Universal symmetry energy contribution to the neutron star equation of state
We discuss the observation that under neutron star conditions of charge neutrality and equilibrium the contribution from the symmetry energy to the equation of state (EoS) follows a universal behaviour. We call this behaviour the conjecture of a Universal Symmetry Energy Contribution (USEC). We find that an USEC holds provided the density dependence of the symmetry energy follows a behaviour that limits the proton fraction to values below the threshold for the direct Urca (DU) cooling process. The absence of DU cooling in typical mass neutron stars appears to be supported by the phenomenology of neutron star cooling data and allows to constrain the behaviour of at high densities. Two classes of symmetry energy functions are investigated more in detail to elucidate the USEC. We derive an analytic formula for the USEC to the neutron star EoS based on the result for the symmetry energy extracted from isobaric analog states of nuclei.
pacs:04.40.Dg, 12.38.Mh, 26.60.+c, 97.60.Jd
The study of matter at extreme densities has been of great importance during the last decades. On the one hand heavy ion collisions (HIC) at relativistic energies in the laboratory probe densities far beyond those found in nuclei. On the other hand, nature provides similar conditions inside neutron stars (NS), which can be studied by means of astronomical observations. The link between these two systems is the nuclear symmetry energy EPJA50 () which is an important ingredient for the physics of neutron stars where matter is highly asymmetric in proton and neutron numbers in contrast to the almost symmetric systems in HIC Tsang:2003td (); Danielewicz:2003dd (); Lattimer:2012xj (). The density dependence of the symmetry energy determines the proton fraction in compact stars and thus the way they cool. In the context of the the NS equation of state (EoS), direct Urca (DU) cooling Lattimer:1991ib () plays a decisive role as we will show in this work. The observation that the variation of is largely compensated by the factor for EoS that do not allow cooling of typical neutron stars by the DU process leads to the conjecture of a Universal Symmetry Energy Contribution (USEC) to the neutron star EoS Klahn:2006ir ().
Two classes of symmetry energy functions are investigated in more detail to elucidate the USEC. Both of them fulfil the constraint from an analysis using isobaric analog states (IAS) by Danielewicz and Lee Danielewicz:2013upa () which revealed that MeV at a reference density fm and is constrained to a rather narrow band at subsaturation densities. The first one follows a power law ansatz (which we refer to as ”MDI type”), , where the IAS constraint limits the admissible values of to the wide range when focussing on variations at saturation density fm only. This range gets narrowed to when also smaller variations at the lower limit for the IAS constraint are respected. This ansatz allows for the DU process at densities below which without further constraints on the stiffness of the nuclear EoS would occur also in typical mass neutron stars and thus apparently violate the USEC. The second one uses a recent parametrization of the density-dependent couplings in the isovector meson channel within the generalized density functional approach to nuclear matter Typel:2014tqa () leading to a moderate increase of the symmetry energy at supersaturation densities; gentle enough to fulfil the DU constraint in the whole range of densities relevant for neutron star interiors and thus in perfect agreement with the USEC. We will denote it as ”DD2-type”.
This paper is organized as follows: In section II we recollect the basic relationships for a general equation of state of compact star matter, i.e., degenerate nuclear matter in -equilibrium with electrons and muons under the condition of electric charge neutrality. We derive a general relationship between the nuclear symmetry energy and the proton fraction in the parabolic approximation for the dependence on the asymmetry . In section III we consider the strong interaction part of the symmetry energy and contrast two classes of symmetry energy models: A) the power-law behaviour and B) the density-dependent RMF model DD2 Typel:2014tqa (). Both fulfill recent analyses of isobaric analogue states Danielewicz:2013upa (). In section IV we show the dependence of neutron star properties on the symmetry energy, discuss the interrelation between these two EoS and the direct Urca process, and determine the conditions under which the symmetry energy contribution to the EoS behaves universal. We derive an analytic formula for this universal contribution as the main result of this work. In the final section we summarize the results and draw conclusions from this work.
