¬elds in a uniformly rotating radiative zone. In principle, the motions imply

a deviation from the strict hydrostatic equation (6.1). From the inertia of

the constructed ¬‚ow, one can compute the deviations of the pressure from

the Archimedean value, which for consistency must be small. Note the order

of approximation: hydrostatic equilibrium ’ thermal imbalance ’ velocity

¬eld ’ dynamical pressure.

Mestel

78

In the sub-adiabatic, radiative envelope of a Cowling-type model, the lo-

cal polytropic index n > 1/(γ ’ 1), yielding A(Ψ) < 0. By (6.9), since

is negligible, the ¬‚ow across a level surface is therefore opposite to ef-

fective gravity when the net local heat supply from radiative transfer is

positive: in a convectively stable domain, with the speci¬c entropy increas-

ing outwards, energy has to be supplied for gas to ¬‚ow against gravity. The

second bracket in (6.10) is positive at the poles and negative at the equator,

and over the bulk of a uniformly rotating envelope, the ˜von Zeipel factor™

(1 ’ „¦2 /2πGρ) ≈ 1. Thus to ¬rst order in the centrifugal perturbation, the

theory yields a quadrupolar-type ¬‚ow, upwards at the poles and downwards

at the equator. Note that ρ cancels out, with the predicted vertical velocity

staying ¬nite as the density becomes small. However, in a rapid rotator,

„¦2 /2πGρ will reach unity before photospheric densities are reached, so that

the theory as developed so far appears to predict a break-up of the circula-

¨

tion into two zones in the low-density surface regions (Gratton 1945; Opik

1951; Mestel 1999).

Even among the class of conservative centrifugal ¬elds, with „¦ = „¦( ),

uniform rotation is special, for it is the the only one that allows ∇2 V to

be constant over a level surface, so that this term makes no contribution

to the circulation velocity. In general, one writes ∇ · F as the sum of its

mean value over a level surface and the variable part (∇ · F ) , responsible

for driving the circulation (e.g. Mestel, p. 465, in Aller & Mclaughlin 1965).

One then ¬nds that there is a term ∝ (∇2 V ) and independent of ρ, yielding

a ¬rst order vertical velocity term (¯/ρ), where ρ = 3M/4πR3 (Baker &

ρ ¯

Kippenhahn 1959).

To sum up the essentials of the theory so far: if » < 1 is a parameter

measuring the ratio of the perturbing (conservative) centrifugal force to

gravity, then the velocities generated over the bulk of the radiative zone are

»(L/M g), yielding a circulation time of the order of the Kelvin-Helmholtz

contraction time increased by the factor 1/». In the low density surface

regions, signi¬cantly higher velocities are predicted. Near the domain of

overshoot from a contiguous convective zone, the theory yields a vertical

velocity becoming large like 1/[n ’ 1/(γ ’ 1)] and a horizontal velocity like

its square. The prediction of regions with singular velocities is a sign that

other contributions to meridional equilibrium “ inertial, viscous, magnetic “

though perhaps negligible over the bulk of the zone, are locally important.

The restriction to conservative centrifugal ¬elds can and indeed should be

relaxed. The papers by Sweet (1950) and by Sweet and Roy (1953) gave a

perturbation procedure for the construction of models of stars and of the

associated circulation velocity ¬elds, under a general prescribed perturbing

Stellar rotation: a historical survey 79

force ¬eld »f per unit mass, where again » is a convenient parameter. The

hydrostatic condition (6.1) becomes

’∇p + ρ∇φ + ρ»f = 0. (6.11)

One writes for φ, and analogously for p, ρ, T ,

φ = φ0 (r) + »φ1 (r, θ) + ....... = φ0 (r) + »[φ11 (r) + φ12 (r, θ)] + ...., (6.12)

π

where φ11 (r) = 0 φ1 (r, θ) sin θdθ, and φ12 (r, θ) is the part that vanishes on

averaging over a sphere. There are eight ¬rst-order quantities to determine.

The r-component of (6.11) imposes two conditions, the θ-component imposes

one, and the equation of state (6.3) and Poisson™s equation (6.2) impose two

each, making seven in all. Imposition of local radiative equilibrium on a

star with a prescribed perturbing force would over-determine the system by

requiring the eight quantities to satisfy nine conditions. It is the vector

character of the equation of support which ensures that in a non-spherically

symmetric system, local thermal balance in general requires energy transport

by a circulation. Equally, as discussed originally by Schwarzschild and later

by Roxburgh and by Clement, there exist circulation-free models, with the

„¦(r, θ)-¬eld ¬xed so as to yield (φ, p, ρ, T ) distributions satisfying radiative

equilibrium to ¬rst order.

