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Gravitational heating of the Moon
let_l~us 1, 412-421 (1963)
Gravitational Heating of the Moon ZDENI~K K O P A L
Department o] Astronomy, University of Manchester, E~glar*d and ,let Propulsion Laboratory, California Institute of Technology Received December 15, 1962 It is pointed out that if the mass of the Moon responds to a varying field of external force as a plastic body characterized by an effective viscosity # of the order close to 10TM gm/cm-sec, the viscous dissipation, into heat, of motion caused by monthly variation in the height of Earth-raised bodily tides would alone be sufficient for raising the temperature of the lunar interior to 1000° or more in the past 4½ thousand million years. The efficiency of this purely gravitational heating process should be proportional to the product t~e, where e denotes the eccentricity of the relative orbit of the Moon; and unless the rising temperature will reduce the effective value of # with sufficient rapidity, any secular increase in e would thus be bound to accelerate this process in the course of time. I NTRODUCTION Extensive investigations of the thermal history of the Moon, pioneered by Urey (1952) and continued by a number of other workers (Kopal, 1961a, 1962a; Levin, 1960, 1962; MacDonald, 1959) have all been based on the assumption t h a t the source of heat now contained in the lunar interior was either primodial, or provided by spontaneous decay of such traces of long-lived radioactive elements (K 4°, Th 23-~, U 2'~5, and U 23s) as are likely to have been present in tile mass of the Moon since the time of its formation. The relative abundances of such elements have been estimated by analogy with chondritic meteorites; and (in the absence of any contrary information) the distribution of the radiogenic heat sources throughout the lunar globe has been assumed to be uniform. Under such circumstances, the wellestablished theory of heat conduction in spheres radioactively heated from within (Lowan, 1933, 1935), permits us to coinpute the temperature profile of the Moon at any particular time of its existence for assumed values of the coefficient K of heat
conduction (usually identified with that of common silicate rocks). I n more recent years, this theory has been extended to take account also of the radiative transfer of heat (MacDonald, 1959). The main outcome of such work has been the realization that, if the radiogenic sources of heat inside the Moon are comparable with those found in chondritic meteorites, and if its mass conducts heat similarly to silicate rocks, the present temperature of the bulk of tim mass of the Moon must be in excess of 1000°K--attaining possibly values as high as 1500-2000°K in the deep interior. BODILY TIDES The aim of the present paper will be to show that, under certain conditions, the radiogenic heat does not constitute the only source capable of raising the internal temperature of the Moon to such values. In order to introduce the circumstances, let us depart from the well-known phenomenon of bodily tides in the E a r t h - M o o n system, raised mutually by one of its components on the other. If we expand the shape of so distorted a component regarded as a fluid 412
413
HEATING OF THE MOON
body in a series of spherical harmonics of the form r = a[1% ~ Y~(a,0,¢)],
(1)
L3
where a denotes the mean radius of a distorted equipotential surface and Y~ (a, 8, ~), the corresponding solid harmonic, it can be shown (cf., e.g., Kopal, 1960), that, correctly to quantities of the first order in superficial distortion,
j = 2,3,4,
(2)
where Aj denotes a constant depending on the structure of the distorted configurationl; m ' / m , the mass ratio of the disturbing to the distorted body; R, the separation of their centers; and X = cos ~ sin 8, the cosine of the angle between the axis joining the centers of the two bodies and an arbitrary radius vector. When we apply the foregoing Eq. (2) to evaluate the magnitude of the bodily tides raised by the Moon on the surface of the Earth, m ' / m = 1/81.37, a / R = 0.01658, and (for a homogeneous configuration), A2 = 2.5. If SO, the second harmonic tidal distortion on the surface of the Earth should be given (for the mean radius a s = 6371 km) by a $ Y 2 = 90.4P:(h) cm.
The observations reveal that the actual amplitude of the second-harmonic bodily tide on the Earth amounts only to 22 cm; the discrepancy between the two values being due partly to the finite concentration of mass inside the Earth (which should diminish the "fluid" value of A2 from 2.5 to approximately 1.96), and partly to the departures of the behavior of the Earth's mass from that appropriate for a fluid body. When, conversely, we apply Eq. (2) to evaluate the height of the bodily tides raised by the Earth on the Moon, m ' / m ----81.37 and a / R - - - - 0 . 0 0 4 5 2 1 - - i n which I For its definition, cf., e.g., Eq. Chapter II, Kopa], 1960).
(1-36)
of
case, on the lunar surface (a ©---- 1738 km) we should expect that, for a fluid homogeneous Moon, a ¢ Y 2 = 33 Ps(k) meters, a c Y 3 = 0.10 P3(~) meters, a© Y4 = 0.00041P4(h) meters,
the higher harmonics being clearly too small to be of any significance. The greater gravitational attraction of the Earth renders the equilibrium bodily tides on the Moon approximately 37 times as high as they are on the Earth, in spite of the smaller fractional size of our satellite; and even if the actual tides amount, as on Earth, to only 24% of their theoretical equilibrium value appropriate for a fluid, the height of lunar second-harmonic bodily tides should still be close to 7.9 P2(2,) meters. FORCED OSCILLATIONS
Moreover, the actual height of these tides should be expected to fluctuate in the course of each month, on account of the fact that the reciprocal of the radius vector R of the relative orbit of the Moon around the Earth varies in accordance with the equation 1
1 -}- ecosv = A(1 -- e~) '
(3)
where the semimajor axis A = 385 561 km; the eccentricity e = 0.0549; and the true anomaly v in the relative orbit is reckoned from the time of the perigee passage. Let us assume hereafter the Moon to be sensibly homogeneous, so that the mass m on the right-hand side of Eq. (2) can be represented by m = mo:(a/a©) 8, m©being the total mass of the Moon and a¢ its external radius. Let, moreover, the constant Aj on the right-hand side of Eq. (2) be replaced by the radial Love number
Aj,
h~ = 1 +
4;= 22j+1 ( j -- -~)'
(4)
to account for the finite rigidity of the distorted homogeneous configuration, characterized (by analogy with the Earth; cf., e.g., Munk and MacDonald, 1960, pp. 28-
414
ZDENEK KOPAL
29) by the coefficient ~ of tidal-effective rigidity close to 2.3, rendering h2 ----0.76. If so, the expression (2) can obviously be rewritten as Yi(a,O,¢)) = hj m s m©
a~-2a¢3 A ~-1 .
1 -f- e cos vV+l 1: J ) Ps(h)
(5)
and, correctly to the first power of orbital eccentricity, v a r y by an amount equal to iIYj(a,8,4)) = h~ m s a i - 2 a ¢ a m¢ A i+~
(e cos v)Pj(X)
(6)
in the course of each month. Since, however, the duration of the sideric month P----27.321661 days, the velocity of this "breathing" of the mass of the Moon, caused by jth harmonic bodily tides raised by fluctuating terrestrial attraction, should possess a radial component U equal to U = 4a(~Yj)/P.
(7)
If we insert for 8Y~ from (6) we find that, for second-harmonic tides, the ratio U
4h,(m¢~(a~,~
2
¥ = - f i \ ~ - - ~ 1 \ - ~ - / e cos ,p~(x)
(8)
leads, on the lunar surface (a = a~), on insertion of appropriate numerical values to a velocity U = 9.2 × 10-~P2(X) cos v cm/see, which remains of the same order of magnitude throughout the most part of the lunar interior. I t m a y be noted that such a velocity--corresponding to 29 nleters/year - - i s by some three orders of magnitude greater than the velocity of convection currents recently envisaged to exist in the lunar interior by several writers (Kopal, 1961b, 1962b,c; Runcorn, 1962). DISSIPATION OF ENERGY
In the face of such a situation, the following question becomes of cardinal iraportance for the study of the internal structure of our satellite: Does the Moon respond to a varying strain invoked by
the monthly Earth-raised tides as an elastic or plastic body? For in the former case, the bodily tides do not accomplish any net work in the course of the orbital cycle; but in the latter case, a tidal-effective viscosity would bring about a degradation of tidal motion into h e a t - - a process which would be bound in influence the thermal equilibrium of lunar interior, in accordance with the well-known consequences of the principle of the conservation of energy. At the present time a definite answer to this question is still lacking. An indication as to what it is likely to be may, however, be obtained by a comparison of the period P of the disturbing force with tile Maxwellian relaxation time t, for a visco-elastic body. As is well known, a rough estimate of the magnitude of t , can be obtained from the formula t, -- ~¢~,
(9)
where ~ denotes the coefficient of viscosity of the respective body; and fl is the coefficient of isothermal volume compression of its material. For silicate rocks (of density close to the mean density of the Moon) the value of fi is known to be close to 1 X 10-12 d y n e / c m 2 (cf., e.g., Birch, 1952). The estimates of the appropriate values of t~ are, on the other hand, still largely uncertain. The velocities of tectonic motions observed in the Earth's crust (such as, for example, the post-glacial uplift of Scandinavia) are indicative of a value of t~ as high as 1022 gin/era.see; while laboratory measurements of the viscosity for specific rocks cluster around / ~ l0 is gm/cm.sec, or even less. The former value, combined with the relatively well-known time t , of the order of 10 TM seconds or some 300 years; the latter, to 106 seconds or about 10 days. If the former applies to the Moon as a whole, the response of the lunar globe to a monthly tidal wave should be essentially elastic and entail no dissipation of energy. On the other hand, a viscosity of 1018 gm/cm-sec would lead to a relaxation time short enough to give rise to visco-elastic flow characterized by appreciable dissipation of motion into heat.