Ii Condition of charge neutrality and equilibrium
Starting point is the energy per nucleon in cold neutron star matter
where is the total baryon density and , are the fractions of protons, electrons and muons, respectively. The dependence of the baryonic part on the asymmetry
is used here to define the symmetry energy as
For the parabolic approximation this results in
The leptonic contribution is a sum of the Fermi gas expressions for the contributing leptons
where . For massless leptons (), this expression goes over to
Under neutron star conditions charge neutrality holds,
The equilibrium with respect to the weak interaction processes and (and similar for muons) for cold neutron stars (temperature below the neutrino opacity criterion MeV) implies
The chemical potentials are defined as
where has been introduced as the partial energy density of species in the system. Due to the charge neutrality relation (9) we can eliminate the electron fraction and denote for brevity the proton fraction as in the following. The total energy (1) at fixed baryon density is then a function of two variables, the proton and electron fractions, which follow from the stationarity conditions
Since electrons in neutron star interiors are ultrarelativistic, , and
We arrive at a system of two equations which determine the proton and muon fractions as functions of the baryon density once the symmetry energy is known
For , the muon fraction is zero and we recover the widely known relationship for so-called matter in equilibrium
Based on these relationships we discuss now two classes of generic behaviour of the high-density symmetry energy and their relation to the phenomenology of neutron stars. Throughout this article we will use for the relativistic density functional approach DD2 with the parametrization of its density dependent meson nucleon couplings as given in Ref. Typel:2009sy ().
Iii Generic examples for
iii.1 MDI-type symmetry energy
We base the discussion of the symmetry energy at high densities on the constraints recently obtained by Danielewicz and Lee Danielewicz:2013upa () by analysing isobaric analogue states of nuclei. They conclude that at a reference density fm the symmetry energy is MeV with a rather narrow error band in the density range , see the red area in Fig. 1.
In that figure we show the symmetry energy behaviour following from the MDI-type Tsang:2003td () power-law ansatz
where the exponent is varied in the range . One may argue that this ansatz would have to be refined by differentiating between the kinetic and potential energy contributions to the symmetry energy which would suggest the behaviour
with MeV and MeV. Comparing this refined ansatz with (20) one recognizes that trivially for there is no difference and that for the densities of our interest, just above the saturation density fm, and a wide range of values (21) can be mapped to (20) by redefining to good accuracy. For example, , and , see Fig. 2.
iii.2 DD2-based symmetry energy
The second example for a generic class of functions describing the high-density behaviour of the nuclear symmetry energy is based on a generalized density functional approach Typel:2009sy (). Recently, variations of its symmetry energy relevant couplings in the meson channel have been introduced in the context of discussing the neutron skin thickness of heavy nuclei Typel:2014tqa (). Subsequently, it was also used in exploring symmetry energy effects in simulations of core-collapse supernovae Fischer:2013eka (). The density dependence of the symmetry energies introduced and labelled in Ref. Typel:2014tqa () are shown in Fig. 3.
We note that the behaviour of the DD2-based symmetry energies bears striking similarities with other microscopic approaches based on realistic NN forces like the celebrated APR EoS Akmal:1998cf (), or the recent Brueckner-Hartree-Fock calculation employing the Argonne V18 NN potential supplemented with three-nucleon forces Logoteta:2015voa ().
iii.3 Derived quantities: , and
In order to characterize the above models for the symmetry energy we compute its standard parameters, the value , the slope and the curvature of the symmetry energy, defined at nuclear saturation density as
In Table 1 we list the corresponding values obtained for the parametrizations of the two types of symmetry energy models used in this work and give also experimental values with their references.
|Roca-Maza:2015eza (); Ban:2010wx ()|
Using our knowledge of the symmetry energy at the Danielewicz-Lee reference point , we derive a relationship between the and parameters from the Taylor expansion of the symmetry energy
Using the fact that this results in
Neglecting the curvature term results in a linear approximation which is in excellent agreement with the numerical results for the relation of both models for the symmetry energy as shown in Fig. 4, see also Horowitz:2014bja ().
We estimate the curvature term analytically for the MDI-type symmetry energy (20) where
Using the relationship
we eliminate from (28) and obtain
With Eq. (27) we obtain the nonlinear relationship
which is shown together with the linear approximation to Eq. (27) in Fig. 4. Note that the linear approximation to (27) has been reported before in Horowitz:2014bja (), while the nonlinear relation (31) is a new result of the present work.
