Math of the Expanding Universe

In this activity, students will learn about the expanding universe and the redshift of lightwaves. They will then calculate the redshift of a supernova, determine its velocity relative to Earth, and find the distance to that object.

• Students should have a general understanding of the electromagnetic spectrum and/or wave properties (i.e., amplitude, wavelength, frequency, and speed).

• Use an overhead camera to show students the stretching elastic demo. Place a ruler parallel to the elastic so students can see the numeric change in distance between wave crests.

Our eyes can only see a small portion of the electromagnetic spectrum – the portion we call visible light. Other portions of the spectrum have wavelengths too long or too short for our eyes to perceive.

Typically, the human eye can detect wavelengths from 380 to 700 nanometers (nm). For comparison, a human hair is about 80,000 nm wide. As the full spectrum of visible light travels through a prism, the wavelengths separate into the colors of the rainbow because each color is a different wavelength, thus refracting, or bending, a different amount. Violet has the shortest wavelength at around 380 nanometers and red has the longest wavelength at around 700 nanometers.

Every type of atom or molecule gives off energy at specific wavelengths. For example, hydrogen atoms emit visible light at 410.2 nm, 434.0 nm, 486.1 nm, and 656.3 nm. This predictable pattern of emitted light is called an emission spectrum.

Close examination of the visible-light spectrum from our Sun and other stars reveals spectral lines that act like fingerprints for atoms and molecules. Identifying these patterns tells scientists what elements and molecules make up a star.

Unlike light leaving the Sun, which takes 8.5 minutes to reach Earth, light leaving distant stars and galaxies can take billions of years to reach our planet. As they travel, the waves that make up the light get stretched as the universe expands – something it’s been doing since the Big Bang.

What this means for light coming from distant galaxies moving away from Earth as the universe expands is that the visible lightwaves you would be able to see with your eyes get stretched into longer wavelengths and shift from visible light toward infrared.

In the case of hydrogen, the four emission lines get shifted toward the redder (longer wavelength) end of the electromagnetic spectrum. Scientists refer to this phenomenon as redshift – and the farther away an object is, the more redshift it undergoes.

Redshift is given a value (z) and is calculated with the following equation:

z = (Wavelength_o - Wavelength_r) / Wavelength_r

• z = redshift
• Wavelength_r = the known wavelength of an element not experiencing redshift
• Wavelength_o = the observed wavelength of that emission line

By determining the redshift of an object, scientists can calculate the speed at which the object is moving away from us (as a result of the universe expanding) and the distance to the object from Earth.

Missions such as the James Webb Space Telescope take advantage of this phenomenon by viewing light in the near- and mid-infrared wavelengths – the wavelengths to which light has shifted after traveling from the most distant, and therefore oldest, stars and galaxies. This allows astronomers to peer into the early history of the universe.

• What implications does redshift have for making observations of the most distant stars and galaxies, where even ultraviolet light may have redshifted beyond the visible spectrum?
• If light has been traveling from a star, supernova or galaxy for one billion years, does that mean the object is currently one billion light years away? Answer: The object would be farther away since the universe is expanding and the object has moved away from Earth during the time the light has been traveling toward Earth.
• Do you think all objects in the universe are redshifted? Answer: As the universe expands galaxy clusters and other distant objects are moving apart from each other, displaying redshift. However, nearby objects, such as planets, and even local galaxies, are gravitationally bound and not displaying redshift as a result of the galaxy expanding. In fact, the Milky Way and the Andromeda galaxies are moving toward each other.
• What would you expect to see if the universe was contracting rather than expanding? Answer: Students would expect to see blue shifted wavelengths, as the light traveling toward the observer on Earth was compressed into shorter wavelengths as the space it was traveling through contracted.

• Students should correctly measure a longer wavelength on the stretched elastic.
• Students should calculate the correct z, v and d values in the problem set.