EVIDENCE FOR THE DISTRIBUTION OF ANGULAR VELOCITY INSIDE THE
SUN AND STARS
L. Mestel We have heard that the solar wind is steadily
removing angular momentum from the solar surface via
magnetic coupling. We now ask how the internal rota-
tion field of the sun responds to this surface stress. We
know that the sun has a deep subphotospheric convec-
tion zone, surrounding a radiative core. We shall assume
that there are only modest variations of angular velocity
within a convective zone, though we should note that
there is at least one model of nonisotropic turbulence
that, in principle, could allow a marked inward variation
[Biermann, 1951, 1958; Kippenhahn, 1963]. We now
ask whether the radiative core also steadily adjusts its
angular velocity to stay more or less in step with the
outer regions, or whether a steep inward angular velocity
gradient is built up, as in Dicke’s [1970, 1971] model,
which has the core rotating some ten times faster than
the convective zone.
One feels that a necessary condition for the persis-
tence of the Dicke model is the absence of even a
modest primeval magnetic field coupling the core and
zone, for torsional hydromagnetic waves would iron out
nonuniformities in rotation in a time much shorter than
the solar lifetime. I personally am doubtful if this is a
plausible assumption; however, I shall act as an
advocatus diaboli and discuss the equilibrium and stabil-
ity of the Dicke model in strictly nonhydromagnetic
terms. The complications that arise are a justification for
the claim I once made that the magnetic field is one of
the great simplifying features of astrophysics.
Howard et al.  and later Bretherton and
Spiegel  suggested that the Dicke model would be
destroyed by a process analogous to Ekman pumping
that is responsible for the rapid “spindown” in a coffee
cup. In an incompressible (or barotropic) fluid the con-
dition of hydrostatic support requires that the centrif-
ugal force be conservative, so that the angular velocity
must be a function only of distance S3 from the axis.
Such a law is inconsistent with the no-slip boundary
condition at the bottom of the cup, so that a dynami-
cally driven circulation is set up, with viscous force
balancing Coriolis force in the thin Ekman boundary
layer. Continuity forces the flow to extend through the
bulk, yielding a very short spindown time.
The treatment of this problem contrasts markedly
with that customary for a nonbarotropic stellar gas,
obeying the law p eupT. A nonconservative field of cen-
trifugal force, such as Dicke’s, can be balanced hydro-
statically by suitable variations of p and T over isobaric
surfaces. The consequent breakdown in radiative equilib-
rium yields buoyancy forces that drive a slow circulation
[Eddington, 1 929 ; Sweet, 1 950; Baker and Kippenhahn,
1959; Mestel, 1965]. The circulation speeds are nor-
mally of the order of the Kelvin-Helmholtz contraction
speed times the factor /jv(£2 2 co)|/g-, where g is the gravi-
tational acceleration. If £2 is slowly varying, this factor is
essentially f2 2 co, but in a region of large rotational shear,
as in the transition between Dicke’s core and the convec-
tion zone, the circulation speeds will be much faster and
will act to reduce the gradient. But before concerning
ourselves with processes dependent on heat transport,
we want to be sure that there is no analog of Ekman
suction, yielding a much shorter spindown time. In fact,
if the angular velocity gradient is too large it is impos-
sible to satisfy the condition of hydrostatic support
without the density gradient becoming locally positive
and so unstable; the thermally driven Eddington-Sweet
circulation is replaced by a dynamically driven flow if
the scale of variation of £2 is less than
d c =“ r[(<d 2 u}/g)(\lr )] 1/2 (1)
where X is the local scale height. This is the analog of the
layer thickness through which Ekman-pumped currents
can travel against the effect of stable stratification
[Holton, 1965] . One is therefore led to consider a
model in which a rapidly rotating core and a slowly
rotating envelop do coexist, with the transition region
between them never smaller than d c . The evolution of
the angular velocity field in the core would be given by
the Eddington-Sweet currents, with the sharp £2 gradient
and any variations of molecular weight playing an impor-
tant role. However, a much shorter spindown time could
result if the transition layer were to become unstable.
One would then arrive at a picture in which the slow but
persistent braking of the star would drive a weak turbu-
lence in the radiative core, which would keep the whole
sun rotating more-or-less uniformly [Spiegel, 1968].
Let us therefore adopt a Dicke-type model, and study
possible instabilities [see Spiegel and Zahn, 1970, for a
recent survey] . If the fluid is inviscid and incompressible
(with £2 necessarily a pure function of cu), a celebrated
criterion due to Rayleigh applies; for stability against
axisymmetric disturbances, we require
( 1 / co 2 )(rf/dco)(£2 2 co 2 ) > 0 (2)
The angular momentum per unit mass must increase
away from the rotation axis. However, there exist some
non axisymmetric unstable modes even if condition (2)
holds [Howard and Gupta, 1962]. Other nonaxisym-
metric instabilities occur if
(d/d^[(l/S)((//dS)(£2w 2 )] =0 (3)
“inflexional instabilities” [Lin, 1955].
The principal modification in astrophysical applica-
tions is the stabilizing effect of a density stratification.
In a zone that is stable against convection the density
gradient is subadiabatic, and energy is required to drive
adiabatic motions against gravity. The Richardson crite-
rion for the stability of shear flow [Chandrasekhar,
1961] sets a lower limit to |£2/(d£2/dd5)|, which turns
out to be of the order of the Holton thickness (l). How-
ever, in the Dicke model £2 is a function of displacement
z parallel to the rotation axis as well as of S3; the sur-
faces of constant angular momentum are not cylinders.
Such models are sometimes subject to rapidly growing
“baroclinic instabilities,” discussed by Hoiland [Ledoux,
1958] and more recently by James and Kahn 
who call them “sliding instabilities” because they involve
motion of gas elements along either isobars or
isentropes. They occur if the local angular momentum
gradient h lies in the shaded region (fig. 1), where p is
the direction of the pressure gradient and s the direction
of the negative entropy gradient. It appears that some
Dicke-type models with the surfaces of constant angular
momentum, as in figure 2, would violate the stability
criterion, but others, as in figure 3, would not.
Figure I. Schematic of angular momentum gradient,
h; p is the direction of the pressure gradient, and s is the
direction of the negative entropy gradient in a star for
the condition of “sliding instabilities’’ to not be met.
The converse takes place when h lies in the shaded
Figure 2. Dicke sun with a surface of constant angular
momentum conceptually illustrated by the folded line
crossing the radiative-convective transition. (The line is a
cut in a surface of rotational symmetry about the axis.
