In the last section, you used magnitude as a substitute for relative distance. But magnitude can also be converted into a direct measure of relative distance. To find the relative distance, first convert the magnitude from logarithmic units to real units. The quantity that magnitude actually measures is radiant flux – the amount of light that arrives at Earth in a given time. The formula for finding the radiant flux F from magnitude m is:
F = 2.51-m.
F is a relative number that compares the arriving radiant flux to a standard: the star Vega in the northern constellation Lyra. But since we are only looking for relative distance, we can use the relative radiant flux. Now, take the square root of the radiant flux. Take the inverse of the square root. The result is a relative distance to the galaxy. To make it easier to understand these relative distances, you should “normalize” them so that the nearest galaxy has a relative distance of 1. Then a galaxy twice as far away as the nearest galaxy will have a relative distance of 2.
To normalize the relative distances, set up a ratio between the relative distances of the nearest galaxy (1) and the second nearest (2) so that d1 / d2 = 1 / x, then solve for x: the normalized distance to galaxy 2. Repeat to find the normalized relative distance to farther galaxies. This technique will accurately measure the true relative distances to galaxies.
Exercise 7: Find the relative distances between the six galaxies whose magnitudes you found in Exercise 1. Use a scientific calculator that can display numbers in scientific notation (that is, as 1.5 million = 1.5E+06).
Use this SkyServer workbook to help you with your calculations.
An alternative way to estimate the distance to a galaxy is to look at its apparent size. The farther away a galaxy is, the smaller its image will appear. To find relative distance using this technique, measure the distance across the galaxy’s image in any convenient units: inches, minutes of a degree, or something else. If you assume that all galaxies have approximately the same true size, the inverse of that number will tell you the relative distance to the galaxy.
Exercise 8: Write the two techniques for finding relative distance as algebraic equations. Derive them using geometrical or physical principles.
Question 2: Suppose the relative distances for a number of galaxies using brightnesses don’t agree with the relative distances using apparent sizes. What would you conclude?
But even if you know the correct relative distance to a galaxy, you still run into the problem you saw in Exercise 5: different galaxies have different properties. Suppose a certain galaxy is twice as large as an average galaxy. On Earth, the only information we could get from the galaxy is what we could see in its image. When we saw the larger image, we would have no way of knowing the galaxy actually was larger: we would assume that it was simply closer to us. Because we lacked knowledge of the galaxy’s true properties, we would misjudge its actual distance from us. Because we misjudged the galaxy’s distance, if we used it in a Hubble diagram, we would not get the correct results. To overcome this problem, we need to look not just at individual galaxies, but also at clusters of galaxies.
Estimating Distances to Clusters
The key to overcoming our lack of knowledge about galaxy properties is to recognize that clusters can be thought of as statistical units or populations of galaxies: small groups might have a dozen member galaxies, and rich clusters could have more than a thousand member galaxies. Even if the properties of individual galaxies vary widely, the average properties of galaxies in the cluster should come close to the average properties of galaxies in the universe. The larger the cluster’s population, the more confident we are that statistical measures are meaningful.
Galaxies in a cluster are effectively all at the same distance (both relative and absolute) from us. This means that their apparent sizes and brightnesses are in the same proportions as their intrinsic, or “true,” sizes and brightnesses. In other words, if galaxy A in a cluster looks 3.5 times brighter than galaxy B, then it really emits 3.5 times as much light. By looking at the galaxies in a cluster, we can get a picture for the variations between galaxies in a population.
Question 3: Why can we assert that galaxies in a cluster are all at the same distance?
Exercise 9: Show that the fractional error in the assumption that galaxies in a spherical cluster are all at the same distance is equal to the cluster’s angular size: the angle of the sky that it takes up when viewed from Earth.
The trick in estimating galaxy distances, however, is knowing which galaxies are actually part of the cluster. Just because two galaxies are in the same area of the sky doesn’t mean that they are in a cluster; they could be in the same general direction, but at very different distances.
The following analogy might help you figure out how to place galaxies in clusters. Suppose that galaxies are like buildings, and groups and clusters are like towns and cities. Suppose you were standing on a very tall platform at Fermilab in Batavia, Illinois. You look out over the large, flat plains of central Illinois with a telescope. Your task is to survey the landscape for buildings, towns, and cities, and make a map that shows their positions with respect to you at the center. You are not allowed to use any information other than what you can see through your telescope.
