0 = dµ/dt ≡ ‚µ/‚t + (v „¦ + v µ ) · ∇µ. (6.25)

Equation (6.25) shows clearly the non-linear feed-back of the µ-distribution,

Stellar rotation: a historical survey 85

set up by the circulation, on the circulation velocity itself. A state of ˜steady

mixing™ has ‚µ/‚t uniform over the whole mixing zone, but non-zero spatial

µ-gradients, determined in the simplest model just by the travel times from

the nuclear-processing convective core to the di¬erent points.

The ¬rst problem is to determine if any plausible parameter range will

allow steady mixing, with µ over the whole star increasing in the time „nucl ,

implying from (6.25) a relative spatial variation of µ of order

∆µ/µ r/v„nucl . (6.26)

Near the core, we anticipate that vµ will oppose the E-V-S velocity v „¦ . We

are looking for conditions under which the circulation can continue, with v

„KH /µ„nucl ≡

less than but still of order v „¦ , so that ∆µ/µ r/v „¦ „nucl

·/µ. In order that |v µ | be indeed below v „¦ , we can demand that the µ-

perturbation be less than the centrifugal: ∆µ/µ < µ, yielding the tentative,

more stringent criterion

(„KH /„nucl )1/2 = · 1/2

µ 1/30, (6.27)

replacing the earlier estimate (6.24). This turns out in fact to be remarkably

close to the result of the detailed treatment (Mestel 1953), which yields

1/30 for the minimum required value for µ, computed at the core/envelope

interface rc .

In itself, the criterion (6.27) is not prohibitive “ the local ratio of centrifu-

gal and gravitational accelerations is certainly small enough for perturba-

tion theory to be applicable. But with „¦ uniform, „¦2 r 3 /GM (r) increases

outwards, in a Cowling model star by about 30, so that the centrifugal pa-

rameter reaches the unacceptably high value of unity at the stellar surface.

We have introduced a weak magnetic ¬eld to o¬set the advection of angu-

lar momentum, and in fact to keep the rotation uniform, so ensuring that

the simple E-V-S contribution v „¦ to the velocity is applicable. The same

constraint on the rotation ¬eld then shows that one cannot overcome the

potential choking by the µ-currents, without simultaneously running into

trouble at the surface. Even more stringent conditions hold in an evolved

star with a burnt-out core and a shell energy source.

It appears that the steady, E-V-S currents that would be set up in a

chemically homogeneous Cowling model star in rapid uniform rotation are

prevented from linking the nuclear-processing convective core with the ra-

diative envelope by the choking e¬ect of the µ-currents. The simplest sug-

gestion is that the star evolves with the convective core steadily increasing in

µ while the envelope retains its initial composition; the E-V-S circulation is

pictured as ¬‚owing through the envelope, but is de¬‚ected horizontally at the

Mestel

86

µ-barrier separating core and envelope. Since the µ-gradient at the barrier

will not in fact be in¬nite, the circulation may still make some penetration

(cf. Huppert & Spiegel 1977). This in turn may lead to the circulation™s

killing itself o¬ through the spread of ˜creeping paralysis™. A µ0 (r)-¬eld with

a negative gradient is steadily set up, extending from the core through the

whole radiative envelope. A slight distortion of the surfaces of constant µ by

the P2 E-V-S ¬‚ow then yields vµ = ’v„¦ . It is not clear how long it would

take for this state to be reached.

As was noted already in 1953, both thermally-determined processes “ the

rotational currents and the µ-choking e¬ect “ are very delicate. The theory

as outlined assumes that the radiative envelope does not su¬er any local

dynamically-driven mixing, acting to smooth out the distribution of matter

set up by the currents (cf. below).

6.4 The angular momentum distribution in a radiative zone

Discussions of the actual rotation law achieved bifurcate sharply, depending

on whether or not magnetic e¬ects are supposed signi¬cant.

6.4.1 Magnetic radiative zones

Already in Mestel (1953), and in many subsequent papers (e.g. Mestel et al.

1988), it is emphasized that the Alfv`n speed along even a very weak poloidal

e

magnetic ¬eld B p can easily exceed the slow circulation speed: hence there

exist steady states in which a very small extra toroidal component B t , main-

tained by a poloidal current density j p = (c/4π)(∇ — B t ), yields a torque

density B p · ∇( Bφ )/4π that can easily o¬set the advection of angular mo-

mentum, leaving the rotation with a correspondingly very small deviation

from Ferraro™s law of isorotation. Note the contrast with Section 6.2, where

the φ-component of the Coriolis acceleration is balanced by a toroidal ther-

mal pressure gradient, whereas a magnetic ¬eld that is axisymmetric can

nevertheless exert a toroidal force through the tension along the twisted

¬eld lines.

