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The equilibrium figures of the Earth and the major planets
ICARUS 1, 442--454 (1963)
The Equilibrium Figures of the Earth and the Major Planets RICHARD
JAMES
Department o] Mathematics, Manchester University o] Technology, England AND ZDENl~K KOPAL
Departme~d o] Astronomy, University o] Manchester, England, and Mathematics Research Center, University o] Wisconsin Received December 27, 1962 The form, exterior potential, and moments of inertia of the terrestrial globe as influenced by its axial rotation have been determined correctly to quantities of second order in superficial distortion, assuming the Earth to be in hydrostatic equilibrium, on the basis of both "standard" and "minimum" distribution of density inside the Earth as deduced by Bullen from seismological evidence. The outcome of our computations reveals that Bullen's "standard" model of the Earth's interior (characterized by a density discontinuity across the interface of the inner core) is--as it should be--compatible with the geometrical terrestrial oblateness, as well as with the moments of inertia of the Earth deduced by astronomical methods; but not entirely so with the properties of the external gravitational field of our planet as inferred from secular perturbations of the orbits of artificial satellites. In particular, the coefficient of the fourth zonal harmonic deduced from satellite motions is found to differ by a factor of two from the one computed on the equilibrium hypothesis--a fact revealing the extent to which the interior of our planet departs from hydrostatic eqtfilibrium (presumably on account of slow convection currents in the mantle). In the second part of the present paper, we have extended our investigation to a similar study of all consequences of the axial rotation of the planets Jupiter and Saturn, on the basis of their models of internal structure worked out recently by de Marcus. These planets are known to rotate so fast that the second-order effects of centrifugal force become of much greater relative importance than is the case for the Earth. The results of our computations are in satisfactory agreement with the observed facts; and the underlying assumption of the mass of these planets being in hydrostatic equilibrium seems to be confirmed as well as we could expect. I t has r e c e n t l y been p r o v e d by one of us ( K o p a l , 1960) t h a t if the level s u r f a c e of a h e t e r o g e n e o u s s e l f - g r a v i t a t i n g body, in h y d r o s t a t i c e q u i l i b r i u m , d i s t o r t e d by a x i a l r o t a t i o n is closely a p p r o x i m a t e d by an expression of the f o r m
r(a,~) = a l l - ~f22(a) + .f~.(a)P,_,(~) + fda)P4(,)
+ • • "],
(1)
a2f"~ + 6D(af'2 + f~) -- 6f2 = (N)[Zal'2(af',
+
9f~)
-- 9Daf'2(af'2 + 2f2)]
a2f"4 + 6D(af'4 4-f4) - 20f4 = (18/35)[2af'2(af'2 + 2f,2) --3D(3a2f'2 ~- 6afff'2 + 7f22)],
(3) w h er c the ] ~ ( a ) ' s are p u r e l y r a d i a l funct i o n s an d , ~ c o s 0 d e n o t e s t h c cosine of th e ' O n p. 70 of Kopal (1960) the factor "2" multiplying ]2 in the first parentheses on the p o l a r distance, the f u n c t i o n s f.,~(a) can be right-hand side of this equation was, unfortushown to s a t i s f y tile f o ll o w in g s i m u l t a n e o u s s y s t e m of s e c o n d - o r d e r differential e q u a t i o n I nately, omitted. 442
EQUILIBRIUM FIGURES OF THE EARTH AND MAJOR PLANETS
subject to the boundary conditions requiring that, at the origin (a = 0) f'2(0) = f'd0) = 0;
(4)
while, On the surface (a-~ al),
5 (~1~a1'~
2f2(al) + alf'~(a,) 't- ~ \'--~-11] 2
3
( Ool2ala~ [5f2(ax) A- a l f ' 2 ( a l ) ]
+~1 [2a12f~2(al)
~-- alf2(al)ff2(al) -]- 12f2~(a1)]
(5) and 4f~(al) -k aff',(al) -- (18/35)[alef~(al) + 5alf~(a,)ff~(al) + 6f~(al)],
(6)
where al = the mean radius of the distorted ' configuration, ml = its mass, G -- gravitation constant, = angular velocity of axial rotation; and
D = p/~
(7)
where
= -~
panda
(8)
denotes :the mean density interior to a. The aim of the present paper will be to report now the outcome of numerical integrations of the foregoing differential equations governing f2 and f4 for the density distributions p(a) characterizing our Earth, as well as the planets Jupiter and Saturn, and by a comparison of this outcome with the actual observations of polar flattening of these celestial bodies to ascertain the consistency of their observed flattening with those computed on the basis of the latest models now available for these planets. THE
EARTH
As far as the Earth is concerned, the best model of its internal structure still appears
443
to be that deduced by Bullen (1940, 1942), and reproduced since in a number of sources (cf. Jacobs, 1956). The respective distribution of density (in gm/cm a) can be found in column 1 of the accompanying Table I, and is diagrammatically shown on Fig. 1, while the corresponding march of the function D is exhibited on Fig. 2. Its nature (containing three distinct discontinuities at the depth of 33, 2898, and 5121 km) makes it impossible to represent this function analytically to the desired accuracy over the entire range and numerical integrations remain obviously the only avenue of approach to its solution. Such integrations were initiated in September 1959 at the Mathematics Research Center, United States Army, by Mr. Gerald B. Thorne, using the IBM 704 automatic computer of the Midwestern Universities Research Association (MURA) at the University of Wisconsin, and completed by the junior author of this paper using the Ferranti Mercury Computer at the University of Manchester. As long as terms of first order only are retained throughout Eqs. (1-6) (i.e., as long as the right-hand sides of these equations are ignored) the problem for computation is simple: The first-order Eq. (1) being linear and homogeneous in the sole dependent variable f2, its integration can be started from an arbitrary value of f2 (0), and its appropriate scale constant adjusted by means of the outer boundary condition (5). When, however, the terms of second order are considered, we face a nonlinear boundary-value problem of the jury type, the solution of which proceeded as follows. Near the center( a : 0), a Taylor series method was used to start the construction of the solution. If the variation of density sufficiently close to the origin can be approximated by = pc(1 -- ha ~ + • . . ) ,
(9)
D = 1 -- (2/5)Xa 2 ~-. • • .
(10)
p
then
Moreover, an insertion of assumed expansions
444
R. JAMES AND Z. KOPAL TABLE I Depth
p
-103f2
"]2
106f4
*]4
-I03f2
112
106f4
*]4
0
2.76
2.2504
0.5627
4.533
1.246
2.2873
0.5214
4.621
1.170
33-
2.76
2.2438
0.5594
4.502
1.239
2.2812
0.5166
4.593
].I57
33 +
3.33
2.2438
0.5594
4.502
1.239
2.2812
0.5166
4.593
1.157
i00
3.38
2.2306
0.5519
4.445
1.218
2.2688
0.5077
4.537
1.132
200
3.47
2.2112
0.5390
4.361
1.186
2.2505
0.4960
4.457
I.I00
300
3.55
2.1920
0.5265
4.278
1.146
2.2334
0.4861
4.378
1.074
400
3.63
2.1731
0.5155
4.198
1.114
2.2148
0.4778
4.302
1.054
413
3.64
2.1707
0.5141
4.188
i. Ii0
2.2125
0.4768
4.292
1.050
500
3.89 ± 0.01
2.1545
0.5067
4.121
1.090
2.1971
0.4710
4.227
1.036
600
4.13 ± 0.01
2.1359
0.5004
4.047
1.072
2.1795
0.4654
4.152
1.024
700
4.33 ± 0.01
2.1173
0.4967
3.970
1.064
2.1619
0.4613
4.079
1.0!6
800
4.49 ± 0.02
2.0988
0.4948
3.897
1.062
2.1443
0.4575
4.006
1.01[ ].010
900
4.60 ± 0.02
2.0800
0.4947
3.822
1.067
2.1266
0.4548
3.933
i000
4.68 ± 0.03
2.0611
0.4957
3.747
1.077
2.1089
0.4525
3.860
1.0!2
1200
4.80 ± 0,03
2.0227
0.4991
3.595
1.107
2.0733
0.4492
3.715
1.021
1400
4.91 ± 0.03
1.9831
0.5013
3.439
1.141
Z.0369
0,4455
3.568
1.037
1600
5.03 ± 0.03
1.9428
0.4988
3.279
1.170
2.0002
0.4397
3.417
1.055
1800
5.13 ± 0.03
1.9021
0.4891
3.067
1.197
1.9632
0.4290
3.265
1.066
2000
5.24 ± 0.03
1.8616
0.4699
2.958
1,178
1.9267
0.4103
3.113
1.061
2200
5.34 ± 0.03
1.8224
0.4381
3.801
1,137
1.8914
0.3794
2.964
1.024
2400
5.44 ± 0.03
1.7856
0.3889
2.655
1.039
1.8585
0.3312
2.824
0.917
2600
5.54 ± 0.03
1.7532
0.3161
2.527
0.85Z
1.8309
0.2592
2.704
0.751
2800
5.63 ± 0.03
1.7280
0.2116
2.432
0.535
1.8094
0.1566
2.616
0.443
2898"
5.68 ± 0.03
1.7192
0.1466
2.402
0.320
1.8030
0.0928
Z.590
0.238
2898 +
9.43 ± 0.35
1.7192
0.1466
2.402
0.320
1.8030
0.0928
2.590
0,238
3000
9.57 ± 0.35
1.7118
0.1454
2.380
0.311
1.7981
0.0880
Z.572
0,224
3200
9,85 ± 0.36
1.6966
0.1453
2.336
0.299
1.7889
0.0800
Z.539
0.200
3400
i0. ii ± 0.37
1.6805
0.1485
Z.Z91
0.295
1.7800
0.0736
2.508
0.180
3600
10.35 ± 0.38
1.6628
0.1557
2.244
0.301
1.7712
0.0686
2.478
0.165
3800
10.56 ± 0.39
1.6428
0.]674
2.193
0.319
1.7620
0.0651
2.448
0.153
4000
10.76 ± 0.40
1.6197
0.1832
2.134
0.352
1.7533
0.0630
2.420
0.145
4200
10.94 ± 0.41
1.5923
0.2038
2.064
0.596
1.7436
0.0621
2.389
0.142
4400
ii.ii ± 0.42
1.5596
0.2256
1.979
0.469
1.7332
0.0619
2.356
0.144
4600
ii.27 ± 0.43
1.5209
0.2391
1.873
0.556
1.7219
0.0612
2.320
0.149
4800
11.41 ± 0.44
1.4779
0.Z305
1.746
0.603
1.7097
0.0559
2.379
0.150
4982
11.54 ± 0.45
1.4479
0.1564
1.630
0.498
1.6996
0.0375
2.340
0.11o
5121-
14.20 ± 3.0
1.4284
0.00769
1.582
0.154
1.6958
0.0000
Z.ZZ4
0.0049
5121 +
16.80 ± 4.6
1.4284
0.00769
1.582
0.254
1.6958
0.0000
2.224
0.0049
5400
16.96 ± 4.8
1.4262
0.00464
1.574
O. lOl
1.6958
0.0000
2.221
0.0030
5700
17.08 ± 4.9
1.4245
0.00221
1.568
0.069
1.6958
0.0000
2.219
0.0014
6000
17.16 ± 5.0
1.4234
0.00067
1.564
0.038
1.6958
0.0000
2.218
0.0004
6371
17.20 ± 5.0
1.4229
0.00000
1.562
0.000
1.6958
0.0000
2.218
0.0000
g/em 5 Probable
15
minimum
I0
5.
