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 planeta ...

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 A Maclaurin spheroid is an oblate spheroid which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. This spheroid is named after the Scottish mathematician Colin Maclaurin, who formulated it for t ...

, which was formulated in 1742, was considered to be the only type of

ellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as th ...

which can be in equilibrium.

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaellipsoid An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a surface that may be defined as th ...

must be equal, leading back to the solution of Maclaurin spheroid. But

Jacobi Jacobi may refer to: * People with the surname Jacobi Mathematics: * Jacobi sum, a type of character sum * Jacobi method, a method for determining the solutions of a diagonally dominant system of linear equations * Jacobi eigenvalue algorithm, ...

realized that

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia

Jacobi formula

For an ellipsoid with equatorial semi-principal axes

a, \ b

and polar semi-principal axis

c

, the angular velocity

\Omega

about

c

is given by :

\frac = 2 abc \int_0^\infty \frac\ , \quad \Delta^2 = (a^2+u)(b^2+u)(c^2+u),

where

\rho

is the density and

G

is the

gravitational constant The gravitational constant (also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant), denoted by the capital letter , is an empirical physical constant involved in ...

, subject to the condition :

a^2 b^2 \int_0^\infty \frac = c^2\int_0^\infty \frac.

For fixed values of

a

and

b

, the above condition has solution for

c

such that :

\frac>\frac + \frac.

The integrals can be expressed in terms of incomplete elliptic integrals. In terms of the Carlson symmetric form elliptic integral

R_

, the formula for the angular velocity becomes :

\frac = \frac (a^ R_(a^,b^,c^,a^) - b^ R_(a^,b^,c^,b^))

and the condition on the relative size of the semi-principal axes

a, \ b, \ c

is :

\frac \frac (R_(a^,b^,c^,a^) - R_(a^,b^,c^,b^)) = \frac c^ R_(a^,b^,c^,c^).

The angular momentum

L

of the Jacobi ellipsoid is given by :

\frac = \frac\frac\sqrt \ , \quad \bar=(abc)^,

where

M

is the mass of the ellipsoid and

\bar

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

a, \ b, \ c

and same mass and with a flow velocity field of :

\mathbf = \zeta \frac,

where

x, \ y, \ z

are Cartesian coordinates on axes

\hat, \ \hat, \ \hat

aligned respectively with the

a, \ b, \ c

axes of the ellipsoid. Here

\zeta

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 (

\nabla\times \mathbf = \zeta \mathbf

). The angular velocity

\Omega

of the Jacobi ellipsoid and vorticity of the corresponding Dedekind ellipsoid are related by :

\zeta = \left( \frac + \frac\right) \Omega.

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

a = b

, 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

\zeta = 2 \Omega

, 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 :

\frac = \frac \left( \frac + \frac\right).

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 Jam ...

, 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 Quadrics Astrophysics Fluid dynamics

History

Jacobi formula

Relationship with Dedekind ellipsoid

See also

References