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On the figure of the moon
ICARUS 7, 275-282 (1967)
:"
On the Figure of the Moon V. S. S A F R O N O V
O. Schmidt Institute of Physics of the Earth, Academy of Sciences, Moscow, U.S.S.R. Communicated by B. J. Levin Received September 15, 1966 The influence of a partially molten state of the central region ("the core") of the Moon, and of the latitude variation of the surface temperature on the lunar figure is considered. Isostatically adjusted models of the Moon regarded to consist of a homogeneous solid outer layer and a homogeneous core with slightly smaller density and higher eUipticity are calculated. In Section I the outer surface and the interface are assumed to be spheroidal. In Section II the effects due to the departures from spheroidal form are considered. Because of a difference of 100-140° in subsurface equatorial and po]ar temperatures of the Moon the thickness of the solid layer can be 60-100 km greater in polar regions than in the equatorial zone, and the core can be much more oblate than the Moon as a whole. With this difference in thickness of the solid layer our model can possess the observed value of (C -- A)/C if the surface layer is 2-3~ denser than the core. This result confirms Levin's suggestion connecting the dynamical flattening of the Moon with the latitude dependence of its temperature. Recently B. J. Levin (1964, 1966) tentatively explained the oblate dynamical figure of the Moon b y the nonisothermal state of its interior. According to the calculations of the lunar thermal history by Levin and Mayeva (1960), Levin (1962), and Mayeva (1964) the Moon seems to possess a semimolten central part ("the core") and a solid outer layer. Moreover, the constant subsurface temperature in polar regions of the Moon should be more than 100 ° lower than in the equatorial zone. Therefore, in the polar regions the solid layer should be thicker and its material denser than in the equatorial zone. Levin's calculations showed that the difference in temperature of the solid layer m a y account for about one-third of the observed value ( C - A ) / C of the ratio of the momenta of the lunar globe. But the decrease of density due to the melting is much greater than that due to thermal expansion, and the difference in thickness of the solid layer should result in a larger value of ( C - A ) / C . Levin believes that the combined action of these two effects may be © 1967 by Academic Press Inc.
sufficient to account for the polar flattening of the Moon as inferred from its librations. This hypothesis has been criticized by Kopal (1965, 1966), who established an analytical solution of the equation of heat conduction for a solid Moon characterized b y a constant value of thermal diffusivity K and heated unidirectioually by the Sun from outside. I n his solution the initial temperature was taken to be zero and the present surface temperature was assumed to v a r y as 250 ° sin '~ 0, where 0 denotes the lunar colatitude (polar distance). Kopal concluded t h a t even if the lunar axis of rotation remained invariable in space throughout most of the astronomical past of our satellite, unidirectional solar radiation could have increased the temperature of the lunar i~terior b y dozens, but not hundreds, of degrees. 1 Nevertheless, Kopal's simple model does not take into account certain features 1It should be noted that, due to an oversight in the expansion of the surface temperature in terms of spherical harmonics, Kopal's temperatures should be multiplied by a factor ~-. 275
276
v . s . SAFRONOV
characteristic of the real Moon, such as the fact that the dependence of K on T in the presence of radioactive heat sources may considerably increase the deviation from spherical symmetry of the temperature distribution. Because the lattice conductivity is inversely proportional to the temperature, the difference AT in average temperature at the poles and the equator of the upper layer may become greater than the difference of subsurface temperature. If so, the results deduced from Kopal's analytical solution of his simplified model are not applicable closely to the real Moon. Of the two temperature effects on the figure of the Moon discussed by Levin we shall consider below only the main one which has not yet been evaluated, namely, the role of different thicknesses of the solid layer in the equatorial and polar zones. In doing so we shall assume that the Moon consists of a homogeneous semimolten "core" of density pl, surrounded by a homogeneous solid outer layer of density p0 which is several per cent higher than m. The thickness of the layer at the poles is d km greater than at the equator. The problem is to find the equilibrium shape of this model. The term "equilibrium" must be precise. Apart from the density inversion along the radius the model cannot be in hydrostatic equilibrium when the surface coincides with the equipotential surface. Because of higher density of material in polar zones and the oblateness of the Moon the gravitational potential at the poles is higher than at the equator. Hence the upper layer must have sufficient strength so that the surface material does not flow from the equator to the poles. The strength of the lunar crust is high enough to satisfy this requirement. But the layer is supported by a semimolten core; and if there are no forces that would prevent the bending of the layer (or if they are relaxed) the Moon should acquire the form corresponding to isostatie equilibrium--the weight of the material in a column from the center to the surface must be the same in any direction. We shall consider such isostatie equilibrium taking place after the complete relaxation of stresses that prevent displacements in the verLical direc-
tion. At first we consider a simpler model, the core and the outer layer of which are confined by spheroidal surfaces. The modification of this result due to deviations of the surfaces from spheroids will be found subsequently. I. SPHEROIDAL BOUNDARIES OF THE OUTER SOLID LAYER
Let us assume that the core is an oblate spheroid with the principal semiaxes (al, cl) and density p~. The solid layer has a density p0 and its outer boundary (i.e., the lunar surface) is a spheroid with semiaxes (a0, co). The gravitational field of such a model may be represented b y the sum of the fields of two spheroids, (a0, Co) of density p0 and (a,, cl) of density p , - - p 0 = - - A p . Let X0, X1 and Z0, Z1 denote gravity accelerations produced by these spheroids along coordinate axes x, z directed along equatorial and polar radii. The condition of isostatic equilibrium for the m a t t e r along these axes
fo°° (Xo + xl)p x = fo" (Zo + z,)pdz
(1)
takes the form
(,o
ap) ;a, (Xo + Xl)dX +
(po
I
Ap) foc' (Zo + Z,)dz
+ po ~o (Zo + Z1)dz.
(2)
Denote Ap/m = ~ and V0, V1 potentials of the spheroids. Then we find
(v0 + vl)o~ - (v0 + v~)oo -- a[(V0 + Vl)c, -
(V0 + V1)I., = 0,
(3)
where the indices a, c mean that the potential is taken at the ends of semiaxes a, c. A more general condition of isostatic equilibrium applicable to any direction ~b of radius-vector r may be written in the same form
(Vo + V~)oo - (Vo + V~)~o -- ~[(V0 + V ~ ) c , -
(V0 + V,)r,] --- 0,
(3')
where r0 and r, are the distances from the center of intersection points of the spheroids and of the radius-vector r at a latitude q~.
