A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under

hydrostatic equilibrium In fluid mechanics, hydrostatic equilibrium, also called hydrostatic balance and 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. I ...

which arises when a self-gravitating,

fluid In physics, a fluid is a liquid, gas, or other material that may continuously motion, move and Deformation (physics), deform (''flow'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are M ...

body of uniform

density Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be u ...

rotates with a constant

angular velocity In physics, angular velocity (symbol or \vec, the lowercase Greek letter omega), also known as the angular frequency vector,(UP1) is a pseudovector representation of how the angular position or orientation of an object changes with time, i ...

. It is named after the German mathematician Carl Gustav Jacob Jacobi.

History

Before Jacobi, the Maclaurin spheroid, which was formulated in 1742, was considered to be the only type of

ellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a Surface (mathemat ...

which can be in equilibrium.

Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiaellipsoid An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional Scaling (geometry), scalings, or more generally, of an affine transformation. An ellipsoid is a quadric surface; that is, a Surface (mathemat ...

must be equal, leading back to the solution of Maclaurin spheroid. But Jacobi 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 is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It ...

, 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 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

R_

, the formula for the angular velocity becomes :

\frac = \frac^2 R_J(a^2,b^2,c^2,a^2) - b^ R_J(a^2,b^2,c^2,b^2) /math>

and the condition on the relative size of the semi-principal axes a, \ b, \ c is

: \frac_J(a^2,b^2,c^2,a^2) - R_J(a^2,b^2,c^2,b^2) =  c^2 R_J(a^2,b^2,c^2,c^2). The angular momentum L of the Jacobi ellipsoid is given by

: \frac = \frac\frac\sqrt \ , \quad r=\sqrt where M is the mass of the ellipsoid and r 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 (or axial vector) 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 an ...

, 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'' (''ApJ'') is a peer-reviewed scientific journal of astrophysics and astronomy, established in 1895 by American astronomers George Ellery Hale and James Edward Keeler. The journal discontinued its print edition and ...

, volume = 141 , date = 1965 , pages = 1043–1055 , bibcode = 1965ApJ...141.1043C , doi= 10.1086/148195 , doi-access= free {{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 Ellipsoids Astrophysics Fluid dynamics Equations of astronomy

History

Jacobi formula

Relationship with Dedekind ellipsoid

See also

References