From Neutron Star Observables to the Equation of State. I. An Optimal Parametrization
DDraft version August 27, 2018
Preprint typeset using L A TEX style emulateapj v. 5/2/11
FROM NEUTRON STAR OBSERVABLES TO THE EQUATION OF STATE.I. AN OPTIMAL PARAMETRIZATION
Carolyn A. Raithel, Feryal ¨Ozel, & Dimitrios Psaltis
Department of Astronomy and Steward Observatory, University of Arizona, 933 N. Cherry Avenue, Tucson, Arizona 85721, USA
Draft version August 27, 2018
ABSTRACTThe increasing number and precision of measurements of neutron star masses, radii, and, in thenear future, moments of inertia offer the possibility of precisely determining the neutron star equationof state. One way to facilitate the mapping of observables to the equation of state is through aparametrization of the latter. We present here a generic method for optimizing the parametrizationof any physically allowed EoS. We use mock equations of state that incorporate physically diverse andextreme behavior to test how well our parametrization reproduces the global properties of the stars,by minimizing the errors in the observables mass, radius, and the moment of inertia. We find thatusing piecewise polytropes and sampling the EoS with five fiducial densities between ∼ − M (cid:12) and the moment of inertia of a 1.338 M (cid:12) neutron star to within less than 10% for 95% ofthe proposed sample of equations of state. INTRODUCTION
A key goal of observations of neutron stars is to de-termine the equation of state of cold, ultradense matter.The equation of state at high densities results from theinteractions of nucleons, quarks, and possibly other con-stituents and fully characterizes the microphysics of theneutron star interior. Over the years, the sample of pro-posed equations of state (EoS) have incorporated vastlydifferent physics and, accordingly, have predicted a va-riety of different global properties for neutron stars (see¨Ozel & Freire 2016 for a recent review). Some early stud-ies employed a purely nucleonic framework (e.g., Baymet al. 1971; Friedman & Pandharipande 1981; Akmalet al. 1998; Douchin & Haensel 2001), while others ex-plored the role of hyperons (e.g., Balberg & Gal 1997),pion condensates (e.g., Pandharipande & Smith 1975),and kaon condensates (e.g., Kaplan & Nelson 1986). Itis even possible, albeit contrived, to construct EoS thatgenerate parallel stable branches of neutron stars withthe same masses but different radii by introducing afirst-order phase transition into the EoS of hybrid neu-tron stars (Glendenning & Kettner 2000; Blaschke et al.2013). More recently, there have been studies that in-corporate the expected quark degrees of freedom at highdensities, either from phenomenological models or deriv-ing from early results of lattice QCD (e.g., Alford et al.2005, 2013; Kojo et al. 2015).Nuclear physics experiments are able to constrain thesevarious EoS only up to densities near the nuclear satu-ration density, ρ sat ∼ . × g cm − (see Lattimer2012). In this regime, there have been attempts to con-vert experimental results more directly into physical con-straints. Even though, at these densities, the interactionsbetween particles can be formally described in termsof static, few-body potentials (see, e.g., Akmal et al.1998; Morales et al. 2002; Gandolfi et al. 2012), there are large uncertainties arising from extrapolations to β -equilibrium and low-temperature matter. Furthermore,at densities well above ρ sat , the interactions between par-ticles can no longer be expanded in these terms.At densities beyond ρ sat , experimentally constrainingthe dense-matter EoS can be accomplished using obser-vations of neutron stars and exploiting the direct map-ping between the EoS and the mass-radius relation (Lind-blom 1992). This can be achieved by comparing mea-sured masses (Demorest et al. 2010; Antoniadis et al.2013) and radii (Guillot et al. 2013; Guillot & Rutledge2014; Heinke et al. 2014; N¨attil¨a et al. 2015; ¨Ozel et al.2016; Bogdanov et al. 2016; see ¨Ozel & Freire 2016 for arecent review) to the predictions of particular EoS. How-ever, this approach is limited in scope, as it serves to con-strain the parameters of already formulated EoS frame-works and may not necessarily span the full range of pos-sibilities. In other words, it may not be capable of recre-ating the EoS from observations in a model-independentway.A separate