Thermal Radiation Laws Lab Version: 2.0 Author: Sean S. Lindsay History: Written August 2019; Last Modified: 26 September 2019

Goal: To become familiar with thermal radiation and two laws describing blackbody radiation: Wien’s Law and the Stefan-Boltzmann Law. By the end of this lab, students are expected to know:

1. What a blackbody and thermal radiation are 2. What blackbody spectrum/curve is and how to obtain information from a plot of a blackbody

spectrum 3. Thermal Radiations Laws: Wien’s Law and Stefan-Boltzmann Law and how they connect to

blackbody radiation 4. How astronomers use: the thermal radiation Laws and photometric filters

Tools used in this lab:

• UNL Astronomy Simulation: Blackbody Curves of Melting: https://astro.unl.edu/classaction/animations/light/meltednail.html

• UNL Astronomy Simulation: Blackbody Curves (NAAP) https://astro.unl.edu/classaction/animations/light/bbexplorer.html

• Microsoft Excel

1. The Electromagnetic Spectrum of Light 1.1 - Electromagnetic spectrum Nearly everything we know about the universe, we have discovered through looking at the light from celestial objects reaching Earth. Without the ability to investigate these objects directly, the light from them that reaches Earth offers our window into understanding everything from their brightness, size, composition, rotational properties, temperatures, densities, and magnetic field to the expansion of the universe itself. In order to extract this information from just the light, it is necessary that we understand the nature of light itself. It is knowing the properties of light that allows us to learn as much as we do about our universe.

When astronomers refer to light, they are not only referring to the light we observe in our daily lives. They are also referring to types of light invisible to the human eye. The light humans can see is referred to as visible (or optical) light. Some forms of life can see beyond this range and see infrared light or ultraviolet light. You are probably familiar with both of these kinds of light. Infrared (IR) light, in quotidian human life, can be thought of as light from heat. It is the energy that you feel as hot when you get near a hot object. Ultraviolet (UV) light is why we wear sunscreen during the summer as it is the light that gives you a sunburn. There are other kinds of light as well, and you have likely heard of most of them, but maybe not realized they are light. The seven kinds of light are radio wave, microwave, infrared, visible, ultraviolet, X-ray, and gamma ray light. Taken together, we call this the electromagnetic spectrum (EM; Fig. 1). More formally, astronomers and physics know light as electromagnetic radiation.

The part of the electromagnetic spectrum humans can see is called the visual spectrum

Why electromagnetic? Electromagnetic refers to the physical nature of light. We now understand that light really is a coupling between an oscillating electric field and magnetic field. It is a fundamental piece of a branch of physics knowns as electromagnetism because it deals with electric and magnetic parts, which are related and inseparable from one another.

Why spectrum? Light can be divided up on a scale between two extremes, from radio waves to gamma rays. What aspect of light defines the scale? The division of the spectrum can be set along the energy, wavelength, or frequency of the light. For energy, we need to understand light as particle. For wavelength and frequency, we must understand light as a wave.

Figure 1. The electromagnetic spectrum of light. 𝛄 is the Greek letter gamma. Image credit: Wiki-Creative Commons

1.2 Light as a Wave

Figure 2. The wavelength of light. Shorter wavelength is the same as higher frequency or higher energy. Wavelength of light is a physical distance. Visible light has wavelengths ranging from about 400 - 800 nm.

Light can be considered as an oscillation in an electric field coupled to a perpendicularly oriented magnetic field. This oscillation is a type of wave called an electromagnetic wave. You are already likely familiar with several types of waves: waves on an ocean, sound waves, and perhaps even seismic waves. Waves are a host of general properties, and light, an electromagnetic wave, is no exception. Some important properties of waves are as follows:

• Amplitude (A): size of the oscillation

• Wavelength(λ): distance between successive identical parts of a wave (SI Unit: meter). λ is the lowercase Greek letter lambda.

• Frequency (ν): number of waves that pass through a point in one second(SI Unit: Hertz or s-1). ν is the Greek letter nu (pronounced “new”).

• Wave Speed(c): the rate of propagation of the wave. (SI Unit: m/s)

Wavelength, or frequency, can be used to arrange the EM from short-to-long wavelength, or equivalently low-to-high frequency, where low frequency means small numbers and high frequency means large numbers. As seen in Figure 1, from short-to-long wavelength (low-to-high frequency) the parts of the EM spectrum are: Radio Waves, Microwaves, Infrared Light, Visible Light, Ultraviolet Light, X-rays, and Gamma Rays. You can think of each wavelength as a distinct “color” of light.

