The theory of the solar gravitational figure is derived including the effects of differential rotation. It is shown that J sub 4 is smaller than J sub 2 by a factor of about 10 rather than being of order J sub 2 squared as would be expected for rigid rotation. The dependence of both J sub 2 and J sub 4 on envelope mass is given. High order p-mode oscillation frequencies provide a constraint on solar structure which limits the range in envelope mass to the range 0.01  M sub E/solar mass 0.04. For an assumed rotation law in which the surface pattern of differential rotation extends uniformly throughout the convective envelope, this structural constraint limits the ranges of J sub 2 and J sub 4 in units of 10 to the -8th power to 10  J sub 2  15 and 0.6  -J sub 4  1.5. Deviations from these ranges would imply that the rotation law is not constant with depth and would provide a measure of this rotation law

Hawkins, G. W.

Ulrich, R. K.

English

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cMIT, CONTRACT 9 5 8 3 9
FINAT, REPORT b ,f
THE SOLAR GRAVITATIONAL FICUR -- J;" AND J4
R.K. ULrLch	 •.
Department of Astronomy, UnLveraity of California, Los Angeles
Ln
artitl
Mrn x j.
r+^,	 C.W. Hawkins	 r
	Department of Astronomy, University of California, Los Angeles
	 {
co
r
ABSTt ACT
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	 3
The theory of the solar gravitational figure is derived including the
44
	
effects of differential rotation. It is shown that J 4 ;'.es smaller than J 2 by a
H v
	
	
factor of about 10 rather thin being of order (Ja) 2 as would be expected for
rigid rotation. The dependence of both J 2 and J4 on envelope mass is given.
bb 4J i7
aHigh order p-mode oscillation frequencies provide a constraint on solar structure,
ai M
	
which limits the range in envelope mass to the range 0.01 < M /Me < 0.04. For
r4	 .-
•• 	
Jf
y a 
	 an nssUilled rotation law i.n which the surface pattern of differential rotation
,H P,
	
=
M
	 exten<!s uniformly throughout the cony^pctive envelope, this structural constraint
h 4limits the range, of J and J in unitg of 10 8 to 10 < J 2 < 15, and 0.6 < - J4
00 x < 1.5. Deviations from these ranges would imply that the rotation law is not00 04M a 
P3 , C4	 constant with depth and would provide a measure of this rotation law.
r^
a1 U' SHq
"This work was performed for the Jet Propulsion Laboratory, California
Institute of Technology sponsored•by the NationaV Aeronautics and
Spacd Administration under Contract NAS7-100." 	
67
Enz
r
t	 w^.	 FBr , r
4 
e	 ..
THE SOLAR GRAVITA'T'IONAL FIGURE
	
J2 AND J4
R.K. Ulrich and G.W. Hawkins
1. INTRODUCTION
The gravitational figure or the sun can be measured directly by sending
a probe carrying a drag-free guidance system to within a few solar radii of the
solar center. The interpretation of such a measurement will require knowledge
of how the solar structure and rotational law influence the multi-pole exprnn-•
sion of the external gravitational field. The study of solar oscillations limits
the variations in J 2
 due to structural uncertainties to less than a factor of 2.
Refinements in the observed frequencies of the non-radial modes of higli order can
we show how the existence of dif=
one will produce J G » J2 An
can substantially increase our
gravitational figure for rotating
stars ,divide into two types.' The theory of rapidly rotation configurations is
further reduce this uncertainty. Tit addition,
ferential rotation throughout the convective z
.icq.,, rate measurement -of J 2 and detection of J4
knowledge of the solar rotation law.
Tile papers dealing with the theory of the
generally limited to the study of polytropic models. Tile classical Clairant-
v Legendre*expansion resulting from this approach is well-de,9crihe by Tassoul
(1978). Applications based on this method are disc:'ussed by James (1964),
Ostriker and Mark (1968) and Hubbard, Slattery and DeVito (1975) . While this
method is very powerful in its ability to treat. models of object s, in whJch cen-
tripetal acceleration and'-gravitational acceleration approach equality, it is
not easily applied to non-polytropic model;:- because of the Fundamental way in
which the polytropic assumption is used.
w,
-3- .
The second type of approach is that described by Col.dretch and Schubert
(1172). In' this approach, the stellar structure equations are analyzed in a
perturbation expansion. Consequently, this method fails for objects rotating
fast enough that the centripetal and gravitational accelerations are comparable.
It has the advantage For application to the solar case that it is not restricted
to polytropic models. Since the sun is rotating slowly, we use the Coldreleh
and Schubert (1972) approach. We present in the following section the theoreti-
cal analysis of a dlffereil:k •3.ally t(itati.ng mode], and derive the equation which
describes the ot:t:t.+polt^ moment of the sun, J 4 . In .5 III we describe the nor-
rotating zero order models and present the results for J 2 and JG in 5 IV.,.
II. TH ORY OF J4
We adopt as a simple rotation law:
0 + S2 2
 u 2`	 (1)
where S2 tj is the equatorial angular rotation rate and p is the cosine or the
colatitude. The term S12 u2 represents the differential rotation. Pitting the
rotation law of lioward, Boyden and Laboite (1980) to Equation (1), we find that
R 2 /R 0	- 0.7. We assume that S2 2 anti 52 0 are constant in thef
onvective envelope
^
and that S2 9 = 0 below the convective envelope. Although th (s latter assumption
I
-may not be valid, S1	 must go to zero at r. = 0 because otherwise there Mould be
a divergence in the shear velocity field.
	
