Jacobi ellipsoid

Last updated
Artistic rendering of Haumea, a dwarf planet with triaxial ellipsoid shape. Haumea Rotation.gif
Artistic rendering of Haumea, a dwarf planet with triaxial ellipsoid shape.

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. [1]

Contents

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. [2] [3] Lagrange in 1811 [4] 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: [5]

"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

The equatorial (a, b) and polar (c) semi-principal axes of a Jacobi ellipsoid and Maclaurin spheroid, as a function of normalized angular momentum, subject to abc = 1 (i.e. for constant volume of 4p/3).
The broken lines are for the Maclaurin spheroid in the range where it has dynamic but not secular stability - it will relax into the Jacobi ellipsoid provided it can dissipate energy by virtue of a viscous constituent fluid. Jacobi-ellipsoid-dimensions-2.svg
The equatorial (a, b) and polar (c) semi-principal axes of a Jacobi ellipsoid and Maclaurin spheroid, as a function of normalized angular momentum, subject to abc = 1 (i.e. for constant volume of 4π/3).
The broken lines are for the Maclaurin spheroid in the range where it has dynamic but not secular stability – it will relax into the Jacobi ellipsoid provided it can dissipate energy by virtue of a viscous constituent fluid.

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. [6] In terms of the Carlson symmetric form 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 [7]

where are Cartesian coordinates on axes aligned respectively with the axes of the ellipsoid. Here is the vorticity, which is uniform throughout the spheroid (). The angular velocity of the Jacobi ellipsoid and vorticity of the corresponding Dedekind ellipsoid are related by [7]

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, [8] but the angular momentum of the Jacobi spheroid is the greater by a factor of [8]

See also

Related Research Articles

In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force F proportional to the displacement x:

<span class="mw-page-title-main">Jerk (physics)</span> Rate of change of acceleration with time

In physics, jerk (also known as jolt) is the rate of change of an object's acceleration over time. It is a vector quantity (having both magnitude and direction). Jerk is most commonly denoted by the symbol j and expressed in m/s3 (SI units) or standard gravities per second (g0/s).

<span class="mw-page-title-main">Navier–Stokes equations</span> Equations describing the motion of viscous fluid substances

The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

<span class="mw-page-title-main">Hydrostatic equilibrium</span> State of balance between external forces on a fluid and internal pressure gradient

In fluid mechanics, hydrostatic equilibrium 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 physics of Earth, the pressure-gradient force prevents gravity from collapsing the planetary atmosphere into a thin, dense shell, whereas gravity prevents the pressure-gradient force from diffusing the atmosphere into outer space. In general, it is what causes objects in space to be spherical.

<span class="mw-page-title-main">Equations of motion</span> Equations that describe the behavior of a physical system

In physics, equations of motion are equations that describe the behavior of a physical system in terms of its motion as a function of time. More specifically, the equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. These variables are usually spatial coordinates and time, but may include momentum components. The most general choice are generalized coordinates which can be any convenient variables characteristic of the physical system. The functions are defined in a Euclidean space in classical mechanics, but are replaced by curved spaces in relativity. If the dynamics of a system is known, the equations are the solutions for the differential equations describing the motion of the dynamics.

<span class="mw-page-title-main">Angular velocity</span> Direction and rate of rotation

In physics, angular velocity, also known as angular frequency vector, is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates around an axis of rotation and how fast the axis itself changes direction.

<span class="mw-page-title-main">Ellipsoid</span> Quadric surface that looks like a deformed sphere

An ellipsoid is a surface that can be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

<span class="mw-page-title-main">Moment of inertia</span> Scalar measure of the rotational inertia with respect to a fixed axis of rotation

The moment of inertia, otherwise known as the mass moment of inertia, angular/rotational mass, second moment of mass, or most accurately, rotational inertia, of a rigid body is a quantity that determines the torque needed for a desired angular acceleration about a rotational axis, akin to how mass determines the force needed for a desired acceleration. It depends on the body's mass distribution and the axis chosen, with larger moments requiring more torque to change the body's rate of rotation by a given amount.

In classical mechanics, Euler's rotation equations are a vectorial quasilinear first-order ordinary differential equation describing the rotation of a rigid body, using a rotating reference frame with angular velocity ω whose axes are fixed to the body. Their general vector form is

In fluid mechanics, the Taylor–Proudman theorem states that when a solid body is moved slowly within a fluid that is steadily rotated with a high angular velocity , the fluid velocity will be uniform along any line parallel to the axis of rotation. must be large compared to the movement of the solid body in order to make the Coriolis force large compared to the acceleration terms.

A fictitious force is a force that appears to act on a mass whose motion is described using a non-inertial frame of reference, such as a linearly accelerating or rotating reference frame. Fictitious forces are invoked to maintain the validity and thus use of Newton's second law of motion, in frames of reference which are not inertial.

