Binary Sources of Gravitational RadiationWe now turn our attention to binary systems. These obviously have a large and varyingquadrupole moment, and have the additional advantage that we actually know that gravitationalradiation is emitted from them in the expected quantities (based on observations of double neutronstar binaries). The characteristics of the gravitational waves from binaries, and what we couldlearn from them, depend on the nature of the objects in those binaries. We will therefore startwith some general concepts and then discuss individual types of binaries.First, let’s get an idea of the frequency range available for a given type of binary. Thereis obviously no practical lower frequency limit (just increase the semimajor axis as much as youwant), but there is a strict upper limit. The two objects in the binary clearly won’t produce asignal higher than the frequency at which they touch. If we consider an object of mass M andradius R, the orbital frequency at its surface is ∼pGM/R3. Noting that M/R3∼ ρ, the density,we can say that the maximum frequency involving an object of density ρ is fmax∼ (Gρ)1/2. Thisis actually more general than just orbital frequencies. For example, a gravitationally bound objectcan’t rotate faster than that, because it would fly apart. In addition, you can convince yourselfthat the frequency of a sound wave through the object can’t be greater than ∼ (Gρ)1/2. Therefore,this is a general upper bound on dynamical frequencies.This tells us, therefore, that binaries involving main sequence stars can’t have frequenciesgreater than ∼ 10−3− 10−6Hz, depending on mass, that binaries involving white dwarfs can’thave frequencies greater than ∼ 0.1 − 10 Hz, also depending on mass, that for neutron stars theupper limit is ∼ 1000 − 2000 Hz, and that for black holes the limit depends inversely on mass (andalso spin and orientation of the binary). In particular, for black holes the maximum imaginablefrequency is on the order of 104(M¯/M) Hz at the event horizon, but in reality the orbit becomesunstable at lower frequencies (more on that later).Now suppose that the binary is well-separated, so that the components can be treated aspoints and we only need take the lowest order contributions to gravitational radiation. Temporarilyrestricting our attention to circular binaries, how will their frequency and amplitude evolve withtime?Let the masses be m1and m2, and the orbital separation be R. We argued in the previouslecture that the amplitude a distance r À R from this source is h ∼ (µ/r)(M/R), where M ≡m1+ m2is the total mass and µ ≡ m1m2/M is the reduced mass. We can rewrite the amplitudeusing f ∼ (M/R3)1/2, to readh ∼ µM2/3f2/3/r∼ M5/3chf2/3/r(1)where Mchis the “chirp mass”, defined by M5/3ch= µM2/3. The chirp mass is named that becauseit is this combination of µ and M that determines how fast the binary sweeps, or chirps, througha frequency band. When the constants are put in, the dimensionless gravitational wave strain– 2 –amplitude (i.e., the fractional amount by which a separation changes as a wave goes by) measureda distance r from a circular binary of masses M and m with a binary orbital frequency fbinis(Schutz 1997)h = 2(4π)1/3G5/3c4f2/3GWM5/3ch1r, (2)where fGWis the gravitational wave frequency. Redshifts have not been included in this formula.The luminosity in gravitational radiation is thenL ∼ 4πr2f2h2∼ M10/3chf10/3∼ µ2M3/R5.(3)The total energy of a circular binary of radius R is Etot= −GµM/(2R), so we havedE/dt ∼ µ2M3/R5µM/(2R2)(dR/dt) ∼ µ2M3/R5dR/dt ∼ µM2/R3.(4)What if the binary orbit is eccentric? The formulae are then more complicated, because onemust then average properly over the orbit. This was done first to lowest order by Peters andMatthews (1963) and Peters (1964), by calculating the energy and angular momentum radiatedat lowest (quadrupolar) order, and determining the change in orbital elements that would occurif the binary completed a full Keplerian ellipse in its orbit. That is, they assumed that to lowestorder, they could have the binary move as if it experienced only Newtonian gravity, and integratethe losses along that path.Before quoting the results, we can understand one qualitative aspect of the radiation when theorbits are elliptical. From our derivation for circular orbits, we see that the radiation is emittedmuch more strongly when the separation is small, because L ∼ R−5. Consider what this wouldmean for a very eccentric orbit (1 − e) ¿ 1. Most of the radiation would be emitted at pericenter,hence this would have the character of an impulsive force. With such a force, the orbit willreturn to where the impulse was imparted. That means that the pericenter distance would remainroughly constant, while the energy losses decreased the apocenter distance. As a consequence, theeccentricity decreases. In general, gravitational radiation will decrease the eccentricity of an orbit.The Peters formulae bear this out. If the orbit has semimajor axis a and eccentricity e, theirlowest-order rates of change arehdadti = −645G3µM2c5a3(1 − e2)7/2µ1 +7324e2+3796e4¶(5)andhdedti = −30415eG3µM2c5a4(1 − e2)5/2µ1 +121304e2¶(6)– 3 –where the angle brackets indicate an average over an orbit. One can show that these rates implythat the quantityae−12/19(1 − e2)µ1 +121304e2¶−870/2299(7)is constant throughout the inspiral.Do we have evidence that these formulae actually work? Yes! Nature has been kind enoughto provide us with the perfect test sources: binary neutron stars. Several such systems are known,all of which have binary separations orders of magnitude greater than the size of a neutron star, sothe lowest order formulae should work. Indeed, the da/dt predictions have been verified to betterthan 0.1% in a few cases. The de/dt predictions will be much tougher to verify, though. Thereason for the difference is that de/dt has to be measured by determining the eccentricity orbitby orbit, whereas da/dt has a manifestation in the total phase of the binary, so it accumulatesquadratically with time. These systems provide really spectacular verification of general relativityin weak gravity. In particular, in late 2003 a double pulsar system was detected, that in additionhas the shortest expected time to merger of any known system (only about 80 million years).Having two pulsars means that extra quantities can be measured (such as the