EQUATION OF STATE









Physical Scenario:



Using a 3D Smoothed Particle Hydrodynamics (SPH) code, I modeled the dynamical instabilities that arise in rapidly rotating Newtonian compressible fluids. The objective of this work (Houser & Centrella 1996) was to study the behavior of compact objects having stiff polytropic equations of state. We considered cases where the polytropic index, n, was equal to 1.5, 1.0 and 0.5. Previously, Williams and Tohline (1987) studied the initial development of the bar instability in similar models with n equal to 0.8, 1.0, 1.3, 1.5, and 1.8; longer evolutions were carried out in Williams & Tohline (1988) for the cases n=0.8 and n=1.8. Their work was done with an Eulerian code in cylindrical coordinates that used the diffusive donor cell advection method which imposed the polytropic equation of state throughout the runs. In addition, they modeled only one hemisphere in the angular coordinate, so that only even toroidal modes were represented. Our simulations do not suffer from these restrictions, and include the calculation of the gravitational radiation.

Here I present numerical simulations of the dynamical instability in rapidly rotating neutron stars. The stars are rotating at approximately 0.47c, where c is the speed of light in a vacuum, and are initially assumed to have a total mass of 1.4 Solar Masses equatorial radius of 10 km. The stars are initially modeled as polytropes with n = 1.5, 1.0, and 0.5. These calculations were done using the 3D SPH hydrodynamics code. The stars are created using approximately 16,000 fluid particles. The code used a purely Newtonian gravitational field, and the gravitational radiation was calculated in the quadrupole approximation. The back reaction of the gravitational radiation was not included.











Results:



All models exhibit the growth of the global m=2 bar mode, with mass and angular momentum being shed from the ends of the bar to form two trailing spiral arms. In general, as n decreases the central bar becomes narrower and more elongated. Once the central core has reached its maximum elongation, it begins to recontract toward a more axisymmetric state. This primary instability is followed by successive episodes of spiral arm ejection and core recontraction, with the number of these episodes increasing for stiffer equations of state. At the end of the simulations, the models with n=1.5 and n=1.0 have settled into a state with a nearly axisymmetric core surrounded by a flattened disk-like halo that contains approximately 10% of the total mass and approximately 30% of the total angular momentum. Since the final cores are still secularly unstable, they are expected to continue evolving under the secular instability (Lai & Shapiro 1995). The model with n=0.5 had a fairly elongated core and was still evolving when that run was terminated.

The development of the instability produces a burst of gravitational radiation. The maximum amplitude of the waveform rh does not vary significantly with the polytropic index, whereas the frequency of the waves decreases somewhat as n decreases. This lowering of the frequency with n reflects the fact that the stiffer polytropes produce more elongated bars, which rotate more slowly; it also results in a decrease in the peak gravitational wave luminosity with n. Since the stiffer models undergo more episodes of spiral arm ejection and core recontraction, they produce longer-lived gravitational wave signals from the dynamical instability, with the total amount of energy and angular momentum emitted in the form of energy and angular momentum emitted in the form of gravitational radiation increasing as n decreases. The nearly axisymmetric final cores (for n=1.5 and n= 1.0) will continue to emit gravitational radiation as they evolve under the secular instability; this has been calculated by Lai and Shapiro (Lai & Shapiro 1995).

The actual values of the gravitational wave quantities depend sensitively on the equatorial radius of the stellar core when the dynamical instability takes place. This in turn depends on the astrophysical scenario in which the instability develops. Consider, for example, the collapse of a rotating stellar core of mass M = 1.4 Solar Masses that is dynamically unstable and is prevented from collapsing further due to centrifugal forces. The equatorial radius of the core at which this centrifugal hangup occurs determines the amplitude and frequency of the resulting gravitational radiation. The simulations presented here use stiff equations of state, which are appropriate only for stellar cores that have collapsed to near neutron star densities.



These simulations were performed on the Cray C90 at the Pittsburgh Supercomputer Center under grant PHY910018P.





MOVIES: n=1.5, n=1.0, and n=0.5





NOTE: The movie shown consist of the projection of approximately 16,000 "fluid" elements or ``particles'' onto the xy plane. In these movies, the axis of rotation is the z axis.



For more information, please see: Houser & Centrella (1996).





This work was supported in part by NSF grant PHY-9208914.











References:

Houser, J.L. & Centrella, J.M. 1996, Phys. Rev. D, 54, 7278.

Lai, D. & Shapiro, S. 1995, Astrophys J., 442, 259.

Williams, H.A. & Tohline, J.E. 1987, Astrophys. J., 315, 594.

Williams, H.A. & Tohline, J.E. 1988, Astrophys. J., 334, 449.