STEHM: Habitability of Small Rocky Planets
- The paper presents STEHM, an atmosphere-retention model that identifies a critical size (~0.8 R⊕) for sustaining multi-gigayear CO2 atmospheres.
- It couples interior structure, stagnant-lid thermal evolution, melt production, and atmospheric escape to simulate sub-Earth habitability.
- The study highlights how variations in volatile inventory, tectonic regime, and orbital distance can critically affect atmospheric retention outcomes.
Searching arXiv for the cited STEHM and related habitability papers to ground the article. {"query":"(Hill et al., 30 Apr 2026) Smaller Than Earth Habitability Model (Rushby et al., 2018) Long Term Planetary Habitability and the Carbonate-Silicate Cycle", "max_results": 5} Searching arXiv for the STEHM paper and its antecedent long-term habitability model. to=arxiv_search 大发娱乐 皇轩=json {"query":"(Hill et al., 30 Apr 2026) OR (Rushby et al., 2018)", "max_results": 10} Smaller Than Earth Habitability Model (STEHM) is a model developed to answer the question, “What is the minimum size of a rocky planet in the habitable zone of a Sun-like star that can hold onto a long-lived (multi-gigayear) atmosphere?” It examines planets from down to under a coupled treatment of interior structure, stagnant-lid thermal evolution, melt production and CO outgassing, and time-dependent atmospheric escape (Hill et al., 30 Apr 2026). In the broader habitability literature, STEHM sits alongside size-dependent climate and carbon-cycle formulations in which equilibrium and are solved as functions of stellar insolation and planet size, including the carbonate-silicate framework developed for planets between $0.5$ and (Rushby et al., 2018).
1. Scientific question and observational role
STEHM was developed because observational surveys such as Kepler, TESS, and future PLATO are finding ever-smaller planets, and many of these objects will lie in the classical liquid-water habitable zone. The central claim of the model is that orbital placement within the habitable zone is not by itself sufficient: if a planet cannot retain a substantial atmosphere, surface water, let alone life, is impossible (Hill et al., 30 Apr 2026).
Within that framing, STEHM is explicitly an atmosphere-retention model for rocky habitable-zone planets around a Sun-like star. Its stated purpose is to identify a lower-size limit for atmosphere retention and thereby help prioritize which small exoplanets are realistic candidates for follow-up atmospheric characterization with facilities such as JWST and the ELTs (Hill et al., 30 Apr 2026).
The model’s default result is a critical radius of approximately at under Earth-like assumptions. Planets in the $0.7$–0 regime are treated as borderline cases whose atmospheric fate depends on formation conditions and early evolution rather than on radius alone (Hill et al., 30 Apr 2026).
2. Boundary conditions, tectonic regime, and coupled modules
STEHM assumes a stagnant-lid tectonic regime: no plate tectonics, and an immobile lithosphere that conducts internal heat. Composition is taken to be Earth-like bulk silicate mantle composition and core-mantle fraction, with carbon and heat-producing elements (U, Th, K) beginning with Earth-scaled abundances, subject to systematic variation (Hill et al., 30 Apr 2026).
The atmosphere is pure CO1. In the model description this is justified as a “best-case” for retention, because CO2 is the heaviest major greenhouse gas and, with its 3 cooling band, is the hardest common molecule to strip thermally (Hill et al., 30 Apr 2026). The host star is a solar analog with XUV evolution given by Ribas et al. (2005), renormalized so present-day solar XUV matches observations, and capped at 4 modern XUV for the first 5 (Hill et al., 30 Apr 2026).
The model architecture is modular. ExoPlex provides static interior structure and enforces mass-radius-gravity self-consistency from 6 down to 7 for a default core radius fraction of 8. A thermal-degassing code following Foley & Smye (2018) provides melt production, crustal growth, and CO9 outgassing. An atmospheric-escape code based on Tian (2009) and Kite et al. (2020) provides Jeans and hydrodynamic escape rates (Hill et al., 30 Apr 2026).
This set of assumptions sharply defines the scope of the model. It is not a general theory of rocky-planet habitability; it is a specific stagnant-lid, pure-CO0, Sun-like-star framework for determining the lower size limit for long-term atmospheric survival.
