---
title: 'STEHM: Habitability of Small Rocky Planets'
url: https://www.emergentmind.com/topics/smaller-than-earth-habitability-model-stehm
type: topic
---

# STEHM: Habitability of Small Rocky Planets

Searching arXiv for the cited STEHM and related habitability papers to ground the article.
{"query":"arXiv:2605.00170 Smaller Than Earth Habitability Model 1805.06943 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":"2605.00170 OR 1805.06943", "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 \(1.0\,R_\oplus\) down to \(0.5\,R_\oplus\) under a coupled treatment of interior structure, stagnant-lid thermal evolution, melt production and CO\(_2\) outgassing, and time-dependent atmospheric escape [2605.00170]. In the broader habitability literature, STEHM sits alongside size-dependent climate and carbon-cycle formulations in which equilibrium \(p_{\rm CO_2}\) and \(T_{\rm surf}\) are solved as functions of stellar insolation and planet size, including the carbonate-silicate framework developed for planets between \(0.5\) and \(2\,R_\oplus\) [1805.06943].

## 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 [2605.00170].

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 [2605.00170].

The model’s default result is a critical radius of approximately \(0.8\,R_\oplus\) at \(1\,\mathrm{AU}\) under Earth-like assumptions. Planets in the \(0.7\)–\(0.8\,R_\oplus\) regime are treated as borderline cases whose atmospheric fate depends on formation conditions and early evolution rather than on radius alone [2605.00170].

## 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 [2605.00170].

The atmosphere is pure CO\(_2\). In the model description this is justified as a “best-case” for retention, because CO\(_2\) is the heaviest major greenhouse gas and, with its \(15\,\mu\mathrm{m}\) cooling band, is the hardest common molecule to strip thermally [2605.00170]. 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 \(10\times\) modern XUV for the first \(0.7\,\mathrm{Gyr}\) [2605.00170].

The model architecture is modular. ExoPlex provides static interior structure and enforces mass-radius-gravity self-consistency from \(1.0\,R_\oplus\) down to \(0.5\,R_\oplus\) for a default core radius fraction of \(0.55\). A thermal-degassing code following Foley & Smye (2018) provides melt production, crustal growth, and CO\(_2\) outgassing. An atmospheric-escape code based on Tian (2009) and Kite et al. (2020) provides Jeans and hydrodynamic escape rates [2605.00170].

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-CO\(_2\), 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
\[
\frac{dM(r)}{dr}=4\pi r^2 \rho(r),\qquad
\frac{dP(r)}{dr}=-\frac{G M(r)\rho(r)}{r^2},\qquad
\rho(r)=f_{\rm EOS}(P,T),
\]
with convergence criterion \(\max_n |\rho_n^{\rm new}-\rho_n^{\rm old}|<10^{-6}\) [2605.00170]. For quick estimates, the model description notes that one often uses
\[
\frac{M}{M_\oplus}\simeq \left(\frac{R}{R_\oplus}\right)^{3.7}
\]
for rocky planets [2605.00170].

The thermal evolution of the convecting mantle is governed by
\[
C_m M_m \frac{dT_m}{dt}=H(t)-Q_{\rm cond}-Q_{\rm deg},
\]
where \(H(t)=\sum_i H_{0,i}e^{-t/\tau_i}\) is radiogenic heating from U\(^{238}\), U\(^{235}\), Th\(^{232}\), and K\(^{40}\). Conductive heat loss through the stagnant lithosphere is
\[
Q_{\rm cond}=4\pi R_p^2 k_m \frac{(T_m-T_s)}{\delta_{\rm lid}},
\]
and the lid thickness follows boundary-layer scaling,
\[
\delta_{\rm lid}\simeq D\left(\frac{Ra}{Ra_c}\right)^{-\beta},
\]
with \(\beta\approx \tfrac14\)–\(\tfrac13\) [2605.00170].