This condition might violate stability against the “sliding
Figure 3. Dicke sun under the condition where
stability is satisfied. The definitions of the figure
elements are the same as those in figure 2.
So far we have assumed adiabatic motions. As soon as
we allow for finite transport processes the whole situa-
tion changes. In stellar interiors the ratio of viscosity to
thermal conductivity is very low (~10 -6 ), so that viscous
effects can often (though not always) be ignored com-
pared with heat flow. Townsend  and Yih
 showed how radiative transfer can remove the
stabilizing effect of stratification, so that for a com-
pletely stable state conditions (2) and (3) replace the
Richardson criterion. The reason is that when temper-
ature perturbations are smoothed out, the stabilizing
effect of buoyancy is simultaneously removed [Moore
and Spiegel, 1964] . More recently, Goldreich and
Schubert  and Fricke  , again ignoring vis-
cosity, have shown that another necessary condition for
the absence of “secular” (dissipation-dependent) insta-
bilities is 00/9 z = 0, or angular momentum constant on
cylinders. We have noted that this is a condition for the
equilibrium of an incompressible star. Fricke [1969a]
summarizes these results by the prescription: to deter-
mine which states of a real star are secularly stable, solve
the problem of the equilibrium and dynamical stability
of the corresponding inviscid, incompressible system.
It is then clear that even Dicke models that are not
subject to baroclinic instabilities are certainly secularly
unstable on the Goldreich-Schubert-Fricke criterion.
However, it is still not generally agreed what asymptotic
state the star reaches, and in what time scale. Colgate
 and Kippenhahn  argue that the
developed weak turbulence that follows from secular
instabilities takes in general at least a Kelvin-Helmholtz
time to alter substantially the overall angular momentum
distribution. More recently, James and Kahn 
have proposed that an arbitrary initial rotation law
rapidly approaches a state with the surfaces of constant
angular momentum either cylinders or isentropes. The
secular instabilities to which such a model is subject are
suppressed by the much more rapid baroclinic instabili-,
ties which they themselves generate. James and Kahn
 also have studied the evolution of the junctions
between the isentropes and cylinders, where the break-
down in radiative equilibrium leads to locally large
Eddington-Sweet velocities; they conclude that the time
for overall redistribution of angular momentum is the
average Eddington-Sweet time, and this would be just
about comparable with the solar lifetime if Dicke’s
internal rotation is correct. However, the subject remains
I have assumed that there are no inward gradients of
mean molecular weight p. It has been known for many
years that a very modest p gradient will suppress the
Eddington-Sweet circulation [ Mestel , 1953, 1957;
Kippenhahn, 1967] , and that the growth of p in the
center of a star is normally able to prevent the circula-
tion from homogenizing the star. Similarly a p gradient
will kill secular instabilities [ Goldreich and Schubert,
1968] . The p gradient that can be built up during the
early solar lifetime clearly depends on the rate at which
instabilities develop and mix matter and angular momen-
tum (see Dicke’s discussion of the lithium problem,
p. 290). Fricke  finds that the maximum oblate-
ness due to internal rotation that can be obtained from a
rotation field satisfying the secular stability require-
ments (including the effect of p gradients) is a factor 4
less than Dicke’s value. But if we can tolerate secularly
unstable rotation laws, because we have grounds for
believing that the consequent angular momentum diffu-
sion time is at least as long as the Kelvin-Helmholtz time,
then a p gradient can be built up that will stabilize the
Dicke model for the much longer nuclear lifetime of the
sun. To return to a point made earlier, those who accept
the arguments but do not like the Dicke model might
very well claim that the conclusion is an argument in
favor of magnetic coupling between core and envelope.
Baker, N.; and Kippenhahn, R.: Zeits.f Astrophys.,
Vol. 48, 1959, p. 140.
Biermann, L,: Zeits. f. Astrophys. , Vol. 28,195 1 , p. 304.
Biermann, L.: Electromagnetic Processes in Cosmical
Physics. Vol. 248, B. Lehnert, ed., Cambridge Univ.
Press, Cambridge, 1958.
Bretherton, F. P.; and Spiegel, E. A.: Astrophys. J . ,
Vol. 153, Pt. 2, 1968, p. L77.
Chandrasekhar, S.: Hydrodynamic and Hydromagnetic
Stability. Oxford Univ. Press, Oxford, 1961. •
Colgate, S. A.: Astrophys. J., Vol. 153, Pt. 2, 1968, p. L8 1 .
Dicke, R. H.: Astrophys. J. , Vol. 159, 1970, p. 1 .
Dicke, R. H.: Ann. Rev. Astr. Astrophys. ,V ol. 8, 1971,
Eddington, A. S.: Mon. Not. R. Astr. Soc., Vol. 90,
1929, p. 54.
Fricke, K.: Zeits. f. Astrophys., Vol. 68, 1968, p. 317.
Fricke, K.: Astron. and Astrophys., Vol. 1, 1969a,
Fricke, K.: Astrophys. Lett., Vol. 63, 19695, p. 219.
Goldreich, P.; and Schubert, G.: Astrophys. J., Vol. 150,
1967, p. 571.
Goldreich, P.; and Schubert, G.: Astrophys. J., Vol. 154,
1968, p. 1005.
Holton, J. R.: J. Atmos. Sci., Vol. 22, 1965, p. 402.
Howard, L. N.; and Gupta, A.: J. Fluid Mech., Vol. 14,
Howard, L. N.; Moore, D. W.; and Spiegel, E. A.: Nature,
Vol. 214, No. 5095, 1967, p. 1297.
James, R. A.; and Kahn, F. D.: Astron. and As trophy s.,
Vol. 5, 1970, p.232.
James, R. A.; and Kahn, F. D.: Astron. and Astrophys.,
Vol. 12, 1971, p. 332.
Kippenhahn, R.: Astrophys. J ., Vol. 137, 1963, p. 664.
Kippenhahn, R.: Zeits, f. Astrophys., Vol. 67, 1967,
Kippenhahn, R.: Astron. and Astrophys., Vol. 2, 1969,
Ledoux, P.: Handbuch derPhysik, Vol. 51, S. Flugge , ed,
(Berlin: Springer-Verlag), 1958, p. 605.
Lin, €. C.: The Theory of Hydrodynamic Stability.
Cambridge Univ. Press, Cambridge, 1955.
Mestel, L.: Mon. Not. R. Astr. Soc., Vol. 1 13, 1953,
Mestel, L.: Astrophys. J., Vol. 126, 1957, p. 550.
