**OPTIMISING
THE ACCELERATION DUE TO GRAVITY ON A PLANET’S SURFACE**

**BY
MICHAEL JEWESS **

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**This
work was published in the Mathematical
Gazette.**

__Abstract __

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 abstract,
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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