277
ON T H E F I G U R E O F T H E MOON
Let us write the well-known expression for t h e potential inside a homogeneous ellipsoid of rotation in polar coordinates
(r, ~)
4 2[ 1 + ~ 1/o2(4_ cos~ ~)] V0r° ~ ~Gp0c0 4 Vor, =
2rGpa~c [ (a ~ _ c2)~/2 (a 2 -
Vi=
c~)arctanl
(
~q~ a r c t a n l
--~cos
2[ 3
C12 2Co2
~ ~-Gpoco t_-
+/o2 (1 _ 5_~o2 3 0 ~+C 1 ' Cos cos2 ~b)]
1+l ~ VI~. ~ --gTrGAo--
1+112
Co
-
C12 12 2--~o2 ~1 c o s 2
r ~ sin ~ ~b(1 -- a r c t a n / ) / , J
1
5 C02
(4) + i-5 ~
c°s~ ~
-
-~ c o s
where
a 2 _ c2
z~ = - -
(5)
C2
Va~, ~
-gTrGApc~ ~ 1+
h 2 ( 4 - cos2~b) • (10)
The potential V~ in a point outside the spheroid m a y be found from expression (4) substituting instead of l the quantity l' a 2 _ c2 1'2 -- C2 -]- U "
(5')
Taking ¢ = ~r/2 we find V0 and V1 in the points Coand c~ of small semiaxis. Substituting this values of the potential into (3') and denoting ci//co
=
h~/lo 2 = (
~
(11)
where u is a positive root of the equation r cos2 ¢ + rs i n 2 ~ a2 + u c~ + u
1 = 0.
(6)
For small 1 this equation yields to the order of small quantity 12 c~ + u ~ r ~ -
c~l2 cos2 ~.
4
[3
4
c~ [
-
2
~ -
3)
26-' - i-2(5~ - 3) -= r2(5 -- 3~3 -- 26) ¢
'
1
•
(8)
At the surface of the spheroid r 2 ~ c~(1 + l 2 cos~ ~).
(
(9)
Substituting this expression for r into (8) we obtain for the spheroids (a0, co) and (al, c~), respectively,
(12)
This condition is fulfilled at any ¢ if the expression in the squared brackets is equal to zero. Then ~ must be equal
r~
+l ~ 1--g~+~-~cos-~b
~ 6~
-- g r3 ~ + 62f2( = 0.
2c 2
+l 2 1-~+]-6~cos~ V . ~ g ~rGP -r
47FGpoC02102COS2~[12
1~
(7)
Developing (4) in ascending powers of 1-', substituting values of I and l' from (5), (5'), and (7), respectively, we find that, to the order of i2,
Vi ~ -~ lrepc 2
we find the equation of isostatic equilibrium along the polar radius and that with a latitude ~bin the form
(13)
The parameter ~ is connected with the difference d of thicknesses of the layer at the poles and the equator by the relation d = Co -- cl
--
1 2 (ao -- al) ~ ~colo ( ~ -- 1),
(14) where 102 is related with tim difference ( C - A ) / C . For a homogeneous spheroid the moments of inertia are A = }M(a 2 + d), C = ~2 M a 2 C - - A = ~ M ( a 2 - d ) = ~Mc21 ~.
(15)
278
v.S.
SAFRONOV
TABLE I THE EXCESS POLAR ']~HICKNESS d OF THE OUTER SOLID LKYER ~ND THE EQUATORIAL BULGE at -- c~ FOR DIFFERENT VALUES OF ~" AND ~':
J: d d~ a, - c, (km)
0.6
0.03
0.01
0.03
o.01
0.03
92 75 1.2
30 75 1.2
94 55 1.3
31 55 1.3
111 38 1.5
36 38 1.5
In our model of two spheroids
c = ~Moco~(1 + lo~) + ~Mlci~(1 + ZI~), C - A = ~Moco~lo2 -b 1Mlci2ll2 ~l° 2MOq-MI~= 2 MoWMI~"2
10~l- ~5~ 2 1 - ~5 (16)
Hence, according to (14) (_/ C - A ( ~ - 1)(1 - ~i-5) co ~ C 1 - ~5~
(17)
For the majority of terrestrial rocks the change of density hp during melting is about 5% and in some cases it reaches 10%. At the depth of the boundary of the core (500 -- 700 kin) this change may be 1.5-2 times smaller and for a partial melting still less. The values of d that provide the observed value ( C - A ) / A = 0.00063 are given in Table I for different l" and for - 0.01 and 0.03(hp = 0.03 and 0.1 gm/ cmS). They range from 30 to 110 kin. The corresponding difference between the equatorial and the polar radii of the Moon is in the range from 1.2 to 1.5 kin. The actual value of d may be approximately evaluated from the relation d~
higher than the difference ATo of lunar surface temperatures at the equator and the poles. On the other hand
(
and to the order of Ios and 112 C--A C
0.8
0.7
0.01
AT (dT/dz) D
where (dT/dz)D is the temperature gradient in the layer at the boundary of the core; and AT, the temperature difference in the layer under the equator and the pole at the depth of the equatorial boundary of the layer. From the numerical integration of the equation of heat conduction Mayeva (1964) has found that owing to the dependence of thermal conductivity on temperature, AT is
dT
)
/dT\ =TM-To < \Tz/av D
'
where TM is melting temperature of silicates; To, the surface temperature at the poles; and D = c i ( 1 - i'), the thickness of the solid layer at the poles. Therefore AT AT d > d m = ((dT/dz))~v = Tu -- T--------~o D AT --
TM
--
To ( 1
--
~-)co.
It can be shown that for the inclination 1° 32' of the lunar equator to the ecliptic the energy received from the Sun by unit ares in the polar regions is 102 times smaller than in the equatorial zone. Therefore the surface temperature at the poles must be about 100°K. Values of d~ found for Tu = 1500°, To = 100°K, AT = 150° are also given in Table I. They are in accordance with the conclusion by Mayeva (1964) that d is about 50-100 km. We see that the coincidence of the values of d which provides the observed (C - A ) / C with d expected from the difference of the equatorial and the polar temperature of the solid layer is reached in the range of values of 5 and ~ given in Table I. In our opinion this result confirms the hypothesis by Levin. IX. I~ONSPHEROIDAL BOUNDARIES OF THE SOLID LAYER
It was shown above that to the order of small quantities l~ and ll~ the body composed of two homogeneous concentric spheroids is in isostatic equilibrium along the polar and any other axes when the ratio
279
ON T H E F I G U R E O F T H E MOON
~T
L I FIG. 1. The models of the Moon consisting of the homogeneous outer solid layer and of the homogeneous partially molten central core with slightly smaller density: A, spheroidal boundaries of the layer; B, nonspheroidal boundaries of the layer. = 12/ll2 satisfies Eq. (13). T h e actual shape of the interface between solid and semimolten substance depends on the temperature distribution on the lunar surface and is in general not a spheroidal one. T h e influence of its departure from spheroid on the shape of the surface of the Moon and on the required value of d m a y be taken into account adding to the spheroids (ao, Co) and (as, c~) two small masses M2 and M r - o n e on the boundary of the core and the other on the surface of the M o o n (Fig. 1). T h e dependence of the subsurface temperatu:e of the M o o n on the latitude ¢ can be assumed in the form
and for its departure At1 f r o m the spheroid r,1 ~ c + ( a - c ) cos 2~b
T~ = To + AT cos ~ ¢,
F o r the intensity of gravity g we can write
(19)
where n is evidently more t h a n ~. Troitsky (1954) assumes, for example, n = ½. If in the first approximation we take a constant t e m p e r a t u r e gradient d T / d r and assume a spheroidal shape of the lunar surface we obtain for the radius of the core rl ----C l ~ - d c o s ~ ¢ ~ - ( a o - - Co) COS2~b (20)
Arl= rl-ra
~d(cosn¢-
cos 2¢),
(21}
n < 2 and Arl > 0. Therefore, the additional mass M2 must be negative because the density between r,1 and r~ is diminishing by Ap as compared with the case of spheroids. F o r the isostatic compensation the corresponding positive mass Ma must be added at the surface of the Moon. T h e thickness Ar0 of the additional layer is evidently given by the relation gopoAro = g~hpArl.
