7.1 Interacting compact binaries

The first example of the use of gravitational radiation in modelling an observed astronomical system was the explanation by Faulkner [157] of how the activity of cataclysmic binary systems is regulated. Such systems, which include many novae, involve accretion by a white dwarf from a companion star. Unlike accretion onto neutron stars, where the accreted hydrogen is normally processed quickly into heavier elements, on a white dwarf the unprocessed material can build up until there is a nuclear chain reaction, which results in an outburst of visible radiation from the system.

Now, in a circular binary system that conserves total mass and angular momentum, a transfer of mass from a more massive to a less massive star will make the orbit shrink, while a transfer in the opposite direction makes the orbit grow. If accretion onto a white dwarf begins with the dwarf as the less massive star, then the stars will draw together, and the accretion will get stronger. This runaway process stops when the stars are of equal mass, and then accretion begins to drive them apart again. Astronomers observed that in this phase accretion in certain very close binaries continued at a more or less steady rate, instead of shutting off as the stars separated more and more. Faulkner pointed out that gravitational radiation from the orbital motion would carry away angular momentum and drive the stars together. The two effects together result in steady accretion at a rate that can be predicted from the quadrupole formula and simple Newtonian orbital dynamics, and which is in good accord with observations in a number of systems.

Binaries consisting of two white dwarfs in very tight orbits will be direct LISA sources: we won’t have to infer their radiation indirectly, but will actually be able to detect it. Some of them will also be close enough to tidally interact with one another, leading in some cases to mass transfer. Others will be relatively clean systems in which the dominant effect will be gravitational radiation reaction.

Observations during the last decade have identified a number of such systems with enough confidence to predict that LISA should see their gravitational waves. These are called verification binaries: if LISA does not see them then either the instrument is not working properly or general relativity is wrong! For a review of verification binaries, see [352].

7.1.1 Resolving the mass-inclination degeneracy

Gravitational-wave–polarization measurements can be very helpful in resolving the degeneracy that occurs in the measurement of the mass and inclination of a binary system. As is well known, astronomical observations of binaries cannot yield the total mass but only the combination m sinι, where ι is the inclination of the binary’s orbit to the line of sight. However, measurement of polarization can determine the angle ι since the polarization state depends on the binary’s inclination with the line of sight.

For instance, consider a circular binary system with total mass M at a distance D. Suppose its orbital angular momentum vector makes an angle ι with the line of sight (the standard definition of the inclination of a binary orbit). The two observed polarizations are given in the quadrupole approximation by Equation (122View Equation). We can eliminate the distance R between the stars that is implicit in the velocity v = R ω (where ω is the instantaneous angular velocity of the orbit, the derivative of the orbital phase function φ (t)) by using the Newtonian orbital dynamics equation 2 3 ω = M ∕R. Then we find

2νM 4νM h+ = -----[πM f(t)]2∕3(1 + cos2 ι) cos[2φ (t)], h × = -----[πM f(t)]2∕3cosι sin[2φ (t)], (132 ) D D
where M is the total mass of the binary and, as before, ν is the symmetric mass ratio m1m2 ∕M 2. The frequency f = ω ∕π is the gravitational wave frequency, twice the orbital frequency. Notice that, consistent with Equation (30View Equation), the masses of the stars enter these equations only in the combination 3∕5 ℳ = ν M.

It is clear that the ratio of the two polarization amplitudes determines the angle ι. In this connection it is interesting to relate the polarization to the orientation. When the binary is viewed from a point in its orbital plane, so that ι = π ∕2, then h× = 0; the radiation has pure + polarization. From the observer’s point of view, the motion of the binary stars projected onto the sky is purely linear; the two stars simply go back and forth across the line of sight. This linear projected motion results in linearly polarized waves. At the other extreme, consider viewing the system down its orbital rotation axis, where ι = 0. The stars execute a circular motion in the sky, and the polarization components h+ and h× have equal amplitude and are out of phase by π ∕2. This is circularly polarized gravitational radiation. So, when the radiation is produced in the quadrupole approximation, the polarization has a very direct relationship to the motions of the masses when projected on the observer’s sky plane. If the radiation is produced by higher multipoles it becomes more complex to make these relations, but it can be done. For example, see [336] for the case of current quadrupole radiation, which is emitted by the r-mode instability discussed in Section 7.3.4.2 below.

While a single detector is linearly polarized, it can still measure the two polarizations if the signal has a long enough duration for the detector to turn (due to the motion of the Earth) and change the polarization component it measures. Alternatively, a network of three detectors can determine the polarization and location of the source even over short observation times.

Such a measurement would lead to a potentially very interesting interplay between gravitational and electromagnetic observations, with applications in the observations of isolated neutrons stars, binary systems, etc. And would lead to synergies, for example, between the LISA and Gaia [290] missions.


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