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Oblate spheroidal coordinates can also be considered as a limiting case of ellipsoidal coordinates in which the two largest semi-axes are equal in length. Oblate spheroidal coordinates are often useful in solving partial differential equations when the boundary conditions are defined on an oblate spheroid or a hyperboloid of revolution.
The planet Jupiter is a slight oblate spheroid with a flattening of 0.06487. The oblate spheroid is the approximate shape of rotating planets and other celestial bodies, including Earth, Saturn, Jupiter, and the quickly spinning star Altair. Saturn is the most oblate planet in the Solar System, with a flattening of 0.09796. [5]
If instead of the Helmholtz equation, the Laplace equation is solved in spheroidal coordinates using the method of separation of variables, the spheroidal wave functions reduce to the spheroidal harmonics. With oblate spheroidal coordinates, the solutions are called oblate harmonics and with prolate spheroidal coordinates, prolate harmonics.
The prolate spheroidal coordinates are produced by rotating the elliptic coordinates about the -axis, i.e., the axis connecting the foci, whereas the oblate spheroidal coordinates are produced by rotating the elliptic coordinates about the -axis, i.e., the axis separating the foci.
An oblate spheroid, highly exaggerated relative to the actual Earth A scale diagram of the oblateness of the 2003 IERS reference ellipsoid, with north at the top. The outer edge of the dark blue line is an ellipse with the same eccentricity as that of Earth.
Tables of numerical values of oblate spheroidal wave functions are given in Flammer, [4] Hanish et al., [16] [17] [18] and Van Buren et al. [19] Asymptotic expansions of angular oblate spheroidal wave functions for large values of c {\displaystyle c} have been derived by Müller., [ 20 ] also similarly for prolate spheroidal wave functions.
Prolate spheroidal coordinates are a three-dimensional orthogonal coordinate system that results from rotating the two-dimensional elliptic coordinate system about the focal axis of the ellipse, i.e., the symmetry axis on which the foci are located. Rotation about the other axis produces oblate spheroidal coordinates.
The unusual cosmic abundance of alpha nuclides has inspired geometric arrangements of alpha particles as a solution to nuclear shapes, although the atomic nucleus generally assumes a prolate spheroid shape. Nuclides can also be discus-shaped (oblate deformation), triaxial (a combination of oblate and prolate deformation) or pear-shaped. [7] [8]