Inertial modes of rigidly rotating neutron stars in Cowling approximation

In this article, we investigate inertial modes of rigidly rotating neutron stars, i.e. modes for which the Coriolis force is dominant. This is done using the assumption of a fixed spacetime (Cowling approximation). We present frequencies and eigenfunctions for a sequence of stars with a polytropic equation of state, covering a broad range of rotation rates. The modes were obtained with a nonlinear general relativistic hydrodynamic evolution code. We further show that the eigenequations for the oscillation modes can be written in a particularly simple form for the case of arbitrary fast but rigid rotation. Using these equations, we investigate some general characteristics of inertial modes, which are then compared to the numerically obtained eigenfunctions. In particular, we derive a rough analytical estimate for the frequency as a function of the number of nodes of the eigenfunction, and find that a similar empirical relation matches the numerical results with unexpected accuracy. We investigate the slow rotation limit of the eigenequations, obtaining two different sets of equations describing pressure and inertial modes. For the numerical computations we only considered axisymmetric modes, while the analytic part also covers nonaxisymmetric modes. The eigenfunctions suggest that the classification of inertial modes by the quantum numbers of the leading term of a spherical harmonic decomposition is artificial in the sense that the largest term is not strongly dominant, even in the slow rotation limit. The reason for the different structure of pressure and inertial modes is that the Coriolis force remains important in the slow rotation limit only for inertial modes. Accordingly, the scalar eigenequation we obtain in that limit is spherically symmetric for pressure modes, but not for inertial modes.

Figure 1

Time evolution (a) and Fourier spectrum (b) of velocity in $\theta$ direction at $r=3\mathrm{km}$, $\theta =\pi /2$, for a simulation of stellar model BU3, which was perturbed using the eigenfunction of inertial mode $i_{22}$. The resolution of the simulation is $150\times 150$ points. The damping is purely numerical.

Figure 2

Uniformity of the global complex phase under ideal circumstances (a) and for a problematic case (b). Gray scaled is the quantity $\cos (\arg (\delta {ϵ}_c)-{\varphi }_0)$.

Figure 3

Eigenfunction of inertial mode $i_{21}$, computed for stellar model BU3. The cuts show the upper half of the meridional plane. (a) Eigenfunction $\delta s$. With the exception of the star surface, the lines represent equidistant isocontours. Negative values correspond to dashed lines, and the thick solid line marks the node of the eigenfunction. The shaded area represents amplitudes below 5% of the maximum one, which is our guess for the accuracy of the node location. The arrows at the surface represent the analytic prediction for the direction of the isopotential lines of $\delta s$ in the slow rotation approximation. (b) Eigenfunction of momentum density components $S^l$ in the meridional plane. The solid lines are the nodes of $\delta s$.

Figure 4

Like Fig. 3, but for the $i_{22}$ mode.

Figure 5

Like Fig. 3, but for the $i_{23}$ mode.

Figure 6

Like Fig. 3, but for the $i_{24}$ mode.

Figure 7

Like Fig. 3, but for the $i_{25}$ mode.

Figure 8

Comparison of the eigenfunction of inertial mode $i_{22}$ at rotation rates 246 Hz (a) and 792 Hz (b). See Fig. 3 for a description of the plot.

Figure 9

Ratio $\nu =\omega /2\Omega$ versus rotation rate $F_R$ for a sequence of stellar models with fixed central density. The shaded region marks the range of ${\nu }_c$ inside the star. The solid lines are fits of the function ${\nu }_0+{\nu }_2{\Omega }^2$ to the data. The coefficients are given in Table .

Figure 10

Dependency of $1/\mu$ on the number of nodes crossing the equatorial plane, for the inertial modes of stellar model BU3 with exactly two nodes crossing the (entire) rotation axis. Plotted is the value in the Newtonian limit, where $\mu$ is computed setting ${\nu }_c=1$, as well as the minimum and maximum values ${\mu }_0$, ${\mu }_1$ in the actual stellar model. The solid line is a linear fit to the Newtonian values, the dotted line is the estimate Eq. (39) computed with ${\nu }_c=1$.