Greenhouse Effect¶

The greenhouse effect is the mechanism by which atmospheric gases trap outgoing thermal infrared radiation and re-emit part of it back toward the surface, raising surface temperatures above the bare-rock radiative equilibrium. It is fundamental to understanding both the current Martian climate and the warming trajectory under any terraforming scenario.

Blackbody radiation and the Stefan-Boltzmann law¶

Any body at temperature \(T\) emits thermal radiation. For a perfect blackbody, the total power emitted per unit area is given by the Stefan-Boltzmann law (Stefan, 1879; Boltzmann, 1884):

\[ F = \sigma T^4 \]

where \(\sigma = 5.670 \times 10^{-8}\,\text{W\,m}^{-2}\,\text{K}^{-4}\) is the Stefan-Boltzmann constant.

Real surfaces are characterised by emissivity \(\varepsilon \in [0, 1]\), so the actual upwelling IR flux is:

\[ F_\text{IR,up} = \varepsilon\,\sigma\,T^4 \]

For bare Martian regolith, \(\varepsilon \approx 0.95\) (Putzig & Mellon, 2007).

Radiative equilibrium temperature¶

Without an atmosphere, a planet's equilibrium temperature \(T_\text{eq}\) is set by the balance between absorbed solar radiation and emitted thermal IR:

\[ T_\text{eq} = \left[\frac{S_{1\,\text{AU}}(1-\alpha)}{4\,\sigma\,r^2}\right]^{1/4} \]

The factor of 4 accounts for the ratio of the planet's cross-sectional area (which intercepts sunlight) to its total surface area (which radiates). For Mars, this gives \(T_\text{eq} \approx 210\,\text{K}\), close to the observed mean surface temperature of about \(210\,\text{K}\) — Mars has a very weak natural greenhouse effect today (Haberle, 1998).

The greenhouse effect¶

Real atmospheres absorb some fraction of the upwelling surface IR and re-emit it both upward and downward. The downwelling component adds energy back to the surface, raising \(T\) above \(T_\text{eq}\). The effective downwelling IR flux from the atmosphere is:

\[ F_\text{IR,down} = \varepsilon_\text{atm}\,\sigma\,T^4 \]

where \(\varepsilon_\text{atm}\) is the effective atmospheric IR emissivity (\(0\) = transparent, \(1\) = fully opaque). The net surface energy budget then includes both fluxes:

\[ F_\text{net} = F_\text{abs} + F_\text{IR,down} - F_\text{IR,up} \]

In tform, the greenhouse amplification is represented by a single dimensionless factor \(\gamma \geq 1\) that scales the effective absorbed solar flux:

\[ \frac{dT}{dt} = \frac{F_\text{net}}{C_\text{eff}} \]

Radiative forcing from greenhouse gases¶

When new greenhouse gases are added to an atmosphere, the change in net downward radiative flux at the tropopause is called the radiative forcing \(\Delta F\) (W m⁻²). Positive \(\Delta F\) warms the surface (IPCC AR6, Chapter 7).

For a gas at concentration \(C\) (in ppb), the forcing scales approximately linearly at low concentrations:

\[ \Delta F = \eta \cdot C \]

where \(\eta\) is the radiative forcing efficiency (W m⁻² ppb⁻¹). This is the quantity tabulated by Marinova et al. (2005) for Mars-specific conditions (where the absence of water-vapour overlap bands and the thinner CO₂ column make \(\eta\) differ significantly from Earth IPCC values).

The total forcing from a mixture of gases is the sum over all species:

\[ \Delta F_\text{total} = \sum_i \eta_i \cdot C_i \]

Global warming potential¶

The Global Warming Potential (GWP) of a gas is its time-integrated forcing relative to CO₂ over a 100-year horizon. For terraforming, GWP is less relevant than atmospheric lifetime: long-lived gases (CF₄ at >50,000 yr, SF₆ at 3,200 yr) are strongly preferred because injected quantities remain effective for geological timescales.