The centripetal acceleration r S12
appears In the equations of momentum ` balance.	 We expand 52 2 as S2^ (1 + a p z )P j
!	
u ii%re:
ORIGINAL PAC	 IS
OFPQoR QUALITY 
r
" a
2$
a ^ 4 _ 1.4
	
(2)
0
rollowing Goldreich and Scliubert (1972) ► we write , the momentum balance equa-
tion in spherical coordinates (r, u) in terms of the gravitational potential ^h ►
the pressure p and the density p. The r and u components of this equation are:
p fir! a - p S1 2 (1'+ au 2 ) r (L - u 2 )	 (3)
P " tt- p n 2 r2 (l	 au 2) ^^,	 (4)
	
ar	 a!i	 Q,
Assuming that the acceleration of the rotation is small compared to the accelera-
tion of gravity, we may expand all quantities into"unperturbed quantities fip, pal
and p 0 and their perturbations ^1 ,
 p,I and p l . The unperturbed equations are scan-
;;
lard and will not, be repeated here. The first-order perturbation eg11 t-ions are:
	
a^
	
aIPQ
	
ap
P O Wt l t p l a r a art - pp,no	 2
	
(1 + ap	 r (l - u>	 (5)
D
P O	
1	
8 p1 + A 
i2 r2 (1 +' ap 	 u .	 (G)
	
„ o a l .	 a^,,	 n	 o
We now expand 1 ► p1 and p l in spherical harmonics, e.g.
s
,cohere PQ (u) is the Legandre polynomial of order R. Using standard recursion
11
relations for the P , the momentum equations have R = 2 and R 	 4 components
3
Ft
wa
R	
a
2s AO ^2 * 
n l2 ar'Q ^
a.2 + AA go r(1)	 ^)
r2 P o
g20
PO 1,2 	 p ]. 2 *	 3
	 (1 + 7Fi)
	
^)
and
R.	 4:	 "'14 +A 	
"0	 ap].4+8a
A $1
2 r	 (10)0 a 	 lei or	 Dr	 35 0 0
p+
 `^ 14 
a I
' l 4 * 33 r 2 P O
 
0	 (11)
The above equations must be supplemented'by the Poisson equation:
a2 fit,, 2 it R t+ 1
ar	 rt	 + r ar	 2	 `^19,	
l 
►n CP1R	 (12)
Equations (H) and (9) and the R
	 2 component of Equation (12) combine to
give the analogue of the standar A Coldreich and Schubert (1972) equation for
X 12 . This equation is:
a2 
x'12	 2 ' ` 12	 0 12	 4n r 2	 dp0	 4	 2 _ r2 d	 2
2	 r Dr r 2	 pig,	 12 dr 	 21 a rp0 r0 	 3 dr	 p 0 0
+ 7 p 0 aS2nJ 	
(13)
This differs from the standard equation in having the extra term involving 4/21
and in replacing n2 with g0 . (1 + 3a/7)
	
