Astron. Astrophys. 347, 821-840 (1999)

Footnotes

1. In plasma physics an instability of the Jeans type is known as the negative-mass instability of a relativistic charged particle ring or the diocotron instability of a nonrelativistic ring that caused azimuthal clumping of beams in synchrotrons, betatrons, and mirror machines (Landau & Neil 1966; Nocentini et al. 1968; Davidson 1992).

2. Common dynamical processes act in the stellar disks of flat galaxies and in a planetary rings system of mutual-gravitating particles (Tremaine 1989).

3. Jog (1996) obtained the criterion for local stability against gravity perturbations in gravitationally coupled stars and gas in a galactic disk by treating the stars and gas as two isothermal fluids. Again, the stability of a disk only with respect to axisymmetric perturbations has been studied.

4. On the other side, the modern observational data convincely indicate the presence of a strong perturbative mechanism disturbing the stellar orbits; see Binney & Tremaine (1987, p. 470) as a review of the problem. The majority of the experts in the field is yield to the opinion that this dynamical relaxation may be explained naturally by collective interactions of stars with unstable density waves.

5. The "sliding brick" technique of Wisdom & Tremaine (1988), Toomre (1990), Toomre & Kalnajs (1991), and Salo (1995) has been used in the past to simulate transport properties of simple fluids under the action of a strong shearing force (Lees & Edwards 1972; Evans & Morriss 1984).

6. In conjunction with the last result, it should be emphasized again that in view of the Lin-Shu type asymptotic theory the analysis presented here provides only an approximate estimation of the local stability criterion.

7. In turn, Toomre (1969) has shown that density waves of the kind originally proposed by Lin and Shu (Lin & Shu 1966; Lin et al. 1969; Shu 1970) cannot be stationary, and a wave theory can explain the phenomena of spiral patterns only if some instability exists which could cause small perturbations to grow to observable amplitudes.

8. One has to recognize, however, that correct N-body simulation of resonant effects is a very difficult problem in stellar dynamics because of lack of fine resolution in the phase space. Perhaps, the better way to study the resonant wave-star interaction involved in the support or damping of the modes is to solve numerically the collisionless Boltzmann equation similar to that by Nishida et al. (1984).

9. According to Polyachenko (1989) the marginal stability condition for Jeans perturbations of an arbitrary degree of axial asymmetry has been available since 1965 (Goldreich & Lynden-Bell 1965), though in a slightly masked form. See Polyachenko & Polyachenko (1997) for a detailed discussion of the problem.

10. Note that the equivalence of the particle orbit theory and the more rigorous Boltzmann kinetic equation approach in the absence of collisions has been demonstrated, and in the astronomical literature is referred to as Jeans' theorem; see Longmire (1963) and Chandrasekhar (1965) for explanations. Similarly, the particle orbit theory reflects both single-particle dynamics and the overall continuity of the system of mutual-gravitating particles. It can be applied only to strongly rarefied particulate systems with practically uncorrelated unperturbed particle motion. The great advantage of using this purely Lagrangian formulation lies in the fact that the equation of continuity and the Poisson equation can be simplified in the asymptotic limit of moderately tightly-wound spiral waves. The analytic calculations presented here are carried out in a new and insightful way, and it is hoped that many workers in the field will find it useful.

11. Since we work within the two-dimensional disk model, the wave vector is perpendicular to the rotation axis, that is, k is given by .

12. To obtain Eq. (2) by using the dispersion relation (27), one first finds the critical wavenumber from the relation . Then this is substituted into the dispersion relation and from the condition the critical velocity dispersion is found.

13. In a plasma, it has already been known that the rate of relaxation toward equilibrium can be greatly enhanced by collective processes (Kulsrud 1972; Alexandrov et al. 1984, p. 408; Krall & Trivelpiece 1986, p. 512).

© European Southern Observatory (ESO) 1999

Online publication: June 6, 1999
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