Secular instability

Secular instability is, in astrophysics and fluid dynamics, a form of instability of a rotating, self-gravitating fluid body that grows only when a dissipative process—most importantly viscosity or the emission of gravitational radiation—is present, and that disappears if the dissipation is removed. It is distinguished from dynamical (or ordinary) instability, which is already present in the idealized, non-dissipative problem and develops on the short dynamical (free-fall) timescale of the body. A secular instability instead grows on the much longer timescale set by the dissipative process, so that a body can be secularly unstable while remaining dynamically stable.

The classic example is the uniformly rotating, homogeneous (incompressible) Maclaurin spheroid, a Newtonian model of a rotating self-gravitating body. When the body rotates rapidly enough that the eccentricity of its meridional section exceeds e ≈ 0.812670—equivalently, when the ratio of rotational kinetic energy to gravitational potential energy exceeds T/|W| ≈ 0.1375—a Jacobi ellipsoid of the same mass and angular momentum has a lower total energy. A small perturbation that breaks the axial symmetry of such a spheroid will, in the presence of dissipation, grow and carry the body toward the lower-energy triaxial figure, while the excess energy is dissipated as heat (viscosity) or radiated as gravitational waves; in a perfect, non-radiating fluid the same perturbation produces only an undamped oscillation.

Terminology and timescales

The adjective secular, from Latin saeculum ("an age" or "a long period of time"), is used in astronomy and celestial mechanics for changes that accumulate slowly over many characteristic periods. In stability theory it denotes an instability that depends on a dissipative process and grows on a long, dissipative timescale rather than on the dynamical timescale of the system. A configuration may therefore be secularly unstable yet dynamically stable: it is stable against perturbations of the ideal (non-dissipative) system, but a slow dissipative process allows it to drift away from equilibrium toward a state of lower energy. Conversely, dynamical instability implies secular instability, and secular stability implies dynamical stability. The term is also used for unrelated phenomena that share this "slow, dissipative" character, such as the secular (thermal) instability of accretion disks.

Mechanism

For a uniformly rotating, incompressible self-gravitating body, the sequence of axisymmetric equilibria (the Maclaurin spheroids) and the sequence of triaxial equilibria (the Jacobi ellipsoids) meet at a point of bifurcation. Beyond this point a Jacobi ellipsoid of the same mass and angular momentum has a lower total energy than the corresponding Maclaurin spheroid, so the axisymmetric figure is no longer the minimum-energy state. A dissipative process that can lower the energy while respecting the relevant conservation laws will therefore drive the body off the spheroidal sequence.

The normal modes of an inviscid Maclaurin spheroid were first obtained by G. H. Bryan in 1889. At the bifurcation eccentricity the relevant non-axisymmetric (bar-shaped) mode has zero frequency—a neutral mode—and it is the action of dissipation on this neutral mode that produces the instability. The neutral perturbation is in fact twofold. Viscosity, which conserves angular momentum but dissipates mechanical energy and tends to enforce uniform rotation, destabilizes the mode whose frequency vanishes in the frame rotating with the body—the perturbation that carries the spheroid onto the rigidly rotating Jacobi sequence. Gravitational radiation, which removes both energy and angular momentum, instead destabilizes the mode whose frequency vanishes in the inertial frame—the perturbation associated with the stationary, internally circulating Dedekind ellipsoid sequence. Because the two driving agents act on different modes, the two instabilities—although both secular and both setting in at the same critical rotation—are physically distinct.

For comparison, the same Maclaurin sequence becomes dynamically unstable to the bar mode only at a much higher rotation, e ≈ 0.952887 (T/|W| ≈ 0.2738), where the perturbation grows exponentially even in a perfect fluid that has no means of losing energy. Dynamical instability implies secular instability, while secular stability implies dynamical stability.

History

The triaxial equilibrium figures relevant to the problem—the Jacobi ellipsoids and the internally circulating Dedekind and Riemann ellipsoids—were established in the 19th century by Jacobi, Dirichlet, Dedekind and Riemann (see Dirichlet's ellipsoidal problem).

The possibility that a rotating fluid mass stable in the absence of friction might be rendered unstable by viscosity was suggested in the 19th century by William Thomson and Peter Guthrie Tait. The conjecture was first proved in 1963 by P. H. Roberts and K. Stewartson, who solved the Navier–Stokes equations for a Maclaurin spheroid of small viscosity and showed that the bar mode, which has zero frequency in the rotating frame at the bifurcation eccentricity e ≈ 0.8127, becomes unstable for larger eccentricities.