Outgassing is tied to melt production. When upwelling mantle crosses the solidus \(T_{\rm sol}(P)\), melt fraction \(F\) produces crust at rate \(\dot M_{\rm crust}\). All melt is assumed to reach the surface (“all-melt degassing”) and carries the incompatible carbon:
\[
\frac{dM_{C,\rm atm}}{dt}=X_C \cdot \dot M_{\rm crust}.
\]
Atmospheric escape is then computed as the sum of thermal and, when appropriate, hydrodynamic loss. The Jeans escape flux is
\[
\phi_{\rm Jeans}=\frac{n_{\rm ex}v_{\rm th}}{2\sqrt{\pi}}(1+\lambda)e^{-\lambda},
\quad
v_{\rm th}=\sqrt{\frac{2kT_{\rm ex}}{m}},
\quad
\lambda=\frac{GM_p m}{kT_{\rm ex}R_{\rm exo}},
\]
and the energy-limited hydrodynamic rate is
\[
\dot M_{\rm hydro}=\eta \pi R_{\rm XUV}^2 \frac{F_{\rm XUV}}{GM_p/R_p},
\]
with \(\eta\lesssim 0.3\) [2605.00170].

The stellar forcing is time dependent:
\[
F_{\rm XUV}(t)=\min\!\left[10F_\odot,\;4.5^{-1}F_{\rm Ribas}(t)\right],
\]
with \(F_{\rm Ribas}\) normalized so that \(F(t=4.6\,\mathrm{Gyr})=F_\odot\) [2605.00170]. 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 [2605.00170].

| Parameter | Default | Variation |
|---|---:|---:|
| Initial mantle C budget | \(1.6\times10^{22}\,\mathrm{mol}\) | \(7\times10^{21}\)–\(2\times10^{22}\); also \(1\times10^{21}\)–\(1\times10^{23}\) |
| HPE abundances | Solar | U: \(0.45\)–\(1.92\times\)Solar; Th: \(0.77\)–\(1.88\times\)Solar; K: \(0.35\)–\(3.63\times\)Solar |
| Initial mantle \(T_0\) | \(1900\,\mathrm{K}\) | \(1500\)–\(2200\,\mathrm{K}\) |
| Core Radius Fraction | \(0.55\) | \(0\)–\(0.7\) |
| Orbital distance | \(1.0\,\mathrm{AU}\) | \(0.75\)–\(1.765\,\mathrm{AU}\) |
| Exobase temperature | \(400\,\mathrm{K}\) | \(800,\,2500\,\mathrm{K}\) |

The model is integrated from \(t=0\) to \(10\,\mathrm{Gyr}\), or until the atmosphere is irreversibly lost and degassing stops, with \(\Delta t\sim 1\,\mathrm{Myr}\) and adaptive stepping to ensure less than \(1\%\) change in \(T_m\) or \(M_{C,\rm atm}\) per step [2605.00170]. At each time step, the sequence is: update \(H(t)\); solve \(dT_m/dt\) via finite difference for \(Q_{\rm cond}\) and \(Q_{\rm deg}\); compute melt production, crustal growth, and outgassing; compute \(F_{\rm XUV}(t)\) and the escape flux; and update \(M_{C,\rm atm}\), recording the atmosphere-loss time if \(M_{C,\rm atm}\to 0\) [2605.00170].

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 \(10^{-3}\) in key variables, on halving \(\Delta t\) [2605.00170]. 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 \(1\,\mathrm{AU}\), \(400\,\mathrm{K}\) exobase temperature, and Solar HPE abundance, STEHM finds that planets \(\geq 0.8\,R_\oplus\) maintain convectively sustained outgassing that balances escape [2605.00170].