approach is to empirically infer the EoSfrom the combination of these radii and mass measure-ments by utilizing a model-independent parametrizationof the EoS. To date, several parametrizations have beenproposed. The EoS can be written as a spectral expan-sion in terms of the enthalpy (Lindblom & Indik 2012,2014). Alternatively, the EoS can be represented as a dis-crete number of segments that are piecewise polytropicor linear (Read et al. 2009a; ¨Ozel & Psaltis 2009; Hebeleret al. 2010; Steiner et al. 2016).Despite these various proposals, only limited system-atic optimizations have been undertaken to determinethe ideal number of segments and functional forms fora parametric EoS. Moreover, these earlier studies usedan existing sample of theoretical EoS as benchmarks andsought to represent only this subset. Finally, many of a r X i v : . [ a s t r o - ph . H E ] D ec these early studies only sought to reproduce the pre-dicted radii to rather large uncertainties by today’s stan-dards, reflecting the available data at the time. However,the current sample of mass and radius measurements, aswell as the anticipated moment of inertia measurementfrom the binary system PSR J0737 − (cid:46) (cid:46) (cid:46) M (cid:12) and the moment of inertia towithin (cid:46)
10% for 95% of the proposed EoS sample. OPTIMIZING THE PARAMETRIC EOS
We parametrize the EoS in terms of n piecewise poly-tropes, spaced between two densities, ρ min and ρ max .We define the dividing density and pressure between eachpiecewise polytrope to be ρ i and P i , respectively. Eachpolytropic segment is then given by P = K i ρ Γ i ( ρ i − ≤ ρ ≤ ρ i ) , (1)where the constant, K i , is determined from the pressureand density at the previous fiducial point according to K i = P i − ρ Γ i i − = P i ρ Γ i i (2)and the polytropic index for the segment, Γ i , is given byΓ i = log ( P i /P i − )log ( ρ i /ρ i − ) . (3)Figure 1 shows an example of piecewise polytropes overthree density segments, with various values of their poly-tropic indices, Γ, to illustrate the behavior of equa-tions (1)-(3). Our primary goal in optimizing theparametrization is to reduce the errors in the predictionof observables (i.e., mass, radius, and moment of inertia)below a threshold that is comparable to the uncertaintiesin present or upcoming observations, while keeping thenumber of polytropic segments to a minimum. For theoptimization process, we produce extreme, albeit phys-ically allowed EoS between 1 − ρ sat to test how well See the Appendix for a discussion of a linearly parametrizedEoS.
Fig. 1.—
Pressure as a function of density for a sample of piece-wise polytropes. The equation of state is divided into polytropicsegments at three fiducial densities that are uniformly spaced inthe logarithm. In each segment, we allow the polytrope in equa-tion (1) to have an index of Γ = 0, 1, or 2 to illustrate their generalbehavior. our parametrizations reproduce observables with vari-ous numbers of polytropic segments included. Once ourparametrization is optimized, we then apply it to morereasonable, physically motivated EoS to test its abilityto recreate those as well.In addition to the number of polytropes to includein the parametrization, there are two other variablesthat we have to optimize: the density at which theparametrization should start and the spacing of thepolytropic segments. For the question of where tostart the parametrization, we explored starting at ρ =10 g cm − as well as at ρ = ρ sat . It is typically as-sumed that the EoS is known up to ρ sat ; however, Lat-timer & Prakash (2001) showed that for a sample ofaround 30 proposed EoS, the predicted pressures varyby a factor of 5 over the range 0 . ρ sat < ρ < ρ sat ,even though these EoS are all meant to be consistentwith nuclear physics experiments in this density regime.This is because the extrapolation of pressures from sym-metric nuclear matter to neutron-rich matter is poorlyconstrained. Meanwhile, densities below ∼ . ρ sat donot significantly affect the global properties of the star.Therefore, we allowed our parametrization to start atboth 0 . ρ sat and ρ sat , in order to explore these two lim-its. As for the question of how to space the polytropicsegments, the preferred option is to space the segmentsevenly in the logarithm of the density. A logarithmicspacing more finely samples the low density region of theEoS, which is