Evolutionary pathways toward survival of a thick CO2- or SO2-rich atmosphere on the lava world TOI-561 b

Abstract: Rocky planets evolve through the exchange of volatiles between their interiors and atmospheres, an interplay still poorly constrained by observations. Remarkably, highly irradiated ultrashort-period (USP) exoplanets may offer a window into this exchange -- some retain low bulk densities compatible with volatile-rich envelopes surrounding rocky interiors, indicating possible secondary atmospheres. TOI-561 b is a prime example, with a bulk density of $4.3\pm0.4$ g cm$<sup>{-3}$ and recent JWST observations favoring a thick volatile atmosphere overlying a dayside magma ocean. Here, we investigate the evolutionary pathways allowing TOI-561 b to retain a substantial atmosphere over gigayears using the PROTEUS coupled interior--atmosphere framework. We explore different core radius fractions, Bond albedos, atmospheric escape efficiencies, mantle redox states, and initial C--H--O--N--S volatile inventories, under in situ evolution and late inward migration. Over half of our simulations leave a bare interior too dense to match observations. Successful cases favor a volatile-rich origin ($\lesssim200$ Earth oceans of hydrogen, S/H $\le10$, and N/H $\le1$), an oxidized mantle ($f$O$_2 \gtrsim \mathrm{IW}+4$), a small iron core ($\le 0.40$ for the core radius fraction), and low escape efficiency ($ε\lesssim 10<sup>{-3}$) in the hydrodynamic escape regime. At present, TOI-561 b is consistent with a global magma ocean beneath a thick (surface pressure $\approx 10<sup>{3}$--$10<sup>{4}$ bar), high mean molecular weight atmosphere ($38$--$60$ g mol$<sup>{-1}$). Two archetypes emerge, differentiated by bulk sulfur content: a CO$_2$-dominated and an SO$_2$-dominated atmosphere. Migration is viable but not required to reproduce the observations. Our study illustrates how interior--atmosphere coupling governs atmospheric retention on irradiated rocky planets.