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Volume 69, 1940
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The Variation of Gravity Within the Earth.

[Read before Auckland, Institute, February 15, 1939; received by the Editor, February 1, 1939; issued separately, September, 1939.]

In two recent papers, Benfield (1937) and Olczak (1938) have given figures for the attraction due to gravity at various points in the interior of the earth, the calculations being based on the writer's solution (1936) of the variation of density within the earth.

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

As in the latter paper, let ρ be the density at a point P distant r from the earth's centre, and M the mass enclosed by a concentric sphere through P. Let α, β be the velocities of P and S seismic waves at P, and γ the constant of gravitation. Then in the course of the writer's solution of the density problem, the following equation had been set up:— dρ/dr=—γMr--22)—4/3β2 and had been applied successively to give the rate of density alteration at each point taken in the earth's interior. The right-hand side contains the factor γMr-2 which had been determined explicitly on each occasion; and this expression of course gives directly the gravitational attraction at distance r from the earth's centre.

The results of Benfield and Olczak were obtained by using the final published density figures of the writer, and accordingly involved a fail amount of additional computation. It may therefore be of interest to give the actual values of the function γMr-2 as found directly by the writer, such values being subject to the minimum of computation errors.

The values to be given are based on the auxiliary material prepared during the writer's revised solution (1937) of the density variation. Allowance has further been made for the improved estimate of the earth's mass at 5.976 × 1027 gms., as given by Olczak. In this connection it is of interest to point out that the same figure for the earth's mass was independently given recently by Jeffreys (1937). A corresponding approximate adjustment for this was made by Benfield and Olczak, but more accurate allowance is possible using the auxiliary figures of the writer's earlier papers.

The values thus obtained by the writer for the gravitational attraction g cms./sec.2 at various depths d km. below the earth's surface are as follows—

d g d g d g
0 982 2000 985 4200 714
100 986 2200 990 4400 654
200 989 2400 1000 4600 592
300 992 2600 1014 4800 528
400 995 2800 1034 5000 463
474 997 2920 1048 5200 398
600 995 3000 1031 5400 331
800 991 3200 986 5600 264
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1000 988 3400 936 5800 196
1200 985 3600 885 6000 127
1400 982 3800 831 6200 59
1600 981 4000 774 6371 0
1800 982
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Graph showing the variation of gravity with depth within the Earth

The general variation of g is exhibited in the accompanying diagram, and is similar to the results of Benfield and Olczak, although the values are for the most part a little higher. This is due partly

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to the use of the writer's revised solution (1937) for the density variation and partly to the more accurate adjustment corresponding to the increased mass adopted for the earth; the effect of the latter is to increase the density more at the greater depths than near the earth's surface. The discontinuities of gradient at depths of 474 and 2920 km. are associated with maxima of 997 and 1048 cms./sec 2 respectively. The minimum value of g in the earth's outer shell is about 981 cms./sec 2, and occurs at a depth of about 1620 km. Inside the central core gravity diminishes to zero by a nearly linear law.

The adoption of an increased mass affects slightly the density figures given by the writer (1937) for the central core. These now range from 9.77 at the boundary to 12.29 at the earth's centre. Above the central core there is virtually no alteration necessary.

A few further remarks are necessary concerning the solution given. First, it is not yet certain from the seismological evidence exactly how sharp is the new discontinuity placed at 474 km. depth. If the transition is fairly gradual, the effect will be essentially to smooth out the discontinuity in the gradient of the gravity curve at this depth, but other features of the curve will be preserved. Any modification made will be fairly small, since, as pointed out by the writer (1938) there are a number of independent lines of evidence which all support the occurrence of a change in composition in the earth at a depth of order 500 km.

Secondly, Gutenberg and Richter (1938) have recently made the suggestion that the central core itself may be composite with possibly two changes of material inside. This is liable to affect slightly the variation of gravity within the central core, but the general solution for the whole earth should not be greatly disturbed, since the mass of the central core is less than a third of the mass of the earth.

References.

Benfield, A. E., 1937. Note on the Variation of Gravity with Depth, Zeitschrift fur Geophysik, Heft ⅘, pp. 157–158.

Bullen, K. E., 1936. The Variation of Density and the Ellipticities of Strata of Equal Density within the Earth, M.N.R.A.S. Geophysical Supplement, vol. 3, no. 9, pp. 395–401.

—– 1937. Note on the Density and Pressure inside the Earth, Transactions of Roy. Soc. of N.Z., vol. 67, pp. 122–124.

—– 1938. Composition of the Earth at a Depth of 500–700 km., Nature, vol. 142, pp. 671–672.

Gutenberg, B. and Richter, C. F., 1938. P′ and the Earth's Core, M.N.R.A.S. Geophysical Supplement, vol. 4, no. 5, pp. 363–372.

Jeffreys, H., 1937. Materials and Density of the Earth's Crust, M.N.R.A.S. Geophysical Supplement, vol. 4, no. 1, p. 59.

Olczak, T., 1938. On the Mass of the Earth and on Gravity within the Earth, Acta Astronomica, sér. c, vol. 3, pp. 81–85.