!
I
I
I
I
I
I
I
I
I.
.5
0
R~t
1%o. 1. I n t e r n a l density distribution of the Earth.
1.00 M inlmum
Probable
0.50
L
I
r
I
I
f
0.5
F1o. 2. Variation of D inside the Earth. 445
i
1.0
R$
446
R. JAMES AND Z. KOPAL
J2(a) = ~ ao4a~s,
(11)
j=0
fda) = ~, [3va2~
(12)
j=o
in E q s . (2) and (3) reveals that ,~ = (2770)(C + 6)X,~o,
(13)
oq = (X/90)[18o~2 + C(3o~ - ~X,~o)] + (2/63)a~.(hao -- 7a2),
(14)
and f30 = (27/35)a02
(15)
~4 = (9/55)[2h~2- lla22 -- 14a0a4 + ~Xaoa2],
(16)
where we have abbreviated C =
3¢°12 7rGp~"
(17)
The parameters ao and flz, constituting the characteristic values of our problem, must be determined in such a way that the end values of ].~,4 and ~,~ satisfy the boundary conditions (5) and (6). This process is equivalent, in fact, to the solution of two simultaneous nonlinear equations in two unknowns. This was done, on the machine, by means of a program (due t o Dr. C. B. Haselgrove of the University of Manchester) based on a generalization, to n dimensions, of the NewtonRaphson method for inverse interpolation. Subsequently, empirical polynomials were used t o represent the variation of #/p~ between: successive discontinuities, and the Runge-Kutta method employed for stepby-step numerical integration. The values of constants occurring in Eqs. (2) and (5) can be specified without difficulty. As, for the Earth, a~ = 6371.221 km, o~, ~ 7.292115 X 10-~ sec-*, and Gm~ = 3.9863 X 10"0° cm3/see2, it follows that
w12a~/Gml = 0.0034500
(18)
3~l~/TrGp~ = 0.0044275
(19)
and for the round value of p~ = 17.2 gm/cm 3. The outcome of our numerical solution is
then contained in columns 3-6 of the accompanying Table I listing, successively, the value of 103f2(a),~lz(a), 10~]~(a), and ~4(a) as functions of the depth below the Earth's surface, when ~ ( a ) denote (as usual) the logarithmic derivatives of f j ( a ) . As m a n y decimals are generally retained as are regarded to be significant, though the last digits are not necessarily exact. Any greater accuracy would, however, require an increased precision in our knowledge of the density distribution inside the Earth. A numerical solution of the second-order effects of terrestrial rotation, on Bullen's model, was previously attempted (to lesser degree of accuracy) by Bullard (1948). He did so by numerical integration of the equations for second-order effects, deduced previously by de Sitter (1924), in terms of the variables ~ and K which are related with our f2 and f~ by means of the equations ~ - ~ = (3/2)f~ + (3/4)L:- + (5/8)f4 + . . . . (20) K = -(27/32)f22 + (35/32)f4 ÷ • • • .
(21) A comparison of Bullard's numerical results as given on p. 188 of his paper referred to above with those listed in our Table I reveals, on the whole, a satisfactory agreement within 2 or 3 units of the last place tabulated by Bullard--with the exception of the region immediately outside the Earth's inner core, where much larger discrepancies have been encountered. Inasmuch as Bullard's numerical work was carried out by hand, before the advent of automatic computing :machines employed in the present study, our results are undoubtedly more accurate and should supersede all previous work. s i n making the respective comparison, the reader should note that the coefficient of sin22¢ in Bullard's equation for the radius-vector on top of p. 187 in his paper is vitiated by a misprint, and should read --(%e" + K ) in place of
~£e~--K. When account is taken of (20) and (21), Bullard's Eqs. (9) and (10) become identical with (3) and (6) of the present paper.
EQUILIBRIUM
FIGURES
OF THE
Once the values of ]j (a) are thus numerically known, their insertion in Eq. (1) gives the respective approximation to the form of the equipotential surface of any layer of constant density and mean radius a inside the Earth, including its free surface. For the latter Eq. (1) now assumes the explicit form
r(al,p) = 6371.22110.9999990 - 0.0022504P2(v) -~ 0.0000045P4(~) -b • • .] km.
(23)
This computed value should be compared with c-~ = 297.0 ± 0.1 for the Hayford international ellipsoid of 1909, or with ¢-~ = 297.10___ 0.36 deduced by Jeffreys (1948) on the basis of all available geodetic as well as astronomical measurements, to see t h a t our assumption of hydrostatic equilibrium appears to be in complete agreement with the observed facts. The exterior potential V(r) of so distorted a terrestrial globe then follows from Eqs. (2-39) of Chapter I I I of Kopal (1960) in the form P2@)
]
where, in accordance with Eqs. (5-27) and (5-28) of the same reference, the constants 3
-
-
,2 ~ 1 2 a 1 3 ~
40 (4.2 + 3) ( 12a,'y 7 (.2 + 2). \ 3 - ~ J
and
MAJOR
k 4 = 7(,; Yb 2)
PLANETS
~ 74
(25)
447
2 + ~2 k,3--~mJ (26)
to the second order in small quantities. Since (~12alS/3Gml,) = 0.0011500 and the surface values of 7j(al) follow from the top line of Table I as 72 = 0.5627 and 74 = 1.126, respectively, it results from the foregoing equations t h a t 10ek4 =
(22)
r(al,0) -- r(al,1) 1 -= r(al,0) 296.8 ± 0.1
r
AND
106k2 = - 1096.1,
Since, at the poles (v = 1), Pz(1) = P~(1) = 1 while, on the equator (~ = 0), P~(0) = - - l / z , and P , ( 0 ) = %, the ellipticity of the Earth's meridional cross section corresponding to Bullen's density profile on the assumption of hydrostatic equilibrium comes out to be
V(r) = Gin-2 1 q-k2
EARTH
3.15;
(27)
or, in terms of other notations sometimes used, J = -3k2 =
0. 001644
(28)
and K = (15/4)k4 = 0.0000118,
(29)
respectively. I t m a y be of interest to compare these constants with their values deduced from secular perturbations of the longitude of the node and the perigee of the orbits of several artificial satellites launched since 1957. The most recent compilation of the relevant data by King-Hele (1962) discloses t h a t 10~k2 = -1082.7 -4- 0.3
(30)
108k4 =
(31)
and 1.7 :i= 0.3,
which appear to differ significantly from the theoretical values of these coefficients as given by Eqs. (27), and would on equilibrium hypothesis correspond to a reciprocal flattening of 298.24 ± 0.03. The origin of this discrepancy is probably real, and should be sought in the departures of the internal structure of the Earth from the state of hydrostatic equil i b r i u m - p r o b a b l y due to the convention currents in the Earth's mantle as envisaged in recent years by Vening Meinesz (1952, 1960), Runcorn (1962), and others. As has been shown recently by one of us (Kopal, 1962), the mean velocity of the hypothetical convection currents sufficient to account for the difference between the
448
R. J A M E S
A N D Z. K O P A L
theoretical (equilibrium) and observed value of k4 is of the order of 10-~ cm/sec. Should, as we suggest, the actual differences between the theoretical and observed kj's be indeed regarded as a measure of the departure of the Earth's interior from hydrostatic equilibrium, it should obviously be futile to convert the values of k2 as deduced from satellite motions into the equivalent "dynamical ellipticity"; for any such conversion implies a tacit assumption of hydrostatic equilibrium, without which the geometry of the surface and internal mass distribution are no longer uniquely related. The term "ellipticity"--without qualification--should, therefore, be reserved in the future to describe the geometrical form of the mean geoid as obtained from geodetic observations; while the term "dynamical ellipticity" is a ntisnomer if applied to bodies that are not in hydrostatic equilibrium, and as such should be relegated into oblivion. With these remarks in mind let us turn to evaluate the theoretical m o m e n t s o] i n e r t i a of the Earth built upon Bullen's model. We are indeed in a position to do so; for once the nmnerical solutions of the second-order boundary value problem (16) smnmarized in columns 3-6 of Table I have been completed, the corresponding moments of inertia of the terrestrial globe about its equatorial and polar axes can be obtained by numerical quadrature of the expressions"
A = (8/3)vpca15(0.0536072
. . .),
(35)
C = (8/3)~p~a~5(0.0537838
. . .),
(36)
corresponding to the value (C -
A)/C
= 0.003282
(37)
