Solar energy

The solar constant is a measure of the solar electromagnetic radiation received per unit area on a surface perpendicular to the rays, at the outer edge of the Earth's atmosphere (1 AU distance from the Sun). Satellite measurements provide precise data for this value.

Due to Earth's elliptical orbit, the distance to the Sun varies throughout the year. In January (perihelion), when Earth is closest to the Sun, extraterrestrial irradiance reaches about $1400 \text{ W/m}^2$. In July (aphelion), at its furthest distance, it drops to roughly $1330 \text{ W/m}^2$.

For standard baseline calculations, the accepted average solar constant is 1367 W/m² (or historically $1377 \text{ W/m}^2$).

Solar irradiance outside Earth's atmosphere is referred to as extraterrestrial radiation ($I_0$). Because the Earth-Sun distance fluctuates annually, $I_0$ varies predictably according to the day of the year.

Extraterrestrial solar spectrum

Earth-Sun distance variation throughout the year

As solar radiation passes through Earth's atmosphere, its intensity is attenuated due to absorption (by ozone, water vapor, $CO_2$) and scattering (Rayleigh and Mie scattering). Consequently, the flux reaching the surface is less than $I_0$.
The atmospheric attenuation is quantitatively modeled using the Beer-Lambert Law, accounting for the optical path length known as Air Mass (AM).

Solar radiation arriving at a terrestrial surface consists of three distinct components:

• Direct (Beam) Radiation ($I_{BC}$): Rays traveling in a straight line from the Sun without being scattered.
• Diffuse Radiation ($I_{DC}$): Sunlight scattered by atmospheric molecules, aerosol particles, and clouds, arriving from all sky directions.
• Reflected Radiation / Albedo ($I_{RC}$): Sunlight reflected off the surrounding ground terrain and nearby structures onto the collector.

Air Mass path length (Standard Testing Condition: AM = 1.5)

Direct, diffuse, and ground-reflected radiation on a tilted surface

Direct Component ($I_{BC}$): Depends heavily on the angle of incidence ($\theta$) between incoming rays and the panel's normal vector. This angle is calculated using solar altitude, azimuth angle, panel tilt ($\beta$), and panel orientation ($\gamma$).

Diffuse Component ($I_{DC}$): Depends on sky conditions and isotropic/anisotropic sky models. A horizontal module receives the maximum sky view factor, while tilting reduces the fraction of the visible sky hemisphere.

Reflected Component ($I_{RC}$): Influenced by the ground reflectivity coefficient (Albedo, $\rho$). Albedo values range from $\approx 0.1$ for dark asphalt to $0.8+$ for fresh snow.

The SPAC (Solar Panel Angle Calculator) application relies on these geometric and physical relationships to calculate optimal module tilt and total expected irradiance.

SPAC calculation algorithm flowchart