“*I’d put my money on the Sun and solar energy, what a source of power! We shouldn’t wait until oil and coal run out before we tackle that.*”

*– Thomas Edison in conversation with Henry Ford and Harvey Firestone, 1931*

*The Sun *

*The Sun*

The Sun is the main source of energy in the earth-atmosphere system, irradiating electromagnetic energy in a wide range of wavelengths. Just 1 second of the total energy output of the Sun would power the US for 9 million years. The Sun generates its energy from the fusion of hydrogen nuclei, forming helium. The Sun fuses 620 million metric tons of hydrogen each second in its core. The heat energy released by fusion in the core can then take up to 10,000 years before it escapes from the sun’s surface in the form of sunlight. Once regarded as a small and relatively insignificant star, the Sun is now thought to be brighter than 85% of the stars in the Milky Way.

^{1}

*Sunlight and the Solar Spectrum*

*Sunlight and the Solar Spectrum*

The light that we see everyday is only a fraction of the total energy emitted by the sun incident on the Earth’s surface. Sunlight is a form of “electromagnetic radiation” and the visible light that we see is a small subset of the electromagnetic spectrum shown below

Sunlight includes all types of solar radiation, not just visible light. The electromagnetic spectrum includes gamma rays, X-rays, ultraviolet light, infrared light and radio waves. In general, short wavelength radiation has more energy (e.g ultraviolet) than longer wavelengths (e.g. infrared).

The idea that light, regardless of source, could be described as waves first gained acceptance in the early 1800s. By the late 1800s, experimental observations made clear that the wavelengths of light emitted by heated objects could not be explained by simple wave-based models of radiation. These discrepancies were resolved by Max Planck in 1900^{2} and by Albert Einstein in 1905^{3}. Planck first proposed that the total energy of light is made up of indistinguishable energy particles, or a quantum energy elements. Einstein correctly distinguished the values of these quantum energy elements. As a result, quantum-mechanics was inaugurated and light was defined as both a “wave-packet” and as an “energy particle” or photon. For their work, Planck and Einstein won the Nobel prize for physics in 1918 and 1921, respectively.

**The Energy of a Photon**

**The Energy of a Photon**

A photon can be defined by either a wavelength, denoted by or equivalently by its energy, denoted by . There is an inverse relationship between the two given by the equation:

(1)

where is Planck’s constant (6.626e-34 joule·s) and c is the speed of light (2.998e8 m/s). The engineering notation “e” is shorthand for base 10. Thus, 2.998e8 = 2.998 x 10^{8}.

An inverse relationship between energy and wavelength implies that light with high energy photons (such as “blue” light) has a short wavelength. Light consisting of low energy photons (such as “red” light) has a long wavelength.

Use of the energy equation is illustrated in the R function code below. When dealing with photons, the common unit of energy is the electron-volt (eV) rather than the joule (J). The required conversion is 1 eV = 1.602e-19 J. Meanwhile, the common unit of measure for wavelength is the micron (e.g. one millionth of a meter, where ).

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# Planck's Energy of a Photon # specify photon wavelength (lambda) Photon.E <- function(lambda){ h <- 6.62606957e-34 # Planck's constant (joule-seconds) c <- 2.99792458e8 # speed of light (m/s) q <- 1.60217657e-19 # convert between joules and eV um <- 1e-6 # convert between meters and microns (um) E <- ((h * c) / (eV * um)) / lambda return(E) } |

Following unit conversions, equation (1) becomes:

(2)

By way of example, if we choose a wavelength in the middle of the visible spectrum (e.g. ), then the energy of a single photon is 2.48 eV.

**Power Density and Photon Counting**

**Power Density and Photon Counting**

The photon is the elementary particle or quantum of light and the photon flux is defined as the number of photons per second per unit area:

(3)

The performance of solar PV power systems is directly influenced by the photon flux, which determines the number of electrons and the current produced by the solar cell. At a given wavelength, the power density is determined by the photon energy for that wavelength times the photon flux. To determine the power density, , in units of W/m², the energy of the photons must be in Joules. The equation is:

(4)

where again q is the standard conversion between electron-volts (eV) and joules (J). Use of the equation is illustrated below:

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# Power Density of Light # specify photon wavelelength (lambda) and flux (phi) Photon.PD <- function(lambda, phi){ q <- 1.60217657e-19 # convert between joules and eV pd <- phi * q * Photon.E(lambda) return(pd) } |

For example, selecting a wavelength in the visible spectrum () and assuming a high photon flux typical of conditions at noon (), then we get a power density of 635.7 W/m^{2}. Alternatively, if we have field sensor measurements of the instantaneous power density for a specific wavelength, equation (4) can be re-arranged to count the number of photons entering a solar energy system.

**Spectral Irradiance**

**Spectral Irradiance**

Spectral irradiance is the common way to characterized light, be it the light of the Sun or a specialized light-bulb. Spectral irradiance, denoted by , is the power density of light for a specified wavelength normalized by the wavelength. Specifically:

(5)

From equations (2) and (4) we get:

(6)

where:

is the spectral irradiance in ;

is the photon flux with # photons in ;

and are the energy and wavelength of the photon in eV and ; and

is the wavelength range over which the power density is normalized.

In practice, spectral irradiance is measured using specialized radiometers. The resulting profile is not smooth or well behaved as the profile will contain unique emitter and absorption bands, depending on the heat/light source.

**Spectral Power Density**

**Spectral Power Density**

International engineering standards for the analysis of solar cell performance require as inputs the photon flux and the spectral irradiance profile. These components are used to measure the total power density emitted by a light source over all wavelengths:

(7)

where is the spectral power density measured in . Integration is achieved using standard integration methods, which sum the power density by discrete wavelength ranges.

(8)

**Blackbody Radiation**

**Blackbody Radiation**

Most light sources, including the Sun, are closely modeled as “blackbody” emitters. A blackbody is defined as an object that absorbes all incoming radiation and emits all energy based on its temperature. According to Planck’s law, all bodies with temperature above 0K emit electromagnetic radiation.^{4} Blackbodies derive their name since they don’t necessarily emit light in the visible range. A blackbody at temperature *T* emits spectral radiation according to the following equation:^{5}

(9)

where

is the spectral irradiance in ;

is Planck’s constant (joule-seconds); is the speed of light (m/s);

is the wavelength of light (;

is temperature (kelvin); and

is Boltzmann’s constant (joule/kelvin);

The following R function ensures correct estimates of blackbody radiation by temperature and wavelength, avoiding typical problems with units of measure.

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Blackbody.Rad <- function(temp, lambda){ c <- 2.998e8 # speed of light (m/s) k <- 1.380e-23 # Boltzmann's constant (joule/K) h <- 6.626e-34 # Planck's constant (joule-sec) term1 <- (2 * pi * h * c^2) term2 <- lambda^5 * (exp((h * c)/(k * temp * lambda)) - 1) rad <- term1 / term2 / 1e6 # radiation (Wm-2 µm-1) return(rad) } |

Meanwhile, it is possible to quickly determine the wavelength at which the radiation profile achieves its maximum by taking the the derivative of equation (9) with respect to , setting the derivative equal to zero, and solving for }. The maximum wavelength where the peak spectral radiation is emitted is defined by:

(10)

This is known as Wein’s Law. The interpretation is simple: the hotter a body, the shorter the peak wavelength of radiation emitted. ^{6} Finally, the total instantaneous radiation emitted by a blackbody object is obtained by substituting equation (9) into equation (7).

*Blackbody Profile of the Sun vs. the Earth*

*Blackbody Profile of the Sun vs. the Earth*

The blackbody radiation function is next applied across a range of wavelengths and for different emitter temperatures. The results define the spectral irradiance profile of the Sun and the Earth.