+ (8/7)f2(1 + ~-,2) + " • .]da. (34) The numerical integrations have revealed that
for Bullen's standard distribution of density inside the E a r t h on the assumption that its entire mass is in hydrostatic equilibrium. This latter assumption can again be tested by comparing Eq. (24) with the values of this constant determined astronomically from the observed terrestrial precession and the motion of the Moon. Thus de Sitter and Brouwer (1938), in their rediscussion of fundamental astronomical constants found 4 that ( C - - A ) / C 0.003272; Clemence (1948) and Jeffreys (1948) later arrived at ( C - - A ) / C = 0.003277; while the latest rediscussion of Jung (1956) yielded ( C - - A ) / C ~ 0 . 0 0 3 2 7 1 ± 0.000036 (the main part of the uncertainty of this ratio arising, not from the difference C - A, but rather from that of C). While this is obviously no place for reopening the whole question of the astronomical determination of (C-- A)/C simultaneously with other fundamental constants, we may note with satisfaction that its value, Eq. (37), determined by us on the assumption of hydrostatic equilibrium inside the Earth, is entirely consistent with its existing astronomical determinations within the limits of their observational errors. Thus neither the computed flattening of Bullen's model of the Earth, nor the difference of the moments of inertia about the principal axes are at variance with the most accurate geodetic and astronomical observations available up to this time, and strengthens our belief that (at least in these respects) the entire mass of the Earth behaves as if it were in hydrostatic equilibrimn throughout the interior. The solution of the boundary-value problem (1-6) discussed in the foregoing section has been based on the most probable distribution of density inside the
:~Deducible h'om Eqs. (8-11) and (8-12) of Chapter III, Kopal (1960).
the
.4 = (8/3)7r £ ": pa4{1 + (1/2)f2[1 + (1/5),7., + (18/7)f2(1 + ~2)] + • • .}da
(32)
pa4{1 - f 2 [ 1 + (1/5),2 fo" (3/7)f_~(1 +-2~2)1~,j + • • .}da,
(33)
~ti'l(|
C = (8/3)~r + so that
C -- A = 4~r fo"' Pa4f'*[1 + (1/5),,
4With Spencer Jones's value of 81.27 ± 0.02 for mass ratio Earth: Moon.
E Q U I L I B R I U M FIGURES OF T H E EARTH AND MAJOR PLANETS
Earth as deduced by Bullen (1940, 1942); and while in most parts of the Earth's mantle these densities appear to be quite well established by the seismological evidence, some uncertainty in their absolute values still persists inside the core. In estimating critically the extent of this uncertainty, Bullen (op. cit.) deduced also two limiting models of the internal constitution of our planet---the lower limit being based on the assumption that the density varies continuously across the interface of the Earth's inner core (at the depth of 5120 km). The absolute densities corresponding to such a model are then obtained by subtracting the uncertainties of the density as indicated in column 2 of Table I from the most probable values of p listed in front of them. The resulting m i n i m u m density distribution and its corresponding D-function is likewise diagrammatically shown on Figs. 1 and 2, and the outcome of a new numerical solution of the boundary-value problem (1-5) repeated with their use 5 is summarized in the remaining columns 7-10 of Table I. An inspection of these results reveals that, although the "standard" and "minimum" distributions of density inside the Earth differ appreciably only in the region of the inner core, the "minimum" surface values of ]j (a) as well as ~j (a) are sufficiently affected by the change to lead to a significant modification in the oblateness of our planet, as well as in its exterior potential or moments of inertia. Thus, for the "minimum" distribution of density considered in the foregoing paragraph, Eq. (1) assumes the specific form r(al,v) = 6371.22110.9999990 0.0022873P~(~) -~ 0.0000046P4(~) + • • -] km, -
(38)
leading to a reciprocal ellipticity e-~ = 292.0 ± 0.1; (39) while the coefficients of the second- and fourth-harmonic terms in the expansion, 5 A n d , of course, w i t h t h e c o n s t a n t (10) replaced b y 0.0062419 for t h e n e w v a l u e of po ~-12.2 g / c m s.
449
Eq. (24), of the exterior potential V(r) become 10ek2 = - 1132.9, 106k4 -3.25;
(40)
or
J = 0.001699, K = 0.0000122,
(41)
respectively. Moreover, the new moments of inertia A = (8/3)~rpca,5(O.0750094 . . . ) , C = (8/3)~rpca15(O.0752610 . . . ) ,
(42)
lead to (C -
A ) / C -- 0.003343.
(43)
A comparison of the foregoing results, Figs. (39) or (43), based on the minimum internal densities of the Earth culminating in pc----12.2 mg/cm 3, with the observed values of these parameters quoted before reveals that the two sets are plainly inconsistent. Our computations reveal, in fact, that if the mass of the Earth possesses a form appropriate for hydrostatic equilibrium the observed reciprocal ellipticity (1 _~ 297.10_ 0.36 (Jeffreys, 1948) corresponds to a central density of the Earth of not less than 17.5 ± 0.4 gm/cm 3, and values of pc as low as 12-13 gm/cm 3 are completely ruled out. MAJOR PLANETS
In conclusion of the present study of the rotational distortion of the Earth in hydrostatic equilibrium, we wish to carry out the same analysis also for the planets Jupiter and Saturn, rotating so fast that the second-order distortion terms in Eqs. (1-6) become of much greater relative importance. The internal structure of these two major planets has recently been investigated in considerable detail by de Marcus (1958), who arrived at the density distributions as reproduced in columns (2) of Tables II and III, and shown on Figs. 3 and 4. A summary of the relevant physical characteristics of these planets is collected in the following tabulation:
450
n.
JAMES
AND
Z.
KOPAL
Jupiter
Mass, T?tl
Saturn
1.902 X 103o gm 6.9861 X 109 cm 30.84 g m / c m 3 1.33 g m / c m ~ 0. 00017683 sec-1 0.084012 0.14520
Mean radius, al Central density, pc Mean density, p~ Angular velocity of rotation ~ ¢~12ala/ G rrtl 3,~/~Gpc
0.5694 X 103° gm 5. 763 X 10° c m 15.62 gra/cm 3 0.71 g m / c m ~ 0.000167659 sec-1 0. 14162 0. O25772
TABLE II
TABLE I I I
C O E F F I C I E N T S OF THE I N T E R N A L D I S T O R T I O N
C O E F F I C I E N T S OF THE I N T E R N A L
OF J U P I T E R
D I S T O R T I O N OF SATURN
a/al
p
1.000 0.00016 0.998 0.032 0.996 0.103 0.994 0.138 0.992 0.162 0.990 0.181 0.9894 + 0.185 0.9894- 0.197 0.98 0.246 0.94 0.367 0.90 0.479 0.86 0.593 0.82 0.714 0.802 + 0.777 0.8021.08 0.80 1.09 0.75 1.31 0.70 1.56 0.65 1.83 0.60 2.12 0.55 2.40 0.50 2.66 0.45 2.90 0.40 3.14 0.35 3.37 0.30 3.58 0.25 3.81 0.20 4.08 0.15 4.40 0.10 19.09 0.05 27.90 0.00 30.84
-- 10~f~
72
102]'4
74
a/al
4.409 4.396 4.484 4.372 4.360 4.348 4.346 4.346 4.292 4.079 3.891 3.723 3.580 3.522 3.522 3.516 3.369 3.233 3.103 2.975 2.843 2.702 2.544 2.361 2.139 1.865 1.534 1.175 0.921 0.830 0.794 0.782
1.381 1.372 1.362 1.352 1.343 1.333 1.331 1.331 1.295 1.152 1.022 0.896 0.766 0.705 0.705 0.701 0.626 0.573 0.538 0.522 0.524 0.547 0.597 0.679 0. 805 0.975 1.158 1.098 0.360 0.138 0.039 0.000
0.202 0.201 0.200 0.199 0.198 0.196 0.196 0.196 0.190 0.168 0.150 0.135 0.123 0.119 0.119 0.118 0.107 0.0981 0.0899 0.0824 0.0754 0.0686 0.0616 0.0541 0.0457 0.0359 0.0247 0.0138 0.0073 0.0059 0.0051 0.0047
3.32 3.30 3.28 3.25 3.23 3.19 3.18 3.18 3.09 2.76 2.45 2.14 1.78 1.60 1.60 1.59 1.39 1.23 1.12 1.04 1.00 1.00 1.05 1.17 1.39 1.77 2.32 2.81 0.85 0.32 0.09 0.00
1.000 0.995 0.990 0.985 0.980 0.975 0.970 + 0.9700.95 0.925 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.5227 + 0.52270.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
T h e a c t u a l n u m e r i c a l i n t e g r a t i o n s of t h e boundary-value problem (1-6) for Jupiter a n d S a t u r n h a s b e e n c a r r i e d o u t on t h e s a m e e l e c t r o n i c c o m p u t e r i n m u c h tile s a m e way as previously for the Earth; the main difference being greater care necessary to
p
0.00016 0.023 0.092 0.125 0.151 0.170 0.185 0.197 0.236 0.268 0.293 0.347 0.397 0.446 0.498 0.552 0.611 0.678 0.719 0.999 1.048 1.163 1.289 2.166 4.16 6.73 !).45 11.93 13.94 15.18 15.62
-- 102f2
72
102f*
7.