415
HEATING OF THE MOON EQUATIONS OF THE PROBLEM
In order to ascertain the actual amount of heat which could thus be produced, let us depart from the equation for the conservation of energy which (on neglect of terms arising from possible convection) assumes the explicit form
pC,(DT/Dt) = Kv2T A- p~ -4- g4,, (10) where p denotes the local density (regarded hereafter as constant) ; T, the temperature; C~, the specific heat at constant volume; ~, the constant coefficient of thermal conduction; ~, rate of liberation of radiogenic heat per unit mass; and ¢, the viscous dissipation function 1/0" 2
+ aoo~ (11) given as a quadratic form of the components aij of the viscous stress tensor. Let us assume, in what follows, that the motion in the lunar interior due to bodily tides is predominantly radial and characterized by the velocity component U, while the angular velocity components V and W are negligible. If so, it can be shown that the only nonvanishing components of the viscous stress tensor become ff aa ~
20U Oa
O'aO ~
10U a ~'
'
0"06= 2 Ua =a~' i a~
1 OU I
(12)
= a sin 00-~"
Remembering, moreover, that (to a sufficient approximation), U is given by Eq. (8) above, the foregoing components of the viscous stress tensor are easily evaluated; and their insertion in Eq. (11) reveals that the function ¢ governing the dissipation of energy of second-harmonic (j = 2) tides assumes the explicit form = ~-k2(3), 4 q- 1) c o s ~ v,
(13)
where we have abbreviated k = 12 ~-~ ( m ~ ' ~ [ a c y \m ~/\ A/"
(14)
Now the average value of h over the whole sphere is equal to 107/315 (or, very
approximately, one-third); while the average value of cos2v over a cycle is one-half. Therefore, if we wish to investigate the global secular variations of temperature in lunar interior, it should be sufficient to replace, in Eq. (10), the function ¢ by its average constant value of -- ~-k~;
(15)
and, by the same argument, replace the operator
D_O 0 Dt -- Ot -}- U--Or on the left-hand side of Eq. (10) by ~/~t, as the average value of U (proportional, to a first approximation, to e cos v) averages out to zero in the course of a cycle. SOLUTION OF THE EQUATIONS
Eq. (10) so simplified then assumes the explicit form
OT ~ 0 (a2 0T~ 3 pC, - ~ q- ~ ~-~ \ - ~ ] -}- pe q- -2 ~k
(16)
and its particular solution relevant to our problem is subject to the boundary condition requiring that the surface temperature T (a©, t) should remain constant--the value of which can [because of the linearity of Eq. (16)] be taken as zero? In view of the linearity of our problem, the desired particular solution Eq. (16) can be expressed as a sum of the form
T(a,t) = T~(a,t) q- Ta(a,t), (17) where T~(a,t) stands for the temperature due to radiogenic heat, while T~ (a,t) represents the possible "gravitational" contribution to internal temperature of the Moo~ due to viscous dissipation of bodily tides. The behavior of the "radiogenic" temperature Tc(a,t) in lunar interior is sufficiently well known from the work of Urey (1952), Levin (1960, 1962), or Kopal (1962a) already referred to. A similar solution of ~In so doing we should, however, r e m e m b e r t h a t this particular solution of our problem will, in effect, furnish the difference T ( a , t ) T(a©,t) between the internal a n d surface temperatures, rather t h a n the absolute temperature itself.
TABLE x=O O.
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5 0 5 5 60 6 5 7 0 8 0 9 0 i0 0
O. 0.0828 O. 1655 0.2483 0.3310 0.4138 0.4966 0.5793 0.6621 0.7449 0.8276 0.9104 0.9932 1.0759 I. 1587 1.3241 1.4895 1.6547 8.3333
I, K = 0 . 0 1 0
0.I000
0.2000
0.3000
0.4000
0.5000
0.6000
0.7000
0.8000
0.8500
0.9000
0.9250
0.9500
0.9750
O. 0.0828 0.1655 0.2483 0,.3311 0.4138 0.4966 0.5793 0.6621 0.7449 0.8276 0 9104 0 9932 1 0759 I 1587 l 3241 1 4894 l 6544 8 2500
O. 0.0828 0.1655 0.2483 0.3311 0.4138 0.4966 0.5793 0.6621 0.7449 0.8276 0.9104 0.9931 1.0759 1.1586 1.3238 1.4887 1.6531 8.0000
O. 0 0828 0 1655 0 2483 0 3311 0 4138 0 4966 0 5793 0 6621 0 7449 0 8276 0 9103 0 9930 l 0756 1 1581 1 3227 1 4865 1 6491 7.5833
O. 0.0828 0.1655 0.2483 0.3311 0.4138 0.4966 0.5793 0.6621 0,7448 0.8274 0 9099 0 9922 i 0743 l 1561 1 3187 1 4795 l 6381 7 0000
O. 0.0828 0.1655 0.2483 0.3311 0.4138 0.4965 0.5792 0.6617 0,7440 0.8260 0.9075 0.9885 1.0690 i. 1488 1.3060 i. 4600 1.6102 6.2500
O. 0.0828 0.1655 0.2483 0.3310 0.4136 0.4960 0.5778 0.6591 0,7395 0.8189 0.8973 0.9744 1.0504 1.1251 1.2706 1.4108 1.5458 5.3333
O. 0.0828 0.1655 0.2482 0.3303 0.4114 0.4910 0.5689 0,6450 0.7190 0.7911 0.8611 0.9292 0.9954 1.0597 1.1831 1.2999 1.4107 4.2500
O. 0 0828 0 1650 0 2448 0 3210 0 3933 0 4618 0 5267 0 5883 0 6469 0 7029 0 7563 0 8075 0 8566 0 9038 0 9929 I 0759 l 1534 3 0000
O. 0.0826 O. 1620 0.2351 0r.3020 0 3635 0 4023 0 4733 0 5228 0 5695 0 6135 0.6553 0.6950 0.7329 0.7692 0.8372 0.9001 0.9586 2.3125
O. 0,0805 O. 1491 0.2070 0.2574 0.3023 0.3429 0.3802 0.4146 0.4467 0.4768 0.5052 0.5320 0.5575 0.5817 0.6271 0.6688 0.7074 1.5833
O. 0.0761 O. 1335 0.1798 0.2191 0.2537 0.2848 0.3131 0.3391 0.3633 0.3859 0.4072 0.4273 0.4463 0.4644 0.4981 0.5291 0.5578 1.2031
O. 0.0653 0.1069 O. 1391 0.1659 0.1893 0.2102 0.2291 0.2464 0.2625 0.2775 0 2915 0 3048 0 3174 0 3293 0 3515 0 3719 0 3908 0 8125
O. 0.0425 0.0642 0.0806 0.0941 O. 1057 O. ll61 0.1255 0.1341 O. 1420 O. 1494 O. 1563 O. 1629 0.i~91 0.1749 0.1859 0.1959 0.2051 0.4114
0 0 5 1 0 1 5
2 0 2 5 3 0 3 5 4 0 4.5 5.0 5.5 6.0 6.5 7.0 8.0 9.0 I0.0 c~C
O. O. 1241 O. 2483 0.3724 O. 4966 O. 6207 O. 7449 O. 8690 0.9932 i. I173 1.2414 1.3655 1.4895 1.6134 I. 7371 1.9836 2.2282 2.4700 8.3333
0 0 1241 0 2483 0 3724 0 4966 0 6207 0 7449 0 8690 0 9932 1 1173 I .2"414 1 3654 1 4894 l 6132 l 7367 1 9827 2.2265 2.4671 8.2500
O. 0.1241 0.2483 0.3724 0.4966 0.6207 0.7449 0.8690 0,9931 1.1172 I..2412 1.3651 1.4887 1.6120 1. 7349 1. 9790 2.2200 2.4569 8.0000
O. 0.1241 0.2483 0.3724 0.4966 0.6207 0.7449 0.8690 0.9930 1. 1169 1.2405 1.3637 1.4865 1.6085 1.7298 1.9695 2.2046 2.4341 7.5833
0 0 1241 0 2483 0 3724 0 4966 0 6207 0 7448 0 8686 0 9922 t 1152 1 2376 1 3591 1 4795 1 5986 1 7164 1 9474 2.1715 2.3882 7.0000
O. 0.1241 0.2483 0.3724 0,4965 0.6205 0.7440 0.8668 0.9885 i. 1090 1.2278 1.3449 1.4600 1.5730 1.6838 1.8989 2.1048 2.3017 6.2500
I I , K = 0.015 O. 0.1241 0.2483 0.3723 0,4960 0.6185 0.7395 0.8582 0.9744 1.0879 1.1985 1.3061 1.4108 1.5125 1.6114 1.8006 1.9791 2.1477 5.3333
t~
o
m
TABLE
~q
3. O. 1241 0.2482 0.3710 0.4910 0.6072 0.7190 0.8263 0.9292 1.0278 I. 1223 1.2129 1.2999 1.3836 1.4640 1.6160 1.7572 1.8890 4.2500
O 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 l 1 3
1240 2448 3756 4618 5579 6469 7299 8075 8804 9492 0142 0759 1345 1904 2949 3907 4792 0000
O. 0.1230 0.2351 0.3334 0.4203 0.4984 0.5695 0.6347 0.6950 0.7512 0.8039 0.8534 0.9001 0.9443 0.9863 1.0645 1.1359 1.2016 2.3125
O. O. 1164 0.2070 0.2805 0.3429 0.3977 0.4467 0.4912 0.5320 0.5697 0.6049 0.6378 0.6688 0.6980 0.7257 0.7770 0.8238 0.8666 1.5833
O. 0.1066 0.1798 0.2370 0.2848 0.3263 0.3633 0.3967 0.4273 0.4554 0.4816 0.5061 0.5291 0.5508 0.5714 0.6094 0.6440 0.6756 1.2031
0 0 0877 0 1391 0 1780 0 2102 0 2379 0 2625 0 2846 0 3048 0 3234 0 3407 0 3568 0 3719 0 3862 0 3997 0 4246 0 4473 0.4680 0.8125
O.
