Solar Flux at Mars¶

This page derives the solar flux model used at each timestep in the Mars climate simulation. The general theory is in Solar Radiation; this page applies it to Mars-specific conditions with calibrated numerical values.

Orbital distance¶

Mars's distance from the Sun at solar longitude \(L_s\) (Allison & McEwen, 2000):

\[ r(L_s) = \frac{a\,(1 - e^2)}{1 + e\cos L_s} \]

With \(a = 1.524\,\text{AU}\) and \(e = 0.0934\):

\(L_s\)	Event	\(r\) (AU)
\(0°\)	N. spring equinox	\(1.517\)
\(71°\)	Aphelion	\(1.666\)
\(180°\)	N. autumn equinox	\(1.517\)
\(251°\)	Perihelion	\(1.381\)

Top-of-atmosphere incident flux¶

The TOA incident shortwave flux on a horizontal surface (Wikipedia: Solar irradiance):

\[ F_\text{TOA} = \frac{S_{1\,\text{AU}}}{r(L_s)^2}\,\max\!\bigl(0,\,\cos\theta_z\bigr) \]

where \(S_{1\,\text{AU}} = 1361\,\text{W\,m}^{-2}\) (Kopp & Lean, 2011) and \(\theta_z\) is the solar zenith angle. The normal-incidence TOA flux at \(L_s \approx 0°\) is:

\[ F_\text{normal} = \frac{1361}{1.517^2} \approx 591.5\,\text{W\,m}^{-2} \]

At perihelion (\(r = 1.381\,\text{AU}\)) this rises to \(\approx 713\,\text{W\,m}^{-2}\), a 21% increase.

Surface incident flux¶

The Martian atmosphere (thin, dusty CO₂) transmits a fraction \(\tau_\text{atm}\) of the TOA flux to the surface (Haberle et al., 1993):

\[ F_\text{sfc} = F_\text{TOA} \cdot \tau_\text{atm} \]

The baseline value \(\tau_\text{atm} = 0.55\) is representative of moderate dust opacity (\(\tau_\text{dust} \approx 0.5\)). During global dust storms, \(\tau_\text{atm}\) can drop below \(0.2\).

Reflected shortwave¶

The surface reflects a fraction \(\alpha\) of incident shortwave (Wikipedia: Albedo):

\[ F_\text{refl} = \alpha \cdot F_\text{sfc} \]

Mars's global mean albedo is \(\alpha \approx 0.25\), though regional values range from \(0.10\) (dark basalt) to \(0.45\) (bright dust, polar caps) (Christensen et al., 2001).

Zenith-angle reference table¶

At \(L_s \approx 0°\) (\(F_\text{normal} = 591.5\,\text{W\,m}^{-2}\)), \(\tau_\text{atm} = 0.55\), \(\alpha = 0.25\):

\(\theta_z\)	\(F_\text{TOA}\)	\(F_\text{sfc}\)	\(F_\text{abs}\)
\(0°\)	\(591.5\,\text{W\,m}^{-2}\)	\(325.3\,\text{W\,m}^{-2}\)	\(244.0\,\text{W\,m}^{-2}\)
\(30°\)	\(512.1\,\text{W\,m}^{-2}\)	\(281.7\,\text{W\,m}^{-2}\)	\(211.3\,\text{W\,m}^{-2}\)
\(60°\)	\(295.8\,\text{W\,m}^{-2}\)	\(162.7\,\text{W\,m}^{-2}\)	\(122.0\,\text{W\,m}^{-2}\)
\(90°\)	\(\approx 0\)	\(\approx 0\)	\(\approx 0\)

At perihelion with \(r = 1.381\,\text{AU}\) the \(0°\) values increase to \(F_\text{TOA} \approx 713\,\text{W\,m}^{-2}\), matching the baseline used in the evolve-1hr diagnostic.

Implementation¶

This model is computed at each integration timestep in the Mars ODE. See src.celestials for the implementation and src.framework.orbital for the orbital distance calculation.