A Jacobi ellipsoid is a triaxial (i.e. scalene) ellipsoid under hydrostatic equilibrium which arises when a self-gravitating fluid body of uniform density rotates with a constant angular velocity. It is named after the German mathematician Carl Gustav Jacob Jacobi.

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 the ...

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 the ...

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

Jacobi Jacobi may refer to: * People with the surname Jacobi (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 eigenva ...

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, 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 , 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