Solar rotation and the perihelion advance of Mercury

ICARUS 8, 492-502 (1968)

Solar Rotation and the Perihelion Advance of Mercury S. P. S. A N A N D AXD G. G. F A H L M A N

Darid Dunlap Observatory, U,dversity of Toronto, Richmond Hill, Ontario, Canada Communicated by Zden6k Kopal Received November 22, 1967 Uniformly rotating polytropic solar models are used to determine the implications of the observed solar oblateness for the perihelion advance of Mercury. In view of the hypothesized rapidly rotating solar interior, we have used the second-order theory of rotating polytropes to study rotational effects up to terms of the order ~4, where ~ is the angular velocity of rotation. It is shown that the observed oblateness necessitates a rapidly rotating core if the surface of the Sun is an equipotential surface. In terms of a simple solar model in which the radiative core and the convective envelope are dynamically distinct regions, it is shown explicitly that the core must be rotating approximately 15 times faster than the surface and also possess an oblateness of 2 X 10 -s. On the other hand, the calculations show that by regarding the core and the envelope as a single dynamical entity, it would be possible to explain the observed oblateness if the angular velocity of the Sun were only about 2.5 times the observed surface velocity. The corresponding quadrupole moment of the Sun and the corresponding perihelion advance of Mercury would be much smaller than that of the decoupled model. Other implications of having a modestly fast rotating core are also briefly discussed. I. INTRODUCTION

R e c e n t l y D i e k e a n d G o l d e n b e r g (1967a) h a v e m e a s u r e d a solar o b l a t e n e s s of t h e o r d e r 5 X 10 -5. T h i s v a l u e is a b o u t five t i m e s g r e a t e r t h a n t h a t e x p e c t e d on t h e basis of t h e o b s e r v e d r o t a t i o n r a t e of t h e solar surface a n d so t h e a b o v e a u t h o r s s u g g e s t e d t h a t t h e cause of t h e o b s e r v e d o b l a t e n e s s is a r a p i d l y r o t a t i n g interior. T h e s e o b s e r v a t i o n s were r e p o r t e d a f t e r a p r e v i o u s a r t i c l e b y D i c k e (1964), in w h i c h he s t a t e d t h a t if a p p r o x i m a t e l y t h e i n n e r half of t h e S u n b y r a d i u s were r o t a t i n g w i t h a p e r i o d of 25.5 h, t h e n t h e o b s e r v e d o b l a t e n e s s w o u l d b e of t h e o r d e r 6 X 10 -5. T h e i m p l i c a t i o n of t h e s e results for g r a v i tational theory may be very important since if it is a s s u m e d t h a t t h e S u n ' s surface is a n e q u i p o t e n t i a l surface, t h e n b y following t h e classical a r g u m e n t s of geop h y s i c s (Jeffreys, 1929), it is e a s y t o show t h a t t h e o b s e r v e d o b l a t e n e s s causes a n 8 % d i s c r e p a n c y in t h e E i n s t e i n p r e d i c t i o n of t h e p e r i h e l i o n a d v a n c e of M e r c u r y .

R o x b u r g h (1967) h a s d i s p u t e d D i c k e ' s i n t e r p r e t a t i o n of t h e o b l a t e n e s s on t h e g r o u n d s t h a t t h e solar surface m a y n o t b e a n e q u i p o t e n t i a l surface l a r g e l y b e c a u s e of t u r b u l e n c e in t h e p h o t o s p h e r e . T h i s a r g u m e n t h a s a l r e a d y b e e n challenged b y D i c k e a n d G o l d e n b e r g (1967b). I n a n earlier p a p e r , R o x b u r g h (1964) a t t e m p t e d to a c c o u n t for M e r c u r y ' s entire p e r i h e l i o n a d v a n c e excess of 43" p e r c e n t u r y by postulating a differentially rotating Sun. H e a s s u m e d t h a t t h e a n g u l a r v e l o c i t y of t h e solar i n t e r i o r is a f l m c t i o n of r a d i u s a n d t h a t it is r o t a t i n g a p p r o x i m a t e l y 100 t i m e s f a s t e r t h a n t h e o u t e r regions of t h e Sun. H o w e v e r , t h e s t a b i l i t y of R o x b u r g h ' s m o d e l is q u e s t i o n a b l e . T h e p r o b l e m of c o n s t r u c t i n g a s t a b l e solar m o d e l is a m a j o r q u e s t i o n raised b y the measured oblateness, assuming that a r a p i d l y r o t a t i n g core is t h e cause. G o l d reich a n d S c h u b e r t (1967) 1 s u g g e s t e d t h a t t h e differential r o t a t i o n p o s t u l a t e d b y 1 Also see Dicke, R. H. 1967, Science 157, 960. 492

