Extraterrestrial Irradiance

Summary

The Extraterrestrial Irradiance model calculates the solar irradiance incident at the top of earth’s atmosphere on a surface perpendicular to the sun’s rays. PlantPredict uses a second-order Fourier series expansion based on earth’s orbital mechanics to compute the extraterrestrial as a function of day of year. This value is essential for calculations, models, and corrections throughout the prediction engine.

Inputs

Name	Symbol	Units	Description
UTC Date-Time	—	datetime	Universal Coordinated Time

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Outputs

Name	Symbol	Units	Description
Extraterrestrial Direct Normal Irradiance	$DNI_{extra}$	W/m²	Solar irradiance at top of atmosphere perpendicular to sun’s rays

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Detailed Description

The extraterrestrial irradiance varies throughout the year due to earth’s elliptical orbit around the sun. The earth-sun distance changes by approximately ±1.7% from its mean value, causing the solar irradiance to vary by approximately ±3.4% (since irradiance varies as the inverse square of distance).

Calculation Method

PlantPredict implements a Fourier series approximation of the extraterrestrial irradiance based on Spencer’s equation.

Step 1: Calculate Day Angle

Compute the fractional day $n_f$ (days) from the UTC date-time:

$$n_f = \text{UTCDate} - \text{January 1 of current year}$$

where $n_f = 0$ for January 1 at 00:00 UTC and $n_f = 365$ (or $366$ for leap years) for December 31 at 24:00 UTC. The value includes the fractional time of day.

The day angle $\zeta$ (in radians) represents the fractional progress through the year:

$$\zeta = \frac{2\pi \, n_f}{365.25}$$

The constant 365.25 accounts for the average year length including leap years.

Step 2: Calculate Extraterrestrial Irradiance

The extraterrestrial direct normal irradiance is computed using a second-order Fourier series expansion:

$$\begin{aligned} DNI_{extra} = G_0 \times \Big(&1.00011 + 0.034221 \cos(\zeta) + 0.00128 \sin(\zeta) \\ &+ 0.000719 \cos(2\zeta) + 0.000077 \sin(2\zeta) \Big) \end{aligned}$$

where:

• $G_0 = 1367$ W/m² is the solar constant (mean extraterrestrial irradiance at 1 astronomical unit = 149,597,870.7 km), per the World Meteorological Organization (WMO) standard
• $\zeta$ is the day angle in radians
• The Fourier coefficients are derived from the Spencer (1971) equation as presented in Duffie & Beckman, Solar Engineering of Thermal Processes

Physical Interpretation

The terms in the Fourier series represent:

1. Constant term (1.00011): Slight adjustment to solar constant
2. Annual variation $(\cos(\zeta), \sin(\zeta))$: Primary effect of earth’s elliptical orbit
3. Semi-annual variation $(\cos(2\zeta), \sin(2\zeta))$: Higher-order orbital effects

The dominant variation is the annual term with amplitude ±3.4%, corresponding to earth’s perihelion (closest approach, ~January 3) and aphelion (farthest point, ~July 4).

Typical Values Throughout the Year

Date	Approximate Day	$DNI_{extra}$ (W/m²)	% Deviation
January 3 (perihelion)	3	~1412	+3.3%
April 3	93	~1362	-0.4%
July 4 (aphelion)	185	~1322	-3.3%
October 4	277	~1368	+0.1%

Downstream Models

The extraterrestrial irradiance $DNI_{extra}$ is used by:

• Clearness Index — atmospheric transmittance calculation
• Transposition Models (Perez, Hay-Davies) — anisotropy index
• Models (DISC, DIRINT) — DNI/DHI estimation from GHI
• Spectral Models — atmospheric optical depth

References

• Spencer, J. W. (1971). Fourier series representation of the position of the sun. Search, 2(5), 172–176.
• Iqbal, M. (1983). An Introduction to Solar Radiation. Academic Press. ISBN: 0-12-373750-8.
• Duffie, J. A., & Beckman, W. A. (2013). Solar Engineering of Thermal Processes (4th ed.). John Wiley & Sons. ISBN: 978-0-470-87366-3. DOI: 10.1002/9781118671603