Lecture Tutorial: Rotation curves of galaxies

In this lecture tutorial, you’ll learn what a rotation curve is, and plot an actual rotation curve for a galaxy.

Introduction to rotation curves

1. To understand what a rotation curve is, first consider the diagram shown below, which represents arunning track with three lanes. Imagine that there is a runner in each lane, and that each runner startsand ends on the same line (shown by the horizontal line near their lane numbers). Is this a fair race?Why or why not?

2. Suppose the runners begin and end the race at exactly the same time (i.e., the race is a tie). Whichlane had the fastest runner? Which had the slowest?

3. A rotation curve is a plot of velocity (speed) against radius (distance from the center of a circle). Usethe axes below to plot the velocity of each runner against their lane number (assuming they all startedand ended the race at the exact same time). Of course, I didn’t tell you the speed of each runner, soyou’ll need to make some guesses. The important thing is to consider how velocity changes with lanenumber. (You can derive more precise numbers in a challenge question).

4. The case you just considered produces a rotation curve known as a “solid body rotation curve”. Whydo you think it has this name?

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5. Can you think of other examples of solid body rotation?

A rotation curve for the solar system

6. The planets of the solar system orbit around the sun, and so we can also plot their speeds against theirdistance to create a rotation curve. Do you think the rotation curve of the planets will be a solid bodyrotation curve? Why or why not?

7. The table below lists the velocity of a few planets in the solar system, and their distances from thesun. Use this table and the axes below to plot the rotation curve of planets in the solar system.

Planet Distance from sun Orbital velocity(relative to Earth) (relative to Earth)

Mercury 0.39 1.61Earth 1 1Jupiter 5.2 0.4Saturn 9.5 0.32Neptune 30.1 0.18Pluto 39.52 0.16

8. Draw a line through the points. This curve is known as a “Keplerian rotation curve” because it is arepresentation of Kepler’s third law of orbital motion. It arises from the fact that the planets are allorbiting around the sun, which exerts a gravitational force on the planets. Now consider the imagebelow, showing the Andromeda galaxy. Using this image, explain why a galaxy might also be expectedto have a “Keplerian rotation curve”.

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Distance [kpc] Velocity [km/s]2 364 546 688 7910 8712 8916 9620 10122 10528 10534 10940 11846 11952 11958 12164 12571 13377 135

A rotation curve for the triangulum galaxy

9. The table above shows the rotation velocities and distances from the center of the triangulum galaxy(also called M33) for both stars and gas clouds. The velocities were derived by measuring the Dopplershift of spectra of the stars and gas clouds. Use the table and the axes below to plot the actual rotationcurve for this galaxy.

10. Does the M33 rotation curve look like solid body rotation, a Keplerian rotation curve, or neither?Explain your answer.

11. Challenge question: What might cause the differences between the observed curve and the expectedKeplerian rotation curve?

Congratulations! You just (re-)discovered dark matter! The rotation curve of galaxies was expected tolook similar to (but slightly different from) Kepler’s third law. The difference between the predictedand observed curve is the main evidence for dark matter.

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12. Challenge question: What is the exact shape of the solid body rotation curve? (Do NOT use theinternet to help you with this question.)

13. Challenge question: How might you use a rotation curve to measure the mass of a galaxy? (Feel freeto use your ipad to look up ideas, but do NOT google this exact question.)

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