Context. The mass-radius relation (MRR) of white dwarf (WD) stars is a fundamental issue. It has been studied and improved from its original version given by S. Chandrasekhar in 1931. The MRR represents a powerful tool to interpret observations of these compact objects. Originally, it was studied by constructing zero-temperature models. After that, many studies were performed considering finite temperature effects, improved versions of the equation of state, and relativistic effects. Aims. The main goal of this paper is to build an MRR employing models of WDs obtained through the evolution of close binary systems (CBSs). This scenario allows for the formation of low-mass helium white dwarfs, thus permitting an extension of the MRR to the low-mass range. On the other hand, our models provide a detailed orbital evolution of the systems, which adds an important ingredient to take into account. Methods. We computed the evolution of a set of CBSs composed by a normal solar-composition donor star together with a compact object (a neutron star or a black hole) that undergo stable mass transfer. We cover a range of initial masses of the donor between 0.50 and 7.00 M⊙ and initial orbital periods from 0.5 to 10 days. For the accreting compact objects we consider a neutron star of 1.40 M⊙ or a black hole of 4.0, 6.0, 8.0, or 10.0 M⊙. Results. We find that, as expected, WDs with moderate or high masses (M > 0.5 M⊙) follow a monotonic MRR with a radius decreasing as a function of its mass and as the objects cool down. However, this monotonic behavior does not occur in the case of low-mass objects, especially in those with rather high effective temperatures, between 30 and 60 kK (1 kK = 103 K). This is so because the outermost hydrogen layer, whose mass fraction depends on the details of the previous evolution, occupies a non-negligible fraction of the stellar radius. Conclusions. The MRR of intermediate or high-mass WDs (M > 0.4 M⊙) formed by binary evolution shows the expected behavior and is in agreement with the MRRs previously presented by other authors. Moreover, this MRR looks similar to that of WDs coming from isolated evolution: the radius of the star decreases as the object cools down. We extended the MRR to the region of low-mass WDs, namely, to masses as low as 0.16 M⊙. Notably, the MRR no longer has a univocal behavior at this point. That is, for a given effective temperature, we obtained models with different values of mass for the same radius. Our analysis shows that the position of a remnant on the MRR at the beginning of the cooling track heavily depends on the details of its previous evolution. Therefore, at M < 0.40 M⊙, for a given mass and effective temperature, there is a wide range of possible radii. This should not be overlooked when using the MRR for low mass WDs.
Pereiras et al. (Thu,) studied this question.