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

The r-modes 77Table 7.1. Gravitational radiation and viscous timescales, in seconds. Negative valuesindicate instability, i.e. a growing rather than damping mode.l m τ gw (s) p gw τ bv (s) p bv τ sv (s)2 2 −20.83 5.93 9.3 × 10 10 1.77 2.25 × 10 83 3 −316.1 7.98 1.89 × 10 10 1.83 3.53 × 10 7the approximate maximum speed √ πG ¯ρ and the temperature in terms of 10 9 K,then it can be shown that [30]1τ = 1( ) pgwτ 1ms gw +1Pτbv(1msP( ) 610 9 K) pbv+ 1T( ) T 2τ sv 10 9 , (7.12)Kwhere the scaling parameters ˜τ sv , ˜τ bv , ˜τ gw and the exponents p gw and p bv haveto be calculated numerically. Some representative values relevant to the r-modeswith 2 l 6 are in table 7.1 [30].The physics of the viscosity is interesting. It is clear from equation (7.12) thatgravitational radiation becomes a stronger and stronger destabilizing influenceas the angular velocity of a star increases, but the viscosity is much morecomplicated. There are two contributions: shear and bulk. Shear viscosity comesmainly from electrons scattering off protons and other electrons. This effect fallswith increasing temperature, just as does viscosity of everyday materials. So acold, slowly rotating star will not have the instability, where a hotter star might.However, at high temperatures, bulk viscosity becomes dominant. This effectarises in neutron stars from nuclear physics. Neutron-star matter always containssome protons and electrons. When it is compressed, some of these react to formneutrons, emitting a neutrino. When it is expanded, some of the neutrons betadecayto protons and electrons, again emitting a neutrino. The emitted neutrinois not trapped in the star; within a short time, of the order of a second or less, itescapes. This irreversible loss of energy each time the star is compressed createsa bulk viscosity. Now, bulk viscosity acts only due to the density perturbation,which is small in r-modes. So the effect of bulk viscosity only dominates at veryhigh temperatures.The balance of the viscous and gravitational effects is illustrated in figure 7.1[30]. This is indicative, but not definitive: much more work is needed on thephysics of viscosity and the structure of the modes at large values of (small P).7.2.2 Nonlinear evolution of the starOur description so far is only a linear approximation. To understand the fullevolution of the r-modes we have to treat the nonlinear hydrodynamical effects