Dedekind Ellipsoids Research Articles

A GENERALIZED FAMILY OF POST-NEWTONIAN DEDEKIND ELLIPSOIDS

We derive a family of Post-Newtonian (PN) Dedekind ellipsoids to first order. They describe non-axially symmetric, homogeneous, and rotating figures of equilibrium. The sequence of the Newtonian Dedekind ellipsoids allows for an axially symmetric limit in which a uniformly rotating Maclaurin spheroid is recovered. However, the approach taken by Chandrasekhar & Elbert (1974) to find the PN Dedekind ellipsoids excludes such a limit. In G\"urlebeck & Petroff (2010), we considered an extension to their work that permits a limit of 1 PN Maclaurin ellipsoids. Here we further detail the sequence and demonstrate that a choice of parameters exists with which the singularity formerly found in Chandrasekhar & Elbert (1974) along the sequence of PN Dedekind ellipsoids is removed.

New exact solutions of magnetic ellipsoidal figures of equilibrium

Abstract In this Letter we report brand new analytic stationary solutions of constant density stars with magnetic field and self-gravity. These solutions include prolate configurations even for purely poloidal magnetic fields as well as oblate configurations. These new analytic solutions are expressed in very simple forms and can be considered as generalized configurations of uniformly rotating constant density spheroids, i.e. Maclaurin spheroids, and of constant density ellipsoids with constant vorticity, i.e. Dedekind ellipsoids. As the axisymmetric Maclaurin spheroids and the triaxial Dedekind ellipsoids have been widely used for the estimation of the effect of rotations and/or internal motions on the self-gravitating bodies, our new analytic solutions may be used widely to estimate the effect of the magnetic fields semi-quantitatively in various contexts hereafter.

THE AXISYMMETRIC CASE FOR THE POST-NEWTONIAN DEDEKIND ELLIPSOIDS

We consider the post-Newtonian approximation for the Dedekind ellipsoids in the case of axisymmetry. The approach taken by Chandrasekhar & Elbert (1974, 1978) excludes the possibility of finding a uniformly rotating (deformed) spheroid in the axially symmetric limit, though the solution exists at the point of axisymmetry. We consider an extension to their work that permits the possibility of such a limit.

Singular Isothermal Disks. I. Linear Stability Analysis

As part of a larger effort to understand how binary and single stars form from the collapse of magnetized molecular cloud cores, we perform a global stability analysis of isopedically magnetized, singular isothermal disks (SIDs). The work described here has precedents in earlier studies of disturbances in power-law disks by Zang in 1976, Toomre in 1977, Lynden-Bell & Lemos in 1993, Syer & Tremaine in 1996, and Goodman & Evans in 1999. We find the analytic criteria for the bifurcation of axisymmetric disks into nonaxisymmetric forms with azimuthal periodicities m = 1 (eccentric displacements), 2 (oval distortions), 3 (triangular distortions), etc. These bifurcations, which occur at zero frequency, are the compressible and differentially rotating analogs of how the classical sequence of incompressible and uniformly rotating Maclaurin spheroids bifurcate (secularly, under dissipative forces) to become Dedekind ellipsoids with figure axes that remain fixed in space. Like Syer & Tremaine and Lynden-Bell & Lemos, we also find that zero-frequency logarithmic spirals are possible scale-free disturbances, but our interpretation of the existence of such steadily propagating wavetrains is different. We give a dynamical instability interpretation based on the onset of swing amplification by overreflection at the corotation circle of prograde spiral density waves the pattern speeds of which have nonzero and positive values. Our analysis yields identical instability criteria as the global normal-modes treatment of Goodman & Evans, and we tentatively also identify dynamical barred-spiral instabilities as the breathing mode limit of two-armed ordinary-spiral instabilities. We prove a general reciprocity theorem, which states that the overreflection factors are identical for spiral density waves launched from cavities interior or exterior to Q-barriers that straddle the corotation circle. This globally valid result supports a unifying interpretation, advocated for many years by C. C. Lin and his colleagues (see, e.g., work by Bertin & Lin): the coexistence of spiral structure in galaxies arising from the instability of internal normal modes in the combined star/gas disk or from driving by external tidal influences associated with the chance passages of companion bodies.