Mestel, L.: Stellar Structure. Vol. 465, L. H. Aller and
D. B. McLaughlin, eds., Chicago Univ. Press, Chicago,
Moore, D. W.; and Spiegel, E. A.: Astrophys. J.,
Vol. 139, 1964, p. 48.
Spiegel, E. A.: Highlights of Astronomy. L. Perek, ed.,
(Reidel: Dordrecht-Holland), 1968.
Spiegel, E. A.; and Zahn, J. P.: Comments on Astro-
physics and Space Physics, Vol. 2, 1970, p. 178.
Sweet, P. A.: Mon. Not. R. Astr. Soc., Vol. 110, 1950,
Townsend, A. A.: J. Fluid Mech., Vol. 4, 1958, p. 361.
Yih, C.—S.: Phys. Fluids, Vol. 4, 1961, p. 806.
R. Kraft I would like to go back to the issue of the Li and Be abundances in the sun, to
remind you of what Mestel said, that in comparison with young stars the solar Li abun-
dance is very low. The solar Be abundance, however, is appropriate in making these
comparisons. One knows that Li can be destroyed at a temperature somewhat higher than
the base of the external convection zone, but that to destroy Be requires still higher
temperature. So one imagines now there must be some way to mix the subadiabatic sub-
convection zone material. And the issue is whether the turbulence that can be set up by
the spindown process may be sufficient.
R. H. Dicke Mestel raised the question of stability that is a source of worry in
connection with a rapidly rotating core in the sun. The instabilities in question are
thermally driven: the Eddington-Sweet thermally driven currents and Goldreich -Schubert,
and Fricke types of mild turbulence also driven by thermal effects. I will take the
following viewpoint: Assume that the Goldreich-Schubert-Fricke instability holds and
then calculate the turbulent transfer of angular momentum out of a star when despinning.
This amounts to a turbulent diffusion of angular momentum. The same turbulent diffu-
sion of angular momentum out of the star implies a turbulent diffusion of Li and Be
downward into the interior, where these elements are burned. The two effects are boot
strapped together: observe the rotation, and you should be able to say what is happening
to the abundances of Li and Be. By observing rotations and abundances of Li and Be we
decide whether or not this instability exists. This is program 1 . 1 have another program
after that, which is to use the “observed” depletion of Li in the sun as a basis for some
conclusions about the present solar wind torque.
I assume that the thermally driven turbulence at the condition of marginal instability,
after averaging Cl sin 2 d over spherical surfaces, leads to O ~ r~ n , where 0 <n < 2. The
Goldreich-Schubert instability results in a function S2.(r) in reasonable accord with the
above equation with n — ClJCl where & 0 refers to the present surface rotation of the
sun. For isotropic diffusion the transport of angular momentum is controlled by the
(dldr)[Dpr*(dTlldr)] = pr 4 ('dCl/dt)
where D is the diffusivity of the turbulent diffusion. Assuming marginal instability Cl(r,t )
is known everywhere in the interior if the surface rotation Q s (t) is known. Integration
The next step is to ask what controls the diffusion of lithium. Let’s designate the
abundance of Li, Be, or whatever the isotope is by the symbol F. The corresponding
diffusion equation is
(d/dr) [J Dpr 2 (d F/dr)] = pr 2 (dF/bt)
The diffusivity D is the same as before. To emphasize the point, we know, or at least
we assume that we know, the time dependence of surface rotation. As the rotation rate of
the surface of the star decreases the variation of the angular velocity of the stellar interior
is known as a function of time and position from the condition of marginal instability.
Instead of the usual interpretation of the diffusion equation, it is interpreted as a
first-order differential equation for D. We solve that differential equation, substitute the
resulting diffusivity D in the diffusion equation for Li (or Be), and solve the differential
equation to obtain the depletion of Li (or Be) at the surface. The stellar rotation and the
depletion of the isotope (Li 6 , Li 7 , or Be 9 ) are boot strapped together. The relation is
F/F* = (£2 S /£2 S *)^- where the asterisks (F* and £2*) refer to original values on the main
sequence, and A is an eigenvalue derived from the solution of the differential equation for
F as an eigenvalue problem. Figure 1 shows £2 for marginal instability calculated from the
Figure 1. £2(r), after averaging over spherical surfaces.
Goldreich-Schubert dispersion relation. Table 1 gives A for three different values of n,
n = 1/2, 1 , and 2. The values of A given in table 1 are all so great that there should be no
lithium or beryllium in the Hyades for which Kraft has measured a rotational slowing by
a factor 2; that is, £2/£2* = 1/2. On the contrary, we do see Li 7 and Be 9 , from which I
conclude that this turbulence does not extend down deep in the star and the Goldreich-
Schubert instability does not occur deep in the star.
Table 1 . A eigenvalue for deep lying mild turbulence
r b = 0.63
2 ( 2 nd mode)
A = 7 r
The next question is whether the instability exists at all or whether it exists only part
way down. Since we don’t at the moment have any other explanation for the depletion of
lithium I’m going to try on for size the idea that the lithium is depleted as a result of
angular momentum being transported by means of this turbulence — angular momentum
flowing out of the star into a stellar wind - but that the turbulence is terminated at a
certain radius (which we will call r c ) because of a slight jump in the mean molecular
weight (Ap ~ 2X10 -3 ). As was noted by Goldreich and Schubert  , such a molecular
weight jump provides a means for turning off the turbulence. Incidentally, when you turn
off the turbulence you also turn off the circulation currents at that point; they both
Figure 2 shows a hypothetical way of obtaining the molecular weight jump. In the
process of stellar contraction in the core, density goes quite high. But increased density
in the core ought to result in increased angular velocity in the core. The curve marked
“after core contraction” shows the high angular velocity of the interior leading to an
angular velocity gradient that may exceed the instability limit. Goldreich-Schubert turbu-
lence and circulation currents may occur inside the core, and if there is any extra helium
as a result of hydrogen burning while this mixing is occurring, extra helium may be mixed
throughout the core while the core’s angular velocity tends to become uniform. As noted
before, the jump in molecular weight required to stabilize is only ~2X10~ 3 , which is very
small. There is a possibility that one ends up with a stabilized boundary atr = 0.55 with
the region 0.55 0 <0.84 being the thermally driven turbulent region. Outside is the
hydrogen convective zone where angular momentum is moved convectively. These are the
assumptions we make.