0.0543 0.0806 O. i001 O. i161 O. 1299 O. 1420 O. 1529 O. 1629 O. 1720 O. 1805 O. 1884 O. 1959 0.2029 0.2095 0.2217 0.2328 0.2430 0.4114
m
0 0 5 1 0
1 5 2 0 2 5 3 0 3.5 4.0 4.5 5.0 5.5 6,0 6.5 7.0 8.0 9.0 10.0
O. 0.1655 0.3310 0.4966 0.6621 0.8276 0.9932 1, 1587 1.3241 1.4895 1. 6547 1.8194
O. 0.1655 0.3311 0.4966 0.6621 0.8276 0.9932 1, 1587 1.3241 1.4894 1.6544 1.8189
1.9836 1.9827 2. 1469 2. 1455 2.3092 2.3071 2.6292 2.6254 2.9416 2.9357 3.2446 3.2361 8.3333 8.2500
O. 0.1655 0.3311 0.4966 0.6621 0.8276 0.9931 1.1586 1.3238 1.4887 1.6531 1.8166 1.9790 2. 1401 2.2995 2.6122 2.9155 3.2078 8.0000
O. 0.1655 0.3311 0.4966 0.6621 0.8276 0.9930 1.1581 1.3227 1.4865 1. 6491 1.8102 1.9695 2. 1268 2.2817 2.5837 2.8741 3. 1519 7.5833
O. 0.1655 0.3311 0.4966 0.6621 0.8274 0 9922 1 1561 1 3187 1 4795 1 6381 1 7941 I 9474 2 0976 2 2446 2.5285 2.7985 3.0546 7.0000
TABLE
III,
O. 0.1655 0.3311 0.4965 0.6617 0.8260 0.9885 1.1488 1.3060 1.4600 1.6102 1.7565 1.8989 2.0372 2.1715 2.4280 2.6689 2,8950 6.2500
O. 0.1655 0.3310 0.4960 0.6591 0.8189 0.9744 1.1251 1.2706 1.4108 1.5458 1.6757 1.8006 1.9208 2.0364 2.2548 2.4572 2,6452 5.3333
TABLE
K=0.020 O. 0.1655 0.3303 0.4910 0.6450 0.7911 0.9292 1.0597 1,1831 1.2999 1.4107 1.5159 1.6160 1.7113 1.8022 1.9720 2. 1275 2,2706 4.2500
O. 0.1650 0.3210 0.4618 0.5883 0.7029 0.8075 0.9038 0.9929 1 0759 1 1534 1 2263
O. 0.1620 0.3020 0.4203 0,5228 0.6135 0.6950 0.7692 0.8372 0.9001 0.9586 1.0132
O. 0.1491 0.2574 0.3429 0,4146 0.4768 0.5320 0.5817 0.6271 0.6688 0.7074 0.7434
O. 0.1069 0.1659 0.2102 0.2464 0.2775 0.3048 0.3293 0.3515 0.3719 0.3908 0.4083
O. 0.0642 0.0941 0.1161 0.1341 0.1494 0.1629 0.1749 0.1859 0.1959 0.2051 0.2137
0.7770 0.8087 0.8385 0.8933 0.9428 0.9877 1.5833
O. 0.1335 0.2191 0.2848 0.3391 0.3859 0.4273 0.4644 0.4981 0.5291 0.5578 0.5845 0.6094 0.6328 0.6548 0.6953 0.7318 0.7649 1.2031
I I 1 1 I 1 3
2949 3597 4210 5346 6377 7317 0000
1.0645 1.1128 1.1584 1.2426 1.3186 1.3879 2.3125
0.4246 0.4399 0.4544 0.4809 0.5048 0.5264 0.8125
0.2217 0.2292 0.2363 0.2493 0.2610 0.2716 0.4114
0.2448 0.4618 0.6469 0.8075 0.9492 1.0759 1.1904 1.2949 1.3907 1.4792 1.5613 1.6377 1.7090 1.7758 1.8973 2.0052 2.1013 3.0000
0.2351 0.4203 0.5695 0.6950 0.8039 0.9001 0.9863 1.0645 1.1359 1.2016 1.2623 1.3186 1.3711 1.4202 1.5094 1.5883 1.6585 2.3125
0.2070 0.3429 0.4467 0.5320 0.~049 0.6688 0.7257 0.7770 0.8238 0.8666 0.9061 0.9428 0.9768 1.0086 1.0663 i. 1173 1.1626 1.5833
0.1798 0.2848 0.3633 0.4273 0,.4816 0.5291 0.5714 0.6094 0.6440 0.6756 0.7048 0.7318 0.7569 0.7804 0.8229 0.8604 0.8938 1.2031
0.1391 0.2102 0.2625 0.3048 0,3407 0.3719 0.3997 0.4246 0.4473 0.4680 0.4871 0.5048 0.5212 0.5365 0.5643 0.5888 0.6106 0.8125
0.0806 0.1161 0.1420 0.1629 0,.1805 0.1959 0.2095 0.2217 0.2328 0.2430 0.2523 0.2610 0.2690 0.2765 0.2901 0.3021 0.3127 0.4114
IV, K = 0 . 0 3 0
O.
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7,0 8.0 9.0 10.0
0.2483 O. 4966 O. 7449 0.9932 1. 2414 1.4895 1.7371 1. 9836 2. 2282 2.4700 2.7081 2.9416 3.1699 3.3922 3.8171 4.2138 4.5815 8.3333
0 2483 0 4966 0 7449 0 9932 1 2414 1 4894 1 7367 1 9827 2 2265 2 4671 2.7038 2.9357 3.1620 3.3822 3.8025 4. 1944 4.5572 8.2500
0.2483 0.4966 0.7449 0.9931 1.2412 1.4887 1.7349 1.9790 2. 2200 2.4569 2.6890 2.9155 3.1358 3.3495 3.7559 4. 1331 4.4813 8.0000
0.2483 0.4966 0.7449 0.9930 1.2405 1.4865 1.7298 1,9695 2. 2046 2.4341 2.6574 2.8741 3.0837 3.2860 3.6684 4.0212 4.3451 7.5833
0.2483 0.4966 0.7448 0.9922 1.2376 1.4795 1.7164 1,9474 2.1715 2.3882 2.5973 2.7985 2.9919 3.1774 3.5253 3.8438 4.1344 7.0000
0.2483 0.4965 0.7440 0.9885 1.2278 1.4600 1.6838 1,8989 2. 1048 2.3017 2.4896 2.6689 2.8398 3.0027 3.3057 3.5805 3.8297 6.2500
0 2483 0 4960 0 7395 0 9744 1 1985 1 4108 1 6114 1 8006 1 9791 2 1477 2.3068 2.4572 2.5995 2.7342 2.9826 3.2060 3.4073 5.3333
0.2482 0.4910 0.7190 0 9292 1 1223 1 2999 1 4640 1 6160 1 7572 1.8890 2.0121 2.1275 2.2359 2.3379 2.5246 2.6912 2.8405 4.2500
--,1
418
ZDEN~.K KOPAL
Eq. (16) for the " g r a v i t a t i o n a l " component of lunar t e m p e r a t u r e leads to an expression of the form
---- 3.154 X 10 ~ sec as our unit of time, and 103 k m -~ l0 s cm as the unit of length, so t h a t the nondimensional value
Tg(a,t) = (~/~)Cr(x,t)
/~ = [3.154 X 10'~/(10s)2]g = 3.154K
(18)
where the constant C = 3 ( a , k ) 2,
(19)
and the function r(x,t) can be expressed as
~-(x,t) = 2
( - 1)~+1{ 1 -- exp [ - K ( n ~ / a c ) 2 t ] }
n=l
×
sin n~rx
(20)
in which we have abbreviated
K = ~/aC,,
and
x = a/a~.
(21)
The corresponding rate of heat flow across the boundary a----a¢, then becomes
OT K Oa
~C(Or) ac
Ox 1"
(22)
~UMERICAL RESULTS I n order to ascertain the actual amount of possible gravitational contribution to the internal t e m p e r a t u r e of the Moon, numerical calculations have been carried out, based on the foregoing equations, for appropriate values of the astronomical and physical p a r a m e t e r s involved. I n accordance with Eqs. (14) and (19),
C = 3
12ac p m ~ A 3 ] '
(23)
which for m e ~ m e = 81.37, a c / A = 0.004521, ac---- 1 . 7 3 8 X 1 0 s cm, P = 2 . 3 6 X 106 sec, e = 0.0549, and h2 = 0.76, is equal to C = 1.789 X 10 -~ crn ~ see -2.