On the Implications of the nth-Order Virial Equations for Heterogeneous and Concentric Jacobi, Dedekind, and Riemann Ellipsoids

view Abstract Citations (3) References (22) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On the Implications of the nth-Order Virial Equations for Heterogeneous and Concentric Jacobi, Dedekind, and Riemann Ellipsoids Filippi, Simonetta ; Ruffini, Remo ; Sepulveda, Alonso Abstract The implications of the nth-order virial equations are analyzed for concentric heterogeneous ellipsoids with a density distribution of the form ρ = ρcf(m2), where m2 = Σ31 =1χ2i, 0 ≤ m2 ≤ 1, and αi are the semiaxes of the external ellipsoid corresponding to m2 = 1. Solutions analogous to Jacobi ellipsoids (with constant angular velocity Ω, without vorticity), to Dedekind ellipsoids (with nonuniform vorticity Ζ and zero angular velocity), and to Riemann ellipsoids (with constant angular velocity and nonuniform vorticity) are explored. It is shown that only the second- and fourth-order virial equations give nontrivial results: all the odd-order virial equations are identically satisfied for ellipsoids rotating around a principal axis of symmetry. The even-order virial equations (sixth, eighth, etc.) are shown to be a consequence of the lowest order equations. The entire family of homogeneous and heterogeneous concentric ellipsoids allowed by the virial equations is presented, confronted, and contrasted with the known cases in the literature. Publication: The Astrophysical Journal Pub Date: April 1996 DOI: 10.1086/177008 Bibcode: 1996ApJ...460..762F Keywords: STARS: INTERIORS; STARS: ROTATION full text sources ADS |

A new criterion for bar-forming instability in rapidly rotating gaseous and stellar systems. 2: Nonaxisymmetric form

We have previously introduced the parameter `alpha' as an indicator of stability to m=2 nonaxisymmetric modes in rotating, self-gravitating, axisymmetric, gaseous and stellar systems. This parameter can be written as a function of the total rotational kinetic energy, the total gravitational potential energy, and as a function of the topology/connectedness and the geometric shape of a system. Here we extend the stability criterion to nonaxisymmetric equilibrium systems, such as ellipsoids and elliptical disks and cylinders. We test the validity of this extension by considering predictions for previously published, gaseous and stellar, nonaxisymmetric models. The above formulation and critical values account accurately for the stability properties of m=2 modes in gaseous Riemann S-type ellipsoids (including the Jacobi and Dedekind ellipsoids) and elliptical Riemann disks as well as in stellar elliptical Freeman disks and cylinders: all these systems are dynamically stable except for the stellar elliptical Freeman disks that exhibit a relatively small region of m=2 dynamical instability.

The equilibrium of a galactic bar. II - Stellar-dynamical counterparts of the s-type Riemann ellipsoids

The family of triaxial stellar systems described in Paper I of this series is generalized to include, in addition to the previous dependence on Jacobi's integral, a dependence of the distribution function on a second isolating integral of the motion of a star in the prevailing gravitational field. The second integral is approximated with the aid of a model of the stellar orbits which is valid in the absence of important resonances and which should be accurate in the systems of relatively small central concentration on which this investigation concentrates. The new stellar systems are stellar-dynamical counterparts of the classical S-type Riemann ellipsoids well known in the study of self-gravitating fluid systems, and, within the framework of stellar dynamics, they are also three-dimensional counterparts of the elliptical disks studied by Freeman and by Hunter. The Riemann-like stellar systems form an extensive family, and they exhibit a rich interplay of effects of the rotations of their triaxial figures and effects of the dependence of their distribution functions on the second integral of the motion. The family includes stellar-dynamical counterparts of the classical Maclaurin spheroids, Jacobi ellipsoids, and Dedekind ellipsoids. The study of triaxial, Riemann-like systems is related to the study ofmore » bar modes of oscillation in corresponding axisymmetric, Maclaurin-like systems. On the basis of this relationship, it is shown that an axisymmetric stellar system having the structure of a uniformly rotating polytrope of index n = 0.5 is dynamically unstable with respect to a bar mode if the ratio of the rotational kinetic energy to the magnitude of the gravitational potential energy exceeds 0.166.« less