6.930 6.870 6.810 6.751 6.693 6.636 6.580 6.580 6.358 6.087 5.823 5.306 4.812 4.336 3.880 3.442 3.027 2.645 2.453 2.453 2.302 1.987 1.712 1.501 1.353 1.252 1.182 1.133 1.10l 1.084 1.079
1.760 1.738 1.718 1.700 1.685 1.673 1.662 1.662 1.636 1.624 1.622 1.621 1.614 1.609 1.613 1.614 1.589 1.509 1.436 1.436 1.426 1.352 1.153 0.826 0.540 0.334 0.195 0.107 0.039 0.006 0.000
0.594 0.583 0.573 0.564 0.555 0.546 0.537 0.537 0.503 0.464 0.426 0.357 0.296 0.242 0.194 0.153 0.117 0.0877 0.0747 0.0747 0.0650 0.0466 0.0324 0.0231 0.0175 0.0140 0.0119 0.0105 0.0096 0.0092 0.0009
3.45 3.38 3.32 3.27 3.22 3.18 3.15 3.15 3.08 3.07 3.08 3.10 3.11 3.10 3.21 3.30 3.35 3.23 3.08 3.08 3.13 3.17 2.91 2.18 1.48 0.95 0.57 0.32 0.13 0.01 0.00
represent their internal density distributions. For Jupiter and Saturn, initial repres e n t a t i o n s o f p / p c o f t h e f o r m o f E q . (9) c o u l d b e u s e d o n l y t o ( a / a l ) = 0.15 a n d 0.10, r e s p e c t i v e l y . 0 u t ~ i d e t h i s c o r e , e m pirical polynomials were used again to
EQUILIBRIUM
I
i
FIGURES
OF THE
EARTH
t
i
i
451
AND MAJOR PLANETS
[
I
1
i
0.8
0.6
10yy
0.4
O.Z
i
I
0.Z
I
~
0.4
I
c
0.6
I
[
0.8 a /a
1
FIG. 3. Internal density distribution and distortion of Jupiter. represent the theoretical variation of p/p~: For Jupiter, polynomials of degrees 3, 2, and 2 were found adequate in the inner, intermediate, and outer shell, respectively; while for Saturn, distinct polynomials had to be used in the ranges 0.1 < a/al < 0.4, 0.4 ~ a/a1 < 0.5227, and 0.5277 < a/a1 < 1. This last range included one discon-
tinuous range of the density, by 7 ~ , at a/a1 ~ 0.97, which was found to exert but very little effect on the solution. In order to test the accuracy of our results, the following two checks have been carried out: (1) The value of the constant X in Eq. (9) was arbitrarily reduced by 1% and the
452
R. J A M E S A N D Z. K O P A L 1.0
I
I
l
I
I
I
4
I
0.8
0.6
! 1
0.4
i
/ / 0.2
Pm/I'c 50
[O.Z
E
i 0.4
r
[
O.6
I
O.8
I I
1.0 a/dl
Fro. 4. Internal density distribution and distortion of Saturn. integration repeated; the corresponding ranges in the terminal values of ]=,,~ or e2,4 were likewise about 1%. Since the actual value of X is defined by de Marcus's model to 3-4 significant figures, we conclude t h a t errors in the representation of the core are likely to affect only the fourth figure in our results.
(2) A linear expression was used for the density distribution in the outer shell of Jupiter, or again a polynomial of fourth degree to represent the density in the interval of 0.1 < a / a l < 0 . 4 in Saturn. This caused no significant change in the terminal values of f2, 4 or 7/.,,4 as given in Tables I I and I I I .
EQUILIBRIUM FIGURES OF THE EARTH AND MAJOR PLANETS
The principal results of the numerical solution of the boundary-value problem (1-6) on the surface ( a = a~) then become: 6
f2 f4 e-1 ~ ~4 ks k4 J K (C -- A ) / C
Jupiter
Saturn
--0.04409 0.00202 15.76 1.381 3.32 --0.01440 0.0005667 0.02160 0.002125 0.0525
--0.06930 0.00594 10.33 1.760 3.45 --0.01797 0.001302 0.02096 0.004883 0.0768
These computed values may now be compared with the measured values of the polar flattening c of Jupiter or Saturn (which is indeed conspicuous to direct observation), or the values of k2,4 (or, which is equivalent, J or K) deduced from the observed rate of apsidal advance or nodal regression of the satellites of these planets. All evidence bearing on such determinations has recently been critically rediscussed by Brouwer and Clemence [1961], who found, empirically the following:
~-1 J K
Jupiter
Saturn
15.34 0.02206 ± 0.00022 0.00253 ± 0.00141
10.21 0.02501 ± 0.00003 0.00386 ± 0.00026
I t m a y be noted t h a t de Marcus (1958) also obtained the theoretical values of the constants e, J, and K for his J o v i a n and Saturnian model by numerical integration of de Sitter's equations for the effects of second-order rotational distortion, which are very different in form from our Eqs. (1-6), b u t should be essentially equivalent to them. de Marcus gave in his paper no details as to the way in which his computations were 'effected. His results agree, on the whole, quite w,~lt with ours; b u t as the latter were evaluated co1:sistently to at least one more decimal place throughout, they are undoubtedly the more accurate ones. The present results for Jupiter should supersede those deduced previously b y Message (1955) on the basis of the density distributions deduced by Ramsey (1951) or Miles and R a m s e y (1952).
453
A comparison of these figures with those evaluated with the aid of our Eqs. ( 1 - 6 ) , on the assumption of hydrostatic equilibrium, for the Jovian and Saturnian models constructed recently by de Marcus reveals an agreement which is, on the whole, satisfactory. The observed values of J for these planets appear to be accounted for within the limits of observational errors; and while the same is not yet quite true of K or c, it can hardly be in doubt that a further improvement of the theoretical models will--unlike the case of the E a r t h - eventually enable us to account for all observed properties of these planets on the assumption that the bulk of their masses is in hydrostatic equilibrium. REFERENCES BROUWRR, D., AND CLEMENCE, G. M. (1961). Orbits and masses of planets and satellites. In " T h e Solar System," (G. P. Kuiper and B. M. Middlehurst, eds.), Vol. III, pp. 72-73. University of Chicago Press. BULLAaD, E. C. (1948). The figure of the Earth. Monthly Notices Roy. Astron. (Geophys. Suppl.) 5, 186. BULLRN, K. E. (1940). Bull. Seism. Soc. Am. 30, 235. BULLRN, K. E. (1942). Bull. Seism. Soc. Am. 32, 19. CLEMENCE, G. M. (1948). On the system of astronomical constants. Astron. J. 53, 169. DR MARCUS, W. C. (1958). The constitution of Jupiter and Saturn. Astron. J. 63, 2. DR SITTER, W. (1924). On the flattening and the constitution of the Earth. Bull. Astron. Inst. Netherlands 2, 97. DR SITTBR, W., AND BaOVWER, D. (1938). On the system of astronomical constants. Bull. Astron. Inst. Netherlands 8, 213. JACOBS, J. A. (1956). The Earth's interior. Handb. der Phys. Berlin 47 (Geophysik I), 372-373 (Tables 3 and 5). JRFF~YS, H. (1948). The figures of the E a r t h and Moon. Monthly Notices Roy. Astron. Soc. (Geophys. Suppl.) 5, 219. JuNo, K. (1956). Figur der Erde. Handb. der Phys. Berlin 47 (Geophysik I), 637. KING-I-IELE, D. G. (1962). The E a r t h ' s gravitational potential, deduced from the orbits of artificial satellites. Geophys. J. Roy. Astron. Soc. 6, 270. KoPAL, Z. (1960). "Figures of Equilibrium of Celestial Bodies." M a t h . Res. Center, U.S.
454
a. JAMES AND Z. KOPAL
Army, Monograph No. 3. University of Wisconsin Press, Madison. KOPAL, Z. (1962). "Thermal History of the Moon and of the Terrestrial Planets: Convection in Planetary Interiors." Jet Propulsion Laboratory, California Institute of Technology, Tech. Rept. No. 32-276. MESSAGE, 1). J. (1955). The second-order theory of the figure of Jupiter. Monthly Notices Roy. Astron. Soc. 115, 550. MILES, B., A~D RAMSSV, W. It. (1952). On the internal structure of Jupiter and Saturn. Monthly Notices Roy. Astron. Soc. l l 2 , 234.
RAiSEr, W. tI. (1952). On the constitution of the major planets. Monthly Notices Roy. Astron. Soc. 111, 427. RvNcoR~r, S. K. (1962). Towards a united theory of continental drift. Nature 193, 311. VENING-MEINESZ, F. A. (1952). Convection currents in the Earth and the origin of the continents. Proc. Koninkl. Ned. Akad. Wetenschap. (Amsterdam) B55, 527, 536, 548. VENING-MEINESZ, F. A. (1960). Continental and ocean-floor topography; mantle convection currents. Proc. Koninkl. Ned. Akad. Wetenschap. (Amsterdam) B63, 410.
The Equilibrium Figures of the Earth and the Major Planets RICHARD
JAMES
Department o] Mathematics, Manchester University o] Technology, England AND ZDENl~K KOPAL
Departme~d o] Astronomy, University o] Manchester, England, and Mathematics Research Center, University o] Wisconsin Received December 27, 1962 The form, exterior potential, and moments of inertia of the terrestrial globe as influenced by its axial rotation have been determined correctly to quantities of second order in superficial distortion, assuming the Earth to be in hydrostatic equilibrium, on the basis of both "standard" and "minimum" distribution of density inside the Earth as deduced by Bullen from seismological evidence. The outcome of our computations reveals that Bullen's "standard" model of the Earth's interior (characterized by a density discontinuity across the interface of the inner core) is--as it should be--compatible with the geometrical terrestrial oblateness, as well as with the moments of inertia of the Earth deduced by astronomical methods; but not entirely so with the properties of the external gravitational field of our planet as inferred from secular perturbations of the orbits of artificial satellites. In particular, the coefficient of the fourth zonal harmonic deduced from satellite motions is found to differ by a factor of two from the one computed on the equilibrium hypothesis--a fact revealing the extent to which the interior of our planet departs from hydrostatic eqtfilibrium (presumably on account of slow convection currents in the mantle). In the second part of the present paper, we have extended our investigation to a similar study of all consequences of the axial rotation of the planets Jupiter and Saturn, on the basis of their models of internal structure worked out recently by de Marcus. These planets are known to rotate so fast that the second-order effects of centrifugal force become of much greater relative importance than is the case for the Earth. The results of our computations are in satisfactory agreement with the observed facts; and the underlying assumption of the mass of these planets being in hydrostatic equilibrium seems to be confirmed as well as we could expect. I t has r e c e n t l y been p r o v e d by one of us ( K o p a l , 1960) t h a t if the level s u r f a c e of a h e t e r o g e n e o u s s e l f - g r a v i t a t i n g body, in h y d r o s t a t i c e q u i l i b r i u m , d i s t o r t e d by a x i a l r o t a t i o n is closely a p p r o x i m a t e d by an expression of the f o r m
r(a,~) = a l l - ~f22(a) + .f~.(a)P,_,(~) + fda)P4(,)
+ • • "],
(1)
a2f"~ + 6D(af'2 + f~) -- 6f2 = (N)[Zal'2(af',
+
9f~)
-- 9Daf'2(af'2 + 2f2)]
a2f"4 + 6D(af'4 4-f4) - 20f4 = (18/35)[2af'2(af'2 + 2f,2) --3D(3a2f'2 ~- 6afff'2 + 7f22)],
(3) w h er c the ] ~ ( a ) ' s are p u r e l y r a d i a l funct i o n s an d , ~ c o s 0 d e n o t e s t h c cosine of th e ' O n p. 70 of Kopal (1960) the factor "2" multiplying ]2 in the first parentheses on the p o l a r distance, the f u n c t i o n s f.,~(a) can be right-hand side of this equation was, unfortushown to s a t i s f y tile f o ll o w in g s i m u l t a n e o u s s y s t e m of s e c o n d - o r d e r differential e q u a t i o n I nately, omitted. 442
EQUILIBRIUM FIGURES OF THE EARTH AND MAJOR PLANETS
subject to the boundary conditions requiring that, at the origin (a = 0) f'2(0) = f'd0) = 0;
(4)
while, On the surface (a-~ al),
5 (~1~a1'~
2f2(al) + alf'~(a,) 't- ~ \'--~-11] 2
3
( Ool2ala~ [5f2(ax) A- a l f ' 2 ( a l ) ]
+~1 [2a12f~2(al)
~-- alf2(al)ff2(al) -]- 12f2~(a1)]
(5) and 4f~(al) -k aff',(al) -- (18/35)[alef~(al) + 5alf~(a,)ff~(al) + 6f~(al)],
(6)
where al = the mean radius of the distorted ' configuration, ml = its mass, G -- gravitation constant, = angular velocity of axial rotation; and
D = p/~
(7)
where
= -~
panda
(8)
denotes :the mean density interior to a. The aim of the present paper will be to report now the outcome of numerical integrations of the foregoing differential equations governing f2 and f4 for the density distributions p(a) characterizing our Earth, as well as the planets Jupiter and Saturn, and by a comparison of this outcome with the actual observations of polar flattening of these celestial bodies to ascertain the consistency of their observed flattening with those computed on the basis of the latest models now available for these planets. THE
EARTH
As far as the Earth is concerned, the best model of its internal structure still appears
443
to be that deduced by Bullen (1940, 1942), and reproduced since in a number of sources (cf. Jacobs, 1956). The respective distribution of density (in gm/cm a) can be found in column 1 of the accompanying Table I, and is diagrammatically shown on Fig. 1, while the corresponding march of the function D is exhibited on Fig. 2. Its nature (containing three distinct discontinuities at the depth of 33, 2898, and 5121 km) makes it impossible to represent this function analytically to the desired accuracy over the entire range and numerical integrations remain obviously the only avenue of approach to its solution. Such integrations were initiated in September 1959 at the Mathematics Research Center, United States Army, by Mr. Gerald B. Thorne, using the IBM 704 automatic computer of the Midwestern Universities Research Association (MURA) at the University of Wisconsin, and completed by the junior author of this paper using the Ferranti Mercury Computer at the University of Manchester. As long as terms of first order only are retained throughout Eqs. (1-6) (i.e., as long as the right-hand sides of these equations are ignored) the problem for computation is simple: The first-order Eq. (1) being linear and homogeneous in the sole dependent variable f2, its integration can be started from an arbitrary value of f2 (0), and its appropriate scale constant adjusted by means of the outer boundary condition (5). When, however, the terms of second order are considered, we face a nonlinear boundary-value problem of the jury type, the solution of which proceeded as follows. Near the center( a : 0), a Taylor series method was used to start the construction of the solution. If the variation of density sufficiently close to the origin can be approximated by = pc(1 -- ha ~ + • . . ) ,
(9)
D = 1 -- (2/5)Xa 2 ~-. • • .