In physics, precisely in the study of the theory of general relativity and many alternatives to it, the post-Newtonian formalism is a calculational tool that expresses Einstein's (nonlinear) equations of gravity in terms of the lowest-order deviations from Newton's law of universal gravitation. This allows approximations to Einstein's equations to be made in the case of weak fields. Higher-order terms can be added to increase accuracy, but for strong fields, it may be preferable to solve the complete equations numerically. Some of these post-Newtonian approximations are expansions in a small parameter, which is the ratio of the velocity of the matter forming the gravitational field to the speed of light, which in this case is better called the speed of gravity. In the limit, when the fundamental speed of gravity becomes infinite, the post-Newtonian expansion reduces to Newton's law of gravity.

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 the shape of Earth in 1742. In fact the figure of the Earth is far less oblate than Maclaurin's formula suggests, since the Earth is not homogeneous, but has a dense iron core. The Maclaurin spheroid is considered to be the simplest model of rotating ellipsoidal figures in hydrostatic equilibrium since it assumes uniform density.

The Mason–Weaver equation describes the sedimentation and diffusion of solutes under a uniform force, usually a gravitational field. Assuming that the gravitational field is aligned in the z direction, the Mason–Weaver equation may be written

In classical mechanics, Poinsot's construction is a geometrical method for visualizing the torque-free motion of a rotating rigid body, that is, the motion of a rigid body on which no external forces are acting. This motion has four constants: the kinetic energy of the body and the three components of the angular momentum, expressed with respect to an inertial laboratory frame. The angular velocity vector of the rigid rotor is not constant, but satisfies Euler's equations. The conservation of kinetic energy and angular momentum provide two constraints on the motion of .

In fluid mechanics, potential vorticity (PV) is a quantity which is proportional to the dot product of vorticity and stratification. This quantity, following a parcel of air or water, can only be changed by diabatic or frictional processes. It is a useful concept for understanding the generation of vorticity in cyclogenesis, especially along the polar front, and in analyzing flow in the ocean.

Acoustic streaming is a steady flow in a fluid driven by the absorption of high amplitude acoustic oscillations. This phenomenon can be observed near sound emitters, or in the standing waves within a Kundt's tube. Acoustic streaming was explained first by Lord Rayleigh in 1884. It is the less-known opposite of sound generation by a flow.

<span class="mw-page-title-main">Wigner rotation</span>

In theoretical physics, the composition of two non-collinear Lorentz boosts results in a Lorentz transformation that is not a pure boost but is the composition of a boost and a rotation. This rotation is called Thomas rotation, Thomas–Wigner rotation or Wigner rotation. If a sequence of non-collinear boosts returns an object to its initial velocity, then the sequence of Wigner rotations can combine to produce a net rotation called the Thomas precession.

In fluid dynamics, Beltrami flows are flows in which the vorticity vector and the velocity vector are parallel to each other. In other words, Beltrami flow is a flow in which the Lamb vector is zero. It is named after the Italian mathematician Eugenio Beltrami due to his derivation of the Beltrami vector field, while initial developments in fluid dynamics were done by the Russian scientist Ippolit S. Gromeka in 1881.

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 inertial frame, is a linear function of the coordinates. Dirichlet's basic idea was to reduce Euler equations to a system of ordinary differential equations such that the position of a fluid particle in a homogeneous ellipsoid at any time is a linear and homogeneous function of initial position of the fluid particle, using Lagrangian framework instead of the Eulerian framework.

References

  1. Jacobi, C. G. (1834). "Ueber die Figur des Gleichgewichts". Annalen der Physik (in German). 109 (8–16): 229–233. Bibcode:1834AnP...109..229J. doi:10.1002/andp.18341090808.
  2. Chandrasekhar, S. (1969). Ellipsoidal figures of equilibrium. Vol. 10. New Haven: Yale University Press. p. 253.
  3. Chandrasekhar, S. (1967). "Ellipsoidal figures of equilibrium—an historical account". Communications on Pure and Applied Mathematics. 20 (2): 251–265. doi:10.1002/cpa.3160200203.
  4. Lagrange, J. L. (1811). Mécanique Analytique sect. IV 2 vol.
  5. Dirichlet, G. L. (1856). "Gedächtnisrede auf Carl Gustav Jacob Jacobi". Journal für die reine und angewandte Mathematik (in German). 52: 193–217.
  6. Darwin, G. H. (1886). "On Jacobi's figure of equilibrium for a rotating mass of fluid". Proceedings of the Royal Society of London. 41 (246–250): 319–336. Bibcode:1886RSPS...41..319D. doi:10.1098/rspl.1886.0099. S2CID   121948418.
  7. 1 2 Chandrasekhar, Subrahmanyan (1965). "The Equilibrium and the Stability of the Dedekind Ellipsoids". Astrophysical Journal . 141: 1043–1055. Bibcode:1965ApJ...141.1043C. doi: 10.1086/148195 .
  8. 1 2 Bardeen, James M. (1973). "Rapidly Rotating Stars, Disks, and Black Holes". In DeWitt, C.; DeWitt, Bryce Seligman (eds.). Black Holes. Houches Lecture Series. CRC Press. pp. 267–268. ISBN   9780677156101.