Last orbits of binary strange quark stars

We present the first relativistic calculations of the final phase of inspiral of a binary system consisting of two stars built predominantly of strange quark matter (strange quark stars). We study the precoalescing stage within the Isenberg-Wilson-Mathews approximation of general relativity using a multidomain spectral method. A hydrodynamical treatment is performed under the assumption that the flow is either rigidly rotating or irrotational, taking into account the finite density at the stellar surface—a distinctive feature with respect to the neutron star case. The gravitational-radiation driven evolution of the binary system is approximated by a sequence of quasiequilibrium configurations at fixed baryon number and decreasing separation. We find that the innermost stable circular orbit (ISCO) is given by an orbital instability both for synchronized and irrotational systems. This contrasts with neutron stars for which the ISCO is given by the mass-shedding limit in the irrotational case. The gravitational wave frequency at the ISCO, which marks the end of the inspiral phase, is found to be $\sim 1400\mathrm{H}\mathrm{z}$ for two irrotational $1.35{\mathrm{M}}_{\odot }$ strange stars and for the MIT bag model of strange matter with massless quarks and a bag constant $B=60\mathrm{M}\mathrm{e}\mathrm{V}\mathrm{f}{\mathrm{m}}^{-3}$. Detailed comparisons with binary neutrons star models, as well as with third order post-Newtonian point-mass binaries are given.

Figure 1

Gravitational mass $M$ versus areal radius $R$ for sequences of static strange quark stars described by the simplest MIT bag model (solid line) and neutron stars described by polytropic EOS with $\gamma =2.5$ and $\kappa =0.0093m_0{n}_{\mathrm{nuc}}^{-1.5}$ (dashed line). The two sequences are crossing at the point $M=1.35{\mathrm{M}}_{\odot }$ and $R=10.677\mathrm{k}\mathrm{m}$ (marked by a circle).

Figure 2

Mass density (top panel) and the adiabatic index $\gamma$ (bottom panel) versus the areal radial coordinate $r$ for a static strange quark star (solid line) and a polytropic neutron star (dashed line), having both a gravitational mass of $1.35{\mathrm{M}}_{\odot }$ and an areal radius $R=10.677\mathrm{k}\mathrm{m}$ (resulting in the compactness parameter $M/R=0.1867$). The vertical dotted line corresponds to the stellar surface.

Figure 3

Binding energy as a function of gravitational-wave frequency along evolutionary sequences of corotating binaries. The solid line denotes strange quark stars, the dashed one neutron stars with polytropic EOS, and the dotted one point-mass binaries in the 3PN approximation [54]. The diamonds locate the minimum of the curves, corresponding to the innermost stable circular orbit; configurations to the right of the diamond are secularly unstable.

Figure 4

Binding energy (top panel) and angular momentum (bottom panel) as a function of gravitational-wave frequency along evolutionary sequences of irrotational binaries. The solid line denotes strange quark stars, the dashed line polytropic neutron stars, and the dotted line point-mass binaries in the 3PN approximation [54]. The diamonds correspond to dynamical orbital instability (the ISCO).

Figure 5

Internal velocity fields of irrotational strange quark stars binaries at the ISCO. upper panel: velocity $\mathit{U}$ in the orbital plane with respect to the “inertial” frame (Eulerian observer); lower panel: velocity field with respect to the corotating frame. The thick solid lines denote the surface of each star.

Figure 6

Coordinate “radius” (half the coordinate size of a star in fixed direction) versus coordinate separation for irrotational quasiequilibrium sequences of binary strange stars (solid line) and neutron stars (dashed line). Upper lines correspond to equatorial radius $R_x$ (radius along the x axis going through the centers of stars in a binary system) and lower lines are polar radius $R_z$ (radius along the rotation axis).

Figure 7

Binding energy along evolutionary sequences of corotational (thick dashed lines) and irrotational (thick solid lines) equal mass of $1.35{\mathrm{M}}_{\odot }$ strange stars and polytropic neutron stars binaries compared with analytical results at the 3rd post-Newtonian order for point-masses by Damour et al. 2000 [55], Damour et al. 2002 [56], and Blanchet 2002 [54].