### The Ellipticities of Surfaces of Equal Density in the Earth's Interior.

[*Read before the Wellington Branch, July 9, 1942; received by the Editor, April 7, 1942; issued separately, September, 1942*.]

1. Figures for the ellipticities of surfaces of equal density within the Earth's interior were previously published by the writer (1936), being based on the solution there given of the problem of the Earth's density variation. Since this date, there have been several improvements made, notably by Jeffreys and by Gutenberg and Richter, to the travel-time tables of seismic waves; and the writer has in consequence found it necessary to make certain modifications to the density values. These would, of course, entail alterations to the previously given ellipticity figures, and Lambert (1939) drew attention to the fact that revised ellipticity figures had not been published in detail by the writer. But, as pointed out by the writer (1937), any alterations which might then be made to the ellipticity figures would be small, and in view of this it was felt best to defer a detailed revision until such time as the seismological data and corresponding density figures had reached a greater degree of finality.

This stage appears now to be reached—see Bullen (1941)—and two recent papers (1940, 1942) contain a fairly complete account of the extent of residual errors likely to be present in the writer's most recent density figures. The degree of precision obtained from this analysis suggests that it is now worth while re-discussing the question of the ellipticity figures.

2. In carrying out the calculations, use was again made of the parameter ζ, where

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ζ=*d* log ε/*d* log *r* (1)

*r* being the distance of a point from the Earth's centre, and ε the ellipticity of the surface of equal density through this point. Let *m* be the mass of the matter enclosed by this surface, and *I* its moment of inertia, and let

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I=*zmr*^{2} (2)

Values of the coefficient *z* are readily obtainable from the figures for the Earth's density distribution, and ζ is then found from the equation

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ζ=5/2(1−3z/2)−1(3)

Equation (3) is an approximate equation given by Darwin in his theory of the figure of the Earth, and the validity of the approximation depends on the range of variation within the Earth of a certain function of ζ. The question of this range of variation, in so far as it is relevant to the present problem, has been considered in the writer's earlier paper (1936) and the validity of the use of equation (3) substantiated. (The writer hopes shortly to publish an alternative and more direct discussion on this point). With values of ζ then determined from (3), the desired values of ε are finally found using (1).

In the following table, values of ε as thus computed are given in terms of the depth *d* km. below the Earth's surface. Corresponding values of the coefficient *z* and the parameter ζ are also included, the latter being sometimes directly useful in other calculations relating to the Earth's interior—see Bullen (1938).

d |
z |
ζ | ε |
---|---|---|---|

0 | 0.334 | 0.56 | 0.00337 |

200 | 0.337 | 0.53 | 0.00331 |

400 | 0.340 | 0.50 | 0.00325 |

600 | 0.342 | 0.49 | 0.00320 |

800 | 0.342 | 0.48 | 0.00315 |

1000 | 0.342 | 0.48 | 0.00309 |

1200 | 0.341 | 0.49 | 0.00303 |

1400 | 0.341 | 0.49 | 0.00298 |

1600 | 0.342 | 0.49 | 0.00292 |

1800 | 0.343 | 0.48 | 0.00286 |

2000 | 0.344 | 0.46 | 0.00280 |

2200 | 0.347 | 0.43 | 0.00274 |

2400 | 0.353 | 0.38 | 0.00269 |

2600 | 0.361 | 0.32 | 0.00264 |

2800 | 0.372 | 0.22 | 0.00260 |

2900 | 0.380 | 0.17 | 0.00260 |

3000 | 0.380 | 0.16 | 0.00257 |

3500 | 0.378 | 0.17 | 0.00250 |

4000 | 0.373 | 0.21 | 0.00242 |

4500 | 0.368 | 0.26 | 0.00229 |

5000 | 0.382 | (0.15) | 0.00213 |

5500 | 0.399 | 0.00 | 0.00212 |

6000 | 0.400 | 0.00 | 0.00212 |

It will be found that within the Earth's outer mantle (i.e., down to a depth of 2900 km.) the values of ε show little difference from the previous results (1936), the maximum change at any depth being 0·00002. Lambert (1939) gave tentative revised ellipticity values, expressing his results as reciprocals of the ellipticity. Within the Earth's mantle these values need corrections up to 6 units in order to fit the figures given in the present paper.

Within the central core, the present results indicate a variation in ε from 0·00260 at the core boundary to 0·00212 at the Earth's centre. The 1936 results, and also the figures of Lambert, showed only very slight variation of ε within the central core. The difference between the latter results and the new figures here given is largely due to the effect on the density solution of the new Lehmann-Gutenberg-Richter discontinuity within the central core.

3. It is possible to give some indication of the greatest residual errors to be expected in the values of ε appearing in the above table. At all depths within the Earth's outer mantle, the greatest error in ε is not likely to exceed 0·00002, so that the figures given for this region of the Earth may be taken as highly accurate. There is greater uncertainty within the central core, as the values of the coefficient *z* are very sensitive to changes in the value taken for the (unknown) density increase across the Lehmann-Gutenberg-Richter discontinuity. The results in the table were computed on the basis of a density increase of 5·0 gm./cm.^{3} across this discontinuity. An increase of

*b* gm./cm.^{3}, where -5<*b*<5, in this value of the density jump would have little effect on the value of ε at the boundary of the central core, but there would be decreases below this level reaching to approximately 0·00009 *b* at the Earth's centre.

4. It is of some importance to seismology to investigate the effect of the revision here made on calculations of the ellipticity corrections to the travel-times of earthquake waves. Within the Earth's mantle, the changes from the 1936 results are so small that the effect on the calculations of ellipticity corrections (1937A, 1938A, 1939) for waves that have not penetrated the central core is quite negligible. In the case of the phase PKP (1938) the effect of the changes in ε will be insignificant for rays that have not penetrated deeply into the central core. The effect will be greatest with PKP for a ray that has passed through the Earth's centre, but it is easy to show that even here the effect will be always rather less than 0·1 second. This is unimportant, being within the limits of accuracy of the ellipticity tables that have been constructed. Similar consideration holds for other phases that have penetrated the Earth's central core. It is therefore unnecessary to modify at all any of the previously published ellipticity corrections to the travel-time of earthquake waves.

#### 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.

—– 1937. Note of the Density and Pressure inside the Earth, *Trans. Roy. Soc. N.Z.*, vol. 69, pp. 122–124.

—– 1937A. The Ellipticity Correction to Travel-times of P and S Earthquake Waves, *M.N.R.A.S., Geophys. Suppl.*, vol. 4, no. 2, pp. 143–157.

—– 1938. Ellipticity Corrections to Waves through the Earth's Central Core, *M.N.R.A.S., Geophys. Suppl.*, vol. 4, no. 5, pp. 317–331.

—– 1938A. Ellipticity Corrections to Earthquake Waves Reflected at the Central Core, *M.N.R.A.S., Geophys. Suppl.*, vol. 4, no. 5, pp. 332–335.

—– 1939. The Ellipticity Correction and Deep Focus Earthquakes, *M.N.R.A.S., Geophys. Suppl.*, vol. 4, no. 6, pp. 469–471.

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

—– 1941. The Elasticity of the Earth's Central Core, *Trans. Roy. Soc. N.Z.*, vol. 71, pp. 164–166.

—– 1942. The Density Variation of the Earth's Central Core, *Bull. Seism. Assn. of Amer., vol. 32, pp. 19–29*.

Lambert, Walter D., 1939. Density, Gravity, Pressure and Ellipticity in the Interior of the Earth, *Physics of the Earth, VII*, McGraw-Hill, pp. 329–344.