In 1969, J. P. Ostriker raised with Chandrasekhar the question of whether the dissipation of energy by gravitational radiation could induce a secular instability of the Maclaurin spheroid at the Maclaurin–Jacobi bifurcation, in the manner of viscosity. In a first analysis, published as a Letter, Chandrasekhar concluded that radiation reaction does not destabilize the particular (Jacobi-like) mode that viscosity makes unstable. He soon recognized, however, that gravitational radiation instead destabilizes the conjugate (Dedekind-like) mode—the one whose frequency vanishes in the inertial frame at the same bifurcation point—and a corrected, fuller treatment establishing this radiation-driven secular instability appeared shortly afterwards.

In two papers published in 1978, John L. Friedman and Bernard F. Schutz developed a Lagrangian perturbation theory of rotating fluids—introducing the notions of trivial and canonical displacements and a conserved canonical energy— and used it to show that the radiation-driven secular instability is a generic feature of all rotating, self-gravitating perfect fluids: every such star is unstable, or marginally unstable, to gravitational radiation for sufficiently large azimuthal mode number m. The radiation-driven instability is now generally called the Chandrasekhar–Friedman–Schutz instability (CFS instability), and the viscosity-driven instability the Roberts–Stewartson instability.

A further historical subtlety concerns higher-order figures: Henri Poincaré found in 1885 that a sequence of pear-shaped figures branches off the Jacobi sequence; it was later shown (Cartan, 1924) that—unlike the Maclaurin–Jacobi bifurcation—this point is one of dynamical, not merely secular, instability.

Driving mechanisms

Viscosity

When the dissipation is shear viscosity, an unstable Maclaurin spheroid evolves quasi-statically along a sequence of figures of decreasing energy at fixed angular momentum and settles onto the corresponding Jacobi ellipsoid. The full nonlinear evolution of a slightly perturbed, viscous Maclaurin spheroid into a Jacobi ellipsoid was demonstrated numerically by W. H. Press and S. A. Teukolsky in 1973.

Gravitational radiation

When the dissipation is the back-reaction of gravitational radiation, a mode that is retrograde in the corotating frame but is dragged into prograde motion in the inertial frame carries negative angular momentum relative to the inertial frame; the emission of gravitational waves then makes the mode energy more negative and so amplifies, rather than damps, the perturbation. This is the mechanism of the Chandrasekhar–Friedman–Schutz instability.

Application to rotating stars

Compressible stars

For compressible (non-uniform-density) rotating stars the situation is more involved, because such stars possess internal energy and can store the heat dissipated by viscosity instead of liberating it immediately; whether the secular bar mode is unstable then depends on the relative magnitudes of the cooling and viscous timescales. Across a wide range of stellar models the secular bar-mode (Jacobi-like) instability nevertheless sets in near T/|W| ≈ 0.14, close to the incompressible value.

Neutron stars and gravitational waves

Secular, gravitational-radiation-driven instabilities are of particular interest for rapidly rotating neutron stars, where they may limit spin rates and act as sources of gravitational waves. The instability of the fundamental (f-) mode requires near-breakup rotation and is suppressed by viscosity for most neutron-star temperatures. In 1998, the axial r-modes—oscillations restored by the Coriolis force and analogous to terrestrial Rossby waves—were shown by Nils Andersson and, independently, by John L. Friedman and Sharon Morsink to be subject to the CFS instability for all rotation rates of a perfect-fluid star, making them potentially important emitters of gravitational radiation. Whether the r-mode instability operates in a given neutron star depends on the competition between gravitational-wave driving and damping by bulk viscosity, shear viscosity, and boundary layers.

See also

• Jacobi ellipsoid
• Dirichlet's ellipsoidal problem
• Chandrasekhar–Friedman–Schutz instability
• Neutron-star oscillation
• Asteroseismology

Further reading

• Chandrasekhar, S. (1969). Ellipsoidal Figures of Equilibrium. New Haven: Yale University Press.
• Lyttleton, Raymond Arthur (1953). The Stability of Rotating Liquid Masses. Cambridge University Press.
• Tassoul, Jean-Louis (1978). Theory of Rotating Stars. Princeton University Press.
• Friedman, John L.; Stergioulas, Nikolaos (2013). Rotating Relativistic Stars. Cambridge University Press.