Table 2 shows the eigenvalue A discussed earlier, but now the turbulence is assumed to
be cut off at r c . The tabulated values are for various assumptions about cutoff radius and
the index n . These have been chosen in such a way as to give reasonable values for the
depletion rate for Li 7 . It is found that the cutoff can never go deep enough to deplete
Be 9 . For a reasonable depletion of Li 7 , Li 6 should be essentially completely burned. If
you reduce Li 7 by a factor of 5 the Li 6 ought to be out of sight.
In attempting to apply this situation to the sun or stars of precisely 1 M Q , one runs
into the problem indicated before. We don’t have observations giving the slowing of
rotations of 1 M Q stars. But we do have the stars that Kraft has observed at 1 .2 M Q , and
we see that their rotations have decreased with time; we also have the lithium abundances
decreasing with time. When you boot strap these two together you find a best fit; you get
the best explanation for the depletion of Li 7 if you take n ~2. For £2»fi 0 , this is
much too large a value of n to be associated with the Goldreich-Schubert threshold, and
some modification of the Goldreich-Schubert effect is required, perhaps by nonrotational
motion of the fluid such as a slight oscillation of the core.
For the sun we are stopped for lack of observations and don’t know what to do. So we
Table 2. A eigenvalue with turbulence quenched at r c
r b = 0.63
take a new approach. After all, we have, or at least we think that we have, a rough value
for the solar wind torque. We can insert that to give the flow of angular momentum inside
the sun, from which we can calculate a present rate of decrease of Li 7 at the surface of
Figure 3 shows lithium abundance in meteorites in the Pleiades, coma cluster, the
Hyades, and the sun. There’s some argument concerning the abundance in the sun, but I
would guess the best value is about [Li 7 ] ~0.8. There is a problem if you take as the
solar wind torque density 6X10 29 dyne cm/sr, which for an isotropic solar wind is
~5X10 3 ° dyne cm total, and attempt to calculate the rate at which Li 7 should be
decreasing. One must decide whether angular momentum is coming from deep inside the
star or only from an outer shell with inner radius r c as a result of the slowing of the shell.
If it comes from slowing of the outer shell alone, the rate of decrease of Li 7 with time is
given by the dashed line (1). If angular momentum arises in the deep solar interior, the
rate of decrease of Li 7 is given by (2).
But I forgot an important point, that if you do have a rapidly rotating core for which
there may be viscous diffusion of angular momentum from the core, you can’t say
exactly how much it is. You can give an upper bound because the initial steepness of the
angular velocity gradient in the young sun cannot exceed a certain amount without also
exceeding the Richardson criterion for instability to ordinary dynamically driven turbu-
lence. The assumption of an initially steep angular velocity gradient provides a takeoff
point for the solution of the viscous diffusion problem to obtain the diffusion of angular
momentum. Adding viscous diffusion as a source of angular momentum gives lines lying
between (1) and (2). You can calculate a value for the flux of angular momentum from
the core, hence a lower bound on the angular rotation of the core, by adding the right
amount of core angular momentum flux to obtain the correct present values for the
Figure 3, [Li 7 ] and log F, logarithmic depletion of Li 7 . The curves (a), (b), and (c)are
integrations applicable to the sun. Curve (d)is an interpretation applicable to Kraft’s stars
of mass 1.2. The associated angular velocity curves (surface angular velocity as function
of time) are given in figure 4.
abundance of lithium and the angular velocity at the sun’s surface. Curves a, b, and c of
figures 3 and 4 give integrations for log F and Cl calculated in this way. Corresponding
lower bounds for the angular velocity of the core are included in figure 5.
Now let me turn the problem around another way. Ask yourself the following:
suppose we know nothing whatever about the solar wind torque, know nothing whatever
about the location of the radius r c except to say that it is somewhere in the zone of Li 7
burning. It is found that to obtain the correct values for the present abundance of Li 7
and surface angular velocity the present solar-wind torque is ~4X10 30 dyne cm if the
torque is proportional to the square of the solar angular velocity. For a torque propor-
tional to the solar angular velocity, the calculated solar-wind torque is roughly a factor
of 2 greater. These are surprising results. The present value of the solar-wind torque
implied by the loss of lithium in the sun is quite insensitive to detailed assumptions and is
quite close to the “observed” solar-wind torque. Another interesting result is that the
maximum value for the angular momentum flux (by viscous diffusion) from a core
rotating rapidly enough to account for the solar oblateness (20 £2 0 ) is 3.5X10 30
dyne cm. The close correspondence with the calculated torque (from lithium depletion),
4X 10 30 dyne cm, and the “observed” torque, 5X 10 30 dyne cm, suggests that the present
source of angular momentum for the solar wind may be viscous diffusion from a rapidly
Figure 4. With various assumptions concerning the initial main-sequence value of the
surface angular velocity, the time dependence is calculated. The curves of figures 3 and 4
assume that the solar-wind torque is proportional to the square of the angular momen-
tum. (a), (b), and (c) are applicable to the sun and (d) refers to Kraft’s stars.
r c /r o
Figure 5. Lower bounds for the angular velocity of the solar core obtained assuming
viscous diffusion of angular momentum from the core. The corresponding angular
momentum flux is that required by integrations such as (a), (b), and (c) of figures 3
and 4. The interpretation of the solar oblateness of Ar/r ~ 5X1 0~ s as the effect of a
rapidly rotating core requires an angular velocity of ~20 £2 0 for r c ~ (1/2 )r 0 .
Goldreich, P.; and Schubert, G.: Differential Rotation in Stars. Ap. J., Vol. 150, 1967,
E. Schatzman There are a number of questions related to this discussion concerning the
transport of matter or momentum, including the question of whether we can apply a
diffusion equation. I would like to give a number of ideas concerning the possibility of an
observational test of this transport inside the sun and possibly in stars, by detailed
analyses of the abundances of certain isotopes at the surface of the sun or in the solar
wind. If we consider the different nuclear reactions that can take place inside the sun,
first there is H 2 burning which can be neglected because it takes place in the very outer
Next there are Li 6 , Li 7 , Be 9 , B 1 0 , and very deep inside the sun He 3 formation by the
reaction Z) 2 + p -> He 3 together with C 1 3 burning which takes place at the very core of
the sun. Now, what we have to do irrespective of whether turbulent transport from the
inside has taken place, is to compare some initial abundances to the observed one. We
don’t know the initial abundances of the sun and can only make guesses, the validity of
which I am not certain. I shall discuss this briefly.