The equation for the R	 4 component of
Equation (12) is similarly derived and is:
i
t	 k	
-- 
k
	
 k 	 l^6-
2
x'14	 34 1.4	 20	 4n r2	 clp0	 ^i	 2
^r2 _ 
+ ar r2 4 14 ' H, `^14 dr * 35 ar n^ ^0
35 x SL Cap 0 $t0
1 	 (14)
Equations (13) and (14) must he solved subject to the boundary conditions
t	 tbat fi i ^ « r. at r	 0 and 
4J^	
r^^'_
1
 as r +	 Numerically, thetic conditions
were satisfied by expanding Q 1^ as CZ r^ fit the innermost mass shell. Equations
(13) and (14) were then in egrated from lie center to the outermost mass paint.
;l
The outer boundary condition was imposed as an equation of condition and .CZ wa s
adjusted through a Newton-Raphson iteration procedure. The distribution of a(r)
was chosen so that a(r) T 0 for r < re and a(r)	 1.4 for r > re where re is
the radius at the inner edge of the convection zone. In crossing this radius,
the following jump condition is applied to 
the414 
solutJon,;
34 14 	 3414	 sn r4 a 0 S1 a (15)0r ^ r e+e	 ar	 r
e
-e ^	 35 Pir
where a is a small distance encompassing the gradient In a. The exterior gravi-
tational field is described by the normalized values of 41Q'
L+1
JR _ 
rCM tip '	
(16)
xil. THE SOLAR MODELS
she static solar model has be^h calculated using the input physics
described in the paper by Bahcal,l et al. (1980). The method differs from
-7-
previous solar modal Calculations that have been clone using the UCLA code in
two respects'.
1) The atmosphere code discussed by Ulrich and Rhodes (1977), Ulrich,
Rhodes and Deubner (1979) and Lubow, Rhodes and Ulrich (1980), has been
extended to include all of the interior physics. 4y adjo a ting Lhe mix-
ing  length and hydrogen abundance, the atmosphere integration has been
extended inward to a radius of 1,5 x 10 9
 cm. The final Interior ,,One
in treated analytically with 4 theory accurate to better than 1/,,percenL.
it
The Interi	 distribution of hydrogen abundance has been talten from tho
current ' 'standard solar model. This method yields neutrino fluxes which
exceed those in the standard model by 10 percent.
2) Partial ionization of all heavy elements is now included in an approxi-
mate way following.Bodenheimer, Forbes, Gould and llenyey (1965). Fur-
ther, details of the calculational methods will be given by Ulrich and
Rhodes (1981).
Two fully converged models were computed: The standard modal and a mocial
with the interior heavy element abundance reduced from Z r- 0.018 Lo Z A 0.005.
The convective envelope abundances were normal. The mix of both cases was that
given by Ro.-is and Aller (1976). The second model is referred to as the low Z
Ri
model. In addition to these models, a series of incomplete- envelopes was ca, cu-
I a ted with a specified mixing length parameter. Th as e envelopes were sto,#:Ped
when either 99 percent of the mass or radlu.q Was U'80d in the inward integration
leaving--r, large residual of the other variable. An approximate analytic conLinu-
ation to the center was added by assuming that M Q r 3 (P	 )	 r2 , (Pr	 0	 0	 P)
IA
.	
a	
•
r2 and (T - T) a r2 . Almost none of the perturbation in is produced in
the poorly treated portion of the model.
	 I
Iv. RESULTS AND DISCUSSION
There are two factors which can influence the values of J 2 and J 4 	l ) the
distribution of mass through the sun and 2) the distribution of 12 (r, 0) 'through
the solar interior. This latter function is observed at r m Ito , but there is
as yet only a Preliminary indication of its variation with r. A range in
structure variation is possible in principle so that both factors must; be con-
sidered in interpreting any potential measurement of J2 and J 4 . Fortunately,
the structural variations are not constrained by the measurements of the eiten-
frequencies of high order non-radial oscillations. The recent observations by
Rhodes, Harvey and Duvall (1981) limit the variations in the"env,lope muss to