(10)
p
then
Moreover, an insertion of assumed expansions
444
R. JAMES AND Z. KOPAL TABLE I Depth
p
-103f2
"]2
106f4
*]4
-I03f2
112
106f4
*]4
0
2.76
2.2504
0.5627
4.533
1.246
2.2873
0.5214
4.621
1.170
33-
2.76
2.2438
0.5594
4.502
1.239
2.2812
0.5166
4.593
].I57
33 +
3.33
2.2438
0.5594
4.502
1.239
2.2812
0.5166
4.593
1.157
i00
3.38
2.2306
0.5519
4.445
1.218
2.2688
0.5077
4.537
1.132
200
3.47
2.2112
0.5390
4.361
1.186
2.2505
0.4960
4.457
I.I00
300
3.55
2.1920
0.5265
4.278
1.146
2.2334
0.4861
4.378
1.074
400
3.63
2.1731
0.5155
4.198
1.114
2.2148
0.4778
4.302
1.054
413
3.64
2.1707
0.5141
4.188
i. Ii0
2.2125
0.4768
4.292
1.050
500
3.89 ± 0.01
2.1545
0.5067
4.121
1.090
2.1971
0.4710
4.227
1.036
600
4.13 ± 0.01
2.1359
0.5004
4.047
1.072
2.1795
0.4654
4.152
1.024
700
4.33 ± 0.01
2.1173
0.4967
3.970
1.064
2.1619
0.4613
4.079
1.0!6
800
4.49 ± 0.02
2.0988
0.4948
3.897
1.062
2.1443
0.4575
4.006
1.01[ ].010
900
4.60 ± 0.02
2.0800
0.4947
3.822
1.067
2.1266
0.4548
3.933
i000
4.68 ± 0.03
2.0611
0.4957
3.747
1.077
2.1089
0.4525
3.860
1.0!2
1200
4.80 ± 0,03
2.0227
0.4991
3.595
1.107
2.0733
0.4492
3.715
1.021
1400
4.91 ± 0.03
1.9831
0.5013
3.439
1.141
Z.0369
0,4455
3.568
1.037
1600
5.03 ± 0.03
1.9428
0.4988
3.279
1.170
2.0002
0.4397
3.417
1.055
1800
5.13 ± 0.03
1.9021
0.4891
3.067
1.197
1.9632
0.4290
3.265
1.066
2000
5.24 ± 0.03
1.8616
0.4699
2.958
1,178
1.9267
0.4103
3.113
1.061
2200
5.34 ± 0.03
1.8224
0.4381
3.801
1,137
1.8914
0.3794
2.964
1.024
2400
5.44 ± 0.03
1.7856
0.3889
2.655
1.039
1.8585
0.3312
2.824
0.917
2600
5.54 ± 0.03
1.7532
0.3161
2.527
0.85Z
1.8309
0.2592
2.704
0.751
2800
5.63 ± 0.03
1.7280
0.2116
2.432
0.535
1.8094
0.1566
2.616
0.443
2898"
5.68 ± 0.03
1.7192
0.1466
2.402
0.320
1.8030
0.0928
Z.590
0.238
2898 +
9.43 ± 0.35
1.7192
0.1466
2.402
0.320
1.8030
0.0928
2.590
0,238
3000
9.57 ± 0.35
1.7118
0.1454
2.380
0.311
1.7981
0.0880
Z.572
0,224
3200
9,85 ± 0.36
1.6966
0.1453
2.336
0.299
1.7889
0.0800
Z.539
0.200
3400
i0. ii ± 0.37
1.6805
0.1485
Z.Z91
0.295
1.7800
0.0736
2.508
0.180
3600
10.35 ± 0.38
1.6628
0.1557
2.244
0.301
1.7712
0.0686
2.478
0.165
3800
10.56 ± 0.39
1.6428
0.]674
2.193
0.319
1.7620
0.0651
2.448
0.153
4000
10.76 ± 0.40
1.6197
0.1832
2.134
0.352
1.7533
0.0630
2.420
0.145
4200
10.94 ± 0.41
1.5923
0.2038
2.064
0.596
1.7436
0.0621
2.389
0.142
4400
ii.ii ± 0.42
1.5596
0.2256
1.979
0.469
1.7332
0.0619
2.356
0.144
4600
ii.27 ± 0.43
1.5209
0.2391
1.873
0.556
1.7219
0.0612
2.320
0.149
4800
11.41 ± 0.44
1.4779
0.Z305
1.746
0.603
1.7097
0.0559
2.379
0.150
4982
11.54 ± 0.45
1.4479
0.1564
1.630
0.498
1.6996
0.0375
2.340
0.11o
5121-
14.20 ± 3.0
1.4284
0.00769
1.582
0.154
1.6958
0.0000
Z.ZZ4
0.0049
5121 +
16.80 ± 4.6
1.4284
0.00769
1.582
0.254
1.6958
0.0000
2.224
0.0049
5400
16.96 ± 4.8
1.4262
0.00464
1.574
O. lOl
1.6958
0.0000
2.221
0.0030
5700
17.08 ± 4.9
1.4245
0.00221
1.568
0.069
1.6958
0.0000
2.219
0.0014
6000
17.16 ± 5.0
1.4234
0.00067
1.564
0.038
1.6958
0.0000
2.218
0.0004
6371
17.20 ± 5.0
1.4229
0.00000
1.562
0.000
1.6958
0.0000
2.218
0.0000
g/em 5 Probable
15
minimum
I0
5.
!
I
I
I
I
I
I
I
I
I.
.5
0
R~t
1%o. 1. I n t e r n a l density distribution of the Earth.
1.00 M inlmum
Probable
0.50
L
I
r
I
I
f
0.5
F1o. 2. Variation of D inside the Earth. 445
i
1.0
R$
446
R. JAMES AND Z. KOPAL
J2(a) = ~ ao4a~s,
(11)
j=0
fda) = ~, [3va2~
(12)
j=o
in E q s . (2) and (3) reveals that ,~ = (2770)(C + 6)X,~o,
(13)
oq = (X/90)[18o~2 + C(3o~ - ~X,~o)] + (2/63)a~.(hao -- 7a2),
(14)
and f30 = (27/35)a02
(15)
~4 = (9/55)[2h~2- lla22 -- 14a0a4 + ~Xaoa2],
(16)
where we have abbreviated C =
3¢°12 7rGp~"
(17)
The parameters ao and flz, constituting the characteristic values of our problem, must be determined in such a way that the end values of ].~,4 and ~,~ satisfy the boundary conditions (5) and (6). This process is equivalent, in fact, to the solution of two simultaneous nonlinear equations in two unknowns. This was done, on the machine, by means of a program (due t o Dr. C. B. Haselgrove of the University of Manchester) based on a generalization, to n dimensions, of the NewtonRaphson method for inverse interpolation. Subsequently, empirical polynomials were used t o represent the variation of #/p~ between: successive discontinuities, and the Runge-Kutta method employed for stepby-step numerical integration. The values of constants occurring in Eqs. (2) and (5) can be specified without difficulty. As, for the Earth, a~ = 6371.221 km, o~, ~ 7.292115 X 10-~ sec-*, and Gm~ = 3.9863 X 10"0° cm3/see2, it follows that
w12a~/Gml = 0.0034500
(18)
3~l~/TrGp~ = 0.0044275
(19)
and for the round value of p~ = 17.2 gm/cm 3. The outcome of our numerical solution is
then contained in columns 3-6 of the accompanying Table I listing, successively, the value of 103f2(a),~lz(a), 10~]~(a), and ~4(a) as functions of the depth below the Earth's surface, when ~ ( a ) denote (as usual) the logarithmic derivatives of f j ( a ) . As m a n y decimals are generally retained as are regarded to be significant, though the last digits are not necessarily exact. Any greater accuracy would, however, require an increased precision in our knowledge of the density distribution inside the Earth. A numerical solution of the second-order effects of terrestrial rotation, on Bullen's model, was previously attempted (to lesser degree of accuracy) by Bullard (1948). He did so by numerical integration of the equations for second-order effects, deduced previously by de Sitter (1924), in terms of the variables ~ and K which are related with our f2 and f~ by means of the equations ~ - ~ = (3/2)f~ + (3/4)L:- + (5/8)f4 + . . . . (20) K = -(27/32)f22 + (35/32)f4 ÷ • • • .
(21) A comparison of Bullard's numerical results as given on p. 188 of his paper referred to above with those listed in our Table I reveals, on the whole, a satisfactory agreement within 2 or 3 units of the last place tabulated by Bullard--with the exception of the region immediately outside the Earth's inner core, where much larger discrepancies have been encountered. Inasmuch as Bullard's numerical work was carried out by hand, before the advent of automatic computing :machines employed in the present study, our results are undoubtedly more accurate and should supersede all previous work. s i n making the respective comparison, the reader should note that the coefficient of sin22¢ in Bullard's equation for the radius-vector on top of p. 187 in his paper is vitiated by a misprint, and should read --(%e" + K ) in place of
~£e~--K. When account is taken of (20) and (21), Bullard's Eqs. (9) and (10) become identical with (3) and (6) of the present paper.
EQUILIBRIUM
FIGURES
OF THE
Once the values of ]j (a) are thus numerically known, their insertion in Eq. (1) gives the respective approximation to the form of the equipotential surface of any layer of constant density and mean radius a inside the Earth, including its free surface. For the latter Eq. (1) now assumes the explicit form
r(al,p) = 6371.22110.9999990 - 0.0022504P2(v) -~ 0.0000045P4(~) -b • • .] km.