(24)
Moreover, the coefficient K of heat conduction of silicate rocks is known to be in the neighborhood of 2 X 10 ~ e r g / g m . d e g (cf. Birch, 1952), so t h a t the coefficient K of thermal diffusivity, as defined by the first one of Eq. (21) becomes (for p = 3.3 g m / cm 3) a quantity of the order of 0.01 cIn2//sec. Let us, in w h a t follows, adopt 10 '~ years
of the coefficient of thermal diffusivity is likely to be between 0.010 and 0.030. Tables I - I V contain then the numerical values of 1 0 2 r ( x , t ) - - w i t h r(x,t) as given by the expansion on the right-hand side of Eq. ( 2 0 ) - - f o r t = 0 (0.5) 7.0 (1.0) 10.0 and oo, expressed in terms of 109 years t a k e n as our unit, and the fractional distance x = 0.0 (01) 0.8 (0.05) 0.90 (0.025) 0.9753; for the normalized coefficients K = 0.010 (Table I) 0.015 (Table I I ) , 0.020 (Table I I I ) and 0.030 (Table I V ) . A graphical representation of the functions 100 r(x,t) is shown on the accompanying Figs. 1-4. Ultimately, the numerical values of the normalized t e m p e r a t u r e gradients 100 (3r/~x)l on the surface are listed in T a b l e V. For the value of C as given by Eq. (23) above and ~ = 2 X 10~ e r g / c m . s e c - d e g , it follows t h a t
Tg(a,t) ~ 7 X 10-13~r(x,t) deg, and a glance at the numerical results listed in Tables I - I V reveals r(x,t) to be a quant i t y of the order of 10 -2. I n consequence, it appears t h a t the viscous dissipation, into heat, of motion actuated by bodily tides leads to the generation of temperatures inside the lunar globe which, for ~----1017 g m / c m - s e c , are of the order of 100°K; and for ~ = 10 is g m / c m - s e c of the order of 1000°K, i.e., comparable with the internal temperatures attained by a gradual release of radiogenic heat in the course of 4.5 billion years. F o r ~ ~ 1017 g m / c m . s e c , the amount of heat produced by viscous dissipation would be too small to be of cosmogonic importance. On the other hand, for/~ >> 10 is g m / cm-sec, the response of the lunar globe to a forced oscillation of tidal origin would again become elastic (as t. > P) and cease, as such, to bring about any dissipation of The column corresponding to x ~ 1 vanishes identically by virtue of the outer boundary condition.
HEATING
OF
THE
MOON
419
5.0
4.C
50
2.0, i
t=lO0
t=8.0
""-
i 1.0 I =4.0 1=50 t=2.0 1=10 i
OI
.I
0.2
0.5
0.4
0.5
I
0.7
0.6
0.8
0.9
1.0 x
x
FIo. 1. The function 100~-(x,D for K -----0.010.
Fro. 3. The function 100r(x,t) for ~: = 0.020.
\
5.0
,.jo
V
I
]
i
'!
0.2
0.3
" "
i
~t.
t = I0.0
2.0
t = ~.0 t=&O
t=6.0
r 1.0
i
t=5.0 I=40 t=5.0
~
~
.
~
,,°, ':i °',
t=2.0 t= fO
0
0.1
0.2
0.3
0,4
05
0.6
0.7
0.8
0.9
1.0
X
' -0
O. I
0.4
0.5 X
10 6
0.7
0.8
0.9
1.0
FIo. 2. The function 100~(x,t) for K---~ 0.015.
Fro. 4. The function 100~(x,t) for K ~-~0.030.
energy over a cycle. However, if ~he actual value of the effective viscosity happens to be of the order of 10 TM g m / c m . s e e (as suggested indeed by experiments with indi-
vidual samples of terrestrial common rocks); the mechanical heating of the Moon throughout its long past could, by itself, have raised the temperature of its
420
ZDEN~K KOPAL
interior to a level comparable with that expected from the radiogenic heating alone. The corresponding rate of heat flow
K(OT/Oa) --- 8 X 10 ~6t~(OT/Ox)~ then turns out, for p. = 10 ~ gin/era.see, to TABLE V
SURFACE TEMPERATURE GRADIENT I02(Or/OX)I
t o.o,o o o~5 ......................... 0. 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 8.0 9.0 10.0
0. -0.0828 -0.1655 -0.2483 -0.3310 -0.4138 -0.4966 -0.5793 --0.6621 --0.7449 --0.8276 --0.9104 --0.9932 --1.0759 --1.1587 --1.3241 --1.4895 -1.6547 --8.3333
0. -0.1241 -0.2483 -0.3724 -0.4966 -0.6207 -0.7449 -0.8690 --0.9932 -1.1173 --1.2414 --1.3655 -1.4895 -1.6134 -1.7371 -1.9836 -2.2282 --2.4700 --8.3333
0.0')0 0. -0.1655 -0.3310 -0.4966 -0.662l -0.8276 -0.9932 -1.1587 --1.324l --1.4895 --1.6547
--1.8H)4 --1.9836 --2.1469 --2.3092 --2.6292 --2.9416 --3.2446 --8.3333
o.oao 0. -0.2483 -0.4966 -0.7449 -0.9932 -1.2414 -1.4895 --1.7371 --1.9836 --2.2282 --2.4700 --2.7081 --2.9416 --3.1699 --3.3922 --3.8171 --4.2138 -4.5815 --8.3333
be a quantity of the order of 10 erg/cm :. see (10 < c a l / c m : - s e c ) at the surface. (.]ONCLUD1N(; ~EMARKN
I t m a y , furthermore, be added t h a t the foregoing figures for the gravitational heat production in elasto-viscous lunar interior represent (for a given adopted value of ~1 the mininmm estimates, as of the motion invoked by bodily tides only the effects of its radial comllonent U were included in the dissipation function (11). In particular, the azimuthal velocity component W due to the tidal lag rnay m a k e fm'ther significant contribution to tile gravitational heat production--possibly comparable with t h a t arising from the radial component it.self. As, however, the theory of tidal lag in viscous spheres produced by the periodic'lily fluctuating difference between the angular velocity of axial rotation and orbital revo-
lution is so far very incomplete, no accurate estimate of gravitational heat production to bc expected from this source is possible at the present time. In conclusion, the question m a y also be asked concerning a possible application of this llrocess to other bodies in the solar system; and of these, the bodily tides raised by the Sun on Mercury and Venus come immediately to mind. As is well known, tile fundamental characteristics of these planets, occurring in Eq. (19) are:
P a e Radius
m®/mp
Mercury
Venlls
87.97 days 57.,(} N 106 k m 0. 2056 2 . 4 X 10~ k m 6.3 X 10s
224.7 days 108.2 X 106 km 0. 00682 6.2 X 103 k m 4.0(,1 X 1()~
When we insert these values in Eq. (19) to evaluate the constants (} (and assume, for the sake of simplicity, that h.,--0.59 as for tile E a r t h ) , tile outcome reveals that Ci~,.c,,~ = 1.42 X 10 -9 cln"/sec 2, Cv ..... = 4.15 X 10 -14 em2/see 2. Accordingly, tile value of C for Mercury proves to be a hundred times, and, for Venus, a hundred thousand times, smaller than tile value Eq. (24) established earlier for our Moon. This reveals t h a t (for the same value of effective viscosity) the heat production by solar tides inside Mercury could have raised its internal temperature hy scarcely more than 10 ° in the past four hillion years; and for Venus the whole phenomenon turns out to t)e utterly insignificant. Therefore, the E a r t h - M o o n system appears to be tile only case in the solar family where gravitational heating due to the dissipation of bodily tides could be of cosmogonical significance. ACKNOWLED(IMENTS The c o m p u t a t i o n s presented in this paper have been carried out at the J e t Propulsion Laboratory's 7090 I B M C o m p u t e r during the writer's stay lhere in the s u m m e r of 1962. T h e c o m p u t a tions were p r o g r a m m e d by Mrs. Maxine Linde under the supervision of Mr, P ' d m e r Poulson,
HEATING
OF T H E
while all graphs presented in this paper were prepared by Mrs. Patricia Conklin. Their contributions to the completion of this work are hereby gratefully acknowledged. REFERENCES BIaCH, F. (1952). Elasticity and constitution of the Earth's interior. J. Geophys. Res. 57, 227. KOPAL, Z. (1960). "Figures of Equilibrium of Celestial Bodies," Section II-1. University of Wisconsin Press, Madison. KOPAL, Z. (1961a). "Thermal History of the Moon and of the Terrestrial Planets I." California Institute of Technology, Jet Propulsion Laboratory Tech. Report No. 32-108. KOPAL, Z. (1961b). Some remarks on the interpretation of the gravitational field of the Earth and the Moon. Proe. 12th Integer. Astronautical Congress, Washington, D. C. KOPAL, Z. (1962a). "Thermal History of the Moon and of the Terrestrial Planets II: Numerical Results." California Institute of Technology, Jet Propulsion Laboratory Tech. Report No. 32-225. KOPAL, Z. (1962b). "Thermal History of the Moon and of the Terrestrial Planets I I I : Convection in Planetary Interiors." California Institute of Technology, Jet Propulsion Laboratory Tech. Report No. 32-276. KOPAL, Z. (1962c). The effects of convection in the
MOON
421
mantle on the gravitational field of the Earth. Proc. o] the Third COSPAR Symposium in Washington (in press). L~VlN, B. Y., AND M~VA, S. V. (1960). Certain computations of the thermal history of the Moon. Doklady Akad. Nauk S~S.S.R. 133, 44. LEVIN, B. Y. (1962). Thermal history of the Moon. In "Proc. of the 14th I.A.U. Symposium on the Moon" (Z. Kopal and Z. K. Mikhailov, eds.), pp. 157-167. Academic Press, London and New York. LOWAN, A. N. (1933). On the cooling of a radioactive sphere. Phys. Rev. 44, 769. LOWAN, A. N. (1935). On the cooling of the Earth. Am. J. Math. 57, 174. MACDONALD, G. J. F. (1959). Calculations of the thermal history of the Earth. Geophys. Res. 64, 1967. MUNK, W. H., AND MACDONALD,G. J. F. (1960). "The Rotation of the Earth." Cambridge University Press, Cambridge. RUNCOaN, S. K. (1962). Convection in the Moon. Nature 195, 1150. UREY, H. C. (1952). "The Planets." pp. 49-57. Yale University Press, New Haven; and subsequent publications, of which Urey's chapter on the "Origin and History of the Moon," in "Physics and Astronomy of the Moon" (Z. Kopal, ed.), pp. 781-523, Academic Press, 1962, should in particular be noted.
