Constraints on a phenomenologically parameterized neutronstar equation of state
Abstract
We introduce a parameterized highdensity equation of state (EOS) in order to systematize the study of constraints placed by astrophysical observations on the nature of neutronstar matter. To obtain useful constraints, the number of parameters should be smaller than the number of neutronstar properties that have been measured or will have been measured in the next several years. And the set must be large enough to accurately approximate the large set of candidate EOSs. We find that a parameterized EOS based on piecewise polytropes with 3 free parameters matches to about 4% rms error an extensive set of candidate EOSs at densities below the central density of stars. Adding observations of more massive stars constrains the higher density part of the EOS and requires an additional parameter. We obtain constraints on the allowed parameter space set by causality and by present and nearfuture astronomical observations. In particular, we emphasize potentially stringent constraints on the EOS parameter space associated with two measured properties of a single star; and we find that a measurement of the moment of inertia of PSR J07373039A can strongly constrain the maximum neutronstar mass. We also present in an appendix a more efficient algorithm than has previously been used for finding points of marginal stability and the maximum angular velocity of stable stars.
pacs:
04.40.Dg, 26.60.Kp, 97.60.JdI Introduction
Because the temperature of neutron stars is far below the Fermi energy of their constituent particles, neutronstar matter is accurately described by the oneparameter equation of state (EOS) that governs cold matter above nuclear density. The uncertainty in that EOS, however, is notoriously large, with the pressure as a function of baryon mass density uncertain by as much as an order of magnitude above nuclear density. The phase of the matter in the core of a neutron star is similarly uncertain: Current candidates for the EOS include nonrelativistic and relativistic meanfield models; models for which neutronstar cores are dominated by nucleons, by hyperons, by pion or kaon condensates, and by strange quark matter (free up, down, and strange quarks); and one cannot yet rule out the possibility that the ground state of cold matter at zero pressure might be strange quark matter and that the term “neutron star” is a misnomer for strange quark stars.
The correspondingly large number of fundamental parameters needed to accommodate the models’ Lagrangians has meant that studies of astrophysical constraints (see, for example, Engvik et al. (1996); Lattimer and Prakash (2001); Lattimer and Prakash (2006); Klähn et al. (2006); Page and Reddy (2006) and references therein) present constraints by dividing the EOS candidates into an allowed and a ruledout list. A more systematic study, in which astrophysical constraints are described as constraints on the parameter space of a parameterized EOS, can be effective only if the number of parameters is smaller than the number of neutronstar properties that have been measured or will have been measured in the next several years. At the same time, the number of parameters must be large enough to accurately approximate the EOS candidates.
A principal aim of this paper is to show that, if one uses phenomenological rather than fundamental parameters, one can obtain a parameterized EOS that meets these conditions. We exhibit a parameterized EOS, based on specifying the stiffness of the star in three density intervals, measured by the adiabatic index . A fourth parameter translates the curve up or down, adding a constant pressure – equivalently fixing the pressure at the endpoint of the first density interval. Finally, the EOS is matched below nuclear density to the (presumed known) EOS. An EOS for which is constant is a polytrope, and the parameterized EOS is then piecewise polytropic. A similar piecewisepolytropic EOS was previously considered by Vuille and Ipser Vuille and Ipser (1999); and, with different motivation, several other authors Zdunik et al. (2006); Bejger et al. (2005a); Haensel and Potekhin (2004); Shibata et al. (2005) have used piecewise polytropes to approximate neutronstar EOS candidates. In contrast to this previous work, we use a small number of parameters chosen to fit a wide variety of fundamental EOSs, and we systematically explore a variety of astrophysical constraints. Like most of the previous work, we aim to model equations of state containing nuclear matter (possibly with various phase transitions) rather than pure quark stars, whose EOS is predicted to be substantially different.
As we have noted, enough uncertainty remains in the pressure at nuclear density, that one cannot simply match to a fiducial pressure at . Instead of taking as one parameter the pressure at a fiducial density, however, one could match to the pressure of the known subnuclear EOS at, say, 0.1 and then use as one parameter a value of for the interval between and . Neutronstar observables are insensitive to the EOS below , because the fraction of mass at low density is small. But the new parameter would indirectly affect observables by changing the value of the pressure at and above nuclear density, for fixed values of the remaining . By choosing instead the pressure at a fixed density , we obtain a parameter more directly connected to physical observables. In particular, as Lattimer and Prakash Lattimer and Prakash (2001) have pointed out, neutronstar radii are closely tied to the pressure somewhat above nuclear density, and the choice is recommended by that relation.
In general, to specify a piecewise polytropic EOS with three density intervals above nuclear density, one needs six parameters: two dividing densities, three adiabatic indices , and a value of the pressure at an endpoint of one of the intervals. Remarkably, however, we find (in Sec. V) that the error in fitting the collection of EOS candidates has a clear minimum for a particular choice of dividing densities. With that choice, the parametrized EOS has three free parameters, and , for densities below g/cm (the density range most relevant for masses ), and four free parameters (an additional ) for densities between g/cm and the central density of the maximum mass star for each EOS.
With the parameterization in hand, we examine in Sect. VI astrophysical constraints on the parameter space beyond the radius relation found by Lattimer and Prakash Lattimer and Prakash (2001). Our emphasis in this first study is on present and very nearfuture constraints: those associated with the largest observed neutronstar mass and spin, with a possible observation (as yet unrepeated) of neutronstar redshift, with a possible simultaneous measurement of mass and radius, and with the expected future measurement of the moment of inertia of a neutron star with known mass. A companion paper Read et al. (2008) will investigate constraints obtainable with gravitationalwave observations in a few years. The constraints associated with the largest observed mass, spin, and redshift have a similar form, each restricting the parameter space to one side of a surface: For example, if we take the largest observed mass (at a 90% confidence level) to be 1.7 , then the allowed parameters correspond to EOSs whose maximum mass is at least 1.7 . We can regard as a function on the 4dimensional EOS parameter space. The subspace of EOSs for which is then described by a 3dimensional surface, and constraint is a restriction to the highmass side of the surface. Similarly, the observation of a 716 Hz pulsar restricts the EOS parameter space to one side of a surface that describes EOSs for which the maximum spin is 716 Hz. Thus we can produce modelindependent extended versions of the multidimensional constraints seen in Lackey et al. (2006).
The potential simultaneous observation of two properties of a single neutron star (for example, moment of inertia and mass) would yield a significantly stronger constraint: It would restrict the parameter space not to one side of a surface but to the surface itself. And a subsequent observation of two different parameters for a different neutron star would then restrict one to the intersection of two surfaces. We exhibit the result of simultaneous observations of mass and moment of inertia (expected within the next decade for one member of the binary pulsar J07373039 Lattimer and Schutz (2005); Bejger et al. (2005b)) and of mass and radius. Gravitationalwave observations of binary inspiral can again measure two properties of a single star: both mass and an observable roughly described as the final frequency before plunge (the departure of the waveform from a pointparticle inspiral); and related work in progress examines the accuracy with which one can extract EOS parameters from interferometric observations of gravitational waves in the inspiral and merger of a binary neutron star system Read et al. (2008).
