without a clear explanation should not be seen as any cause for alarm or

despondency, but should instead be welcomed, in the Goughian spirit, as an

occasion for celebration, providing new open questions for future generations

of scientists and mathematicians to address and new opportunities to expand

our knowledge of the physics of the Sun and stars.

Acknowledgments This is an appropriate place to put on record my thanks

to Douglas Gough for the continuing scienti¬c inspiration he has provided

throughout my research career as both student and post-doc. His zeal for

understanding, his exuberant enthusiasm, and his penetrating insight have

made it an honour to have collaborated with him.

I must also o¬er thanks to my daughter Adelena who, by being born at

the appointed time and no earlier, allowed me to attend the meeting at

Chateau de Mons where the oral version of this paper was presented.

This research was partially supported by the European Commission

through the TMR programme (European Solar Magnetometry Network,

contract ERBFMRXCT980190) and by the Norwegian Research Council™s

grant 121076/420, “Modeling of Astrophysical Plasmas”.

References

Cargill, P. J., Spicer, D. S., & Zalesak, S. T. 1997, ApJ, 488, 854

Carlsson, M. & Stein, R. F. 1997, ApJ, 481, 500

Carlsson, M. 1999, in Wilson, A., ed., Ninth European Meeting on Solar Physics:

Magnetic Fields and Solar Processes, ESA SP-448, ESA Publications

Division, Noordwijk, the Netherlands, p. 183

Judge, P. G., Tarbell, T. D., & Wilhelm, K. 2001, ApJ, 554, 424

Korevaar, P. 1989, A&A, 226, 209

McIntosh, S. W., Bogdan, T. J., Cally, P. S., Carlsson, M., Hansteen, V. H.,

Judge, P. G., Lites, B. W., Peter, H., Rosenthal, C. S., & Tarbell, T. D. 2001,

ApJ, 548, L237

Mcintosh, S. W. & Judge, P. G. 2001, ApJ, 561, 420

Rosenthal, C. S., Bogdan, T. J., Carlsson, M., Dorch, S. B. F., Hansteen, V.,

McIntosh, S., McMurry, A., Nordlund, ˚. & Stein, R. F. 2002, ApJ, 564, 508

A

Shibata, K. 1983, PASJ, 350, 263

Skartlien, R. 2000, ApJ, 536, 465

II Stellar rotation and magnetic ¬elds

6

Stellar rotation: a historical survey

LEON MESTEL

Astronomy Centre, University of Sussex, Falmer, Brighton BN1 9QJ, UK

Prologue

By now I am reconciled to the contemplation of my former undergraduate

pupils becoming senior citizens. For Douglas I have always had a particularly

soft spot “ it was so good for my ego to have daily commerce with a man

of the same Napoleonic stature as my own. For decades I have watched his

progress, grati¬ed at the successful trailing of my St. John™s College gown.

I look forward to his reaching three-score-and-ten, con¬dent that like others

known to him, he will be treating retirement as a purely notional concept.

6.1 Radiative zones: the Eddington-Vogt-Sweet theory

In a rotating star, with magnetic forces assumed negligible, the equation of

hydrostatic support in standard notation is

’∇p/ρ + ∇φ + „¦2 = 0, (6.1)

where φ is the gravitational potential, yielding the gravitational acceleration

g = ∇φ and satisfying Poisson™s equation

∇2 φ = ’4πGρ; (6.2)

p and ρ are the pressure and density; G is Newton™s gravitational constant;

and in general the angular velocity „¦ = „¦( , z) or „¦(r, θ) in cylindrical and

spherical polars respectively. In a non-degenerate star, p and ρ are related

by the equation of state

p = pg + pr = (R/µ)ρT + aT 4 /3, (6.3)

where T is temperature, µ is the mean molecular weight, and R and a are

the gas constant and radiation density constant. In all but the most massive

stars, with µ lying between 1/2 (pure hydrogen) and 4/3 (pure helium), the

75

Mestel

76

radiation pressure pr is a small correction, and so it will be ignored in most

of the subsequent discussion.

The astrophysicist™s approach to the problem, going back to von Zeipel,

Eddington and Chandrasekhar, and developed later by Sweet and many oth-

ers, is to treat „¦( , z) initially as an arbitrary imposed function, yielding

a centrifugal perturbation which distorts the (p, ρ, T ) ¬elds from their nor-

mal spherical symmetry. It is recognized that the case „¦ = „¦( ) “ which

includes uniform rotation “ is special, for then there exists a centrifugal

potential V , and the curl of (6.1) yields ∇p — ∇ρ = 0 (˜barotropy™), whence

p = p(Ψ), ρ = ρ(Ψ) = dp/dΨ, (6.4)

with the joint potential Ψ = φ + V de¬ning the level surfaces. If for the

moment the star is assumed to be chemically homogeneous, then T = T (Ψ)

also.

