On Cooling of Neutron Stars With Stiff Equation of State Including Hyperons
Abstract
The existence of high mass () pulsars PSR J16142230 and PSR J03480432 requires the compact star matter to be described by a stiff equation of state (EoS). Presence of hyperons in neutron stars leads to a softening of the EoS that results in a decrease of the maximum neutronstar mass below the measured values of masses for PSR J16142230 and PSR J03480432 pulsars, if one exploits ordinary relativistic meanfield (RMF) models (hyperon puzzle). However, within a RMF EoS with a scaled hadron effective masses and coupling constants the maximum neutronstar mass remains above even when hyperons are included. Also other important constraints on the equation of state, e.g. the flow constraint from heavyion collisions are to be fulfilled. We demonstrate how a satisfactory explanation of all existing observational data for the temperatureage relation is reached within the ânuclear medium coolingâ scenario with a relativisticmeanfield EoS with a scaled hadron effective masses and coupling constants including hyperons.
I Introduction
Equation of state (EoS) of the cold hadronic matter should:

satisfy experimental information on properties of dilute nuclear matter;

empirical constraints on global characteristics of atomic nuclei;

constraints on the pressure of the nuclear mater from the description of particle transverse and elliptic flows and the production in heavyion collisions, cf. Danielewicz:2002pu (); Lynch:2009vc ();

allow for the heaviest known pulsars PSR J16142230 (of mass ) Fonseca:2016tux () and PSR J0348+0432 (of mass ) Antoniadis:2013pzd ();

allow for an adequate description of the compact star cooling Blaschke:2004vq (), most probably without direct Urca (DU) neutrino processes in the majority of the known pulsars detected in soft rays Klahn:2006ir ();

yield a massradius relation comparable with the empirical constraints including recent gravitation wave LIGOVirgo detection TheLIGOScientific:2017qsa ();

being extended to nonzero temperature (for where is the critical temperature of the deconfinement), appropriately describe supernova explosions, protoneutron stars, and heavyion collision data, etc.
The most difficult task is to satisfy simultaneously the heavyioncollision flow and the maximum neutronstar mass constraints. The fulfillment of the flow constraints Danielewicz:2002pu (); Lynch:2009vc () requires a rather soft EoS of isospinsymmetric matter (ISM), whereas the EoS of the betaequilibrium matter (BEM) should be stiff in order to predict the maximum mass of a neutron star to be higher than the measured mass Antoniadis:2013pzd () of the pulsar PSR J0348+0432, being the heaviest among the known pulsars.
Ii Equation of state and pairing gaps
In standard RMF models hyperons and isobars may appear in neutronstar cores already for , which results in a decrease of the maximum neutronstar mass below the observed limit. The problems were named the hyperon puzzle SchaffnerBielich:2008kb (); Djapo:2008au (). Within the RMF models with the fielddependent hadron effective masses and coupling constants the hyperon puzzle is resolved, see Maslov:2015msa (); Maslov:2015wba (). Here we use the MKVORbased models from these works. Most of other constraints on the EoS including the flow constraints are also appropriately satisfied. In Fig 1 we demonstrate the neutron star mass as a function of the central density for the MKVOR model without hyperons and for the MKVORH model with includes hyperons, cf. Fig. 20 and 25 in Maslov:2015wba (). For MKVOR model the maximum neutronstar mass reaches and the DU reaction is allowed for . For MKVORH model the maximum neutronstar mass is . The DU reactions on hyperons , become allowed for . The DU reactions with participation of , and become allowed f or . However, the neutrino emissivity in these processes is not as high as for the standard DU processes on nucleons due to a smaller coupling for the hyperons. Below we use MKVOR and MKVORH EoSs for calculations of the cooling history of neutron stars.
We adopt here all cooling inputs such as the neutrino emissivities, specific heat, crust properties, etc., from our earlier works performed on the basis of the HHJ equation of state (EoS) Blaschke:2004vq (); Grigorian:2005fn (); Blaschke:2011gc (), a stiffer HDD EoS Blaschke:2013vma () and even more stiffer DD2 and DD2vex EoSs Grigorian:2016leu () for the hadronic matter. These works exploit the nuclear medium cooling scenario where the most efficient processes are the medium modified Urca (MMU) processes, and , medium modified nucleon bremstrahlung (MNB) processes , , , and the pairbreakingformation (PBF) processes and . The latter processes are allowed only in supefluid matter.
The results are rather insensitive to the value of the pairing gap since the neutron pairing does not spread in the interior region of the neutron star. We use the same values as we have used in our previous works. Within our scenario we continue to exploit tiny values of the pairing gap. For calculation of the proton pairing gaps we use the same models as in Grigorian:2016leu () but now we exploit EoS of the MKVORH model. The corresponding gaps are shown on the left panel of in Fig. 2.
With the increase of the density in the MKVORH model the hyperons are the first to appear at the density , and then the hyperons appear at . We take the values of the gaps from the calculations TT00 (); TN06 (). The model TT1 uses the bare NDsoft model by the Nijmegen group for interaction and model TTGm uses results of Gmatrix calculations by Lanskoy and Yamamoto LanskYam () at density . The other 3 models include threenucleon forces TNI6u forces for several pairing potentials: NDSoft, Ehime and FGA. On the right panel we show the hyperon pairing gaps which we exploit in this work. are considered unpaired.
The quantity
(1) 
in a dense neutronstar matter (for ) has a minimum for , where is the neutron Fermi momentum. For the minimum occurs for . The value has the meaning of the squared effective pion gap. Of key importance is that we use here the very same density dependence of the effective pion gap as in our previous works, e.g., see Fig. 2 of Grigorian:2016leu (). To be specific we assume a saturation of the pion softening for . We plot this pion gap in Fig. 3.
Iii Results
On the left panel in Figure 4 we show the cooling history of neutron stars calculated using the EoS of MKVOR model without inclusion of hyperons. The demonstrated calculations employ the proton gap following the EEHO model shown in Figure 2, and the solid curve in Figure 3 was used for the effective pion gap.
Hyperons are taken following the TNFGA parameter choice. With the pion gaps given by the solid and dashed curves and with proton gaps following the EEHO, EEHOR, CCDK, CCYms, and T curves we also appropriately describe the cooling history of neutron stars within our scenario.
Iv Conclusion
Thus we have demonstrated that the presently known cooling data can be appropriately described within our nuclear medium cooling scenario, under the assumption that different sources have different masses.
Acknowledgements.
The research was supported by the Ministry of Education and Science of the Russian Federation within the state assignment, project No 3.6062.2017/BY. The work was also supported by Slovak grant VEGA1/0469/15. We acknowledge as well the support of the Russian Science Foundation, project No 171201427.References
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