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Estimating Distances to Galaxies
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.
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.
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.
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.
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, but you have much better and more extensive data available to you.
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 the result you seek in Section IV.
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:
1) the apparent
sizes of face-on galaxies with high-contrast spiral patterns
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.