A physical mechanism is proposed that couples the Sun's dynamo magnetic field to its gravitational potential energy. The mechanism involves the isotropic field pressure resulting in a lifting force on the convective envelope, thereby raising its potential energy. Decay of the field due to solar activity allows the envelop to subside and releases this energy, which can augment the otherwise steady solar luminosity. Equations are developed and applied to the Sun for several field configurations. The best estimate model suggests that uniform luminosity variations as large as 0.02% for half a sunspot cycle may occur. Brief temporal variations or the rotation of spatial structures could allow larger excursions in the energy released

Endal, A. S.

Schatten, K. H.

English

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INFLUENCE OF 14AGNETIC PRESSURE ON STELLAR STRUCTURE:
A MECHANISM FOR SOLAR VARIABILITY
(NASA-TM-81129) INFLUENCE OF MAGNETIC	 N80-30267
PRESSURE ON STELLAR STRUCTURE: A MECHANISM
FOR SOLAR VARIABILITY (NASA) 1u p
HC A02/MF A01	 CSC:L 03B	 'Inclas
G3/92 28406
F
f'
Kenneth H. Schatten
Laboratory for Planetary Atmospheres
NASA/Goddard Space Flight Center
Greenbelt, MD 20771
and
Andrew S. Endal
Department of Physics and Astronomy
Louisiana State University
Baton Rouge, LA 70803	 •_	 '"
r
,* NASA - Summer Faculty Fellow at Goddard Space Flight Center
Abstrict .
We propose a physical mechanism that couples the sun's dynamo
magnetic field to its gravitational potential energy. The mechanism
involves the isotropic field pressure resulting in a lifting force
on the convective envelope, thereby-raising its potential energy.
Decay of the field due to solar activity allows-the envelope to
subside and releases this energy, which can augment the otherwise
steady solar luminosity. Equations are developed and applied to
the sun for several field configurations. Our "best estimate" model
suggests that uniform luminosity variations as large as-0.02% for
half a sunspot cycle may occur. Brief temporal variations or the
rotation of spatial structures could allow larger excursions in the
energy released.
•	
l
F
Introduction
Recent studies of terrestial climate variations by Mitchell et al.
(1979) and Dicke (1978, 1979) suggest a link with the state of the
sun's activity cycle (see White (1979) for reviews of the subject).
Such a link would result if the solar dynamo-could modify the total
energy output of the sun and this has prompted theoretical investigations
into possible effects of the d.%-namo upon the solar luminosity. Thomas
(1979) , has examined possible luminosity changes associated with magnetic
bouyancy in sunspots; Spiegel and Weiss (1980) have considered the
effect of magnetic fields on the temperature structure of the convection
zone; and Scha l,ten and Sofia (1980) have suggested that magnetic fields
may alter the depth of the convection zone. These investigations were
concerned with the effects of magnetic fields on energy transport. In
this paper, we consider the direct effect of the isotropic pressure
exerted by the dynamo field on the gravitational potential energy of the .
solar convection zone.	 ' I
The standard theory of the solar dynamo involves a buried toroidal
magnetic field which alternately is wound up by differential rotation
and develops into solar activity. This magnetic field is assumed to exert'an
isotropic pressure which can lift the convection zone and increase its
gravitational energy. This potential energy increase will occur during
the field amplification phase of the solar cycle. When the field breaks up
into activity (solar flares, sunspots, etc.), the convection zone
subsides and releases the stored potential energy. This energy will be
shown to be much greeter than the energy released by solar activity.
When added to the otherwise steady solar luminosity, the result is a
solar-cycle modulation of the total energy output of the sun. In the
.following section we develop a simple model for this process. The
estimated amplitude of a uniform luminosity modulation is ti0.02%,
i.e., about 10 3 times greater than that associated with surface activity.
1	 1
3
i	 I
Potential Energy Changes Due to the Dynamo Field
We are interested in obtaining the uplift and energy change which
results from the dynamo field. To do this, we first need to estimate
the strength of the toroidal field at the base of the convection zone.
This can be done by combining observations of the surface field with
physical limits on dynamically stable field configurations.
The observations of Howard (1977) and Sheeley (1976) provide the
polar magnetic flux to be ^ p = 1.2 x 1022 Mx, with an upper limit of
2 x 1022 Mx. Figure 1 indicates the manner in which the field threads
to the base of the convection zone. The toroidal field subtends a
latitude range 0 in each hemisphere and the field lines are inclined
at an angle a with respect to a latitude circle. We identify 0 with
the range in which sunspots occur, while a can be inferred from the
tilt of sunspot pairs. We use N37 0 = 0.65 radian and ati60 - 0.1
radian Allen, 1963). This gives
@=20 t = 2^p p.cot a/27r c 2.3 x 1022 Mx,	 (1)
where 0 is the total magnetic flux through both hemispheres and 0t
the toroidal flues in each hemisphere.
The flux condition governing the field through the element of area
shown in figure 1 is
B Ar R  0 = Ot t	 (2)
where Ar is the radial thickness of the field layer and R. the radius
at the base of the convection zone.
We can obtain limits on the toroidal field by considering some
simple stability conditions. The equation for radial force balance is
dr
RC
where P is the gas pressure, Mi the mass internal to the convection Zone,
and p the gas density. A Rayleigh-Taylor interchange instability can
I	 i
4
develop if a positive radial gradient in the gas density occurs Similarly,
a positive gradient of temperature would prevent the outward flow of thermal
energy through the field layer and lead to a thermal instability. If both
the density and temperature gradients.must be less than or equal to zero,
the same condition applies to the gas pressure gradient. Thus, equation (3)
limits the field gradient to
-1 dB2 	GM i 	dP GMi
P,	 (4)
Sir dr
	
