Jupiter internal structure: the effect of different equations of state (Corrigendum)
Résumé
After publication of Miguel et al. (2016), we found an error in the calculation of Jupiter internal structure when using REOS3b-H and REOS3sc-H equations of state (EOSs). The lowest pressure in those tables (see appendix in Miguel et al. 2016) was P = 31.623 bars, while our calculations of Jupiter's internal structure started at P = 1 bar. This difference lead to an extrapo-lation of the EOS of hydrogen, which caused an overestimation of Jupiter's interior calculations. We fixed this problem by using the SCvH-H EOS (Saumon et al. 1995) for pressures lower than 6 kbar and merged it with REOS3b-H and REOS3sc-H (correspondingly) for higher pressures. There was also a small offset of ∼2% in the specific entropy at low pressures (P < 1 kbar) between the REOS3b and REOS3sc tables with respect to the SCvH table, caused by a small difference in the s 0 parameter chosen to calculate the specific entropy. In Miguel et al. (2016), this parameter was determined calculating s − s 0 from equation 14 at T = 29 500 K and ρ = 0.036 g cc −1 and then using the value of s in SCvH EOS at the same conditions. This temperature was much higher than the border condition in our interior models (at 165 K) and the intrinsic differences between the tables at those conditions lead to the offset in the specific entropy. In order to fix this issue, we performed new calculations at T = 150 K and ρ = 0.3 g cc −1 , which corrected the offset in S between the EOSs. We also improved our method for calculating the gravity harmonics. Following Nettelmann (2017) and using results calculated with the more accurate CMS method (Hubbard 2013) as a comparison, we corrected the coefficients of the theory of figures of fourth order and improved the numerical handling of density discontinuities. With these improvements, the results agree with Nettelmann et al. (2012) and Hubbard & Militzer (2016). The differences from previous calculations are of the Full appendix tables are only available at CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsarc.u-strasbg.fr/viz-bin/qcat?J/A+A/618/C2 order of 6 × 10 −7 in J 2 , 1.4 × 10 −8 in J 4 , 2 × 10 −6 in J 6 and 6 × 10 −8 in J 8. We thus ran our simulations again and show the corrected figures below. We also include the corrected EOS tables (see Appendix A). For comparison purposes, the MH13+SCvH table for pure hydrogen was also recalculated by accounting for the entropy of mixing. Along a fixed adiabat, Jupiter's internal temperature is found to be 10% smaller with the new REOS3b table compared to the previous calculation. Nevertheless, the trend remains the same, with REOS3b models being the warmest and MH13 ones the coldest. The core masses found with REOS3b are now 5 < M core < 16 M Earth , i.e., the lower limit is about 2 M Earth smaller than previous calculations. The mass of heavy elements found with REOS3b is significantly lower than previously, being now 13 < M Z < 27 M Earth instead of 20 − 40 M Earth previously. The space of solutions of M core and M Z found with MH13 is almost the same as before with 1 < M Z < 8 M Earth and 10 < M core < 17 M Earth compared with our previous estimations 1-7 M Earth and 11-17 M Earth , correspondingly. Due to the changes in the calculation of gravity harmonics, the results using SCvH also changed. The core masses remain unchanged but M Z changed from 18-44 M Earth to 23-36 M Earth. The new results still show that the internal structure of Jupiter strongly depends on the accuracy of the EOSs. The differences between the MH13 and REOS3 solutions remain significant, in particular in terms of the total mass of heavy elements in the envelope. The location at which helium rain occurs still controls to a large extent the inferred core mass. The main conclusions of the paper therefore remain unchanged. We wish to apologise for this oversight. We thank Burkhard Militzer for noticing the EOS issue. We also thank William B. Hubbard and Nadine Nettelmann who shared codes and algorithms that allowed an accurate comparison of gravitational moments. Article published by EDP Sciences C2, page 1 of 6
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