with the direct Urca process. The upper curve shows cooling of low-mass stars via neutron-neutron bremsstrahlung neutrino emission (as in Figs. \[cool4\]c and d). The lower curve presents fast cooling of a maximum-mass star with direct Urca process open in the inner core. Any value of $T_{\rm s}^\infty$ between these two curves can be explained by a cooling of a star with some intermediate mass. As discussed above, this scenario explains all the data and predicts the existence of very cold neutron stars.

Fig. \[exotica\]c refers to the scenario with pion-condensation. The upper cooling curve is the same as in Fig. \[exotica\]b (because low-mass stars have no inner cores), but the lower curve goes essentially higher (because the neutrino emission due to pion condensation is lower than due to direct Urca process). This scenario also explains all the sources and predicts the existence of cold stars, but not so cold as in the direct Urca case.

Fig. \[exotica\]d is a similar plot for the scenario with kaon-condensation in the inner core. High-mass stars cool slower than in the previous scenarios, but all the data are still explained and a population of stars slightly colder than the observed ones is predicted.

