Solar Radiation: The Fundamentals

Solar radiation is the electromagnetic radiation emitted from the surface of the Sun. This is the principal natural source of energy on Earth, which sustains life and influences the Earth’s climate. The world’s dependence on electricity has led to increased harnessing of this (almost) perpetual source of energy.

Solar radiation is well documented through the fundamental laws of radiation, mathematics and more recently, the measurement of the intensity and spectral distribution of this radiation.

Radiation Laws and The Sun

The intensity and spectral distribution of radiation of a specific emitting body may be approximated from the example of a blackbody. A blackbody is an idealised surface that absorbs electromagnetic radiation across all wavelengths, irrelevant of the angle of incidence, while reflecting no radiation. As a result, the emitting body will emit maximum energy at all wavelengths and all directions.

Since an emitting body’s intensity and spectral distribution is solely dependent on temperature and the surface’s material properties, it can be described through the fundamental radiation laws of Planck, Stefan-Boltzmann and Wien’s displacement [1].

The Sun’s (gaseous) surface may also be approximated as a blackbody emitter, with an effective surface temperature of approximately 5800 K. Considering that the approximate temperature and area of the Sun’s surface is known, the solar radiant flux traversing through an element of area (unit area) at the Earth’s mean distance from the Sun, can be calculated from the fundamental radiation laws. This is called the solar constant and this value always assumes it is the annual average radiant flux at the mean Sun-Earth distance. The mathematical derivation of this can be found in Chapter 1 and 2 of Liou [1].

(Solar) Radiation Terminology

The (correct) terminology in the solar radiation research field is based on basic radiometric quantities. To truly understand the differences and similarities in terms such as irradiance and luminance, the concept of a solid angle and the representation of radiant energy in polar coordinates is necessary.

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The ratio of an area $\sigma$ of a spherical surface with a distance $r$ squared from a point is called the solid angle and is written as

\begin{equation} \Omega = \frac{\sigma}{r^2}, \end{equation}

with units steradian (sr). The differential solid angle is

\begin{equation} d\Omega = \frac{d\sigma}{r^2} = \sin \theta\,d\theta\,d\phi, \end{equation}

with $\theta$ the zenith angle and $\phi$ the azimuth angle in polar coordinates.

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Now, considering the area element $dA$ is crossed by a differential amount of (radiant) energy $dE_\lambda$, in a specific interval of wavelengths ($\lambda$ to $\lambda + d\lambda$) for time $dt$ and the directions of the energy are limited to a differential solid angle ($d\Omega$), the monochromatic intensity or monochromatic radiance is \begin{equation} I_\lambda = \frac{dE_\lambda}{cos\theta\,d\Omega\,d\lambda,dt\,dA}, \end{equation} with units energy per area per time per wavelength per steradian. The normal component of the monochromatic radiance integrated over the (hemispherical) solid angle is the monochromatic flux density or monochromatic irradiance \begin{equation} F_\lambda = \int_\Omega I_\lambda \cdot \cos \theta d\Omega. \end{equation} The broadband irradiance or total flux density integrated over the entire electromagnetic spectrum is \begin{equation} F = \int_0^\infty F_\lambda d\lambda. \end{equation} This is often simply referred to as irradiance. The basic radiometric quantities as used here (throughout) are summarised in the table below.

Symbol	Quantity	Unit
$E$	Energy	Joule	J
$f$	Flux (luminosity)	Joule per second or Watt	J·s-1 or W
$F$	Flux density (irradiance)	Watt per square meter	W·m-2
$I$	Intensity (radiance) Brightness (luminance)	Watt per square meter per steradian	W·m-2·sr-1

Broadband Extraterrestrial Radiation

The solar constant is the annual average solar radiation that arrives from the extraterrestrial area above the Earth’s atmosphere and therefore marginally varies through the year and during the 11-year solar cycle. However, due to the ellipticity of the Earth’s orbit, the distance between the Sun and Earth varies throughout the year, causing some fluctuations in the average radiation perceived at the top of the atmosphere [4]. The solar constant will therefore be a relative minimum (1321.7 W·m^-2) at the aphelion and relative maximum (1412.5 W·m^-2) at the perihelion [3].

However, exploring enough solar radiation literature will indicate that various values are assumed for the solar constant. The most often assumed modelled solar constant (at mean Sun-Earth distance) is 1366.1 W·m^-2, as from the ASTM E-490 standard [3]. However, in 2018 Gueymard proposed a new value of 1361.1 W·m^-2 based on 42-years broadband radiation measurements [10].

