Energy lost by a radiating system

This expression illustrates two things: first, that L is dimensionless in geometrized units and, second, that it is almost always easier to compute in geometrized units, and then convert back at the end. The conversion is

So for the binary pulsar system described in § 9.3, if its orbit were circular, we would have ω = 2π/P = 7.5049 × 10−13 m−1 and

in geometrized units. We can, of course, convert this to watts, but a more meaningful procedure is to compare this with the Newtonian energy of the system, which is (defining the orbital radius r = 12 l0),

The physical question is: How long does it take to change this? Put differently, the energy radiated in waves must change the orbit by decreasing its energy, which makes |E| larger and hence ω larger and the period smaller. What change in the period do we expect in, say, one year?

From Eq. (9.139), by taking logarithms and differentiating, we get

Since dE/dt is just −L, we can solve for dP/dt:

dP/dt = (3 PL)/(2E) ≈ −15 PM−1(M )8/3

which is dimensionless in any system of units. It can be reexpressed in seconds per year:

This estimate needs to be revised to allow for the eccentricity of the orbit, which is considerable: e = 0.617. The correct formula is derived in Exer. 49, § 9.7. The result is that the true rate of energy loss is some 12 times our estimate, Eq. (9.138). This is such a large factor because the stars’ maximum angular velocity (when they are closest) is larger than the mean value we have used for , and since L depends on the angular velocity to a high power, a small change in the angular velocity accounts for this rather large factor of 12. So the relativistic prediction is:

The observed value as of 2004 is (Weisberg and Taylor 2005)

The effect has been observed in other binaries as well (Lorimer 2008).

9.5 A s t ro p h y s i c a l s o u rce s o f g ra v i t a t i o n a l wa v e s

Overview

Physicists and astronomers have made great efforts to understand what kinds of sources of gravitational waves the current detectors might be able to see. This has been motivated partly by the need to decide whether the large investment in these detectors is justified, and partly because the sensitivity of the detectors depends on the accuracy with which waveforms can be predicted, so that they can be recognized against detector noise. While the astrophysics of potential sources is beyond the scope of this book, it is useful to review the basic classes of sources, learn what general relativity says about them, and understand why they might be interesting to observe. We shall consider four groups of sources: binary systems, spinning neutron stars, gravitational collapse, and the Big Bang.

Binary systems

We have seen how to compute the expected wave amplitude from a binary system in Eqs. (9.98) and (9.99), as well as in Exers. 29 and 39 in § 9.7. There are a number of known binary systems in our Galaxy which ought, by these equations, to be radiating gravitational waves in the frequency band observable by LISA, and with amplitudes well above LISA’s expected instrumental noise. When LISA begins its observations, therefore, scientists will be looking for these signals as proof that general relativity is correct at this basic level, as well as that the spacecraft is operating properly.

The ground-based detectors will not, however, be looking for signals from long-lived binary systems. The reason is evident if we combine some of our previous computations. We have calculated what the gravitational wave luminosity of an equal-mass binary is

(Eq. (9.136)) and what its binding energy is (Eq. (9.139)). We used the last two equations to derive the lifetime of the Hulse–Taylor binary pulsar. We can generalize this to find the lifetime of any equal-mass circular binary system, expressing it in terms of the masses M of the stars and the frequency of the orbit f = ω/2π :

What the second equation tells us is that, if LIGO observes a binary system composed of stars of solar mass, it has only a few seconds to make the observation. During this time, the signal increases in frequency, something the gravitational wave scientists call a ‘chirp’. There are no realistic systems in the frequency range of ground-based detectors that are long-lived. Instead, these detectors look for inspiral events ending in the merger of the two objects, which itself might produce a burst of radiation. For neutron stars, the merger happens when the signal frequency reaches about 2 kHz, a number that is sensitive to the neutron star equation of state (see the next chapter). If the objects in the binary are black holes of mass 10 M , the final frequency is similar, and it scales inversely with the masses. Advanced LIGO and VIRGO should see several neutron-star mergers per year, and while the event rate for black holes is harder to predict, it is likely to be similar.

It is particularly interesting from the point of view of general relativity to observe the merger of two black holes. This can be simulated numerically, and by comparing the observed and predicted waveforms we have a unique test of general relativity in the strongest possible gravitational fields. This is also the only direct way to observe a black hole: after a merger of black holes or neutron stars has led to a single black hole, that hole will oscillate for a short time until is radiates away all its deformities and settles down as a smooth Kerr black hole (see Ch. 11). This ‘ringdown radiation’ carries a distinctive signature that will distinguish the black hole from any neutron star or other material system.

LISA, observing between 0.1 mHz and 10 mHz, will follow the coalescence and merger of black holes around 106M . Astronomers know that such black holes exist in the centers of most galaxies, including our own Milky Way (Merritt and Milosavljevic 2005), as we discuss in Ch. 11. LISA will have sufficient sensitivity to see such mergers anywhere in the universe, even back to the time of the formation of the first stars and galaxies. Its observations may be very informative about the way galaxies themselves formed and merged in the early universe.

Notice that, if we observe a chirp signal well enough to measure its inspiral timescale τ from the rate of change of the signal’s frequency, then we can infer the mass M of the system from Eq. (9.147). LISA by itself, or a network of ground-based detectors working together, can measure the degree of circular polarization of the waves. This contains the information about the inclination of the orbit to the line of sight, and allows us to compute from the observed amplitude of the waves some standard amplitude, such as the amplitude the same system would be radiating if it were oriented face-on. Then we can go back