i.e., “How do turbulent ¬‚uids with angular momentum like to rotate?”

Nevertheless, I never expected to know the internal rotation of the Sun

within my lifetime and I have immense admiration for Douglas “ and the

helioseismic fraternity “ for having persisted in analysing solar pulsations

until that became possible. Such is the real meat of good science.

References

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26

Continuum equations for stellar dynamics

EDWARD A. SPIEGEL

Department of Astronomy

Columbia University, New York, NY 10027, USA

JEAN-LUC THIFFEAULT

Department of Applied Physics and Applied Mathematics

Columbia University, New York, NY 10027, USA

The description of a stellar system as a continuous ¬‚uid represents a con-

venient ¬rst approximation to stellar dynamics, and its derivation from the

kinetic theory is standard. The challenge lies in providing adequate clo-

sure approximations for the higher-order moments of the phase-space density

function that appear in the ¬‚uid dynamical equations. Such closure approx-

imations may be found using representations of the phase-space density as

embodied in the kinetic theory. In the classic approach of Chapman and

Enskog, one is led to the Navier“Stokes equations, which are known to be

inaccurate when the mean free paths of particles are long, as they are in

many stellar systems. To improve on the ¬‚uid description, we derive here

a modi¬ed closure relation using a Fokker“Planck collision operator. To

illustrate the nature of our approximation, we apply it to the study of gravi-

tational instability. The instability proceeds in a qualitative manner as given

by the Navier“Stokes equations but, in our description, the damped modes

are considerably closer to marginality, especially at small scales.

26.1 A kinetic equation

If we have a system of N stars, with N very large, and wish to study its

large-scale dynamics, we have to choose the level of detail we can pro¬tably

treat. Even if we could know the positions and velocities of all N stars

for all times, we would be mainly interested in the global properties that

are implied by this information. For just such reasons, many investigators

prefer to ¬nd an approach that leads directly to a macroscopic description

of the dynamics. As Ogorodnikov (1965) has put it, “In order to exhibit

more clearly the kinematics of highly rare¬ed media, and of stellar systems

in particular, it is useful to make a comparison with the motion of a ¬‚uid.”

However, traditional methods for deriving ¬‚uid equations are e¬ective only

377

Spiegel & Thi¬eault

378

for media in which the mean free paths of constituent particles are very short

compared to all macroscopic scales of interest. This condition is not met

in many stellar systems and plasmas and so we here describe an approach

that is e¬ective for deriving ¬‚uid equations for rare¬ed media such as stellar

systems.

The ¬rst problem we must face is to decide what the kinetic description

of a stellar dynamical system ought to be. Since an N -body description is

not what we want to work with, even if we could, since that approach would

have us computing the complicated trajectory of the stellar system through a

phase space with large dimension. So we go straight to the description of the

system in the six-dimensional phase space whose coordinates are the spatial

coordinates (xi ) and the velocities (v i ) of the N stars, where i = 1, 2, 3.

A plot of the locations of each of the N stars in the six-dimensional phase

space at some given time, would reveal a swarm of points whose detailed

description would also be too complicated for us, at least in a ¬rst look at

the problem. So instead, we concentrate on an ensemble mean of such a

description and seek an equation for the density distribution of this mean.

That equation, on which we base this work, is an evolution or continuity

equation for the probability density in the six-dimensional phase space.

Since we treat the stars as points, the true density F in the six-dimensional

phase space is a summation of delta functions at suitable locations. This

density is advected by a six-dimensional phase velocity that Hamilton™s equa-

tions tell us is solenoidal. Hence the total time derivative of the density is

DF = 0 , (26.1)

where the comoving derivative in phase space is de¬ned as

D := ‚t + v · ∇x + a · ∇v (26.2)

and the subscripts x and v on the gradient symbols indicate that they

are gradients with respect to position and velocity respectively. As usual,

™

v = x where the dot means total time derivative and the quantity a is the

gravitational acceleration per unit mass; we assume that all the stars in the

system have the same mass, m. The gravitational acceleration is given by

the gradient of the gravitational potential per unit mass, which is a solution

of Poisson™s equation,

F d3 v .

∆¦ = 4π G (26.3)

Let f (x, v, t) be the ensemble mean of F . Then the total density will be

ˆ ˆ ˆ

F = f + f where f represents the ¬‚uctuations about the ensemble mean; f

Continuum equations for stellar dynamics 379

will involve the same summation of delta functions as does F plus a smooth

background distribution with negative mass density arranged so that the

ˆ

ensemble average of f is zero. We similarly split the gravitational potential

ˆ

¦ into an ensemble mean part φ plus a ¬‚uctuating part φ. Then, if we take

the ensemble average of (26.1), we obtain

ˆ ˆ

‚t f + v · ∇xf + (∇xφ) · ∇vf = ’ (∇xφ) · ∇vf , (26.4)

where

f d3 v .

∆φ = 4πG (26.5)

The terms on the left hand side of (26.4) describe how the mean phase

density, or distribution function, f , streams through the single-particle phase

space. The right side represents the mean in¬‚uence on the evolution of

f exerted by the average of the ¬‚uctuation-interaction term. The latter

represents the self-interactions of the system caused by ¬‚uctuating e¬ects

and may be thought of as representing the in¬‚uence of collective modes

(such as waves or quasiparticles) on the motions of the individual particles.

It is typical that the chance of close approaches of two stars in many stellar

systems is small. Because of this, one commonly made approximation is to

neglect the right side of (26.4) completely and so work with what is called

the collisionless Boltzmann (or Vlasov) equation. More reasonably perhaps,

one may try to give an expression for the way the self-interaction term a¬ects

the ¬‚ow of the phase density through the phase space.

As in Boltzmann theory, we shall suppose that the right side of (26.4)

may be expressed as a functional of f itself so that the kinetic equation is

deterministic. That is, we assume that the kinetic equation takes the form

‚t f + v · ∇xf + (∇xφ) · ∇vf = C[f ] , (26.6)

where C[·] may be called a collision term in keeping with the terminology

of kinetic theory even though it does not arise from direct binary collisions.

This approach may be acceptable because it appears that the ¬‚uid descrip-

tion that we seek is not very sensitive to the details of the right side of

(26.4). On the other hand, we must admit that this hope is founded on a

very limited range of trial forms since the job of deriving the consequences

of each form is laborious. Our aim here is to adopt one standard form for

the collision term and use it to go on to a coarser description of the stellar