(Received 1946 August 30) Summary

Transcription

1 THE FIGURE OF THE EARTH E. C. Bullard, F.R.S. (Received 1946 August 30) Summary A purely numerical method has been devised for the treatment of Darwin de Sitter's theory of the figure of a rotating earth in hydrostatic equilihrium. This has been applied to the density distribution suggested by Bullen. De Sitter's numerical constants A, K, are found to have the values X,=o*ooo16fo~ooo18, ~ 1 ~ X 6IO-'. 8 The ellipticity, on the assumption of hydrostatic equilibrium, is found to be -1=297'338&0'050. I. It is well known* that if the Earth is in hydrostatic equilibrium its figurecan be inferred to a close approximation from the observed value of the precessional constant, without a detailed knowledge of the internal distribution of density. The agreement of the observed figure with that calculated in this way is so good that it is desirable to include the small influence of the internal density distribution the small terms of the order of the square of the ellipticity that are neglected in the simple theory. This was done by Darwin 7 others over forty years ago, the work further extended by de Sitter.1 Their papers contain the complete theory of the matter. As, however, they had no detailed knowledge of the density distribution only limits could be set to the quantities concerned. Recently Bullen 8 has provided an estimate of the variation of density with depth, based mainly on seismological evidence. This density distribution enables the relation between the precessional constant the ellipticity, also the departure of the figure of equilibrium from an ellipsoid, to be calculated with more than sufficient accuracy for comparison with the observed values. It is the purpose of this paper to make these calculations. Bullen in an earlier paper 11 has calculated the variation of ellipticity with depth using preliminary density values. He does not however use the second order theory so cannot investigate the relation between the ellipticity the precessional constant to the approximation required by the accuracy of the observed values. Darwin's theory is analytical, that is to say the density distribution is expressed as an algebraic function the solution is obtained as a power series. These methods are ill adapted to dealing with a density distribution, such as Bullen's, that is expressed by a table of values. The method used in this paper for the solution of Darwin's equations is entirely numerical can be applied to any arbitrary density distribution. * H. Jeffreys, The Emth, chnptcr I?, Cambridge, t G. Darwin, M.N., 60, , $ W. de Sitter, Bull. astr. Insts. Netherlds., 2, , K. E. Bullen, Bull. seis. SOC. Amer., 30, , 1940 ; 32, 19-29, K. E. Bullen, M.N., Geophys. Suppl., 3, , 1936.

2 The Figure of the Earth '87 2. Method for the ellipticity.-let the surfaces of equal density be r=b[~ -~sin~++((9~~-~)sin~z+], where r is the distance from the centre, b the equatorial radius, + the geographical latitude, Q the ellipticity K a quantity expressing the departure from an ellipsoid. Both B K are functions of b, the equatorial radius. Surface values will be represented by el, K ~, similarly for other quantities. De Sitter shows that the theory is most simply developed in terms of the mean radius, b(r -&), of a quantity E' given by El= - 5~~142 + (1) Following Radau he takes a variable 7' given by E' dp ' where p is the mean radius expressed as a fraction of that of the outer surface. He shows than q' satisfies where I. 6 I05 F($) ={I + 4.1' - ~ q ' ~ + -(I - S/D)Q}/ 4 3 In these expressions D is the mean density within a radius 8, 6 is the density at radius p1 is w2r:lfm,. where w is the Earth's angular velocity, Mits mass f the constant of gravitation.* The numerical value of p1 is * If 6 is known as a function of p then D can be calculated from A good first approximation to 7' can be obtained from (3) by assuming F to be unity, from which it never departs by as much as 7 x IO-~. An approximation to E', can then be obtained by integrating (2) with the boundary condition :1?1;=5p1/2-26; + IOpf/2I + &/7-6~1pJ7. (6) The values of E' obtained by this integration can be used in (4) (5) to get a better value of F. This improved F can be used in (3) to give a second approximation to q' so on till the necessary accuracy has been obtained. Actually the second approximation is all that is necessary. E is then found from (I) using the values of K found below. 3. Results for Ellipticity.-The calculations described above have been carried out using the densities given in Table I1 of Bullen's 1940 paper Table V of his I942 paper. The values obtained for the mean density for q', F E are given in Tahle I in Figs. I 2. They have been computed to one more place of decimals than is given in the table. (4) * We use de Sitter's notation, D 6 are expressed as fractions of the mean density of the whole Earth. t W. de Sitter D. Brouwer, Bull. as&. Insts. Nether@., 8, , 1938, equation (40) with the Constants adopted below,

