Modeling differential rotations of compact stars in equilibriums

Outcomes of numerical relativity simulations of massive core collapses or binary neutron star mergers with moderate masses suggest formations of rapidly and differentially rotating neutron stars. Subsequent fall back accretion may also amplify the degree of differential rotation. We propose new formulations for modeling the differential rotation of those compact stars, and present selected solutions of differentially rotating, stationary, and axisymmetric compact stars in equilibrium. For the cases when rotating stars reach break-up velocities, the maximum masses of such rotating models are obtained.

Figure 1

Rotation curves $\mathrm{\Omega }(\varpi )$ for the rotation laws (1), (4), and (5) in the Newtonian limit. Panel (a): $j$-constant law (1) (whose $\mathrm{\Omega }(\varpi )$ is the left expression in Eq. (6) with $q=1$). Panel (b): Eq. (4) with $A=0.1$ ($\mathrm{\Omega }(\varpi )$ is the left expression in Eq. (6)). Panel (c): same as panel(b), Eq. (4) but with ${\mathrm{\Omega }}_{\text{eq}}/{\mathrm{\Omega }}_{\mathrm{c}}=0.5$. Panel (d): a new rotation law (5) with a negative sign ($\mathrm{\Omega }(\varpi )$ is the right expression in Eq. (6) with a negative sign). Panel (e): a new rotation law (5) with a positive sign ($\mathrm{\Omega }(\varpi )$ is the right expression in Eq. (6) with a positive sign). For the cases of panel(c)-(e), the parameter $A$ is determined by setting ${\mathrm{\Omega }}_{\text{eq}}/{\mathrm{\Omega }}_{\mathrm{c}}=0.5$, where ${\mathrm{\Omega }}_{\mathrm{c}}$ and ${\mathrm{\Omega }}_{\text{eq}}$ are the angular velocities at the rotation axis and at the equatorial surface, respectively. In each panel, the order of the labels in the legends (from top left to bottom right) correspond the order of the curves in the plot (from top to bottom).

Figure 2

Selected rotation curves $\mathrm{\Omega }(\varpi )$ for the rotation laws (8) and (9) in the Newtonian limit. Left and right panels are the same curves in linear and log-log scales, respectively. The parameters $A$ and $B$ are determined by setting ${\mathrm{\Omega }}_{\text{eq}}/{\mathrm{\Omega }}_{\mathrm{c}}=0.5$ and ${\mathrm{\Omega }}_{\max }/{\mathrm{\Omega }}_{\mathrm{c}}=2$, where ${\mathrm{\Omega }}_{\max }$ is the maximum angular velocity at a given point on the equatorial plane.

Figure 3

Equilibriums of differentially rotating compact stars. The set of model parameters for the rotation law for each pair of panels is described in Table 2. The pairs of panels from top to bottom in the left column, then from top to bottom in the right column correspond to Models I to V and DR in Table 2, respectively. The left panel of each pair shows the contours of the rest mass density $\rho$ (black solid curves), the color map for the angular velocity $\mathrm{\Omega }{R}_0$, and the deformation sequence of the surface of the rotating star $R_s(\theta ,\varphi )$ (green dashed curves) on the $xz$ (meridional) plane. In the right panels, plotted are the normalized rest mass density profiles $\rho /{\rho }_c$ and the normalized rotation curves $\mathrm{\Omega }/{\mathrm{\Omega }}_{\mathrm{c}}$ along the equatorial axis ($x=\varpi \sin \theta$ with $\theta =\pi /2$).

Figure 4

$M_{\mathrm{ADM}}$ is plotted with respect to the maximum rest mass density ${\rho }_c$ for differentially rotating Models I to V in Table 2. For reference, Model DR in Table 2 [11], the uniformly rotating Model UR and the spherically symmetric (TOV) model are also plotted. The curves for Models II, III and UR correspond to their maximum deformation sequences. Those of Models IV, V, DR correspond to sequences with a fixed axis ratio $R_z/R_0=0.25$, and Model I with $R_z/R_0=0.5$. Each solution in Table 3 and the maximum mass of the TOV solution Table 1 is marked by a circle on each curve.