SOLAR ROTATION

Dicke is itself highly unstable and would be smeared out b y its own turbulence in a short time. Howard, Moore, and Spiegel 2 (1967) raised an objection to another of Dicke's ideas. Dicke assumed t h a t the surface of the proto-Sun once rotated at higher angular velocities than is the case now, and that angular momentum was transferred away from the surface b y the action of the magnetic field in the solar wind. This slowing down of the solar rotation, which must spread throughout the convective regions, need not have penetrated to the radiative core. Howard et al. (1967) have suggested this is not so and under the conditions assumed b y Dicke, they have shown that there is a rotational coupling between the two solar regions causing the core to spin down. A model incorporating the ideas of Dicke is described b y Deutsch (1967). Essentially it consists of three layers: (1) the radiative core is assumed to be differentially rotating at a high angular velocity; (2) the outer convective envelope is assumed to be in near solid body rotation at the observed surface velocity; (3) a thin transition zone is needed between the fast rotating core and the slow rotating envelope. Clearly the problem of building up a detailed solar model including a significant differential rotation is very complex and so it was felt that it would be useful to consider a simpler model in order to gain some insight into the problem of solar rotation. Employing a rotating solar polytrope Winer (1966) used Chandrasekhar's (1933) first order theory of rotating polytropes to discuss the perihelion advance of Mercury. We have used the second order theory developed b y one of the authors (Anand, 1968) to further investigate this problem. The main aim of this paper is to apply this second order theory of rotating polytropes to the Sun in order to investigate the perihelion advance of Mercury as a function of the visual solar oblateness. The plan of this paper is as follows: The mathematics of the problem are formulated in Section II. In Section III, we have taken Also see Dicke, R. H. 1967, Astrophys. J. 149, L121.

493

into account second order effects and calculated the perihelion advance of Mercury as a function of solar oblateness using first, the observed surface rotation to find the expected oblateness of 0.05". In Section IV we present the conclusions drawn from our results. The Appendix contains the details of the calculation of the potential function due to an oblate polytropic mass. I I . FORMULATION OF THE PROBLEM

To calculate the perihelion advance of Mercury due to solar rotation, we have to know the expressions for the orbital perturbation function and the solar oblateness up to the second order in the rotational perturbation parameter v (v = ¢o2/2rG~, where o~ is the angular velocity of rotation and ~, is the central density of the Sun). We derive these physical quantities under the following subheadings.

A. External Gravitational Potential of an Axisymmetric Polytrope In general, one can write the external gravitational potential, V, due to an axisymmetric mass distribution in the following way:

V -

GM + G (C - A)P2(cos 02) r2

F.)3

-t- ~ KP4(cos 02), r2

(1)

where

M = 2r fo" fo e~°) pr'~ sin 01 dr1 dO1, (2) C - , 4

=

2 ~ r f ° ]o f m - ~ ) prl 4~'2(cos " 01) X sin 01drld01,

(3)

K = 27r ~o foR(~)prt6Pt(cos 01) X sin 01drld01.

(4)

In the above, M is the total mass, C and A are the principal moments of inertia about the x axis and the z axis, r is the radial coordinate, 0 is the polar angle, P2(cos 0~) and P,(cos 02) are Legendre polynomials of order two and four; the subscript 1 refers to points internal to the mass distribution causing the gravitational

494

S.P.S.

ANAND AND G. G. FAttLMAN

field, whereas the subscript 2 refers to points external to that mass. In our investigation, we will consider only these first three terms in the gravitational potential. Winer (1966) did not use the third term of Eq. (1) because, in first order theory, that term does not contribute to the perihelion advance. For details of the derivation of Eq. (1), see Brouwer and Clelnence (1961). Introducing polytropic variables in Eq. (3) and (4) we get

or

(~b'2)]~=~ + v2[(a(2B2~b2 + 2f2) -- ~4(B2¢/2 + f'2)]~=~,} (n > 1~;

C -- A = --~X2~raS{vA2[(a(2¢2 -

K = --~X27ra 7{v2[(5(4B4~b4 + 4f4) -- ~6(B4¢'4 + f'4)]t=a + v~v~[q~(q~O ' + 2A2~b~)]~=hl (n = 1), or

K = -- }X2~aVv2[~ 5(4B 4¢4 + f44) ~4Onp2(/z) d~ dla,

(5) ~O"P~(u) d~ du,

(6)

(n > 1).

The Values of all the radial functions at = ~1 are tabulated by Anand (1968).

where r~ = a$,

a ~ = (n -t- 1)~X-l+l/n/4wG,

# = cosO~, p = XO ~, p ~ gpl+l/n.

(7) (s) (9)

The equation of the boundary, E(u), is given by 2(u) = ~1 + v[qo + q~P2(u)] + v~[to + hP2(u) + ttPdu)],

B. Visual Oblateness of Rotating Solar Polytropes

We define the visual oblateness, ~, as twice the ratio of the difference to the sum of the equatorial and polar radii. The equation of the boundary (10) then gives the oblateness in the form (14)

= A51v -Jr- A52v 2,

(10)

where ~ is the nondimensional radius of a spherical polytrope and qo, q:, to, t2, and t~ are constants which are tabulated by Anand (1968). The density function, 0(~,~), has the following expansion up to second order in v: O(~,t~) = O(() + v[¢0(~) + A2~b2(()P2(u)] + v~[f0(~) + f2(~)P2(u) + f,(~)P~(u) + B~¢2(~)P2(t~) + B,~b,(~)P,(~)], (11) where ¢0, ¢:, ¢4, f0, f~, and f, are radial functions and A2, B~, and B~ are constants (for details, see Anand, 1968). On substituting Eq. (11) into Eq. (5) and (6), C - - A and K can be calculated as shown in the Appendix. The resulting expressions are C -- A = --~X2~rot~{vA2[~a(2¢2 -- ~¢'2)]~=~ -~- v~[~a(2B2,¢2 -+ 2f2 ) - - ~4(B2.¢'2 -I- f'e)]~=e~

