(Received 1946 August 30) Summary

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T H E FIGURE OF T H E EARTH E.

C.

Bullard, F.R.S.

(Received 1946 August 30)

Summary

A purely numerical method has been devised for the treatment of Darwin and 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, and K , are found to have the values X,=o*ooo16fo~ooo18,

~ 1 ~ 6 8 X IO-'.

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 and 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

and others over forty years ago, and 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 and the ellipticity, and 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 and so cannot investi- gate the relation between the ellipticity and 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 and 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 and can be applied to any arbitrary density distribution.

*

H. Jeffreys, The Emth, chnptcr I ? , Cambridge, 1929.

t

G. Darwin, M.N., 60, 82-124, 1899.

$ W. de Sitter, Bull. astr. Insts. Netherlds., 2 , 97-108, 1924.

11 K. E. Bullen, M.N., Geophys. Suppl., 3, 395-401, 1936.

K. E. Bullen, Bull. seis. SOC. Amer., 30, 235-250, 1940 ; 32, 19-29, 1942.

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The Figure of the Earth

'87

2. Method for the ellipticity.-Let the surfaces of equal density be r = b [ ~ - ~ s i n ~ + + ( ( 9 ~ ~ - ~ ) s i n ~ z + ] ,

where r is the distance from the centre, b the equatorial radius,

+

the geographical latitude, Q the ellipticity and K a quantity expressing the departure from an ellipsoid. Both B and K are functions of b, the equatorial radius. Surface values will be represented by el, K ~ , and similarly for other quantities.

De Sitter shows that the theory is most simply developed in terms of the mean radius, b(r - &), and of a quantity E' given by

Following Radau he takes a variable 7' given by

E l = € - 5 ~ ~ 1 4 2

+

(1)

E' dp

'

where

p

is the mean radius expressed as a fraction of that of the outer surface. He shows than q' satisfies

(4) I . 6

F($)

={I

+

4.1'

- ~ q

+

-(I ' ~

-

S/D)Q}/

4 3

I05 where and

I n these expressions D is the mean density within a radius

8,

6 is the density at radius and p1 is w2r:lfM,. where w is the Earth's angular velocity, M it s mass and

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

0.00345000 0*00000002.

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 I O - ~ . 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) and (5) to get a

better value of F. This improved F can be used in (3) to give a second approximation to q' and 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 and Table V of his I942 paper. The values obtained for the mean density and for

q', F and E are given in Tahle I and in Figs. I and 2. They have been computed

to one more place of decimals than is given in the table.

*

We use de Sitter's notation, D and 6 are expressed as fractions of the mean density of the

t

W. de Sitter and D. Brouwer, Bull. as&. Insts. Nether@., 8, 213--231, 1938, equation (40)

whole Earth.

with the Constants adopted below,

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I 88 Depth km. 0 33 33 I 0 0 200 3- 400 413 5 0 0 600 700 800 900 I O O O I 2 0 0 1400 I 600 I 800 2000 2200 2400 2600 2800 2900 2900 3000 3200 34- 3600 3800 4- 4200 4400 4600 4800 4982 5121 5121 5400 5700 6000 6371 Density gm./cm.a 2.76 2.76 3'32 3 *38 3 '47 3'55 3 '63 3 *a4 3'89 4'13 4'33 4'49 4-60 4.68 4-80 4-91 5 '03 5-13 5 '24 5 '34 5'44 5 '54 5 *68 9 '43 9'57 5 '63 9'85 10'1 I 10.35 10.56 10.76 10.94 I I 27 1 1 - 4 1 11.54 14.20 16-80 16-96 17-08 17.16 17-20 1 1 - 1 1 Mean Density gm./cm.8 5'53 5'57 5'57 5 '64 5 '75 5-87 5 -98 6 -00 6.10 6-21 6.32 6 -42 6'53 6.63 6.86 7-1 I 7'39 7-71 8-08 8.50 8.99 9'58 10.28 10.70 1070 10.81 I I -03 11.25 I I '49 11'75 I 2 -06 12-42 12.89 13.54 14-49 15.84 16.96 16.96 17-06 17-13 17.18 17-20

E.

C. Bullard TABLE I

v'

0.565 0.557 0.557 0.550 0'539 0.528 0.515 0.514 0'499 0.506 0.496 0.496 0.4.98 0'495 0.501 0.501 0.497 0.488 0.470 0'439 0.391 0.319 0.215 0.149 0.149 0.148 0.148 0 ' 1 5 2 0.160 0.172 0.188 0.207 0.227 0.238 0.216 0.121 0.008 0.008 0.005 0'002 0'001 0'000 F 0.99964 0.99972 0'99972 0.99979 0.99988 0.99997 I -00006 I .oooo7 1.00013 1*00018 I '00020 I '00020 1.ooo19 1-00018 1 . 0 0 0 1 ~ 1a014 1.00016 I -00030 I'O0044 I.oO058 1-00062 1*00041 1.00019 I -0003 I 1*00030 I '0003 I 1-00032 1.-35 I '00039 I '-43 I *-49 I -00054 I'-55 I'oO021 I '00047 1.00014 0 '99994 I '00Ooo 1'00000 I 'OOoaO I '00000 I 'Ooooo c K 10-8 0*003364 68 0.003354 67 0.003354 67 0.003334 66 0.=3305 64 0.003276 62 0.003248 60 0-003244 60 0~003220 58 0'003192 57 0.003165 55 0.003 I37 54 0'003109 52 0*003081 5 1 0'003023 47 0~002963 44 0.002903 40 0.002842 35 0.002782 31 0.002667 21 0~002618 17 0~002580 14 0.002567 13 0*002567 13 0.002556 13 omoz5 3 2 I 2 0.002508 12 0.002480 12 0.00245 0 I 2 0'002723 26 0.002414 I 2 0'002372 I 2 0.002322 I 1 0~002203 6 0-002265 9 0'002 I5 5 2 0.002141 I 0-002141 I om02 I 35 0 0~002138 0 0'002 I33 0 0-002 I 32 0 by guest on July 14, 2016 http://gsmnras.oxfordjournals.org/ Downloaded from

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The F@e

of

the Earth ₁₈₉ 16 4 Fro. 1. 0004 0003 Y 0001 0000 80 x 10-0 60 K 40 20 0 3000 6371 Depth km. FIG.' 2.