the region that most affects the result-ing neutron star observables (Lattimer & Prakash 2001;Read et al. 2009a; ¨Ozel & Psaltis 2009). For complete-ness, we also explored a second possibility: spacing thefiducial densities between each polytropic segment lin-early.We found that the combination of starting theparametrization at ρ sat and spacing the fiducial densitiesevenly in the logarithm resulted in the smallest errors inmass and radius. We therefore start the first polytropeat ρ = ρ sat and space the remaining fiducial densitiesevenly in the logarithm between ρ and 7.4 ρ sat . We setthe last point, ρ n = 7 . ρ sat , following the results of Readet al. (2009a) and ¨Ozel & Psaltis (2009) who found thatthe pressure at this density determines the neutron starmaximum mass and that pressures at higher densities donot significantly affect the overall shape of the resultingmass-radius curve. We determined the pressure corre-sponding to each fiducial density by sampling whicheverEoS we were parametrizing, i.e., P i = P EoS ( ρ i ). For ρ ≤ ρ , we connected our parametrization to a low-density EoS.We varied the total number of fiducial densities above ρ sat from 3 to 12. Clearly, as the number of polytropesused to represent the EoS increases, the errors in the ob-servables are expected to reduce. Our goal in the remain-der of this paper is to determine the minimum number offiducial densities required to reproduce the mass, radius,and moment of inertia of a neutron star to within desiredobservational uncertainties. FROM EOS TO OBSERVABLES
In order to determine how well our parametrizationswere able to reproduce the observations predicted bya given EoS, we used the Tolman-Oppenheimer-Volkoff(TOV) equations and solved them to find the mass, ra-dius, and moment of inertia.The TOV equations give the pressure, P , and the en-closed mass, M , of the star as a function of radius, ac-cording to d P d r = − Gc ( (cid:15) + P )( M + 4 πr P/c ) r − GM r/c (4)and d M d r = 4 πr (cid:15)c , (5)where the energy density, (cid:15) , is given by d (cid:15)ρ = − P d ρ . (6)To get the full relation between energy density andmass density, we can integrate equation (6) for Γ (cid:54) = 1 to (cid:15) ( ρ ) = (1 + a ) ρc + K Γ − ρ Γ , (7)where a is an integration constant. Along any densitysection of the EoS, requiring continuity at either end-point determines a such that equation (7) becomes (cid:15) ( ρ ) = (cid:20) (cid:15) ( ρ i − ) ρ i − − P i − ρ i − (Γ i − (cid:21) ρ + K i Γ i − ρ Γ i , ( ρ i − ≤ ρ ≤ ρ i ) (8)where K i and Γ i are determined as in equations (2) and(3).Similarly, for the case of Γ=1, equation (6) becomes (cid:15) ( ρ ) = (cid:15) ( ρ i − ) ρ i − ρ + K i ln (cid:18) ρ i − (cid:19) ρ − K i ln (cid:18) ρ (cid:19) ρρ i − ≤ ρ ≤ ρ i . (9)We used equations (8) or (9) to relate an EoS to theenergy density, and then used that energy density to in-tegrate the TOV equations outwards from the center ofthe star. The radius at which the pressure becomes neg-ligible gives the total mass and radius of the star.In order to calculate the moment of inertia, we simul-taneously solved equations (4) and (5) with two coupled differential equations for the relativistic moment of iner-tia, d I d r = 8 π (cid:15) + P ) c f jr − GMrc , (10)and dd r (cid:18) r j d f d r (cid:19) + 4 r d j d r f = 0 , (11)where f ( r ) ≡ − ω ( r )Ω , j ≡ e − ν/ (1 − GM/rc ) / , ω ( r )is the rotational frequency of the local inertial frame atradius r , and Ω is the spin frequency of the star. Theboundary conditions for the second-order partial differ-ential equation (11) are (cid:20) dfdr (cid:21) r =0 = 0 (12)and f ( r = R NS ) = 1 − Gc IR . (13)To solve these coupled equations, we integrated equa-tions (10) and (11) outwards from the center of the star,using equation (12) as one boundary condition, and iter-ated it to find the value of f for which equation (13) isvalid.In this way, we determined the mass, radius, and mo-ment of inertia for a given equation of state. GENERATING MOCK EQUATIONS OF STATE
In order to be as general as possible and go beyondthe current sample of proposed equations of state, wetested our parametrization on a sample of mock EoS thatincorporated extreme but physically allowed behavior,with the hypothesis that if our parametrization couldaccurately capture these extreme cases, then it would beable to reproduce more reasonable EoS as well.
