Do triaxial supramassive compact stars exist?

We study quasiequilibrium solutions of triaxially deformed rotating compact stars—a generalization of Jacobi ellipsoids under relativistic gravity and compressible equations of state (EOSs). For relatively stiff (piecewise) polytropic EOSs, we find supramassive triaxial solutions whose masses exceed the maximum mass of the spherical solution, but are always lower than those of axisymmetric equilibriums. The difference in the maximum masses of triaxial and axisymmetric solutions depends sensitively on the EOSs. If the difference turns out to be only about 10%, it will be strong evidence that the EOS of high density matter becomes substantially softer in the core of neutron stars. This finding opens a novel way to probe phase transitions of high density nuclear matter using detections of gravitational waves from new born neutron stars or magnetars under fallback accretion.

Figure 1

The rest mass $M_0$ is plotted against the square of the eccentricity (in proper length) $e^2≔1-({\overline{R}}_z/{\overline{R}}_x{)}^2$ for axisymmetric (red dots) and triaxial (blue crosses) solutions of uniformly rotating compact stars. Solid (red) and dashed (blue) envelope curves are polynomial fits to the extrapolated limiting solutions. Left to right panels correspond to the results of polytropic EOSs $\mathrm{\Gamma }=3$ and 4, and piecewise polytropic EOS $({\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2)=(4,2.5)$, respectively. Solutions above the horizontal dashed line in each panel are supramassive, $M_0>M_{\max }^{\mathrm{SPH}}$. In each panel, the left fitted curve to the triaxial solutions (blue dashed) corresponds to the bifurcation points, and the right to the mass shedding (Roche) limits.

Figure 2

The gravitational (Arnowitt-Deser-Misner) mass $M$ is plotted against the central density ${\rho }_c$ for the same model with the right panel of Fig. 1, $({\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2)=(4,2.5)$. Solid and dashed curves are axisymmetric sequences with $J=\text{constant}$, and $M_0=\text{constant}$, respectively. Turning points of these curves are indicated by a dotted curve. Top (red) and bottom (green) circles are the maximum of the gravitational mass for the axisymmetric and spherical stars, respectively. For a reference, we draw evolutionary tracks (thick curves with arrows) of newly born magnetars under the fallback accretion modeled by Piro and Ott, Eqs. (14)–(16) in [13]. For simplicity, a spin equilibrium (Eq. (17) in [13]) is always assumed.

Figure 3

An illustrative gravitational waveform from a compact star under fallback accretion. A diminishing periodic wave from a triaxially deformed rotating star is followed by a burst wave from a collapse.