(22)
Hence Aro ~ ~'Ar~ ~-- ~'d(cosn¢ -- cos 2~)
(23)
and M s ~-. - - M ~ / v .
(24)
280
v.s.
SAF~ONOV
T h e e q u a t i o n of i s o s t a t i c e q u i l i b r i u m will b e s i m i l a r to (3) b u t i n s t e a d of (V0 + V~) w e m u s t w r i t e n o w Vo A- V~ -t- V~ -t- Va, w h e r e V: a n d Va a r e p o t e n t i a l s p r o d u c e d b y t h e m a s s e s M s a n d Ma. As t h e s e masses a r e s m a l l in c o m p a r i s o n w i t h M0 t h e p o t e n t i a l s V~ a n d Va in s q u a r e b r a c k e t s of (3) w h i c h a r e m u l t i p l i e d b y s m a l l f a c t o r ~ m a y be neglected. T h e y give t o g e t h e r a n e g a t i v e c o r r e c t i o n in d of a b o u t 1 kin. T o e s t i m a t e Vs a n d Va in t h e p o i n t s Co, a0 we c a n use t h e w e l l - k n o w n expression r e l a t i n g e x t e r n a l pot e n t i a l w i t h m o m e n t s of i n e r t i a (Menzel, 1955) y ~ G M _~_GA - ~ - B - t - C - 3 1 r 2r ~
GMs
G(Cs - A~),
Co
COa
Vs~° ~
GMdd 2Co
z = r sin q~ and
dM2 ~- --ApArlri 2 COS~d~bd~. Therefore C2
-
As = f r12(cos2 ~ cos2h - sin s oh)riM2
vdAp f+~/2 r14(cos~ ~b -- cos 2 ~) J -~/2
)< (2 -- 3 cos s ~b) cos ~bd~
vclSllSAp --/o,~/2(cos ~ ¢ -- cos ~ ~)
w h e r e owing to al -- cl >> a0 -- Co w e h a v e t a k e n d ~ a~ - c, ~ c11~2/2. I n t h e s a m e w a y
dM3 = p0r02Ar0cos ~d4~dX ~ - dM~/~, Ca -- A3 = f r02(coss ~ cos 2 X -- sin s ch)dM~
3G(C~ - A~) 2c0a
(26) (27)
-- ~--3(C2 -- As),
(33)
~ - 45 @ 5 ( 1 _ ~._3)i~"
(34)
a n d finally Ks~-K~
and
8
V~o -- V~oo ~
GM~los 2c0
3G(Ca - A~) 2c0~
(28)
T h e expressions (27) a n d (28) will give a d d i t i o n a l t e r m s in (12) w h e r e w e shall t a k e = 0. T h e first t e r m s in t h e r i g h t - h a n d p a r t s of (27) a n d (28) a r e s m a l l in c o m p a r i son ~4th t h e first t e r m of (12) a n d m a y b e n e g l e c t e d ( t h e y f o r m a p a r t of i t e q u a l to g[(M~ + Ma)/Mo] < 10-a). T h e r a t i o s of t h e s e c o n d t e r m s in (27) a n d (28) to t h e first t e r m of (12) are, r e s p e c t i v e l y ,
Ks =
(31)
X (2 -- 3 cos s ~b) cos ~bd~b = ~rApc15112I, (32)
'
Vsoo = GM---2- G(Cs - A~) a0 2a0~
VSco --
x = r cos ~b cos k, y = r c o s ~ s i n h ,
(25)
w h e r e I is t h e m o m e n t of i n e r t i a r e l a t i v e to Che axis d i r e c t e d a l o n g r a d i u s - v e c t o r r. T h e n g2vo *~"
In the polar coordinates
T h e c o n d i t i o n of i s o s t a t i c e q u i l i b r i u m now becomes 2~-1~ -2 + 3 -- 5~= 5 -- 3~-a -- 2if -- 11.25I(1 -- ~-~)
F o r t h e e v a l u a t i o n of d we m u s t r e p l a c e in (14) t h e p a r a m e t e r l0s b y (C - A ) / C . F r o m (15) a n d (29) 2 Cs - As ~ -- ~ K s ( C 0 -- Ao), C3 -- A3 ~ -
15 (C~ - A~) 2 MocoSlo~
K~ =
15 (C~ - A~) 2 MocoSlos
(29)
A d d i t i o n a l t e r m s K~ a n d Ka will e n t e r i n t o t h e s q u a r e b r a c k e t s of (12). I n o r d e r to e s t i m a t e t h e m we m u s t e v a l u a t e t h e differences of t h e m o m e n t s of i n e r t i a C - - A
Cs = f (x s -t- yS)dM~, As = f
(~ + ze) dM~. (30)
(35)
2-~ g ~ ( C o -
Ao).
(36)
A d d i n g t h e s e differences into t h e r i g h t p a r t of (16) w e find C -- A C
+ K~) l02 [1 -- ,~(Ks 2 2 1 - - ~-5
iti'~]
and l02
2
C - A C 1 -
1 - ~-5 8~-5~[1 + 3.75I(~ -3 - - 1)] (37)
ON
THE
FIGURE
OF
THE
MOON
281
T A B L E II P A R A M E T E R S OF T H E M O D E L FOR n = 1 / 2
AND n = 1
0.01 i':
0.6
0.03
0.7
0.8
179 0.15 0.0011 181 1.9
189 0.14 0.0012 191 2.2
126 0.10 0. 0009 128 1.5
143 0.10 0.0010 145 1.8
0.6
0.7
0.8
66 0.060 0.0011 68 1.85
66 0.056 0.0012 68 2.0
72 0.054 0.0014 74 2.4
42 0.041 0.0008 43 1.4
42 0.036 0.0009 44 1.6
48 0.036 0.0011 50 1.8
n = 1/2 d (km)
185
112/2 ~ c~1 lo~/2 .-~ c~o
al - o (kin) ao -- co (kin)
18 0.0010 187 1.75
d (kin)
125
O.
