Advances in stellar interior modeling are being driven by new data from large-scale surveys and high-precision photometric and spectroscopic observations. Here we focus on single stars in normal evolutionary phases; we will not discuss the many advances in modeling star formation, interacting binaries, supernovae, or neutron stars. We review briefly: 1) updates to input physics of stellar models; 2) progress in two and three-dimensional evolution and hydrodynamic models; 3) insights from oscillation data used to infer stellar interior structure and validate model predictions (asteroseismology). We close by highlighting a few outstanding problems, e.g., the driving mechanisms for hybrid gamma Dor/delta Sct star pulsations, the cause of giant eruptions seen in luminous blue variables such as eta Car and P Cyg, and the solar abundance problem.
Context: The standard cooling models of neutron stars predict temperatures $T<10^{4}$ K for ages $t>10^{7}$ yr. However, the likely thermal emission detected from the millisecond pulsar J0437-4715, of spin-down age $t_s \sim 7\times10^9$ yr, implies a temperature $T\sim 10^5$ K. Thus, a heating mechanism needs to be added to the cooling models in order to obtain agreement between theory and observation. Aims: Several internal heating mechanisms could be operating in neutron stars, such as magnetic field decay, dark matter accretion, crust cracking, superfluid vortex creep, and non-equilibrium reactions ("rotochemical heating"). We study these mechanisms in order to establish which could be the dominant source of thermal emission from old pulsars. Methods: We show by simple estimates that magnetic field decay, dark matter accretion, and crust cracking mechanism are unlikely to have a significant effect on old neutron stars. The thermal evolution for the other mechanisms is computed using the code of Fern\'andez and Reisenegger. Given the dependence of the heating mechanisms on the spin-down parameters, we study the thermal evolution for two types of pulsars: young, slowly rotating "classical" pulsars and old, fast rotating millisecond pulsars. Results: We find that magnetic field decay, dark matter accretion, and crust cracking do not produce detectable heating of old pulsars. Rotochemical heating and vortex creep can be important both for classical pulsars and millisecond pulsars. More restrictive upper limits on the surface temperatures of classical pulsars could rule out vortex creep as the main source of thermal emission. Rotochemical heating in classical pulsars is driven by the chemical imbalance built up during their early spin-down, and therefore strongly sensitive to their initial rotation period.
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