On Cooling of Neutron Stars With Stiff Equation of State Including Hyperons
The existence of high mass () pulsars PSR J1614-2230 and PSR J0348-0432 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 neutron-star mass below the measured values of masses for PSR J1614-2230 and PSR J0348-0432 pulsars, if one exploits ordinary relativistic mean-field (RMF) models (hyperon puzzle). However, within a RMF EoS with a scaled hadron effective masses and coupling constants the maximum neutron-star mass remains above even when hyperons are included. Also other important constraints on the equation of state, e.g. the flow constraint from heavy-ion collisions are to be fulfilled. We demonstrate how a satisfactory explanation of all existing observational data for the temperature-age relation is reached within the ânuclear medium coolingâ scenario with a relativistic-mean-field EoS with a -scaled hadron effective masses and coupling constants including hyperons.
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;
yield a mass-radius relation comparable with the empirical constraints including recent gravitation wave LIGO-Virgo detection TheLIGOScientific:2017qsa ();
being extended to non-zero temperature (for where is the critical temperature of the deconfinement), appropriately describe supernova explosions, proto-neutron stars, and heavy-ion collision data, etc.
The most difficult task is to satisfy simultaneously the heavy-ion-collision flow and the maximum neutron-star mass constraints. The fulfillment of the flow constraints Danielewicz:2002pu (); Lynch:2009vc () requires a rather soft EoS of isospin-symmetric matter (ISM), whereas the EoS of the beta-equilibrium 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 neutron-star cores already for , which results in a decrease of the maximum neutron-star mass below the observed limit. The problems were named the hyperon puzzle SchaffnerBielich:2008kb (); Djapo:2008au (). Within the RMF models with the field-dependent hadron effective masses and coupling constants the hyperon puzzle is resolved, see Maslov:2015msa (); Maslov:2015wba (). Here we use the MKVOR-based 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 neutron-star mass reaches and the DU reaction is allowed for . For MKVORH model the maximum neutron-star 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 pair-breaking-formation (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 ND-soft model by the Nijmegen group for interaction and model TTGm uses results of G-matrix calculations by Lanskoy and Yamamoto LanskYam () at density . The other 3 models include three-nucleon forces TNI6u forces for several pairing potentials: ND-Soft, Ehime and FG-A. On the right panel we show the hyperon pairing gaps which we exploit in this work. are considered unpaired.
in a dense neutron-star 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.
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 TN-FGA 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.
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 VEGA-1/0469/15. We acknowledge as well the support of the Russian Science Foundation, project No 17-12-01427.
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