Unstable normal modes of low $T/W$ dynamical instabilities in differentially rotating stars

We investigate the nature of low $T/W$ dynamical instabilities in differentially rotating stars by means of linear perturbation. Here, $T$ and $W$ represent rotational kinetic energy and the gravitational binding energy of the star. This is the first attempt to investigate low $T/W$ dynamical instabilities as a complete set of the eigenvalue problem. Our equilibrium configuration has “constant” specific angular momentum distribution, which potentially contains a singular solution in the perturbed enthalpy at a corotation radius in linear perturbation. We find the unstable normal modes of differentially rotating stars by solving the eigenvalue problem along the equatorial plane of the star, imposing the regularity condition on the center and the vanished enthalpy at the oscillating equatorial surface. We find that the existing pulsation modes become unstable due to the existence of the corotation radius inside the star. The feature of the unstable mode eigenfrequency and its eigenfunction in the linear analysis roughly agrees with that in three-dimensional hydrodynamical simulations in Newtonian gravity. Therefore, our normal mode analysis in the equatorial motion proves valid to find the unstable equilibrium stars efficiently. Moreover, the nature of the eigenfunction that oscillates between corotation and the surface radius for unstable stars requires reinterpretation of the pulsation modes in differentially rotating stars.

Figure 1

Eigenfunctions of the first five $m=2$ normal modes of differentially rotating $n=1$ polytropic stars in the cylindrical model. Labels $f$ and $p_N$ denote the $f$ mode and $p$ modes with node number $N$ in Table 2, respectively. Note that node numbers counted between corotation ($r_{\mathrm{cr}}/r_e=0.486$, 0.222, 0.132, 0.0564 for $N=0,1,2,3$ normal modes) and the equatorial surface radius are used to identify $f$ and $p$ modes.

Figure 2

Same as Fig. 1, but for first three $m=1$ normal modes of $n=3$ polytropic stars. Note that the corotation radius for $N=0$, 1, and 2 normal modes are $r_{\mathrm{cr}}/r_e=0.190$, 0.122, and 0.0288.

Figure 3

The $m=2$ scalar potential density diagnostics in the equatorial plane. Note that $U_2$ represents the $m=2$ scalar potential density weighted average in the equatorial plane. The labels (a), (b), and (c) denote the evolution time $t=59.7P_c$, $74.6P_c$, and $89.5P_c$, respectively. We find features similar to the cylindrical model (Fig. 1) in which no nodes are present.

Figure 4

Same as Fig. 3 but for $m=1$. Note that $U_1$ represent the $m=1$ scalar potential density-weighted average in the equatorial plane. The labels (a), (b), and (c) denote the evolution time $t=52.0P_c$, $58.5P_c$, and $64.9P_c$, respectively. We find features similar to the cylindrical model (Fig. 2) in which no nodes are present.