Accretion onto a small black hole at the center of a neutron star

We revisit the system consisting of a neutron star that harbors a small, possibly primordial, black hole at its center, focusing on a nonspinning black hole embedded in a nonrotating neutron star. Extending earlier treatments, we provide an analytical treatment describing the rate of secular accretion of the neutron star matter onto the black hole, adopting the relativistic Bondi accretion formalism for stiff equations of state that we presented elsewhere. We use these accretion rates to sketch the evolution of the system analytically until the neutron star is completely consumed. We also perform numerical simulations in full general relativity for black holes with masses up to nine orders of magnitude smaller than the neutron star mass, including a simulation of the entire evolution through collapse for the largest black hole mass. We construct relativistic initial data for these simulations by generalizing the black hole puncture method to allow for the presence of matter, and evolve these data with a code that is optimally designed to resolve the vastly different length scales present in this problem. We compare our analytic and numerical results, and provide expressions for the lifetime of neutron stars harboring such endoparasitic black holes.

Figure 1

The analytical and numerical values of the accretion rate ${\dot{M}}_{\mathrm{BH}}^{\star }$ as a function of $M_{\mathrm{BH}}/M$, computed for our fiducial $\mathrm{\Gamma }=2$ neutron star model (see Table 2 for details). On the left we express the accretion rate in geometrized units, while on the right we express it in units of solar masses $M_{\odot }$ per year (since the former is dimensionless, no mass scale is needed for this conversion). The solid (blue) line represents the analytical solution (44) based on Bondi accretion for stiff EOSs, while (red) triangles and (green) circles represent numerical measurements of the accretion rate based on the growth of the black hole horizon and on the rest-mass flux, respectively (see Secs. 3c2 and 3c3). The dashed (red) line represents the minimum accretion rate (45) for a $\mathrm{\Gamma }=2$ polytrope. (See also Table 3 for a detailed listing of these results.)

Figure 2

Profiles of the initial rest-mass density ${\rho }_0$ as a function of isotropic radius $r$, for our fiducial neutron star model (see Table 2) with different black hole puncture masses $\mathcal{M}$. Here and throughout we choose $m=-6$ in (48) unless stated otherwise. Even for tiny black hole masses, the logarithmic radial variable allows us to resolve the vastly different length scales of the black hole and the neutron star.

Figure 3

The rest-mass density ${\rho }_0$ as a function of isotropic radius $r$ for different values of the conformal exponent $m$ in (48) for $\tilde{\mathcal{M}}={10}^{-3}$, so ${\tilde{M}}_{\mathrm{BH}}(0)=1.267\times {10}^{-3}$, and $M_{\mathrm{BH}}(0)/M(0)=8.03\times {10}^{-3}$. Although the initial density profiles, shown in the inset, clearly depend on $m$, they all evolve to the same density profile, shown in the large plot, once a quasiequilibrium has been reached. Here and in several of the following graphs we suppress the innermost few grid points well inside the horizon, since they are affected by numerical noise caused by the puncture singularity at the origin.

Figure 4

Growth of the black hole’s irreducible mass (59) as a function of time, for ${\tilde{M}}_{\mathrm{BH}}(0)=1.26\times {10}^{-3}$ (the solid orange line). The dashed-dotted (green) line shows a linear fit, whose slope we identify with the accretion rate ${\dot{M}}_{\mathrm{BH}}$. For comparison, we also include as the dashed (red) line the irreducible mass of a black hole with the same initial mass $M_{\mathrm{BH}}(0)$ in vacuum, i.e., without a neutron star.

Figure 5

Profiles of the rest-mass density ${\rho }_0$, lapse $\alpha$ and shift vector ${\beta }^r$ in an evolution of our fiducial neutron star with a black hole of mass ${\tilde{M}}_{\mathrm{BH}}(0)=1.267\times {10}^{-6}$ at different (coordinate) times. At the later times, all functions have settled down into a (quasi) equilibrium. Note also that the density and lapse “plateau” in a region $M_{\mathrm{BH}}\ll r\ll R$, where they take the nearly constant values ${\rho }_{0\star }\simeq {\rho }_{0c}=0.2$ and ${\alpha }_{\star }\simeq 0.623$ (marked by the horizontal dotted lines). Note also that the shift is very close to zero in this region. The black circles denote the location of the black hole horizon.

Figure 6

Profiles of the flux (70) at different instants of time, for a black hole with initial mass ${\tilde{M}}_{\mathrm{BH}}(0)=1.267\times {10}^{-10}$, $\mathcal{M}=1\times {10}^{-10}$ embedded in our fiducial neutron star model. In the outer parts of the star, the nonzero flux reflects a numerical adjustment of the star, resulting from the fact that the initial data are not in perfect equilibrium on the numerical grid. In the inner part, the flux approaches a value that becomes independent of both space and time, resulting in an equilibrium accretion flow onto the black hole. For comparison we also included, as the faint lines, profiles for an evolution of the same neutron star but without a black hole, which shows the same behavior in the outer parts of the star, but very different behavior in the vicinity of the black hole.

Figure 7

Comparison of our numerical fluid flow profiles (dashed red lines) with those obtained from integrating the Bondi equations (solid blue lines), for our fiducial neutron star model hosting a black hole of initial mass ${\tilde{M}}_{\mathrm{BH}}(0)=1.267\times {10}^{-6}$, at time $t=1.59\times {10}^3{M}_{\mathrm{BH}}$ (see text for details). The open black circles denote the location of the black hole horizon at $r=2M_{\mathrm{BH}}$, where $r$ is the areal radius, while the solid green dots mark the location of the critical radius (which, in the Newtonian limit, reduces to the location at which the fluid flow becomes supersonic).

Figure 8

The black hole mass $M_{\mathrm{BH}}$ as a function of coordinate time $t$ in a simulation leading to complete collapse. The solid line represents the numerical values of the black hole mass $M_{\mathrm{BH}}$, starting with the initial black hole mass ${\tilde{M}}_{\mathrm{BH}}(0)=0.0126$ and $M_{\mathrm{BH}}(0)/M_{\mathrm{ADM}}=0.0761$. At late times, after the black hole has consumed the entire neutron star, its mass agrees to high accuracy with the initial total gravitational mass $M_{\mathrm{ADM}}$. The dashed line shows the analytical estimate (b16) that takes into account effects of stellar evolution, while the dotted line shows the estimate (35) that ignores these effects. In this comparison we have adopted the value ${\alpha }_{\star }\simeq 0.623$, even though for these simulations the black-hole mass is too large to identify a clean “plateau” like the one shown in Fig. 5 (see text for details).

Figure 9

The numerical solution $u$ for our fiducial neutron-star model and $\tilde{\mathcal{M}}={10}^{-5}$ with $n=-6$ (dashed line), together with the approximate solution (c9) (solid line). The two vertical dotted lines mark the approximate locations of the apparent horizon and the stellar surface.