Nonlinear radial oscillations of neutron stars

The effects of nonlinear oscillations in compact stars are attracting considerable current interest. In order to study such phenomena in the framework of fully nonlinear general relativity, highly accurate numerical studies are required. A numerical scheme specifically tailored for such a study is based on formulating the time evolution in terms of deviations from a stationary equilibrium configuration. Using this technique, we investigate over a wide range of amplitudes nonlinear effects in the evolution of radial oscillations of neutron stars. In particular, we discuss mode coupling due to nonlinear interaction, the occurrence of resonance phenomena, shock formation near the stellar surface as well as the capacity of nonlinearities to stabilize perturbatively unstable neutron star models.

Figure 1

The convergence factor $q$ of $\Delta \sigma$, $\Delta N$ and $\xi$ as functions of time. The obtained results are consistent with second-order convergence corresponding to $q=4$.

Figure 2

The evolution of the first four eigenmode coefficients $A_i$ obtained for model 1 and an initial displacement corresponding to the fundamental mode with a surface amplitude of ${\xi }_1=10\mathrm{m}$.

Figure 3

The envelopes of the amplitudes of the second and fourth eigenmode coefficients $A_i$ obtained for model 2 and an initial displacement corresponding to the second mode with a surface amplitude of ${\xi }_2=10\mathrm{m}$. Note that the excitation of mode $i=4$ is so strong that it leads to a visible temporary decrease in $A_2(t)$.

Figure 4

The maximal eigenmode coefficients of the first six eigenmodes as functions of the amplitude of the initial perturbation in form of the fundamental mode. The curves represent a linear function in ${\xi }_1$ for the fundamental mode and quadratic power laws for all other modes.

Figure 5

The maximal eigenmode coefficients of the first ten eigenmodes as functions of the amplitude of the initial perturbation in form of the second (left panel) and the third (right panel) mode.

Figure 6

The maximal eigenmode coefficients of the first ten eigenmodes as functions of the amplitude of the initial perturbation in form of the second eigenmode (left panel) or the third eigenmode (right panel) as obtained for model 2.

Figure 7

The evolution of the fourth eigenmode coefficient for different perturbation amplitudes.

Figure 8

The coefficient of the fundamental eigenmode. The dashed line corresponds to an initial expansion of the star, the solid line to a compression.

Figure 9

The frequency of the fundamental mode as function of the initial perturbation. The frequencies have been obtained by a Fourier analysis of the nonlinear evolution.

Figure 10

Snapshots of the evolution of the velocity $w$ for a moderate amplitude $B=2\times {10}^{-4}$ of the initial perturbation (left panel) and of $\Delta \sigma$ obtained for an amplitude $B={10}^{-3}$.