damping rate considerably closer to marginality.

Coexisting with the real root associated with the instability, there is also a

pair of unconditionally damped roots. The real part of these roots is plotted

in Fig. 26.2. At k = 0 one of these roots is marginal (real part of growth

rate equal to zero) and is an isopycnal mode (constant density). However,

this mode is never destabilised and its growth rate immediately decreases

as k increases away from zero.

For small k the two damped roots are distinct and real, but they come

together at larger k and become a complex-conjugate pair with nonzero

imaginary part (not plotted), indicating oscillatory behaviour. The Navier“

Stokes case (dashed line) is seen to have heavily damped complex roots

at large wavenumber.† But for our system of equation (solid line), the

growth rate actually increases a little, looking as though it may be headed

for a Hopf bifurcation (i.e., overstability, where the real part of the growth

† In the Navier“Stokes system the complex roots come together again at very large k, but the

growth rate continues to decrease with k.

Continuum equations for stellar dynamics 389

0

0.5

1

1.5

Re γ

2

2.5

3

3.5

0 1 2 3

0.5 1.5 2.5

k2

Fig. 26.2. Real part of the two complex roots γ as a function of wavenumber k,

for Navier“Stokes (dashed line) and the equations derived in Section 26.3 (solid).

Large wavenumbers are much more strongly damped for Navier“Stokes than in the

system derived here.

rate becomes positive at nonzero imaginary part), before leveling o¬ at an

asymptotic value of the damping rate given by ’53/165 ’0.3212. Indeed,

it can be shown that there is no Hopf bifurcation for any realisable parameter

values in our equations, but the fact that the complex modes are somewhat

“destabilised” by the new terms is intriguing (this is also true to a lesser

extent for the real mode described above). This destabilisation has its source

in the k2 coe¬cient of the γ 3 term in the dispersion relation (26.57), which is

not present in Navier“Stokes, but its physical signi¬cance is not yet apparent

to us.

26.5 Conclusion

The basic approach in this as in other derivations of ¬‚uid equations from ki-

netic theory is to write the general moment equations (26.17)“(26.19). These

are the ¬‚uid equations and, to complete them, we need closure relations for

the pressure tensor and heat ¬‚ux. This is an issue astronomers are familiar

with from the study of radiative transfer. For the purpose, we could invent

a phenomenological approximation as Eddington did in radiative transfer or

Spiegel & Thi¬eault

390

we may pursue approximate solutions of the kinetic theory as Hilbert did

by expanding in the collision time. The Hilbert expansion was developed

by Chapman and Enskog in deriving the Navier“Stokes equations (Uhlen-

beck & Ford, 1963) and we have pursued that line as well following earlier

work (Chen, 2000; Chen et al., 2000, 2001) based on the relaxation model of

kinetic theory (Bhatnager et al., 1954; Welander, 1954). However, in that

latter work, as here, we depart from the Chapman“Enskog approach in an

essential way in not using results from lower orders to to simplify the results

in the current order.

To express this idea in equations, let us consider what happens in general

in such problems in the ¬rst order. Once we have expressed the one-particle

distribution function as f = f0 (1 + „ •) where „ is the (small) collision time,

we are led to an equation for • in the form

L• = Df0 + O(„ ) , (26.59)

where L is the linearisation of the collision operator. In general, L is self-

adjoint, as it is in the present study. Then Lψ ± = 0 implies ψ ± L• dv = 0

and so we must have

ψ ± [Df0 ] dv = O(„ ) . (26.60)

Since ψ ± represents the collisionally invariant quantities, the ¬‚uid equations

to the current order may serve as the solvability condition (26.60).

In the Chapman“Enskog procedure, this solvability condition is taken to

be a lower order version of the ¬‚uid equations, here the Euler equation, and

it is used to simplify the right side of (26.59). Then, the results are used in

the general ¬‚uid equations. For both of these two conditions to be satis¬ed,

we require „ to be very small indeed.

What we are doing here is to say that, to the ¬rst order, the ¬‚uid equations

themselves are a realisation of condition (26.60) and that it is redundant to

apply the same condition twice, once with O(„ ) retained and once with it

omitted, as one does in the Chapman“Enskog method. Rather, we simply

use the full condition (26.60) as a compatibility condition. It is for this

reason that, in the ¬rst order theory, we allow ourselves to di¬er from the

Chapman“Enskog results by terms O(„ 2 ). In particular, we have for the

trace of the pressure tensor

„™

Tr P = 3 p 1 ’ + O(„ 2 ) .

S (26.61)

2Cv

Continuum equations for stellar dynamics 391

This result di¬ers from the exact trace (3p) by O(„ 2 ) and this, we suggest

is allowed in a ¬rst order-theory.

We may add that in comparing the results of this approach to experiments

on ultrasound we ¬nd that they do better than the usual Navier“Stokes ver-

sion. Here we have an interesting example of a dictum of J. B. Keller: “Two

theories may have the same accuracy but di¬erent domains of validity.”

We regret that though, in honour of Douglas Gough™s birthday, we have

gone to second order in this approach (for only the relaxation model so

far), we could not ¬t the derivations into the space we were allotted in this

volume. So those results will have to be presented elsewhere. We are happy

to report that, in that next order, the trace of our pressure tensor di¬ers

from the exact trace by O(„ 3 ). For now we must be content with mentioning

that result and presenting our best wishes to Douglas on his birthday.

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27

Formation of planetary systems

DOUGLAS N. C. LIN

Department of Astronomy & Astrophysics, University of California,

Santa Cruz, CA 95064, USA