Conventions: We use cgs units, denoting restmass density by , and (energy density)/ by . We define restmass density as where g and is the baryon number density. In Sec. III, however, we set to simplify the equations and add a footnote on restoring .
Ii Candidates
A test of how well a parametrized EOS can approximate the true EOS of cold matter at high density is how well it approximates candidate EOSs. We consider a wide array of candidate EOSs, covering many different generation methods and potential species. Because the parametrized EOS is intended to distinguish the parts of parameter space allowed and ruled out by present and future observations, the collection includes some EOSs that no longer satisfy known observational constraints. Many of the candidate EOSs were considered in Refs. Lattimer and Prakash (2001); Bejger et al. (2005b); Lackey et al. (2006); and we call them by the names used in those papers.
For plain nuclear matter, we include:

three relativistic mean field theory EOSs (MS12 and one we call MS1b, which is identical to MS1 except with a low symmetry energy of 25 MeV Müller and Serot (1996)).
We also consider models with hyperons, pion and kaon condensates, and quarks, and will collectively refer to these EOSs as models.

one neutrononly EOS with pion condensates (PS Pandharipande and Smith (1975)),

two relativistic mean field theory EOSs with kaons (GS12 Glendenning and SchaffnerBielich (1999)),

one effective potential EOS including hyperons (BGN1H1 Balberg and Gal (1997)),

one relativistic mean field theory EOS with hyperons and quarks (PCL2 Prakash et al. (1995)), and

four hybrid EOSs with mixed APR nuclear matter and colourflavorlocked quark matter (ALF14 with transition density and interaction parameter given by , ; , ; , ; and , respectively Alford et al. (2005)).
The tables are plotted in Fig. 1 to give an idea of the range of EOSs considered for this parameterization.
Iii Piecewise polytrope
A polytropic EOS has the form,
(1) 
with the restmass density and the adiabatic index, and with energy density fixed by the first law of thermodynamics,^{1}^{1}1In this section, for simplicity of notation, . To rewrite the equations in cgs units, replace and in each occurrence by and . Both and have units g/cm.
(2) 
For of the form (1), Eq. (2) has the immediate integral
(3) 
where is a constant; and the requirement implies and the standard relation
(4) 
The parameterized EOSs we consider are piecewise polytropes above a density , satisfying Eqs. (1) and (3) on a sequence of density intervals, each with its own and : An EOS is piecewise polytropic for if, for a set of dividing densities , the pressure and energy density are everywhere continuous and satisfy
(5) 
Then, for ,
(6) 
with
(7) 
The specific enthalpy^{2}^{2}2A note on terminology: When the entropy vanishes, the specific enthalpy, , and Gibbs free energy, , coincide. For nonzero entropy, it is the term or, equivalently, that appears in the first law of thermodynamics, where is the chemical potential. Because always has the meaning of enthalpy, and because for isentropic stars and isentropic flows the constancy of injection energy and Bernoulli’s law, respectively, are commonly stated in terms of enthalpy (see, for example Pippard (1964)), we refer to as the specific enthalpy, rather than the specific Gibbs free energy or the chemical potential. is defined as and is given in each density interval by
(8) 
The internal energy is then
(9) 
and the sound velocity is
(10) 
Each piece of a piecewise polytropic EOS is specified by three parameters: the initial density, the coefficient , and the adiabatic index . However, when the EOS at lower density has already been specified up to the chosen , continuity of pressure restricts to the value
(11) 
Thus each additional region requires only two additional parameters, and . Furthermore, if the initial density of an interval is chosen to be a fixed value for the parameterization, specifying the EOS on the density interval requires only a single additional parameter.
Iv Fitting methods
As noted above, in choosing a parameterization for the EOS space, our goal was to maintain high enough resolution, with a small number of parameters, that a point in parameter space can accurately characterize the EOS of neutronstar matter. A measure of this resolution is the precision with which a point in parameter space can fit the available candidate EOSs. We describe in this section how that measure is defined and computed for a choice of parameter space – that is, for a choice of the set of free parameters used to specify piecewise polytropes.
There is general agreement on the lowdensity EOS for cold matter, and we adopt the version (SLy) given by Douchin and Haensel Douchin and Haensel (2001). Substituting an alternative lowdensity EOS from, for example, Negele and Vautherin Negele and Vautherin (1973), alters by only a few percent the observables we consider in examining astrophysical constraints, both because of the rough agreement among the candidate EOSs and because the low density crust contributes little to the mass, moment of inertia, or radius of the star.
Each choice of a piecewise polytropic EOS above nuclear density is matched to this lowdensity EOS. The way in which the match is done is arbitrary, and, again, the small contribution of the lowdensity crust to stellar observables means that the choice of match does not significantly alter the relation between astrophysical observables and the EOS above nuclear density. In our choice of matching method, the first (lowestdensity) piece of the piecewise polytropic curve is extended to lower densities until it intersects the lowdensity EOS, and the lowdensity EOS is used at densities below the intersection point. This matching method has the virtue of providing monotonically increasing EOSs without introducing additional parameters. The method accommodates a region of parameter space larger than that spanned by the collection of candidate EOSs. It does, however, omit EOSs with values of and that are incompatible, for which the slope of the vs curve is too shallow to reach the pressure from the lowdensity part of the EOS.
The accuracy with which a piecewise polytrope , approximates a candidate EOS is measured by the root mean square or rms residual of the fit to tabulated points :
(12) 
In each case, we compute the residual only up to the maximum density that can occur in a stable star – the central density of the maximum mass nonrotating model based on the candidate EOS. Because astrophysical observations can, in principle, depend on the highdensity EOS only up to the value of for that EOS, only the accuracy of the fit below for each candidate EOS is relevant to our choice of parameter space.
For a given parameterization, we find for each candidate EOS the smallest value of the residual over the corresponding parameter space and the parameter values for which it is a minimum. In particular, the MINPACK nonlinear leastsquares routine LMDIF, based on the LevenbergMarquardt algorithm, is used to minimize the sum of squares of the difference between the logarithm of the pressuredensity points in the specified density range and the logarithm of the piecewise polytrope formula, which is a linear fit in each region to the curve of the candidate EOS. The nonlinear routine allows the dividing points between regions to be varied.
Even with a robust algorithm, the nonlinear fitting with varying dividing densities is sensitive to initial conditions. Multiple initial parameters for free fits are constructed using fixedregion fits of several possible dividing densities, and the global minimum of the resulting residuals is taken to indicate the best fit for the candidate EOS.
We begin with a single polytropic region in the core, specified by two parameters: the index and a pressure at some fixed density. Here, with a single polytrope, the choice of that density is arbitrary; for more than one polytropic piece, we will for convenience take that density to be the dividing density between the first two polytropic regions. Changing the value of moves the polytropic curve up or down, keeping the logarithmic slope fixed. The lowdensity SLy EOS is fixed, and the density where the polytropic EOS intersects SLy changes as changes. The polytropic index is determined by Eq. (11). This is referred to as a one free piece fit.
