Lf1 = D ln f0 , (26.21)

˜

where f1 := f1 /f0 and

˜ ˜ ˜

Lf := ’c · ∇cf + R T ∇c · (∇c f ) . (26.22)

The right-hand side of (26.21) may be written out as

c2 c2

n 1 3

D’ =’ c · Dc + ’ D ln T + D ln ρ .

+ ln

(2πRT )3/2

2R T RT 2R T 2

(26.23)

We note that

Dc = ’∇φ ’ Du/Dt ’ c · ∇u, where D/Dt := ‚t + u · ∇ . (26.24)

The operator L maps polynomials in c to polynomials of the same degree

˜

so, given the form of (26.23), we may seek a solution f1 to (26.21) as a cubic

in c. We write

˜

f1 = a + bi ci + +cij ci cj + dijk ci cj ck , (26.25)

where a, b, c and d are functions of x and t, and symmetric in their in-

dices; repeated indices are summed. Inserting (26.25) into the left-hand

side of (26.21), we obtain

˜

Lf1 = ’3dijk ci cj ck ’ 2cij ci cj + (6R T dill ’ bi )ci + 2R T cll . (26.26)

There is no a term because it is annihilated by the Fokker“Planck collision

operator. We now equate coe¬cients of c between (26.26) and (26.23), and

¬nd

1

dijk = ’ (δij ∇xk ln T + δik ∇xj ln T + δkj ∇xi ln T ) , (26.27)

18R T

1 D ln T

cij = ’ (∇xj ui + ∇xi uj ) + δij , (26.28)

4R T Dt

1 1 Dui

bi = ’ ∇xi ln T ’ ∇xi φ + ’ ∇xi ln ρ . (26.29)

6 RT Dt

Since a is still unspeci¬ed, we may use it to satisfy the matching condi-

f1 d3 v = 0, resulting from mass conservation. Only the terms

tion (26.16)

Spiegel & Thi¬eault

384

even in c contribute, and we ¬nd

f1 d3 v = n(R T cll + a) = 0 , (26.30)

which allows us to solve for a in terms of the trace of c.

The pressure tensor and heat ¬‚ux (26.20) are then obtained from f1 by

performing straightforward Gaussian integrals, and we get

Qk = 1 pR T (5bk + 21R T dkll ) .

Pij = p δij + 2pR T cij , (26.31)

2

The a term is absent from the pressure because we used the density matching

condition (26.30). From (26.31) and (26.28), we ¬nd that the pressure tensor

can be written

D ln T

P = p I ’ 2µ E ’ µ + 2∇·u I (26.32)

3

Dt

to ¬rst order in „ , where the viscosity µ := 1 p „ , and

2

∇xj ui + ∇xi uj ’ ∇ · u δij

1 2

E ij := (26.33)

2 3

is the rate-of-strain tensor in traceless form.

From (26.31), (26.27), and (26.29), to ¬rst order in „ , the heat ¬‚ux is

1 Du

q = ’· ∇T ’ 3· T ∇ ln p + + ∇φ , (26.34)

RT Dt

where the thermal conductivity · := (5/6)p „ R.

These results di¬er from those of the usual Navier“Stokes equations for

which P = p I ’ 2µ E and q = ’· ∇T . To get some understanding of the

import of the additional terms found here we introduce the speci¬c entropy

S = Cv ln p ρ’5/3 , (26.35)

where Cv := 3R/2 is the speci¬c heat at constant volume. Since

DS D ln T

+ 2∇·u ,

™

S := = Cv (26.36)

3

Dt Dt

we ¬nd that

„™

P =p 1’ S I ’ 2µ E + O(„ 2 ), (26.37)

2Cv

q = ’· ∇T + 3(· T /p)∇ · T + O(„ 2 ), (26.38)

with T := (P ’ p I). If we put these results into S, we obtain

™

2Cv

S=’ [T : ∇u + ∇ · q] .

™ (26.39)

3p

Continuum equations for stellar dynamics 385

™

Hence S can be seen to be O(„ ) and so the additional terms in the pressure

tensor are O(„ 2 ). A similar argument may be made for the new terms in

the heat ¬‚ux. Though these terms do not appear in the conventional ¬‚uid

equations, they can be quite signi¬cant when the mean free paths are long.

If we eliminate the entropy using

1™ p 5ρ

™ ™

S= ’ (26.40)

Cv p 3ρ

we encounter the combination p(t) ’ 1 „ p(t) which are the ¬rst two terms of

2™

a Taylor series of p in „ /2. We may then write

P = p(t ’ 1 „ ) I + µ ∇ · u I ’ 2µ E + O(„ 2 ) .

10

(26.41)

2 3

We see that our procedure has taken account of the physical fact that the

medium senses what particles were doing one collision time prior to the

present time but is not yet aware of what they are doing at the present.

26.4 The Jeans instability

As an application of the equations of motion (26.17)“(26.19), we will use

them together with the pressure tensor and heat ¬‚ux derived in Section 26.3

to examine the Jeans criterion for gravitational instability. This instabil-

ity, describing the gravitational collapse of a homogeneous medium, was

¬rst investigated by Jeans (1929). He found that perturbations above a

critical wavelength (the Jeans length) were unstable to gravitational col-

lapse, but that shorter wavelengths were una¬ected due to the large-scale

nature of the gravitational force. The Jeans length is the ratio of the adi-

abatic sound speed aS to the gravitational frequency 4πGρ. Pacholczyk

& Stod´lkiewicz (1959) and Kato & Kumar (1960) investigated the e¬ect

o

of viscosity and thermal conductivity on the instability and found that the

collapse occurs above a critical wavelength given by the ratio of the isother-

mal sound speed aT to the gravitational frequency. That critical wavelength

is slightly larger than in the ideal case, which involves the adiabatic sound

speed. Their interpretation for the increase in critical wavelength is that

temperature gradients are adverse to the collapse, and a nonzero thermal

conductivity allows the smoothing out of these gradients through very slow

displacements of the medium; the mode is thus isothermal.

We will now investigate how the gravitational instability occurs in our

set of equations. We take as our equilibrium a medium at rest and with

uniform density ρ0 and temperature T0 . We expand each ¬‚uid variable into

Spiegel & Thi¬eault

386