Prolate spheroid



A prolate spheroid, or prolatum, is a spheroid in which the polar axis is greater than the equatorial diameter. Prolate spheroids stand in contrast to oblate spheroids. The prolate spheroid is defined by the equation $$\mu = c$$ for some arbitrary constant c, in prolate spheroidal coordinates.

Properties
The surface area of a prolate spheroid is discussed under spheroid.

The volume of a prolate spheroid is calculated by $$V = \frac{4}{3}\pi a^2 b$$

Uses
The prolate spheroid is the shape of the ball in several sports, such as in Rugby union, Rugby league, and Australian rules football. In American football and Canadian football, a more pointed prolate spheroid is used (one resembling a vesica piscis in cross section).

Several moons of the Solar system approximate prolate spheroids in shape, though they are actually scalene. Examples are Mimas, Enceladus, and Tethys (satellites of Saturn) and Miranda (a satellite of Uranus). In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via tidal forces when they orbit a massive body in a close orbit. The most extreme example is Jupiter's moon Io, which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing spectacular volcanism. It should be noted that the axis of the prolate spheroid does not run through the satellite's poles in this case, but through the two points on its equator directly facing toward and away from the primary.

It is also used to describe the shape of some nebulae (nebulas) such as the Crab Nebula.

The most common shapes for the density distribution of protons and neutrons in an atomic nucleus are spherical, prolate and oblate spheroidal. Deformed shapes occur as a result of the competition between electromagnetic repulsion between protons, surface tension and quantum shell effects.