We then consider two polytropic regions within the core, specified by the four parameters , as well as three polytropic regions specified by the six parameters , where, in each case, . Again changing translates the piecewisepolytropic EOS of the core up or down, keeping its shape fixed. While some EOSs are well approximated by a single highdensity polytrope, others require three pieces to capture the behaviour of phase transitions at high density.
The six parameters required to specify three free polytropic pieces seems to push the bounds of what may be reasonably constrained by a small set of astrophysical measurements. We find, however, that the collection of candidate EOSs has a choice of dividing densities for which the residuals of the fit exhibit a clear minimum. This fact allows us to reduce the number of parameters by fixing the densities that delimit the polytropic regions of the piecewise polytrope. A three fixed piece fit, using three polytropic regions but with fixed g/cm and g/cm, is specified by the four parameters , , , and . The choice of and is discussed in Section V.2. Note that the density of departure from the fixed lowdensity EOS is still a fitted parameter for this scheme.
V Best fits to Candidate EOS
v.1 Accuracy of alternative parameterizations
The accuracy of each of the alternative choices of parameters discussed in the last Section, measured by the rms residual of Eq. (12), is portrayed in Table 1. The Table lists the average and maximum rms residuals over the set of 34 candidate EOSs.
Type of fit  All  

Mean RMS residual  
One free piece  0.0386  0.0285  0.0494 
Two free pieces  0.0147  0.0086  0.0210 
Three fixed pieces  0.0127  0.0098  0.0157 
Three free pieces  0.0071  0.0056  0.0086 
Standard deviation of RMS residual  
One free piece  0.0213  0.0161  0.0209 
Two free pieces  0.0150  0.0060  0.0188 
Three fixed pieces  0.0106  0.0063  0.0130 
Three free pieces  0.0081  0.0039  0.0107 
For nucleon EOSs, the fourparameter fit of two free polytropic pieces models the behaviour of candidates well; but this kind of fourparameter EOS does not accurately fit EOSs with hyperons, kaon or pion condensates, and/or quark matter. Many require three polytropic pieces to capture the stiffening around nuclear density, a subsequent softer phase transition, and then final stiffening. On the other hand, the six parameters required to specify three free polytropic pieces exceeds the bounds of what may be reasonably constrained by the small set of modelindependent astrophysical measurements. As mentioned in the introduction, however, an alternative four parameter fit can be made to all EOSs if the transition densities are held fixed for all candidate EOSs. The choice of fixed transition densities, and the advantages of such a parameterization over the two free piece fit, are discussed in the next subsection.
The hybrid quark EOS ALF3, which incorporates a QCD correction parameter for quark interactions, exhibits the worstfit to a onepiece polytropic EOS with residual 0.111, to the threepiece fixed region EOS with residual 0.042, and to the threepiece varying region EOS with residual 0.042. It has a residual from the twopiece fit of 0.044, somewhat less than the worst fit EOS, BGN1H1, an effectivepotential EOS that includes all possible hyperons and has a twopiece fit residual of 0.056.
v.2 Fixed region fit
A good fit with a minimal number of parameters is found for three regions with a division between the first and second pieces fixed at g/cm and a division between the second and third pieces fixed at g/cm. The EOS is specified by choosing the adiabatic indices in each region, and the pressure at the first dividing density, . A diagram of this parameterization is shown in Fig. 2. For this 4parameter EOS, best fit parameters for each candidate EOS give a residual of or better, with the average residual over 34 candidate EOSs of .
The dividing densities for our parameterized EOS were chosen by minimizing the rms residuals over the set of 34 candidate EOSs. For two dividing densities, this is a twodimensional minimization problem, which was solved by alternating between minimizing average rms residual for upper or lower density while holding the other density fixed. The location of the best dividing points is fairly robust over the subclasses of EOSs, as illustrated in Fig. 3.
With the dividing points fixed, taking the pressure to be the pressure at , is indicated by empirical work of Lattimer and Prakash Lattimer and Prakash (2001) that finds a strong correlation between pressure at fixed density (near this value) and the radius of neutron stars. This choice of parameter allows us to examine (in Sec. VI.5) the relation between and the radius; and we expect a similar correlation between and the frequency at which neutronstar inspiral dramatically departs from pointparticle inspiral for neutron stars near this mass.
The following considerations dictate our choice of the fourparameter space associated with three polytropic pieces with two fixed dividing densities. First, as mentioned above, we regard the additional two parameters needed for three free pieces as too great a price to pay for the moderate increase in accuracy. The comparison, then, is between two fourparameter spaces: polytropes with two free pieces and polytropes with three pieces and fixed dividing densities.
Here there are two key differences. Observations of pulsars that are not accreting indicate masses below 1.45 (see Sec. VI), and the central density of these stars is below for almost all EOSs. Then only the three parameters of the fixed piece parameterization are required to specify the EOS for moderate mass neutron stars. This class of observations can then be treated as a set of constraints on a 3dimensional parameter space. Similarly, because maximummass neutron stars ordinarily have most matter in regions with densities greater than the first dividing density, their structure is insensitive to the first adiabatic index. The three piece parameterization does a significantly better job above because phase transitions above that density require a third polytropic index . If the remaining three parameters can be determined by pulsar observations, then observations of more massive, accreting stars can constrain .
The best fit parameter values are shown in Fig. 4 and listed in Table 3 of Appendix C. The worst fits of the fixed region fit are the hybrid quark EOSs ALF1 and ALF2, and the hyperonincorporating EOS BGN1H1. For BGN1H1, the relatively large residual is due to the fact that the best fit dividing densities of BGN1H1 differ strongly from the average best dividing densities. Although BGN1H1 is well fit by three pieces with floating densities, the reduction to a fourparameter fit limits the resolution of EOSs with such structure. The hybrid quark EOSs, however, have more complex structure that is difficult to resolve accurately with a small number of polytropic pieces. Still, the bestfit polytrope EOS is able to reproduce the neutron star properties predicted by the hybrid quark EOS.
Vi Astrophysical constraints on the parameter space
Adopting a parameterized EOS allows one to phrase each observational constraint as a restriction to a subset of the parameter space. In subsections A–D we find the constraints imposed by causality, by the maximum observed neutronstar mass and the maximum observed neutronstar spin, and by a possible observation of gravitational redshift. We then examine, in subsection E, constraints from the simultaneous measurement of mass and moment of inertia and of mass and radius. We exhibit in subsection F the combined constraint imposed by causality, maximum observed mass, and a future momentofinertia measurement of a star with known mass.
In exhibiting the constraints, we show a region of the 4dimensional parameter space larger than that allowed by the presumed uncertainty in the EOS – large enough, in particular, to encompass the 34 candidate EOSs considered above. The graphs in Fig. 4 display the ranges , , , and . Also shown is the location in parameter space of each candidate EOS, defined as the set of parameters that provide the best fit to that EOS. The shaded region in the top graph corresponds to incompatible values of and mentioned in Sect. IV: These are values for which the pressure is so large and so small that no curve can start from the lowdensity EOS above neutron drip and reach at unless the slope exceeds .