3 I 88 E. C. Bullard TABLE I Depth km. Density gm./cm.a Mean Density gm./cm.8 v' F c K I '32 3 *38 3 '47 5'53 5'57 5'57 5 '64 5 ' ' ' * = '55 3 '63 3 *a4 3'89 4' ' I I.oooo * ~ ' IOOO I I 600 I '33 4' ' '24 5 '34 5'44 5 '54 5 '63 5 *68 9 '43 9'57 9'85 10'1 I I I ' I 7' ' I I I I '49 11'75 I ' ' ' I '00020 I ' ooo ~ 1a I'oO021 I I'O0044 I.oO * I I 1*00030 I '0003 I I '00039 I '-43 I *-49 I I'-55 I ' I ' * ' ~ ' ~ ~ * omoz5 32 I I I2 0' I I ~ ' '99994 I '00Ooo 1'00000 I 'OOoaO 0'002 I I I 0~ om02 I '001 0'000 I '00000 I 'Ooooo 0'002 I I 32 0

4 The of the Earth Y Fro Depth km. FIG.' x 10-0 The ellipticity of the outer surface obtained by this process ( or rpg7.3) is of no great significance, it is merely the ellipticity which would be taken up by a liquid body with the assumed density distribution. As one of the data used in getting the density distribution is the ratio of the Earth's radius of gyration to its diameter, as this can only be obtained by an assumption about the ellipticity, we have simply reproduced one of our assumptions. Any reasonable value for the ellipticity could be obtained by this process by a small adjustment of the densities. What is required is to calculate the ellipticity from the precessional constant with allowance for the slight effect of the internal density distribution. The latter is involved only through the small quantity A, given by 60 K This can be calculated when?; has been found. The precessional constant the mass of the Moon give the quantity H = (C - A)/C, where C A are the moments of inertia with respect to polar

5 I90 E. C. Bullard equatorial axcs. H, A, K~ then give the ellipticity. The relations are somewhat complicated, but may be simplified by putting approximate numerical values in all the small terms. De Sitter Brouwer have done this, assuming i for the precessional constant get w = o & 3.2 x IO-~, (8) 6-1 = [ (u - v) ~ % $], where w, 2, u, v, x # are given by, H = ( I + w), p- =81*53(1+2), rl = (1+ u) metres, g, = ( I + v) cm./sec.2, K = 0~ IO-~X, A, = 0~00040 g, being the value of gravity at latitude sin-, d& p the ratio of the Moon s mass to that of the Earth. The value of A, obtained from (7) is A, = 0~ With Spencer Jones s t value for the mass of the Moon (p-l= _ ) equation (8) gives Using these values of H A, the value of K, found below Spencer Jones s values of rl g, we get $, 1, = , E = The value for A, is considerably less than de Sitter s result, which is o.ooo# f 0~00015 (p.e.). The reason for de Sitter s high value is that equation (4) of his 1924 paper overestimates the average value of F,. In the average required F is multiplied by 1Bp, so that the values in the outer part of the Earth are heavily weighted relative to those in the inner parts ; in the outer 1500 km. F, - I never exceeds 2.1 x IO-~ compared with de Sitter s assumed average of 5 x IO-~. There also seems to be an algebraic mistake in the unnumbered equations immediately following his (25), but this has not greatly affected the result. 4. The uncertainty of the ellipticity.-the effect on Al of adding to the Earth a shell of thickness Ag of density A8 greater than the assumed density can be calculated from the expressions given in Section 3. Any slight change in the density distribution from that assumed can then be allowed for by approximating to it by a series of such shells. Let the effect of such a shell on A, be AA,/A, = kajla8, then the values of k for various values of the shell radius were found to be + #, H = 0* * g. Depth of shell (km.) o k * W. de Sitter D. Brouwer, Rull. astr. Insts. Netherlak., 8, , t H. Spencer Jones, Mem. R.A.S., 66, 60, Using X~=O*OOO++ putting KI zero, Spencer Jones (loc. cit.) gets e-1= but this appears to be fi numerical error, with his data I get 297.~01.

