4.4 Quasi-equilibrium and pre-merger simulations

NS-NS binaries may be well approximated by QE configurations up until they reach separations comparable to the sizes of the binary components themselves, that latter phase lasting a fraction of a second after an inspiral of millions of years or more. The eventual merger will occur after the binary undergoes one of two possible orbital instabilities. If the total binary energy and angular momenta reach a minimum at some separation, which defines the ISCO, the binary becomes dynamically unstable and plunges toward merger. Alternately, if the NS fills its Roche lobe (typically the lower density NS) mass will transfer onto the primary and the secondary will be tidally disrupted. The parameters of some NS-NS systems could technically allow for stable mass transfer, in which mass loss from a lighter object to a heavier one leads to a widening of the binary separation. This does occur for some binaries containing white dwarfs, but every dynamical calculation to date using full GR or even approximate GR has found that the rapid inspiral rate leads to inevitably unstable mass transfer and the prompt merger of a binary.

Many of the results later confirmed using relativistic QE sequences were originally derived in Newtonian and PN calculations, particularly as explicit extensions of Chandrasekhar’s body of work (see [65]). Chandrasekhar’s studies of incompressible fluids were first extended to compressible binaries by Lai, Rasio, and Shapiro [156, 155Jump To The Next Citation Point, 158Jump To The Next Citation Point, 157Jump To The Next Citation Point, 159Jump To The Next Citation Point], who used an energy variational method with an ellipsoidal treatment for polytropic NSs. They established, among other results, the magnitude of the rapid inspiral velocity near the dynamical stability limit [156], the existence of a critical polytropic index (n ≈ 2) separating binary sequences undergoing the two different terminal instabilities [155Jump To The Next Citation Point], the role played by the NS spin and viscosity and magnitude of finite-size effects in relation to 1PN terms [158Jump To The Next Citation Point, 157Jump To The Next Citation Point], and the development of tidal lag angles as the binary approaches merger [159Jump To The Next Citation Point]. They also determined that for most reasonable EOS models and non-extreme mass ratios, as would pertain to NS-NS mergers, an energy minimum is inevitably reached before the onset of mass transfer through Roche lobe overflow. The general results found in those works were later confirmed by [201Jump To The Next Citation Point], who used a SCF technique [131, 132], finding similar locations for instability points as a function of the adiabatic index of polytropes, but a small positive offset in the radius at which instability occurred. Similar results were also found by [311, 312Jump To The Next Citation Point], but with a slight modification in the total system energy and decrease in the orbital frequency at the onset of instability.

The first PN ellipsoidal treatments were developed by Shibata and collaborators using self-consistent fields [270Jump To The Next Citation Point, 269, 279Jump To The Next Citation Point, 281Jump To The Next Citation Point, 299] and by Lombardi, Rasio, and Shapiro [177Jump To The Next Citation Point]. Both groups found that the nonlinear gravitational effects imply a decrease in the orbital separation (increase in the orbital frequency) at the instability point for more compact NS. This result reflects a fairly universal principle in relativistic binary simulations: as gravitational formalisms incorporate more relativistic effects, moving from Newtonian gravity to 1PN and on to CF approximations and finally full GR, the strength of the gravitational interaction inevitably becomes stronger. The effects seen in fully dynamical calculations will be discussed in Section 6, below.

The first fully relativistic CTS QE data for synchronized NS-NS binaries were constructed by Baumgarte et al. [26, 25Jump To The Next Citation Point], using a grid-based elliptic solver. Their results demonstrated that the maximum allowed mass of NSs in close binaries was larger than that of isolated NSs with the same (polytropic) EOS, clearly disfavoring the “star-crushing” scenario that had been suggested by [327Jump To The Next Citation Point, 187Jump To The Next Citation Point] using a similar CTS formalism (but see also the error in these latter works addressed in [104Jump To The Next Citation Point], discussed in Section 6.3 below). Baumgarte et al. also identified how varying the NS radius affects the ISCO frequency, and thus might be constrained by GW observations. Using a multigrid method, Miller et al. [192Jump To The Next Citation Point] showed that while conformal flatness remained valid until relatively near the ISCO, the assumption of syncronized rotation broke down much earlier. Usui et al. [319Jump To The Next Citation Point] used the Green’s function approach with a slightly different formalism to compute relativistic sequences and determined that the CTS conditions were valid up until extremely relativistic binaries were considered.

The first relativistic models of physically realistic irrotational NS-NS binaries were constructed by the Meudon group [51Jump To The Next Citation Point] using the Lorene multi-domain pseudo-spectral method code. Since then, the Meudon group and collaborators have constructed a wide array of NS-NS initial data, including polytropic NS models [125, 303Jump To The Next Citation Point, 304Jump To The Next Citation Point], as well as physically motivated NS EOS models [36Jump To The Next Citation Point] or quark matter condensates [170Jump To The Next Citation Point]. Irrotational models have also been constructed by UryÅ« and collaborators [313Jump To The Next Citation Point, 317Jump To The Next Citation Point] for use in dynamical calculations, and nuclear/quark matter configurations have been generated by Oechslin and collaborators [212Jump To The Next Citation Point, 209Jump To The Next Citation Point]. A large compilation of QE CTS sequences constructed using physically motivated EOS models including FPS (Friedman–Pandharipande) [222], SLy (Skyrme Lyon) [83Jump To The Next Citation Point], and APR [3Jump To The Next Citation Point] models, along with piecewise polytropes designed to model more general potential cases (see [237Jump To The Next Citation Point]), was published in [305Jump To The Next Citation Point].

