4.3 Neutron-star binaries

Neutron-star binaries, like any relativistic binary system, cannot exist in a true equilibrium configuration since they must emit gravitational radiation. But, as is true for black-hole binary data, for orbits outside the innermost stable circular orbit, the gravitational radiation reaction time scale is much longer than the orbital period and it is a reasonable approximation to consider the stars to be in a quasiequilibrium state.

A binary configuration obviously lacks the azimuthal symmetry that was assumed in the discussions of stationary solutions of Einstein’s equations in Section 2.4 and hydrostatic equilibrium in Section 4.1. Fortunately, the condition of hydrostatic equilibrium requires only the presence of a single, timelike Killing vector. With the assumption that gravitational radiation is negligible, we can assume that the matter is in some equilibrium state as viewed from the reference frame that is rotating along with the binary. That is, if the binary has a constant orbital angular velocity of Ω, then the time vector in the rotating frame t′μ is a Killing vector and it is related to the time vector in the rest frame of the binary μ t by

t′μ = tμ + Ωξμ, (122 )
where μ ξ is a generator of rotations about the rotation axis13. Bonazzola et al[18Jump To The Next Citation Point] refer to t′μ as a helicoidal Killing vector.

Two equilibrium states for the matter have been explored in the literature. The simplest case is that of co-rotation, where the 4-velocity of the matter is proportional to t′μ. In this case, the matter is at rest in the frame of reference rotating with the binary system, the corotating reference frame. The second equilibrium state is that of counter-rotation, where there is no rotation in the rest frame of the binary. We will explore these two cases further below.

Stationarity of the gravitational field, unlike hydrostatic equilibrium, requires the presence of separate timelike and azimuthal Killing vectors. For the case of a binary system, there is no unique definition of quasiequilibrium. The earliest results on constructing quasiequilibrium solutions of Einstein’s equations stem from work by Wilson and Mathews [104Jump To The Next Citation Point105Jump To The Next Citation Point106Jump To The Next Citation Point], and others have explored similar schemes [11Jump To The Next Citation Point18Jump To The Next Citation Point12Jump To The Next Citation Point]. Although written in slightly different forms, the system of equations for the gravitational fields in all of these schemes are fundamentally identical. While they were developed before the conformal thin-sandwich decomposition of Section 2.3, the conformal thin-sandwich decomposition (see Eq. (51View Equation)) offers the easiest way to interpret this approach. We consider ourselves to be in the corotating reference frame so that our time vector is ′μ t. To make the transition back to the rest frame of the binary as easy as possible, we write the shift vector of our 3 + 1 decomposition as

Bi = βi + Ω ξi, (123 )
so that i β remains as the shift vector of the 3 + 1 decomposition made with respect to the rest frame of the binary system.

The primary assumptions are that the conformal 3-metric &tidle;γij is flat, the initial-data slice is maximal so that K = 0, and &tidle;uij = 0. We see from (42View Equation) that the last assumption implies that the conformal 3-geometry is instantaneously stationary as seen in the corotating reference frame. The final choice that must be made is for the conformally rescaled lapse &tidle;α. An elliptic equation for the lapse can be obtained by demanding that the trace of the extrinsic curvature K also be instantaneously stationary in the corotating reference frame. This is the so-called maximal slicing condition on the lapse. For the particular assumptions we have made here, this equation can be written

[ 7 ] &tidle;∇2 (&tidle;αψ7 ) = (α&tidle;ψ7) -ψ −8A&tidle;ijA&tidle;ij + 2πG ψ4 (ρ + 2S ) . (124 ) 8

It is interesting to note that, for a conformally flat 3-geometry,

(&tidle;𝕃B )ij = (&tidle;𝕃 β)ij, (125 )
so Ω does not appear in the equations for the gravitational fields except in the matter terms and possibly in boundary conditions. Equations (51View Equation) and (124View Equation) can be solved for the gravitational fields on an initial-data hypersurface, given values for the matter terms and appropriate boundary conditions.

If we choose the matter so that it is in hydrostatic equilibrium with respect to the pseudo-Killing vector t′μ, then these equations for the gravitational fields will yield data that are in quasiequilibrium in the sense that &tidle;γij and K are both instantaneously stationary.

For corotating binaries, the matter is at rest in the corotating reference frame of the binary. It is rigidly rotating and hydrostatic equilibrium is specified by solving the relativistic Bernoulli equation (118View Equation), with dΩ = 0, self-consistently with the equations for the gravitational fields.