To find the constraints on the parameterized EOS imposed by the maximum observed mass and spin, one finds the maximum mass and spin of stable neutron stars based on the EOS associated with each point of parameter space. A subtlety in determining these maximum values arises from a break in the sequence of stable equilibria – an island of unstable configurations – for some EOSs. The unstable island is typically associated with phase transitions in a way we now describe.
Spherical Newtonian stars described by EOSs of the form are unstable when an average value of the adiabatic index falls below 4/3. The strongerthanNewtonian gravity of relativistic stars means that instability sets in for larger values of , and it is ordinarily this increasing strength of gravity that sets an upper limit on neutronstar mass. EOSs with phase transitions, however, temporarily soften above the critical density and then stiffen again at higher densities. As a result, configurations whose inner core has density just above the critical density can be unstable, while configurations with greater central density can again be stable. Models with this behavior are considered, for example, by Glendenning and Kettner Glendenning and Kettner (2000), Bejger et al. Bejger et al. (2005a) and by Zdunik et al. Zdunik et al. (2006) (these latter authors, in fact, use piecewise polytropic EOSs to model phase transitions).
For our parameterized EOS, instability islands of this kind can occur for , when and . A slice of the fourdimensional parameter space with constant and is displayed in Fig. 5. The shaded region corresponds to EOSs with islands of instability. Contours are also shown for which the maximum mass for each EOS has the constant value (lower contour) and (upper contour).
An instability point along a sequence of stellar models with constant angular momentum occurs when the mass is maximum. On a massradius curve, stability is lost in the direction for which the curve turns counterclockwise about the maximum mass, regained when it turns clockwise. In the bottom graph of Fig. 5, massradius curves are plotted for six EOSs, labeled AF, associated with six correspondingly labeled EOSs in the top figure. The sequences associated with EOSs B, C and E have two maximum masses (marked by black dots in the lower figure) separated by a minimum mass. As one moves along the sequence from larger to smaller radius – from lower to higher density, stability is temporarily lost at the first maximum mass, regained at the minimum mass, and permanently lost at the second maximum mass.
It is clear from each graph in Fig. 5 that either of the two local maxima of mass can be the global maximum. On the lower boundary (containing EOSs A and D), the lower density maximum mass first appears, but the upperdensity maximum remains the global maximum in a neighborhood of the boundary. Above the upper boundary (containing EOS F), the higherdensity maximum has disappeared, and near the upper boundary the lowerdensity maximum is the global maximum.
vi.1 Causality
For an EOS to be considered physically reasonable, the adiabatic speed of sound cannot exceed the speed of light. Perfect fluids have causal time evolutions (satisfy hyperbolic equations with characteristics within the light cone) only if (the phase velocity of sound) is less than the speed of light. An EOS is ruled out by causality if for densities below the central density of the maximummass neutron star for that EOS. An EOS that becomes acausal beyond at density higher than this can always be altered for to a causal EOS. Because the original and altered EOS yield identical sequences of neutron stars, causality should not be used to rule out parameters that give formally acausal EOSs above .
We exhibit the causality constraint in two ways, first by simply requiring that each piecewise polytrope be causal at all densities and then by requiring only that it be causal below . The first, unphysically strong, constraint, shown in Fig. 6, is useful for an intuitive understanding of the constraint: The speed of sound is a measure of the stiffness of the EOS, and requiring causality eliminates the largest values of and .
Fig. 7 shows the result of restricting the constraint to densities below , with the speed of sound given by Eq. (10). A second surface is shown to account for the inaccuracy with which a piecewise polytropic approximation to an EOS represents the speed of sound. In all but one case (BGN1H1) the fits to the candidate EOSs overpredict the maximum speed of sound, but none of the fits to the candidate EOSs mispredict whether the candidate EOS is causal or acausal by more than 11% (fractional difference between fit and candidate). We adopt as a suitable causality constraint a restriction to a region bounded by the surface mean, corresponding to the mean plus one standard deviation in the error between for the candidate and best fit EOSs.
In the lower parts of each graph in Fig. 7, where dyne/cm, the bounding surface has the character of the first causality constraint, with the restriction on each of the three variables and becoming more stringent as the other parameters increase, and with restricted to be less than about . In this lowpressure part of each graph, the surface is almost completely independent of the value of : Because the constraint takes the form (for ) and is so low, the constraint rules out values of only at or beyond the maximum we consider.
In the upper part of each graph, where dyne/cm, unexpected features arise from the fact that we impose the causality constraint only below the maximum density of stable neutron stars – below the central density of the maximummass star.
The most striking feature is the way the constraint surface turns over in the upper part of the top graph, where dyne/cm, in a way that allows arbitrarily large values of . This occurs because, when is large, the density of the maximummass star is small, and a violation of causality typically requires high density. That is, when the density is low, the ratio in Eq. (10) is small. As a result, in the top graph, remains too small to violate causality before the maximum density is reached. In the bottom graph, with , is now large enough in Eq. (10) that the EOS becomes acausal just below the transition to . This is the same effect that places the upper limit on seen in the second graph of Fig. 6.
A second feature of the upper parts of each graph is the exact independence of the bounding surface on . The reason is simply that in this part of the parameter space the central density of the maximum mass star is below , implying that no stable neutron stars see .
Finally, we note that in both graphs, for small (the right of the graph), the EOSs yield the sequences mentioned above, in which an island of instability separates two stable sequences, each ending at a local maximum of the mass. Requiring to satisfy causality for both stable regions rules out EOSs below the lower part of the bifurcated surface.
vi.2 Maximum Mass
A stringent observational constraint on the EOS parameter space is set by the largest observed neutronstar mass. Unfortunately, the highest claimed masses are also subject to the highest uncertainties and systematic errors. The most reliable measurements come from observations of radio pulsars in binaries with neutron star companions. The masses with tightest error bars (about 0.01 ) cluster about 1.4 Lattimer and Prakash (2007). Recent observations of millisecond pulsars in globular clusters with nonneutron star companions have yielded higher masses: Ter 5I and Ter 5J Ransom et al. (2005), M5B Freire et al. (2008), PSR J1903+0327 Champion et al. (2008), and PSR J04374715 Verbiest et al. (2008) all have 95% confidence limits of about 1.7 , and the corresponding limit for NGC 6440B Freire et al. (2008) is about 2.3 . However these systems are more prone to systematic errors: The pulsar mass is obtained by assuming that the periastron advance of the orbit is due to general relativity. Periastron advance can also arise from rotational deformation of the companion, which is negligible for a neutron star but could be much greater for pulsars which have white dwarf or main sequence star companions. Also the mass measurement is affected by inclination angle, which is known only for the very nearby PSR J04374715. And with the accumulation of observations of these eccentric binary systems (now about a dozen) it becomes more likely that the anomalously high figure for NGC 6440B is a statistical fluke. Fig. 8 shows the constraint on the EOS placed by the existence of 1.7 neutron stars, which we regard as secure. Also shown in the figure are the surfaces associated with maximum masses of 2.0 and 2.3 .