n=l l~2/2 .~ c~ lo2/2 ~ ~o
O. O.
al - cl (kin) ao -- co (km)
12 0008
126 1.4
T h e r e f o r e f r o m (14) d_
C-A
Co
C
(i'$1 -
1)(1-
~i-5)
@ ~ [ 1 + 3.75I(~ --3 - - 1)1
(3s) H e r e ~ is to b e t a k e n a c c o r d i n g t o (35) a n d I is t h e i n t e g r a l in (32). F o r n = ½, I = 0.14 a n d f o r n = 1, I = 0 . 0 7 . 5 T h e v a l u e s o f d f o u n d f r o m (38), t h e e l l i p t i c i t i e s of t h e
d i f f e r e n t v a l u e s of ~ a n d ~'. I n T a b l e I I I d e v i a t i o n s (rso - - co) of t h e s p h e r o i d (ao, Co) f r o m t h e s p h e r e of r a d i u s co a n d d e v i a t i o n s of t h e l u n a r s u r f a c e f r o m t h e s p h e r o i d (Aro) a n d f r o m t h e s p h e r e (Aroo) a r e g i v e n f o r = 0.03, ~- = 0.7 a n d f o r d i f f e r e n t v a l u e s of ~. I t is s e e n f r o m T a b l e I I t h a t t h e e l l i p t i c i t y ai of t h e c o r e is a p p r o x i m a t e l y i n v e r s e l y
TABLE III THE ~:
D E V I A T I O N S OF T H E S U R F A C E FROM S P H E R I C A L AND S P H E R O I D A L F O R M 0
15
30
r~ -- c0 (km) 2~ro = r3 -- rso (km) Aroo = ro -- co (kin)
2.00 0 2.00
1.86 0.07 1.93
1.50 0.25 1.75
r~o -- co (kin) hro = ro -- r,o (km) hroo = ro - - co (km)
1.56 0 1.56
1.45 0.03 1.48
1.17 0.10 1.27
45
60
75
N0
0.50 0.65 1.15
0.13 0.61 0.74
0 0 0
0.39 0.22 0.61
0.10 0.17 0.27
0 0 0
n = 1/2 1.00 0.47 1.47 n=l
s p h e r o i d s (a ~ 1 ~ / 2 ) , a n d t h e d i f f e r e n c e s of t h e i r p r i n c i p a l s e m i a x e s (a - - c) a r e g i v e n i n Table II for n= ½ and n = 1 and for 2 These values of I are found assuming r = c. This approximation is good for the outer spheroid as l0 is very small. But for the inner spheroid the correct value of I is somewhat greater and leads accordingly to a smaller value of d (by about 10% at ~ = 0.01). In Table II this correction is taken into account.
0.78 0.18 0.96
p r o p o r t i o n a l to t h e r e l a t i v e d i f f e r e n c e ~ of d e n s i t i e s of t h e solid l a y e r a n d of t h e core. T h e e l l i p t i c i t y a0 of t h e M o o n as a w h o l e a n d a c c o r d i n g l y t h e d i f f e r e n c e of t h e e q u a t o r i a l a n d t h e p o l a r r a d i i a 0 - Co of t h e M o o n depend on ~ very weakly and are determ i n e d m a i n l y b y t h e s h a p e of t h e core. I n t h e c a s e of t h e s p h e r o i d a l c o r e (n = 2) a0 - - Co ~ 1.2-1.3 k m a n d t h e l u n a r s u r f a c e is also s p h e r o i d a l . A t n = 1, a 0 - Co
282
v . s . SAFRONOV
1.5-1.6 kin. At n = ½, a0 - co ~ 2 k m and seems to be appreciably greater t h a n is found f r o m the measurements of the lunar disk. T h e observational d a t a on the geometric figure of the Moon and also on the dependence (19) of the lunar surface t e m perature on the latitude are still very uncertain. Besides, variation of the t e m p e r a ture gradient d T / d r in the solid layer with makes n in (20) and in Tables I I and I I I different from n in (19). For these reasons a n y speculations about possible deviation of the M o o n f r o m isostatic equilibrium would be premature. Inhomogeneity of the Moon connected with the existence of the less dense and more oblate core influences not only the shape of the M o o n but also its external gravitational field. Therefore, it should be possible to s t u d y this inhomogeneity and to find precisely the parameters of the lunar model considered above with the aid of observational d a t a on the motion of artificial satellites of the Moon. CONCLUSION
The calculations of isostatically adjusted lunar models consisting of a homogeneous core and a homogeneous outer solid layer and possessing the observed value ( C - A ) / C give the relation between the difference of densities of the layer and the core and the difference of thickness of the layer in the polar and the equatorial regions. For the 60-100 k m difference of thickness expected from the 100-140 ° difference of the equatorial and the polar subsurface t e m p e r a t u r e the outer layer m u s t be 2 - 3 % denser t h a n the core. T h e requirement is in accordance with the idea a b o u t the semimolten state of the lunar core.
This result supports B. J. Levin's h y p o t h esis accounting for the observed difference ( C - A ) / C of the M o o n b y latitude dependence of its subsurface temperature. In order to reveal possible departure of the M o o n from isostasy, more definite d a t a on the shape of the Moon, on its external gravitational field, on the variation with latitude of its m e a n surface temperature, and on the t e m p e r a t u r e gradient in the solid layer are necessary. I:~EFERENCES
KOPAL, Z. (1965). Effects of thermal expansion on the moments of inertia of the Moon. Icarus 4~ 166-172. KOPAL,Z. (1966). "An Introduction to the Study of the Moon," Astrophysics and Space Science Library, Vol. 4, pp. 100-104. D. Reidel Publ. Co., Dordrecht, Holland. LEvIN, B. J. (1962). Thermal history of the Moon. In "The Moon," Intern. Astron. Union Symp. No. 14 (Z. Kopal and Z. K. Mikhailov, eds.). Academic Press, New York and London. L~.VIN, B. J. (1964). A new hypothesis for the nonequilibrium figure of the Moon. Nature 202, 1201-1202. LEVIN, B. J. (1967). "Thermal Effects on the Figure of the Moon." Proc. Roy. Soc. London A,296, 266-269. LEVIN, B. J., AND MAYEVA,S. V. (1960). Nektorye
rastchety termicheskoi istorii Luny. Dokl. Akad. Nauk SSSR 133, 44. MA~vn, S. V. (1964). Nektorye rastchety termicheskoi istorii Marsa i Luny. Dokl. Akad. Nauk SSSR 159, 294. M~z~L, D. H. (ed.) (1955). "Fundamental Formulas of Physics." Prentice-Hall, New York. PIZZETTI, P. (1913). "Principii della teoria meccanica, della figura dei planefi" (E. Spoerri, ed.), Pisa. TaoiTsxv, V. S. (1954). K teorii radioizluchenija Luny. Astron. Zh. 31, 511.