| Radius | Atmospheric outcome | End state |
|---|---|---|
| \(1.0\,R_\oplus\) | Atmosphere maintained | \(137\,\mathrm{bar}\) CO\(_2\) |
| \(0.9\,R_\oplus\) | Atmosphere maintained | \(79\,\mathrm{bar}\) CO\(_2\) |
| \(0.8\,R_\oplus\) | Atmosphere maintained | \(22\,\mathrm{bar}\) CO\(_2\) |
| \(0.7\,R_\oplus\) | Lost by \(\sim 0.6\,\mathrm{Gyr}\), briefly regained, then lost again | \(\sim 0.08\,\mathrm{bar}\) temporary recovery |
| \(0.6\,R_\oplus\) | Lost early | \(\sim 0.4\,\mathrm{Gyr}\) loss time |
| \(0.5\,R_\oplus\) | Lost very early | \(\sim 0.03\,\mathrm{Gyr}\) loss time |

Initial carbon inventory is the most influential parameter. The critical radius is reported as approximately
\[
R_{\rm crit}\approx
\begin{cases}
0.9\,R_\oplus, & C_{\rm init}=7\times10^{21}\,\mathrm{mol} \\
0.8\,R_\oplus, & C_{\rm init}=2\times10^{22}\,\mathrm{mol} \\
0.6\,R_\oplus, & C_{\rm init}=1\times10^{23}\,\mathrm{mol} \\
1.0\,R_\oplus, & C_{\rm init}=1\times10^{21}\,\mathrm{mol}.
\end{cases}
\]
The model description emphasizes that orders-of-magnitude differences from Earth values are required to make a significant difference to atmospheric longevity [2605.00170].

Other parameters produce smaller shifts. Low-HPE cases leave the critical radius at \(\sim 0.8\,R_\oplus\) but reduce end-state CO\(_2\) by \(20\)–\(30\%\); high-HPE cases slightly increase CO\(_2\) and prolong degassing, without changing the threshold. A hot-start case at \(2200\,\mathrm{K}\) gives the same \(\sim 0.8\,R_\oplus\) threshold, whereas a cold-start case at \(1500\,\mathrm{K}\) delays outgassing until stellar XUV is weaker and extends the threshold to \(\sim 0.7\,R_\oplus\), with \(14\,\mathrm{bar}\) end state; the model notes that the origin of cold starts is uncertain [2605.00170].

Core structure also matters. A no-core end member, maximizing mantle volume together with HPE and C inventory, permits planets \(\geq 0.7\,R_\oplus\) to regain atmospheres, including \(0.7\,\mathrm{bar}\) at \(0.7\,R_\oplus\). A high-CRF case of \(0.70\) yields slightly lower CO\(_2\), but the threshold remains \(\sim 0.8\,R_\oplus\) [2605.00170]. Orbital distance shifts the retention boundary as well: at the outer CHZ (\(1.676\,\mathrm{AU}\)) or OHZ (\(1.765\,\mathrm{AU}\)), planets \(\geq 0.7\,R_\oplus\) retain atmospheres because XUV is weaker, while at the inner CHZ (\(0.95\,\mathrm{AU}\)) or OHZ (\(0.75\,\mathrm{AU}\)) only planets \(\geq 0.8\,R_\oplus\) retain atmospheres, with pressures of \(0.4\)–\(10\,\mathrm{bar}\) [2605.00170].

## 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 [1805.06943]. There, the biogeochemical carbon cycle is represented by a balance between total volcanic-tectonic CO\(_2\) outgassing and silicate-weathering CO\(_2\) drawdown:
\[
F_{\rm out}(\mathrm{CO_2}) = F_{\rm weather}(p_{\rm CO_2},T_{\rm surf},R_E).
\]
Internal heat flux controls spreading, subduction, volcanism, and ridge degassing through
\[
Q(t,M)=Q_\oplus \left(\frac{M}{M_\oplus}\right)^{0.452}\left[1+\left(\frac{t}{t_0}\right)\right]^{-0.7},
\]
and the total CO\(_2\) outgassing flux is
\[
F_{\rm out}(\mathrm{CO_2})=F_{{\rm out},\oplus}\left(\frac{Q(t,M)}{Q_\oplus}\right)^2.
\]
Silicate weathering is written as
\[
F_{\rm weather}=F_{{\rm weather},\oplus}
\left(\frac{p_{\rm CO_2}}{p_{{\rm CO_2},\oplus}}\right)^{\beta}
\exp\!\bigl[B(T_{\rm surf}-T_\oplus)\bigr]R_E k,
\]
with \(\beta=0.3\), \(B=13.7\,\mathrm{K^{-1}}\), \(T_\oplus=288\,\mathrm{K}\), and \(k=0.5\) for the sub-Earth formulation [1805.06943].

