Extended Data Fig. 9: The minimum value of the efficiency parameter αλCE required for WD 1856 b to form via common-envelope evolution as a function of the progenitor stellar mass.
The two dashed curves show the minimum αλCE values from our analytic calculation (equation (11)) required for a 15MJ object to eject the primary star’s envelope. The purple dashed curve is taken directly from equation (11), and the brown dashed curve results if the progenitor star has lost 0.1M☉ in a stellar wind by the time of the common envelope. The three solid curves show the minimum αλCE computed directly from MIST tracks in three different situations: before the star reaches the AGB (red), before more than 30% of the star’s envelope mass has been lost (black), and at any point in the star’s evolution, regardless of the mass lost (blue). Stars in the grey region at low masses evolve too slowly for the system to have left the main sequence more than 5.85 Gyr ago and are not viable solutions. For values of αλCE > 1 (horizontal grey line), one must invoke the internal energy of the star to help to unbind the envelope during the common-envelope phase. Before mass is lost during the AGB phase, it is difficult for WD 1856 b to eject the common envelope, but it is possible that WD 1856 b could have ejected its progenitor’s envelope if the common-envelope phase began after the progenitor reached the AGB. We have smoothed the lower two curves to remove some unphysical scatter that is probably due to numerical artefacts in the model grids.