:"
On the Figure of the Moon V. S. S A F R O N O V
O. Schmidt Institute of Physics of the Earth, Academy of Sciences, Moscow, U.S.S.R. Communicated by B. J. Levin Received September 15, 1966 The influence of a partially molten state of the central region ("the core") of the Moon, and of the latitude variation of the surface temperature on the lunar figure is considered. Isostatically adjusted models of the Moon regarded to consist of a homogeneous solid outer layer and a homogeneous core with slightly smaller density and higher eUipticity are calculated. In Section I the outer surface and the interface are assumed to be spheroidal. In Section II the effects due to the departures from spheroidal form are considered. Because of a difference of 100-140° in subsurface equatorial and po]ar temperatures of the Moon the thickness of the solid layer can be 60-100 km greater in polar regions than in the equatorial zone, and the core can be much more oblate than the Moon as a whole. With this difference in thickness of the solid layer our model can possess the observed value of (C -- A)/C if the surface layer is 2-3~ denser than the core. This result confirms Levin's suggestion connecting the dynamical flattening of the Moon with the latitude dependence of its temperature. Recently B. J. Levin (1964, 1966) tentatively explained the oblate dynamical figure of the Moon b y the nonisothermal state of its interior. According to the calculations of the lunar thermal history by Levin and Mayeva (1960), Levin (1962), and Mayeva (1964) the Moon seems to possess a semimolten central part ("the core") and a solid outer layer. Moreover, the constant subsurface temperature in polar regions of the Moon should be more than 100 ° lower than in the equatorial zone. Therefore, in the polar regions the solid layer should be thicker and its material denser than in the equatorial zone. Levin's calculations showed that the difference in temperature of the solid layer m a y account for about one-third of the observed value ( C - A ) / C of the ratio of the momenta of the lunar globe. But the decrease of density due to the melting is much greater than that due to thermal expansion, and the difference in thickness of the solid layer should result in a larger value of ( C - A ) / C . Levin believes that the combined action of these two effects may be © 1967 by Academic Press Inc.
sufficient to account for the polar flattening of the Moon as inferred from its librations. This hypothesis has been criticized by Kopal (1965, 1966), who established an analytical solution of the equation of heat conduction for a solid Moon characterized b y a constant value of thermal diffusivity K and heated unidirectioually by the Sun from outside. I n his solution the initial temperature was taken to be zero and the present surface temperature was assumed to v a r y as 250 ° sin '~ 0, where 0 denotes the lunar colatitude (polar distance). Kopal concluded t h a t even if the lunar axis of rotation remained invariable in space throughout most of the astronomical past of our satellite, unidirectional solar radiation could have increased the temperature of the lunar i~terior b y dozens, but not hundreds, of degrees. 1 Nevertheless, Kopal's simple model does not take into account certain features 1It should be noted that, due to an oversight in the expansion of the surface temperature in terms of spherical harmonics, Kopal's temperatures should be multiplied by a factor ~-. 275
276
v . s . SAFRONOV
characteristic of the real Moon, such as the fact that the dependence of K on T in the presence of radioactive heat sources may considerably increase the deviation from spherical symmetry of the temperature distribution. Because the lattice conductivity is inversely proportional to the temperature, the difference AT in average temperature at the poles and the equator of the upper layer may become greater than the difference of subsurface temperature. If so, the results deduced from Kopal's analytical solution of his simplified model are not applicable closely to the real Moon. Of the two temperature effects on the figure of the Moon discussed by Levin we shall consider below only the main one which has not yet been evaluated, namely, the role of different thicknesses of the solid layer in the equatorial and polar zones. In doing so we shall assume that the Moon consists of a homogeneous semimolten "core" of density pl, surrounded by a homogeneous solid outer layer of density p0 which is several per cent higher than m. The thickness of the layer at the poles is d km greater than at the equator. The problem is to find the equilibrium shape of this model. The term "equilibrium" must be precise. Apart from the density inversion along the radius the model cannot be in hydrostatic equilibrium when the surface coincides with the equipotential surface. Because of higher density of material in polar zones and the oblateness of the Moon the gravitational potential at the poles is higher than at the equator. Hence the upper layer must have sufficient strength so that the surface material does not flow from the equator to the poles. The strength of the lunar crust is high enough to satisfy this requirement. But the layer is supported by a semimolten core; and if there are no forces that would prevent the bending of the layer (or if they are relaxed) the Moon should acquire the form corresponding to isostatie equilibrium--the weight of the material in a column from the center to the surface must be the same in any direction. We shall consider such isostatie equilibrium taking place after the complete relaxation of stresses that prevent displacements in the verLical direc-
tion. At first we consider a simpler model, the core and the outer layer of which are confined by spheroidal surfaces. The modification of this result due to deviations of the surfaces from spheroids will be found subsequently. I. SPHEROIDAL BOUNDARIES OF THE OUTER SOLID LAYER
Let us assume that the core is an oblate spheroid with the principal semiaxes (al, cl) and density p~. The solid layer has a density p0 and its outer boundary (i.e., the lunar surface) is a spheroid with semiaxes (a0, co). The gravitational field of such a model may be represented b y the sum of the fields of two spheroids, (a0, Co) of density p0 and (a,, cl) of density p , - - p 0 = - - A p . Let X0, X1 and Z0, Z1 denote gravity accelerations produced by these spheroids along coordinate axes x, z directed along equatorial and polar radii. The condition of isostatic equilibrium for the m a t t e r along these axes
fo°° (Xo + xl)p x = fo" (Zo + z,)pdz
(1)
takes the form
(,o
ap) ;a, (Xo + Xl)dX +
(po
I
Ap) foc' (Zo + Z,)dz
+ po ~o (Zo + Z1)dz.
(2)
Denote Ap/m = ~ and V0, V1 potentials of the spheroids. Then we find
(v0 + vl)o~ - (v0 + v~)oo -- a[(V0 + Vl)c, -
(V0 + V1)I., = 0,
(3)
where the indices a, c mean that the potential is taken at the ends of semiaxes a, c. A more general condition of isostatic equilibrium applicable to any direction ~b of radius-vector r may be written in the same form
(Vo + V~)oo - (Vo + V~)~o -- ~[(V0 + V ~ ) c , -
(V0 + V,)r,] --- 0,
(3')
where r0 and r, are the distances from the center of intersection points of the spheroids and of the radius-vector r at a latitude q~.