The most extensive set of results calculated using the waveless/near-zone helical symmetry condition appear in [316Jump To The Next Citation Point], with equal-mass NS-NS binary models constructed for the FPS, Sly, and APR EOS in addition to Γ = 3 polytropes. Results spanning all of these QE techniques are summarized in Table 2.

Table 2: A summary of various studies focusing on QE sequences of NS-NS binaries. Please refer to Section 6 for a discussion of papers that focus on dynamical simulations instead. Gravitational schemes include Newtonian gravity (‘Newt.’), lowest-order post-Newtonian theory (‘PN’), conformal thin sandwich (‘CTS’) including modified forms of the spatial metric (‘Mod. CTS’), and waveless/near-zone helical symmetry techniques. Numerical methods include ellipsoidal formalisms (‘Ellips.’), self-consistent fields (‘SCF’), numerical grids (‘Grid’), multigrids, and multipatch, Green’s function techniques (‘Green’s’), spectral methods (‘Spectral’), or SPH relaxation (‘SPH’). With regard to EOS models, ‘WD’ refers to the exact white dwarf EOS assuming a cold degenerate electron gas [64]. The ‘Physical’ EOS models include the FPS [222], SLy [83Jump To The Next Citation Point], and APR [3Jump To The Next Citation Point] nuclear EOS models, along with their parameterized approximations and other physically motivated models. The compactness 𝒞 = M ∕R refers to the value for a NS in isolation before it is placed in a binary, and plays no role in Newtonian physics. The mass ratio q = M2 ∕M1 is defined to be less than unity, and ‘spin’ refers to either synchronized or irrotational configurations.
Author Ref. Grav. Method EOS Compactness Mass ratio Spin
Lai [155] Newt. Ellips. 7 5 Γ = 5,3,2,∞ N/A 1.0 Syn.
Lai [158] Newt. Ellips. 5 Γ = 3,2,3 N/A 1.0 Syn./Irr.
Lai [157] Newt. Ellips. 7 5 Γ = 5,3,3,∞ N/A 0.2 – 1.0 Syn./Irr.
New [201Jump To The Next Citation Point] Newt. SCF Γ = 53,2,3,WD N/A 1.0 Syn.
UryÅ« [312] Newt. SCF Γ = 53,2, 177 ,2,3,∞ N/A 1.0 Irr.
Shibata [270] PN Grid Γ = 2 𝒞 = 0.08 –0.12 1.0 Syn.
Shibata [279] PN Grid Γ = 3 𝒞 = 0.03 1.0 Syn.
Shibata [281] PN Ellips. Γ = 5,2, 7,3,5,∞ 3 3 𝒞 = 0–0.03 1.0 Syn.
Lombardi [177] PN Ellips. Γ = 2,3 𝒞 = 0.12 –0.25 1.0 Syn./Irr.
Baumgarte [25] CTS Multigrid Γ = 2 𝒞 = 0.05 –0.2 1.0 Syn.
Usui [319] Mod. CTS Green’s Γ = 2,3,∞ 𝒞 = 0.05 –0.25 1.0 Syn.
UryÅ« [313Jump To The Next Citation Point] CTS Green’s Γ = 2 𝒞 = 0.1 –0.19 1.0 Syn./Irr.
UryÅ« [317] CTS Green’s 9 Γ = 5,2,2.25,2.5,3 𝒞 = 0.1 –0.19 1.0 Irr.
Bonazzola [51] CTS Spectral Γ = 2 𝒞 = 0.14 1.0 Syn/Irr.
Taniguchi [303Jump To The Next Citation Point] CTS Spectral Γ = 2 𝒞 = 0.12 –0.18 0.9 – 1.0 Syn./Irr.
Taniguchi [304] CTS Spectral Γ = 1.8,2.25,2.5 𝒞 = 0.08 –0.18 0.83 – 1.0 Syn./Irr.
Miller [192Jump To The Next Citation Point] CTS Multigrid Γ = 2 𝒞 = 0.15 1.0 Syn.
Bejger [36] CTS Spectral Phys. 𝒞 = 0.14 –0.19 1.0 Irr.
Limousin [170] CTS Spectral Quark 𝒞 = 0.19 1.0 Syn./Irr.
Oechslin [212Jump To The Next Citation Point] CTS SPH Quark 𝒞 = 0.12 –0.20 1.0 Irr.
Taniguchi [305] CTS Spectral Physical 𝒞 = 0.1 –0.3 0.7 – 1.0 Irr.
UryÅ« [316] WL/NHS Multipatch Γ = 3, Physical 𝒞 = 0.13 –0.22 1.0 Irr.

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