Fig. 2.—
Cumulative distribution of polytropic indices, Γ, cal-culated at all tabulated densities in a given range in a sample of49 proposed EoS. We found that the majority of polytropic indiceslie between Γ ∼ ∼ For our mock EoS, we assumed the true EoS of neu-tron stars to be well known up to the nuclear saturationdensity. We then divided the high-density regime, from1-8 ρ sat , into 15 segments that were evenly-spaced in thelogarithm of the density, so that each segment would besmall relative to the overall range of densities. The log-arithmic sampling also sampled the lower density regionbetter and thus allowed more variability in the regionthat most affects the global properties of the star (Lat-timer & Prakash 2001; Read et al. 2009a; ¨Ozel & Psaltis2009). In each of the 15 segments, we allowed our mockequation of state to be one of two extremes, which we de-scribe below (see § dPd(cid:15) ≡ (cid:16) c s c (cid:17) ≤ , (14)where c s is the local sound speed.We can use this relation to derive the correspondingbound on Γ by noting that we can write equation (14) as dPd(cid:15) = dPdρ (cid:18) d(cid:15)dρ (cid:19) − = (cid:16) c s c (cid:17) . (15)For a polytrope, dP/dρ = Γ P/ρ , and the mass density-energy density relation of equation (6) can be expandedto d(cid:15)dρ = 1 ρ ( (cid:15) + P ) . (16)The polytropic index can therefore be written asΓ = c s c (cid:15) + PP ≤ (cid:15) + PP ≡ Γ luminal . (17)In order to ensure that our upper extreme did not violatecausality, we set our upper polytropic index to be theminimum of Γ = 5 and Γ luminal .With this set of steps, we obtained the pressure, massdensity, and energy density for each segment of our mockequations of state. The mock EoS start at ρ sat , so oncethe pressure at this density is determined, the above re-lations will uniquely determine the rest of the behaviorof each mock EoS. We introduced further freedom in ourmock EoS by allowing two significantly different pres-sures at the starting point, ρ sat . This is motivated bythe fact that such a bifurcation is also seen among theset of proposed EoS. We used the EoS SLy (Douchin &Haensel 2001) and PS (Pandharipande & Smith 1975)to determine the lower and higher starting pressures, re-spectively. With 15 density-segments, two options for the poly-tropic behavior along each segment, and two options forthe starting pressure, our algorithm produced 2 × =65,536 different mock EoS against which we could testour parametrization. However, we excluded any mockEoS that were too soft to produce a 1.9 M (cid:12) neutron star(Demorest et al. 2010; Antoniadis et al. 2013), reducingour final sample size to 53,343. The Mock EoS
The 53,343 extreme mock equations of state thatreached 1.9 M (cid:12) are shown in Figure 3. All possible tra- Fig. 3.—
The grid of 53,343 extreme, mock equations of state,starting at nuclear saturation density with a pressure correspond-ing to either the EoS SLy (in red) or PS (in blue). One samplemock EoS is shown bolded in black. Each mock EoS is composedof 15 segments that are spaced evenly in the logarithm between 1and 8 ρ sat . Each segment is a polytrope with either Γ=1 or theminimum of Γ=5 and Γ luminal . Only EoS that reach 1.9 M (cid:12) areshown here, leading to the absence of segments in the lower rightcorner of the parameter space. Fig. 4.—
Mass-radius curves corresponding to the 53,343 mockEoS shown in Figure 3. The red curves are those that startedat ρ sat with the corresponding pressure of the EoS SLy; the bluecurves use the nuclear saturation pressure of EoS PS. By includingboth starting pressures in our sample, we are able to densely samplea realistic range of radii of ∼ M max ∼ . R ( M (cid:12) / km), derivedby Lindblom (1984). The small discrepancy between this line andthe observed cutoff in our M-R curves can be attributed to thedifferent EoS that was assumed in the low-density region of theLindblom (1984) analysis. Fig. 5.— (a)-(c):
Cumulative distributions of the differences in radii between our parametrization and the full EoS for all mock EoSshown in Figure 4. The different colors represent the number of fiducial densities above ρ sat (i.e., the number of polytropic segmentsincluded in the parametrization). The radius residuals are measured at 1.4, 1.6, and 1.8 M (cid:12) , respectively. The vertical dashed lines marka residual of 0.5 km; the horizontal dashed lines mark the 95% level of the cumulative distribution. We find that our goal of residuals ≤ (d): Cumulative distribution of the difference in maximum mass between our parametrizationand the full EoS. The lines and colors are as for the other three panels, but here the vertical dashed line is shown at 0.1 M (cid:12) , correspondingto our desired maximum residual. This goal is also approximately achieved with 5 fiducial densities. jectories through this grid are indeed included, with theconstraint that pressure must always increase with den-sity (i.e., only monotonic behavior in this grid is allowed).The broadening of the mock EoS at high pressures is aresult of our requirement that the upper polytropic limitmust be the minimum of Γ=5 and Γ luminal .The corresponding mass-radius curves, calculated ac-cording to the method described in §