3. Interior structure, thermal evolution, outgassing, and escape
STEHM does not use a simple analytical mass-radius law in its primary implementation. Instead, ExoPlex solves the one-dimensional hydrostatic equations
1
with convergence criterion 2 (Hill et al., 30 Apr 2026). For quick estimates, the model description notes that one often uses
3
for rocky planets (Hill et al., 30 Apr 2026).
The thermal evolution of the convecting mantle is governed by
4
where 5 is radiogenic heating from U6, U7, Th8, and K9. Conductive heat loss through the stagnant lithosphere is
0
and the lid thickness follows boundary-layer scaling,
1
with 2–3 (Hill et al., 30 Apr 2026).
Outgassing is tied to melt production. When upwelling mantle crosses the solidus 4, melt fraction 5 produces crust at rate 6. All melt is assumed to reach the surface (“all-melt degassing”) and carries the incompatible carbon: 7 Atmospheric escape is then computed as the sum of thermal and, when appropriate, hydrodynamic loss. The Jeans escape flux is
8
and the energy-limited hydrodynamic rate is
9
with 0 (Hill et al., 30 Apr 2026).
The stellar forcing is time dependent: 1 with 2 normalized so that 3 (Hill et al., 30 Apr 2026). In operational terms, atmospheric longevity in STEHM is the competition between declining interior outgassing and declining stellar XUV-driven escape.
4. Parameterization and numerical integration
The primary parameters varied in STEHM are the initial mantle carbon budget, heat-producing element abundances, initial mantle temperature, core radius fraction, orbital distance, and exobase temperature (Hill et al., 30 Apr 2026).
| Parameter | Default | Variation |
|---|---|---|
| Initial mantle C budget | 4 | 5–6; also 7–8 |
| HPE abundances | Solar | U: 9–$0.5$0Solar; Th: $0.5$1–$0.5$2Solar; K: $0.5$3–$0.5$4Solar |
| Initial mantle $0.5$5 | $0.5$6 | $0.5$7–$0.5$8 |
| Core Radius Fraction | $0.5$9 | 0–1 |
| Orbital distance | 2 | 3–4 |
| Exobase temperature | 5 | 6 |
The model is integrated from 7 to 8, or until the atmosphere is irreversibly lost and degassing stops, with 9 and adaptive stepping to ensure less than 0 change in 1 or 2 per step (Hill et al., 30 Apr 2026). At each time step, the sequence is: update 3; solve 4 via finite difference for 5 and 6; compute melt production, crustal growth, and outgassing; compute 7 and the escape flux; and update 8, recording the atmosphere-loss time if 9 (Hill et al., 30 Apr 2026).
Sensitivity analyses vary each primary parameter independently across its literature range while holding the others at default values, with convergence defined by negligible changes, less than 0 in key variables, on halving 1 (Hill et al., 30 Apr 2026). This design isolates parameter influence rather than sampling a joint uncertainty distribution.
5. Critical radius, atmospheric outcomes, and parameter dependence
Under the default Earth-like case at 2, 3 exobase temperature, and Solar HPE abundance, STEHM finds that planets 4 maintain convectively sustained outgassing that balances escape (Hill et al., 30 Apr 2026).
| Radius | Atmospheric outcome | End state |
|---|---|---|
| 5 | Atmosphere maintained | 6 CO7 |
| 8 | Atmosphere maintained | 9 CO$0.7$0 |
| $0.7$1 | Atmosphere maintained | $0.7$2 CO$0.7$3 |
| $0.7$4 | Lost by $0.7$5, briefly regained, then lost again | $0.7$6 temporary recovery |
| $0.7$7 | Lost early | $0.7$8 loss time |
| $0.7$9 | Lost very early | 00 loss time |
Initial carbon inventory is the most influential parameter. The critical radius is reported as approximately
01
The model description emphasizes that orders-of-magnitude differences from Earth values are required to make a significant difference to atmospheric longevity (Hill et al., 30 Apr 2026).