+ v2~4[qo(q20' + A~b2) + q:~bo]~=~ + } v ~ [ q 2 ( q ~ O ' + 2Ae~b2)]~=~}

(n = 1),

(13)

(12)

where

A~2-

1 1 [

-~

5

3t~A-~t4

3qoq2--~q22] ~ •

(15) C. Calculation of a Planetary Perihelion Advance Due to Higher Order Terms in the Potential V

The orbital perturbation given by Sterne (1960) as

I

oR.,

equation

is

tan li

d-t = (~'a)-I/2 cot ~b- ~ e -[- c o s ~

Oi )

(16) where ~ is the planetary longitude of perihelion measured with respect to the line of nodes in the solar equatorial plane; = G M , M being the mass of the Sun, sin ¢ = e, the eccentricity of the planetary orbit; i is the inclination of the planetary orbit with respect to the solar equatorial

495

SOLAR ROTATION

plane; - R ~ i s the perturbing potential; and a is the planetary semimajor axis. In our case, the perturbation functions

Ow/Oe = s i n w ( 2 + e c o s w )

a/r = (1 q- e cos w) sec 2 ~.

a (C--A)(1-

R~ =

(29)

Combining all the preceding, we finally obtain Equation (16) in the form:

3cos 20)

=~'~2(1-3sin ra

(28)

The expression relating r, a, and e is

are

R4

see 24.

2~),

(17)

G K ( - - 3 -4- 30 cos 2 0 -- 35 cos 4 0)

= 8r2---~

= ~'3q(--3r ~ q-30sin2¢~--35sin4~)8 ,

(18)

de5 (m + 1)3,m dw - am(1 - - e2) m

X [1{(1 q-ecosw)(cosw)[--Pm(~)] + ~ 1

where f~ is the planetary heliocentric latitude and r is now the planetary heliocentric radial coordinate. We also have 7~-= ( C - - A ) / M , y~= K/M, sin 5 = sin i sin u,

(19) (20)

where u is related to the planetary true anomaly, w, b y u = w -t- &.

(21)

In order to get Eq. (16) into a usable form for this problem, it is necessary to manipulate the planetary orbital elements as follows: For the details of certain quoted results the reader is referred to a standard text on celestial mechanics; e.g., Brouwer and Clemence (1961). dco ds~ dt dw dt dw

(22)

w)m_t (1 -~- e c o s

sin w(2 q- e cos w) O [--Pro(u)]t

-t- ~

1

1 0

]

tan ~ i ~ [--Pm(t~)]~.

(30)

If the right-hand side of Eq. (30) is regarded as a function of w only, we can integrate around the orbit as measured b y true anomaly w; and so we find the perihelion advance per period, due to the perturbation R~ given by the expression (m -t- 1)%~ a=~ = a~(1 _ d) m {11 + I~ + I~},

(31)

where

£

I1=

12 =

2~ 1

fo 2~

e (1 + e cos w) ~ cos w[--Pm(y)] dw,

1 m+le

1

(1 + e cos w)"-*

X sin w(2 -t- e cos w) 0 [ - P r o ( , ) ] dw,

where dt

r2

dw - (~-a)1/~ see 4. Generally, the perturbation R,,, can be written

(23) function,

R,, = (~'y,,/r'~+~)[-Pm(t~)J,

(24)

where = sin i sin (w -/- ~).

(25)

(26)

where Or/Oe= --a cos w,

(1 + e cos

w)

m-1

1.0 X tan ~ , ~ [--Pm(~)] dw.

(32)

The integrations expressed in Eq. (32) are straightforward but rather tedious and so only the results will be quoted. (a) m = 2: I1 = ~r -- lTr sin s i[1 -t- 2 sin 2 &], I2 = ---~lr sin 2 i cos 2&,

We m a y also write OR,~ ORm Or OR,~Ow Oe - Or Oe -1- --Ow -~e'

Ia = fo 2~ 1 m+l

(27)

1

Ia = --~r sin 2 i 1 + sec~"

(33)

Therefore the perihelion advance due to R2 is

496

S. P. S. ANAND / k i d G. G. FAHLMAN

3(C

A)

-

A~& -- MAC(1 -- ee)'. l + sec i Note that if terms of the order sin 2 i are neglected, the result is the same as Eq. (11) of Winer (1966). (b) m = 4. In the case of Mercury, we neglect terms of order sin ~ i (sin~ i ~ 10-a v, the perturbation parameter) to obtain sin s i[1 + 2 sin s + ½e2(1 + 4 sins ~)], I2 = ~ r sin s i cos 2~(14 + 5ee),

A,

All = ~

[2Ae¢2(}x)-- ~1 2~h2(~1)1,

Ar, = ~

{2[B2qJe(~0+ fe(~0] -- }l[Be~b'2(~0 + f'2(~0]},

A21 = ~

{q0[q2O'(}0+ A2¢e(~l)]

+ qe~0(~0 }, {q2[q20'(~1) + 2A2~b~(~0]},

A.22 = ~

I~ = --~r(4 + 3e-') + ~

I~-

37r sin2 i 2(l+seci)

[

1 +~

3 ee(1 + 2 sin~ ~)]"

(35) 5vK X { -~-[

I_3(4+3e

X

~) +

{4[B4¢4(~1)+ f4(}0] -- ~l[B6b'4(~l) + f'4(~1)1}, (39)

A41 = A2e.

