Phase conversion dissipation in multicomponent compact stars

We propose a mechanism for the damping of density oscillations in multicomponent compact stars. The mechanism is the periodic conversion between different phases, i.e., the movement of the interface between them, induced by pressure oscillations in the star. The damping grows nonlinearly with the amplitude of the oscillation. We study in detail the case of r-modes in a hybrid star with a sharp interface, and we find that this mechanism is powerful enough to saturate the r-mode at very low saturation amplitude, of order ${10}^{-10}$, and is therefore likely to be the dominant r-mode saturation mechanism in hybrid stars with a sharp interface.

Figure 1

Schematic model: two incompressible phases in a cylinder with piston, in a gravitational field. An oscillation of the external pressure on the piston leads to interconversion of the two phases and hence movement of the piston.

Figure 2

Pressure gradients in the cylinder of Fig. 1. The solid line $\overline{p}\left(x\right)$ is the pressure profile in long-term equilibrium. The dashed line is a snapshot of the system at a moment when the piston has moved inward a distance $\delta x_{\mathrm{p}}$, the pressure everywhere has risen, and the phase boundary has moved out a distance $\delta x_{\mathrm{b}}$ as the low-density phase in part of the region with $p>p_{\mathrm{crit}}$ has converted to the high-density phase.

Figure 3

Schematic plot of the pressure as a function of baryon chemical potential in $\beta$-equilibrated $\left({\mu }_{\mathrm{K}}=0\right)$ nuclear matter and quark matter. At a given pressure, the phase with lower ${\mu }_{\mathrm{B}}$ is thermodynamically favored.

Figure 4

Conversion of nuclear matter into quark matter. (b) Spatial variation of the $K$-fraction parameter $a$ [Eq. (26)] in the conversion region where the pressure is above $p_{\mathrm{crit}}$ (see Fig. 3). (a) Corresponding path in the $({\mu }_{\mathrm{B}},{\mu }_{\mathrm{K}})$ plane of chemical potentials. The quark matter isobar (red curve passing through Q and ${\mathrm{Q}}^{*})$ is at the same pressure as the equilibrated nuclear matter (point $\mathrm{N})$, and the arrows follow increasing pressure except from N to ${\mathrm{Q}}^{*}$ to $\mathrm{Q}$ where pressure is constant (traversing increasing $x$ in the right panel).

Figure 5

Conversion of quark matter into nuclear matter. (b) Spatial variation of the $K$-fraction parameter $b$ [Eq. (46)] in the conversion region where the pressure is below $p_{\mathrm{crit}}$ (see Fig. 3). (a) Corresponding path in the $({\mu }_{\mathrm{B}},{\mu }_{\mathrm{K}})$ plane of chemical potentials. The nuclear matter isobar (blue curve passing through N and ${\mathrm{N}}^{*})$ is at the same pressure as the equilibrated quark matter (point $\mathrm{Q})$, and the arrows follow increasing pressure except from N to ${\mathrm{N}}^{*}$ to $\mathrm{Q}$ where pressure is constant (traversing increasing $x$ in the right panel).

Figure 6

Diagram showing how the ideal boundary position (dashed line) and the real boundary position (solid [blue] line) vary in time. The ideal boundary is where the phase boundary would be if the phase conversion process equilibrated instantaneously, and it is determined by the instantaneous external pressure [Eq. (9)]. The real boundary is always “chasing” the ideal boundary, with velocity given by Eq. (42) and (59) where $\delta z\left(t\right)$ is its distance from the ideal boundary. The real boundary coincides with the ideal boundary twice per cycle, at $t=t_1$ and $t=t_2$.

Figure 7

(a) Dissipated power due to phase conversion $P_{\mathrm{dis}}$ (thick solid red [gray] curve) as a function of r-mode amplitude $\alpha$ for a specific example hybrid star (see text). (b) The same quantity where the vertical axis now shows the ratio $P_{\mathrm{dis}}/{\alpha }^2$. At first $P_{\mathrm{dis}}$ is proportional to ${\alpha }^3$ at very low amplitude (dashed line), then at some intermediate amplitude varies less quickly, with a maximum in $P_{\mathrm{dis}}/{\alpha }^2$, and finally changes to ${\alpha }^2$ at higher amplitude. Also shown is gravitational radiation power $P_{\mathrm{gr}}$ (thin solid blue [gray] straight line), which is proportional to ${\alpha }^2$ at all amplitudes. The r-mode amplitude will stop growing when dissipation balances radiation, at the first point of intersection between the two curves. This defines the saturation amplitude ${\alpha }_{\mathrm{sat}}$.

Figure 8

R-mode saturation amplitude (red [gray] solid curve) and its low-amplitude analytical approximation (black dashed curve) as a function of the radius of the quark matter core ${\overline{R}}_{\mathrm{b}}$ divided by the star radius $R$ in a family of hybrid stars. For ${\overline{R}}_{\mathrm{b}}/R<{({\overline{R}}_{\mathrm{b}}/R)}_{\mathrm{crit}}\approx 0.38$, damping is too weak to saturate the r-mode. At ${\overline{R}}_{\mathrm{b}}/R\gtrsim 0.75$ the hybrid star is unstable against gravitational collapse. The mass fraction of the core is in the range $0.12\lesssim M_{\mathrm{core}}/M_{\mathrm{star}}\lesssim 0.68$ for all the configurations shown on the red solid curve.