(23)
This computed value should be compared with c-~ = 297.0 ± 0.1 for the Hayford international ellipsoid of 1909, or with ¢-~ = 297.10___ 0.36 deduced by Jeffreys (1948) on the basis of all available geodetic as well as astronomical measurements, to see t h a t our assumption of hydrostatic equilibrium appears to be in complete agreement with the observed facts. The exterior potential V(r) of so distorted a terrestrial globe then follows from Eqs. (2-39) of Chapter I I I of Kopal (1960) in the form P2@)
]
where, in accordance with Eqs. (5-27) and (5-28) of the same reference, the constants 3
-
-
,2 ~ 1 2 a 1 3 ~
40 (4.2 + 3) ( 12a,'y 7 (.2 + 2). \ 3 - ~ J
and
MAJOR
k 4 = 7(,; Yb 2)
PLANETS
~ 74
(25)
447
2 + ~2 k,3--~mJ (26)
to the second order in small quantities. Since (~12alS/3Gml,) = 0.0011500 and the surface values of 7j(al) follow from the top line of Table I as 72 = 0.5627 and 74 = 1.126, respectively, it results from the foregoing equations t h a t 10ek4 =
(22)
r(al,0) -- r(al,1) 1 -= r(al,0) 296.8 ± 0.1
r
AND
106k2 = - 1096.1,
Since, at the poles (v = 1), Pz(1) = P~(1) = 1 while, on the equator (~ = 0), P~(0) = - - l / z , and P , ( 0 ) = %, the ellipticity of the Earth's meridional cross section corresponding to Bullen's density profile on the assumption of hydrostatic equilibrium comes out to be
V(r) = Gin-2 1 q-k2
EARTH
3.15;
(27)
or, in terms of other notations sometimes used, J = -3k2 =
0. 001644
(28)
and K = (15/4)k4 = 0.0000118,
(29)
respectively. I t m a y be of interest to compare these constants with their values deduced from secular perturbations of the longitude of the node and the perigee of the orbits of several artificial satellites launched since 1957. The most recent compilation of the relevant data by King-Hele (1962) discloses t h a t 10~k2 = -1082.7 -4- 0.3
(30)
108k4 =
(31)
and 1.7 :i= 0.3,
which appear to differ significantly from the theoretical values of these coefficients as given by Eqs. (27), and would on equilibrium hypothesis correspond to a reciprocal flattening of 298.24 ± 0.03. The origin of this discrepancy is probably real, and should be sought in the departures of the internal structure of the Earth from the state of hydrostatic equil i b r i u m - p r o b a b l y due to the convention currents in the Earth's mantle as envisaged in recent years by Vening Meinesz (1952, 1960), Runcorn (1962), and others. As has been shown recently by one of us (Kopal, 1962), the mean velocity of the hypothetical convection currents sufficient to account for the difference between the
448
R. J A M E S
A N D Z. K O P A L
theoretical (equilibrium) and observed value of k4 is of the order of 10-~ cm/sec. Should, as we suggest, the actual differences between the theoretical and observed kj's be indeed regarded as a measure of the departure of the Earth's interior from hydrostatic equilibrium, it should obviously be futile to convert the values of k2 as deduced from satellite motions into the equivalent "dynamical ellipticity"; for any such conversion implies a tacit assumption of hydrostatic equilibrium, without which the geometry of the surface and internal mass distribution are no longer uniquely related. The term "ellipticity"--without qualification--should, therefore, be reserved in the future to describe the geometrical form of the mean geoid as obtained from geodetic observations; while the term "dynamical ellipticity" is a ntisnomer if applied to bodies that are not in hydrostatic equilibrium, and as such should be relegated into oblivion. With these remarks in mind let us turn to evaluate the theoretical m o m e n t s o] i n e r t i a of the Earth built upon Bullen's model. We are indeed in a position to do so; for once the nmnerical solutions of the second-order boundary value problem (16) smnmarized in columns 3-6 of Table I have been completed, the corresponding moments of inertia of the terrestrial globe about its equatorial and polar axes can be obtained by numerical quadrature of the expressions"
A = (8/3)vpca15(0.0536072
. . .),
(35)
C = (8/3)~p~a~5(0.0537838
. . .),
(36)
corresponding to the value (C -
A)/C
= 0.003282
(37)
+ (8/7)f2(1 + ~-,2) + " • .]da. (34) The numerical integrations have revealed that
for Bullen's standard distribution of density inside the E a r t h on the assumption that its entire mass is in hydrostatic equilibrium. This latter assumption can again be tested by comparing Eq. (24) with the values of this constant determined astronomically from the observed terrestrial precession and the motion of the Moon. Thus de Sitter and Brouwer (1938), in their rediscussion of fundamental astronomical constants found 4 that ( C - - A ) / C 0.003272; Clemence (1948) and Jeffreys (1948) later arrived at ( C - - A ) / C = 0.003277; while the latest rediscussion of Jung (1956) yielded ( C - - A ) / C ~ 0 . 0 0 3 2 7 1 ± 0.000036 (the main part of the uncertainty of this ratio arising, not from the difference C - A, but rather from that of C). While this is obviously no place for reopening the whole question of the astronomical determination of (C-- A)/C simultaneously with other fundamental constants, we may note with satisfaction that its value, Eq. (37), determined by us on the assumption of hydrostatic equilibrium inside the Earth, is entirely consistent with its existing astronomical determinations within the limits of their observational errors. Thus neither the computed flattening of Bullen's model of the Earth, nor the difference of the moments of inertia about the principal axes are at variance with the most accurate geodetic and astronomical observations available up to this time, and strengthens our belief that (at least in these respects) the entire mass of the Earth behaves as if it were in hydrostatic equilibrimn throughout the interior. The solution of the boundary-value problem (1-6) discussed in the foregoing section has been based on the most probable distribution of density inside the
:~Deducible h'om Eqs. (8-11) and (8-12) of Chapter III, Kopal (1960).
the
.4 = (8/3)7r £ ": pa4{1 + (1/2)f2[1 + (1/5),7., + (18/7)f2(1 + ~2)] + • • .}da
(32)
pa4{1 - f 2 [ 1 + (1/5),2 fo" (3/7)f_~(1 +-2~2)1~,j + • • .}da,
(33)
~ti'l(|
C = (8/3)~r + so that
C -- A = 4~r fo"' Pa4f'*[1 + (1/5),,
4With Spencer Jones's value of 81.27 ± 0.02 for mass ratio Earth: Moon.
E Q U I L I B R I U M FIGURES OF T H E EARTH AND MAJOR PLANETS
Earth as deduced by Bullen (1940, 1942); and while in most parts of the Earth's mantle these densities appear to be quite well established by the seismological evidence, some uncertainty in their absolute values still persists inside the core. In estimating critically the extent of this uncertainty, Bullen (op. cit.) deduced also two limiting models of the internal constitution of our planet---the lower limit being based on the assumption that the density varies continuously across the interface of the Earth's inner core (at the depth of 5120 km). The absolute densities corresponding to such a model are then obtained by subtracting the uncertainties of the density as indicated in column 2 of Table I from the most probable values of p listed in front of them. The resulting m i n i m u m density distribution and its corresponding D-function is likewise diagrammatically shown on Figs. 1 and 2, and the outcome of a new numerical solution of the boundary-value problem (1-5) repeated with their use 5 is summarized in the remaining columns 7-10 of Table I. An inspection of these results reveals that, although the "standard" and "minimum" distributions of density inside the Earth differ appreciably only in the region of the inner core, the "minimum" surface values of ]j (a) as well as ~j (a) are sufficiently affected by the change to lead to a significant modification in the oblateness of our planet, as well as in its exterior potential or moments of inertia. Thus, for the "minimum" distribution of density considered in the foregoing paragraph, Eq. (1) assumes the specific form r(al,v) = 6371.22110.9999990 0.0022873P~(~) -~ 0.0000046P4(~) + • • -] km, -
(38)
leading to a reciprocal ellipticity e-~ = 292.0 ± 0.1; (39) while the coefficients of the second- and fourth-harmonic terms in the expansion, 5 A n d , of course, w i t h t h e c o n s t a n t (10) replaced b y 0.0062419 for t h e n e w v a l u e of po ~-12.2 g / c m s.
449
Eq. (24), of the exterior potential V(r) become 10ek2 = - 1132.9, 106k4 -3.25;
(40)
or
J = 0.001699, K = 0.0000122,
(41)
respectively. Moreover, the new moments of inertia A = (8/3)~rpca,5(O.0750094 . . . ) , C = (8/3)~rpca15(O.0752610 . . . ) ,
(42)
lead to (C -
A ) / C -- 0.003343.
(43)
A comparison of the foregoing results, Figs. (39) or (43), based on the minimum internal densities of the Earth culminating in pc----12.2 mg/cm 3, with the observed values of these parameters quoted before reveals that the two sets are plainly inconsistent. Our computations reveal, in fact, that if the mass of the Earth possesses a form appropriate for hydrostatic equilibrium the observed reciprocal ellipticity (1 _~ 297.10_ 0.36 (Jeffreys, 1948) corresponds to a central density of the Earth of not less than 17.5 ± 0.4 gm/cm 3, and values of pc as low as 12-13 gm/cm 3 are completely ruled out. MAJOR PLANETS
In conclusion of the present study of the rotational distortion of the Earth in hydrostatic equilibrium, we wish to carry out the same analysis also for the planets Jupiter and Saturn, rotating so fast that the second-order distortion terms in Eqs. (1-6) become of much greater relative importance. The internal structure of these two major planets has recently been investigated in considerable detail by de Marcus (1958), who arrived at the density distributions as reproduced in columns (2) of Tables II and III, and shown on Figs. 3 and 4. A summary of the relevant physical characteristics of these planets is collected in the following tabulation:
450
n.
JAMES
AND
Z.
KOPAL
Jupiter
Mass, T?tl
Saturn
1.902 X 103o gm 6.9861 X 109 cm 30.84 g m / c m 3 1.33 g m / c m ~ 0. 00017683 sec-1 0.084012 0.14520
Mean radius, al Central density, pc Mean density, p~ Angular velocity of rotation ~ ¢~12ala/ G rrtl 3,~/~Gpc
0.5694 X 103° gm 5. 763 X 10° c m 15.62 gra/cm 3 0.71 g m / c m ~ 0.000167659 sec-1 0. 14162 0. O25772
TABLE II
TABLE I I I
C O E F F I C I E N T S OF THE I N T E R N A L D I S T O R T I O N
C O E F F I C I E N T S OF THE I N T E R N A L
OF J U P I T E R
D I S T O R T I O N OF SATURN
a/al
p
1.000 0.00016 0.998 0.032 0.996 0.103 0.994 0.138 0.992 0.162 0.990 0.181 0.9894 + 0.185 0.9894- 0.197 0.98 0.246 0.94 0.367 0.90 0.479 0.86 0.593 0.82 0.714 0.802 + 0.777 0.8021.08 0.80 1.09 0.75 1.31 0.70 1.56 0.65 1.83 0.60 2.12 0.55 2.40 0.50 2.66 0.45 2.90 0.40 3.14 0.35 3.37 0.30 3.58 0.25 3.81 0.20 4.08 0.15 4.40 0.10 19.09 0.05 27.90 0.00 30.84
-- 10~f~
72
102]'4
74
a/al
4.409 4.396 4.484 4.372 4.360 4.348 4.346 4.346 4.292 4.079 3.891 3.723 3.580 3.522 3.522 3.516 3.369 3.233 3.103 2.975 2.843 2.702 2.544 2.361 2.139 1.865 1.534 1.175 0.921 0.830 0.794 0.782
1.381 1.372 1.362 1.352 1.343 1.333 1.331 1.331 1.295 1.152 1.022 0.896 0.766 0.705 0.705 0.701 0.626 0.573 0.538 0.522 0.524 0.547 0.597 0.679 0. 805 0.975 1.158 1.098 0.360 0.138 0.039 0.000
0.202 0.201 0.200 0.199 0.198 0.196 0.196 0.196 0.190 0.168 0.150 0.135 0.123 0.119 0.119 0.118 0.107 0.0981 0.0899 0.0824 0.0754 0.0686 0.0616 0.0541 0.0457 0.0359 0.0247 0.0138 0.0073 0.0059 0.0051 0.0047
3.32 3.30 3.28 3.25 3.23 3.19 3.18 3.18 3.09 2.76 2.45 2.14 1.78 1.60 1.60 1.59 1.39 1.23 1.12 1.04 1.00 1.00 1.05 1.17 1.39 1.77 2.32 2.81 0.85 0.32 0.09 0.00
1.000 0.995 0.990 0.985 0.980 0.975 0.970 + 0.9700.95 0.925 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.5227 + 0.52270.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
T h e a c t u a l n u m e r i c a l i n t e g r a t i o n s of t h e boundary-value problem (1-6) for Jupiter a n d S a t u r n h a s b e e n c a r r i e d o u t on t h e s a m e e l e c t r o n i c c o m p u t e r i n m u c h tile s a m e way as previously for the Earth; the main difference being greater care necessary to
p
0.00016 0.023 0.092 0.125 0.151 0.170 0.185 0.197 0.236 0.268 0.293 0.347 0.397 0.446 0.498 0.552 0.611 0.678 0.719 0.999 1.048 1.163 1.289 2.166 4.16 6.73 !).45 11.93 13.94 15.18 15.62
-- 102f2
72
102f*
7.
