Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

70 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
derived an approximate formula for the ‘ellipticity’ of the Earth:
a value not very much different from the actual value of about 0.0034
known today angular velocity, R: (mean) radius, M: mass of the
Earth, G: Newton’s constant of gravitation]. However, the first empirical
proof of the flattening of the Earth at the poles was provided by geodetic
measurements in Lapland performed by Maupertius and Clairaut in
1736/1737. This was the end of a controversy between the schools of
Newton and the Cassini family, the latter had claimed that the Earth is
flattened at the equator.
It should also be mentioned that Kant [7] interpreted the Milky Way
system as a rotating disk in balance between centrifugal and gravitational
forces.
The first exact solution describing a rotating fluid body in Newton’s
theory of gravitation was found by Maclaurin [8]. The Maclaurin
spheroids are characterized by the following relation between the
(constant) angular velocity the (constant) mass–density and the
eccentricity
It is interesting to note, that this reduces to Newton’s formula (1) for
small In the opposite limit, with
(fixed),
the Maclaurin disk with radius is obtained. It has a surface mass–
density depending on the distance from the center,
and satisfies the following relation between angular velocity, mass, and
radius: