Tutorial: Stefan-Boltzmann Law

(Adapted from Lecture-Tutorials for Introductory Astronomy, ©CAPER Team, Preliminary Edition, 2002)

Part I: Size, Temperature, and Luminosity

Beginning astronomy students often think that they must come up with totally new and different explanations for things that go on in other places of the Universe besides Earth. This is not the case! If you are not venturing near a black hole nor attempting to go nearly as fast as the speed of light, the physics we know here on Earth applies to the rest of the solar system, the Galaxy, and the Universe. In this tutorial you will be led through the steps to understanding the Stefan-Boltzmann Law:

The amount of energy put out per second (the number of watts) is proportional to the surface area of the sphere (4 pi times the radius squared) and the temperature raised to the 4th power.

You are comparing the ability of a grouping of electric hot plates (shown below as burners on a “stove top”) of different sizes and temperatures to bring identical pot of water to a boil. The pots are all as large as the largest hot plate. The temperatures of the hot plates are coded: the lighter the shade of gray, the higher the temperature.

1. For each pair of hot plates (read horizontally), circle the one that will boil the water more quickly. Is there a set of burners for which there is no way to tell? If so, which ones?

2. If you use two hot plates of the same size, can you assume that the one that boils water first is at the higher temperature? (This is meant to be obvious.) Which lettered example(s) above supports your answer?

4. If you use two hot plates of different sizes, can you assume that the one that boils water first is at a higher temperature? Which lettered example(s) above supports your answer?

5. Two students are discussing their answers to question 4:

First student: In 1d, the hot plate on the left boils water quicker than the one on the right, even though it is smaller. The hot plate’s higher temperature is what makes it boil water more quickly.

Second student: But, the size of the hot plate also plays a part in making it feel hot. If the hot plate on the left were the size of a pinhead, the water would take a long time to boil! I bet that if the size difference were great enough, the one at the lower temperature could boil the water first.

You are called in to mediate this argument. What do you say? Whom do you agree with and why?

The time it takes for the water to come to a boil is determined by the rate at which the element transfers energy to the pot. This rate is related to both the size and the temperature of the hot plate. For stars, the rate at which energy is emitted is called luminosity, and it is an intrinsic characteristic (part of its true nature) of the star. Similar to the above example, a star’s luminosity can be increased by:

  • increasing its temperature; or
  • increasing its surface area (increasing its radius or size).

This relationship between size, surface temperature, and luminosity is usually referred to as the Stefan-Boltzmann law. Our knowledge of this law allows us to compare the sizes and temperatures of stars. For example, let’s say we know a star’s luminosity and its temperature. What can we determine? What if we have some estimate of the star’s radius and its temperature, what can we calculate?

6. If two hot plates are at the same temperature and one boils water more quickly, what can you conclude about the sizes of the hot plates?

7. Likewise, if two stars have the same surface temperature and one is more luminous, what can you conclude about the sizes of the stars?

Part II: Application to the H-R Diagram

8. Compare the following pairs of stars which may have similar temperatures or luminosities or radii. If the quantities are very close, then consider them identical for purposes of this comparison.

Pair / More luminous / Hotter / Larger
Arcturus & tau Ceti
Sun & alpha Centauri A
Barnard’s Star & Aldebaran
white dwarf & Barnard’s Star
Spica and Canopus
Spica & Sirius

9. Which pairs of stars were the hardest to figure out? Why do you think this was so?