![Haumea Rotation](https://upload.wikimedia.org/wikipedia/commons/9/9c/Haumea_Rotation.gif)
A Jacobi ellipsoid is a
triaxial (i.e. scalene) ellipsoid under
hydrostatic equilibrium
In fluid mechanics, hydrostatic equilibrium (hydrostatic balance, hydrostasy) is the condition of a fluid or plastic solid at rest, which occurs when external forces, such as gravity, are balanced by a pressure-gradient force. In the planetary ...
which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. It is named after the
German
German(s) may refer to:
* Germany (of or related to)
** Germania (historical use)
* Germans, citizens of Germany, people of German ancestry, or native speakers of the German language
** For citizens of Germany, see also German nationality law
**Ge ...
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasiona ...
.
History
Before Jacobi, the
Maclaurin spheroid, which was formulated in 1742, was considered to be the only type of
ellipsoid which can be in equilibrium.
Lagrange in 1811 considered the possibility of a tri-axial ellipsoid being in equilibrium, but concluded that the two equatorial axes of the
ellipsoid must be equal, leading back to the solution of Maclaurin spheroid. But
Jacobi realized that
Lagrange's demonstration is a sufficiency condition, but not necessary. He remarked:
"One would make a grave mistake if one supposed that the spheroids of revolution are the only admissible figures of equilibrium even under the restrictive assumption of second degree surfaces" (...) "In fact a simple consideration shows that ellipsoids with three unequal axes can very well be figures of equilibrium; and that one can assume an ellipse of arbitrary shape for the equatorial section and determine the third axis (which is also the least of the three axes) and the angular velocity of rotation such that the ellipsoid is a figure of equilibrium."
Jacobi formula
![Jacobi-ellipsoid-dimensions-2](https://upload.wikimedia.org/wikipedia/commons/d/d5/Jacobi-ellipsoid-dimensions-2.svg)
For an ellipsoid with equatorial semi-principal axes
and polar semi-principal axis
, the angular velocity
about
is given by
:
where
is the density and
is the
gravitational constant, subject to the condition
:
For fixed values of
and
, the above condition has solution for
such that
:
The integrals can be expressed in terms of
incomplete elliptic integrals.
In terms of the
Carlson symmetric form In mathematics, the Carlson symmetric forms of elliptic integrals are a small canonical set of elliptic integrals to which all others may be reduced. They are a modern alternative to the Legendre forms. The Legendre forms may be expressed in terms ...
elliptic integral
, the formula for the angular velocity becomes
:
and the condition on the relative size of the semi-principal axes
is
:
The angular momentum
of the Jacobi ellipsoid is given by
:
where
is the mass of the ellipsoid and
is the ''mean radius'', the radius of a sphere of the same volume as the ellipsoid.
Relationship with Dedekind ellipsoid
The Jacobi and Dedekind ellipsoids are both equilibrium figures for a body of rotating homogeneous self-gravitating fluid. However, while the Jacobi ellipsoid spins bodily, with no internal flow of the fluid in the rotating frame, the Dedekind ellipsoid maintains a fixed orientation, with the constituent fluid circulating within it. This is a direct consequence of
Dedekind's theorem.
For any given Jacobi ellipsoid, there exists a Dedekind ellipsoid with the same semi-principal axes
and same mass and with a
flow velocity field of
:
where
are Cartesian coordinates on axes
aligned respectively with the
axes of the ellipsoid. Here
is the
vorticity
In continuum mechanics, vorticity is a pseudovector field that describes the local spinning motion of a continuum near some point (the tendency of something to rotate), as would be seen by an observer located at that point and traveling along wi ...
, which is uniform throughout the spheroid (
). The angular velocity
of the Jacobi ellipsoid and vorticity of the corresponding Dedekind ellipsoid are related by
:
That is, each particle of the fluid of the Dedekind ellipsoid describes a
similar elliptical circuit in the same period in which the Jacobi spheroid performs one rotation.
In the special case of
, the Jacobi and Dedekind ellipsoids (and the Maclaurin spheroid) become one and the same; bodily rotation and circular flow amount to the same thing. In this case
, as is always true for a rigidly rotating body.
In the general case, the Jacobi and Dedekind ellipsoids have the same energy,
but the angular momentum of the Jacobi spheroid is the greater by a factor of
:
See also
*
Maclaurin spheroid
*
Riemann ellipsoid
Georg Friedrich Bernhard Riemann (; 17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first ri ...
*
Roche ellipsoid
F. Hoffmann-La Roche AG, commonly known as Roche, is a Swiss multinational healthcare company that operates worldwide under two divisions: Pharmaceuticals and Diagnostics. Its holding company, Roche Holding AG, has shares listed on the SIX ...
*
Dirichlet's ellipsoidal problem In astrophysics, Dirichlet's ellipsoidal problem, named after Peter Gustav Lejeune Dirichlet, asks under what conditions there can exist an ellipsoidal configuration at all times of a homogeneous rotating fluid mass in which the motion, in an iner ...
*
Spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...
*
Ellipsoid
References
{{Reflist, refs=
[{{cite journal
, last = Chandrasekhar
, first = Subrahmanyan
, author-link = Subrahmanyan Chandrasekhar
, title = The Equilibrium and the Stability of the Dedekind Ellipsoids
, journal = ]Astrophysical Journal
''The Astrophysical Journal'', often abbreviated ''ApJ'' (pronounced "ap jay") in references and speech, is a peer-reviewed scientific journal of astrophysics and astronomy, established in 1895 by American astronomers George Ellery Hale and ...
, volume = 141
, date = 1965
, pages = 1043–1055
, bibcode = 1965ApJ...141.1043C
, url = http://adsabs.harvard.edu/full/1965ApJ...141.1043C
, doi= 10.1086/148195
[{{cite book
, last = Bardeen
, first = James M.
, author-link = James M. Bardeen
, editor-last1 = DeWitt
, editor-first1 = C.
, editor-last2 = DeWitt
, editor-first2 = Bryce Seligman
, title = Black Holes
, series = Houches Lecture Series
, publisher = CRC Press
, date = 1973
, pages = 267–268
, chapter = Rapidly Rotating Stars, Disks, and Black Holes
, isbn = 9780677156101
, chapter-url = https://books.google.com/books?id=sUr-EVqZLckC&pg=PA268
]
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