The ellipticity of the outer surface obtained by this process (0.003364 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, and 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

This can be calculated when

?;

has been found.

The precessional constant and the mass of the Moon give the quantity

H

= (C

-

A)/C, where

C

and

A are

the

moments

of

inertia with

respect

to polar

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I90 E. C. Bullard

and equatorial axcs. The relations are

somewhat complicated, but may be simplified by putting approximate numerical values in all the small terms. De Sitter and Brouwer” have done this, and assuming 5493”.156 i- 0.175 for the precessional constant get

and

where w, 2, u, v ,

x

and

#

are given by,

H, A, and K~ then give the ellipticity.

w

-

0.67472 = o & 3.2 x IO-~, ₍₈₎ 6-1 = 296.753[1- 0.1874(u - v ) - 0.8138~

+

0.1696% - 0-8098$], H = 0.003279423( I

+

w), rl = 6371260(1+ u ) metres, g, = 979770( I

+

v ) cm./sec.2, K = 0~00000050

+

IO-~X, A, = 0~00040

+

#,

p-’=81*53(1+2),

g, being the value of gravity at latitude sin-,

d&

and p the ratio of the Moon’s mass to that of the Earth.

The value of A, obtained from (7) is

A, = 0~00016.

With Spencer Jones’s

t

value for the mass of the Moon (p-l= 81.271 _+ 0.021)

equation (8) gives

Using these values of H and A,, the value of K , found below and Spencer Jones’s values of rl and g, we get $,

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,.

I n 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 I O - ~ compared with de Sitter’s assumed average of 5 x I O - ~ .

There also seems to be an algebraic mistake in the unnumbered equations im- mediately 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 and 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.

then the values of

k

for various values of the shell radius were found to be H = 0*00327237 0*0000005g.

€1,

= 297.338, E = 0.00336317.

Let the effect of such a shell on A, be

AA,/A, = kAjlA8,

Depth of shell (km.) o 500 1000 2000 3000 4000 5000 6000 6371

k

0.0 140 160 76 _-42 -45 -21 -1.6 0.0

*

W. de Sitter and D. Brouwer, Rull. astr. Insts. Netherlak., 8, 213-231, 1938.

t

H. Spencer Jones, Mem. R.A.S., 66, 60, 1941.

3 Using X~=O*OOO++ and putting K I zero, Spencer Jones (loc. cit.) gets e-1=296.776 but this

appears to be fi numerical error, with his data I get 297.~01.

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The Figure of the Earth ₁₉₁ In calculating these the effect of various small terms has been neglected and

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, and 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 standard 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 4482

and 3.0 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. and 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 and 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 and moment of inertia are not altered. The above three errors combined give uncertainties of 0~00018 in A, and 0.027 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 and we have

and

Al = 0~00016 & 0~00018

e-l = 297.338 k 0.050.

This

ellipticity may be compared with those obtained without the assumption of hydrostatic equilibrium from the variation of gravity with latitude and from the motion of the Moon.

296.17 & 0.68

*

and

296.72 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

These give

(9) which may be deduced from Darwin’s equation (23).

The solution required is that which is finite at the origin and satisfies Darwin’s (42)

at the outer surface. are also zero there.

Expansion in series shows that the only solutions that are finite at the origin Such sblutions may be written

K = A K ~ + K ~ , (11)

7

* H. Jeffreys, M.N., Geophys. Suppl., 5, 65-66, 1943.

t

This i s calculated from the data given by Spencer Jones, loc. cit., p. 63. G I5

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192 The Figure of the Eurth

where K A is a complimentary function obtained by numerical integration of

(9)

with the right-hand side put equal to zero and xB is a particular integral obtained by numerical integration of the full equation. These integrations were started from the series expansion, valid near the origin,

K = Ap2

+

6.091 x 10-8

p.

The numerical integrations were performed by the methods given in a Nautical Substitution in (10) then gives A, and K can be The results are given in Table I and in Almanac Office publication."

calculated for all depths from (11). 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 10-8.

This is rather larger than de Sitter's result of

50

x I O - ~ . 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 and moment of inertia constant, increases K (Darwin shows that K = 140 x I O - ~ when the core is reduced to a point mass). These results show K to be relatively insensitive to the exact density distribution and 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 and N~ is

r = b[r - 0.00336317 sina

t#

+

639 x 10- sin-a

241.

₍₁₂₎

The corresponding gravity formula is inserting numerical values gives

(12) and (13) are the expressions that would be obtained on an earth with Bullen's density distribution and 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) and (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 and 0.002 cm./sec.2 to g.

g =g,[r - (el - %pl

-

yp:

+

8 q p 1 - &l) sine

4

-

&:

+

3 ~ ~ ) sin2

241,

g =g,[I

-

0.00529317 sin24 - 787 x 10- sina 241. ₍₁₃₎

Department of Geodesy and Geophysics, Cambridge :

1946 August 28.

* interpolation and Allied Tables, 2nd Edition, pp. 942-3, 1942.