Gravitational Heating of the Moon ZDENI~K K O P A L
Department o] Astronomy, University of Manchester, E~glar*d and ,let Propulsion Laboratory, California Institute of Technology Received December 15, 1962 It is pointed out that if the mass of the Moon responds to a varying field of external force as a plastic body characterized by an effective viscosity # of the order close to 10TM gm/cm-sec, the viscous dissipation, into heat, of motion caused by monthly variation in the height of Earth-raised bodily tides would alone be sufficient for raising the temperature of the lunar interior to 1000° or more in the past 4½ thousand million years. The efficiency of this purely gravitational heating process should be proportional to the product t~e, where e denotes the eccentricity of the relative orbit of the Moon; and unless the rising temperature will reduce the effective value of # with sufficient rapidity, any secular increase in e would thus be bound to accelerate this process in the course of time. I NTRODUCTION Extensive investigations of the thermal history of the Moon, pioneered by Urey (1952) and continued by a number of other workers (Kopal, 1961a, 1962a; Levin, 1960, 1962; MacDonald, 1959) have all been based on the assumption t h a t the source of heat now contained in the lunar interior was either primodial, or provided by spontaneous decay of such traces of long-lived radioactive elements (K 4°, Th 23-~, U 2'~5, and U 23s) as are likely to have been present in tile mass of the Moon since the time of its formation. The relative abundances of such elements have been estimated by analogy with chondritic meteorites; and (in the absence of any contrary information) the distribution of the radiogenic heat sources throughout the lunar globe has been assumed to be uniform. Under such circumstances, the wellestablished theory of heat conduction in spheres radioactively heated from within (Lowan, 1933, 1935), permits us to coinpute the temperature profile of the Moon at any particular time of its existence for assumed values of the coefficient K of heat
conduction (usually identified with that of common silicate rocks). I n more recent years, this theory has been extended to take account also of the radiative transfer of heat (MacDonald, 1959). The main outcome of such work has been the realization that, if the radiogenic sources of heat inside the Moon are comparable with those found in chondritic meteorites, and if its mass conducts heat similarly to silicate rocks, the present temperature of the bulk of tim mass of the Moon must be in excess of 1000°K--attaining possibly values as high as 1500-2000°K in the deep interior. BODILY TIDES The aim of the present paper will be to show that, under certain conditions, the radiogenic heat does not constitute the only source capable of raising the internal temperature of the Moon to such values. In order to introduce the circumstances, let us depart from the well-known phenomenon of bodily tides in the E a r t h - M o o n system, raised mutually by one of its components on the other. If we expand the shape of so distorted a component regarded as a fluid 412
413
HEATING OF THE MOON
body in a series of spherical harmonics of the form r = a[1% ~ Y~(a,0,¢)],
(1)
L3
where a denotes the mean radius of a distorted equipotential surface and Y~ (a, 8, ~), the corresponding solid harmonic, it can be shown (cf., e.g., Kopal, 1960), that, correctly to quantities of the first order in superficial distortion,
j = 2,3,4,
(2)
where Aj denotes a constant depending on the structure of the distorted configurationl; m ' / m , the mass ratio of the disturbing to the distorted body; R, the separation of their centers; and X = cos ~ sin 8, the cosine of the angle between the axis joining the centers of the two bodies and an arbitrary radius vector. When we apply the foregoing Eq. (2) to evaluate the magnitude of the bodily tides raised by the Moon on the surface of the Earth, m ' / m = 1/81.37, a / R = 0.01658, and (for a homogeneous configuration), A2 = 2.5. If SO, the second harmonic tidal distortion on the surface of the Earth should be given (for the mean radius a s = 6371 km) by a $ Y 2 = 90.4P:(h) cm.
The observations reveal that the actual amplitude of the second-harmonic bodily tide on the Earth amounts only to 22 cm; the discrepancy between the two values being due partly to the finite concentration of mass inside the Earth (which should diminish the "fluid" value of A2 from 2.5 to approximately 1.96), and partly to the departures of the behavior of the Earth's mass from that appropriate for a fluid body. When, conversely, we apply Eq. (2) to evaluate the height of the bodily tides raised by the Earth on the Moon, m ' / m ----81.37 and a / R - - - - 0 . 0 0 4 5 2 1 - - i n which I For its definition, cf., e.g., Eq. Chapter II, Kopa], 1960).
(1-36)
of
case, on the lunar surface (a ©---- 1738 km) we should expect that, for a fluid homogeneous Moon, a ¢ Y 2 = 33 Ps(k) meters, a c Y 3 = 0.10 P3(~) meters, a© Y4 = 0.00041P4(h) meters,
the higher harmonics being clearly too small to be of any significance. The greater gravitational attraction of the Earth renders the equilibrium bodily tides on the Moon approximately 37 times as high as they are on the Earth, in spite of the smaller fractional size of our satellite; and even if the actual tides amount, as on Earth, to only 24% of their theoretical equilibrium value appropriate for a fluid, the height of lunar second-harmonic bodily tides should still be close to 7.9 P2(2,) meters. FORCED OSCILLATIONS
Moreover, the actual height of these tides should be expected to fluctuate in the course of each month, on account of the fact that the reciprocal of the radius vector R of the relative orbit of the Moon around the Earth varies in accordance with the equation 1
1 -}- ecosv = A(1 -- e~) '
(3)
where the semimajor axis A = 385 561 km; the eccentricity e = 0.0549; and the true anomaly v in the relative orbit is reckoned from the time of the perigee passage. Let us assume hereafter the Moon to be sensibly homogeneous, so that the mass m on the right-hand side of Eq. (2) can be represented by m = mo:(a/a©) 8, m©being the total mass of the Moon and a¢ its external radius. Let, moreover, the constant Aj on the right-hand side of Eq. (2) be replaced by the radial Love number
Aj,
h~ = 1 +
4;= 22j+1 ( j -- -~)'
(4)
to account for the finite rigidity of the distorted homogeneous configuration, characterized (by analogy with the Earth; cf., e.g., Munk and MacDonald, 1960, pp. 28-
414
ZDENEK KOPAL
29) by the coefficient ~ of tidal-effective rigidity close to 2.3, rendering h2 ----0.76. If so, the expression (2) can obviously be rewritten as Yi(a,O,¢)) = hj m s m©
a~-2a¢3 A ~-1 .
1 -f- e cos vV+l 1: J ) Ps(h)
(5)
and, correctly to the first power of orbital eccentricity, v a r y by an amount equal to iIYj(a,8,4)) = h~ m s a i - 2 a ¢ a m¢ A i+~
(e cos v)Pj(X)
(6)
in the course of each month. Since, however, the duration of the sideric month P----27.321661 days, the velocity of this "breathing" of the mass of the Moon, caused by jth harmonic bodily tides raised by fluctuating terrestrial attraction, should possess a radial component U equal to U = 4a(~Yj)/P.
(7)
If we insert for 8Y~ from (6) we find that, for second-harmonic tides, the ratio U
4h,(m¢~(a~,~
2
¥ = - f i \ ~ - - ~ 1 \ - ~ - / e cos ,p~(x)
(8)
leads, on the lunar surface (a = a~), on insertion of appropriate numerical values to a velocity U = 9.2 × 10-~P2(X) cos v cm/see, which remains of the same order of magnitude throughout the most part of the lunar interior. I t m a y be noted that such a velocity--corresponding to 29 nleters/year - - i s by some three orders of magnitude greater than the velocity of convection currents recently envisaged to exist in the lunar interior by several writers (Kopal, 1961b, 1962b,c; Runcorn, 1962). DISSIPATION OF ENERGY
In the face of such a situation, the following question becomes of cardinal iraportance for the study of the internal structure of our satellite: Does the Moon respond to a varying strain invoked by
the monthly Earth-raised tides as an elastic or plastic body? For in the former case, the bodily tides do not accomplish any net work in the course of the orbital cycle; but in the latter case, a tidal-effective viscosity would bring about a degradation of tidal motion into h e a t - - a process which would be bound in influence the thermal equilibrium of lunar interior, in accordance with the well-known consequences of the principle of the conservation of energy. At the present time a definite answer to this question is still lacking. An indication as to what it is likely to be may, however, be obtained by a comparison of the period P of the disturbing force with tile Maxwellian relaxation time t, for a visco-elastic body. As is well known, a rough estimate of the magnitude of t , can be obtained from the formula t, -- ~¢~,
(9)
where ~ denotes the coefficient of viscosity of the respective body; and fl is the coefficient of isothermal volume compression of its material. For silicate rocks (of density close to the mean density of the Moon) the value of fi is known to be close to 1 X 10-12 d y n e / c m 2 (cf., e.g., Birch, 1952). The estimates of the appropriate values of t~ are, on the other hand, still largely uncertain. The velocities of tectonic motions observed in the Earth's crust (such as, for example, the post-glacial uplift of Scandinavia) are indicative of a value of t~ as high as 1022 gin/era.see; while laboratory measurements of the viscosity for specific rocks cluster around / ~ l0 is gm/cm.sec, or even less. The former value, combined with the relatively well-known time t , of the order of 10 TM seconds or some 300 years; the latter, to 106 seconds or about 10 days. If the former applies to the Moon as a whole, the response of the lunar globe to a monthly tidal wave should be essentially elastic and entail no dissipation of energy. On the other hand, a viscosity of 1018 gm/cm-sec would lead to a relaxation time short enough to give rise to visco-elastic flow characterized by appreciable dissipation of motion into heat.
