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This work was published in the Mathematical Gazette.




The Earth (more precisely, the “geoid” thereof) is known to approximate closely to a slightly oblate spheroid whose unique axis coincides with the Earth’s axis of rotation.  (By “spheroid” is meant is an ellipsoid of revolution, ie one with two semi-axes equal;  a (slightly) oblate one has these two semi-axes (slightly) longer than the unique one.)  To the nearest km, the diameter of the “geoid” pole-to-pole is 43 km less than the equatorial diameter of 12 756 km.  There is a reduction of practical significance (0.527 %) in the acceleration of free fall at sea level between the poles and the equator, and therefore in the weight of objects.  Of this, 0.345 % derives directly from the rotation of the Earth;  the balance of 0.182 % results from the purely gravitational effect of the Earth’s deviation from sphericity.


It was Newton in Principia who famously made the first step towards this modern understanding of the shape and gravity of the Earth.  He obtained qualitatively correct results on the assumption that the Earth was of uniform density.  A preliminary step in his calculation was to compute (with good accuracy) the acceleration due to gravity at the poles and equator of a slightly oblate non-rotating spheroid of uniform density.


To obtain good quantitative agreement with the values quoted in the first paragraph of this section, Newton’s successors abandoned the assumption of uniform density (the real Earth has a metallic core and a mantle and crust of much less dense oxides).  Nevertheless, from a mathematical point of view, the gravitational properties of a non-rotating body of uniform density are worth pursuing.


The present paper poses and solves two optimisation problems which were not considered by Newton and which, as far as the author is aware, have not been considered by others.  They concern how a hypothetical planet-builder can make best use of a given amount of incompressible uniform material if he wishes to obtain, if only at one or two points on the surface, the maximum acceleration due to gravity.  The planet-builder is also granted planet rigidity;  he can form the planet into the shape he chooses, without worrying that the planet will under its own gravity revert to being a sphere.  The first optimisation problem constrains the planet-builder to form the material into a spheroid (of any degree of oblateness);  the second imposes no constraint on the form, and therefore is a “calculus of variations” problem.  The solution to the first problem is a surprisingly oblate spheroid, with the acceleration due to gravity maximised at the two poles.  The solution to the second problem is a particular ovoid of revolution, also surprisingly flattened, with the acceleration due to gravity highest at the pole at the blunt end.


Keywords:  Isaac Newton; potential theory, shape of the earth, Principia, calculus of variations, spheroid, ovoid, polar flattening.




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