The shaded regions shown in Fig. 4 follow from the requirements that (i) the model for the symmetry energy shall not leave the region of the Danielewicz-Lee constraint shown in Figs. 1 - 3 and (ii) that central densities in typical neutron stars shall not exceed the threshold density for the rapid direct Urca cooling process, see next section.
Iv Results for neutron star properties
iv.1 Neutron star masses and radii vs.
Given the equation of state of neutron star matter, the structure and global properties of compact stars are obtained from solving the Tolman-Oppenheimer-Volkoff (TOV) equations Tolman:1939jz (); Oppenheimer:1939ne ()
Starting with a central energy density and pressure as boundary value at , these equations are integrated out to the distance where the pressure vanishes , defining the radius , the mass and the baryon number of the star. Varying the central energy density one obtains a sequence of star configurations for a given EoS, which then uniquely corresponds to a mass-radius curve Lindblom:1992 (), as shown in Fig. 5 for the classes of EoS under investigation in this work. The EoS for symmetric matter is given by the DD2 EoS and for we use either the MDI-type or the DD2-type functions. The grey shaded region in Fig. 5 is bordered by the sequences for the DD2- and the DD2+ symmetry energies which mark the corner points for the grey band highlighted in the plane of Fig. 4. As for the description of the neutron star crust which plays an important role for the determination of the radii, we have taken the SLy EoS composed of different parts, including the well established results from hp () and dh (). The crust table, together with this crust EoS full description, can be found in ioffe (). Further detailed discussion of the relation between symmetry energy and neutron star radius can be found, e.g., in AlvarezCastillo:2012rf ().
In order to quantify the relationship between neutron star radii and the nuclear symmetry energy, we first introduce the notion of a mean baryon density for a neutron star configuration characterized by its mass and baryon density profile,
It is remarkable that despite the large spread in radii for stars of the typical binary radio pulsar mass (mean ) their variation in mean baryon densities is rather well centered around the saturation density. Thus if we describe a star by the DD2 EoS and ask for which gravitational mass the mean density equals the saturation density fm of that model, we find a mass of , see Fig. 6.
iv.2 Direct Urca process constraint
With the two generic classes of symmetry energy behaviour introduced in the previous section we can now solve Eqs. (16) and (17) for the proton and muon fractions under neutron star conditions. The proton fraction plays an important role in the neutron star phenomenology as it determines whether the fastest neutrino cooling process, the direct Urca (DU) process , can occur or not Lattimer:1991ib (). If the central density of a neutron star exceeds the critical value and triggers the DU process this causes a dramatic drop of the core temperature due to rapid energy loss by neutrino emission. After the typical transport timescale of about 100 years the cooling wave reaches the surface of the star and the photon luminosity drops rapidly making the star practically invisible. This process can therefore not be operative in typical neutron stars as we do observe cooling neutron stars much older than 1000 years with surface temperatures that are not compatible with the DU cooling scenario. For a detailed discussion of the DU process constraint see, e.g., Refs. Klahn:2006ir (); Blaschke:2006gd ().
The key relation for deriving the DU threshold condition is the triangle inequality for the Fermi momenta of neutron, proton and electron involved in the process (neutrino momenta are small in comparison to these and can be safely neglected) leading to the condition
which can be formulated in terms of proton and muon fractions as , being equivalent to
For densities below the muon threshold one easily recovers the classical result for the DU threshold without muons Lattimer:1991ib (). Inserting the density dependent muon fraction one obtains the density dependence of the DU threshold shown in Figs. 7, 8 and 9. The density dependence of the particle fractions and the DU- and muon thresholds for the proton fraction are shown in Fig. 7 for the MDI-type parametrization with .
For the proton fraction without muons (i.e., for ) is density independent. Its value can be found from solving Eq. (19) with and amounts to . As we see from Fig. 8 (upper panel), the onset of muons does practically not change this. The muon fraction reaches for asymptotically large , see Eq. (17).