6 The Figure of the Earth 191 In calculating these the effect of various small terms has been neglected k may be in error by up to 10 per cent. We wish to use these values of k to calculate the uncertainty in A, ; to do this the uncertainty in 6 is needed as a function of the radius, an estimate of how far the errors in density at different radii are independent. From information given by Bullen it is considered reasonable to assume a stard error of 0.03gm./cm.~ in 6 for the part of the Earth outside the core, 0.2 inside the core down to the discontinuity at a depth of from there to the centre. An error of 0.03 throughout the part outside the core would give a 21 per cent. error in Al, a 0.2 error in the outer core would give 50 per cent. a 3.0 error in the inner core a IOO per cent. error. From this it is clear that A1 is uncertain by its whole value that the main part of the uncertainty comes from the uncertainty in the density near the centre. This conclusion is not changed if the density changes are subject to the condition that the total mass moment of inertia are not altered. The above three errors combined give uncertainties of 0~00018 in A, in e-l. As the uncertainty in e-l due to the uncertainty in the mass of the Moon is 0.042, the accuracy achieved is sufficient we have Al = 0~00016 & 0~00018 e-l = k This ellipticity may be compared with those obtained without the assumption of hydrostatic equilibrium from the variation of gravity with latitude from the motion of the Moon. These give & 0.68 * k 0.65 t There is no significant discrepancy. In view of the observed departures from a figure of equilibrium (e. g. the ellipticity of the equator), this close agreement could not have been predicted. 5. The calculation of the departure from an ellipsoid.-the quantity K expressing the departure of the level surfaces from exact ellipsoids satisfies the relation (9) which may be deduced from Darwin s equation (23). The solution required is that which is finite at the origin satisfies Darwin s (42) at the outer surface. Expansion in series shows that the only solutions that are finite at the origin are also zero there. Such sblutions may be written 7 K=AK~+K~, (11) * H. Jeffreys, M.N., Geophys. Suppl., 5, 65-66, t This is calculated from the data given by Spencer Jones, loc. cit., p. 63. G I5

7 192 The Figure of the Eurth where KA is a complimentary function obtained by numerical integration of (9) with the right-h side put equal to zero xb is a particular integral obtained by numerical integration of the full equation. the series expansion, valid near the origin, K = Ap x 10-8 p. These integrations were started from The numerical integrations were performed by the methods given in a Nautical Almanac Office publication." Substitution in (10) then gives A, K can be calculated for all depths from (11). The results are given in Table I in Fig. 2. The relation which may be deduced from Darwin's equation (20), was used as a check on the correctness of the arithmetic. The value of K at the surface is K~ = 68 x This is rather larger than de Sitter's result of 50 x IO-~. This increase was to be expected as de Sitter has taken the core to have a radius of 0.8 that of the Earth, whereas it is now known to be only 0.54 of the radius. Decreasing the radius of the core, whilst adjusting the densities to maintain the total mass moment of inertia constant, increases K (Darwin shows that K = 140 x IO-~ when the core is reduced to a point mass). These results show K to be relatively insensitive to the exact density distribution the result obtained is unlikely to be appreciably affected by uncertainties in Bullen's densities. The form of the outer surface using the values obtained for el N~ is r = b[r sina t# x 10- sin-a 241. (12) The corresponding gravity formula is g =g,[r -(el-%pl- yp: + 8qp1 -&l) sine 4 - -&: + 3 ~ sin2 ~ 241, ) inserting numerical values gives g =g,[i sin x 10- sina 241. (13) (12) (13) are the expressions that would be obtained on an earth with Bullen's density distribution exactly in hydrostatic equilibrium throughout. As has been pointed out above, the values of the coefficients of sin24 obtained for the actual Earth agree closely with (12) (13). There is no experimental evidence as to the sin22+ terms, the part of them depending on K contributes at its maximum 4.3 metres to the figure cm./sec.2 to g. Department of Geodesy Geophysics, Cambridge : 1946 August 28. * interpolation Allied Tables, 2nd Edition, pp , 1942.