Since all of the candidate highmass pulsars are spinning slowly enough that the rotational contribution to their structure is negligible, the constraint associated with their observed masses can be obtained by computing the maximum mass of nonrotating neutron stars. Corresponding to each point in the parameter space is a sequence of neutron stars based on the associated parameterized EOS; and a point of parameter space is ruled out if the corresponding sequence has maximum mass below the largest observed mass. We exhibit here the division of parameter space into regions allowed and forbidden by given values of the largest observed mass.
We plot contours of constant maximum mass in Fig. 8. Because EOSs below a maximum mass contour produce stars with lower maximum masses, the parameter space below these surfaces is ruled out. The error in the maximum mass between the candidate and best fit piecewise polytropic EOSs is mean (the magnitude of the mean error plus one standard deviation in the error over the 34 candidate EOSs), so the parameters that best fit the true EOS are unlikely to be below this surface.
The surfaces of Fig. 8 have minimal dependence on , indicating that the maximum mass is determined primarily by features of the EOS above . In Fig. 8 we have set to the least constraining value in the range we consider – to the value that gives the largest maximum mass at each point in space. Varying causes the contours to shift up, constraining the parameter space further, by a maximum of dyne/cm. The dependence of the contour on is most significant for large values of where the average density of a star is lower. The dependence on decreases significantly as decreases.
As discussed above, some of the EOSs produce sequences of spherical neutron stars with an island of instability separating two stable sequences, each with a local maximum of the mass. As shown in Fig. 5, this causes a contour in parameter space of constant maximum mass to split into two surfaces, one surface of parameters which has this maximum mass at the lower local maximum and another surface of parameters which has this maximum mass at higher branches. Since such EOSs allow stable models up to the largest of their local maxima, we use the least constraining surface (representing the global maximum mass) when ruling out points in parameter space.
vi.3 Gravitational redshift
We turn next to the constraint set by an observed redshift of spectral lines from the surface of a neutron star. We consider here only stars for which the broadening due to rotation is negligible and restrict our discussion to spherical models. The redshift is then , and measuring it is equivalent to measuring the ratio . With no independent measurement of mass or radius, the associated constraint again restricts the parameter space to one side of a surface, to the EOSs that allow a redshift as large as the largest observed shift. ^{3}^{3}3 One could also imagine a measured redshift small enough to rule out a class of EOSs. The minimum redshift for each EOS, however, occurs for a star whose central density is below nuclear density. Its value, , thus depends only on the EOS below nuclear density. (See, for example Haensel et al.Haensel et al. (2002).) For spherical models, the configuration with maximum redshift for a given EOS is ordinarily the maximummass star. By increasing or , one stiffens the core, increasing the maximum mass, but also increasing the radius at fixed mass. The outcome of the competition usually, but not always, yields increased redshift for larger values of these three parameters; that is, the increased maximum mass dominates the effect of increased radius at fixed mass for all but the largest values of .
Cottam, Paerels, and Mendez Cottam et al. (2002) claim to have observed spectral lines from EXO 0748676 with a gravitational redshift of . With three spectral lines agreeing on the redshift, the identification of the spectral features with iron lines is better founded than other claims involving only a single line. The identification remains in doubt, however, because the claimed lines have not been seen in subsequent bursts Cottam et al. (2008). There is also a claim of a simultaneous massradius measurement of this system using Eddingtonlimited photospheric expansion xray bursts Ozel (2006) which would rule out many EOSs. This claim is controversial, because the 95% confidence interval is too wide to rule out much of the parameter space, and we believe the potential for systematic error is understated. However, the gravitational redshift is consistent with the earlier claim of 0.35. Thus we treat as a tentative constraint. We also exhibit the constraint that would be associated with a measurement of .
Our parameterization can reproduce the maximum redshift of tabulated EOSs to 3.2% (mean+1). Figure 9 displays surfaces of constant redshift and for the least constraining value of in the range we consider. Surfaces with different values of are virtually identical for dyne/cm, but diverge for higher pressures when is small (). In the displayed parameter space, points in front of the surface, corresponding to stiffer EOSs in the inner core, are allowed by the potential measurement. From the location of the and surfaces, it is clear that, without an upper limit on , an observed redshift significantly higher than is needed to constrain the parameter space. In particular, most of the parameter space ruled out by is already ruled out by the constraint displayed in Fig. 8.
vi.4 Maximum Spin
Observations of rapidly rotating neutron stars can also constrain the EOS. The highest uncontroversial spin frequency is observed in pulsar Ter 5AD at 716 Hz Hessels et al. (2006). There is a claim of 1122 Hz inferred from oscillations in xray bursts from XTE J1239285 Kaaret et al. (2007), but this is controversial because the statistical significance is relatively low, the signal could be contaminated by the details of the burst mechanism such as fallback of burning material, and the observation has not been repeated.
The maximum angular velocity of a uniformly rotating star occurs at the Kepler or massshedding limit, , with the star rotating at the speed of a satellite in circular orbit at the equator. For a given EOS, the configuration with maximum spin is the stable configuration with highest central density along the sequence of stars rotating at their Kepler limit. An EOS thus maximizes rotation if it maximizes the gravitational force at the equator of a rotating star – if it allows stars of large mass and small radius. To allow high mass stars, the EOS must be stiff at high density, and for the radius of the highmass configuration to be small, the EOS must be softer at low density, allowing greater compression in the outer part of the star Stergioulas et al. (2002); Glendenning (1992). In our parameter space, a high angular velocity then restricts one to a region with large values of and , and small values of and .
As with the maximum mass, the maximum frequency is most sensitive to the parameter , but the frequency constraint complements the maximum mass constraint by placing an upper limit on over the parameter space, rather than a lower limit.
To calculate the maximum rotation frequencies for our parameterized EOS, we used the opensource code rns for axisymmetric rapid rotation in the updated form rns2.0 Stergioulas (2000). For a given EOS, the model with maximum spin is ordinarily close to the model with maximum mass, but that need not be true for EOSs that yield two local mass maxima. The resulting calculation of maximum rotation requires some care, and the method we use is described in Appendix B. The error incurred in using the parameterized EOS instead of a particular model is 2.7% (mean+1).
Spin frequencies of 716 Hz and even the possible 1122 Hz turn out to be very weak constraints because both are well below the Kepler frequencies of most EOSs. Thus we plot surfaces of parameters giving maximum rotation frequencies of 716 Hz in Fig. 10 and 1300 Hz and 1500 Hz in Fig. 11. The region of parameter space above the maximum observed spin surface is excluded. In the top figure, maximum mass stars have central densities below so there is no dependence on . In the bottom figure the least constraining value of is fixed. The surface corresponding to a rotation of 716 Hz only constrains the parameter space that we consider ( dyne/cm) if . The minimum observed rotation rate necessary to place a firm upper limit on is roughly 1200 Hz for . The surface Hz for is also displayed in Fig. 11 to demonstrate that much higher rotation frequencies must be observed in order to place strong limits on the parameter space.