When comparing the behaviour of the proton fractions for the MDI-type models in Fig. 8 with those of the DD2-based models in Fig. 9 one observes a striking difference. While for the former the DU constraint is violated for all already for densities below , the DU process is excluded for all DD2-based models with exception of the model DD2++ which has a proton fraction that touches the DU constraint for densities above .
The question arises for a comparison with the central densities of neutron stars depending on their mass. We show these densities in Tab. 2 for five neutron star masses: where the DD2 EoS which was employed for and the results for the symmetry energy models used in this work are given. For those models where there is no entry for in the table, the DU threshold is never reached so that the constraint can not be violated. For the model the threshold density exists, but it is high enough that the central densities even in the most massive stars of do not exceed it. This statement, however, has to be taken with the caveat that we have used here the rather stiff of the DD2 model. For the and models the opposite conclusion applies: the DU constraint is violated in all cases, even for the very lightest stars. The entries given by bold numbers denote a violation of the DU constraint. The DD2++ model case is very interesting: here the DU constraint is fulfilled for typical neutron stars with masses smaller than and violated for the more massive ones. This border, however, could be lifted completely once we adopt a further stiffening of , e.g., by a moderate excluded volume modification. Then also the most massive stars would not exhibit DU cooling Alvarez-Castillo:2016 ().
iv.3 Universal symmetry energy contribution
With the models for the density dependence of the symmetry energy discussed in sect. III and the density dependent particle fractions under neutron star conditions we turn now to the discussion of the symmetry energy contribution (6) to the neutron star equation of state (1) and its comparison with the lepton contribution (3).
In Fig. 10 we show the results for the MDI-type models in the range . We observe that the leptonic contribution to the EoS while being fixed at the reference density is otherwise wildly varying and strongly density dependent. The models with interfere with the symmetry energy contribution for densities below and exceed it in this range. One may characterize the situation for these parameter values of the MDI-type model by saying that the neutron star behaves like a white dwarf: variations of the EoS in terms of isospin asymmetry are dominated by the lepton component!
In the case of the DD2-based EoS shown in Fig. 11 the situation is quite different. Here the variation of the lepton contribution stays bounded and always at about half the symmetry energy contribution or even below that. The variation of the leptonic contribution which is most pronounced between is almost invisible in the symmetry energy contribution. Therefore, we claim that for symmetry energy models which do not violate the DU constraint as the DD2-based one, the symmetry energy contribution to the neutron star EoS behaves universal! (Note that the behaviour of the symmetry energy for the DD2++ model with its gentle violation of the direct Urca constraint above marks the border of this universal behaviour.)
We would like to give an analytic understanding of this USEC. We start with the reference density fm of the Danielewicz-Lee analysis Danielewicz:2013upa () for which the symmetry energy MeV is known. We insert these values on the r.h.s. of Eq. (19) and solve the resulting cubic equation for the proton fraction for which we obtain
Next we discuss the situation at asymptotically large baryon densities (), where according to Eq. (17) holds . This results in
The statement of universality of this contribution would mean that it is independent of the variation of the proton fraction
which entails and thus . Inserting these values in (42) gives
with a value at that is rather close to the exact one (40) and a quite astonishing overall agreement with the numerical solution, see Fig. 12. Although the curve follows the trend of the USEC quite well up to the highest densities relevant for neutron stars, there is an overall deviation of about MeV.
In order to improve the situation and to arrive at an analytical solution for the USEC to the neutron star EoS we follow the observation from Fig. 10 and Fig. 11 that the muon fraction stays overall closely below the massless limit case of . Let us discuss first the deviation from the asymptotic solution for . For this purpose we introduce the deviation from the asymptotic solution as and insert this into Eq. (17) to get
Expanding the l.h.s. to lowest order in , we obtain the relationship
Inserting this asymptotic behaviour for the ratio in Eq. (41) we obtain
Applying the argument of universality, i.e. vanishing variation , we find the relation
which now yields a density dependent solution for the proton fraction. In the limit , i.e. for massless muons or for infinite density, the solution is recovered. As we are looking for the density dependence in the vicinity of this asymptotic solution, we introduce the small auxiliary quantity which after insertion in (48) fulfils the equation
Since , the replacement is in order,. This reduces (49) to the quadratic equation
Since , we work with the approximate solution
Inserting this and into (47) we arrive at
where in the last step Taylor expansions have been used. Since according to (51) depends on the density via the muon threshold as defined in Eq. (18), the result (52) represents a density dependent correction of the behaviour (44) which is obtained in the limit . The result (52) is shown as the dotted line in Fig. 12 which excellently describes the universal behaviour of the symmetry energy contribution to the neutron star EoS for densities above .