The wavelength is a measure of distance between the crests of the wave, and therefore is measured as a physical distance. The frequency is counting the number of wavelengths passing a point every second. For frequency, we use the unit Hertz (Hz). Hertz is a measure of the number of cycles per second, and so in this case it is the number of wavelengths passing every second. When frequency is multiplied by the wavelength, you then have a total number of wavelengths, a total distance, passing every second. Note that this is how much distance in how much time, or rather a speed. Indeed, the speed of a wave is found by multiplying the wavelength by the frequency. For light, the speed is always the same, c = 3.0 x 108 meters per second (m/s), or roughly 186,000 miles per second. This speed is the same regardless of wavelength (or equivalently frequency), and is therefore simply referred to as the speed of light and indicated by a lower-case letter c. This provides the connection between the speed of light, the wavelength of light, and the frequency of light, which can be summarized by the equation

𝑐 = 𝜆𝜈

where c is the speed of light, λ is the wavelength, and 𝝂 is the frequency.

A note on the speed of light: It is usually set by properties of the medium through which the wave is moving. Electromagnetic waves in a vacuum move at a constant speed in all reference frames: c = 300,000,000 meters per second (186,000 miles per second). The part of the electromagnetic spectrum humans can see is called the visible spectrum, or sometimes the optical spectrum. From long to short wavelengths, it is divided in six colors: red, orange, yellow, green, blue, and violet (ROYGBV) with wavelengths centered near 680 nm for red, 610 nm for orange, 580 nm for yellow, 540 nm for green, 470 nm for blue, and 410 for violet. The visible spectrum is shown in Fig. 1. Task 1: Explore Blackbody Radiation

Open the University of Nebraska – Lincoln Astronomy Simulation “Blackbody Curves of Melting.”1 With this simulation, you will observe how an object’s temperature relates to the light (radiation) it emits. Remember that all objects with a temperature emit radiation, and that dense objects emit thermal radiation, which in an ideal case is called blackbody radiation.

On the left, the simulation displays a nail being heated by an electric current passing through it. Like an electric stovetop, as the metal in the nail becomes hotter, it will begin to glow a dull red. As it gets hotter, it will glow yellow, and eventually white. Prior to glowing red, the nail is still emitting light, but the majority of that light is in the infrared portion of the electromagnetic spectrum, which the human eye cannot see.

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On the right, the simulation displays the light the nail is emitting as a function of wavelength. This is the nail’s blackbody spectrum. The x-axis is the wavelength of light. For the visible spectrum, you can think of this as the color of light. The y-axis is the intensity of light. You can think of intensity as how bright, or how many photons of light are being emitted. Hence, a blackbody curve allows you to look at how bright each wavelength of light is being emitted by an object at a certain temperature. The small graph that is inset in the top-left shows the blackbody curve for the wavelength range from 0 to 10 µm. The main figure is zoomed in around the visible portion of the electromagnetic spectrum, which is highlighted by the visible colors. It covers a wavelength range of 300 to 800 nm (0.3 – 0.8 µm).

You begin the simulation by pressing the “Start” button. 1) This question has you observe and characterize how the blackbody spectrum of the nail changes

with increasing temperature. Run the simulation until the nail breaks. In part (a) note how the overall size of the nail’s blackbody curve changes with increasing temperature. In part (b) you will examine how the peak of the blackbody curve (the highest point of the curve) changes with increasing temperature. a) As the temperature increases, how does the size of the blackbody curve change?

b) As the temperature increases, how does the location of the peak of the blackbody (the point where the highest intensity is) change?

2) Continue the simulation as the two pieces of the nail cool down. a) As the temperature decreases, how does the size of the blackbody curve change?

b) As the temperature decreases, how does the peak wavelength location change?

Task 2: The Thermal Radiation Laws: Wien’s Law and Stefan-Boltzmann’s Law Open the University of Nebraska – Lincoln Astronomy Simulation “Blackbody Curves (NAAP). – Blackbody Explorer” With this simulation, you can create a blackbody curve for any temperature between T = 3,000 K and T = 25,000 K. The simulation allows you to create multiple blackbody curves on the same set of axes, identify the wavelength where the blackbody curve has the highest intensity (a.k.a., the “peak wavelength”), see the total area under the blackbody curve, and examine photometric filters. Using the temperature slider, examine blackbody curves for the following temperatures and fill out the peak wavelength and area under the curve in the table below. For the “Peaks in what part of the EM spectrum,” all peaks will occur either in the Infrared, Visible, or Ultraviolet portion of the electromagnetic spectrum. If Visible, put the color of light instead of just “Visible.”

Table 1. Blackbody Curve Data

Temp [K] Peak Wavelength [µm]

Blackbody Area [W/m2]

Peaks in what part of EM spectrum?