In units of log ZVjj = 12, where is the abundance of hydrogen, the initial value of
logA , 7 = 3. Assuming earth abundances, then the initial value for Li 5 in these units
~1.9. Now, if we consider the spallation ratio, if produced by spallation of carbon or
nitrogen, the value would be about 2.7, that is to say, about one-half the abundance of
Li 7 .
In regard to Be 9 there are some difficulties. Again using earth abundance for Be, a
value of ~1 is obtained whereas using the spallation ratio yields ~1.7. For B 10 the
spallation ratio should be 3.3. These numbers are to be compared to what observations?
For lithium, we can take three for the initial value and the observed value is depleted
by a factor of a hundred; this can be explained by turbulent transport from the lower
boundary of the convective zone to the place where Li is being burned. Li 6 is not
observed and probably has an abundance less than one-twentieth of Li 7 , that is,
log Ay 6 ~0.3. Using the values 1 and 0.3, we have compatibility with the turbulent
process in which the time scale is proportional to the square of the distance over which
the transport takes place. For Be 9 with an observed value of 0.7-1 , depending on interpre-
tation of the profile of the spectral lines of Be, based upon the earth abundance, no
burning exists, in which case we would have the case raised by Professor Dicke. On the
other hand, using the spallation value for the initial concentration of Be we note deple-
tion by an appreciable factor, which could also be explained by a turbulent transport. For
Be 1 0 we know that log A<2.7, given by the limit of visibility of the spectral lines. This is
a depletion by a small factor, if any, perhaps 4, and this is also compatible with transport.
But the real clue concerning this problem rests with He 3 and the C 13 . He 3 has not
been observed spectroscopically, but we have solar wind observations and I want to refer
here to Professor Geiss’ measurements on the surface of the moon for which he reports
a value of He 4 /He 3 ~ 2X10 3 . Now, what is the initial value? Perhaps it corresponds
to the very lowest value which has been obtained in meteorites, which is ~4 or 5X 10 3 . So
there is a possibility that the present He 3 concentration is larger than the He 3 concen-
tration in the solar wind say a few million years ago. This can be interpreted also as due
to turbulent transport and in fact we have two ways of estimating the rate at which the
turbulent transport takes place. One is by considering the rate at which the He 3 concen-
tration increases with time at the surface of the sun, and the other one is the absolute
value of the present abundance of He 3 at the surface of the sun, if it is assumed that He 3
is being produced at the center by thermonuclear reactions. Now, this represents one of
the possibilities for testing the turbulent transport from the center to the surface. And
just from orders of magnitude we also obtain a turbulent diffusion coefficient d ~10 3 .
C 1 3 is also interesting because if we take the earth abundance ratio C 1 2 /C 1 3 ~ 80, do
we observe in the sun the same or possibly a smaller ratio? This cannot be considered as
settled. Suppose C 12 /C 13 > 80 can be explained by C 13 burning at the center of the
sun because the C 12 /C 13 ratio in the carbon cycle is about 4. This is an increase and
seems to go the other way around, but we have to remember that the carbon is essentially
turned into nitrogen during the carbon cycle, which means finally the destruction of
carbon in favor of nitrogen and consequently a greater destruction of C 1 3 than C 12 . If
the ratio is larger than 80, this could possibly give an indication of the presence of
turbulent transport from the center to the surface of the sun. I don’t mean at all that this
is a demonstration which has taken place because as you can judge, there are a number of
difficulties concerning the initial abundances which are present.
A. Ingersoll I want to discuss the question of whether the oblateness measurements
that Dicke and Goldenberg  made do indicate that the core of the sun is rotating
rapidly, or whether there is an equally attractive alternate possibility. Dicke and
Goldenberg looked at the shape of the sun in visible light, and there are really three ways
that the sun might look oblate in visible light. The first possibility is that the equipoten-
tials, gravitational plus centrifugal, are oblate, which would be the case if the interior of
the sun were rotating rapidly. The second and third are variations of the possibility that
the solar equator is somehow hotter than the poles. If the equator were hotter, it would
also be brighter, and this might be confused with an oblateness because of the limitations
of seeing in the earth’s atmosphere.
I divide this hotter-equator possibility into two categories because the first of these,
the one considered and rejected by Dicke and Goldenberg, is that the equator of the sun
is hotter at all depths by a certain amount of AT. This would be like saying that the
equivalent temperature of the sun is greater at the equator than it is at the poles, or that
the radiant flux is greater at the equator than it is at the poles. Their measurements
suggest that this is an unlikely possibility, although I do not feel that it can be
conclusively ruled out.
The second possibility, which Spiegel and I have proposed [Ingersoll and Spiegel,
1971] , is that the equator of the sun is hotter only in the chromosphere but not in the
photosphere. This possibility is much easier to confuse with a real oblateness. To show
why this is so, I must digress to define certain aspects of the Dicke-Goldenberg experi-
ment. They took an image of the sun and projected it onto a perfectly circular occulting
disk, slightly smaller than the solar image. The radial angular distance from the edge of
the disk to the mean solar limb is 6 , and they did their experiments for 6 ~ 6.5 ", 12.8 ",
and 19.1 ". In each case, they scanned around the edge of the disk, measuring all the light
that was coming from beyond the occulting disk, and looked for an increase in flux at the
equator relative to that at the poles. This difference in flux is the signal they used to infer
the solar oblateness. The important thing about this quantity 6 is that for each of the
three possibilities that I mentioned earlier, there is a different relationship between signal
amplitude and 5 .
First, if the sun is truly oblate, then the signal is approximately independent of how
much sun is in the field of view, and therefore, the signal amplitude is proportional to
6° - that is, independent of 6. In this case the signal simply depends on the difference
between the equatorial and polar radii of the sun, and not on how much sun is occulted.
Next, if the equivalent temperature of the sun is greater at the equator than at the poles,
then the signal amplitude is proportional to the fraction of the solar disk in the field of
view — that is, to 5 1 . From the data taken at the three values of 6 , Dicke and Goldenberg
concluded that this was very unlikely. What Spiegel and I pointed out is that if the
equator is hotter than the poles, but only in an optically thin part of the sun’s atmo-
sphere, then the dependence on 6 is intermediate between these two and is proportional
to S 1/2 . Here we postulate an equatorial temperature, excess in parts of the sun’s
atmosphere that can be seen even on the extreme limb - that is, in the very top of the
photosphere and in the chromosphere. In this case, each emitter in the field of view
contributes as much to the signal as any other, and the number of emitters in the field of
view is simply proportional to the solar surface area exposed from the edge of the
occulting disk to the limb, and this is proportional to 6 1/2 .