Similarly, the terms representing the perihelion advance can be written

e) + sin si

1 e(1 + 4 s i n 2 ~ ) ] 1 -I- 2sin2~ + ~e

+~6e°s2~(14+5e

A31 = ~1q

ACo = A61v + A6~v 2,

GO)

where A 61 = C l l A n D-2 M

3 1 2 1 + see i

D2

D4

A6"2 = CnA12 - ~ + C31A31 - ~

[

1+~

+ ~1~ (C11Ae1+ C..Aee) --~ The total perihelion advance per period is given by the sum A~ = Ae~ + A~.

(37)

] I [ . OBLATENESS AND PERIHELION ADVANCE FOR THE CASE OF A I:)OLYTROPIC SUN AND THE PLANET MERCURY

-4- C 4 1 A 4 1 - ~ ] -

The D's depend only on the elements of Mercury's orbit D2

3~r a2(1 -- e2)2

For ease in computation the expressions for C - A and K will be written in the form C -- A = CllAHV + CllA12v 2 K = C~lA31v 2 + ~ln(C41A4~v2),

X D4

2V ~1n(C11Aelve "~ C2eA 2eve)

15

{1 - -

7r

= ~ - a4(1 -- e2)~

sin : i I 1 7 -

CI1 = --§2*rRS~, C~I = --~2~R~,

C~e = {Cn, C41 = ~C~I,

1]}

sinei 1 -- (4 !

+ l+seci

+

3e2)

(38)

where ~1~ is the standard Kronecker symbol, the C's depend only on solar parameters, and the A's only on the polytropic index n.

(41)

+e 2-(1-

6sin s& 1

sin ~-&) + 1 + s e c i

497

SOLAR ROTATION TABLE I COEFFICIENTS APPEARING IN THE ~-XPRESSIONS FOR PERIHELION ADVANCE AND OBLATENESS n A~t As: A~I A6:

1.0 3.750 -12.377 1.263 1.461

X 10 -3 X 10-~

1.5

2.0

3.0

5.780 --4.077 1 . 2 6 7 X 10 -~ 1 . 6 5 1 X 10 -~

9.821 8.341 1 . 2 4 5 X 10 -3 3 . 1 8 5 X 10 - z

41.841 5 . 0 7 6 X 10 -3 1 . 1 5 6 X 10 -3 1 . 5 6 2 X 10 -1

I n the above, M represents the mass of the Sun, ~ the m e a n density of the Sun, a n d R, the solar radius. We have already shown [Eq. (14)] t h a t the oblateness can be written as: (r = ABly + A~2v~,

(43)

where A6I and A62 have been defined in Eq. (15). All the solar system data used in calculating the C and D coefficients was obtained from Allen (1963), whereas the polytropic constants were evaluated using the data given b y Anand (1968). I n the above we have considered polytropic models with a fixed central density. N o t e t h a t to second order in v, at least ( C - A ) and K are unaffected b y a n y changes in the central density due to rotation. The oblateness, however, will be affected in the second order term. The values of A51, A52, A61, and A6~ are tabulated in Table I. The following cases are considered: (a) The observed solar rotation rate, oJ = 2.87 X 10-6 radians per second (at latitude 16 ° ) was used to calculate the p a r a m e t e r v for each polytropic index n = 1, 1.5, 2, 3, 3.5. These values of v were then used to calculate b o t h the solar oblateness and the perihelion advance of Mercury. These results are shown in Table II. N o t e t h a t the oblateness measured in seconds of

3.5 1. 192 -5.642 2.212 4.664

X X X X

10 ~ 104 10 -~ 10 -1

arc actually measures the angular difference between the equatorial and polar radii as seen from the Earth. As can be seen, the calculated a are much smaller t h a n the observed oblateness. T h e effect of this oblateness on the perihelion advance of Mercury is also small, the values of A& being all m u c h less t h a n the reported error in the observations. We note t h a t if the Sun were rotating uniformly throughout with its surface velocity, then regardless of the observed oblateness, the mass quadrupole m o m e n t arising from a rotational distortion would cause a perihelion advance of the a m o u n t indicated in Table II. This interpretation would then necessitate an alternate explanation, other t h a n rotational distortion, for the observed oblateness. (b) Dicke's observed value of the solar oblateness, a = 5 X 10-5 was used to calculate v directly from Eq. (43). These values of v were then used to calculate the perihelion advance of Mercury from Eq. (40). T h e values of the angular velocity, ~0~, at the surface of the Sun which would be necessary to explain the observed oblateness on the basis of a uniformly rotating Sun are also calculated. The results are shown in Table I I I . An inspection of these results shows t h a t the observed oblateness could in fact be explained if the Sun were uniformly rotating