R2c	 dr R2c
Assuming the field outside our toroidal volume is small, we replace
( dB2/drJ-AB2/Ar
 by 82/Ar and find
B2/Ar<8tr GM i p/Rc2	(5)
Using the flux condition (2), we get
1/3	 2	 1/3
B<	
8nG'1^	 t and nr> 	 t (6)
Rc 0 	 (87ffrGMipOf)
For the conditions at the bottom of the convection zone in the solar
model of Endal and Sofia (1980), B is limited to 2 x 105 G and Ar must be
at 1 ,wast 13 km.
We can now determine how far the magnetic field will lift the solar
convection zone and raise its potential energy. A magnetic field B,
contained in a plasma of volume V and temperature T, exerts the same
pressure as a hypothetical mass AM given by
2	 (7)
AM B ^ '87rn^ kT
where n is the number of particles per unit mass and, for the toroidal
volume element, V = 8Rc
2
 (Ar) 0. Since this will be a small perturbation
on the global structure, we can picture the magnetic field as displacing
this hypothetical mass AM at the bottom of the convection zone. This is
5
^s..f-
I	 i
equivalent to lifting he entire convection zone through a distance
AM	 20 
i3 
-1 
Ar, •	
(8)h s	 — 
41r^ 1r
where 0 is the ratio of gas to magnetic pressure. The increase in
the gravitational potential energy due to this uplift is
GM  Mcz 20 a-1 Ar,	 (9)AE • gMcz h n R 
c 
2	 n
where Mcz is the mass of the convection zone.
With 0t = 1022 Mx and 8 and Ar at the limits (6), AE is 1.5 x 1038
ergs; this can provide a 0.02% increase in the sun's luminosity for half
a solar cycle. This is ,the "maximum standard" shown in Table 1. The
term maximum reflects that we are using the most favorable values allowed
by the limits (6). We are not including possihle short temporal variations
or spatial structure which could increase the magnitude somewhat.
The energy change AE is proportional to the four-thirds power of the
polar flux. This is illustrated by the second set of models in Table 1,
where we have varied the polar flux to the maximum value (in modern times)
given by Howard (1977) and to half the Sheeley (1976) value. A third set
of models illustrates the effect of varying the depth of the convection zone.
The last set of models is calculated assuming the plasma 8 is 40 and 4 x 105,
rather than relying on the stability condition, which appears to limit 0
to a value closer to 4 x 103 . Although the interchange instability is
strong, it is possible that the magnetic field may damp the resulting plasma
motions so that the stability criteria may be violated for a short time.
Note that, for a given E; Ar is determined by the flux condition (2).
One last calculation will provide an additional check on our model.
The energy to inflate the convection zone must first be applied to the
magnetic field, as it is the pressure of the field which does the lifting.
6 1
.r^
The field receives its energy from differential rotation; can the differential
rotation supply 1.5 x 10 38 erg in 5 years or 2 x 1031 erg s al ? If we use a
formula relating the magnetic pressure in a torque taw with a differential
rotation, we find
Power
.
 n T4w • (82/87r) RC  4w.
	
