Volume 73, 1943-44
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### The Radau-Darwin Approximation in the Theory of the Figure of the Earth.

[Received by the Editor, September 2, 1943; issued separately, March, 1944].

In a recent estimation (1942)* of the ellipticities of surfaces of equal density within the Earth, the writer made use of an approximate equation first used by Radau (1885) and Darwin (1900) in problems on the theory of the figure of the Earth. Some further argument is necessary to justify the use of this approximation.

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Let ∊ be the ellipticity of the surface of equal density (within the Earth) whose mean radius is r, m the mass of the matter enclosed by this surface, and ρ the density at points of this surface. For present purposes it is sufficiently accurate to assume that the distribution of mass within the Earth is consistent with a hydrostatic state. The equation d2∊/dr2 + 8πρr2/m d∊/dr - {1 - 4πρr3/m} 6∊/y2 = 0 (1) is then found (see Jeffreys, 1929) to give the variation of ∊ with r to sufficient accuracy.

When the density variation is known to a sufficient degree of precision, as is the case now (see Bullen, 1940 and 1942A), it is theoretically possible by numerical integration of (1) to determine ∊ as a function of r throughout the Earth. Direct use of the equation (1) would, however, involve much heavy labour, and this can be avoided by the use of the Radau-Darwin approximation, to be described shortly. But a point of logic arises in connection with the use of this approximation in that the approximation itself is not valid unless the true values of ∊ obey certain restrictions. Actually the values of ∊ as foxind in Paper I are such as to obey these restrictions; but since these values were calculated using the approximation, it would be arguing in a circle to infer from this result alone the validity of the approximation. The purpose of the present paper is to give the additional argument necessary to show that the Radau-Darwin approximation, and hence also the various results that its use entails, are indeed valid. The writer has considered this point previously (1936), but the following argument is more precise.

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As a preliminary to making his approximation, Radau modified the equation (1) by introducing parameters η and ρo, where η = r/∊ d∊/dr, (2) and ρo is the mean density of the matter within the equal-density surface of ellipticity ∊; thus, to a sufficient approximation,

[Footnote] * This paper will be hereinafter referred to as Paper I.

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ρor3 = ∫or3ρr2dr, (3) and m = 4/3πρor3(4)

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Replacement of ∊ and m in (1) by η and ρo leads (without further approximation) to “Radau's equation” d/dr {ρor5 (1 + η)½} = ρor4 ψ (η), (5) where ψ (η) = (1 + ½ η - 1/10 η 2) (1 + η) (6)

Use of the equation (5) would give sufficiently accurate values of ∊ but the process of calculation would still be severe. Radau's approximation was to put ψ (η) equal to unity in (5); and rests on the circumstance that ψ (η) is equal to unity within 8 parts in 10,000, if η lies in the range 0  η 0.6. If, therefore, it can be established that ∊ does in actual fact lie within this range inside the Earth, the approximation is justified.

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It will be convenient to re-write (5) in the form ∫orρor4 ψ (η) dr = 1/5 ρor5 (1 + η½. (7)

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The corresponding approximate equation is then ∫orρor4dr = 1/5 ρor5 (1 + η)½, (8) the latter being equivalent to the equation (3) of Paper I, which was the basis of the writer's recent calculation of values of ∊. This equivalence may be readily inferred from Darwin's work on the theory of the figure of the Earth, or deduced in the present notation as follows. As in Paper I, introduce z where zmr2 = I, (9)

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I being the moment of inertia of the matter inside the equal-density surface of ellipticity ∊. Then since, approximately, I = ∫or8/3πρor4dr, (10) we have by (4) and (9) zρor5 = 2 ∫orπρor4dr. (11)

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From (3) we can deduce that 3 ∫orρr4dr = ρor5 - ∫or 2ρo(r)r4dr. (12)

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From (11) and (12), we obtain (1 - 3z/2) ρor5 = 2 ∫or ρor4dr. (13)

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Finally, by (8) and (13) we obtain (1 - 3z/2) = 2/5 (1 + η)½, (14) which agrees with equation (3) of Paper I. Hence use of this equation (14), as was made in Paper I, is equivalent to using (8) to calculate values of η and ∊.

The validity of the approximation in question will therefore be established if we can show that the use of the approximate equation (8) in place of (7) introduces sufficiently small errors in ∊. We

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shall now use η1 to denote values of η as determined using (8), and shall keep η to denote the more accurate values that satisfy (7). Thus we re-write (8) as ∫orρor4dr = 1/5 ρor5 (1 + η1)½. (15)

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From the fact that the density increases steadily from the Earth's outer surface to the centre it can be shown using (1)—see Jeffreys (1929)—that ∊ must increase steadily from the centre to the outer surface. Hence by (2), η cannot be negative at any point within the whole Earth. Now the maximum value of ψ(η) as given by (6), for η 0, is found to be 1.00074. Hence by (7) 1/5 ρor5 (1 + η1)½ < 1.00074 ∫or ρor4dr. Hence by (15), (1 + η)½<1.00074 (1 + η1)½;
i.e., 1 + η<1.0015 (1 + η1);
i.e., η - η<0.0015 (1 + η1) (16)
It follows that if the values of η as calculated using (8) need a positive correction, this correction will be less than 0.0015 (1 + η1) for all values of r. Now the maximum value of η1 as determined in Paper I is 0.56. Hence the maximum positive correction to be applied to η1 to give η is less than 0.003. Thus if any of the previously calculated values needs a positive correction, this will be such that η still lies in the range 0 η  0.6.

If on the other hand, η1 needs a negative correction for any value of r, the corresponding value of η will still lie in this same range, since as already pointed out, η cannot be negative.

As we have thus shown that throughout the Earth η must lie in the range 0 η  0.6, it follows that the Radau-Darwin approximation is valid. This conclusion, moreover, substantiates the reliability of the calculations in Paper I, as well as the various other applications of the approximation, including notably Darwin's estimation of the connection between the moment of inertia of the Earth and the ellipticity of its outer surface.

#### References.

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

— 1940. The Problem of the Earth's Density Variation. Bull. Seism. Soc. of Amer., vol. 30, pp. 235–250.

— 1942. The Ellipticities of Surfaces of Equal Density in the Earth's Interior. Trans. Roy. Soc. N.Z., vol. 72, pp. 141–143.

— 1942A. The Density Variation of the Earth's Central Core. Bull. Seism. Soc. of Amer., vol. 32, pp. 19–29.235

Darwin, G. H., 1900. The Theory of the Figure of the Earth Carried to the Second Order of Small Quantities. M.N.R.A.S., 60, 82–124.

Jeffreys, H., 1929. The Earth. Cambridge University Press, chapter xii, pp. 203–231.

Radau, R., 1885. Sur la Loi des Densités à l'Inérieur de la Terre. Comptes Rendus, 100, 972–974.