Very old isolated compact objects as dark matter probes

Very old isolated neutron stars and white dwarfs have been suggested to be probes of dark matter. To play such a role, two requests should be fulfilled, i.e., the annihilation luminosity of the captured dark matter particles is above the thermal emission of the cooling compact objects (request-I) and also dominate over the energy output due to the accretion of normal matter onto the compact objects (request-II). Request-I calls for very dense dark matter medium and the critical density sensitively depends on the residual surface temperature of the very old compact objects. The accretion of interstellar/intracluster medium onto the compact objects is governed by the physical properties of the medium and by the magnetization and rotation of the stars and may outshine the signal of dark matter annihilation. Only in a few specific scenarios both requests are satisfied and the compact objects are dark matter burners. The observational challenges are discussed and a possible way to identify the dark matter burners is outlined.