Solar Radiation¶
Solar radiation is the primary energy input to any planetary climate system. This page covers the geometry and physics of how solar energy reaches a planetary surface, from the top of atmosphere down to the ground. These relations are applied in the Mars model but hold generically for any planet.
Solar constant and the inverse-square law¶
The solar irradiance at a distance \(r\) from the Sun follows the inverse-square law:
where \(S_{1\,\text{AU}} = 1361\,\text{W\,m}^{-2}\) is the solar constant — the total solar irradiance at 1 AU (Kopp & Lean, 2011) — and \(r\) is in AU.
At Mars's semi-major axis (\(a = 1.524\,\text{AU}\)), the time-averaged flux is approximately \(\bar{F} \approx 586\,\text{W\,m}^{-2}\), dropping to \(\sim 493\,\text{W\,m}^{-2}\) at aphelion and rising to \(\sim 718\,\text{W\,m}^{-2}\) at perihelion.
Top-of-atmosphere flux¶
The top-of-atmosphere (TOA) incident shortwave flux on a horizontal surface depends on the solar zenith angle \(\theta_z\) (Wikipedia: Air mass (solar energy)):
The \(\max(0,\cdot)\) term enforces that there is no downward solar flux below the horizon (\(\theta_z > 90°\)).
Atmospheric transmittance¶
A fraction of TOA radiation is absorbed and scattered by the atmosphere before reaching the surface. The surface incident shortwave flux is:
where \(\tau_\text{atm} \in [0, 1]\) is the effective atmospheric transmittance. For current Mars (thin, dusty atmosphere), a representative value is \(\tau_\text{atm} \approx 0.55\) (Haberle et al., 1993).
Dust storms substantially reduce \(\tau_\text{atm}\), causing surface cooling even while the upper atmosphere warms — a key phenomenon in Martian meteorology.
Surface albedo and absorbed flux¶
The surface reflects a fraction \(\alpha\) of incident shortwave radiation (Wikipedia: Albedo). The net absorbed shortwave flux is:
Mars's global mean albedo is \(\alpha \approx 0.25\), though it varies from \(\sim 0.10\) in dark basaltic regions to \(\sim 0.45\) over bright dust deposits and polar caps (Christensen et al., 2001).
Numerical example at Mars baseline¶
Using \(S_{1\,\text{AU}} = 1361\,\text{W\,m}^{-2}\), \(r \approx 1.38\,\text{AU}\) (near \(L_s = 0°\)), \(\tau_\text{atm} = 0.55\), \(\alpha = 0.25\):
| \(\theta_z\) | \(F_\text{TOA}\) | \(F_\text{sfc}\) | \(F_\text{abs}\) |
|---|---|---|---|
| \(0°\) | \(713.2\,\text{W\,m}^{-2}\) | \(392.3\,\text{W\,m}^{-2}\) | \(294.2\,\text{W\,m}^{-2}\) |
| \(30°\) | \(617.7\,\text{W\,m}^{-2}\) | \(339.7\,\text{W\,m}^{-2}\) | \(254.8\,\text{W\,m}^{-2}\) |
| \(60°\) | \(356.6\,\text{W\,m}^{-2}\) | \(196.1\,\text{W\,m}^{-2}\) | \(147.1\,\text{W\,m}^{-2}\) |
| \(90°\) | \(\approx 0\) | \(\approx 0\) | \(\approx 0\) |
Implementation¶
These equations are computed at each timestep in the Mars model. See Solar Flux — Mars for the Mars-specific derivation, and src.framework.orbital for the orbital distance calculation.