(10)
In the maximum standard model, this can supply 3.5 x 1031 erg s"1 , for
4w = 10"6 s' 1 , which is more than adequate. Note'that most of the
7.5 x 1038 erg released when the convection zone settles must go into
luminosity, rather than magnetic activity, as solar activity does not
nearly involve this much energy. Further, we have ignored the ti3 x 1037
ergs of magnetic energy, 8 2V/8n, which will add approximately 20% to
4E in the maximum standard case.
I
a...
7
xr
Discussion
Although our table 1 provides results for several sets of parameter
values, we feel the maximum standard model is close to what would be
expected from a large solar cycle. Most of the other models are provided
solely to show the sensitivity of luminosity changes to parameter variations.
The reasoning is that this standard-is based upon an accurate convection
zone model, thus the base level of the convection zone is thought to be
reliable. Secondly, the model is based upon the Rayleigh-Taylor instability
being approached; it is felt that this instability is approached as solar
cycles do break up when magnetic buoyancy is reached. It is possible that
another instability may limit the growth of the field prior to onset of the
Rayleigh-Taylor instability. In this case, the maxim m 8 2/Ar would be
reduced by a factor f, relative to the value given be equation (4). However, 	 }
the energy release AE would only be reduced by f l/3 so our computed energy
release is rather insensitive to this possibility. Thus, the values
cited in the maximum standard are thought be be approached, with the true
value falling closer to a larger $ (a somewhat smaller energy output).
Another consideration is the period of these luminosity oscillations.
The references cited in the introduction discuss, for the most part, a 22
year (Hale double cycle) variation in climate, whereas we have discussed
the matter in a way which would suggest'an 11-year variation. We feel that
should a subsurface directed magnetic field be frozen-in to the core
material, it could modulate the 11-year solar cycle fields in such a way
as to result in an offset of field stren;ths. This could result in the
double cycle (22 year) predominating. This matter'deserves further study.
Another point to note is that numerous 11-year variations are found! See
Herman and Goldberg (1917) for a review of the subject, or Nastrom and
Belmont (1980) for recent findings.
F - __
Conclusions
We have found that a reasonable solar cycle magnetic field can
uplift the convection zone. When the cycle deteriorates during the
declining phase, the resulting collapse of material may provide temporal
variations up to about 0.02% of the solar luminosity. Larger energy
excursions may occur associated with the nonuniform release of energy
in the declining phase, however, it is difficult to estimate the size of
these irregularities. The value calculated is about 103 times larger
than the average energy released by conventional forms of solar activity.
An is year (or possibly a 22 year) variation may result.
9
Tabl e 1
z
Solar Convection Zone Uplift Models and
Associated Energy Changes
Model p NO	 Rc (km) B(G) ^ir(km) AE(ergs) AL(Lo)
1.	 Maximum
Standard 1.222	5.45 13 1.538 0.02%
2.	 Flux
•	 Variation
(max.) 222	 * 2;85 18 3.038 0.05%
(min.) 0.622.	 * 1.8 8 G.038 0.01%
3.	 Varying
Field Depth
(max.) *	 4.05 65 6 1.839 0.297,**
(min.) *	 6.05 1.45 17 237 0.01%
4.	 B-1 = 25-2
(max.) *	 * 2:36	 h T 1.539 0.2%**
6-1	 2.5 e
(min.) *	 * 2.34 130 1.537 2-3t
(Note: Exponents refer to powers of 10, thus 55 = 5 x 105)
* indicates the maximum standard value is used.
**these values are unrealistic and serve only to show the sensitivity of the
model to parameter variations. .
10
iJ
Acknowledgement
We acknowledge useful, interesting discussions with Sabatino Sofia.
FIGURE CAPTIONS
Figure 1 ' Shown is the geometry of*the threading of the sun's polar
magnetic field to the base of the convection zoner where it
is twisted into a toroidal field. The area at the left has
a width, Rc 0 0 and a thickness or, the field lines make an
angle, a, with respect to a circle of latitude. The geometry
is used to estimate the magnitude of the sun's subsurface
field. The field is spread into a toroidal volume, Y=402
Ar(20).
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Allen, C. W., Astrophysical Quantities, Athlone Press, Univ. of London,
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Dicke, , R. H., Is there a chronometer hidden deep in the Sun?, Nature, 276,
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Dicke, R. H., Solar luminosity and the sunspot. cycle, Nature, 280, 24-279
1979.
Endal, A. S., and Sofia S., Rotation in Solar-Type stars. I Evolutionary
models for the spin-down of the sun, Astrophys. J., submitted, 1980.
Herman, J. R. and R. A. Goldberg, Sun Weather and Climate, NASA SP 426,
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Howard, R. H., Large-Scale solar magnetic fields, Ann. Rev. of Astron. and
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Mitchell, J. M., Jr., Stockton, C. W. and Meko, D. M., Solar Terrestrial
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Influence of magnetic pressure on stellar structure: A Mechanism for solar variability

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