415
HEATING OF THE MOON EQUATIONS OF THE PROBLEM
In order to ascertain the actual amount of heat which could thus be produced, let us depart from the equation for the conservation of energy which (on neglect of terms arising from possible convection) assumes the explicit form
pC,(DT/Dt) = Kv2T A- p~ -4- g4,, (10) where p denotes the local density (regarded hereafter as constant) ; T, the temperature; C~, the specific heat at constant volume; ~, the constant coefficient of thermal conduction; ~, rate of liberation of radiogenic heat per unit mass; and ¢, the viscous dissipation function 1/0" 2
+ aoo~ (11) given as a quadratic form of the components aij of the viscous stress tensor. Let us assume, in what follows, that the motion in the lunar interior due to bodily tides is predominantly radial and characterized by the velocity component U, while the angular velocity components V and W are negligible. If so, it can be shown that the only nonvanishing components of the viscous stress tensor become ff aa ~
20U Oa
O'aO ~
10U a ~'
'
0"06= 2 Ua =a~' i a~
1 OU I
(12)
= a sin 00-~"
Remembering, moreover, that (to a sufficient approximation), U is given by Eq. (8) above, the foregoing components of the viscous stress tensor are easily evaluated; and their insertion in Eq. (11) reveals that the function ¢ governing the dissipation of energy of second-harmonic (j = 2) tides assumes the explicit form = ~-k2(3), 4 q- 1) c o s ~ v,
(13)
where we have abbreviated k = 12 ~-~ ( m ~ ' ~ [ a c y \m ~/\ A/"
(14)
Now the average value of h over the whole sphere is equal to 107/315 (or, very
approximately, one-third); while the average value of cos2v over a cycle is one-half. Therefore, if we wish to investigate the global secular variations of temperature in lunar interior, it should be sufficient to replace, in Eq. (10), the function ¢ by its average constant value of -- ~-k~;
(15)
and, by the same argument, replace the operator
D_O 0 Dt -- Ot -}- U--Or on the left-hand side of Eq. (10) by ~/~t, as the average value of U (proportional, to a first approximation, to e cos v) averages out to zero in the course of a cycle. SOLUTION OF THE EQUATIONS
Eq. (10) so simplified then assumes the explicit form
OT ~ 0 (a2 0T~ 3 pC, - ~ q- ~ ~-~ \ - ~ ] -}- pe q- -2 ~k
(16)
and its particular solution relevant to our problem is subject to the boundary condition requiring that the surface temperature T (a©, t) should remain constant--the value of which can [because of the linearity of Eq. (16)] be taken as zero? In view of the linearity of our problem, the desired particular solution Eq. (16) can be expressed as a sum of the form
T(a,t) = T~(a,t) q- Ta(a,t), (17) where T~(a,t) stands for the temperature due to radiogenic heat, while T~ (a,t) represents the possible "gravitational" contribution to internal temperature of the Moo~ due to viscous dissipation of bodily tides. The behavior of the "radiogenic" temperature Tc(a,t) in lunar interior is sufficiently well known from the work of Urey (1952), Levin (1960, 1962), or Kopal (1962a) already referred to. A similar solution of ~In so doing we should, however, r e m e m b e r t h a t this particular solution of our problem will, in effect, furnish the difference T ( a , t ) T(a©,t) between the internal a n d surface temperatures, rather t h a n the absolute temperature itself.
TABLE x=O O.
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5 0 5 5 60 6 5 7 0 8 0 9 0 i0 0
O. 0.0828 O. 1655 0.2483 0.3310 0.4138 0.4966 0.5793 0.6621 0.7449 0.8276 0.9104 0.9932 1.0759 I. 1587 1.3241 1.4895 1.6547 8.3333
I, K = 0 . 0 1 0
0.I000
0.2000
0.3000
0.4000
0.5000
0.6000
0.7000
0.8000
0.8500
0.9000
0.9250
0.9500
0.9750
O. 0.0828 0.1655 0.2483 0,.3311 0.4138 0.4966 0.5793 0.6621 0.7449 0.8276 0 9104 0 9932 1 0759 I 1587 l 3241 1 4894 l 6544 8 2500
O. 0.0828 0.1655 0.2483 0.3311 0.4138 0.4966 0.5793 0.6621 0.7449 0.8276 0.9104 0.9931 1.0759 1.1586 1.3238 1.4887 1.6531 8.0000
O. 0 0828 0 1655 0 2483 0 3311 0 4138 0 4966 0 5793 0 6621 0 7449 0 8276 0 9103 0 9930 l 0756 1 1581 1 3227 1 4865 1 6491 7.5833
O. 0.0828 0.1655 0.2483 0.3311 0.4138 0.4966 0.5793 0.6621 0,7448 0.8274 0 9099 0 9922 i 0743 l 1561 1 3187 1 4795 l 6381 7 0000
O. 0.0828 0.1655 0.2483 0.3311 0.4138 0.4965 0.5792 0.6617 0,7440 0.8260 0.9075 0.9885 1.0690 i. 1488 1.3060 i. 4600 1.6102 6.2500
O. 0.0828 0.1655 0.2483 0.3310 0.4136 0.4960 0.5778 0.6591 0,7395 0.8189 0.8973 0.9744 1.0504 1.1251 1.2706 1.4108 1.5458 5.3333
O. 0.0828 0.1655 0.2482 0.3303 0.4114 0.4910 0.5689 0,6450 0.7190 0.7911 0.8611 0.9292 0.9954 1.0597 1.1831 1.2999 1.4107 4.2500
O. 0 0828 0 1650 0 2448 0 3210 0 3933 0 4618 0 5267 0 5883 0 6469 0 7029 0 7563 0 8075 0 8566 0 9038 0 9929 I 0759 l 1534 3 0000
O. 0.0826 O. 1620 0.2351 0r.3020 0 3635 0 4023 0 4733 0 5228 0 5695 0 6135 0.6553 0.6950 0.7329 0.7692 0.8372 0.9001 0.9586 2.3125
O. 0,0805 O. 1491 0.2070 0.2574 0.3023 0.3429 0.3802 0.4146 0.4467 0.4768 0.5052 0.5320 0.5575 0.5817 0.6271 0.6688 0.7074 1.5833
O. 0.0761 O. 1335 0.1798 0.2191 0.2537 0.2848 0.3131 0.3391 0.3633 0.3859 0.4072 0.4273 0.4463 0.4644 0.4981 0.5291 0.5578 1.2031
O. 0.0653 0.1069 O. 1391 0.1659 0.1893 0.2102 0.2291 0.2464 0.2625 0.2775 0 2915 0 3048 0 3174 0 3293 0 3515 0 3719 0 3908 0 8125
O. 0.0425 0.0642 0.0806 0.0941 O. 1057 O. ll61 0.1255 0.1341 O. 1420 O. 1494 O. 1563 O. 1629 0.i~91 0.1749 0.1859 0.1959 0.2051 0.4114
0 0 5 1 0 1 5
2 0 2 5 3 0 3 5 4 0 4.5 5.0 5.5 6.0 6.5 7.0 8.0 9.0 I0.0 c~C
O. O. 1241 O. 2483 0.3724 O. 4966 O. 6207 O. 7449 O. 8690 0.9932 i. I173 1.2414 1.3655 1.4895 1.6134 I. 7371 1.9836 2.2282 2.4700 8.3333
0 0 1241 0 2483 0 3724 0 4966 0 6207 0 7449 0 8690 0 9932 1 1173 I .2"414 1 3654 1 4894 l 6132 l 7367 1 9827 2.2265 2.4671 8.2500
O. 0.1241 0.2483 0.3724 0.4966 0.6207 0.7449 0.8690 0,9931 1.1172 I..2412 1.3651 1.4887 1.6120 1. 7349 1. 9790 2.2200 2.4569 8.0000
O. 0.1241 0.2483 0.3724 0.4966 0.6207 0.7449 0.8690 0.9930 1. 1169 1.2405 1.3637 1.4865 1.6085 1.7298 1.9695 2.2046 2.4341 7.5833
0 0 1241 0 2483 0 3724 0 4966 0 6207 0 7448 0 8686 0 9922 t 1152 1 2376 1 3591 1 4795 1 5986 1 7164 1 9474 2.1715 2.3882 7.0000
O. 0.1241 0.2483 0.3724 0,4965 0.6205 0.7440 0.8668 0.9885 i. 1090 1.2278 1.3449 1.4600 1.5730 1.6838 1.8989 2.1048 2.3017 6.2500
I I , K = 0.015 O. 0.1241 0.2483 0.3723 0,4960 0.6185 0.7395 0.8582 0.9744 1.0879 1.1985 1.3061 1.4108 1.5125 1.6114 1.8006 1.9791 2.1477 5.3333
t~
o
m
TABLE
~q
3. O. 1241 0.2482 0.3710 0.4910 0.6072 0.7190 0.8263 0.9292 1.0278 I. 1223 1.2129 1.2999 1.3836 1.4640 1.6160 1.7572 1.8890 4.2500
O 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 l 1 3
1240 2448 3756 4618 5579 6469 7299 8075 8804 9492 0142 0759 1345 1904 2949 3907 4792 0000
O. 0.1230 0.2351 0.3334 0.4203 0.4984 0.5695 0.6347 0.6950 0.7512 0.8039 0.8534 0.9001 0.9443 0.9863 1.0645 1.1359 1.2016 2.3125
O. O. 1164 0.2070 0.2805 0.3429 0.3977 0.4467 0.4912 0.5320 0.5697 0.6049 0.6378 0.6688 0.6980 0.7257 0.7770 0.8238 0.8666 1.5833
O. 0.1066 0.1798 0.2370 0.2848 0.3263 0.3633 0.3967 0.4273 0.4554 0.4816 0.5061 0.5291 0.5508 0.5714 0.6094 0.6440 0.6756 1.2031
0 0 0877 0 1391 0 1780 0 2102 0 2379 0 2625 0 2846 0 3048 0 3234 0 3407 0 3568 0 3719 0 3862 0 3997 0 4246 0 4473 0.4680 0.8125
O.
