John L. Friedman

Secular instability of rotating Newtonian stars

The effect of gravitational radiation and of viscosity on the stability of rotating self-gravitating fluids is considered. Previous cirteria governing secular stability to radiation are shown to fail as a result of the trival displacements introduced in a previous paper (1). The required modification is obtained by describing perturbations in terms of canonical displacements (displacements orthogonal to the trivials). There nevertheless remain physical perturbations having angular dependence e/sup i/mphi which, for sufficiently large m, make the stability functional (the canonical energy E/sub c/) negative, and it follows that all rotating stars are unstable or marginally unstable to gravitational radiation. In the case of stability against viscosity, the corresponding stability criterion is shown to involve the the canonical energy in a rotating frame, E/sub c/,R, a functional invariant under gauge transformations associated with the trival displacements. By using the functional E/sub c/,R to analyze local stability, it is found that a star is locally stable against viscosity if and only if the special entropy increases outward (in the sense of decreasing pressure). Finally, the behavior of normal modes is discussed and used to elucidate the generic radiation-induced instability; and certain orthogonality properties are derived.

Axial Instability of Rotating Relativistic Stars

Perturbations of rotating relativistic stars can be classified by their behavior under parity. For axial perturbations (r-modes), initial data with negative canonical energy is found with angular dependence eim for all values of m ≥ 2 and for arbitrarily slow rotation. This implies instability (or marginal stability) of such perturbations for rotating perfect fluids. This low m-instability is strikingly different from the instability to polar perturbations, which sets in first for large values of m. The timescale for the axial instability appears, for small angular velocity Ω, to be proportional to a high power of Ω. As in the case of polar modes, viscosity will again presumably enforce stability except for hot, rapidly rotating neutron stars. This work complements Anderssons numerical investigation of axial modes in slowly rotating stars.

Comparing models of rapidly rotating relativistic stars constructed by two numerical methods

We present the first direct comparison of codes based on two different numerical methods for constructing rapidly rotating relativistic stars. A code based on the Komatsu-Eriguchi-Hachisu (KEH) method (Komatsu et al. 1989), written by Stergioulas, is compared to the Butterworth-Ipser code (BI), as modified by Friedman, Ipser and Parker. We compare models obtained by each method and evaluate the accuracy and efficiency of the two codes. The agreement is surprisingly good. A relatively large discrepancy recently reported (Eriguchi et al. 1994) is found to arise from the use of two different versions of the equation of state. We find, for a given equation of state, that equilibrium models with maximum values of mass, baryon mass, and angular momentum are (generically) all distinct and either all unstable to collapse or are all stable. Our implementation of the KEH method will be available as a public domain program for interested users.

Constraints on a phenomenologically parametrized neutron-star equation of state

We introduce a parametrized high-density equation of state (EOS) in order to systematize the study of constraints placed by astrophysical observations on the nature of neutron-star matter. To obtain useful constraints, the number of parameters must be smaller than the number of EOS-related neutron-star properties measured, but large enough to accurately approximate the large set of candidate EOSs. We find that a parametrized EOS based on piecewise polytropes with 3 free parameters matches, to about 4% rms error, an extensive set of candidate EOSs at densities below the central density of

Generic instability of rotating relativistic stars

1.4{M}_{\ensuremath{\bigodot}}

Matter effects on binary neutron star waveforms

stars. Adding observations of more massive stars constrains the higher-density part of the EOS and requires an additional parameter. We obtain constraints on the allowed parameter space set by causality and by present and near-future astronomical observations with the least model dependence. Stringent constraints on the EOS parameter space are associated with the future measurement of the moment of inertia of PSR J0737-3039A combined with the maximum known neutron-star mass. We also present in an appendix a more efficient algorithm than has previously been used for finding points of marginal stability and the maximum angular velocity of stable stars.