Because it is computationally expensive to use rns to evaluate the maximum rotation frequency for a wide range of values in a 4parameter space, one can also use an empirical formula. Haensel and Zdunik Haensel and Zdunik (1989) found that the maximum stable rotation for a given EOS can be found from the maximummass spherically symmetric model for that EOS with mass and radius :
(13) 
In other words the maximum rotation is proportional to the square root of the average density of the star.
The original calculation of Haensel and Zdunik gave . An overview of subsequent calculations is given by Haensel et al. in Haensel et al. (1995), reporting values of for a range of EOS sets and calculation methods including those of Friedman et al. (1986); Lattimer et al. (1990); Cook et al. (1994). If we calculate maximum rotations with rns as described above, using the 34 tabulated EOSs, we find . The corresponding best fit parameterized EOSs give .
vi.5 Moment of inertia or radius of a neutron star of known mass
The moment of inertia of the more massive component, pulsar A, in the double pulsar PSR J07373039 may be determined to an accuracy of 10% within the next few years Lattimer and Schutz (2005) by measuring the advance of the system’s periastron, and implications for candidate EOSs have been examined in Bejger et al. (2005b); Lattimer and Schutz (2005); Morrison et al. (2004). As noted earlier, by finding both mass and moment of inertia of the same star one imposes a significantly stronger constraint on the EOS parameter space than the constraints associated with measurements of mass or spin alone: The latter restrict the EOS to the region of parameter space lying on one side of a surface, the region associated with the inequality or with . The simultaneous measurement, on the other hand, restricts the EOS to a single surface. That is, in an dimensional parameter space, the full ndimensional set of EOSs which allow a 1.338 model, and those EOSs for which that model has moment of inertia form the ()dimensional surface in parameter space given by . (We use here the fact that the 44 Hz spin frequency of pulsar A is slow enough that the moment of inertia is nearly that of the spherical star.) Moreover, for almost all EOSs in the parameter space, the central density of a star is below the transition density . Thus the surfaces of constant moment of inertia have negligible dependence on , the adiabatic index above , and the EOS is restricted to the twodimensional surface in the  space given by .
This difference in dimensionality means that, in principle, the simultaneous equalities that give the constraint from observing two features of the same star are dramatically stronger than the inequalities associated with measurements of mass or spin alone. In practice, however, the twodimensional constraint surface is thickened by the error of the measurement. The additional thickness associated with the error with which the parametrized EOS can reproduce the moment of inertia of the true EOS is smaller, because the parameterized EOS reproduces the moment of inertia of the 34 candidate EOSs to within 2.8% (mean).
In Fig. 12 we plot surfaces of constant moment of inertia that span the range associated with the collection of candidate EOSs. The lower shaded surface represents g cm. This surface has very little dependence on because it represents a more compact star, and thus for a fixed mass, most of the mass is in a denser state . The structures of these stars do depend on , and the corresponding dependence of on is shown by the separation between the surfaces in Fig. 12. The middle outlined surface represents g cm, and is almost a surface of constant . The top outlined surface represents g cm. This surface has little dependence on , because a star with an EOS on this surface would be less compact and thus most of its mass would be in a lower density state .
If the mass of a neutron star is already known, a measurement of the radius constrains the EOS to a surface of constant mass and radius, in the 4dimensional parameter space. The thickness of the surface is dominated by the uncertainty in the radius and mass measurements, since our parameterization produces the same radius as the candidate EOSs to within 1.7% (mean). We plot in Fig. 13 surfaces of constant radius for a star that span the range of radii associated with the collection of candidate EOSs. As with the moment of inertia, the radius depends negligibly on as long as the radius is greater than 11 km. For smaller radii, the variation with is shown by the separation between the surfaces in Fig. 13.
Very recently analyses of timeresolved spectroscopic data during thermonuclear bursts from two neutron stars in lowmass xray binaries were combined with distance estimates to yield and km or and km for EXO 1745248 Ozel et al. (2008) and and km for 4U 160852 Guver et al. (2008), both with error bars of about 1 km in . These results are more model dependent than the eventual measurement of the moment of inertia of PSR J07376069A, but the accuracy of the measurement of remains to be seen.
vi.6 Combining constraints
The simultaneous constraints imposed by causality, a maximum observed mass of 1.7, and a future measurement of the moment of inertia of PSR J07373039A, restrict the parameter space to the intersection of the allowed regions of Figs. 7, 8, and 12. We show in Fig. 14 the projection of this jointly constrained region on the subspace. This allows one to see the cutoffs imposed by causality that eliminate large values of and and (in the top figure) the cutoffs imposed by the existence of a model that eliminates small values of and .
We noted above that measuring the moment of inertia of a 1.338 star restricts the EOS at densities below to a twodimensional surface in the space. In the full 4dimensional parameter space, the corresponding surfaces of constant and of Fig. 14 are then three dimensional and independent of . Their projections onto the subspace are again threedimensional and independent of , their thickness due to the unseen dependence of the mass and moment of inertia on . For small moments of inertia there is negligible dependence on so the allowed volume in Fig. 14 is thin. The thickness of the allowed volume increases as the moment of inertia increases because the dependence on also increases.
In Fig. 15 we explore a relation between the moment of inertia of PSR J07373039A and the maximum neutron star mass, in spite of the fact that the maximum mass is significantly greater than 1.338 . For three values of the moment of inertia that span the full range associated with our collection of candidate EOSs, we show joint constraints on and including causality and maximum neutron star mass. For g cm, is nearly unconstrained, while is required to lie in a small range between the causality constraint and the reliable observations of stars with mass 1.7 . For larger values of , is more constrained and is less constrained. However, the highest values of are associated with the highest maximum neutron star masses. Thus, if a neutron star mass of about is confirmed, it implies that is about g cm. Conversely iff is measured first and is about g cm, it implies that the maximum neutron star mass is less than about 1.9 .
The allowed range for as a function of the moment of inertia of J07373039A is shown in Fig. 16. The entire shaded range is allowed for a maximum mass. The medium and darker shades are allowed for a maximum mass. Only the range with the darker shade is allowed if a star is confirmed. It should be noted that for small moments of inertia, this plot overstates the uncertainty in the allowed parameter range. As shown in Fig. 14, the allowed volume in space for a small moment of inertia observation is essentially two dimensional. If the moment of inertia is measured to be this small, then the EOS would be better parameterized with the linear combination instead of two separate parameters and .
Vii Discussion
We have shown how one can use a parameterized piecewise polytropic EOS to systematize the study of observational constraints on the EOS of cold, highdensity matter. We think that our choice of a 4parameter EOS strikes an appropriate balance between the accuracy of approximation that a larger number of parameters would provide and the number of observational parameters that have been measured or are likely to be measured in the next several years. The simple choice of a piecewise polytrope, with discontinuities in the polytropic index, leads to suitable accuracy in approximating global features of a star. But the discontinuity reduces the expected accuracy with which the parameterized EOS can approximate the local speed of sound. One can largely overcome the problem by using a minor modification of the parameterized EOS in which a fixed smoothing function near each dividing density is used to join the two polytropes.
