Are neutron stars crushed? Gravitomagnetic tidal fields as a mechanism for binary-induced collapse

Numerical simulations of binary neutron stars by Wilson, Mathews, and Marronetti indicated that neutron stars that are stable in isolation can be made to collapse to black holes when placed in a binary. This claim was surprising as it ran counter to the Newtonian expectation that a neutron star in a binary should be more stable, not less. After correcting an error found by Flanagan, Wilson and Mathews found that the compression of the neutron stars was significantly reduced but not eliminated. This has motivated us to ask the following general question: Under what circumstances can general-relativistic tidal interactions cause an otherwise stable neutron star to be compressed? We have found that if a nonrotating neutron star possesses a current-quadrupole moment, interactions with a gravitomagnetic tidal field can lead to a compressive force on the star. If this current quadrupole is induced by the gravitomagnetic tidal field, it is related to the tidal field by an equation-of-state-dependent constant called the gravitomagnetic Love number. This is analogous to the Newtonian Love number that relates the strength of a Newtonian tidal field to the induced mass quadrupole moment of a star. The compressive force is almost never larger than the Newtonian tidal interaction that stabilizes the neutron star against collapse. In the case in which a current quadrupole is already present in the star (perhaps as an artifact of a numerical simulation), the compressive force can exceed the stabilizing one, leading to a net increase in the central density of the star. This increase is small ($\lesssim 1\%$) but could, in principle, cause gravitational collapse in a star that is close to its maximum mass. This paper also reviews the history of the Wilson-Mathews-Marronetti controversy and, in an appendix, extends the discussion of tidally induced changes in the central density to rotating stars.

Figure 1

Internal velocity field $\mathit{v}$ of a nonrotating neutron star with a current-quadrupole moment induced by the tidal field of an orbiting companion. The arrows denote the velocity vectors and are generated by Eqs. (3.11, 3.35). The velocity field is stationary in a coordinate frame rotating at the binary’s orbital angular velocity (which points perpendicular to the equatorial plane of the star).

Figure 2

Same as Fig. 1 but showing only the induced velocity field on a slice through the equatorial plane. The velocity current loops set up in the star are easily seen.

Figure 3

The total gravitomagnetic tidal acceleration of a nonrotating neutron star interacting with a binary companion. Only a slice through the equatorial plane is shown. The arrows represent the acceleration vectors in an inertial reference frame centered on the star’s center of mass and are given by Eqs. (3.18, 3.35). Their magnitude vanishes at the center of the star. As in Figs. 1, 2 the acceleration field rotates at the binary’s orbital angular velocity. The radially inward pointing acceleration indicates that the gravitomagnetic tidal interaction causes compression.

Figure 4

Change in central density for the revised Wilson-Mathews simulations as a function of proper distance between the neutron star centers. The lower set of points (Table III of 14) corresponds to stars that are 12% from their maximum mass (in isolation). These stars were compressed, but did not collapse before the last stable orbit. A fit to these points (dashed, blue curve) indicates the scaling $\delta {\rho }_c/{\rho }_c\propto {\alpha }^{2.0}$. The upper set of points (Table IV of 14) corresponds to stars that are 8.6% from their maximum mass (in isolation). In this case the stars did collapse (not shown here) and the orbit remained stable. A fit to these points (solid, red curve) indicates the scaling $\delta {\rho }_c/{\rho }_c\propto {\alpha }^{1.4}$. In both cases, a “realistic” equation of state was used (see 10, 14).

Figure 5

Critical orbital separation for compression to overwhelm stabilization. The solid (blue) curve shows, as a function of mass ratio, the critical orbital separation $d_{\mathrm{crit}}$ (in units of the NS radius) where the compression and stabilization terms in Eq. (3.37) are equal. The short-dashed (green) curve shows the tidal disruption limit $d_{\mathrm{tidal}}/R$. The dashed-dotted (red) curve shows the ISCO $d_{\mathrm{isco}}/R$ while the long-dashed (black) curve shows the event horizon $d_{\mathrm{horizon}}/R$ of a nonspinning black hole with mass $M^{\prime }$. This plot shows that in an inspiralling binary, either the tidal disruption limit, the ISCO, or the event horizon is reached before the critical separation where compression dominates. This is also true for rapidly spinning black holes: as the spin parameter $a/M^{\prime }$ varies, the lines corresponding to the ISCO and horizon would shift up or down, but they would always remain above the curve for $d_{\mathrm{crit}}$. These curves assume a neutron star with $M=1.4M_{\odot }$, $R=10\mathrm{km}$, and a $\Gamma =2$ polytropic equation of state.

Figure 6

Change in central density of a neutron star with a preexisting velocity field [Eq. (4.6) with the cosine term set to $+1$]. The size of the current quadrupole is fixed at $V_{\mathcal{S}}=0.1(M/R{)}^{1/2}$. All curves are for a NS with mass $M=1.4M_{\odot }$ and radius $R=10\mathrm{km}$. From left to right, the first two curves are for a NS companion with $M^{\prime }/M=1$ (blue, solid), 3 (red, short-dashed) and are terminated at the tidal disruption radius. The next two (black) curves are for a black hole companion with $M^{\prime }/M=5$ (dot-dashed), 10 (long-dashed) and are terminated at the innermost stable circular orbit. For the value of $V_{\mathcal{S}}$ used here, compression dominates over stabilization before tidal disruption or orbit instability occurs.

Figure 7

Same as Fig. 6 except that the size of the current-quadrupole moment is smaller, $V_{\mathcal{S}}=0.01(M/R{)}^{1/2}$. The compression is much smaller in this case and, for three of the stars, is eventually dominated by the tidal stabilization. For the $M^{\prime }/M=3$ binary the neutron star central density decreases by $\lesssim 0.5\%$ before tidal disruption.