Resolution of a paradox: SDSS J1257+5428 can be explained as a descendant of a cataclysmic variable with an evolved donor

Context. The binary system SDSS J1257+5428 consists of an extremely low-mass white dwarf (WD) with a mass ranging from ∼0.1 to ∼0.24 M ⊙ , along with a more massive WD companion of approximately 1 M ⊙ that is significantly hotter. Recently, a tertiary WD orbiting this binary was discovered, setting a lower limit for the total age (∼4 Gyr) of the triple and providing further constraints on SDSS J1257+5428 that could be used to constrain its formation pathways. Up to now, the existence of this system has been described as paradoxical since tested models for its formation cannot account for its properties. Aims. Here we investigate under which conditions SDSS J1257+5428 could be understood as a descendant of a cataclysmic variable with an evolved donor star, which is a scenario that has never been explored in detail. Methods. We used the rapid BSE code for pre-common-envelope (CE) evolution and the detailed MESA code for post-CE evolution to carry out binary evolution simulations and searched for potential formation pathways for SDSS J1257+5428 that lead to its observed characteristics. For the post-CE evolution, we adopted a boosted version of the CARB model, which has been successfully used to explain the properties of close binary stars hosting evolved solar-type stars. Results. We find that SDSS J1257+5428 can be explained as a post-cataclysmic-variable system if (i) the progenitor of the extremely low-mass WD was initially a solar-type star that evolved into a subgiant before the onset of mass transfer and underwent hydrogen shell flashes after the mass transfer stopped, (ii) the massive WD was highly or entirely rejuvenated during the cataclysmic variable evolution, and (iii) magnetic braking was strong enough to make the evolution convergent. In this case, the torques due to magnetic braking need to be stronger than those provided by the CARB model by a factor of ∼100. Conclusions. We conclude that SDSS J1257+5428 can be reasonably well explained as having originated from a cataclysmic variable that hosted an evolved donor star and should no longer be regarded as paradoxical. If our formation channel is correct, our findings provide further support that stronger magnetic braking acts on progenitors of (i) close detached WD binaries, (ii) close detached millisecond pulsar with extremely low-mass WDs, (iii) AM CVn binaries, and (iv) ultra-compact X-ray binaries, in comparison to the magnetic braking strength required to explain binaries hosting main-sequence stars and single main-sequence stars.