We see that highmass neutron stars are likely to provide the strongest constraints from a single measurement. The work dramatizes the significantly more stringent constraints associated with measurements like this, if two (or more) physical features of the same star can be measured, and an dimensional parameter space is reduced by one (or more) dimension(s), to within the error of measurement. In particular, a moment of inertia measuremement for PSR J07373039 (whose mass is already precisely known) could strongly constrain the maximum neutron star mass.
The effect of EOSdependent tidal deformation can modify the gravitational waves produced by inspiraling neutron stars. This modification is largely dependent on the radius of the neutron star. Flanagan and Hinderer Flanagan and Hinderer (2008) investigate constraints on an EOSdependent tidal parameter, the Love number, from observations of early inspiral. A companion to this paper Read et al. (2008) uses the parametrized EOS in numerical simulations to examine the future constraint associated with expected gravitationalwave observations of late inspiral in binary neutron stars.
Finally, we note that the constraints from observations of different neutron star populations constrain different density regions of the EOS. For moderate mass stars such as those found in binary pulsar systems, the EOS above g/cm is unimportant. For nearmaximum mass stars, the EOS below g/cm has little effect on neutron star properties. This general behavior is independent of the details of our parameterization.
Acknowledgements.
We thank P. Haensel for helpful suggestions at the start of this work. J. Lattimer and M. Alford generously provided EOS tables from Lattimer and Prakash (2001) and Alford et al. (2005). Other tables are from the LORENE C++ library (http://www.lorene.obspm.fr). The work was supported in part by NSF Grants PHY 0503366 and PHY0555628, by NASA Grant ATP0300010027, and by the Penn State Center for Gravitationalwave Physics under NSF cooperative agreement PHY0114375.References
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Appendix A Evaluating mass, radius, and moment of inertia
The moment of inertia of a rotating star is the ratio , with the asymptotically defined angular momentum. In finding the moment of inertia of spherical models, we use Hartle’s slowrotation equations Hartle (1967), adapted to piecewise polytropes in a way we describe below. The metric of a slowly rotating star has to order the form
(14)  
where and are the metric functions of the spherical star, given by
(15)  
(16)  
(17)  
(18) 
The framedragging is obtained from the component of the Einstein equation in the form
(19) 
where is the angular velocity of the star measured by a zeroangularmomentum observer and
(20) 
The angular momentum is obtained from , which has outside the star the form .
In adapting these equations, we roughly follow Lindblom Lindblom (1992), replacing as a radial variable by a generalization of the Newtonian enthalpy. ^{4}^{4}4Lindblom, however, uses instead of as his radial variable. Because of the form of the piecewise polytrope, is a more convenient choice here. This variable is also used by Haensel and Potekhin in Haensel and Potekhin (2004). Because is monotonic in , one can integrate outward from its central value to the surface, where .
For the piecewise polytropes of Sec. III, the equation of state given in terms of is
(21)  
(22)  
(23) 
where is the polytropic index.
This replacement exploits the first integral of the equation of hydrostatic equilibrium to eliminate the differential equation (16) for ; and the enthalpy, unlike and , is smooth at the surface for a polytropic EOS. Eqs. (1719) are then equivalent to the firstorder set
(24)  
(25)  
(26)  
(27) 
where .
The integration to find the mass, radius, and moment of inertia for a star with given central value proceeds as follows: Use the initial conditions and arbitrarily choose a central value of . Integrate to the surface where , to obtain the radius and mass . The angular momentum is found from the radial derivative of the equation
(28) 
evaluated at , namely
(29) 
and is then given by
(30) 
These values of and are each proportional to the arbitrarily chosen , implying that the moment of inertia is independent of .
Appendix B Stability of rotating models
The massshed limit gives a maximally rotating equilibrium model for each central energy density , but, as in the spherical case, these equilibrium models are not guaranteed to be stable to perturbations.
Overall stability in uniformly rotating models is governed by the stability of the model to pseudoradial perturbations. As in the spherical case, there can exist alternating regions of stable and unstable rotating models along a sequence of fixed . A criterion for the onset of instability is developed by Friedman, Ipser and Sorkin in Friedman et al. (1988): The critical points that potentially indicate a change in stability are extrema of massenergy under variation in both baryon mass and angular momentum ; and can be determined by extremizing on sequences of constant or extremising on sequences of constant . Universally valid searches for limiting stability, as in for example Cook et al. (1992), have therefore required explicitly covering the set of models with sequences of constant rest mass and extremizing on each one, or vice versa—a computationally expensive procedure.
For most only EOSs, the maximally rotating stable model is close to the point on the massshed limit with maximal massenergy, and this model has been used for an estimate of maximal rotation in surveys of large numbers of EOSs. However, this is not always the case. An example is in EOS L of Cook et al. (1992), or the parameterized EOS of Fig. 17.
Consider the twoparameter family of rotating neutron stars as a surface in  space. The central energy density and axis ratio are suitable parameters for this surface. At points where is maximum along constant sequences, the vector tangent to the sequence points in the direction. Models with limiting stability are found where the tangent plane to the surface of equilibrium models contains a vector in the direction, .
Given the parameterization of the surface in terms of and , the normal vector to the surface is along
(31) 
with component along .
(32) 
which is zero at the critical line between stable and unstable equilibriums on the surface . A covariant statement of this condition for marginal stability is .
The maximally rotating model for a given EOS may be determined, without finding sequences of constant and , by considering a sequence of central energy densities . First, increase the axis ratio until the Kepler limit is found, as in the example program main.c of rnsv2.0. Second, vary and around this point to estimate the partial derivatives of Eq. 32. The sign of will change as the Kepler limit sequence crosses the stability limit.
Appendix C Analytic fits to tabulated EOSs
As another measure of the ability of the parameterized EOS to fit candidate EOSs from the literature, we examine how well the parameterized EOS reproduces neutron star properties predicted by the candidate EOSs. We use an analytic form of the (SLy) lowdensity EOS that closely matches its tabulated values. With rms residual less than , for SLy is approximated between g/cm and g/cm by four polytropic pieces. The four regions correspond roughly to a nonrelativistic electron gas, a relativistic electron gas, neutron drip, and the density range from neutron drip to nuclear density. Using the notation of Sect. III, the analytic form of the SLy EOS is set by the values of and listed in Table 2. The parameters for the three piece polytropic highdensity EOS, the corresponding residuals, as well as the observable properties of these EOSs and the error in using the best fit parameterized EOS instead of the tabulated EOS are listed in Table 3. The parameterized EOS systematically overestimates the maximum speed of sound.
