Determine the depth-dependent internal heat flux

Derive the depth-dependent internal heat flux in evolving sub-Neptune envelopes by accounting for planetary contraction, interior cooling, tidal heating, and the latent and gravitational energy released by silicate condensation.

Background

The atmospheric structure calculations assume that the heat flux through the deep non-convective region equals the planet's emergent radiative flux. The paper explains that this assumption may fail during early evolution, when energy is partitioned among envelope contraction and interior cooling, and when additional energy sources operate at depth.

A time-dependent treatment is needed to determine the actual luminosity entering the radiative-gradient calculation. Because that flux controls the super-adiabaticity and extent of deep non-convective regions, resolving it is important for predicting planetary radii and evolution.

References

The actual value of $L$ at depth cannot be adequately analyzed without a time-evolving model, which we leave for future work.

— Magnesium silicate condensation in sub-Neptune envelopes: the fundamental link between chemistry, structure, and observables  (2608.24873 - Misener et al., 25 Aug 2026) in Section 'Atmospheric structure and evolution: open questions'

However, it is not clear whether SiO(s) would really be expected at such depths, as its tabulated equilibrium constants were studied in the context of stellar winds, with pressures $\sim 10{-4}$~bar and temperatures $\lesssim 1800$~K.

— Magnesium silicate condensation in sub-Neptune envelopes: the fundamental link between chemistry, structure, and observables  (2608.24873 - Misener et al., 25 Aug 2026) in Section 'Sensitivity to condensate speciation'