3000 5000 7500 10000 12,500 15,000 20,000 25,000

3) Open the Microsoft Excel Template for this lab. Fill out the values you recorded in the appropriate

Excel columns. You will see the data you enter be placed on the two graphs below the entered data. The left graph plots the peak wavelength in microns vs the temperature in Kelvin of the blackbody. The right graph plots the total area under the blackbody curve in Watts per square meter vs the temperature in Kelvin of the blackbody. These plots demonstrate the relationship between an object’s temperature and what wavelength of light they emit most of and how much total energy they emit. a) You will now explore the relationship between blackbody temperature and at what wavelength

the blackbody curve reaches its maximum value, i.e., the peak wavelength. i) On the left graph (peak wavelength vs temperature), right click on one of the data points and

choose “Add Trend Line.” You will see the following options for what kind of function to fit to the data: Exponential, Linear, Logarithmic, Polynomial, and Power. Try fitting the data with each function. For Polynomial, try order 2, a quadratic equation. Which function provides the best fit to the data? Best fit to data is with a ______________________________ function.

ii) Choose the option “Display Equation on chart” to get the equation relating the peak wavelength to the temperature. Write the equation in the space below. Is the peak wavelength proportional to or inversely proportional to the temperature?

iii) The graph’s y-axis is peak wavelength measured in microns [µm]. Let’s label that 𝜆!. The graph’s x-axis is temperature measured in Kelvin [K]. Let’s label that T. Rewrite the equation replacing y with 𝜆! and x with T. This equation is known as Wien’s Displacement Law, often just “Wien’s Law,” which relates the temperature of a blackbody to the peak wavelength of that object’s blackbody spectrum. Congratulations, you just discovered one of the two thermal radiation laws! Let’s now find the other one.

b) You will now explore the relationship between blackbody temperature and the total area under the blackbody curve. The total area under the blackbody curve is really a measure of all the light an object is emitting every second. Particles of light come in discrete packets (quanta) called photons, such that a photon of a specific wavelength carries a specific amount of energy. Thus, the area under the blackbody curve is just counting up all the photons over all of the wavelengths, or in other words, it is counting up all the energy the blackbody is emitting. This total is given in amount of energy measured in Joules per second [J/s], or Watts [W], being emitted by every square meter (m2), an area, of the blackbody’s surface. i) On the total area vs temperature graph, right click on one of the data points and choose “Add

Trend Line.” Again, try all the options including a quadratic, i.e., a polynomial order 2 equation. Best fit to data is with a ______________________________ function.

ii) Choose the option “Display Equation on chart” to get the equation relating the total area under the curve (energy per square meter) to the temperature. You may need to change the format of the number to “scientific” to see the value of the constant. This can be done by double left-clicking the equation box and selecting the tab that looks like a bar graph. Write the equation in the space below.

iii) The graph’s y-axis is total energy per second per area measured in W/m2, a quantity called Flux, F. Astronomers call the total power (energy per second) an object emits over the entire surface is called Luminosity, L. Luminosity is measured in Watts [W]. So, the y-axis is the object’s luminosity divided by the total surface area of the object, A. Let’s, therefore, replace y with "

#.

The graph’s x-axis is temperature measured in Kelvin [K]. Let’s label that T. Rewrite the equation replacing y with "

# and x with T.

iv) Astronomers always want to know the luminosity (total energy per second) an object like a star emits. To get total energy, we need to multiply both sides of the equation by the area, A. On the left-hand side of the equation that leaves us with L, and on the right-hand side, we multiply everything by A. Rewrite the equation from (iii) as L equals… This equation is known as Stefan-Boltzmann’s Law, which relates the temperature of a blackbody to the total amount of energy that object is emitting, or rather, its luminosity. The constant in this equation is called the Stefan-Boltzmann constant, which is usually written as 𝜎$% instead of actually writing out the number.

v) The true value of the Stefan-Boltzmann constant is 𝜎$% = 5.67 × 10&' W m-2 K-4. Compare the number you got to that. Is it the same? If not, you may want to check your “Blackbody Area” data in Table 1.

Task 3: Application to the Stars This task moves away from the UN – Lincoln Astronomy Simulations and it has you apply the two thermal radiation laws you developed in the previous section to actual spectra2 of stars. Below are the spectra of several different stars, each one with a different temperature. The stars’ spectra are not perfect blackbodies because of the cooler gas above the “surface” of the star absorbing light at specific wavelengths. Understanding what causes these absorptions at specific wavelengths is the goal of a future lab. For now, you just need to focus on the general blackbody shape you can still discern. 4) To the best of your ability, identify and record the peak wavelength of each of the stars’ spectra.

Using that peak wavelength, apply Wien’s Law from Question 4a to determine the surface temperature of the star.