6.930 6.870 6.810 6.751 6.693 6.636 6.580 6.580 6.358 6.087 5.823 5.306 4.812 4.336 3.880 3.442 3.027 2.645 2.453 2.453 2.302 1.987 1.712 1.501 1.353 1.252 1.182 1.133 1.10l 1.084 1.079
1.760 1.738 1.718 1.700 1.685 1.673 1.662 1.662 1.636 1.624 1.622 1.621 1.614 1.609 1.613 1.614 1.589 1.509 1.436 1.436 1.426 1.352 1.153 0.826 0.540 0.334 0.195 0.107 0.039 0.006 0.000
0.594 0.583 0.573 0.564 0.555 0.546 0.537 0.537 0.503 0.464 0.426 0.357 0.296 0.242 0.194 0.153 0.117 0.0877 0.0747 0.0747 0.0650 0.0466 0.0324 0.0231 0.0175 0.0140 0.0119 0.0105 0.0096 0.0092 0.0009
3.45 3.38 3.32 3.27 3.22 3.18 3.15 3.15 3.08 3.07 3.08 3.10 3.11 3.10 3.21 3.30 3.35 3.23 3.08 3.08 3.13 3.17 2.91 2.18 1.48 0.95 0.57 0.32 0.13 0.01 0.00
represent their internal density distributions. For Jupiter and Saturn, initial repres e n t a t i o n s o f p / p c o f t h e f o r m o f E q . (9) c o u l d b e u s e d o n l y t o ( a / a l ) = 0.15 a n d 0.10, r e s p e c t i v e l y . 0 u t ~ i d e t h i s c o r e , e m pirical polynomials were used again to
EQUILIBRIUM
I
i
FIGURES
OF THE
EARTH
t
i
i
451
AND MAJOR PLANETS
[
I
1
i
0.8
0.6
10yy
0.4
O.Z
i
I
0.Z
I
~
0.4
I
c
0.6
I
[
0.8 a /a
1
FIG. 3. Internal density distribution and distortion of Jupiter. represent the theoretical variation of p/p~: For Jupiter, polynomials of degrees 3, 2, and 2 were found adequate in the inner, intermediate, and outer shell, respectively; while for Saturn, distinct polynomials had to be used in the ranges 0.1 < a/al < 0.4, 0.4 ~ a/a1 < 0.5227, and 0.5277 < a/a1 < 1. This last range included one discon-
tinuous range of the density, by 7 ~ , at a/a1 ~ 0.97, which was found to exert but very little effect on the solution. In order to test the accuracy of our results, the following two checks have been carried out: (1) The value of the constant X in Eq. (9) was arbitrarily reduced by 1% and the
452
R. J A M E S A N D Z. K O P A L 1.0
I
I
l
I
I
I
4
I
0.8
0.6
! 1
0.4
i
/ / 0.2
Pm/I'c 50
[O.Z
E
i 0.4
r
[
O.6
I
O.8
I I
1.0 a/dl
Fro. 4. Internal density distribution and distortion of Saturn. integration repeated; the corresponding ranges in the terminal values of ]=,,~ or e2,4 were likewise about 1%. Since the actual value of X is defined by de Marcus's model to 3-4 significant figures, we conclude t h a t errors in the representation of the core are likely to affect only the fourth figure in our results.
(2) A linear expression was used for the density distribution in the outer shell of Jupiter, or again a polynomial of fourth degree to represent the density in the interval of 0.1 < a / a l < 0 . 4 in Saturn. This caused no significant change in the terminal values of f2, 4 or 7/.,,4 as given in Tables I I and I I I .
EQUILIBRIUM FIGURES OF THE EARTH AND MAJOR PLANETS
The principal results of the numerical solution of the boundary-value problem (1-6) on the surface ( a = a~) then become: 6
f2 f4 e-1 ~ ~4 ks k4 J K (C -- A ) / C
Jupiter
Saturn
--0.04409 0.00202 15.76 1.381 3.32 --0.01440 0.0005667 0.02160 0.002125 0.0525
--0.06930 0.00594 10.33 1.760 3.45 --0.01797 0.001302 0.02096 0.004883 0.0768
These computed values may now be compared with the measured values of the polar flattening c of Jupiter or Saturn (which is indeed conspicuous to direct observation), or the values of k2,4 (or, which is equivalent, J or K) deduced from the observed rate of apsidal advance or nodal regression of the satellites of these planets. All evidence bearing on such determinations has recently been critically rediscussed by Brouwer and Clemence [1961], who found, empirically the following:
~-1 J K
Jupiter
Saturn
15.34 0.02206 ± 0.00022 0.00253 ± 0.00141
10.21 0.02501 ± 0.00003 0.00386 ± 0.00026
I t m a y be noted t h a t de Marcus (1958) also obtained the theoretical values of the constants e, J, and K for his J o v i a n and Saturnian model by numerical integration of de Sitter's equations for the effects of second-order rotational distortion, which are very different in form from our Eqs. (1-6), b u t should be essentially equivalent to them. de Marcus gave in his paper no details as to the way in which his computations were 'effected. His results agree, on the whole, quite w,~lt with ours; b u t as the latter were evaluated co1:sistently to at least one more decimal place throughout, they are undoubtedly the more accurate ones. The present results for Jupiter should supersede those deduced previously b y Message (1955) on the basis of the density distributions deduced by Ramsey (1951) or Miles and R a m s e y (1952).
453
A comparison of these figures with those evaluated with the aid of our Eqs. ( 1 - 6 ) , on the assumption of hydrostatic equilibrium, for the Jovian and Saturnian models constructed recently by de Marcus reveals an agreement which is, on the whole, satisfactory. The observed values of J for these planets appear to be accounted for within the limits of observational errors; and while the same is not yet quite true of K or c, it can hardly be in doubt that a further improvement of the theoretical models will--unlike the case of the E a r t h - eventually enable us to account for all observed properties of these planets on the assumption that the bulk of their masses is in hydrostatic equilibrium. REFERENCES BROUWRR, D., AND CLEMENCE, G. M. (1961). Orbits and masses of planets and satellites. In " T h e Solar System," (G. P. Kuiper and B. M. Middlehurst, eds.), Vol. III, pp. 72-73. University of Chicago Press. BULLAaD, E. C. (1948). The figure of the Earth. Monthly Notices Roy. Astron. (Geophys. Suppl.) 5, 186. BULLRN, K. E. (1940). Bull. Seism. Soc. Am. 30, 235. BULLRN, K. E. (1942). Bull. Seism. Soc. Am. 32, 19. CLEMENCE, G. M. (1948). On the system of astronomical constants. Astron. J. 53, 169. DR MARCUS, W. C. (1958). The constitution of Jupiter and Saturn. Astron. J. 63, 2. DR SITTER, W. (1924). On the flattening and the constitution of the Earth. Bull. Astron. Inst. Netherlands 2, 97. DR SITTBR, W., AND BaOVWER, D. (1938). On the system of astronomical constants. Bull. Astron. Inst. Netherlands 8, 213. JACOBS, J. A. (1956). The Earth's interior. Handb. der Phys. Berlin 47 (Geophysik I), 372-373 (Tables 3 and 5). JRFF~YS, H. (1948). The figures of the E a r t h and Moon. Monthly Notices Roy. Astron. Soc. (Geophys. Suppl.) 5, 219. JuNo, K. (1956). Figur der Erde. Handb. der Phys. Berlin 47 (Geophysik I), 637. KING-I-IELE, D. G. (1962). The E a r t h ' s gravitational potential, deduced from the orbits of artificial satellites. Geophys. J. Roy. Astron. Soc. 6, 270. KoPAL, Z. (1960). "Figures of Equilibrium of Celestial Bodies." M a t h . Res. Center, U.S.
454
a. JAMES AND Z. KOPAL
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