3, are shown inFigure 4. As Figure 4 demonstrates, starting the mockEoS at the two different pressures corresponding to thetwo families of proposed EoS allowed us to fully span therange of reasonable radii. With this choice, we achieveda dense sampling of mass-radius curves that span radiifrom ∼ M (cid:46) M (cid:12) . Furthermore, the mockmass-radius curves include curves that shallowly slopeupwards, that are nearly vertical, and that bend back-wards, indicating that we have sampled a wide range ofpossible underlying behavior.The trend of increasing maximum mass with radius isa result of our causality constraint. Lindblom (1984) de-rived the maximum gravitational redshift of a neutronstar as a function of mass by assuming that the equation of state is trusted up to 3 × g cm − and configuringthe resulting mass of the star to maximize the redshift.As in our analysis, that study required that the relation-ship between pressure and density inside the star notviolate causality. After converting their relationship be-tween the mass and maximum gravitational redshift to amass-radius relationship, we find that the correspondingrelationship of M max ∼ . R ( M (cid:12) / km), shown as thedashed line in Figure 4, is very close to what is seen inour mock EoS. The small differences between this rela-tionship and that in our mock EoS can be attributed tothe different EoS assumed up to the first fiducial den-sity: Lindblom (1984) assumed the EoS BPS, while weassumed either SLy or PS. Furthermore, they assumedtheir EoS up to 3 × g cm − , while we assumed SLyor PS only up to ρ sat ∼ . × g cm − . Determining the goodness of the parametricrepresentation
We quantified how well our parametrization repre-sented the full EoS and chose the optimal number ofsampling points by comparing the radii of our results to
Fig. 6.—
Cumulative distribution of the differences in momentof inertia for a star of mass M A = 1 . M (cid:12) , calculated betweenour parametrization and the full EoS. The different colors repre-sent the number of fiducial densities above ρ sat that we includedin our parametrization (i.e., the number of polytropic segments).The vertical dashed line marks a residual of 10% for a hypotheti-cal moment of inertia measurement of 10 g cm ; the horizontaldashed lines mark the 95% level of the cumulative distribution.With 5 fiducial densities, the moment of inertia residuals are lessthan 0.17 × g cm in 95% of cases. We therefore find that,depending on the exact value of the upcoming measurement ofthe moment of inertia for Pulsar A in the double pulsar system,5-6 fiducial densities may be needed to reproduce I A to the 10%accuracy level. those found using the full EoS at three fiducial masses, M =1.4, 1.6, and 1.8 M (cid:12) . We also calculated ∆ M max ≡| M max (full) − M max (parametric) | to determine how wellour parametrization reproduced the maximum mass pre-dicted by the full EoS. Finally, we calculated the differ-ence in the moment of inertia, ∆ I A , predicted by ourparametrization and by the full EoS for a star of mass M = 1 . M (cid:12) , i.e., the mass of the Pulsar A in thebinary system PSR J0737 − R < M max < . M (cid:12) . Because the moment ofinertia for Pulsar A has not yet been measured, we didnot impose strict requirements on ∆ I A , but still includedthis observable in our results to see qualitatively how wellit compares to some reasonable predictions for I A . Results of parametrization of mock EoS
Figure 5 shows the cumulative distribution of residualsin radius and mass for when various numbers of poly-tropic segments were included in the parametrization forour full sample of 53,343 extreme mock EoS. We foundthat our goal of ∆
R < ρ sat (i.e., includ-ing 5 polytropes). Specifically, using 5 fiducial densities,we found that for 95% of the mock EoS, ∆ R = 0.50, 0.44,and 0.48 km at M = 1.4, 1.6 and 1.8 M (cid:12) , respectively.In addition, a parametrization with 5 fiducial densitiesreproduced M max to within 0.12 M (cid:12) in 95% of the cases.We also calculated the difference in the moment of in-ertia for a neutron star with the same mass as Pulsar A in the double pulsar system, i.e., M A = 1 . M (cid:12) . The cu-mulative distribution of these residuals as a function ofthe number of polytropes included in the parametriza-tions is shown in Figure 6. By sampling the EoS at 5fiducial densities, we found that the residuals in momentof inertia are less than 0 . × g cm in 95% of thecases. The moment of inertia for Pulsar A is expectedto be on the order of 10 g cm and is expected to bemeasured with 10% accuracy (Kramer & Wex 2009). Asa result, depending on the exact value of the forthcom-ing moment of inertia measurement, 5-6 fiducial densitiesmay be required to recreate the moment of inertia to the10% accuracy level.Even a parametrization with just 3 fiducial densitiesreproduced the radii of ∼
