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EVIDENCE FOR THE DISTRIBUTION OF ANGULAR VELOCITY INSIDE THE 
SUN AND STARS 


INTRODUCTION 

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. [1967] and later Bretherton and 
Spiegel [1968] 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 


287 



[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 [1970] 
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 
region. 



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 
instability. ”) 



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. 


288 



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 [1958] and Yih 
[1961] 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 [1967] and Fricke [1968] , 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 
[1968] and Kippenhahn [1969] 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 [1970] 
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 
[1971] 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 
controversial. 

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 [19695] 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. 


REFERENCES 

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, 
p. 297. 

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, 
p. 388. 

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, 
1962, p.463. 


289 



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, 
p. 271. 

Kippenhahn, R.: Astron. and Astrophys., Vol. 2, 1969, 
p. 309. 

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, 
p. 716. 


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, 
1965. 

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, 
p. 548. 

Townsend, A. A.: J. Fluid Mech., Vol. 4, 1958, p. 361. 

Yih, C.—S.: Phys. Fluids, Vol. 4, 1961, p. 806. 


COMMENTS 

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 
diffusion equation 


(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 
gives D(r,t). 


290 



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 


n 

Li 6 

r b = 0.63 

Li 7 

0.58 

Be 9 

0.47 

1/2 

237 

140 

46 

1 

174 

100 

31.3 

2 

222 

120 

33 

2 ( 2 nd mode) 

1460 

780 

212 


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 [1967] , 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 
terminate. 

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 


n 

r c 

Li 6 

r b = 0.63 

Li 7 

0.58 

2 

0.578 

5.265 

0.901 


.57 

6.01 

1.633 


.56 

5.97 

2.372 

2nd mode 

1 

52.3 

20.5 

3rd mode 

T 

145.0 

55.8 


.55 

8.0 

3.013 


.53 

10.33 

4.376 


.50 

14.57 

6.772 

1 

0.578 

9.64 

1.761 

2nd mode 


70.3 

20.98 


.57 

10.9 

3.30 


.56 

12.5 

4.48 


.55 

14.2 

5.61 

2nd mode 


100.5 

43.5 

1/2 

0.578 

18.6 

3.48 

2nd mode 


134.9 

40.36 


.57 

20.75 

6.45 


.56 

23.63 

8.70 



> 

it 



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 
the sun. 

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 
rotating core. 




STELLAR AGE 

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 . 




REFERENCE 

Goldreich, P.; and Schubert, G.: Differential Rotation in Stars. Ap. J., Vol. 150, 1967, 
p. 571. 


COMMENTS 

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 
layers. 

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 

JLrl 

~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. 

COMMENTS 

A. Ingersoll I want to discuss the question of whether the oblateness measurements 
that Dicke and Goldenberg [1967] 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 


299 




DISCUSSION 


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. 


REFERENCES 

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 
that’s possible. 

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 


300 



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). 


COMMENTS 

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. [1965] , 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 [1969] . 

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., 
1970] . 

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 
plasma comes. 

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 
radius. 



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. 

REFERENCES 

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. 

COMMENTS 

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 [1935] 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 



P97 


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. 


305 


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. 

REFERENCES 

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. 


306