Copyright © Peter Wakefield Sault 2003-2005
All rights reserved worldwide
The shape of the Earth is an oblate spheroid,
which for most practical purposes can be approximated as an ellipsoid.
Due to the Earth's spin it is wider across the Equator than across the Poles, the
difference in diameters amounting to some 27 miles.
The first step in approximating a great circle circumference is to
ascertain the minor radius, R. The major radius (a) is
always the equatorial radius of the Earth. That great circle tilted 0°
is the Equator itself (EW, coloured red in Figure D-1) and is here
treated as though it were a true circle. As the angle of tilt increases,
the minor radius - which is that from the centre of the Earth to the
vertex of the great circle - decreases. The length of the radius is
calculated by taking it as an intermediate radius of a meridian (EVN,
coloured green in Figure D-1), an ellipse which passes through
the North and South Poles and whose major and minor radii are known,
being the equatorial (a) and polar (b) radii respectively,
using the following formula, where q is the
angle of tilt to the Equator (i.e. the dihedral angle between the plane of the
equator and that of the tilted great circle):-
The circumference P of the tilted great circle (VW, coloured blue
in Figure D-1) is then calculated by substituting R into
Ramanujan's second approximation (which, for ellipses comparable in size and
eccentricity to great circles of the earth, is accurate to within a Bohr radius):-
Ignoring topographical features, the Moon is an almost perfect sphere.
It has a very slight bulge, amounting to no more than a few metres, in the side
which faces the Earth. This bulge, which dates from the solidification of the
Moon before the features which we see resulting from meteoric bombardment
appeared, is sufficient to keep the same side facing towards the Earth at
all times. It is a curious accident that
the apparent angular diameter of the Moon as viewed from the Earth, about ½°, is almost
identical to that of the Sun, allowing the Solar Corona to be viewed at the
periphery of the Lunar disk during total eclipses of the Sun.
The Moon's mean radius, k, is expressed in units of Earth's equatorial radius.
The Moon's orbit about the Earth is an ellipse of eccentricity 0.0549
inclined at 5°8' to the Ecliptic, the plane of the Earth's orbit.
The points where the Moon's orbit crosses the Ecliptic are the nodes,
which move westward, taking 18.6 years to go all the way round the Earth.
The point where the Moon is nearest the Earth, the perigee,
moves eastward, taking 8.8 years for a complete circuit.
The movement of the Moon against the stars is, therefore, quite complicated and variable.
Nevertheless, the Moon remains within the zodiacal band along the Ecliptic.
The highest latitude where the Moon ever passes directly overhead is 28°35',
comprising the sum of the inclinations of the terrestrial Equator and
the orbit of the Moon, 23°27' and 5°8' respectively, to the Ecliptic.
Copyright © Peter Wakefield Sault 2003-2005All rights reserved worldwide
Music: Nikolai RIMSKY-KORSAKOV (1844-1908), Scheherezade, Op.35 (symphonic suite), The Sea and Sinbad's Ship.
MIDI realization J.Nishio
THE CLASSICAL ARCHIVES