277
ON T H E F I G U R E O F T H E MOON
Let us write the well-known expression for t h e potential inside a homogeneous ellipsoid of rotation in polar coordinates
(r, ~)
4 2[ 1 + ~ 1/o2(4_ cos~ ~)] V0r° ~ ~Gp0c0 4 Vor, =
2rGpa~c [ (a ~ _ c2)~/2 (a 2 -
Vi=
c~)arctanl
(
~q~ a r c t a n l
--~cos
2[ 3
C12 2Co2
~ ~-Gpoco t_-
+/o2 (1 _ 5_~o2 3 0 ~+C 1 ' Cos cos2 ~b)]
1+l ~ VI~. ~ --gTrGAo--
1+112
Co
-
C12 12 2--~o2 ~1 c o s 2
r ~ sin ~ ~b(1 -- a r c t a n / ) / , J
1
5 C02
(4) + i-5 ~
c°s~ ~
-
-~ c o s
where
a 2 _ c2
z~ = - -
(5)
C2
Va~, ~
-gTrGApc~ ~ 1+
h 2 ( 4 - cos2~b) • (10)
The potential V~ in a point outside the spheroid m a y be found from expression (4) substituting instead of l the quantity l' a 2 _ c2 1'2 -- C2 -]- U "
(5')
Taking ¢ = ~r/2 we find V0 and V1 in the points Coand c~ of small semiaxis. Substituting this values of the potential into (3') and denoting ci//co
=
h~/lo 2 = (
~
(11)
where u is a positive root of the equation r cos2 ¢ + rs i n 2 ~ a2 + u c~ + u
1 = 0.
(6)
For small 1 this equation yields to the order of small quantity 12 c~ + u ~ r ~ -
c~l2 cos2 ~.
4
[3
4
c~ [
-
2
~ -
3)
26-' - i-2(5~ - 3) -= r2(5 -- 3~3 -- 26) ¢
'
1
•
(8)
At the surface of the spheroid r 2 ~ c~(1 + l 2 cos~ ~).
(
(9)
Substituting this expression for r into (8) we obtain for the spheroids (a0, co) and (al, c~), respectively,
(12)
This condition is fulfilled at any ¢ if the expression in the squared brackets is equal to zero. Then ~ must be equal
r~
+l ~ 1--g~+~-~cos-~b
~ 6~
-- g r3 ~ + 62f2( = 0.
2c 2
+l 2 1-~+]-6~cos~ V . ~ g ~rGP -r
47FGpoC02102COS2~[12
1~
(7)
Developing (4) in ascending powers of 1-', substituting values of I and l' from (5), (5'), and (7), respectively, we find that, to the order of i2,
Vi ~ -~ lrepc 2
we find the equation of isostatic equilibrium along the polar radius and that with a latitude ~bin the form
(13)
The parameter ~ is connected with the difference d of thicknesses of the layer at the poles and the equator by the relation d = Co -- cl
--
1 2 (ao -- al) ~ ~colo ( ~ -- 1),
(14) where 102 is related with tim difference ( C - A ) / C . For a homogeneous spheroid the moments of inertia are A = }M(a 2 + d), C = ~2 M a 2 C - - A = ~ M ( a 2 - d ) = ~Mc21 ~.
(15)
278
v.S.
SAFRONOV
TABLE I THE EXCESS POLAR ']~HICKNESS d OF THE OUTER SOLID LKYER ~ND THE EQUATORIAL BULGE at -- c~ FOR DIFFERENT VALUES OF ~" AND ~':
J: d d~ a, - c, (km)
0.6
0.03
0.01
0.03
o.01
0.03
92 75 1.2
30 75 1.2
94 55 1.3
31 55 1.3
111 38 1.5
36 38 1.5
In our model of two spheroids
c = ~Moco~(1 + lo~) + ~Mlci~(1 + ZI~), C - A = ~Moco~lo2 -b 1Mlci2ll2 ~l° 2MOq-MI~= 2 MoWMI~"2
10~l- ~5~ 2 1 - ~5 (16)
Hence, according to (14) (_/ C - A ( ~ - 1)(1 - ~i-5) co ~ C 1 - ~5~
(17)
For the majority of terrestrial rocks the change of density hp during melting is about 5% and in some cases it reaches 10%. At the depth of the boundary of the core (500 -- 700 kin) this change may be 1.5-2 times smaller and for a partial melting still less. The values of d that provide the observed value ( C - A ) / A = 0.00063 are given in Table I for different l" and for - 0.01 and 0.03(hp = 0.03 and 0.1 gm/ cmS). They range from 30 to 110 kin. The corresponding difference between the equatorial and the polar radii of the Moon is in the range from 1.2 to 1.5 kin. The actual value of d may be approximately evaluated from the relation d~
higher than the difference ATo of lunar surface temperatures at the equator and the poles. On the other hand
(
and to the order of Ios and 112 C--A C
0.8
0.7
0.01
AT (dT/dz) D
where (dT/dz)D is the temperature gradient in the layer at the boundary of the core; and AT, the temperature difference in the layer under the equator and the pole at the depth of the equatorial boundary of the layer. From the numerical integration of the equation of heat conduction Mayeva (1964) has found that owing to the dependence of thermal conductivity on temperature, AT is
dT
)
/dT\ =TM-To < \Tz/av D
'
where TM is melting temperature of silicates; To, the surface temperature at the poles; and D = c i ( 1 - i'), the thickness of the solid layer at the poles. Therefore AT AT d > d m = ((dT/dz))~v = Tu -- T--------~o D AT --
TM
--
To ( 1
--
~-)co.
It can be shown that for the inclination 1° 32' of the lunar equator to the ecliptic the energy received from the Sun by unit ares in the polar regions is 102 times smaller than in the equatorial zone. Therefore the surface temperature at the poles must be about 100°K. Values of d~ found for Tu = 1500°, To = 100°K, AT = 150° are also given in Table I. They are in accordance with the conclusion by Mayeva (1964) that d is about 50-100 km. We see that the coincidence of the values of d which provides the observed (C - A ) / C with d expected from the difference of the equatorial and the polar temperature of the solid layer is reached in the range of values of 5 and ~ given in Table I. In our opinion this result confirms the hypothesis by Levin. IX. I~ONSPHEROIDAL BOUNDARIES OF THE SOLID LAYER
It was shown above that to the order of small quantities l~ and ll~ the body composed of two homogeneous concentric spheroids is in isostatic equilibrium along the polar and any other axes when the ratio
279
ON T H E F I G U R E O F T H E MOON
~T
L I FIG. 1. The models of the Moon consisting of the homogeneous outer solid layer and of the homogeneous partially molten central core with slightly smaller density: A, spheroidal boundaries of the layer; B, nonspheroidal boundaries of the layer. = 12/ll2 satisfies Eq. (13). T h e actual shape of the interface between solid and semimolten substance depends on the temperature distribution on the lunar surface and is in general not a spheroidal one. T h e influence of its departure from spheroid on the shape of the surface of the Moon and on the required value of d m a y be taken into account adding to the spheroids (ao, Co) and (as, c~) two small masses M2 and M r - o n e on the boundary of the core and the other on the surface of the M o o n (Fig. 1). T h e dependence of the subsurface temperatu:e of the M o o n on the latitude ¢ can be assumed in the form
and for its departure At1 f r o m the spheroid r,1 ~ c + ( a - c ) cos 2~b
T~ = To + AT cos ~ ¢,
F o r the intensity of gravity g we can write
(19)
where n is evidently more t h a n ~. Troitsky (1954) assumes, for example, n = ½. If in the first approximation we take a constant t e m p e r a t u r e gradient d T / d r and assume a spheroidal shape of the lunar surface we obtain for the radius of the core rl ----C l ~ - d c o s ~ ¢ ~ - ( a o - - Co) COS2~b (20)
Arl= rl-ra
~d(cosn¢-
cos 2¢),
(21}
n < 2 and Arl > 0. Therefore, the additional mass M2 must be negative because the density between r,1 and r~ is diminishing by Ap as compared with the case of spheroids. F o r the isostatic compensation the corresponding positive mass Ma must be added at the surface of the Moon. T h e thickness Ar0 of the additional layer is evidently given by the relation gopoAro = g~hpArl.
(22)
Hence Aro ~ ~'Ar~ ~-- ~'d(cosn¢ -- cos 2~)
(23)
and M s ~-. - - M ~ / v .
(24)
280
v.s.
SAF~ONOV
T h e e q u a t i o n of i s o s t a t i c e q u i l i b r i u m will b e s i m i l a r to (3) b u t i n s t e a d of (V0 + V~) w e m u s t w r i t e n o w Vo A- V~ -t- V~ -t- Va, w h e r e V: a n d Va a r e p o t e n t i a l s p r o d u c e d b y t h e m a s s e s M s a n d Ma. As t h e s e masses a r e s m a l l in c o m p a r i s o n w i t h M0 t h e p o t e n t i a l s V~ a n d Va in s q u a r e b r a c k e t s of (3) w h i c h a r e m u l t i p l i e d b y s m a l l f a c t o r ~ m a y be neglected. T h e y give t o g e t h e r a n e g a t i v e c o r r e c t i o n in d of a b o u t 1 kin. T o e s t i m a t e Vs a n d Va in t h e p o i n t s Co, a0 we c a n use t h e w e l l - k n o w n expression r e l a t i n g e x t e r n a l pot e n t i a l w i t h m o m e n t s of i n e r t i a (Menzel, 1955) y ~ G M _~_GA - ~ - B - t - C - 3 1 r 2r ~
GMs
G(Cs - A~),
Co
COa
Vs~° ~
GMdd 2Co
z = r sin q~ and
dM2 ~- --ApArlri 2 COS~d~bd~. Therefore C2
-
As = f r12(cos2 ~ cos2h - sin s oh)riM2
vdAp f+~/2 r14(cos~ ~b -- cos 2 ~) J -~/2
)< (2 -- 3 cos s ~b) cos ~bd~
vclSllSAp --/o,~/2(cos ~ ¢ -- cos ~ ~)
w h e r e owing to al -- cl >> a0 -- Co w e h a v e t a k e n d ~ a~ - c, ~ c11~2/2. I n t h e s a m e w a y
dM3 = p0r02Ar0cos ~d4~dX ~ - dM~/~, Ca -- A3 = f r02(coss ~ cos 2 X -- sin s ch)dM~
3G(C~ - A~) 2c0a
(26) (27)
-- ~--3(C2 -- As),
(33)
~ - 45 @ 5 ( 1 _ ~._3)i~"
(34)
a n d finally Ks~-K~
and
8
V~o -- V~oo ~
GM~los 2c0
3G(Ca - A~) 2c0~
(28)
T h e expressions (27) a n d (28) will give a d d i t i o n a l t e r m s in (12) w h e r e w e shall t a k e = 0. T h e first t e r m s in t h e r i g h t - h a n d p a r t s of (27) a n d (28) a r e s m a l l in c o m p a r i son ~4th t h e first t e r m of (12) a n d m a y b e n e g l e c t e d ( t h e y f o r m a p a r t of i t e q u a l to g[(M~ + Ma)/Mo] < 10-a). T h e r a t i o s of t h e s e c o n d t e r m s in (27) a n d (28) to t h e first t e r m of (12) are, r e s p e c t i v e l y ,
Ks =
(31)
X (2 -- 3 cos s ~b) cos ~bd~b = ~rApc15112I, (32)
'
Vsoo = GM---2- G(Cs - A~) a0 2a0~
VSco --
x = r cos ~b cos k, y = r c o s ~ s i n h ,
(25)
w h e r e I is t h e m o m e n t of i n e r t i a r e l a t i v e to Che axis d i r e c t e d a l o n g r a d i u s - v e c t o r r. T h e n g2vo *~"
In the polar coordinates
T h e c o n d i t i o n of i s o s t a t i c e q u i l i b r i u m now becomes 2~-1~ -2 + 3 -- 5~= 5 -- 3~-a -- 2if -- 11.25I(1 -- ~-~)
F o r t h e e v a l u a t i o n of d we m u s t r e p l a c e in (14) t h e p a r a m e t e r l0s b y (C - A ) / C . F r o m (15) a n d (29) 2 Cs - As ~ -- ~ K s ( C 0 -- Ao), C3 -- A3 ~ -
15 (C~ - A~) 2 MocoSlo~
K~ =
15 (C~ - A~) 2 MocoSlos
(29)
A d d i t i o n a l t e r m s K~ a n d Ka will e n t e r i n t o t h e s q u a r e b r a c k e t s of (12). I n o r d e r to e s t i m a t e t h e m we m u s t e v a l u a t e t h e differences of t h e m o m e n t s of i n e r t i a C - - A
Cs = f (x s -t- yS)dM~, As = f
(~ + ze) dM~. (30)
(35)
2-~ g ~ ( C o -
Ao).
(36)
A d d i n g t h e s e differences into t h e r i g h t p a r t of (16) w e find C -- A C
+ K~) l02 [1 -- ,~(Ks 2 2 1 - - ~-5
iti'~]
and l02
2
C - A C 1 -
1 - ~-5 8~-5~[1 + 3.75I(~ -3 - - 1)] (37)
ON
THE
FIGURE
OF
THE
MOON
281
T A B L E II P A R A M E T E R S OF T H E M O D E L FOR n = 1 / 2
AND n = 1
0.01 i':
0.6
0.03
0.7
0.8
179 0.15 0.0011 181 1.9
189 0.14 0.0012 191 2.2
126 0.10 0. 0009 128 1.5
143 0.10 0.0010 145 1.8
0.6
0.7
0.8
66 0.060 0.0011 68 1.85
66 0.056 0.0012 68 2.0
72 0.054 0.0014 74 2.4
42 0.041 0.0008 43 1.4
42 0.036 0.0009 44 1.6
48 0.036 0.0011 50 1.8
n = 1/2 d (km)
185
112/2 ~ c~1 lo~/2 .-~ c~o
al - o (kin) ao -- co (kin)
18 0.0010 187 1.75
d (kin)
125
O.