For counterrotating binaries, the matter is not rotating in the rest frame of the binary. Counterrotating equilibrium configurations can be obtained by assuming the matter to have irrotational flow [99Jump To The Next Citation Point92Jump To The Next Citation Point]. As long as the flow is isentropic, we can express the enthalpy (117View Equation) as14

𝜀 +-P- h = ρ0 , (126 )
where ρ0 is the rest-mass density. For irrotational flow, the vorticity of the fluid (cf. Ref. [99Jump To The Next Citation Point]) is zero. Combining this with the Euler equation, we find that the 4-velocity of the fluid can be expressed as
hu μ = ∇ μφ, (127 )
where φ is the velocity potential, or flow field. Equation (127View Equation), together with the normalization condition μ u uμ = − 1, automatically satisfies the Euler equation and we are left with the continuity equation which must be satisfied,
∇ μ(ρ0uμ) = 0. (128 )
The continuity equation (128View Equation), the normalization condition, and Eq. (127View Equation) yield
∘ -------------- ∇2φ = − (∇ μφ)∇ μln(ρ0∕h) with h = − (∇ νφ )(∇νφ ). (129 )

Stationarity (or quasistationarity) in the corotating reference frame requires that

hu t′μ = − C, (130 ) μ
where C is a positive constant. Now, in terms of the 3 + 1 decomposition with Bi as the shift vector (see Eq. (123View Equation)), we find that the Bernoulli equation is written as
h2 = − (¯∇i φ)¯∇i φ + 12 (C + Bj ¯∇jφ )2. (131 ) α
The flow field φ must satisfy
( ) ( ) ( ) ∇¯2 φ − Bi ¯∇ λ-- − λ-K = − ¯∇iφ − λ-Bi ¯∇ ln αρ0- , i α2 α α2 i h (132 ) λ ≡ C + Bi ¯∇ φ, i
subject to the boundary condition15 at the surface of the flow,
( ) | ¯ i λ-- i ¯ || ∇ φ − α2B ∇iρ0| = 0. (133 ) surf
Solving Eqs. (131View Equation) and (132View Equation) self-consistently with the equations for the gravitational fields yields a counterrotating binary in hydrostatic equilibrium.

As mentioned above, the earliest work on neutron-star binaries was carried out by Wilson and Mathews [104105]. Wilson et al[106] describe their approach for generating initial data for equilibrium neutron-star binaries. In these early works, the equation of hydrostatic equilibrium was not used. Rather, an initial guess for the density profile was chosen and the full hydrodynamic system was evolved with viscous damping until equilibrium was reached. During each step of the hydrodynamic evolution, the equations for the gravitational fields were resolved. The resulting data represented neither strictly co- nor counter-rotating binary neutron stars. This work led to the controversial result [76] that each neutron star in the binary may become radially unstable and collapse prior to the merger of the pair of stars. While an error was found in this work [4977] with the result that the signature of collapse is significantly weaker, the controversy has not yet been completely resolved.

The first use of corotating hydrostatic equilibrium with the Wilson–Mathews approach for specifying the gravitational fields was by Cook et al[46] for the test case of an isolated neutron star. This approach was then used to study corotating neutron-star binaries by Baumgarte et al[1112] and by Marronetti et al[72]. Interestingly, turning-point methods for detecting secular instabilities [959653] can be applied to the case of corotating binaries [13].

Corotating binary configurations are relatively easy to construct. However, it is believed that the viscosity of neutron-star matter is not large enough to allow for synchronization of the spin with the orbit [6214]. But if the initial spins of the neutron stars are not too large, close binaries should be well approximated by irrotational models. Bonazzola et al[18] (as corrected by Asada [7]) developed the first approach for constructing counterrotating binary configurations. However, simpler formulations of irrotational flow were developed independently by Teukolsky [99] and Shibata [92], and Gourgoulhon [55] showed that all three approaches were equivalent. Numerical solutions of the equations for irrotational flow coupled to the equations for the gravitational fields are more difficult to construct than those for corotation because of the boundary condition (133View Equation) on the flow field that must be applied on the surface of each neutron star. This boundary condition is particularly difficult to implement because the location of the surface of the star is not known a priori, and will move as the equations are being solved. The first models of irrotational binary neutron stars were constructed by Bonazzola et al[20], Marronetti et al[73], and UryÅ« and collaborators [101102]. A description of the numerical methods used by Bonazzola et al. can be found in Refs. [5758].

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