0.0543 0.0806 O. i001 O. i161 O. 1299 O. 1420 O. 1529 O. 1629 O. 1720 O. 1805 O. 1884 O. 1959 0.2029 0.2095 0.2217 0.2328 0.2430 0.4114
m
0 0 5 1 0
1 5 2 0 2 5 3 0 3.5 4.0 4.5 5.0 5.5 6,0 6.5 7.0 8.0 9.0 10.0
O. 0.1655 0.3310 0.4966 0.6621 0.8276 0.9932 1, 1587 1.3241 1.4895 1. 6547 1.8194
O. 0.1655 0.3311 0.4966 0.6621 0.8276 0.9932 1, 1587 1.3241 1.4894 1.6544 1.8189
1.9836 1.9827 2. 1469 2. 1455 2.3092 2.3071 2.6292 2.6254 2.9416 2.9357 3.2446 3.2361 8.3333 8.2500
O. 0.1655 0.3311 0.4966 0.6621 0.8276 0.9931 1.1586 1.3238 1.4887 1.6531 1.8166 1.9790 2. 1401 2.2995 2.6122 2.9155 3.2078 8.0000
O. 0.1655 0.3311 0.4966 0.6621 0.8276 0.9930 1.1581 1.3227 1.4865 1. 6491 1.8102 1.9695 2. 1268 2.2817 2.5837 2.8741 3. 1519 7.5833
O. 0.1655 0.3311 0.4966 0.6621 0.8274 0 9922 1 1561 1 3187 1 4795 1 6381 1 7941 I 9474 2 0976 2 2446 2.5285 2.7985 3.0546 7.0000
TABLE
III,
O. 0.1655 0.3311 0.4965 0.6617 0.8260 0.9885 1.1488 1.3060 1.4600 1.6102 1.7565 1.8989 2.0372 2.1715 2.4280 2.6689 2,8950 6.2500
O. 0.1655 0.3310 0.4960 0.6591 0.8189 0.9744 1.1251 1.2706 1.4108 1.5458 1.6757 1.8006 1.9208 2.0364 2.2548 2.4572 2,6452 5.3333
TABLE
K=0.020 O. 0.1655 0.3303 0.4910 0.6450 0.7911 0.9292 1.0597 1,1831 1.2999 1.4107 1.5159 1.6160 1.7113 1.8022 1.9720 2. 1275 2,2706 4.2500
O. 0.1650 0.3210 0.4618 0.5883 0.7029 0.8075 0.9038 0.9929 1 0759 1 1534 1 2263
O. 0.1620 0.3020 0.4203 0,5228 0.6135 0.6950 0.7692 0.8372 0.9001 0.9586 1.0132
O. 0.1491 0.2574 0.3429 0,4146 0.4768 0.5320 0.5817 0.6271 0.6688 0.7074 0.7434
O. 0.1069 0.1659 0.2102 0.2464 0.2775 0.3048 0.3293 0.3515 0.3719 0.3908 0.4083
O. 0.0642 0.0941 0.1161 0.1341 0.1494 0.1629 0.1749 0.1859 0.1959 0.2051 0.2137
0.7770 0.8087 0.8385 0.8933 0.9428 0.9877 1.5833
O. 0.1335 0.2191 0.2848 0.3391 0.3859 0.4273 0.4644 0.4981 0.5291 0.5578 0.5845 0.6094 0.6328 0.6548 0.6953 0.7318 0.7649 1.2031
I I 1 1 I 1 3
2949 3597 4210 5346 6377 7317 0000
1.0645 1.1128 1.1584 1.2426 1.3186 1.3879 2.3125
0.4246 0.4399 0.4544 0.4809 0.5048 0.5264 0.8125
0.2217 0.2292 0.2363 0.2493 0.2610 0.2716 0.4114
0.2448 0.4618 0.6469 0.8075 0.9492 1.0759 1.1904 1.2949 1.3907 1.4792 1.5613 1.6377 1.7090 1.7758 1.8973 2.0052 2.1013 3.0000
0.2351 0.4203 0.5695 0.6950 0.8039 0.9001 0.9863 1.0645 1.1359 1.2016 1.2623 1.3186 1.3711 1.4202 1.5094 1.5883 1.6585 2.3125
0.2070 0.3429 0.4467 0.5320 0.~049 0.6688 0.7257 0.7770 0.8238 0.8666 0.9061 0.9428 0.9768 1.0086 1.0663 i. 1173 1.1626 1.5833
0.1798 0.2848 0.3633 0.4273 0,.4816 0.5291 0.5714 0.6094 0.6440 0.6756 0.7048 0.7318 0.7569 0.7804 0.8229 0.8604 0.8938 1.2031
0.1391 0.2102 0.2625 0.3048 0,3407 0.3719 0.3997 0.4246 0.4473 0.4680 0.4871 0.5048 0.5212 0.5365 0.5643 0.5888 0.6106 0.8125
0.0806 0.1161 0.1420 0.1629 0,.1805 0.1959 0.2095 0.2217 0.2328 0.2430 0.2523 0.2610 0.2690 0.2765 0.2901 0.3021 0.3127 0.4114
IV, K = 0 . 0 3 0
O.
0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7,0 8.0 9.0 10.0
0.2483 O. 4966 O. 7449 0.9932 1. 2414 1.4895 1.7371 1. 9836 2. 2282 2.4700 2.7081 2.9416 3.1699 3.3922 3.8171 4.2138 4.5815 8.3333
0 2483 0 4966 0 7449 0 9932 1 2414 1 4894 1 7367 1 9827 2 2265 2 4671 2.7038 2.9357 3.1620 3.3822 3.8025 4. 1944 4.5572 8.2500
0.2483 0.4966 0.7449 0.9931 1.2412 1.4887 1.7349 1.9790 2. 2200 2.4569 2.6890 2.9155 3.1358 3.3495 3.7559 4. 1331 4.4813 8.0000
0.2483 0.4966 0.7449 0.9930 1.2405 1.4865 1.7298 1,9695 2. 2046 2.4341 2.6574 2.8741 3.0837 3.2860 3.6684 4.0212 4.3451 7.5833
0.2483 0.4966 0.7448 0.9922 1.2376 1.4795 1.7164 1,9474 2.1715 2.3882 2.5973 2.7985 2.9919 3.1774 3.5253 3.8438 4.1344 7.0000
0.2483 0.4965 0.7440 0.9885 1.2278 1.4600 1.6838 1,8989 2. 1048 2.3017 2.4896 2.6689 2.8398 3.0027 3.3057 3.5805 3.8297 6.2500
0 2483 0 4960 0 7395 0 9744 1 1985 1 4108 1 6114 1 8006 1 9791 2 1477 2.3068 2.4572 2.5995 2.7342 2.9826 3.2060 3.4073 5.3333
0.2482 0.4910 0.7190 0 9292 1 1223 1 2999 1 4640 1 6160 1 7572 1.8890 2.0121 2.1275 2.2359 2.3379 2.5246 2.6912 2.8405 4.2500
--,1
418
ZDEN~.K KOPAL
Eq. (16) for the " g r a v i t a t i o n a l " component of lunar t e m p e r a t u r e leads to an expression of the form
---- 3.154 X 10 ~ sec as our unit of time, and 103 k m -~ l0 s cm as the unit of length, so t h a t the nondimensional value
Tg(a,t) = (~/~)Cr(x,t)
/~ = [3.154 X 10'~/(10s)2]g = 3.154K
(18)
where the constant C = 3 ( a , k ) 2,
(19)
and the function r(x,t) can be expressed as
~-(x,t) = 2
( - 1)~+1{ 1 -- exp [ - K ( n ~ / a c ) 2 t ] }
n=l
×
sin n~rx
(20)
in which we have abbreviated
K = ~/aC,,
and
x = a/a~.
(21)
The corresponding rate of heat flow across the boundary a----a¢, then becomes
OT K Oa
~C(Or) ac
Ox 1"
(22)
~UMERICAL RESULTS I n order to ascertain the actual amount of possible gravitational contribution to the internal t e m p e r a t u r e of the Moon, numerical calculations have been carried out, based on the foregoing equations, for appropriate values of the astronomical and physical p a r a m e t e r s involved. I n accordance with Eqs. (14) and (19),
C = 3
12ac p m ~ A 3 ] '
(23)
which for m e ~ m e = 81.37, a c / A = 0.004521, ac---- 1 . 7 3 8 X 1 0 s cm, P = 2 . 3 6 X 106 sec, e = 0.0549, and h2 = 0.76, is equal to C = 1.789 X 10 -~ crn ~ see -2.