On the existence and structure of inhomogeneous analogs of the Dedekind and Jacobi ellipsoids

view Abstract Citations (35) References (7) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS On the existence and structure of inhomogeneous analogs of the Dedekind and Jacobi ellipsoids Ipser, J. R. ; Managan, R. A. Abstract The existence and structure of fully nonaxisymmetric configurations of rapidly rotating inhomogeneous fluid bodies in gravitational equilibrium are investigated. A vector-pointing formalism is established for describing equilibrium configurations of inviscid fluid obeying a barotropic equation of state in which the density is constant along velocity streamlines, and used to investigate the existence of inhomogeneous Dedekind ellipsoids, in which a stationary surface is maintained in the inertial frame by internal motions with constant vorticity parallel to the shortest axis, and Jacobi ellipsoids, which rotate uniformly. A numerical method is used to construct Jacobi configurations with polytropic equations of state, demonstrating the existence of fully nonaxisymmetrical inhomogeneous equilibrium figures. In agreement with other analyses, however, calculations indicate that the Jacobi configurations do not exist for polytropic indexes greater than or equal to 0.8, and hence the Dedekind configurations do not exist at all, at least when the density vanishes at the surface. Publication: The Astrophysical Journal Pub Date: November 1981 DOI: 10.1086/159383 Bibcode: 1981ApJ...250..362I Keywords: Ellipsoids; Existence Theorems; Gravitational Fields; Rotating Fluids; Stellar Rotation; Astrodynamics; Equilibrium; Astrophysics full text sources ADS |

Stability of dedekind ellipsoids

The stability of Dedekind ellipsoids, which are characterized by their being fixed in space and preserving their shape through internal motion of the fluid when there are ellipsoidal perturbations of the surface, has been considered by Riemann and Chandrasekhar [1–4]. On the basis of an energy criterion, that the potential energy be minimal, Riemann showed that these ellipsoids are stable but for perturbation of only some of the variables. Chandrasekhar studied the stability on the basis of linearized virial equations. He showed that the Dedekind ellipsoids are stable under the condition that restrictions in the form of certain equations are imposed on the perturbing frequency σ. In the present paper, the stability is studied exactly, without linearization of the equations, on the basis of Routh's theorem [5] with respect to all variables under the assumption that after the perturbation the ellipsoid must preserve its shape. It is shown that the Dedekind ellipsoids are stable for ellipsoidal perturbations with respect to all variables without any restrictions on the frequency σ.

The deformed figures of the Dedekind ellipsoids in the post-Newtonian approximation to general relativity - Corrections and amplifications

view Abstract Citations (4) References (4) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The deformed figures of the Dedekind ellipsoids in the post-Newtonian approximation to general relativity: corrections and amplifications. Chandrasekhar, S. ; Elbert, D. D. Abstract Two errors in the analysis of an earlier paper on the same subject are corrected. It is found that as a consequence of the corrections, the solution to the post-Newtonian equations (appropriate to determining the deformed figures of the Dedekind ellipsoid) now diverges at a point where the axes of the ellipsoid are in the ratios 1:0.3370:0.2850. In addition, the fourth-harmonic oscillations of the Dedekind ellipsoid are considered. It is found that it becomes dynamically unstable when its axes are in the ratios 1:0.3121:0.2680. Publication: The Astrophysical Journal Pub Date: February 1978 DOI: 10.1086/155906 Bibcode: 1978ApJ...220..303C Keywords: Ellipsoids; Newton Theory; Relativity; Velocity Distribution; Boundary Value Problems; Gravitational Fields; Harmonic Oscillation; Linear Equations; Pressure Distribution; Astrophysics; Neutron Stars; Relativistic Astrophysics; Background Radiation; Gravitation Theory full text sources ADS |

Variational aspects of relativistic field theories, with application to perfect fluids

