11.5. Angular Momentum Transport

It is easy to show that to minimize its total energy, an isolated galaxy tends to transfer its angular momentum outwards (cf. Lynden-Bell & Kalnajs 1972, hereafter LBK72). In fact, for a given total angular momentum, the least energy state corresponds to solid body rotation. But galaxies are in general in differential rotation, with their angular velocity decreasing with radius. The exchange will tend to give angular momentum to the outer particles with the least , and with the higher specific angular momentum.

LBK72 remarked that only trailing waves can transfer angular momentum outwards, while leading waves have the opposite effect; this explains why trailing waves are preponderant in observed galaxies. Since angular momentum transfer, mediated by non-axisymmetric instabilities, is the motor of secular evolution of galaxies, and of the formation of bars and resonant rings, it is fundamental to summarize here the essential phenomena.

LBK72 stated that, for a steady wave, stars can exchange angular momentum at resonances only: they emit angular momentum at inner Lindblad resonance, while they absorb angular momentum at corotation and outer Lindblad resonance. LBK72 also showed that, while stars do not gain or lose angular momentum on average away from resonances, they are able however to transport angular momentum as lorries in their orbiting around the galactic center. When they are at large radii, they gain angular momentum, while they lose some at small radii: even if the net balance is zero, they carry angular momentum radially, and the sense of this lorry transport is opposite to that of the spiral wave. This phenomenon can then damp the wave, if the amplitude is strong enough. LBK72 noticed that this damping phenomenon became negligible for small wavelengths, i.e. when kr 1.

In a recent investigation, Zhang (1996) introduces an equivalent ``dissipation'' for the stellar component, corresponding to small-angle scatterings of neighboring stars in the spiral arms. This process transforms ordered motions into disordered ones (resulting in increased velocity dispersion, or large epicycle amplitudes), and when collective effects are taken into account, secular modifications of the stellar orbits can result: they can lose energy and angular momentum inside corotation. If this dissipation is taken into account, together with the phase-shift between the stellar density and the potential of the spiral wave, gravity torques are exerted by the wave on the stars, and stars gain or lose angular momentum, even away from resonances.

This angular momentum transfer between the wave and the stars helps the wave to grow in the linear regime, but soon due to the collective dissipation process, it will contribute to damping and saturation. The wave is not quite steady. Zhang (1996) has shown that the stellar density azimuthal profile steepens with time, indicating the presence of large-scale gravitational shocks. The result of the secular evolution is a global redistribution of mass: stars inside corotation lose angular momentum and move to smaller radii, which steepens the radial density profile in the disk. This has been observed during many N-body simulations of a stellar disk that has obtained a long-lived spiral pattern (e.g. Donner & Thomasson 1994).

Expressions for the Torque and Phase-shift

It is obvious that the torque exerted by the wave on particles is a second order term, since the tangential force ðV₁ / ð is first order, and has a net effect only on the non-axisymmetric term of the surface density ₁(r, ). The torque T(r) applied by the wave on the disk matter in an annular ring of width dr, can be expressed by:

T (r) = r dr -₁(r, ) ðV₁ / ð d 9

(Zhang 1996). In this formula, it is easy to see that the torque vanishes if the density and potential spiral perturbations are in phase. But there must be a phase-shift in the general case, according to the Poisson equation. The potential is non-local, and is influenced by the distant spiral arms. So the sign and amplitude of the phase-shift depends on the radial density law of the perturbation. It has been shown by Kalnajs (1971) that the peculiar radial law in r^-3/2 for an infinite spiral perturbation provides exactly no phase-shift. The phase-shift is such that the spiral density leads the potential if the radial falloff is slower than r^-3/2, and the reverse if it is steeper.

Now in a self-consistent disk, the phase-shift given by the Poisson equation must agree with that given by the equations of motion. Through the computation of linear periodic orbits in the rotating frame, Zhang (1996) has found that the forcing consists of two terms in quadrature, and that the phase-shift of the orbit orientation with respect to the forcing potential has the expression:

tan (m ) - 2 / [( - _p) kr] 10

This shows that in the WKBJ approximation, the phase-shift is negligible (kr 1), and that it changes sign at corotation. The fact that the torque is a non-linear effect, and vanishing in the WKBJ approximation, may explain why it was neglected before, but it has a quite important effect in galactic conditions. Inside corotation, we expect that the density leads the potential, and the contrary outside corotation.

The amplitude of the density-potential phase-shift can be estimated through N-body simulations: Figure 67 shows the results of a Fourier analysis of the m = 2 component, in a purely stellar simulation (2-D polar grid particle mesh with 10⁵ particles). In this simulation, initial parameters were chosen such as to stabilize the stellar disk with respect to bar formation through the presence of an analytical bulge component of mass equal to the mass of the self-gravitating stellar disk. A bar eventually developed, but was delayed until 2 x 10⁹ yrs, i.e. 20 dynamical times. In the mean time, a spiral wave developed, and remained for several dynamical times. Its power spectrum revealed a well-defined pattern speed, at least up to corotation ( 7 kpc). We can see in Figure 67 that the phase-shift between stellar density and potential is quite high, up to 28° inside corotation, where it is most meaningful.

Figure 67. Results of the Fourier analysis of the stellar density (crosses) and potential (full line) in a purely stellar N-body simulation, while a spiral structure rotates in the disk. Plotted here are the phases of the pattern at each radius. The density leads the potential almost everywhere inside corotation. The perturbation is weaker outside corotation.

The mechanism of the angular momentum transfer for the gas is the same as that previously described: the gas settles in a spiral structure which is not in phase with the potential Figure 68). Gravity torques are exerted by the wave on the gas. The dissipation here is of course different, since the gas radiates away its energy. The gas component is then maintained cool and responsive to new gravitational instabilities. This is the source of more drastic secular evolution, with the possibility of the whole gas component inflowing towards the center.

Figure 68. Same as Figure 67 for an N-body simulation taking into account gas and stars, and where a bar develops. The phases of the gas pattern are indicated by asterisks, while crosses indicate the stars, and the full line the potential. Now the potential is in phase with the stellar density in the bar, and leads the density in the spiral outside corotation.