Other parameters produce smaller shifts. Low-HPE cases leave the critical radius at 02 but reduce end-state CO03 by 04–05; high-HPE cases slightly increase CO06 and prolong degassing, without changing the threshold. A hot-start case at 07 gives the same 08 threshold, whereas a cold-start case at 09 delays outgassing until stellar XUV is weaker and extends the threshold to 10, with 11 end state; the model notes that the origin of cold starts is uncertain (Hill et al., 30 Apr 2026).
Core structure also matters. A no-core end member, maximizing mantle volume together with HPE and C inventory, permits planets 12 to regain atmospheres, including 13 at 14. A high-CRF case of 15 yields slightly lower CO16, but the threshold remains 17 (Hill et al., 30 Apr 2026). Orbital distance shifts the retention boundary as well: at the outer CHZ (18) or OHZ (19), planets 20 retain atmospheres because XUV is weaker, while at the inner CHZ (21) or OHZ (22) only planets 23 retain atmospheres, with pressures of 24–25 (Hill et al., 30 Apr 2026).
6. Carbonate-silicate antecedent and size-dependent habitable-zone shifts
A related sub-Earth habitability formulation is reconstructed from Rushby et al. in the long-term carbon-cycle framework of planetary habitability (Rushby et al., 2018). There, the biogeochemical carbon cycle is represented by a balance between total volcanic-tectonic CO26 outgassing and silicate-weathering CO27 drawdown: 28 Internal heat flux controls spreading, subduction, volcanism, and ridge degassing through
29
and the total CO30 outgassing flux is
31
Silicate weathering is written as
32
with 33, 34, 35, and 36 for the sub-Earth formulation (Rushby et al., 2018).
In that model, geophysical and hydrological parameters are scaled to planet radius. For 37,
38
and for 39,
40
The runoff factor is
41
with 42, while the radiative-convective climate model returns
43
for an N44-H45O-CO46 atmosphere (Rushby et al., 2018).
The reported temperature deviations relative to an Earth twin at the same insolation are size dependent. At 47, 48 gives 49 and 50 gives 51. At 52, the corresponding values are 53 and 54 (Rushby et al., 2018). These deviations are described as almost linear in 55 and larger in magnitude at lower insolation, where the CO56 greenhouse effect is larger (Rushby et al., 2018).
Habitable-zone boundaries in that framework are defined by 57 for the outer edge and 58 for the inner edge. The inner edge is essentially size independent at 59–60, whereas the outer edge moves inward for smaller 61: 62 at 63, 64 at 65, and 66 at 67, with 68 per Earth radius near 69 (Rushby et al., 2018).
7. Interpretation, scope limits, and relation to classical habitable-zone reasoning
STEHM identifies a default critical radius of 70 at 71, and quantifies how initial volatile inventory, HPE budget, mantle temperature, and core size shift that limit between 72 and 73 (Hill et al., 30 Apr 2026). Within its own assumptions, it therefore provides a ranking criterion for small habitable-zone planets: many 74–75 planets are poor targets for transmission or emission spectroscopy because they are expected to have little or no atmosphere, whereas 76–77 objects are conditional candidates whose formation and early evolution histories become decisive (Hill et al., 30 Apr 2026).
The model also states its own limitations. It includes no plate tectonics or weathering feedback; the atmosphere is pure CO78; and non-thermal loss, magnetic fields, and magma-ocean atmospheric escape are not included (Hill et al., 30 Apr 2026). Conversely, the carbonate-silicate framework emphasizes that radiative-convective habitable-zone limits based on user-specified greenhouse abundances ignore the biogeochemical plausibility of those abundances (Rushby et al., 2018).
Taken together, these two model classes separate two distinct constraints on habitability. Atmosphere retention constrains whether a rocky habitable-zone planet can sustain a long-lived atmosphere at all, while long-term carbon-cycle regulation constrains which atmospheric compositions are geochemically maintainable once an atmosphere exists. This suggests that being in the classical liquid-water habitable zone is a necessary but not sufficient condition in both the atmospheric-retention and carbon-cycle senses (Hill et al., 30 Apr 2026, Rushby et al., 2018).