a
0.01 < M /hto < 0.04. As a measure of the importance of structure in determining
J 2
 and J4 , we Have assumed that St (r, 0) = S2 (R, 0) for r > r e and that $1 (r, 0)
H (R) for r < re . We have used the rotation law given by Howard, Boyden and
La Bonte (1980) in calculating J 2 and J4 . Our results are given I-a Figure 1.
The above I'mits on M /M translate into permitted ranges in J and J
e o	 2	 4
1.0x10 7
. <J 2 < 1.S x10-7
6	 x1.0T9<- J4<1.5x10_g	
(17)
Measured values of J 2 or J4 outside of these ranges would indicate a deviation
from our simple assumptions for 0 (r, 0).
The subsurface rotation rate can be measured with the method described by
Rhodes, Deubner and Ulrich (1979) and by Deubner, Rhodes and Ulrich (1979).
FP P;; t
•
-9-
This second paper gives someb first results which show n (r, 0) increasing as r
decreases. 16this result is borne out by subsequent observations, we would
expect larger values for J 	 Ii2 trod
	 41 . The theory or the solar dynamo depe ► do
on such a variation of SI with r (Stix 3.977). Ultimately, it may be poss-ible
to probe the rotation law using , the oscillations; however, there are two critj.--
cal uncertainties which must be resolved before we will know the ef-fectivness
of the oscillation method: 1) the oscillations mus-t,  be adequately long lived
and 2) it: Must be possible to identify correctly individual modes of oscilla-
tion for all spatial scales. Also, the oscillations cannot provide a good
measure of the rotation rate off the solar equator because foreshortening limits
the viewing -Legion. Thus, the, measurement of both J 2 and 34 would provide very
important constraints on $1 (r, 0), which are different from the constraints
imposed by the oscillations.
We would like to thank Jim Underwood for his helpful comments on the manu-
script.
tow
REFERENCES
nnlicall, J.N. , Huebner, W.V., Lubow, S.H., Magee, N.H. , Merts, A. ia. , l'^hrker, P.D.,
Ror.snyai, il, , Ulrich, R.K. and Argo, M.F. 1980, I?hyn. Rev., t,ett ers, 45, 945,
bodenhaimer, t'., Forbes, J.E., Gould, N.L. and Henyey, L.G. 1965, A^. J., 1111,
101.9.
Aeubw r, V.-L., Ulricht K.K. and Rhodes, R.J., Jr. Astr^on. Asrroal ., 72, 177.
Col.dreich, P. and Schubert, G. 1968, Ap. J., 154, 1005.
llubbard, W.11., Slattery, W.L. and peVito, C.L. 1975, Ap .L., 399, 504.,Mw
James, R.A. 1964 A2. J., 140, 552.
Lubow, S.L., Rhodes, R.J., Jr. and Ulrich, R.K. 1980, in."Nonradinl And Non-
linear Stellar Pulsation," ed, H.A. Bill and W. Dziembowski (Springers
Heidelberg), p. 300.
0striker, J.P. and Mark, J.W.-K. 1968,A 2_. J., 151, 1075.
Rhodes, R.J., Jr. Ueubner, V.--L. and Ulrich, R.K. 1979, Ap. J., 227, 629.
Rhodes, E.J., Jr., Harvey, J. and Duvall, T. 1981, in preparation.
Ross, J.I's. and All.er, L.H. 1976, Science, 191 ,, 1223.
Stix, M. 1976, in "Basic Mechanisms of Solar Activity," I.A.U. Symposium No. 71,
ed. V. Bumba and J. Kl.eozek (Reidel Dordrecht, Holland), p. 367.
`1'assoul, J.-L. 1978, "Theory of Rotating Stars" (Princeton University Press;
Princeton, NJ).
Ulrich, R.K and Rhodes, E.3 	 Jr. 19,77, A p . J., 2218,, 901.
Ulrich, R.K, and Rhodes, C.J., Jr. 1981, in preparation.
Ulrich, R.K., Rhodes,, R.J.,. Jr. apd neubner, F.-L. 1979, Ap. J., 2227 , 638.
FJOURE (,Arrm
j-*jZqra t. The depandanca of J 2 and J4 on cV"cuL,:td ,*, envelope mus Me/1,100
Corresponclina valties of L/It are indb^.;YNd CAI Ole LOP. Of the rigure,
These vahies arc dependent 
on 
n number of ColopIlLaLlontil deviLls and are lent).
significant than M /1`1	 Tho clinracterisLics of the standard model atid Oic low0
model are discussed In Clio text.
r.
-c
I/W
10	 1.4	 1,004 2.5 4.0
J2
-7►
i
m0
..Q
-9
I	 x
Ira!
O STANDARD MODEL
O .LOW Z MODEL
--- INCOMPLETE
ENVELOPES
Iuy
 brief ivocep
Figure 1 i
I
C^Ri tC L PAC14'
!	 OF PG()il


The solar gravitational figure: J2 and J4

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