n=l l~2/2 .~ c~ lo2/2 ~ ~o
O. O.
al - cl (kin) ao -- co (km)
12 0008
126 1.4
T h e r e f o r e f r o m (14) d_
C-A
Co
C
(i'$1 -
1)(1-
~i-5)
@ ~ [ 1 + 3.75I(~ --3 - - 1)1
(3s) H e r e ~ is to b e t a k e n a c c o r d i n g t o (35) a n d I is t h e i n t e g r a l in (32). F o r n = ½, I = 0.14 a n d f o r n = 1, I = 0 . 0 7 . 5 T h e v a l u e s o f d f o u n d f r o m (38), t h e e l l i p t i c i t i e s of t h e
d i f f e r e n t v a l u e s of ~ a n d ~'. I n T a b l e I I I d e v i a t i o n s (rso - - co) of t h e s p h e r o i d (ao, Co) f r o m t h e s p h e r e of r a d i u s co a n d d e v i a t i o n s of t h e l u n a r s u r f a c e f r o m t h e s p h e r o i d (Aro) a n d f r o m t h e s p h e r e (Aroo) a r e g i v e n f o r = 0.03, ~- = 0.7 a n d f o r d i f f e r e n t v a l u e s of ~. I t is s e e n f r o m T a b l e I I t h a t t h e e l l i p t i c i t y ai of t h e c o r e is a p p r o x i m a t e l y i n v e r s e l y
TABLE III THE ~:
D E V I A T I O N S OF T H E S U R F A C E FROM S P H E R I C A L AND S P H E R O I D A L F O R M 0
15
30
r~ -- c0 (km) 2~ro = r3 -- rso (km) Aroo = ro -- co (kin)
2.00 0 2.00
1.86 0.07 1.93
1.50 0.25 1.75
r~o -- co (kin) hro = ro -- r,o (km) hroo = ro - - co (km)
1.56 0 1.56
1.45 0.03 1.48
1.17 0.10 1.27
45
60
75
N0
0.50 0.65 1.15
0.13 0.61 0.74
0 0 0
0.39 0.22 0.61
0.10 0.17 0.27
0 0 0
n = 1/2 1.00 0.47 1.47 n=l
s p h e r o i d s (a ~ 1 ~ / 2 ) , a n d t h e d i f f e r e n c e s of t h e i r p r i n c i p a l s e m i a x e s (a - - c) a r e g i v e n i n Table II for n= ½ and n = 1 and for 2 These values of I are found assuming r = c. This approximation is good for the outer spheroid as l0 is very small. But for the inner spheroid the correct value of I is somewhat greater and leads accordingly to a smaller value of d (by about 10% at ~ = 0.01). In Table II this correction is taken into account.
0.78 0.18 0.96
p r o p o r t i o n a l to t h e r e l a t i v e d i f f e r e n c e ~ of d e n s i t i e s of t h e solid l a y e r a n d of t h e core. T h e e l l i p t i c i t y a0 of t h e M o o n as a w h o l e a n d a c c o r d i n g l y t h e d i f f e r e n c e of t h e e q u a t o r i a l a n d t h e p o l a r r a d i i a 0 - Co of t h e M o o n depend on ~ very weakly and are determ i n e d m a i n l y b y t h e s h a p e of t h e core. I n t h e c a s e of t h e s p h e r o i d a l c o r e (n = 2) a0 - - Co ~ 1.2-1.3 k m a n d t h e l u n a r s u r f a c e is also s p h e r o i d a l . A t n = 1, a 0 - Co
282
v . s . SAFRONOV
1.5-1.6 kin. At n = ½, a0 - co ~ 2 k m and seems to be appreciably greater t h a n is found f r o m the measurements of the lunar disk. T h e observational d a t a on the geometric figure of the Moon and also on the dependence (19) of the lunar surface t e m perature on the latitude are still very uncertain. Besides, variation of the t e m p e r a ture gradient d T / d r in the solid layer with makes n in (20) and in Tables I I and I I I different from n in (19). For these reasons a n y speculations about possible deviation of the M o o n f r o m isostatic equilibrium would be premature. Inhomogeneity of the Moon connected with the existence of the less dense and more oblate core influences not only the shape of the M o o n but also its external gravitational field. Therefore, it should be possible to s t u d y this inhomogeneity and to find precisely the parameters of the lunar model considered above with the aid of observational d a t a on the motion of artificial satellites of the Moon. CONCLUSION
The calculations of isostatically adjusted lunar models consisting of a homogeneous core and a homogeneous outer solid layer and possessing the observed value ( C - A ) / C give the relation between the difference of densities of the layer and the core and the difference of thickness of the layer in the polar and the equatorial regions. For the 60-100 k m difference of thickness expected from the 100-140 ° difference of the equatorial and the polar subsurface t e m p e r a t u r e the outer layer m u s t be 2 - 3 % denser t h a n the core. T h e requirement is in accordance with the idea a b o u t the semimolten state of the lunar core.
This result supports B. J. Levin's h y p o t h esis accounting for the observed difference ( C - A ) / C of the M o o n b y latitude dependence of its subsurface temperature. In order to reveal possible departure of the M o o n from isostasy, more definite d a t a on the shape of the Moon, on its external gravitational field, on the variation with latitude of its m e a n surface temperature, and on the t e m p e r a t u r e gradient in the solid layer are necessary. I:~EFERENCES
KOPAL, Z. (1965). Effects of thermal expansion on the moments of inertia of the Moon. Icarus 4~ 166-172. KOPAL,Z. (1966). "An Introduction to the Study of the Moon," Astrophysics and Space Science Library, Vol. 4, pp. 100-104. D. Reidel Publ. Co., Dordrecht, Holland. LEvIN, B. J. (1962). Thermal history of the Moon. In "The Moon," Intern. Astron. Union Symp. No. 14 (Z. Kopal and Z. K. Mikhailov, eds.). Academic Press, New York and London. L~.VIN, B. J. (1964). A new hypothesis for the nonequilibrium figure of the Moon. Nature 202, 1201-1202. LEVIN, B. J. (1967). "Thermal Effects on the Figure of the Moon." Proc. Roy. Soc. London A,296, 266-269. LEVIN, B. J., AND MAYEVA,S. V. (1960). Nektorye
rastchety termicheskoi istorii Luny. Dokl. Akad. Nauk SSSR 133, 44. MA~vn, S. V. (1964). Nektorye rastchety termicheskoi istorii Marsa i Luny. Dokl. Akad. Nauk SSSR 159, 294. M~z~L, D. H. (ed.) (1955). "Fundamental Formulas of Physics." Prentice-Hall, New York. PIZZETTI, P. (1913). "Principii della teoria meccanica, della figura dei planefi" (E. Spoerri, ed.), Pisa. TaoiTsxv, V. S. (1954). K teorii radioizluchenija Luny. Astron. Zh. 31, 511.