Figure 1 is our reworking of the Dicke and Goldenberg data. We have plotted signal
8 1/2 (ARCSEC 1/2 )
0 US r-— 1 1 1 1 1
0 A 4 9 t6 25
ARCSEC TO EXPOSED UMB 8
Figure 1 Signal amplitude versus 5 1/2 after correction
for surface rotation. Units are BAr/r, where B is relative
brightness at the occulting disk, and Ar/r is measured
oblateness [Dicke and Goldenberg, 1967/. Error bars
give the square root of variance for each 5. Curves illus-
trate three possible dependences on 5.
amplitude versus 6 1/2 , for 6 = 6.5 ", 12.8 ", and 19.1 ", which are the three values of 6
used in the experiments. The three lines drawn represent the three possibilities: signal
amplitude « 5°, 6 1/2 , S 1 . Actually, the signal due to a true oblateness would not be
exactly <*d° , but would depend on the brightness at the edge of the occulting disk, and
this brightness increases slightly with 6 . So a true oblateness is consistent with these data.
Dicke and Goldenberg ruled out the parabola, signal « 5 *. The curve shown corresponds
to AT e is 5° K —that is, to a 5° excess in the equivalent temperature of the sun at the
equator relative to that at the poles. Obviously, it would be very interesting to measure
that somehow — I suppose by sending a satellite over the poles. The line on the graph
labeled 6 1/2 corresponds to what Spiegel and I suggested, with
t 0 AT s 0.3° K , r Q « 0.1
Here AT is the required temperature difference between equator and poles, which is
restricted, we assume, to an optically thin layer. And r 0 is the value of the optical depth
at the level below which this temperature difference is assumed to vanish. The restriction
t q « 0.1 simply ensures that this layer is optically thin. Examination of figure 1 shows
that this possibility fits the Dicke and Goldenberg data quite well.
Now if Spiegel and I are correct in our interpretation, and if the chromosphere really is
hotter at the equator than it is at the poles, the heat source for the equatorial chromo-
sphere must be greater than the heat source for the polar chromosphere by a specific
amount. This excess mechanical flux upward at the equator must be whatever is necessary
to supply the excess emission implied by the relation t q AT sr0.3° K. The required excess
flux is AF ~ 2.5X10 7 ergs/cm 2 /sec, which is comparable to what many people believe is
the total mechanical and hydromagnetic energy flux into the chromosphere. So if our
interpretation is correct, then we have to be prepared either for a mechanical heating of
the chromosphere, which is larger than what most people believe, or a variation in this
heating from equator to pole, which is comparable in magnitude to the heating itself.
Dicke, R. H.; and Goldenberg, H. N.: Solar Oblateness and General Relativity . Phys. Rev.
Ltrs.,V ol. 18, 1967, p.313.
Ingersoll, A.; and Spiegel, E.: Temperature Variation and the Solar Oblateness. Ap. J.,
Vol.163, 1971, p.375.
R. H. Dicke There are three points I would make. First, the question was raised as to
whether a general temperature difference of the photosphere between the equator and
the pole could account for the observations. The measurements were made with three
different amounts of limbs exposed, which lead to a light flux ratio of approximately 1 .0
to 2.5 between the smallest and the greatest amount. Under an oblate sun hypothesis
these two signals have a ratio of about 1.0 to 1.2 and when we renormalize (correct the
signal of the biggest exposure by a factor of 1 .2 downward), the observations are satisfac-
tory. I can’t believe that they would be satisfactory if we had reduced the signal by a
factor 2.5. There would then be a sizable discrepancy in those three curves. I don’t think
On the question of a hot layer, I think one must go far above an optical depth of 0.1
to make the scheme work. For levels above 0.01 you need at least a 40° temperature
difference between the equator and the pole. For this case, I think that the signal could
be sufficiently close to what we observed that this might be a satisfactory way of
accounting for the signals. On the other hand, one has to make a physically reasonable
statement. There are two requirements to be satisfied. One is the requirement of energy
balance for the necessary steady state — the problem of getting excess energy at the
equator into the particular layer, the upper photosphere, to heat it up enough to give the
excess radiation. And the other requirement is one of dynamic balance for the necessary
steady state. There may be several ways this can be done; the one that’s been suggested
by the authors, which is to require that the angular velocity increase outward in the
upper photopshere with a scale height of about 1 ,500 km, may well be in difficulty with
what is known observationaily about the rotation of the sun at various levels. So I would
say that insofar as the observations are concerned it is possible that one could account for
them in this way, but I haven’t seen a coherent physical statement of how such a physical
state would be maintained or dynamically balanced.
A. Ingersoll The first point Dicke raised was that he didn’t feel that the data could
be consistent with a temperature difference between equator and poles that extended
deep into the atmosphere of the sun. Now, that really hinges on whether you feel that the
parabola can be made to fit the three data points, the parabola being the solid line in the
graph I showed earlier.
R. Dicke I don’t know how you got these points. The paper didn’t list them - the
paper didn’t even give the normalization ratios that you would have had to know to
compute these points; the ratios weren’t in the paper.
A. Ingersoll We assumed that the values of 5 and the values of the photospheric
brightness at the edge of the occulting disk were those which you gave in your paper. We
used the limb darkening curve you gave in your paper -
R. Dicke We didn’t give a limb darkening curve.
A. Ingersoll Well, not in Dicke and Goldenberg [Phys. Rev. Letters, 18, 31, 1967] ,
but in Dicke [Ap. J., 159, 1, 1970] from which we took these values.
R. Dicke But those were not observations, but a theoretical limb darkening curve
from a theoretical paper.
A. Ingersoll Let me put it this way: All the data we got for making this graph came
from various papers you have written; we consulted no others for this.
Now, the second point, I guess, was the question of the dynamical balance. If we are
to accept the fact that the parabola does not fit the data, then the temperature difference
between the equator and pole is concentrated only in the chromosphere, and it is true
that you need to balance the forces implied by this horizontal temperature difference.
The most likely way is that angular velocity should be increasing with height. We calcu-
late that if angular velocity increases by ~5 percent in 100 km over some 100-km region
near the temperature minimum, that would be enough. So there’s another observation
that should be made in order to test this observation.
E. Schatzman There is a very well-known solar oblateness in the meter wavelength
that corresponds to a structure of the corona, but very high in the corona. The oblateness
is considerable. So might there be a relation between your assumption concerning the
chromosphere and what has been observed at meter wavelength?