The curl of (6.11) yields the ¬rst order equation

‚ 4πGρ0 ‚φ12 4πG ‚ ‚fr

∇2 φ12 ’ (rρ0 fθ ) ’ ρ0

= . (6.13)

‚θ (’φ0 ) ‚θ φ0 ‚r ‚θ

Once this is solved, e.g. by expansion in Legendre functions, then ρ12 , p12

and T12 can be found; substitution into the steady-state form of (6.8) then

yields the ¬rst order velocity components

cv ρ0 T0 d{log(T0 /ργ’1 )}/dr (vr )1 = »(ρ ’ ∇ · F )12 , (6.14)

0

with (vθ )1 ¬xed by continuity. The method can in principle be continued to

higher order in ». Note that the mean of (ρ ’ ∇ · F ) over a sphere vanishes

only to ¬rst order, for e.g. convection of ¬rst-order perturbations to the

thermal ¬eld by the ¬rst-order velocity ¬eld contributes second-order terms

to the energy balance (Sweet & Roy 1953).

The structure of the predicted circulation ¬elds is very sensitive to vari-

ations in the assumed perturbing ¬eld. Of particular interest is the case of

a non-uniform rotation ¬eld „¦(r), which yields vr = p(r)P2 (cos θ), with the

deviation from the Eddington-Sweet case determined by the derivatives of

„¦. Even if µ(r) ≡ „¦2 r/|g| 1 everywhere, the basic assumption of the

Eddington-Sweet approach “ that the star adjusts its (p, ρ, φ, T ) ¬eld to

Mestel

80

satisfy hydrostatic equilibrium “ will break down locally if the scale of vari-

ation d of „¦ is too small a fraction of r. Multiply equation (6.1) by ρ and

take the curl:

∇ρ — g + ∇ρ — „¦2 + ρ∇ — („¦2 ) = 0. (6.15)

If indeed µ 1 but also d r locally “ e.g. in the thin domain between

a rapidly rotating stellar core and a slowly rotating envelope “ then prima

facie, the dominant terms reduce (6.15) to

‚ log ρ/‚θ ≈ ’[r 2 („¦2 ) /g] sin θ cos θ, (6.16)

yielding

‚ log ρ/‚r ≈ ‚(log ρ(r, 0))/‚r ’ sin2 θ r 2 („¦2 ) /2g . (6.17)

Thus over at least part of the domain of rapidly varying „¦, the last term

in (6.17) will be of order „¦2 r 2 /gd2 , while ’‚ log ρ(r, 0)/‚r is of the order of

the unperturbed inverse scale-height 1/l = |ρ0 /ρ0 |. To avoid an unstable,

outwardly increasing density ¬eld, it appears that d cannot be less than dc

given by

(dc /l)2 µ(r/l) (6.18)

In fact, the largest term dropped in the reduction of (6.15) to (6.16) is

„¦2 r sin θ(‚ρ/‚r), and this will not remain small; but as long as ‚ρ/‚r < 0

this term will increase ‚ log ρ/‚θ still further; and if ‚ρ/‚r > 0, we in

any case have instability. Thus we expect that a local „¦-gradient close to

„¦/dc would in fact be rapidly reduced by spontaneous dynamically-driven

|„¦/„¦ |

motions. However, a rotating star with µ < 1 and d dc can

prima facie be kept in hydrostatic equilibrium by appropriate variations in

the (p, ρ, T ) ¬elds.

If also d r, then the local thermally-driven circulation speeds will be

large compared with the standard estimates (Sakurai 1975, Zahn 1992).

From the equation of state (6.3) (with radiation pressure neglected), the

density perturbation ρ12 (r 2 ρ0 /g0 )(„¦2 ) requires a corresponding temper-

ature perturbation T12 (T0 /ρ0 )ρ12 . The dominant term in ’∇ · F is then

(e.g. Mestel, p. 474, in Aller & Mclaughlin 1965) ’(L/4πr 2 T0 )‚ 2 T12 /‚r 2 ,

(L/4πρg2 )|(„¦2 ) | (L/M g)(„¦2 r/g)(¯/ρ)(r/d)3 ;

yielding local values vr ρ

and by continuity, vθ ∝ |(„¦ 2 ) | ∝ (r/d)4 . These comparatively high speeds

would be an important part of the input into the equation determining the

actual „¦-¬eld (cf. Section6.4).

Stellar rotation: a historical survey 81

6.2 Comparison with geophysical theory

Before considering further the astrophysical problem, it is instructive to con-

trast the above treatment with that of meteorologists studying the dynamics

of the gas in the earth™s atmosphere (e.g. Pedlosky 1982, chapters 1 and 2).

In a frame rotating with the angular velocity „¦c of the earth™s solid crust,

the ˜relative ¬‚uid velocities™ u, of order U , satisfy

ρ (du/dt + 2„¦c — u) = ’∇p + ρ∇¦ + F, (6.19)

where F is the viscous force density, ¦ = φ + |„¦c — r|2 /2 is again the sum of

the gravitational and centrifugal potentials, and the acceleration is the sum

of the relative and Coriolis accelerations. Because of the large scales L of the

relative motions, the Rossby number U/2„¦c L is small “ the accelerations of

the winds are small compared with the Coriolis acceleration, so in this sense