Peak Wavelength in nm: Peak Wavelength in nm:

Temperature: Temperature:

2 All stellar spectra from Pickles et al., PASP 110,863, 1998. Database hyperlink

Work Work

Peak Wavelength in nm: Peak Wavelength in nm:

Temperature: Temperature:

Work Work

5) Using your temperatures from Question 5, calculate the energy per unit area for each of the

four stars. To do this, you will use the version of the Stefan-Boltzmann Law that you determined in question 5b-iii. Your answers will be in the units Watts per square-meter. Show your work. a) Energy per second per square-meter for the G0V Star: _________________________.

b) Energy per second per square-meter for the K4V Star: _________________________.

c) Energy per second per square-meter for the K0V Star: _________________________.

d) Energy per second per square-meter for the F5V Star: _________________________.

6) In this final task, you determine the total luminosity (energy output every second) of the four stars and compare them to our Sun. You will have to use your answers from Question 6, the version of the Stefan-Boltzmann Law from Question 5b-iv, and the luminosity of the Sun, which is Luminosity of the Sun: LSun = 3.828 x 1026 Watts. The total surface area of a sphere is Area = 4πR2, where R is the radius of the star. To get total luminosity, you need to multiply your answers from Question 6 by the total surface area of the star. Show your work in the space provided below the table.

Star Radius [m] R

Surface Area 4πR2

Luminosity [W] LStar

𝑳𝒔𝒕𝒂𝒓𝑳𝑺𝒖𝒏7

G0V 7.6 x 108 m K4V 4.5 x 108 m K0V 5.5 x 108 m F5V 8.7 x 108 m

Name: Lab Instructor: Lab Section No.:

Thermal Radiation Laws Take-home Questions

These questions reinforce the concepts learned in the “Thermal Radiation Laws” lab. Complete these questions and turn them into your lab instructor at the beginning of your next lab meeting. 1. In a graph of a blackbody spectrum, what information is shown by the x-axis?

2. In a graph of a blackbody spectrum, what information is shown by the y-axis?

3. As the temperature of an object increases, how does the location of the peak of the blackbody spectrum change?

4. As the temperature of an object increases, how does the total amount of area under the blackbody spectrum change?

5. What is the name of the thermal radiation law that connects the temperature of an object to the peak wavelength of that object’s blackbody spectrum? Is the peak wavelength proportion to, or inversely proportional to, the temperature? In addition to answering that question, also write the equation for this thermal radiation law.

6. What is the name of the thermal radiation law that connects the temperature of an object to the energy per second per square meter, i.e., the flux, an object emits? Is the flux an object emits proportional to, or inversely proportional to, the temperature? What power of temperature does the flux depend on? In addition to the previous questions, write the equation for this thermal radiation law.

7. Determine the peak wavelength of the blackbody spectra for the following objects. Give you answers in microns [µm] and nanometers [nm].

a. A human with temperature, T = 310 K 𝜆![µ𝑚] = _______________µ𝑚 𝜆![µ𝑚] = _______________𝑛𝑚 Work:

b. The planet Venus with temperature, T = 735 K 𝜆![µ𝑚] = _______________µ𝑚 𝜆![µ𝑚] = _______________𝑛𝑚 Work:

c. The Sun with surface temperature, T = 5,800 K 𝜆![µ𝑚] = _______________µ𝑚 𝜆![µ𝑚] = _______________𝑛𝑚 Work:

d. The white dwarf star, Sirius B with surface temperature, T = 25,000 K 𝜆![µ𝑚] = _______________µ𝑚 𝜆![µ𝑚] = _______________𝑛𝑚 Work:

8. Use the Stefan-Boltzmann Law to determine the following properties of the white dwarf star, Sirius B, with T = 25,000 K and radius, R = 5,840 km = 5.84 x 106 m. 𝜎$% = 5.67 × 10&' W m-2 K-4

a. The total flux, F of Sirius B. Your answer will be in Watts per square meter [W/m2].

b. The luminosity, L, (total energy per second) emitted by Sirius B. Recall the surface area of a sphere is 𝐴𝑟𝑒𝑎 = 4𝜋𝑅(. Use the radius in meters for this question. Your answer will be in Watts (energy per second)

c. Given the luminosity of our Sun is L = 3.828 x 1026 W, does Sirius B have a larger or smaller luminosity? What is the ratio of the luminosity of Sirius B to the luminosity of the Sun? 𝐿$)*)+,%

𝐿$+.7 = __________________.

d.

9. Using the graph below, draw two generic blackbody spectra for a Hot Object and a Cold Object. Label the axes of the graph and indicate which blackbody spectrum is the Hot Object, and which is the Cold Object.