6.80110e09  1.58425  2.44034e+07 
1.06186e06  1.28733  3.78358e+11 
5.32697e+01  0.62223  2.62780e+12 
3.99874e08  1.35692  – 
EOS  residual  %  %  %  %  %  %  
PAL6  34.380  2.227  2.189  2.159  0.0011  0.693  1.37  1.477  0.47  0.374  0.51  1660  0.97  1.051  2.03  10.547  0.54 
SLy  34.384  3.005  2.988  2.851  0.0020  0.989  1.41  2.049  0.02  0.592  0.81  1810  0.10  1.288  0.08  11.736  0.21 
AP1  33.943  2.442  3.256  2.908  0.019  0.924  9.94  1.683  1.60  0.581  2.79  2240  1.05  0.908  2.57  9.361  1.85 
AP2  34.126  2.643  3.014  2.945  0.0089  1.032  0.42  1.808  1.50  0.605  0.33  2110  0.02  1.024  2.34  10.179  1.57 
AP3  34.392  3.166  3.573  3.281  0.0091  1.134  2.72  2.390  1.00  0.704  0.57  1810  0.14  1.375  1.59  12.094  0.96 
AP4  34.269  2.830  3.445  3.348  0.0068  1.160  1.45  2.213  1.08  0.696  0.22  1940  0.05  1.243  1.36  11.428  0.90 
FPS  34.283  2.985  2.863  2.600  0.0050  0.883  2.29  1.799  0.03  0.530  0.67  1880  0.11  1.137  0.03  10.850  0.12 
WFF1  34.031  2.519  3.791  3.660  0.018  1.185  7.86  2.133  0.29  0.739  2.21  2040  0.30  1.085  0.10  10.414  0.02 
WFF2  34.233  2.888  3.475  3.517  0.017  1.139  7.93  2.198  0.14  0.717  0.71  1990  0.03  1.204  0.59  11.159  0.28 
WFF3  34.283  3.329  2.952  2.589  0.017  0.835  8.11  1.844  0.48  0.530  2.26  1860  0.59  1.160  0.25  10.926  0.12 
BBB2  34.331  3.418  2.835  2.832  0.0055  0.914  7.75  1.918  0.10  0.574  0.97  1900  0.47  1.188  0.17  11.139  0.29 
BPAL12  34.358  2.209  2.201  2.176  0.0010  0.708  1.03  1.452  0.18  0.382  0.29  1700  1.03  0.974  0.20  10.024  0.67 
ENG  34.437  3.514  3.130  3.168  0.015  1.000  10.71  2.240  0.05  0.654  0.39  1820  0.44  1.372  0.97  12.059  0.69 
MPA1  34.495  3.446  3.572  2.887  0.0081  0.994  4.91  2.461  0.16  0.670  0.05  1700  0.18  1.455  0.41  12.473  0.26 
MS1  34.858  3.224  3.033  1.325  0.019  0.888  12.44  2.767  0.54  0.606  0.52  1400  1.67  1.944  0.09  14.918  0.06 
MS2  34.605  2.447  2.184  1.855  0.0030  0.582  3.96  1.806  0.42  0.343  2.57  1250  2.25  1.658  0.46  14.464  2.69 
MS1b  34.855  3.456  3.011  1.425  0.015  0.889  11.38  2.776  1.03  0.614  0.56  1420  1.38  1.888  0.64  14.583  0.32 
PS  34.671  2.216  1.640  2.365  0.028  0.691  7.36  1.755  1.53  0.355  1.45  1300  2.39  2.067  3.06  15.472  3.72 
GS1^{1}^{1}1The tables for GS1, GS2, and H6 do not go up to the central density of the maximum mass star. For most observables, the EOS can be safely extrapolated to higher density with minimal error. However, the maximum speed of sound is highly sensitive to how this extrapolation is done. Thus, we only use the maximum speed of sound up to the last tabulated point when comparing the values for the table and fit.  34.504  2.350  1.267  2.421  0.018  0.695  0.49  1.382  1.00  0.395  0.64  1660  9.05  0.766  3.13  ^{2}^{2}2GS1 has a maximum mass less than .  
GS2^{1}^{1}1The tables for GS1, GS2, and H6 do not go up to the central density of the maximum mass star. For most observables, the EOS can be safely extrapolated to higher density with minimal error. However, the maximum speed of sound is highly sensitive to how this extrapolation is done. Thus, we only use the maximum speed of sound up to the last tabulated point when comparing the values for the table and fit.  34.642  2.519  1.571  2.314  0.026  0.592  16.10  1.653  0.30  0.339  7.71  1340  3.77  1.795  3.33  14.299  0.07 
BGN1H1  34.623  3.258  1.472  2.464  0.029  0.878  7.42  1.628  0.39  0.437  3.55  1670  2.08  1.504  0.56  12.901  1.96 
GNH3  34.648  2.664  2.194  2.304  0.0045  0.750  2.04  1.962  0.13  0.427  0.37  1410  0.04  1.713  0.38  14.203  0.28 
H1  34.564  2.595  1.845  1.897  0.0019  0.561  2.81  1.555  0.92  0.311  1.47  1320  1.46  1.488  1.45  12.861  0.03 
H2  34.617  2.775  1.855  1.858  0.0028  0.565  1.38  1.666  0.77  0.322  0.55  1280  1.29  1.623  0.82  13.479  0.29 
H3  34.646  2.787  1.951  1.901  0.0070  0.564  7.05  1.788  0.79  0.343  1.07  1290  0.88  1.702  1.18  13.840  0.31 
H4  34.669  2.909  2.246  2.144  0.0028  0.685  4.52  2.032  0.85  0.428  1.01  1400  1.28  1.729  1.18  13.774  1.34 
H5  34.609  2.793  1.974  1.915  0.0050  0.596  1.65  1.727  1.00  0.347  0.82  1340  1.55  1.615  1.31  13.348  0.68 
H6^{1}^{1}1The tables for GS1, GS2, and H6 do not go up to the central density of the maximum mass star. For most observables, the EOS can be safely extrapolated to higher density with minimal error. However, the maximum speed of sound is highly sensitive to how this extrapolation is done. Thus, we only use the maximum speed of sound up to the last tabulated point when comparing the values for the table and fit.  34.593  2.637  2.121  2.064  0.0087  0.598  11.71  1.778  0.07  0.346  8.65  1310  5.33  1.623  2.19  13.463  0.37 
H7  34.559  2.621  2.048  2.006  0.0046  0.630  1.82  1.683  1.12  0.357  0.57  1410  1.52  1.527  2.33  12.992  0.23 
PCL2  34.507  2.554  1.880  1.977  0.0069  0.600  1.74  1.482  0.79  0.326  2.25  1440  1.87  1.291  3.27  11.761  1.15 
ALF1  34.055  2.013  3.389  2.033  0.040  0.565  18.59  1.495  0.53  0.386  3.52  1730  2.44  0.987  0.40  9.896  0.22 
ALF2  34.616  4.070  2.411  1.890  0.043  0.642  1.50  2.086  5.26  0.436  0.62  1440  1.01  1.638  6.94  13.188  3.66 
ALF3  34.283  2.883  2.653  1.952  0.017  0.565  11.29  1.473  0.06  0.358  2.46  1620  1.79  1.041  0.76  10.314  0.25 
ALF4  34.314  3.009  3.438  1.803  0.023  0.685  14.78  1.943  0.93  0.454  0.59  1590  0.52  1.297  2.38  11.667  1.20 
Mean error  5.68  0.71  0.74  0.43  1.26  0.37  
Standard deviation of error  5.52  0.96  2.42  2.25  1.57  1.29 