Orbital Solar Array Power (3D)

What is modelled

A solar array in a circular orbit around the Earth. Above the atmosphere sunlight is 1361 W/m² and never blocked by weather, but the satellite spends part of each orbit in the Earth's shadow and its cells age under radiation. The page computes eclipse geometry, the angle between panel and Sun, the cell temperature and efficiency, the power over one orbit, how it fades over the years, the battery needed for the eclipse and a comparison with a ground array.

Equations

• Orbit: a = R + h, period T = 2π√(a³/μ), speed √(μ/a), with R = 6371 km and μ = 398 600 km³/s².
• Eclipse (cylindrical shadow). With beta the angle between the Sun direction and the orbit plane, eclipse occurs only if |β| < β* = arcsin(R/a), and then fe = (1/π) arccos(√(h² + 2Rh) / (a cosβ)). At 408 km and β = 0 this is about 39 % (36 of 92.6 minutes); in geostationary orbit at most about 5 %.
• Solar flux: S = 1361/r² W/m² with Sun distance r = 1 − 0.0167 cos(2π(d − 3)/365.25) AU (about 1322 to 1414 W/m² over the year).
• Pointing: two-axis tracking gives cosθ = 1; a one-axis rotary joint (SADA) perpendicular to the orbit plane gives cosθ = cosβ; a fixed zenith-facing panel gives cosθ = cosβ cosν (only positive values), about 1/π of the tracked average. A pointing error multiplies by cos(error).
• Cell temperature from radiative balance (absorptance 0.9, front + back emissivity 1.7, plus 150 W/m² of Earth albedo and infrared): 0.9 (S C cosθ + 150)(1 − η) = 1.7 σ T4; efficiency η(T) = η0(1 + k(T − 301 K)), solved by iteration since η depends on T. A concentrator with ratio C raises the cell flux C times (optical efficiency 0.9) while the cell area shrinks to A/C, so the cells run hotter and lose efficiency.
• Array power P = η(T) S cosθ A (1 − d)years and zero in eclipse; the orbit average is found by sampling.
• Battery: eclipse energy E = Pload tecl, nominal capacity E/(DoD · 0.95), cycles per year = orbits per year.

Reading the 3D picture

One unit of the 3D scene is 1000 km, the Earth and the orbit are to scale, the satellite is magnified (slider) so that you can see it. The yellow arrow points to the Sun, the green arrow is the panel normal; the angle between them is theta. The translucent blue cylinder is the Earth shadow used in the eclipse formula; the orbit ring turns blue inside it. Wings glow orange in proportion to the computed cell temperature.

Checks run in the page

• Eclipse fraction from the closed form against marching a satellite around the orbit and testing the Earth-shadow cylinder.
• Orbital period from Kepler's formula against a leapfrog integration of Newtonian gravity.
• Zero residual of the cell energy balance, convergence of the orbit average, and (for tracked arrays) average = (1 − f_e) × sunlit power.

Simplifications

Circular orbits, a spherical Earth with a sharp cylindrical shadow (no penumbra, no oblateness), the Sun direction fixed during one orbit (in reality beta drifts over weeks as the orbit precesses and the seasons change; the slider sets it), a single effective degradation rate per technology (typical orders of magnitude, they depend strongly on altitude and shielding), no shadowing by the spacecraft itself, no cell mismatch or wiring losses, no power conditioning efficiency, and an energy comparison that ignores cost, launch mass and any transmission to the ground. The ground figure is simply 1000 W/m² × cell efficiency × the capacity factor you choose.

What to try

• Raise beta past beta*: the eclipse disappears and the orbit-average power rises.
• Switch to a fixed panel and watch the average power drop to about 1/π of the tracked value.
• Choose silicon and age it ten years; then add the concentrator and see why hot cells lose efficiency.