To complete the system, one must introduce the energy equation; and as

is well known, it is at this point that the problem is qualitatively altered

(von Zeipel, in Eddington 1926). If the centrifugal ¬eld is conservative, then

the radiative ¬‚ux F through a medium of opacity κ(p, ρ) can be written

F = ’(4acT 3 /3κρ)∇T = ’(4acT 3 /3κρ)(dT /dΨ) ∇Ψ ≡ ’f (Ψ)∇Ψ,

(6.5)

where c is the speed of light. This variation of F over a level surface,

proportional to the e¬ective gravity ∇Ψ, is justly described as ˜von Zeipel™s

theorem™. When „¦ is independent of , ∇2 V = 2„¦2 , and the ˜von Zeipel

paradox™ for a uniformly rotating star results if one attempts, as in a non-

rotating star, to balance the local radiative e¬„ux of energy by local energy

liberation, at a rate dependent just on the local variables ρ(Ψ) and T (Ψ);

i.e. if from (6.5) and (6.2) one writes

4acT 3 dT

d

ρ (ρ, T ) = ∇ · F = ’ (∇Ψ)2 (6.6)

dΨ 3κρ dΨ

4acT 3 dT

+’ (2„¦2 ’ 4πGρ) .

3κρ dΨ

All the quantities in (6.6) are functions of Ψ and so constant on a level

surface, except for (∇Ψ)2 . Thus the coe¬cient of (∇Ψ)2 must vanish sep-

arately, so that von Zeipel™s theorem (6.5) now holds with the coe¬cient

f (Ψ) forced to be constant over the whole domain. The remaining terms

then yield the manifestly spurious result

∝ 1 ’ („¦2 /2πGρ) . (6.7)

Stellar rotation: a historical survey 77

Von Zeipel™s paradox is a mathematical peculiarity. It is unacceptable

not only because of the unphysical result (6.7), but even more from the

implied non-uniform behaviour as „¦ ’ 0. In that limit, Ψ ≡ φ, and there

is no requirement that the ¬rst term in ∇ · F should vanish independently

of the second. In fact in a domain in thermal equilibrium but without any

active nuclear sources, such as the radiative envelope of a Cowling model

star, the two terms on the right of (6.6) are equal and opposite, whereas von

Zeipel™s argument implies that a small uniform rotation forces the ¬rst to

vanish, so demanding that the second be balanced by the spurious energy

source (6.7). However, as pointed out by Vogt and by Eddington (1929),

the correct immediate conclusion from this reductio ad absurdum is not that

the energy equation alone is able to restrict the class of allowed rotation

¬elds, but rather that the net e¬„ux ∇ · F of radiant energy from unit

volume is balanced jointly by the local nuclear energy generation, plus the

energy transported by a thermally-driven circulation v in meridian planes “

in Eddington™s words, ˜home products plus smuggled goods™.

The energy equation is written

cv ρdT /dt ’ (p/ρ)dρ/dt = ρ ’ ∇ · F (6.8)

with d/dt the derivative following the motion v and cv the speci¬c heat

at constant volume. In a hypothetical steady state, and for the special

conservative centrifugal ¬eld with „¦ constant, (6.8) becomes

ρA(Ψ)(v.∇Ψ) = ρ + f (Ψ)(2„¦2 ’ 4πGρ) + f (Ψ)(∇Ψ)2 (6.9)

where f (Ψ) is de¬ned in (6.5) and A(Ψ) = T ds/dΨ = cv T d[log(T /ργ’1 )]/dΨ

with s the speci¬c entropy and γ the usual ratio of speci¬c heats. Division

of (6.9) by |∇Ψ| and application of the condition of zero net ¬‚ow of gas

across a level surface relates f (Ψ) and f (Ψ); substitution back into (6.9)

then yields for the velocity component normal to the level surface

dS/|∇Ψ|

ρA(Ψ)v · ∇Ψ = f (Ψ)(4πGρ)(1 ’ „¦2 /2πGρ) ’ ρ |∇Ψ|2 ’1 .

|∇Ψ|dS

(6.10)

Equations (6.9), (6.10) and the continuity equation ∇ · ρv = 0, combined