ner_tags: O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Tag I-Tag I-Tag O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O B-Citation O O B-Citation O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Citation O O O O O O O O O O O O O B-Citation O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula B-Citation O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O B-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula O O O O O O O O O O O O B-Citation I-Citation I-Citation I-Citation O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B-Software O O B-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula I-Formula
tokens: show a comparison between the new and the old interpolation method to approximate the temporal evolution of the mass (Fig. A1 ) and the Helium core radius (Fig. A2 ) of a star with mass M ZAMS = 70 M ⊙ and metallicity Z = 2.0 × 10 −2 . It is apparent that the new interpolation method works much better than the previous one, especially for estimating the values of the physical parameters that are important for evolved stars (such as the Helium core radius). It is worth noting that the new version of sevn can also interpolate tables at different metallicity. That is, if the pre-evolved tracks are not available for the metallicity Z s , sevn first evaluates and , where Z 1 ≡ Z s − Δ Z , Z 2 ≡ Z s + Δ Z and Δ Z is step in the metallicity grid. Then, the value is where and . APPENDIX B: THE FITTING FORMULA FOR PPISNE AND PISNE The new version of sevn includes a fitting formula to derive M rem accounting for PPISNe and PISNe. This formula has been obtained by fitting the masses of the compact remnants shown in tables 1 and 2 of Woosley ( 2017 ) as a function of the final Helium mass fraction and of the Helium core mass M He,f of the stars. Using the following fitting formulas, we get the parameter α P so that M rem = α P M rem, no psn , where M rem, no psn is the mass of the compact remnant we would obtain without PPISN/PISN (see equation 1 ). First, we define the following quantities We then express α P as a function of , , and M He,f : APPENDIX C: COMPARISON WITH <xref ref-type="bibr" rid="bib62">W17</xref> In this appendix, we compare our results with those presented in Woosley ( 2017 ). Fig. C1 is a detail of Fig. 2 , where we also plot the compact remnant masses taken from table 2 of W17 (black points). W17 explores the evolution of stars at Z ≃ 0.1 Z ⊙ in the mass range 70 M ZAMS / M ⊙ 150, also varying the amount of mass-loss through stellar winds to mimic the results expected from stars at lower metallicity. The fitting formula we implemented in sevn gives results in agreement with those of W17 . We obtain a comparable maximum BH mass (this work:∼55 M ⊙ , W17 : 58 M ⊙ ) and a similar mass range where a dearth of compact remnants is observed (this work: 55 ≲ M rem / M ZAMS ≲ 120, W17 : 58 M rem / M ⊙ 133). The differences between this work and W17 are due to the parameters M He,f and used for the PPISNe and PISNe fitting formula (see Appendix B ). Indeed, the values of M He,f and M fin , for different M ZAMS and metallicity, strongly depend on the adopted stellar evolutionary models. Figure C1. A detail of Fig. 2 that shows the ranges 40 ≤ M ZAMS / M ⊙ ≤ 160 and 0 ≤ M rem / M ⊙ ≤ 75. Black points are the compact remnant masses taken from table 2 of Woosley ( 2017 ). APPENDIX D: STELLAR EVOLUTION TABLES Tables D1 , D2 and D3 show the final parameters of various progenitor stars and the mass of their compact remnants when PPISNe and PISNe are turned on and when PPISNe and PISNe are switched off in sevn , for Z = 2.0 × 10 −2 , 2.0 × 10 −3