TABLE II OBLATENESS AND PERIHELION ADVANCE BASED ON OBSERVED SOLAR ROTATION n

1.0

1.5

2.0

3.0

3.5

v

4 . 2 4 0 X 10 -6 1 . 5 9 X 10 -5 1 . 5 8 X 10 -2

2 . 3 2 8 × 10 -6 1 . 3 5 X 10 -5 1 . 3 3 X 10 -2

1 . 2 2 3 X 10 -~ 1 . 2 0 X 10 -5 1 . 1 9 X 10 -2

2 . 5 7 4 X 10 -7 1 . 0 8 X 10 -5 1 . 0 7 X 10 -2

9 . 122 X 10 -S 1 . 0 9 X 10 -5 1 . 0 8 X 10 - 3

0. 025

0. 017

a (seconds of arc at earth) ~ (seconds of arc per century)

0.46

0.25

0.13

498

S. P. S. A N A N D A N D G. G. F A H L M A N

TABLE III PERIHELION ADVANCE AND DERIVED ~URFACE ROTATION RATE BASED ON OBSERVED OBLATENESS n

1.0

v A/~ ( s e c o n d s of a r c p e r century) ~, (radians per second)

1 . 3 4 5 )< 10 -5 1.46

8.724 0.95

X 10 -6

5.1341 0.55

X 10 -8

1.205 0.20

X 10 -8

4 . 2 3 2 X 10 -7 0. 080

5.111

5.555

>( 10 -8

5.880

X 10 -~

6.209

X 10 -8

6.181

X 10 -8

1.5

at a rate approximately 2.3 times greater than the observed surface velocity. The corresponding perihelion advance would be small for all n. If we assume the effective polytropic index of the Sun to be n = 3, t h e n the perihelion advance is only 0.20" :per century, a value 17 times smaller than t h a t claimed b y Dicke. Of course we do not observe a rotation rate of nearly 6 X 10-6 rad/sec on the solar surface and furthermore, as Dicke (1967b) points out, the solar envelope can be considered to be rotating nearly uniformly at its surface value down to depths of at least a few hundred kilometers. Hence it is difficult to apply the results in Table I I I directly, but it is felt that they do indicate the possibility of explaining the observed oblateness without invoking an extremely rapidly rotating core. If the outer surface of the Sun is considered to be an equipotential surface then, in the absence of shear stresses in the envelope, we m a y apply a different analysis to find the relationship between the mass quadrupole moment and the observed oblateness. Briefly, the classical equipotential surface theory, as developed by Jefferys, states that if the distorted surface of a uniformly rotating body is an equipotential surface, then one m a y derive, independently of the density distribution in the body, the following simple relationship between the effective quadrupole moment, the visual oblateness, and the ratio of the centrifugal to gravitational force at the equator of the surface: 3C-2

A

MR 2

-

¢

1~2R 3 2 GM

(44)

The quantity on the left-hand side is what Dicke has termed the gravitational

2.0

3.0

3.5

)< 10 - e

oblateness, A¢ since, in first order theory, it is that term rather than a which appears explicitly in the expression for the perihelion advance. Although this equation is based on a first order perturbation theory, second order effects on the solar surface should not cause a large error in the quadrupole moment. It can be shown that to first order, the polytropic theory developed here will reduce to Eq. (44). However, we will show that this equation can be given a somewhat wider interpretation than ordinary polytropic theory allows for. Now Dicke (1964) has stated that in order for the Sun to have maintained a very fast rotating core, it is necessary for the solar interior to be essentially decoupled from the outer convective regions. The model thus suggested essentially consists of a fast rotating core surrounded b y a tenuous, dynamically distinct envelope which because of its small mass, exerts only a negligible influence on the motion of the core. The observed shape of the outer surface is determined solely b y the shape of the gravitational field of the core and the centrifugal force of the uniformly rotating envelope. Hence if A~b is the gravitational oblateness of the core; a the observed oblateness of the outer envelope and ~,, the ratio of centrifugal to gravitational force on the equator of the surface, then we m a y write = AO -~ ½~,.

(45)

Since the mass of the outer envelope is entirely negligible as far as the external gravitational field is concerned, Eq. (45) is formally identical to Eq. (44) but has a slightly wider interpretation. If we now take the fast rotating part of the Sun to be the inner 85~v of the Sun b y radius, i.e., the radiative core, then we m a y apply the