In a last step we recall that the symmetry energy is known exactly at the Danielewicz-Lee reference density and by construction all symmetry energies go through this point. Therefore, the exact value (40) of MeV shall be part of the USEC that we construct now by an extrapolation from the asymptotic behaviour through with an exponential ansatz
where the slope parameter regulates how fast the asymptotic solution is reached. This analytic formula (53) for the USEC to the neutron star EoS is the main result of this work. Its behaviour is illustrated in Fig. 12 in comparison to the contributions based on the DD2-type symmetry energies. In Fig. 12 we show the USEC (53) for the two choices of and which span the variation of the USEC in the vicinity of due to variations of the symmetry energy functional. At larger densities , these variations become unimportant and the asymptotic behaviour of is reached.
We have developed general relationships for the nuclear symmetry energy and their characterizing parameters , and in the vicinity of the saturation density that are based on the knowledge of the value for the symmetry energy from IAS at the density .
We have investigated two examples for parametrizations of the high-density behaviour of the symmmetry energy: a MDI-type form and a DD2-type form. They are equivalent in the region of the saturation density where the parameters are defined and both are fixed to the Danielewicz-Lee point at .
The proximity of and allows to define the symmetry energy parameters in terms of the results of the IAS analysis by Danielewicz and Lee. Both types of parametrizations for the generic high-density behaviour of the symmetry energy show a universal response to parameter changes in the region of the saturation density and result in the same correlation.
The resulting consequences for neutron star matter are twofold. Bulk properties of neutron stars which very much depend on integral quantities can be given as functions of quantities at the representative mean baryon density which for stars of the quite typical mass turns out to be equal to the saturation density for the DD2 EoS. These properties are insensitive to the specific form of the high-density behaviour of the symmetry energy.
Quantities which depend on the fact whether a threshold density is passed or not are very sensitive to the high-density behaviour of and as an example we consider the direct Urca process. It is remarkable that the critical proton fraction for the onset of the direct Urca process turns out to be a good measure for the discrimination between two types of high-density behaviour: as long as the DU process is not switched on the symmetry energy contribution to the neutron star EoS behaves in a universal fashion!
We have derived a general formula for the universal high-density behaviour of this contribution which applies for a rich class of symmetry energy functions including the DD2-type ones and the APR EoS. We present an extrapolation of the analytic formula from high to low densities, joining the asymptotic solution with the exact value at the IAS reference density.
The result of our study allows to extract the symmetric part of the nuclear EoS from NS phenomenology, e.g., from a measurement of the M-R relationship. Vice-versa, the measurement of the high-density behaviour of the symmetric EoS, e.g., in heavy-ion collision experiments would allow to predict the EoS and thus the M-R relation for neutron stars.
We acknowledge discussions with Jim Lattimer in an early stage of this work and with Pawel Danielewicz during his recent visit at University of Wroclaw. We are grateful to Gerd Röpke, Stefan Typel and Hermann Wolter for careful reading and detailed comments on the manuscript. We thank Stefan Typel for providing the EoS data for the DD2 models. This work was supported by the Polish National Science Center (NCN) under grant No. UMO-2014/13/B/ST9/02621 (D.E.A-C. and D.B.) and UMO-2013/09/B/ST2/01560 (T.K.).
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- (32) In order to avoid complications with the muon threshold that actually is very close to the reference density anyway, we may take the freedom to choose a muon mass that would lead to a coincidence of muon threshold and reference density () entailing MeV. Compared to the physical muon mass MeV, this 10% error is acceptable for the sake of clarity of our arguments.