an equilibrium piece and a small perturbation,

ρ = ρ0 (1 + •), T = T0 (1 + θ), p = ρRT = p0 (1 + ),=•+θ .

(26.42)

The gravitational potential φ in (26.18) is obtained from the Poisson equa-

tion

∆φ = 4πG (ρ ’ ρ0 ) , (26.43)

where we choose ’ρ0 as a background “neutralising” density in order to

have a proper uniform equilibrium about which to expand; the equilibrium

velocity then vanishes. The density ’ρ0 is a repulsion term and may be

regarded as a Newtonian analogue of Einstein™s cosmological constant. To

leave it out as Jeans did and jump straight to the linearised equation (26.47)

below is expedient but questionable.

The linearised equations of motion (26.17)“(26.19) and Poisson equation

(26.43) are

‚t • + ∇ · u = 0, (26.44)

ρ0 ‚t u + ∇ · P = ’ρ0 ∇φ, (26.45)

+ p0 ∇ · u + ∇ · q = 0,

3

2 p0 ‚t θ (26.46)

∆φ = 4πGρ0 •. (26.47)

The pressure tensor P is given by (26.32) and the heat ¬‚ux q by (26.34). We

take the divergence of the velocity equation (26.45) and use the continuity

equation (26.44) to eliminate ∇ · u,

ρ0 ‚t • ’ ∇∇ : P = 4πGρ2 • ,

2

(26.48)

0

where we also used the Poisson equation (26.47) to eliminate φ. We then

need to take two divergences of the linearised pressure tensor,

∇∇ : P = p0 ∆(• + θ) + µ 2‚t ∆• ’ ‚t ∆θ . (26.49)

We introduce the isothermal sound speed aT , the kinematic viscosity ν, and

the thermal di¬usivity κ, through

p0 µ · ·

a2 := , ν := , κ := =5 . (26.50)

T

ρ0 ρ0 ρ0 Cp Rρ0

2

Then, on inserting (26.49) into (26.48), we obtain

‚t • = a2 ∆(• + θ) + ν 2‚t ∆• ’ ‚t ∆θ + 5 kJ a2 • ,

2 2

(26.51)

T T

3

where the Jeans wavenumber is given by

4πGρ0 3 4πGρ0

2

kJ := = . (26.52)

a2 5 a2

S T

Continuum equations for stellar dynamics 387

Next, we take the divergence of the linearised heat ¬‚ux,

∇ · q = ’·T0 ∆θ ’ 3·T0 a’2 ‚t ∇ · u + ∆φ + a2 ∆(• + θ) , (26.53)

T

T

and insert this into the temperature equation (26.46),

‚t θ ’ 2 ‚t • ’ 5 κ a’2 4a2 ∆θ ’ 3 ‚t • ’ 5 kJ a2 • ’ a2 ∆•

2 2

= 0 , (26.54)

T T T

T

3 3 3

where again we eliminated ∇ · u using the continuity equation (26.44), and

used the de¬nition (26.50) of the thermal di¬usivity κ.

Equations (26.51) and (26.54) are the equations required to derive a dis-

persion relation for the gravitational instability. It is convenient to use the

viscous time µ/p0 as unit of time and aT µ/p0 as unit of length. Recycling

the same symbols for the dimensionless quantities turns equations (26.51)

and (26.54) into

‚t ’ ∆ ’ 2‚t ∆ ’ 5 kJ • + (‚t ’ 1)∆θ = 0,

2 2

(26.55)

3

3‚t ’ θ + ’2‚t + 10 ‚t ’ 5 kJ ’ ∆

2 2

40

3∆ • = 0. (26.56)

3

On letting •, θ ∼ exp(ikx + γt), we ¬nd the dispersion relation

k4 + 1 ’ kJ k2 ’ kJ γ

+ 2k2 γ 3 + k2 γ 2 + 2 2

3 22 22 10

5 15 3 3

k2 ’ kJ k2 = 0, (26.57)

2

2 5

+ 3 3

which may be compared to the expression obtained from Navier“Stokes by

Kato & Kumar (1960),

k4 + k2 ’ kJ γ + k2 ’

γ3 + k2 γ 2 + 2

kJ k2 = 0 ,

2

3 22 8 2 5

(26.58)

5 15 9 3 3

with the Fokker“Planck values for the viscosity and thermal di¬usivity

inserted into their result. In each case the system is marginally stable

with γ = 0 at k2 = (5/3)kJ , and is damped for larger k. This is illus-

2

trated in Fig. 26.1. For k = 0, both dispersion relations predict a growth

rate of γ(k = 0) = 5/3 kJ in dimensionless units, which with dimensions

is 4πGρ0 . This is consistent with the fact that dissipation is unimportant

at large scales, so the growth rate at k = 0 involves only the gravitational

time.

The asymptotic growth rate as k ’ ∞ is ’3/4 for Navier“Stokes and

’1/11 for our system, independent of k. (Multiply by µ/p0 to recover

dimensions.) Thus, at large wavenumbers, the modes tend to be uniformly

damped, both in our case and for Navier“Stokes. This is because at large k

the ¬‚uid behaves like a Stokes ¬‚ow, where we can ignore the inertial and

gravitational terms completely, and the balance is between the Laplacian

of the pressure and the viscosity, which have the same number of spatial

Spiegel & Thi¬eault

388

1.5

1

γ 0.5

0

0.5

0 1 2 3 4 5

k 2

Fig. 26.1. Growth rate γ as a function of wavenumber k, for Navier“Stokes (dashed

line) and the equations derived in Section 26.3 (solid). The growth rate of this mode

is real and, in both cases, damping sets in above the isothermal Jeans wavenum-

2

ber (5/3)kJ (here we have taken kJ = 1 in dimensionless units).

derivatives; hence the lack of dependence on k. For our case (the equations