80% of our extreme mock EoSto within 0.5 km. However, requiring ∆ M max < . M (cid:12) and ∆ I/I (cid:46)
10% requires more points. We thereforeconclude that, given the most recent observational un-certainties and the continuing prospects for even smallererrors in the near future, a parametrization that sam-ples the EoS at 5 fiducial densities is optimal. Specif-ically, we recommend spacing the five fiducial densi-ties evenly in the logarithm of the density, such that( ρ , ρ , ρ , ρ , ρ , ρ ) = (1 . , . , . , . , . , . ρ sat . ADDING PHASE TRANSITIONS TO THE MOCK EOS
We also considered more diverse equations of state byallowing there to be a first-order phase transition in themock EoS described in §
4. We allowed only one phasetransition per mock EoS, but allowed the phase transi-tion to start in any of our 15 density segments and to lastanywhere between 1 and 15 segments. For the remainingsegments, the mock EoS was polytropic with an indexof Γ=1 or the minimum of Γ=5 and Γ luminal , as above.With the addition of these phase transitions, there arenow 2 N × N ( N + 1) / N segments of theEoS, for each possible starting pressure. For 15 segmentsand our two starting pressures (corresponding to the EoSSLy and PS at ρ sat ), this corresponds to 7,864,320 pos-sibilities for the mock EoS.We randomly sampled ∼ ρ sat , i.e., the optimal number of poly-tropic segments found in § M (cid:12) , respectively. These errors arecomparable to those from the mock EoS without phasetransitions. We also found that 95% of the differences inmaximum mass were less than 0.22 M (cid:12) , which is a largererror than in our previous, less extreme sample of mockEoS. However, 70% of the errors in maximum mass werestill less than 0.1 M (cid:12) for this sample, indicating thatthis parametrization reasonably recreated the maximummass for many of our most extreme sample of mock EoS.The error distribution for the moment of inertia was al-most identical to the distribution for the sample withoutphase transitions. We found that a parametrization with5 fiducial densities reproduced the moment of inertia to Fig. 7.—
Same as Figure 5 but for 176,839 randomly sampled mock EoS, each of which include a single first-order phase transition.Approximately half of this sample of mock EoS starts at a pressure corresponding to the EoS SLy at ρ sat , while the other half starts atthe pressure predicted by the EoS PS at ρ sat . We find that a parametrization with five fiducial densities above ρ sat (i.e., five polytropicsegments) is sufficient to reproduce the radii at our three fiducial masses to within less than 0.5 km in 95% of cases. The errors in maximummass are significantly worse than for the sample of mock EoS without phase transitions: 95% of this sample has ∆ M max < . M (cid:12) or less. within 0.17 × g cm for 95% of the mock EoS withphase transitions.The tail end of all four distributions in Figure 7 ex-tended to higher errors than did the tail in Figure 5.This is because the arbitrarily-placed phase transitionsin this sample of mock EoS can make the resulting mass-radius curve have very sharp turn-overs. If a sharp turn-over occurs near one of the fiducial masses at which wemeasure radius residuals, we will infer an artificially higherror. However, the distributions in Figure 7 show thatthe probability of the true EoS falling in this tail is small( (cid:46)
5% with 5 fiducial densities). We therefore find that,for the vast majority of cases, a parametrization thatsamples the EoS at 5 fiducial densities is sufficient to re-produce radius and maximum mass observables to within0.5 km and 0.1-0.2 M (cid:12) , even if the EoS contains a first-order phase transition at an arbitrary density and overan arbitrary range. APPLICATION OF THE PARAMETRIZATION TOPHYSICALLY-MOTIVATED EOS
Even though we optimized our parametrization usingEoS that span a much wider range of possibilities thanthe currently proposed ones, we nevertheless explored how well this parametrization reproduced the physically-motivated EoS found in the literature. To this end, weapplied our optimized parametrization (5 fiducial den-sities above ρ sat ) to a sample of 42 proposed EoS, which incorporate a variety of different physical pos-sibilities and calculation methods. (The tabular datafor these EoS are compiled in Cook et al. 1994; Lat-timer & Prakash 2001; Read et al. 2009a; and ¨Ozel& Freire 2016). Our sample included purely nucleonicequations of state, such as: relativistic (BPAL12 andENG) and nonrelativistic (BBB2) Brueckner-Hartree-Fock EoS; variational-method EoS (e.g. FPS andWFF3); and a potential-method EoS (SLy). We also in-clude models which incorporate more exotic particles, in-cluding, for example, a neutron-only EoS with pion con-densates (PS), a relativistic mean-field theory EoS with This sample is smaller than the 49 EoS that we previouslycited because we exclude from this subsample any calculated EoSthat become acausal, except for the EoS AP4. Even though AP4reaches a local sound speed of c s ∼ . c by densities of ρ ∼ ρ sat and becomes more acausal thereafter, we do include this EoS, as itis commonly used and included in the literature. We also excludefrom this sample two EoS that are not calculated to high enoughdensities to accommodate our parametrization at 7.4 ρ sat . Fig. 8.—
