Orbital Mechanics¶
Planetary orbits are described by Keplerian orbital elements. For a planet on an elliptical orbit, the key elements are the semi-major axis \(a\), orbital eccentricity \(e\), and the true anomaly \(\nu\) (the angle from perihelion). In tform the true anomaly is parameterised through the solar longitude \(L_s\), which is the standard convention in Mars science (Allison & McEwen, 2000).
Orbital distance¶
The instantaneous Sun–planet distance \(r\) as a function of solar longitude \(L_s\) follows directly from the vis-viva equation and the geometry of an ellipse (Wikipedia: Elliptic orbit):
| Symbol | Meaning | Unit |
|---|---|---|
| \(r\) | Sun–planet distance | AU |
| \(a\) | Semi-major axis | AU |
| \(e\) | Orbital eccentricity | dimensionless |
| \(L_s\) | Solar longitude (true anomaly from perihelion) | rad |
For Mars: \(a = 1.524\,\text{AU}\), \(e = 0.0934\) (NASA Mars Fact Sheet). Mars's high eccentricity causes a ~19% variation in solar flux between perihelion (\(L_s = 251°\)) and aphelion (\(L_s = 71°\)), driving strong seasonal asymmetry.
Solar longitude and Martian seasons¶
\(L_s\) runs from \(0°\) (northern spring equinox) to \(360°\). Key events:
| \(L_s\) | Event |
|---|---|
| \(0°\) | Northern spring equinox |
| \(90°\) | Northern summer solstice |
| \(180°\) | Northern autumn equinox |
| \(251°\) | Perihelion (closest to Sun, southern summer) |
| \(270°\) | Northern winter solstice |
Because perihelion coincides with southern summer, the southern hemisphere receives more intense (but shorter) summers than the north — a major driver of the asymmetric CO₂ polar cap exchange.
Solar zenith angle¶
The solar zenith angle \(\theta_z\) is the angle between the local vertical and the direction to the Sun. It determines how much of the solar beam is intercepted per unit horizontal area (Wikipedia: Solar zenith angle):
| Symbol | Meaning |
|---|---|
| \(\phi\) | Geographic latitude |
| \(\delta\) | Solar declination (function of \(L_s\) and axial tilt) |
| \(h\) | Hour angle (local solar time) |
At \(\theta_z = 0°\) the Sun is directly overhead; at \(\theta_z = 90°\) it is on the horizon and the direct flux is zero.
Axial tilt and declination¶
The solar declination \(\delta\) oscillates with the axial tilt \(\varepsilon\) over the course of the year:
Mars has \(\varepsilon = 25.19°\) (NASA Mars Fact Sheet), slightly larger than Earth's \(23.45°\), contributing to pronounced seasonal temperature swings.
Implementation¶
Orbital distance and zenith angle are computed in src.framework.orbital (OrbitalParameters.distance_from_sun) and used by the Mars physics model to set the instantaneous solar flux at each timestep.