Editing Spheroid (section)

==Occurrence and applications==
The most common shapes for the density distribution of protons and neutrons in an [[atomic nucleus]] are [[spherical]], prolate, and oblate spheroidal, where the polar axis is assumed to be the spin axis (or direction of the spin [[angular momentum]] vector). Deformed nuclear shapes occur as a result of the competition between [[Electromagnetic force|electromagnetic]] repulsion between protons, [[surface tension]] and [[Quantum mechanics|quantum]] [[Nuclear shell model|shell effects]].

Spheroids are common in [[3D cell culture]]s.
Rotating equilibrium spheroids include the [[Maclaurin spheroid]] and the [[Jacobi ellipsoid]].
[[Spheroid (lithic)|Spheroid]] is also a shape of archaeological artifacts.

===Oblate spheroids===
[[File:Jupiter_OPAL_2024.png|thumb|The planet [[Jupiter]] is a slight oblate spheroid with a [[flattening]] of 0.06487]]
The oblate spheroid is the approximate shape of rotating [[planet]]s and other [[astronomical object|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.<ref>{{Cite web |title=Spheroid - Explanation, Applications, Shape, Example and FAQs |url=https://www.vedantu.com/maths/spheroid |access-date=2024-11-26 |website=VEDANTU |language=en}}</ref> See [[planetary flattening]] and [[equatorial bulge]] for details.

[[Age of Enlightenment|Enlightenment]] scientist [[Isaac Newton]], working from [[Jean Richer]]'s pendulum experiments and [[Christiaan Huygens]]'s theories for their interpretation, reasoned that Jupiter and [[Earth]] are oblate spheroids owing to their [[centrifugal force]].<ref>{{Cite book |url=https://books.google.com/books?id=RKXeEAAAQBAJ&pg=PA91 |title=Background to Discovery: Pacific Exploration from Dampier to Cook |date=1990 |publisher=[[University of California Press]] |isbn=978-0-520-06208-5 |editor-last=Howse |editor-first=Derek |series= |location= |pages=91}}</ref><ref>{{cite journal|last=Greenburg|first=John L.|date=1995|title=Isaac Newton and the Problem of the Earth's Shape|journal=History of Exact Sciences|volume=49|issue=4|pages=371–391|publisher=Springer|doi=10.1007/BF00374704|jstor=41134011|s2cid=121268606}}</ref><ref>{{Cite web |last=Choi |first=Charles Q. |date=2007-04-12 |title=Strange but True: Earth Is Not Round |url=https://www.scientificamerican.com/article/earth-is-not-round/ |access-date=2025-03-02 |website=[[Scientific American]] |language=en}}</ref> Earth's diverse cartographic and geodetic systems are based on [[reference ellipsoid]]s, all of which are oblate.

===Prolate spheroids===
[[File:Gilbert_rugby_ball_on_grass.jpg|thumb|left|A [[rugby ball]].]]
The prolate spheroid is the approximate shape of the ball in several sports, such as in the [[rugby ball]].

Several [[moons]] of the Solar System approximate prolate spheroids in shape, though they are actually [[triaxial ellipsoid]]s. Examples are [[Saturn]]'s satellites [[Mimas (moon)|Mimas]], [[Enceladus (moon)|Enceladus]], and [[Tethys (moon)|Tethys]] and [[Uranus]]'s satellite [[Miranda (moon)|Miranda]].

In contrast to being distorted into oblate spheroids via rapid rotation, celestial objects distort slightly into prolate spheroids via [[tide|tidal forces]] when they orbit a massive body in a close orbit. The most extreme example is Jupiter's moon [[Io (moon)|Io]], which becomes slightly more or less prolate in its orbit due to a slight eccentricity, causing intense [[volcanism]]. The major 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. This combines with the smaller oblate distortion from the synchronous rotation to cause the body to become triaxial.

The term is also used to describe the shape of some [[nebula]]e such as the [[Crab Nebula]].<ref name="Trimble1973">{{ citation | last1 = Trimble | first1 = Virginia Louise | author1-link = Virginia Trimble | title = The Distance to the Crab Nebula and NP 0532 | date = October 1973 | journal = Publications of the Astronomical Society of the Pacific | volume = 85 | issue = 507 | page = 579 | bibcode = 1973PASP...85..579T | doi = 10.1086/129507 | doi-access = free }}</ref> [[Fresnel zone]]s, used to analyze wave propagation and interference in space, are a series of concentric prolate spheroids with principal axes aligned along the direct line-of-sight between a transmitter and a receiver.

The [[atomic nucleus|atomic nuclei]] of the [[actinide]] and [[lanthanide]] elements are shaped like prolate spheroids.<ref>{{Cite web|url=https://www.britannica.com/science/nuclear-fission/Fission-theory|title=Nuclear fission - Fission theory|website=Encyclopedia Britannica}}</ref> In anatomy, near-spheroid organs such as [[testicle|testis]] may be measured by their [[Anatomical terms of location#Axes|long and short axes]].<ref>[https://books.google.com/books?id=safNmcP3lakC&pg=PA559 Page 559] in: {{cite book|title=Introduction to Vascular Ultrasonography|author=John Pellerito, Joseph F Polak|edition=6|publisher=Elsevier Health Sciences|year=2012|isbn=9781455737666}}</ref>

Many submarines have a shape which can be described as prolate spheroid.<ref name="scientific american">{{cite web |url=http://www.scientificamerican.com/article/football-science-shapes/ |title=What Do a Submarine, a Rocket and a Football Have in Common? |work=[[Scientific American]] |date=8 November 2010 |access-date=13 June 2015 }}</ref>

===Dynamical properties===
{{see also|Ellipsoid#Dynamical properties}}
For a spheroid having uniform density, the [[moment of inertia]] is that of an ellipsoid with an additional axis of symmetry. Given a description of a spheroid as having a [[major axis]] {{mvar|c}}, and minor axes {{mvar|a {{=}} b}}, the moments of inertia along these principal axes are {{mvar|C}}, {{mvar|A}}, and {{mvar|B}}. However, in a spheroid the minor axes are symmetrical. Therefore, our inertial terms along the major axes are:<ref>{{cite web |url=http://mathworld.wolfram.com/Spheroid.html |title=Spheroid. |work=MathWorld--A Wolfram Web Resource. |access-date=16 May 2018 |author=Weisstein, Eric W.}}</ref>

:<math>\begin{align}
A = B &= \tfrac15 M\left(a^2+c^2\right), \\
C &= \tfrac15 M\left(a^2+b^2\right) =\tfrac25 M\left(a^2\right),
\end{align}</math>

where {{mvar|M}} is the mass of the body defined as

:<math> M = \tfrac43 \pi a^2 c\rho.</math>