Nonaxisymmetric Neutral Modes in Rotating Relativistic Stars

All rotating perfect fluid configurations having two-parameter equations of state are shown to be dynamically unstable to nonaxisymmetric perturbations in the framework of general relativity. Perturbations of an equilibrium fluid are described by means of a Lagrangian displacement, and an action for the linearized field equations is obtained, in terms of which the symplectic product and canonical energy of the system can be expressed. Previous criteria governing stability were based on the sign of the canonical energy, but this functional fails to be invariant under the gauge freedom associated with a class of trivial Lagrangian displacements, whose existence was first pointed out by Schutz and Sorkin [12]. In order to regain a stability criterion, one must eliminate the trivials, and this is accomplished by restricting consideration to a class of “canonical” displacements, orthogonal to the trivials with respect to the symplectic product. There nevertheless remain perturbations having angular dependenceeimφ (φ the azimuthal angle) which, for sufficiently largem, make the canonical energy negative; consequently, even slowly rotating stars are unstable to short wavelength perturbations. To show strict instability, it is necessary to assume that time-dependent nonaxisymmetric perturbations radiate energy to null infinity. As a byproduct of the work, the relativistic generalization of Ertels theorem (conservation of vorticity in constant entropy surfaces) is obtained and shown to be Noetherrelated to the symmetry associated with the trivial displacements.

Upper Limits Set by Causality on the Rotation and Mass of Uniformly Rotating Relativistic Stars

Using an extended set of equations of state and a multiple-group multiple-code collaborative effort to generate waveforms, we improve numerical-relativity-based data-analysis estimates of the measurability of matter effects in neutron-star binaries. We vary two parameters of a parameterized piecewise-polytropic equation of state (EOS) to analyze the measurability of EOS properties, via a parameter {\Lambda} that characterizes the quadrupole deformability of an isolated neutron star. We find that, to within the accuracy of the simulations, the departure of the waveform from point-particle (or spinless double black-hole binary) inspiral increases monotonically with {\Lambda}, and changes in the EOS that did not change {\Lambda} are not measurable. We estimate with two methods the minimal and expected measurability of {\Lambda} in second- and third- generation gravitational-wave detectors. The first estimate, using numerical waveforms alone, shows two EOS which vary in radius by 1.3km are distinguishable in mergers at 100Mpc. The second estimate relies on the construction of hybrid waveforms by matching to post-Newtonian inspiral, and estimates that the same EOS are distinguishable in mergers at 300Mpc. We calculate systematic errors arising from numerical uncertainties and hybrid construction, and we estimate the frequency at which such effects would interfere with template-based searches.

Extracting equation of state parameters from black hole-neutron star mergers. I. Nonspinning black holes

We study nonaxisymmetric perturbations of rotating relativistic stars modeled as perfect-fluid equilibria. Instability to a mode with angular dependence exp (im) sets in when the frequency of the mode vanishes. The locations of these zero-frequency modes along sequences of rotating stars are computed for the first time in the framework of general relativity. We consider models of uniformly rotating stars with polytropic equations of state, finding that the relativistic models are unstable to nonaxisymmetric modes at significantly smaller values of rotation than in the Newtonian limit. Most strikingly, the m = 2 bar mode can become unstable even for soft polytropes of index N ≤ 1.3, while in Newtonian theory it becomes unstable only for stiff polytropes of index N ≤ 0.808. If rapidly rotating neutron stars are formed by the accretion-induced collapse of white dwarfs, instability associated with these nonaxisymmetric, gravitational-wave driven modes may set an upper limit on neutron-star rotation. Consideration is restricted to perturbations that correspond to polar perturbations of a spherical star. A study of axial perturbations is in progress.

On the stability of axisymmetric systems to axisymmetric perturbations in general relativity. I. The equations governing nonstationary, and perturbed systems

Causality alone suffices to set a lower bound on the period of rotation of relativistic stars as a function of their maximum observed mass. That is, by assuming a one-parameter equation of state (EOS) that satisfies vsound 0.282+0.196(M/M-1.442). The limit does not assume that the EOS agrees with a known low-density form for ordinary matter, and if one adds that assumption, the minimum period is raised by a few percent: for a match at n = 0.1 fm-3, we find P(ms)>0.295 + 0.203(M/M-1.442). The minimizing EOS yields models with a maximally soft exterior supported by a maximally stiff core. An analogous upper limit set by causality on the maximum mass of rotating neutron stars requires a low-density match, and the limit depends on the matching density, m. We recompute it, obtaining a slightly revised value, M6.1(2×1014 g cm-3/m)1/2 M☉.