(24)
Moreover, the coefficient K of heat conduction of silicate rocks is known to be in the neighborhood of 2 X 10 ~ e r g / g m . d e g (cf. Birch, 1952), so t h a t the coefficient K of thermal diffusivity, as defined by the first one of Eq. (21) becomes (for p = 3.3 g m / cm 3) a quantity of the order of 0.01 cIn2//sec. Let us, in w h a t follows, adopt 10 '~ years
of the coefficient of thermal diffusivity is likely to be between 0.010 and 0.030. Tables I - I V contain then the numerical values of 1 0 2 r ( x , t ) - - w i t h r(x,t) as given by the expansion on the right-hand side of Eq. ( 2 0 ) - - f o r t = 0 (0.5) 7.0 (1.0) 10.0 and oo, expressed in terms of 109 years t a k e n as our unit, and the fractional distance x = 0.0 (01) 0.8 (0.05) 0.90 (0.025) 0.9753; for the normalized coefficients K = 0.010 (Table I) 0.015 (Table I I ) , 0.020 (Table I I I ) and 0.030 (Table I V ) . A graphical representation of the functions 100 r(x,t) is shown on the accompanying Figs. 1-4. Ultimately, the numerical values of the normalized t e m p e r a t u r e gradients 100 (3r/~x)l on the surface are listed in T a b l e V. For the value of C as given by Eq. (23) above and ~ = 2 X 10~ e r g / c m . s e c - d e g , it follows t h a t
Tg(a,t) ~ 7 X 10-13~r(x,t) deg, and a glance at the numerical results listed in Tables I - I V reveals r(x,t) to be a quant i t y of the order of 10 -2. I n consequence, it appears t h a t the viscous dissipation, into heat, of motion actuated by bodily tides leads to the generation of temperatures inside the lunar globe which, for ~----1017 g m / c m - s e c , are of the order of 100°K; and for ~ = 10 is g m / c m - s e c of the order of 1000°K, i.e., comparable with the internal temperatures attained by a gradual release of radiogenic heat in the course of 4.5 billion years. F o r ~ ~ 1017 g m / c m . s e c , the amount of heat produced by viscous dissipation would be too small to be of cosmogonic importance. On the other hand, for/~ >> 10 is g m / cm-sec, the response of the lunar globe to a forced oscillation of tidal origin would again become elastic (as t. > P) and cease, as such, to bring about any dissipation of The column corresponding to x ~ 1 vanishes identically by virtue of the outer boundary condition.
HEATING
OF
THE
MOON
419
5.0
4.C
50
2.0, i
t=lO0
t=8.0
""-
i 1.0 I =4.0 1=50 t=2.0 1=10 i
OI
.I
0.2
0.5
0.4
0.5
I
0.7
0.6
0.8
0.9
1.0 x
x
FIo. 1. The function 100~-(x,D for K -----0.010.
Fro. 3. The function 100r(x,t) for ~: = 0.020.
\
5.0
,.jo
V
I
]
i
'!
0.2
0.3
" "
i
~t.
t = I0.0
2.0
t = ~.0 t=&O
t=6.0
r 1.0
i
t=5.0 I=40 t=5.0
~
~
.
~
,,°, ':i °',
t=2.0 t= fO
0
0.1
0.2
0.3
0,4
05
0.6
0.7
0.8
0.9
1.0
X
' -0
O. I
0.4
0.5 X
10 6
0.7
0.8
0.9
1.0
FIo. 2. The function 100~(x,t) for K---~ 0.015.
Fro. 4. The function 100~(x,t) for K ~-~0.030.
energy over a cycle. However, if ~he actual value of the effective viscosity happens to be of the order of 10 TM g m / c m . s e e (as suggested indeed by experiments with indi-
vidual samples of terrestrial common rocks); the mechanical heating of the Moon throughout its long past could, by itself, have raised the temperature of its
420
ZDEN~K KOPAL
interior to a level comparable with that expected from the radiogenic heating alone. The corresponding rate of heat flow
K(OT/Oa) --- 8 X 10 ~6t~(OT/Ox)~ then turns out, for p. = 10 ~ gin/era.see, to TABLE V
SURFACE TEMPERATURE GRADIENT I02(Or/OX)I
t o.o,o o o~5 ......................... 0. 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 8.0 9.0 10.0
0. -0.0828 -0.1655 -0.2483 -0.3310 -0.4138 -0.4966 -0.5793 --0.6621 --0.7449 --0.8276 --0.9104 --0.9932 --1.0759 --1.1587 --1.3241 --1.4895 -1.6547 --8.3333
0. -0.1241 -0.2483 -0.3724 -0.4966 -0.6207 -0.7449 -0.8690 --0.9932 -1.1173 --1.2414 --1.3655 -1.4895 -1.6134 -1.7371 -1.9836 -2.2282 --2.4700 --8.3333
0.0')0 0. -0.1655 -0.3310 -0.4966 -0.662l -0.8276 -0.9932 -1.1587 --1.324l --1.4895 --1.6547
--1.8H)4 --1.9836 --2.1469 --2.3092 --2.6292 --2.9416 --3.2446 --8.3333
o.oao 0. -0.2483 -0.4966 -0.7449 -0.9932 -1.2414 -1.4895 --1.7371 --1.9836 --2.2282 --2.4700 --2.7081 --2.9416 --3.1699 --3.3922 --3.8171 --4.2138 -4.5815 --8.3333
be a quantity of the order of 10 erg/cm :. see (10 < c a l / c m : - s e c ) at the surface. (.]ONCLUD1N(; ~EMARKN
I t m a y , furthermore, be added t h a t the foregoing figures for the gravitational heat production in elasto-viscous lunar interior represent (for a given adopted value of ~1 the mininmm estimates, as of the motion invoked by bodily tides only the effects of its radial comllonent U were included in the dissipation function (11). In particular, the azimuthal velocity component W due to the tidal lag rnay m a k e fm'ther significant contribution to tile gravitational heat production--possibly comparable with t h a t arising from the radial component it.self. As, however, the theory of tidal lag in viscous spheres produced by the periodic'lily fluctuating difference between the angular velocity of axial rotation and orbital revo-
lution is so far very incomplete, no accurate estimate of gravitational heat production to bc expected from this source is possible at the present time. In conclusion, the question m a y also be asked concerning a possible application of this llrocess to other bodies in the solar system; and of these, the bodily tides raised by the Sun on Mercury and Venus come immediately to mind. As is well known, tile fundamental characteristics of these planets, occurring in Eq. (19) are:
P a e Radius
m®/mp
Mercury
Venlls
87.97 days 57.,(} N 106 k m 0. 2056 2 . 4 X 10~ k m 6.3 X 10s
224.7 days 108.2 X 106 km 0. 00682 6.2 X 103 k m 4.0(,1 X 1()~
When we insert these values in Eq. (19) to evaluate the constants (} (and assume, for the sake of simplicity, that h.,--0.59 as for tile E a r t h ) , tile outcome reveals that Ci~,.c,,~ = 1.42 X 10 -9 cln"/sec 2, Cv ..... = 4.15 X 10 -14 em2/see 2. Accordingly, tile value of C for Mercury proves to be a hundred times, and, for Venus, a hundred thousand times, smaller than tile value Eq. (24) established earlier for our Moon. This reveals t h a t (for the same value of effective viscosity) the heat production by solar tides inside Mercury could have raised its internal temperature hy scarcely more than 10 ° in the past four hillion years; and for Venus the whole phenomenon turns out to t)e utterly insignificant. Therefore, the E a r t h - M o o n system appears to be tile only case in the solar family where gravitational heating due to the dissipation of bodily tides could be of cosmogonical significance. ACKNOWLED(IMENTS The c o m p u t a t i o n s presented in this paper have been carried out at the J e t Propulsion Laboratory's 7090 I B M C o m p u t e r during the writer's stay lhere in the s u m m e r of 1962. T h e c o m p u t a tions were p r o g r a m m e d by Mrs. Maxine Linde under the supervision of Mr, P ' d m e r Poulson,
HEATING
OF T H E
while all graphs presented in this paper were prepared by Mrs. Patricia Conklin. Their contributions to the completion of this work are hereby gratefully acknowledged. REFERENCES BIaCH, F. (1952). Elasticity and constitution of the Earth's interior. J. Geophys. Res. 57, 227. KOPAL, Z. (1960). "Figures of Equilibrium of Celestial Bodies," Section II-1. University of Wisconsin Press, Madison. KOPAL, Z. (1961a). "Thermal History of the Moon and of the Terrestrial Planets I." California Institute of Technology, Jet Propulsion Laboratory Tech. Report No. 32-108. KOPAL, Z. (1961b). Some remarks on the interpretation of the gravitational field of the Earth and the Moon. Proe. 12th Integer. Astronautical Congress, Washington, D. C. KOPAL, Z. (1962a). "Thermal History of the Moon and of the Terrestrial Planets II: Numerical Results." California Institute of Technology, Jet Propulsion Laboratory Tech. Report No. 32-225. KOPAL, Z. (1962b). "Thermal History of the Moon and of the Terrestrial Planets I I I : Convection in Planetary Interiors." California Institute of Technology, Jet Propulsion Laboratory Tech. Report No. 32-276. KOPAL, Z. (1962c). The effects of convection in the
MOON
421
mantle on the gravitational field of the Earth. Proc. o] the Third COSPAR Symposium in Washington (in press). L~VlN, B. Y., AND M~VA, S. V. (1960). Certain computations of the thermal history of the Moon. Doklady Akad. Nauk S~S.S.R. 133, 44. LEVIN, B. Y. (1962). Thermal history of the Moon. In "Proc. of the 14th I.A.U. Symposium on the Moon" (Z. Kopal and Z. K. Mikhailov, eds.), pp. 157-167. Academic Press, London and New York. LOWAN, A. N. (1933). On the cooling of a radioactive sphere. Phys. Rev. 44, 769. LOWAN, A. N. (1935). On the cooling of the Earth. Am. J. Math. 57, 174. MACDONALD, G. J. F. (1959). Calculations of the thermal history of the Earth. Geophys. Res. 64, 1967. MUNK, W. H., AND MACDONALD,G. J. F. (1960). "The Rotation of the Earth." Cambridge University Press, Cambridge. RUNCOaN, S. K. (1962). Convection in the Moon. Nature 195, 1150. UREY, H. C. (1952). "The Planets." pp. 49-57. Yale University Press, New Haven; and subsequent publications, of which Urey's chapter on the "Origin and History of the Moon," in "Physics and Astronomy of the Moon" (Z. Kopal, ed.), pp. 781-523, Academic Press, 1962, should in particular be noted.