(ed.). Gravitational waves (IOP, 2001)(422s).

76 Source calculationswhen looking at one wheel from the side. However, in this case the pattern rotateswith the angular velocity 2/3 of equation (7.10). Since the pattern of line-ofsightmomenta repeats itself every half rotation period, the gravitational wavesare circularly polarized with frequency 4/3. Seen along the x-axis, the wheelalong the x-axis contributes nothing, but the other wheel contributes fully, so theradiation amplitude in this direction is half that going out the rotation axis. Seenalong a line at 45 ◦ to the x-axis, the line-of-sight momenta of the wheels on thefront part of the star cancel those at the back, so there is no radiation. Thus,along the equator there is a characteristic series of maxima and zeros, leadingto a standard m = 2 radiation pattern. This pattern also rotates around the star,but the radiation in the equator remains linearly polarized because there is onlythe ⊗ component, not the ⊕. Again, the radiation frequency is twice the patternspeed because the radiation goes through a complete cycle in half a wave rotationperiod.This discussion cannot go into the depth required to understand the r-modesfully. There are many issues of principle: what happens beyond linear order in; what happens if the star is described in relativity and not Newtonian gravity;what is the relation between r-modes and the so-called g-modes that can havesimilar frequencies; what happens when the amplitude grows large enough thatthe evolution is nonlinear; what is the effect of magnetic fields on the evolution ofthe instability? The literature on r-modes is developing rapidly. We have includedreferences where some of the most basic issues are discussed [17, 18, 29–31], butthe interested student should consult the current literature carefully.7.2.1 Linear growth of the r-modesWe have seen how the r-mode becomes unstable when coupled to gravitationalradiation, and now we turn to the practical question: is it important? This willdepend on the balance between the growth rate of the mode due to relativisticeffects and the damping due to viscosity.When coupled to gravitational radiation and viscosity, the mode has acomplex frequency. If we define I(σ ) := 1/τ, then τ is the characteristicdamping time. When radiation and viscosity are treated as small effects, theircontributions to the eigenfrequencies add, so we have that the total damping isgiven by1τ() = 1 + 1 ,τ GR τ v1τ v= 1 τ s+ 1 τ b, (7.11)where 1/τ GR ,1/τ v are the contributions due to gravitational radiation emissionand viscosity, and where the latter has been further divided between shearviscosity ( 1 τ s) and bulk viscosity ( 1 τ b).If we consider a ‘typical’ neutron star with a polytropic equation of statep = kρ 2 (for which k has been chosen so that a 1.5M ⊙ model has a radiusR = 12.47 km), and if we express the angular velocity in terms of the scale for