The electromagnetic energy arriving at the top of the atmosphere (TOA) is called the extraterrestrial irradiance. This TOA irradiance is a function of the solar constant ($S$) and the day of year ($DOY$), and is already corrected for the ellipticity of the Earth’s orbit, according to the ASCE formulation [5]

\begin{equation}\label{eq:F0} F_0 = S\left[1 + 0.033 \cdot \cos \left( \frac{2 \pi}{365} \cdot DOY \right) \right]. \end{equation}

Similarly, the extended Fourier representation by Spencer [6]

\begin{equation}\label{eq:F0_spec} \begin{aligned} F_0 &= S \left( 1.00011 + 0.034221 \cos x + 0.00128 \sin x \\ &- 0.000719 \cos 2 x + 0.000077 \sin 2x \right), \end{aligned} \end{equation} where \begin{equation}\label{eq:x} x = \frac{2 \pi}{365} \left( DOY – 1 \right), \end{equation}

according to Reno et al. [2]. However, according to Sandia National Laboratories [7] and NREL [8], $x = \frac{2 \pi}{365} \cdot DOY$. Furthermore, the TOA radiation is actually a function of the solar constant at mean Sun-Earth distance $S$, the mean Sun-Earth distance $R_{AV}$ and the actual Sun-Earth distance $R$ [8] \begin{equation} F_0 = S \cdot \left( \frac{R_{AV}}{R} \right)^2. \end{equation} There are several formulations very similar to eq. (\ref{eq:F0}) found in literature, e.g. that of Gregg and Carder [9] \begin{equation}\label{eq:F0_gc} F_0 = S \bigg[ 1 + e \cdot \cos \Big(2\pi(DOY – 3)/365 \Big) \bigg]^2, \end{equation}

where $e$ is the orbital eccentricity and proposed as a constant, $e = 0.0167$.

The broadband extraterrestrial irradiance is, however, a reasonably predictable parameter and therefore well documented in literature. Formulations range from tabulated values as in the ASHRAE model to fundamental formulations based on the Earth eccentricity, latitude and the declination and hour angles, which are all within reasonable accuracy [11].

[1] K. N. Liou, Introduction to Atmospheric Radiation, Elsevier, 2002. [Online]. Available: https://app.knovel.com/hotlink/toc/id:kpIARE0015/introduction-atmospheric/introduction-atmospheric

[2] M. J. Reno, C. W. Hansen, J. S. Stein, “Global Horizontal Irradiance Clear Sky Models: Implementation and Analysis,” SANDIA Report, SAND2012-2389, 2012. [Online]. Available: https://pdfs.semanticscholar.org/5df9/9811445709d5249100120e1dc9298db5446e.pdf

[3] NREL, “2000 ASTM Standard Extraterrestrial Spectrum Reference E-490-00”, NREL.gov, https://www.nrel.gov/grid/solar-resource/spectra-astm-e490.html (Accessed 3 July 2020).

[4] D. D. Dionysiou, G. L. Puma, J. Schneider and D. Bahnemann, “Extraterrestrial Irradiance and Spectrum” in Photocatalysis – Applications, Royal Society of Chemistry, 2016. [Online] Available: https://app.knovel.com/hotlink/pdf/id:kt0110N5I1/photocatalysis-applications/extraterrestrial-irradiance

[5] I. A.Walter, R. G. Allen, R. Elliott, M. Jensen, D. Itensu, B. Mecham, T. Howell, R. Snyder, P. Brown, S. Echings et al., “ASCE’s standardized reference evapotranspiration equation,” in Watershed management and operations management 2000, 2000, pp. 1-11. [Online]. Available:
https://epic.awi.de/id/eprint/42362/1/ascestzdetmain2005.pdf

[6] J. Spencer, “Fourier series representation of the position of the sun,” Search, vol. 2, no. 5, p. 172, 1971.

[7] Sandia National Laboratories, “PV Performance Modelling Collaborative: Extraterrestrial radiation.” [Online]. Available: https://pvpmc.sandia.gov/modeling-steps/1-weather-design-inputs/irradiance-and-insolation-2/extraterrestrial-radiation/

[8] NREL, “Solar radiation basics, SOLPOS.” http://solardat.uoregon.edu/SolarRadiationBasics.html (Accessed 3 July 2020).

[9] W. W. Gregg and K. L. Carder, “A simple spectral solar irradiance model for cloudless maritime atmospheres,” Limnology and oceanography, vol. 35, no. 8, pp. 1657-1675, 1990.

[10] C. A. Gueymard, “A reevaluation of the solar constant based on a 42-year total solar irradiance time series and a reconciliation of spaceborne observations,” Solar Energy, vol. 168, pp. 2-9, 2018. DOI: 10.1016/j.solener.2018.04.001

[11] L. T. Wong and W. K. Chow, “Solar radiation model,” Applied Energy, vol. 69, pp. 191-224, 2001.