499

SOLAR ROTATION

TABLE IV QUANTITIES RELATING TO THE FAST ROTATING CORE n

1

A6,' A6~1 v ~o (radians/sec) T (period in days) a~

9. 1224 1.0556 4.3366 1.2201 5.9 1.6260

1.5

2.0

3.0

X 10 -4 9. 1561 )< 10-4 8.9953 X 10-4 >( 10-2 1.1926 X 10-2 2.301 )< 10-2 )< 10-5 4.3203 X 10-5 4.395 X 10-5 X 10-5 1.6433 X 10-5 2.287 )< 10-5 3.2 4.4 × 10-4 2.4970 )< 10-4 1.9822 X 10-3

p o l y t r o p i c t h e o r y to e s t i m a t e t h e r o t a t i o n a l v e l o c i t y a n d p h y s i c a l o b l a t e n e s s of t h e core. T h e g r a v i t a t i o n a l o b l a t e n e s s of t h e core is 4 × 10 -~ a n d b y first o r d e r t h e o r y , w h i c h is sufficiently a c c u r a t e for t h i s calculation, t h e c o r r e s p o n d i n g p e r i h e l i o n a d v a n c e of M e r c u r y is 3.4" p e r c e n t u r y . W e use t h i s v a l u e of ~ t o c a l c u l a t e v f r o m E q . (40) a f t e r n o t i n g t h a t t h e coefficients A6I a n d A62 m u s t b e m o d i f i e d t o t a k e i n t o a c c o u n t t h e f a c t t h a t we a r e n o w d e a l i n g w i t h t h e core of t h e S u n only. W i t h t h i s v a l u e of v, w e n o w c a l c u l a t e o~o t h e a n g u l a r v e l o c i t y of t h e core, a n d also t h e p h y s i c a l o b l a t e n e s s ac f r o m E q . (43). T h e s e results, g i v e n in T a b l e IV, show e x p l i c i t l y t h a t a v e r y f a s t r o t a t i n g core is n e c e s s a r y t o e x p l a i n t h e r a t h e r m o d e s t o b s e r v e d surface o b l a t e n e s s . I n p a r t i c u l a r , w e see t h a t if w e a s s u m e t h e effective p o l y t r o p i c index of t h e core to b e n = 3, t h e n t h e core m u s t b e r o t a t i n g w i t h a p e r i o d of 1A d a n d h a v e a n o b l a t e n e s s of n e a r l y 2 X 10 -a. D i c k e h a s s u g g e s t e d t h a t t h e core is r o t a t i n g w i t h a p e r i o d of 1.8 d w h i c h is c o n s i s t e n t w i t h t h e p o l y t r o p i c t h e o r y d e v e l o p e d here. I t should, however, be e m p h a s i z e d t h a t t h e s t a b i l i t y of such a c o n f i g u r a t i o n is q u e s t i o n a b l e . (c) As a check on t h e i m p o r t a n c e of second o r d e r effects, we h a v e t a k e n t h e v a l u e of t h e o b l a t e n e s s to b e 0.3", w h i c h

8.349 1.129 4.710 5.161 1.4 1.982

)< X X X

3.5

10-4 10-1 10-5 10-5

1.598 3.370 2.464 6.269 1.2 X 10-3 2.970

X X )< )<

10-3 10-1 10-5 10-5

X 10-3

was u s e d b y W i n e r , a n d r e p e a t e d t h e calculations leading to Table III. The results a r e g i v e n in T a b l e V. W e n o t e t h a t t h e results for t h e p e r i h e l i o n a d v a n c e a r e n e a r l y twice W i n e r ' s results. T h e r e a s o n for t h i s is t h a t W i r i e r r e d u c e d ~ m e a s u r e d a t t h e solar surface as a n o n d i m e n s i o n a l r a t i o , t o t h e a n g u l a r difference b e t w e e n t h e e q u a torial and polar radii as measured at the E a r t h b y m u l t i p l y i n g b y t h e r a t i o of t h e d i a m e t e r of t h e S u n t o t h e A s t r o n o m i c a l U n i t a n d t h e n c o n v e r t i n g r a d i a n s t o seconds. T h e correct c o n v e r s i o n f a c t o r is t h e r a t i o of t h e solar r a d i u s t o t h e a. u. W i t h t h i s in m i n d we find t h a t t h e second o r d e r effects a c t u a l l y cause a n a p p r o x i m a t e 5 % decrease in W i n e r ' s results. IV. CONCLUSIONS T h e foregoing c a l c u l a t i o n s show t h a t t h e o b s e r v e d solar o b l a t e n e s s is five t i m e s greater than the oblateness which would be p r e d i c t e d if t h e S u n were r o t a t i n g t h r o u g h o u t w i t h its surface r o t a t i o n a l v e l o c i t y . O n t h e a s s u m p t i o n t h a t t h e surface is a n equip o t e n t i a l surface, it is t h e r e f o r e n e c e s s a r y t o a s s u m e t h a t t h e i n t e r i o r of t h e S u n is rot a t i n g f a s t e r t h a n t h e surface. B e c a u s e of t h e l a c k of a useful t h e o r y t o d e a l w i t h d i f f e r e n t i a l l y r o t a t i n g stars, t h e q u e s t i o n of how fast t h e i n t e r i o r m u s t b e

TABLE V PERIHELION ADVANCE AND DERIVED SURFACE ROTATION RATE BASED ON ~ = 0 . 3 " n

v A~ (seconds of arc per century) ~ (radians per second)