In that model, geophysical and hydrological parameters are scaled to planet radius. For \(R\leq 1\,R_\oplus\),
\[
\frac{M}{M_\oplus}=\left(\frac{R}{R_\oplus}\right)^{3.268},
\]
and for \(R>1\,R_\oplus\),
\[
\frac{M}{M_\oplus}=\left(\frac{R}{R_\oplus}\right)^{3.65}.
\]
The runoff factor is
\[
R_E=\left(\frac{R}{R_\oplus}\right)^2
\left[1-C\frac{A_{\rm ocean}}{A_{\rm planet}}\right],
\]
with \(C=E_{\rm land}/E_{\rm ocean}\approx 0.176\), while the radiative-convective climate model returns
\[
T_{\rm surf}=\mathcal{R}[S,p_{\rm CO_2}]
\]
for an N\(_2\)-H\(_2\)O-CO\(_2\) atmosphere [1805.06943].

The reported temperature deviations relative to an Earth twin at the same insolation are size dependent. At \(S=1\,S_\oplus\), \(R=2\,R_\oplus\) gives \(\Delta T_{\rm surf}\approx +5\,\mathrm{K}\) and \(R=0.5\,R_\oplus\) gives \(\Delta T_{\rm surf}\approx -2\,\mathrm{K}\). At \(S=0.75\,S_\oplus\), the corresponding values are \(\approx +12\,\mathrm{K}\) and \(\approx -8\,\mathrm{K}\) [1805.06943]. These deviations are described as almost linear in \((R/R_\oplus-1)\) and larger in magnitude at lower insolation, where the CO\(_2\) greenhouse effect is larger [1805.06943].

Habitable-zone boundaries in that framework are defined by \(T_{\rm surf}=273\,\mathrm{K}\) for the outer edge and \(T_{\rm surf}=343\,\mathrm{K}\) for the inner edge. The inner edge is essentially size independent at \(\approx 1.13\)–\(1.15\,S_\oplus\), whereas the outer edge moves inward for smaller \(R\): \(\simeq 0.95\,S_\oplus\) at \(0.5\,R_\oplus\), \(\simeq 0.75\,S_\oplus\) at \(1.0\,R_\oplus\), and \(\simeq 0.45\,S_\oplus\) at \(2.0\,R_\oplus\), with \(\partial S_{\rm outer}/\partial R\approx -0.2\) per Earth radius near \(R=1\) [1805.06943].

## 7. Interpretation, scope limits, and relation to classical habitable-zone reasoning

STEHM identifies a default critical radius of \(\sim 0.8\,R_\oplus\) at \(1\,\mathrm{AU}\), and quantifies how initial volatile inventory, HPE budget, mantle temperature, and core size shift that limit between \(0.6\) and \(1.0\,R_\oplus\) [2605.00170]. Within its own assumptions, it therefore provides a ranking criterion for small habitable-zone planets: many \(0.5\)–\(0.8\,R_\oplus\) planets are poor targets for transmission or emission spectroscopy because they are expected to have little or no atmosphere, whereas \(0.7\)–\(0.8\,R_\oplus\) objects are conditional candidates whose formation and early evolution histories become decisive [2605.00170].

The model also states its own limitations. It includes no plate tectonics or weathering feedback; the atmosphere is pure CO\(_2\); and non-thermal loss, magnetic fields, and magma-ocean atmospheric escape are not included [2605.00170]. 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 [1805.06943].

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 [2605.00170][1805.06943].

Source: https://www.emergentmind.com/topics/smaller-than-earth-habitability-model-stehm