5.3 Microphysical numerical techniques

5.3.1 Neutron star physics and equations of state

One of the largest uncertainties in the input physics of NS-NS merger simulations is the true behavior of the nuclear matter EOS. To date, EM observations have yielded relatively weak constraints on the NS mass-radius relation, with the most precise simultaneous measurement of both as of now resulting from observations of Type 1 X-ray bursts from accreting NSs in three different sources [220]. In each case, the NS mass was found to lie in the range 1.3M ⊙ ≲ MNS ≲ 2M ⊙ and the radius 8 km ≲ RNS ≲ 12 km, implying a NS compactness

( ) ( ) −1 𝒞 ≡ GMNS---= 0.1476 MNS-- -RNS--- ≈ 0.16– 0.37. (28 ) RNSc2 M ⊙ 10 km
A more stringent constraint on the NS EOS is provided by observations of the Shapiro time delay in the binary millisecond pulsar PSR J1614–2230, which was found to have a mass MNS = 1.97 ± 0.04M ⊙ [81Jump To The Next Citation Point], which would rule out extremely soft EOS models incapable of supporting such a massive NS against collapse. As we discuss in more detail below, GW observations are likely to eventually yield tighter constraints than our current EM-based ones, though BH-NS mergers, which can undergo stronger tidal disruptions than NS-NS mergers at frequencies closer to LIGO and other GW observatories’ maximum frequency sensitivity band, may prove to be more useful for the task than NS-NS mergers.

Given the large theoretical uncertainties in describing the proper physical NS EOS, many groups have chosen the simplest possible parameterization: a polytrope (see Eq. 12View Equation). Under this choice, the enthalpy h takes the particularly simple form

h ≡ 1 + 𝜀 + P ∕ρ = 1 + Γ 𝜀. (29 )
Initial data are generally assumed to follow the relation
Γ P = K ρ , (30 )
where K is constant across the fluid. In the presence of shocks, the value of K for a particular fluid element will increase with time. We note that the Whisky group [17Jump To The Next Citation Point, 18Jump To The Next Citation Point] uses the term “polytropic” to refer to simulations in which Eq. 30View Equation is enforced throughout, which implies adiabatic evolution without shock heating, and use the term “ideal fluid” to describe an EOS that includes the effects of shock heating and enforces Eq. 29View Equation.

Since the temperatures of NSs typically yield thermal energies per baryon substantially below the Fermi energy, one may treat nearly all NSs as effectively cold, except for the most recently born ones. During the merger process for NS-NS binaries, the matter will remain cold until the two NSs are tidally disrupted and a disk forms, at which point the thermal energy input and substantially reduced fluid densities require a temperature evolution model to properly model the underlying physics. In light of these results, some groups adopt a two-phase model for the NS EOS (see, e.g., [286Jump To The Next Citation Point]), where a cold, zero-temperature EOS, evaluated as a function of the density only, encodes as much information about as we possess about the NS EOS, and the hot phase depends on both the density and internal energy, typically in a polytropic way,

P(ρ, 𝜀) = Pcold(ρ) + Phot(ρ,𝜀) [Phot(ρ,𝜀) = (Γ − 1)ρ(𝜀 − 𝜀cold)]. (31 )

There are a number of physically motivated EOS models that have been implemented for merger simulations, whose exact properties vary depending on the assumptions of the underlying model. These include models for which the pressure is tabulated as a function of the density only: FPS [222], SLy [83Jump To The Next Citation Point], and APR [3Jump To The Next Citation Point]; as well as models including a temperature dependence: Shen [268Jump To The Next Citation Point, 267Jump To The Next Citation Point] and Lattimer–Swesty [162]. A variety of models have been used to study the effects of quarks, kaons, and other condensates, which typically serve to soften the EOS, leading to reduced maximum masses and more compact NSs [223, 119, 230, 120, 23, 5].

Given the variance among even the physically motivated EOS models, it has proven useful to parameterize known EOS models with a much more restricted set of parameters. In a series of works, a Milwaukee/Tokyo collaboration determined that essentially all current EOS models could be fit using four parameters, so that their imprint on GW signal properties could be easily analyzed [237, 238, 184]. Their method assumes that the SLy EOS describes NS matter at low densities, and that the EOS at higher densities can be described by a piecewise polytropic fit with breaks at ρ = 1014.7 and 1015 g ∕cm3. The four resulting parameters are P = P (ρ = 1014.7) 1, the pressure at the first breakpoint density, which normalizes the overall density scale, as well as Γ 1,Γ 2,Γ 3, the adiabatic exponents in the three regions. Their results indicate that advanced LIGO should be able to determine the NS radius to approximately 1 km at an effective distance of 100 Mpc, which would place tight constraints on the value of P1 in particular.


  Go to previous page Go up Go to next page