7.2 Axisymmetric characteristic hydrodynamic simulations

The approach initiated by Font and Papadopoulos has been applied in axisymmetry to pioneering studies of gravitational waves from relativistic stars. The gravitational field is treated by the original Bondi formalism using the axisymmetric code developed by Papadopoulos [221, 142]. Because of the twist-free property of the axisymmetry in the original Bondi formalism, the fluid motion cannot have a rotational component about the axis of symmetry, i.e., the fluid velocity is constrained to the (r,𝜃) plane. In his thesis work, Siebel [269] extensively tested the combined hydrodynamic-gravity code in the nonlinear, relativistic regime and demonstrated that it accurately and stably maintained the equilibrium of a neutron star.

As a first application of the code, Siebel, Font, Müller, and Papadopoulos [270] studied axisymmetric pulsations of neutron stars, which were initiated by perturbing the density and 𝜃-component of velocity of a spherically-symmetric equilibrium configuration. The frequencies measured for the radial and non-radial oscillation modes of the star were found to be in good agreement with the results from linearized perturbation studies. The Bondi news function was computed and its amplitude found to be in rough agreement with the value given by the Einstein quadrupole formula. Both computations involve numerical subtleties: The computation of the news involves large terms, which partially cancel to give a small result, and the quadrupole formula requires computing three time derivatives of the fluid variables. These sources of computational error, coupled with ambiguity in the radiation content in the initial data, prevented any definitive conclusions. The total radiated mass loss was approximately 10–9 of the total mass.

Next, the code was applied to the simulation of axisymmetric supernova core collapse [271Jump To The Next Citation Point]. A hybrid equation of state was used to mimic stiffening at collapse to nuclear densities and shock heating during the bounce. The initial equilibrium state of the core was modeled by a polytrope with index Γ = 4∕3. Collapse was initiated by reducing the polytropic index to 1.3. In order to break spherical symmetry, small perturbations were introduced into the 𝜃-component of the fluid velocity. During the collapse phase, the central density increased by five orders of magnitude. At this stage the inner core bounced at supra-nuclear densities, producing an expanding shock wave, which heated the outer layers. The collapse phase was well approximated by spherical symmetry but non-spherical oscillations were generated by the bounce. The resulting gravitational waves at null infinity were computed by the compactified code. After the bounce, the Bondi news function went through an oscillatory build up and then decayed in an ℓ = 2 quadrupole mode. However, a comparison with the results predicted by the Einstein quadrupole formula no longer gave the decent agreement found in the case of neutron star pulsations. This discrepancy was speculated to be due to the relativistic velocities of ≈ 0.2 c reached in the core collapse as opposed to 10−4 c for the pulsations. However, gauge effects and numerical errors also make important contributions, which cloud any definitive interpretation. This is the first study of gravitational wave production by the gravitational collapse of a relativistic star carried out with a characteristic code. It is clearly a remarkable piece of work, which offers up a whole new approach to the study of gravitational waves from astrophysical sources.

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