In principle, a small town in the relative foreground could look like a large city that is farther away: a one-story building will have the same apparent height as a ten-story building that happens to be ten times farther away. But it is unlikely that you would confuse the small town for the large city – there are enough other bits of information at your disposal to get these populations of buildings in their correct relative positions.
Question 4: What are some of those clues and cues? Would any of those techniques apply to estimating relative distances for galaxies in space?
Exercise 10: Look at SDSS images for the following clusters:
| Cluster name | Run | Camcol | Field |
|---|---|---|---|
| Abell 2255 | 1356 | 2 | 32 |
| Abell 603 | 756 | 3 | 748 |
| Abell 1865 | 752 | 3 | 468 |
| Unnamed | 1339 | 3 | 11 |
| Unnamed | 756 | 1 | 728 |
For each cluster, think about how we know that the galaxies are actually part of the same cluster. What properties are similar between galaxies in the same cluster? What properties show a wide range? How might you be able to tell – using just these images – if any particular galaxy is actually in the cluster, as opposed to being at a different distance along the same line-of-sight?
Once you know which galaxies are members of clusters, you can compare the properties of individual galaxies in different clusters. For example, the average size of a galaxy in one cluster should be about the same as the average size of a galaxy in another cluster. Or, the brightest galaxy in one cluster should have about the same true brightness as the brightest galaxy in another cluster.
There is no “best” way to measure relative distances to galaxies, but some may be arguably more effective than others. Edwin Hubble and his co-worker Milton Humason tried a number of schemes, mostly related to the value of the apparent brightness of the brightest galaxies in rich clusters, and other astronomers tried others. But today, you have much better and more extensive data available to you!
Question 5: What would tell astronomers that one approach was better than another?
Any technique for measuring distances will have associated errors – that’s why we speak of estimates. As long as there are enough galaxies in our 3-D map, a structure in the map may be apparent even if your distances are only approximate. You do want to avoid a type of error called a systematic bias, that is, where all the distances are incorrect by an unknown factor that depends on distance. If your distance estimates suffer from this problem, you won’t get a useful Hubble diagram.
Because the properties of galaxies vary so widely, you should use the same measure of relative distance for each cluster you examine. Some examples of things that astronomers have tried as indicators of relative distances are:
- the apparent sizes of face-on galaxies with high-contrast spiral patterns
- the apparent size of the rings in ring galaxies
- the apparent sizes of edge-on disk galaxies
- the apparent brightnesses of the brightest galaxy in a rich cluster of galaxies
- the apparent brightness of the 10th-ranked (or 3rd-ranked, or 5th-ranked) galaxy in a rich cluster of galaxies
- the apparent size of the cluster itself
- the value of a feature in the histogram of the apparent brightnesses (or sizes) of galaxies in a cluster
Relative Distances for Sample Galaxies
Now that you have identified some of the ways to determine relative distances to galaxies and clusters, you are ready to use these ways to find the relative distance to some real galaxy clusters.
Exercise 11: Look at the SDSS image below. The image shows three galaxy clusters in the same area of the sky.
Look closely at the image and decide which galaxies belong to which clusters. Make some notes for yourself about which galaxies belong where.
Click on the image for a larger view (opens in a new window).
Use this SkyServer workbook to record your data.
Exercise 12: Now, find the relative distances to the galaxies you studied in Exercise 11.
Open the Navigation Tool window. The tool will open in the area of the image shown above: RA = 178.27, Dec = 1.025. Use the zoom buttons (the magnifying glasses and blue rectangles on the left side of the window) to zoom in our out. Click on the NWSE buttons to shift the part of the sky shown in the main window. Use the zoom and NWSE buttons until what you see in the main Navigation Tool window looks like what you saw in Exercise 11. Then, click the “specObjs” checkbox. The main window will reload with red squares around all galaxies for which the SDSS has measured a spectrum.
Measure one property of each galaxy you see marked by a red square. From your measurement, calculate the relative distance of each galaxy with respect to the closest one (which would have a relative distance of 1). Record your galaxy measurements as a table with the following format: object ID, right ascension, declination, measurement, relative distance. Then, click “Add to Notes” to save each galaxy to your notebook.
Use the same workbook you used in the last exercise.
Exercise 13: Repeat Exercise 12 for the same clusters using a different measured quantity leading to another estimate of relative distance. Add two columns to the right edge of your table for your second measurement and second relative distance. How do your independent estimates of the cluster distances compare? Which is better? Why?