary Evolution. [10 points] The following is an example of a process that occurs in the evolution of close binary systems from their initial, main-sequence congurations to nal compact-body cong- urations (WD/WD, NS/NS, BH/BH, etc): A 16 M , late in its life, evolves into a red giant phase which has a 4M compact core and a 12M pued-up envelope. The radius of the envelope is R 200R , but most of its mass is contained inside R1 100R , which you can idealize as having uniform density. Suppose that this star has a 3M companion, and when the big star expands into its red giant phase, the outer part of its envelope engulfs this companion. As the companion orbits inside the giant stars envelope, it stirs the envelope up, gradually feeding the orbital energy into the envelopes gas, causing gas to get ejected and causing the companion to slowly spiral inward. Ultimately the companion reaches a radius small enough that the energy it has injected into the giants envelope is enough to eject al l of the enve- lopes mass. The result, then, is a 3M companion orbiting a 4M compact star (the red giants remnant core). Estimate the separation between these two stars. Problem 8. Eddington Limit. [15 points] A compact star accretes gas from a dense surrounding medium (e.g., in the common-envelope evolution described above). Idealize the accretion as spherically symmetric. When the gas hits the surface of the compact star, its energy of infall is converted into heat and thence into outgoing radiation. This radiation ows out through the accreting gas and, if the gas was not already highly ionized, the radiation ionizes it. It is a good approximation to treat the infalling 2 gas as fully ionized. Then the radiation can scatter o electrons (with scattering cross section 8r0 /3 where ro is the classical electron radius). In this scattering, on average, all of the photons momentum is transferred to the electron, giving the electron an outward kick. The collective inuence of all these kicks on electrons, gives a net outward force per unit mass on the infalling gas. (a) Derive a formula for the net outward force per unit mass of gas in terms of the gass mass density and the outowing ux of radiation energy F . (b) Explain why the radiations similar outward force on the protons, per unit mass of gas, is negligible compared to that on the electrons. (c) Explain why the inward force of gravity acting on the protons, per unit mass of gas, is far larger than that acting on the electrons. (d) With the outward radiation force acting on the electrons and the inward grav- itational force acting on the protons, a charge separation develops between the electrons. This charge separation produces an electric eld that locks the elec- trons and protons together. Estimate very roughly how much charge separation occurs and show that it is totally negligible compared to the size of the star. (e) Because the outowing radiation ux is F = L/(4R2 ) where R is radius and L is the stars luminosity, produced by accretion, the outward radiation force per unit mass on the gas scales as 1/R2 the same scaling as the gravitational force per unit mass. Therefore, there is a critical luminosity LEdd (called the Eddington Luminosity) above which the radiation force overwhelms the gravitational force, turning o the...