Cumulative distribution of the residuals measured between the full EoS and the parametric version for 42 proposed EoS. Theparametrization uses 5 fiducial densities above ρ sat (i.e., it includes 5 polytropic segments). Left panel:
Residuals in radius, as calculatedat 1.4, 1.6, and 1.8 M (cid:12) . The vertical dashed line indicates residuals of 0.5 km, while the horizontal line shows the 95% inclusion level (this95% inclusion line is identical in all three panels). We find that the residuals are less than 0.10, 0.12, and 0.09 km at 1.4, 1.6, and 1.8 M (cid:12) respectively for 95% of the proposed EoS. Middle panel:
Differences in maximum mass. The vertical dashed line indicates residuals of0.1 M (cid:12) . We find that the errors in maximum mass are less than 0.04 M (cid:12) for 95% of the proposed EoS. Right panel:
Differences in themoment of inertia for a star of mass M A = 1 . M (cid:12) . The vertical dashed line indicates residuals of 10% for a hypothetical moment ofinertia measurement of 10 g cm . We find that 95% of the proposed EoS have residuals in the moment of inertia of 0 . × g cm or smaller. hyperons and quarks (PCL2), and an effective-potentialEoS with hyperons (BGN1H1).In applying our parametrization to these proposedEoS, we no longer connected to SLy or PS for ρ < ρ sat .Instead, we assumed that each EoS is known up to ρ sat , and we therefore used the full EoS that we wereparametrizing for the low-density regime. For ρ > ρ sat ,we applied our parametrization as above. After applyingour parametrization, we calculated the resulting residu-als in radii at the three fiducial masses, the maximummass, and the moment of inertia. The residuals for ouroptimized, 5-polytrope parametrization are shown in Fig-ure 8.The errors in applying the parametrization to theproposed, physically motivated EoS were much lowerthan for the more extreme, mock EoS. Our optimizedparametrization reproduced the radii of ∼
95% of the pro-posed EoS to within 0.10, 0.12, and 0.09 km (at 1.4, 1.6,and 1.8 M (cid:12) , respectively) and the maximum masses of95% of the EoS to within 0.04 M (cid:12) . As examples, weshow in Figure 9 the full mass-radius relation as well asthe one calculated from our parametrized EoS for severalproposed EoS: AP4 (nucleonic), ALF4 (quark hybrid),SLy (nucleonic), and BGN1H1 (includes hyperons). Asseen here, the differences in mass-radius space are ex-tremely small. CONCLUSIONS
In this paper, we investigated an optimal parametriza-tion of the neutron star equation of state that can be usedto interpret neutron star observations. We found thata parametric EoS with five polytropic segments evenly For every EoS, we only calculated and included the radiusresdiual at a given mass if the EoS actually reaches that mass.If, for example, an EoS only produces masses up to 1.7 M (cid:12) , westill included the radius residuals at 1.4 and 1.6 M (cid:12) and simplyexcluded the data point at 1.8 M (cid:12) . Fig. 9.—
Top:
Mass-radius relations for the EoS AP4, AL4,BGN1H1, and SLy as solid lines. The dashed lines show ourparametrization of each, with five fiducial densities above ρ sat . Thedifferent symbols represent the mass and radius of a star with acentral density equal to each fiducial density. Bottom:
Pressure asa function of mass density for these EoS. The symbols represent thelocation of each fiducial density. We find that this parametrizationreproduces the EoS to very high accuracy in mass-radius space. spaced in logarithm between 1 and 7.4 ρ sat was suffi-cient to reproduce the radii of proposed EoS to within0.12 km and the maximum mass to within 0.04 M (cid:12) in95% of cases. This parametrization was also able to re-produce the radii of our more extreme, mock EoS towithin 0.5 km, suggesting that even if a more extremeEoS is proposed or realized in nature, our parametriza-tion will be robust enough to reproduce it well.The radii of approximately fifteen neutron stars havealready been measured, for most of which the massesare also known (Guillot et al. 2013; Guillot & Rutledge2014; Heinke et al. 2014; N¨attil¨a et al. 2015; ¨Ozel et al.2016; Bogdanov et al. 2016). Even though the uncer-tainties in the individual radius measurements are of or-der ∼ P of 0.013, which corresponds to a frac-tional error in pressure of ∼ − Acknowledgements.