By investigating perturbations of classical field theories based on variational principles we develop a variety of relations of interest in several fields, general relativity, stellar structure, fluid dynamics, and superfluid theory. The simplest and most familiar variational principles are those in which the field variations are unconstrained. Working at first in this context we introduce the Noether operator, a fully covariant generalization of the socalled canonical stress energy tensor, and prove its equivalence to the symmetric tensor T μν . By perturbing the Noether operator's definition we establish our fundamental theorem, that any two of the following imply the third (a) the fields satisfy their field equations, (b) the fields are stationary, (c) the total energy of the fields is an extremum against all perturbations. Conversely, a field theory which violates this theorem cannot be derived from an unconstrained principle. In particular both Maxwell's equations for F μν and Euler's equations for the perfect fluid have stationary solutions which are not extrema of the total energy [(a) + (b) (c)]. General relativity is a theory which does have an unconstrained variational principle but the definition of Noether operator is more ambiguous than for other fields. We define a pseudotensorial operator which includes the Einstein and Landau-Lifschitz complexes as special cases and satisfies a certain criterion on the asymptotic behavior. Then our extremal theorem leads to a proof of the uniqueness of Minkowski space: It is the only asymptotically flat, stationary, vacuum solution to Einstein's equations having R 4 global topology and a maximal spacelike hypersurface. We next consider perfect fluid dynamics. The failure of the extremal-energy theorem elucidates why constraints have always been used in variational principles that lead to Euler's equations. We discuss their meaning and give what we consider to be the “minimally constrained” principle. A discussion of one constraint, “preservation of particle identity,” from the point of view of path-integral quantum mechanics leads to the conclusion that it is inapplicable to degenerate Bose fluids, and this gives immediately the well-known irrotational flow of such fluids. Finally, we develop a restricted extremal theorem for the case of perfect fluids with self-gravitation, which has the same form as before except that certain perturbations are forbidden in (c). We show that it is a generalization of the Bardeen-Hartle-Sharp variational principle for relativistic stellar structure. It may be useful in constructing nonaxisymmetric stellar models (generalized Dedekind ellipsoids). We also give the Newtonian versions of the main results here, and we show to what extent the extremal theorems extend to fields that may not even have a variational principle.

The deformed figures of the Dedekind ellipsoids in the post-Newtonian approximation to general relativity

The effects of general relativity, in the post-Newtonian approximation, on the Dedekind figures of equilibrium of homogeneous masses are determined. It is shown how the post-Newtonian figures can be obtained by first altering the velocity field in the Dedekind ellipsoid appropriately, and then subjecting it to a suitable Lagrangian displacement cubic in the coordinates. The solution exhibits a singularity at a point where the axes of the Dedekind ellipsoid are in the ratios 1:0.6158:0.4412. However, in contrast to what happens along the Jacobian sequence, the occurrence of the singularity along the Dedekind sequence is not associated with the onset of any instability at that point by a strict Newtonian-like dynamic perturbation.

The Equilibrium and the Stability of the Roche-Riemann Ellipsoids

view Abstract Citations (24) References (4) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Equilibrium and the Stability of the Roche-Riemann Ellipsoids Aizenman, Morris L. Abstract The classical problem of Roche (wherein a fluid mass maintains an ellipsoidal form while in synchro- nous Keplerian orbit about a rigid sphere) is extended to the case where motions of uniform vorticity exist within the ellipsoid. It is shown that the introduction of such motions gives rise to new types of equilibri- um configurations (the Roche-Riemann ellipsoids). The case where the vorticity vector ~ lies parallel to the angular velocity vector ~ is considered first, and the problem is then generalized to the case where ~ and ~ lie in one of the principal planes of the ellipsoid. The existence of a minimum distance between the ellipsoid and the point mass (commonly referred to as the Roche limit) is found to be a function of the semimajor axes of the ellipsoid; in fact, this distance maybe zero. The Roche and the Dedekind ellipsoids are shown to be special cases of the Roche-Riemann ellipsoids. The stability of the ellipsoids is determined, and oniy some of the allowed equilibrium configurations are stable with respect to small perturbations. The results of these computations are illustrated by means of diagrams in the (a2/al, a3/ai)-plane. The existence of curves of bifurcation from ellipsoids with ~ parallel to ~ to ellipsoids where ~ and ~ lie in a principal plane is demonstrated, and it is found that there is an exchange of stability along these curves. I. INTRODUCTIO Publication: The Astrophysical Journal Pub Date: August 1968 DOI: 10.1086/149683 Bibcode: 1968ApJ...153..511A full text sources ADS |

The Equilibrium and the Stability of the Dedekind Ellipsoids.

view Abstract Citations (6) References (7) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS The Equilibrium and the Stability of the Dedekind Ellipsoids. Chandrasekhar, S. Abstract Figures ot equilibrium of liquid masses with internal motions of uniform vorticity define the Dedekind sequence. It has been known for a long time that the Dedekind sequence of stationary ellipsoids is congruent to the Jacobian sequence of uniformly rotating ellipsoids. It is shown in this paper that the characteristic frequencies of oscillation of a Dedekind ellipsoid, belonging to the second harmonics, are identical with those of a Jacobi ellipsoid having the same figure; but that the points at which instability sets in by a mode of oscillation belonging to the third harmonics are different along the two sequences. Publication: The Astrophysical Journal Pub Date: April 1965 DOI: 10.1086/148195 Bibcode: 1965ApJ...141.1043C full text sources ADS |