Several conclusions are immediately apparent when we look closely at the scaling:

- The radiation profile of the Sun significantly dwarfs that of the Earth in both magnitude and high-frequency energy;
- The Sun and the Earth have a very small area of overlap between their profiles; and
- The range of wavelengths in the solar profile is relatively narrow (), while that of the Earth is 19 times wider ().

Meanwhile, the spectral power density, which defines the integrated, instantaneous radiation emitted across all wavelengths, equals 63.2 million W/m^{2} for the Sun and only 450 W/m^{2} for the Earth.

The R code for the blackbody radiation data and basic charts appears below:

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# Blackbody Radiation Exercise # function inputs and spectral irradiance temp1 <- 5780 # Sun temperature (K) temp2 <- 300 # Earth temperature lambda <- seq(0.05, 60, by = 0.0025) / 1e6 rad1 <- BlackBody.Rad(temp1, lambda) rad2 <- BlackBody.Rad(temp2, lambda) # consolidate data rad.df <- data.frame(wavelength = lambda * 1e6, type = NA, Sun.rad = rad1 / 1e8, Earth.rad = rad2) # define wavelength types by row uv <- which(rad.df[, "wavelength"] < 0.4) vis <- which(rad.df[, "wavelength"] >= 0.4 & rad.df[, "wavelength"] <= 0.7) ir <- which(rad.df[, "wavelength"] > 0.7) rad.df[uv, "type"] <- "Ultraviolet" rad.df[vis, "type"] <- "VisibleLight" rad.df[ir, "type"] <- "InfraRed" # plot the solar profile ggplot(rad.df, aes(wavelength, Sun.rad)) + geom_line(color = "green", size = 1.3) + geom_area(aes(ymin = 0, ymax = Sun.rad, fill = factor(type)), alpha = 0.40) + scale_x_continuous(limits = c(0, 3), breaks = seq(0, 3, by = 0.25)) + scale_y_continuous(limits = c(0, 0.90), breaks = seq(0, 0.90, by = 0.10)) + scale_fill_manual(name = "Basic Spectrum", breaks = c("Ultraviolet","VisibleLight","InfraRed"), values = c("thistle", "slategray3", "grey95")) + labs(title = expression(atop(bold("Solar Blackbody Radiation Profile"), atop("T = 5,780 K",""))), x = "Wavelength (microns)", y = "Spectral Intensity (W/m-2/micron/1e8)") # plot the Sun vs the Earth profiles xbreaks <- c(0.1, 0.25, 0.5, 1, 2.5, 5, 10, 20, 30, 40, 50, 60) xlabels <- c("0.10", "0.25", "0.50", "1", "2.5", "5", "10", "20", "30", "40", "50", "60") ggplot(rad.df, aes(wavelength, Sun.rad * 1e2)) + geom_line(color = "green", size = 1.2) + geom_area(aes(ymin = 0, ymax = Sun.rad), fill = "grey95", alpha = 0.45) + geom_line(aes(wavelength, Earth.rad), color = "green", size = 1) + geom_area(aes(wavelength, Earth.rad, ymin = 0, ymax = Earth.rad), fill = "thistle", alpha = 0.45) + scale_x_log10(breaks=xbreaks, labels = xlabels) + labs(title = expression(atop(bold("Blackbody Radiation Profiles"), atop("Sun (T = 5,780 K) vs. Earth (T = 300 K)", "X-axis scaled to log10"))), x = "Wavelength (microns)", y = "Spectral Intensity (W/m-2/micron)") + annotate("text", x = 3, y = 70, label = "Solar Profile Scaled Down By 1e6", fontface = 3, color = "firebrick") # find wavelength with peak blackbody radiation sun.max <- 2897 / temp1 earth.max <- 2897 / temp2 # find integrated radiation intensity over all wavelengths wl.delta <- 0.0025 sun.total <- sum(rad.df[, "Sun.rad"]) * wl.delta * 1e8 earth.total <- sum(rad.df[, "Earth.rad"]) * wl.delta |