We thank Gordon Baym foruseful comments on the manuscript. We gratefullyacknowledge support from NASA grant NNX16AC56G.
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For completeness, we also explored a parametrization that uses linear segments between a number of density points0to represent the EoS. As in the case of our polytropic parametrization, we started this parameterization at ρ sat andspaced the segments evenly in the logarithm of the density. The EoS along each linear segment is given by P = m i ρ + b i ( ρ i − ≤ ρ ≤ ρ i ) , (A1)where continuity at the endpoints implies m i = P i − P i − ρ i − ρ i − (A2)and b i = P i − − (cid:18) P i − P i − ρ i − ρ i − (cid:19) ρ i − . (A3)Using this linear relationship for pressure to integrate the differential equation (6), we find that the energy densityin this case is given by (cid:15) ( ρ ) = (1 + a ) ρc + mρ log ρ − b, (A4)where a is an integration constant. By requiring continuity in the energy density at the endpoints of each segment,we can solve for the integration constant such that (cid:15) ( ρ ) = (cid:18) (cid:15) i − + b i ρ i − − m i log ρ i − (cid:19) ρ + m i ρ log ρ − b i , ( ρ i − ≤ ρ ≤ ρ i ) . (A5)We used equation (A5) to relate the linear EoS to the energy density and then used that energy density to integratethe TOV equations and solve for the total mass, radius, and moment of inertia of the neutron star.We applied a five-segment parametrization to the ∼ M (cid:12) as well as the errors in the maximum mass. In this way, we can directly comparethese results with those from the five-polytrope optimal parametrization in Figure 10. We find that the polytropicparametrization performs modestly better. However, the linear parametrization is still able to recreate the radii of thefull EoS to within (cid:46) . ∼
80% of the extreme, mock EoS.Even though our mock EoS are composed of polytropic segments, there are multiple mock EoS segments perparametrization segment. Moreover, the mock EoS segments are offset from the parametrization segments. Wetherefore do not expect that the mock EoS should significantly bias the performance of a polytropic parametrizationover a linear parametrization.In order to see how well the linear parametrization performs for more physically motivated EoS, we also appliedit to the sample of 42 proposed EoS from §
6. We compare these results to those of the polytropic parametrizationin Figure 11. We find again that the polytropic parametrization performs better than the linear parametrization,although the differences between the two parametrizations are most significant at radius and maximum mass errorsthat are well below observational uncertainties.It is likely that the same levels of errors could be achieved by the linear parametrization if more than five segmentswere included; however, for five segments, the polytropic parametrization performs modestly better in most cases.A polytropic parametrization is also the more natural choice for the neutron star EoS. We, therefore, recommend afive-segment polytropic parametrization over a linear one.1
Fig. 10.— (a)-(c):
Cumulative distributions of the differences in radii between a polytropic parametrization and the full EoS (solidline) and the differences between a linear parametrization and the full EoS (dashed line). These differences were calculated for all mockEoS shown in Figure 4. Both parametrizations contain five fiducial densities (i.e., five segments). The radius residuals are measured at1.4, 1.6, and 1.8 M (cid:12) , respectively. The vertical dashed lines mark a residual of 0.5 km; the horizontal dashed lines mark the 95% levelof the cumulative distribution. We find that a polytropic parametrization results in smaller errors, but that the linear parametrizationstill achieves the desired radius residual of 0.5 km for ∼
80% of our extreme, mock EoS. (d):
Cumulative distribution of the differencein maximum mass between each parametrization and the full EoS. The lines and linestyles are as for the other three panels, but herethe vertical dashed line is shown at 0.1 M (cid:12) , corresponding to our desired maximum residual. We find that the linear and polytropicparametrizations perform comparably well in recreating the neutron star maximum mass. Fig. 11.—
Same as Figure 10 but for 42 proposed, physically-motivated EoS. We find that a polytropic parametrization results in smallererrors, but that both parametrizations are able to recreate the radii and maximum masses of the full EoS to well below the expectedobservational uncertainties of 0.5 km and 0.1 M (cid:12)(cid:12)