1.0

1.5

2.0

3.0

3.5

8.069 X 10-6 5.234 X 10-5 3. 080 X 10-5 7.224 X 10-6 2. 542 X 10-6 8.76 5.71 3.30 0.72 0.48 1.252

X 10-5 1.361 X 10-5 1.440 X 10-5 1.520 X 10.5

1.515 X 10.5

500

S. P. S. ANAND AND G. G. FAHLMAN

rotating can not be readily answered. If a perturbation theory is being considered, then it would be advisable to go to second order terms at least since we have shown t h a t for the fast rotating polytropic models, second order effects are significant, especially in view of the smallness of the quantities of interest. The results show explicitly t h a t if we regard the Sun as consisting of a radiative core surrounded b y a virtually massless envelope with little dynamical interaction between the two regions, then the core must indeed be in very rapid rotation in order to explain the observed oblateness. The outer layers have been reduced nearly to rest b y some sort of braking action, probably connected with the solar wind 3 (Deutsch, 1967) while the inner regions have maintained a primordial fast rotation. I n the absence of shear stresses, the envelope arranges itself so that its outer surface always forms an equipotential surface in the field formed jointly b y its own rotation and the gravitational field of the core. The shape of this outer surface thus betrays the hidden rotation of the core and allows us to calculate its velocity and oblateness as shown in Table IV. On the other hand, the results of Table H I show t h a t if we regard the core and the envelope as sharing the same dynamical motion (in this case, solid b o d y rotation) then it is not at all necessary to assume a very fast rotating core in order to account for the observed oblateness. The main difficulty here seems to be t h a t the slow rotation of the surface can be regarded as extending at least to a depth of a few hundred kilometers into the solar envelope. However, it would be of some interest to construct a solar model incorporating a modest differential rotation extending smoothly from the core to the outer surfaces through the envelope to determine whether or not it could account for most of the observed surface oblateness without necessarily implying a significantly large gravitational quadrupole moment. One implication of these results is t h a t 3 Also see D i c k e 1964 a n d B r a n d t , J. C. 1966,

Astrophys. J. 144, 1221.

the total angular m o m e n t u m of the Sun would not be changed m u c h from the presently accepted values. I t has been argued b y Dicke (1964) t h a t a fast rotating core would of course increase the solar angular m o m e n t u m and if a very fast rotation were assumed, then the Sun would have an angular m o m e n t u m compatible with the upper main sequence stars. I t can be shown (McNally, 1965) t h a t if the angular m o m e n t u m of the planets were added to t h a t of the Sun, then the total angular m o m e n t u m of the solar system is also compatible with the upper main sequence stars. I n fact, this result can be used to postulate the existence of a large nmnber of planetary systems which would revolve around lower main sequence stars (van Den Heuvel, 1966). Thus a not so fast rotating core can be just as compatible with angular m o m e n t u m arguments as a fast rotating core. A decided advantage to having the core rotating only slightly faster t h a n the surface is t h a t the stability problems are not so severe. I n particular it is possible to construct a plausible differential rotation which satisfies the stability criterion t h a t the angular m o m e n t u m per unit mass should monotonically increase outwards from the axis of rotation. The steep gradient of angular velocity in the decoupled model proposed b y Dicke violates this criterion and forms the basis of most of the instability arguments. Such considerations would be obviated if it can be shown t h a t a suitable distribution of angular m o m e n t u m throughout the whole Sun could account dynamically for most of the observed oblateness. At the present time, work is being undertaken to develop a solar model incorporating a modest differential rotation distributed smoothly between the core and the envelope (Naylor, Fahlman, and Anand, 1968). ACKNOWLEDGMENTS

The authors wish to thank Professors M. J. Clement and Robert Roeder for their useful comments and also Mr. Mark Naylor for his assistance in the numerical computation. One of the authors (G. G. F.) wishes to acknowledge a very fruitful discussion on this problem with Professor R. H.

SOLAR ROTATION ]:)icke. This work was supported by th: National Research Council of Canada.

c - A =

The expression for C - - A is given by Eq. (5) of the text as f40"P2(u) d~ d~; (il) now consider the following integral

f

The second integral on the right-hand side of Eq. (A2) gives (vB12

-~- v2B~)[On$ 4

1

"4- nOn-lv~(~bo + A2¢EP2)]~ffih + ½v~B~[nO,-~O'~ ~ + 4~0"]~=~,,

(A3)

B,2 = qo -4- q2P2 B ~ = to + t~P2(u) + t,P4(u).

(A4)

where

In writing Eq. (A3), all terms up to order v2 have been included. Since at ~ = }1, 0(~) = 0, then from Eq. (A3) it follows that

~ Z(t~)One4d~ = 0

+

"1oe`

+ n(n -

1)0"-2(.~A2~2 + ¢oA~¢~)f4 d~ 1 •

(A8) The radial functions ~b2 and f2 satisfy the following equations given b y Anand (1967): 1 d (

d¢~

1 d (

df~

¢~

[z(") o-~ ~d5 (A2)

=

'l(vA

X n0"-1~2~ 4 d~ + v~ fo e` nO"-if2

APPENDIX

f z(,) 0 , ~ d~

501

2

--}- n ( n - 1)0 "-2 )< (~A22¢~2 -4- ¢0A2¢2). (A9) B y substituting the left-hand side of Eq. (A9) into the corresponding integrals appearing in Eq. (A8) and integrating the result by parts twice, we obtain C - A = -§h2~as{v[~a(2A2¢2 - ~A2¢/2)]~=~ -4- v2[~3(2B2¢2 A- 2f2) -- (4(B~¢'2 -4- f'2)]~=~,}. (A10) (ii) n = 1.

c - A =

f,

fo o°e

-t- ½B~220']~=~,P2(tt) dt~

= v2[B~2~(~b0+ A2¢2P~) -t- ½B~2O'~]~=~, (n = 1).