R. H. Dicke It seems to me that the postulate of the increasing angular velocity does
fit observations; that is, one sees angular velocity increase with height in the chromo-
sphere. The sign is correct for the chromosphere and consequently may be correct for the
upper photosphere where the balance is actually needed if the upper photosphere is to be
extended on the equator with a higher temperature. So it’s not a question of whether the
idea is qualitatively wrong but whether in fact it is quantitatively right. (Ed. note: See
comment by Livingston, p. 304).
C. P. Sonett We have carried out extensive calculations regarding a mechanism for early
electrical heating of meteorite parent bodies with the view to obtaining clues about the
early solar system especially the question of the pristine solar spin rate and evolving
conditions in the solar nebula just after condensation of the primary objects. The pro-
posed mechanism and the calculations which have been carried out are based upon the
following observational evidence. Certain classes of meteorites, particularly the iron-
nickels and achondrites, has been exhaustively studied for evidence of cooling from
elevated temperatures [Wood, 1964; Goldstein and Short, 1967] . The iron-nickels show
evidence for cooling rates which range approximately from l-10°/million years indicating
that at the time of the cooling cycle these objects were at depths within parent bodies to
several hundred km radius. Some error might accrue in these estimates on the basis that
for the nickel-irons the diffusion of Ni across grain boundaries between kamacite and
taenite, both of which are Ni-Fe phases, varies from the values used because of “doping”
of the matrix by trace elements which can adversely affect diffusion coefficients. However,
the basic phenomenon cannot be avoided by this argument; only the rates can be
modified, which means that the parent body sizes would have to be adjusted. On the
other hand, it has been argued that because Si grains are found within a metallic matrix,
that primordial condensation is required to form the meteorite bodies. Here we assume
that the parent body heating mechanism is correct. There are compelling reasons for
believing that, for example, the Widmanstatten patterns in the irons could only be
produced by a well-behaved cooling from an elevated temperature.
The time setting for the cooling cycle is early in the chronology of the solar system.
This is established, at least for Weekeroo Station, by Wasserburg et al.  , who dated
Si inclusions as about 4.6 billion years old. Thus, at least on this basis, the heating and
cooling episodes are very early. To explain a heating episode for parent bodies of the
restricted sizes postulated, since the event appears to have taken place very early, requires
either fossil radionuclides or some exotic form of heating. Long-lived radioactives are
ruled out because their energy-deposition rates are too low for the short time scales
proposed. Similarly, accretional heating released by the potential through which objects
fall in accreting would be ruled out because of the small size of the bodies and the small
gravitational energy Sonett  .
The classical means of heating of parent bodies has been based on a class of extinct
isotopes thought to have been present during the formative period of the solar system.
That such isotopes were present is clear from both the presence of Xe 129 from the decay
of I 129 , Xe components from Pu 244 fission decay and the appearance of fission tracks in
meteoritic matter. Although the existence is verified for these cases, the speculated level
of activity assignable to these isotopes is far below that required for the heating cycle.
Other nuclides have been popular candidates in the past. Perhaps the most prevalent has
been Al 26 hypothesized to have arisen in spallations associated with the early sun. How-
ever, the most recent tests show no evidence for this isotope [Schramm et al., 1970] ,
and thus the hypothesis is not well supported.
In view of the lack of strong evidence for radioisotopic heating, the study of the fossil
residues remains a fundamental requisite for understanding of the cosmochemical forma-
tive processes leading to the condensation into material bodies, but the source of the
heating cycle appears to require a separate explanation.
It seems likely that the early sun was spinning rapidly and that it was endowed with at
least a modest magnetic field. These conditions arise quite naturally from the trapping of
field in the Hayashi contraction and the spinup due to condensing angular momentum
from the primordial cloud. If we associate the contractive period with the precursor phase
of an early star prior to a f Tauri efflux of mass, then conditions are quite naturally
established for the establishment of strong electric fields in the expanding cloud, a result
of the combination of high spin, magnetic field, and plasma outflow [Sonett et al.,
The conditions just described can lead to strong electrical currents flowing through
planetary objects, the circuit being completed through the surrounding “solar wind.”
Electromagnetically the interaction is classified as transverse magnetic (TM) and has been
discussed extensively in the literature [Sonett and Colburn, 1967; Schubert and
Schwartz, 1969] . It’s application to the present cases, forming in effect a linear unipolar
generator, requires that the electrical impedance along the current streamlines through
the body be sufficiently small so that strong currents can flow. On the other hand, too
low an impedance will result in the formation of a steady-state magnetohydrodynamic
bow shock wave ahead of the body facing into the direction from which the flow of
To provide an appropriate impedance, we invoke the well-known exponential depen-
dence of the bulk electrical conductivity of rocky matter on the reciprocal temperature.
Extensive calculations have been made involving parameterization of the problem. Signifi-
cant heating due to Joule losses from the current system are found. It is clear that
because the currents close through the surface of the body that the crustal temperature is
a crucial aspect of the heating, and that a sufficiently elevated temperature is required. To
provide this it is only necessary to consider further the general properties of T Tauri
objects, which are often endowed with an infrared excess attributed to dust-induced
opacity. We term the enclosure a hohlraum and invoke an interior surface temperature to
this enclosing matter; thus, the planetary object “sees” a background temperature suffi-
cient to maintain an adequate bulk electrical conductivity [Sonett et al., 1970] .
Although this all may appear as unduly complicated, the effects required appear to be
commonly hypothesized or observed in what are thought to be early stars. Their paramet-
ric association, numerically adjusted to provide significant heating, has shown that only
quite modest requirements must be placed on the system to provide the heating cycle.
We now turn the problem around to discuss the spindown issue. It is clear that a
rapidly spinning sun must eventually be braked so that the present epoch spin rate is
achieved [Dumey, Chap. 4,p. 282], Although the calculations referenced- use an exponen-
tially decreasing field and magnetic braking, some other shaping of the field decay is
equally appropriate and angular momentum can also be shed by the outflowing gas. Thus,
in the present calculations, the field and spin damping are represented in an integral sense
only, and the instantaneous rates cannot easily be determined. However, the evidence is
strong that some form of heating other than fossil nuclides is required if the heating cycle
continues to be maintained as a viable requirement.
The electrical problem is complicated by the additional presence of a TE (transverse
electric) mode of interaction, which simply stated is due to eddy current generated from
the action of B, the time rate of change of the interplanetary magnetic field seen in the
frame comoving with the planet [Schubert and Schwartz, 1969] . The tendency would be
to associate this mode more with turbulence in the outflowing gas which in turn is
reflected in magnetic field disturbances. This mode also has the simplification that the
hohlraum is not required as the current system is toroidal closing wholly within the
planet. Calculations are in progress to determine the efficiency of the TE mode and the
coupled action of both the TE and TM modes together with a modest addition of
radioactives, which are known to inhibit the later stages of the heating by the TM mode.