In principle, the postulated magnetic ¬eld will contribute also to hydro-

static support. However, even the strongest observed stellar magnetic ¬elds

“ such as the 34,000 G ¬eld of Babcock™s star HD215441 “ when supposed

to increase inwards at a plausible (or even implausible) rate, have a mean

energy density far less than that of the rotational kinetic energy. Because

the circulation speeds v „¦ are normally so slow, it is very easy to satisfy

Stellar rotation: a historical survey 87

simultaneously the two inequalities

ρ„¦2 r 2 /2 B 2 /8π ρ(v „¦ )2 /2. (6.28)

In a simple, axisymmetric system, over the bulk of the star, the magnetic

torque can e¬ectively dominate the ˜toroidal dynamics™ in the time available,

without there being any signi¬cant modi¬cation of the ˜poloidal dynamics™,

which remains essentially just hydrostatic pressure balance against gravity

and centrifugal force.

Prima facie, the inequality (6.28) could break down near the surface of a

radiative zone, where we have seen that the generalized E-V-S theory yields

terms ∝ 1/ρ, either to ¬rst or to second order in the centrifugal parameter.

However, at such low densities the neglect of the magnetic contribution

to the poloidal dynamics will also break down. Approximate models of

uniformly rotating magnetic stars by Moss (see Mestel 1999) have ∇ — B

adjusting itself so that (in an obvious notation), (∇F )„¦ + (∇F )B falls o¬

more rapidly than ρ: the Lorentz force kills o¬ the embarrassing 1/ρ terms,

yielding a net circulation speed that not only stays ¬nite but vanishes as

ρ ’ 0. One would like to see more work on these lines, but it can justly be

claimed that this introduction of a weak magnetic ¬eld enables one to speak

of steady E-V-S circulation models that are both thermally and dynamically

self-consistent throughout the radiative zone.

The rather sharp dichotomy between the poloidal and toroidal dynamics

is a consequence of the assumption of axial symmetry. A plausible model

for the observed strongly magnetic stars is the oblique rotator, with the

axis of the large-scale ¬eld inclined to the axis of uniform rotation, and

this has some properties in common with a top. The Lorentz forces ex-

erted by the ¬eld of total ¬‚ux F cause small but ¬nite density perturbations

that are at least approximately symmetric about the magnetic axis. To

keep the angular momentum vector invariant in space, there must be su-

perposed on the basic rotation „¦ the Eulerian nutation, a rotation about

the magnetic axis analogous to the geophysicist™s Chandler wobble, of or-

der ω (F 2 /π 2 GM 2 )„¦ „¦. However, the density-pressure ¬eld contains

also the much larger perturbations due to the centrifugal forces, which are

symmetric about the rotation axis. To maintain hydrostatic equilibrium in

a radiative domain, the changes in the (ρ, p) ¬eld due to the Eulerian nu-

tation must be o¬set by nearly divergence-free internal motions (Spitzer, in

Lehnert 1958, p. 169; Mestel 1999 and references therein). These dynami-

cally forced, oscillatory ˜ξ-motions™ have the period of the Eulerian nutation

which is much longer than the free oscillation periods of the star, but can

be shorter than the Kelvin-Helmholtz or the nuclear time-scales even if the

Mestel

88

total magnetic ¬‚ux yields surface ¬eld strengths below the detectable limit.

In a rapid rotator, the amplitudes of the ξ-motions are large and so could

cause signi¬cant interchange of material between a convective core and a ra-

diative envelope, so destroying the adverse µ-gradient that is impeding the

non-oscillatory E-V-S ¬‚ow. If so, then this cooperation between thermally-

driven rotational currents and rotationally driven dynamical motions could

have a signi¬cant e¬ect over the leisurely time-scale of main-sequence stellar

evolution.

The discussion so far has assumed that the star has already been brought

into a state of near uniform rotation, which it can maintain magnetically

against advection of angular momentum by the slow circulation. In an

aligned magnetic rotator, an initial non-uniform rotation generates torsional

oscillations, with exchange of energy between the rotational motion and

the toroidal magnetic ¬eld generated by the shear. Mestel & Weiss (1987)

estimate that even with a generous allowance for an e¬ective macroresistivity

due to the hydromagnetic instability of a dominantly toroidal ¬eld, a poloidal