(A5) Hence we may distinguish two cases for

d.

A- 2~r~a5 f -~ B,2~14[(¢0 -4- A2¢2P2)

(n > 1)

X

(All)

The first integral appearing in (All) is the same integral evaluated in the case n > 1. Recalling the definition of B12 and using Eq. (A7), we may easily evaluate the second integral. We finally obtain C -- A = -~2~raS{v[~3(2A2¢2 - ~A 2¢' 2)]~=~,

C-A.

+ v~[~a(2B2¢2 + 2f~) - ~4(B2¢'~ + f'2)]~=~, + v~14[qo(q~O' + A2~b2) + q2¢~0]~ + {-~14[q2(q~O' + 2Ad~2)]~=~, (A12)

(i) n > 1.

(h6) For reference, we make note of the well known result: f - ~ P,,,(#)P,~(tt) dtt =

6f~=_{nO,_,f

2

2n -4- 1 6....

(A7)

Substituting for 0 given by Eq. (11) and using Eq. (A7) we find that, to second order in v, Eq. (A6) reduces to

The expression for K is given by Eq. (6) of the text as K

2~rha ~ f --1 f0"z(,u)~6Onp4(~) d~ d~. (A13)

Following the same procedure as given in Eq. (A2) to (A4), but this time with the integral ~ ( " ) 0"~ ~ d~,

(A14)

502

S. P. S. ANAND AND G. G. FAHLMAN

we find t h a t foz(~) O",}6 d~ = 0

(n > 1)

= v~B12}16[~o + A2~b2P2]~h + ½B~22v2}~eO'(}~) (n = 1). (A15)

T h u s we are able to distinguish the following two cases for K : (iii) n > 1. g = 2T'hO~7 f - - 1 f0~l One6 d~P4(j.,t)dj£

(A16)

Again substituting for O a n d using Eq. (A7), we find K = - 2 7 r h a 7 -~ 2 v 2 { f o ~ ~6nO~-lB4¢~4d~

- 1) + n(n ----5---

gg ~~ ] ~ d~. (A17)

o . _ ~ l S A O . ,1

T h e radial functions ~'4 and f4 satisfy (Anand, 1967)

~d}

~ d } ] --

}~ = - - n O " - ' f ~

+ n ( n -- 1) 0,_ ~ 18 3-~ A~2~ ~.

Proceeding as with C - A ,

(A18)

we obtain

K = --§h2~ra~v2[}~(4B#4 -4- 4f~) -- }~(B,$'~ + f'*)]~=w

(A19)

(iv) n = 1. U ---- 2"/l'~Ot'f1-1 f0 ~1 0"~ 6 d~P4(g)d~

+ -J7 £ - 1 ½Bi22~160,(~)p~(, ) d/zl

(A20)

T h e first integral in (A20) was evaluated in Case (iii). T h e additional terms can be calculated quite straightforwardly and we finally obtain K = -- §h27ra 7 {v2[}54(B4~b4 -}- f4) -- }6(B4~b'4 + f'4)]~=h + ~,v~}16[q~(q~O' + 2A2~k2)]~,}.

(A21)

t~EFERENCES ALLEN, C. W. (1963). "Astrophysical Quantities," Athlone Press, London. A~AND, S. P. S. (1968). Astrophys. J. 153, in press. BROUWER, D., AND CLEMENCE, G. M. (1961). "Methods of Celestial Mechanics." Academic Press, New York. CHANDRASEr~HAR,S. (1933). Monthly Notices Roy. Astron. Soc. 93, 390. DEUTSCH, A. J. (1967). Science 156, 236. DicKs, R. H. (1964). Nature 202, 432. DICKS, R. H., AND GOLDEN'BERG,H. M. (1967a). Phys. Rev. Letters 18, 313. DICKE, 1:~. H., AND GObDRERO,H. (1967b). Nature 214, 1294. GOLDREICH, P., AND SCHUBERT, G. (1967). Science 156, 1101. HOWARD,L. N., MOORE,D. W., AND SPIEGEL,E. A. (1967). Nature 214, 1297. ,]'EFFREYS,I-I. (1929). "The Earth," 2nd ed., Chap. XII. Cambridge Univ. Press, London and New York. McNAbLY, D. (1965). The Observatory 85, 166. NAYLOR, M., FAHLMAN, G. G., AND ANAND, S. P. S. (1968). To be published. ROXBURGH, I. W. (1964). Icarus 3, 92. ROXBURGH,I. W. (1967). Nature 213, 1077. SCHWARZSCHILD, M. (1965). "Structure and Evolution of the Stars." Dover, New York, (Originally Published by Princeton University Press in 1958). S~RNE, T. E. (1960). "An Introduction to Celestial Mechanics," Chap. 4. Interscience, New York. VAN DEN HEUVEL, E. P. J. (1966). The Observatory 86, 113. WINSR, I. M. (1966). Monthly Notices Roy. Astron. Soc. 132, 401.