Figure 1 shows the heating of small bodies as a function of their radius. The peak
temperatures are achieved in times of the order of 0.5 million years for the larger cases,
Figure 1. Peak core temperature versus parent body
while for the very small objects of 10- to 25-km radius the peak heating of the core is
achieved in a much shorter time, so that a relaxation begins to take place and the interior
cools long before the overall induction process is completed.
The clue to the high solar spin rate is carried in the requirement for the large electric
field at the site of the parent body which then leads to the large induction. The very
substantial electric field in turn is strongly dependent upon the winding up of the inter-
planetary field into a tight Parker spiral because of the large spin. Thus, the evidence for
early heating of parent bodies leads quite naturally to the condition where the sun was
endowed with a high spin rate during its early histroy. Although the specific spin rate at a
given time cannot be presently foretold, it is clear that the elements of the overall
theoretical treatment leading to better understanding of this are intrinsically carried in
the thermal history of these small bodies, provided that electrical heating proves to
withstand the test of further examination.
Goldstein, J. I.; and Short, J. M.: Cooling Rates of 27 Iron and Stony-Iron Meteorites.
Geochim. Cosmochim. Acta.,W ol. 31, 1967, p. 1001.
Schramm, D. N.; Tera, F.; and Wasserburg, G. J.: The Isotopic Abundance of 26 Mgand
Limits on 26 A1 in the Early Solar System. Earth and Planet. Science Letter, Vol. 10,
1970, p. 44.
Schubert, G.; and Schwartz, K.: A Theory for the Interpretation of Lunar Surface
Magnetometer Data. The Moon, Vol. 1, 1969, p. 106.
Sonett, C. P.: Fractionation of Iron: A Cosmogonic Sleuthing Tool. I. Radioisotope
Heating. Comm. Astrophys. Space Phys., Vol. 1 , 1969, pp. 6, 41 .
Sonett, C. P.; and Colburn, D. S.: Establishment of a Lunar Unipolar Generator and
Associated Shock and Wake by the Solar Wind. Nature, Vol. 216, 1967, p. 340.
Sonett, C. P.; Colburn, D. S.; Schwartz, K.; and Keil, K.: The Melting of Asteroidal-Sized
Bodies by Unipolar Dynamic Induction From a Primordial T Tauri Sun. Astrophys.
and Sp. Sci., Vol. 7, 1970, p. 446.
Wasserburg, G. J.; Gurnett, D. S.; and Frondel, C.: Strontium-Rubidium Age of an Iron
Meteorite. Science, Vol. 150, 1965, p. 1814.
Wood, J. A.: The Cooling Rates and Parent Bodies of Several Iron Meteorites. Icarus,
Vol. 3, 1964, p. 429.
W. C. Livingston Spectroscopic observations made on the solar disk near the equatorial
limbs consistently indicate an increase of angular velocity as we pass outward through the
sun’s atmosphere. The chromosphere, as revealed by H a , appears to rotate 5 to 8 percent
faster than the underlying photosphere as represented by the metallic lines [ Livingston ,
1969] . Because manifestations of magnetism such as sunspots, filaments (or promi-
nences), and plages corotate with the spectroscopic photosphere, it has been suggested
that we are observing in Ha the “superrotation” of neutral matter that can flow indepen-
dent of magnetic constraints. (The term superrotation is borrowed from aerodynamics
where it is used to describe an analogous condition in the atmosphere of the earth and
Venus [King-Hele, 1970; Gierasch, 1970].) As a working hypothesis we propose the
existence of an east to west wind whose lower boundary is the photosphere and whose
upper extent is unknown.
Seeking additional evidence of this superrotating wind, in 1968 we began to obtain
prominence spectra. The structure of prominences undoubtedly is dominated by mag-
netic forces, so one would not expect to find any degree of superrotation in these objects.
However, some early work by Evershed  suggested some peculiarities in their
spectroscopic rotation rates.
Our spectra are taken with the slit placed normal to the limb and generally at a
position angle such that the height of the Ha emission is at a maximum. Records are
taken in both Ha and Ca + K. Figure 1 Elustrates a phenomenon often found on our
spectra. At the top of the line, corresponding to the upper edge of the prominence, the
weakened emission typically becomes diffuse and exhibits an abrupt displacement in
•wavelength, indicating line-of-sight motions differing from the main body below. By
analogy with smoke escaping from the confines of a stack, we picture gas escaping from
the magnetic confines of a prominence to be picked up and accelerated by a wind.
Indeed, in the majority of cases within our limited sample this displacement is in agree-
ment with an east to west superrotating wind [ Livingston , 1971] . Further examples are
Figure 1. Spectrum of solar prominence in Q-& K 3933, slit perpendicular to the limb.
Magnetic constraints of the main body of the prominence is analogous to the smoke stack
with gas escaping at the top and caught up in the prevailing wind.
given in figure 2. The magnitude of the displacement ranges from a few km/sec to as
much as 50 km/ sec.
Figure 2. Examples similar to figure 1 taken in both Ha and Ca + K. In all cases (except
the ambiguous last) the Doppler displacement of the upper emission is compatible with a
hypothetical east-west wind.
In summary, both disk and prominence spectra suggest the existence of systematic east
to west flow patterns at the chromospheric level. Whether or not such a superrotating
surface wind of neutral gas would interact with the solar wind plasma remains to be
studied. We can note that any interaction would be in the forward direction and counter
to the backward drag of the Archimedes spiral.
Evershed, J.: The Solar Rotation and Shift Towards the Red Derived From the H and K
Lines in Prominences. Mon. Node. Roy. Astron. Soc.,V ol. 95, 1935, p. 503.
Gierasch, P. J.: The Four-Day Rotation in the Stratosphere of Venus: A Study of
Radiative Driving. Icarus, V ol. 13, 1970, p. 25.
King-Hele, D. G.: Super-rotation of the Upper Atmosphere at Heights of 150-170 km.
Nature, Vol. 226, 1970, p. 439.
Livingston, W. C.: On the Differential Rotation With Height in the Solar Atmosphere.
Solar Phys., Vol. 9, 1969, p. 448.
Livingston, W. C.: Solar Rotation: Direct Evidence From Prominences for a Westward
Wind. Submitted to Solar Phys . , 1971.