Rest mass contribution

The Helmholtz free energy of homogeneous nucleonic matter is denoted \(f_{\mathrm{Hom}}(n_n^{\prime},n_p^{\prime},T)\) in Du et al. (2022) where the primes refer to the local nucleonic densities in the gaseous or low-density phase. The rest mass energy density corresponding to this part of the full free energy density is then \(m_n n_n^{\prime} + m_p n_p^{\prime}\). The rest mass energy density is also omitted from the free energy density of nuclei, referred to in Du et al. (2022) as \(\sum_i f_i\). The rest mass energy density associated with the nuclear contribution is then \(\sum_i N_i n_i m_n + \sum_i Z_i n_i m_p\). Multiplying \(f_{\mathrm{Hom}}\) by \(\xi\) and then combining these two contributions to the free energy, we find that the total rest mass energy density (which is not included in the published tables) is:

\[f_{\mathrm{rest}} \equiv \xi n_n^{\prime} m_n + n_p^{\prime} \xi m_p + \sum_i N_i n_i m_n + \sum_i Z_i n_i m_p\]

then by Eq. 2 in Du et al. (2022) this is equal to

\[f_{\mathrm{rest}} = n_B (1-Y_e) m_n + n_B Y_e m_p \, .\]

Dividing this by \(n_B\) gives the contribution which has been subtracted from Fint as described in Table Format.