At quite uncertain times and
-- James Clerk Maxwell
In the Pleiades lab you learned about H-R, or color-magnitude, diagrams. A color-magnitude diagram is a plot of apparent magnitude versus the (B-V) color index of a group of stars. An H-R diagram is usually a plot of absolute magnitude versus spectral type or color index. Note that they are basically the same thing, a plot of brightness versus stellar temperature, measured in different ways. When you plot these values for stars that are in a cluster, a close spatial group of stars, you can safely assume that these stars have roughly the same distance modulus, similar ages and similar initial chemical composition. The last two come from the fact that these stars formed out of one clump of interstellar material, all at roughly the same time. The location of stars in the color-magnitude diagram is then primarily due to the mass of the individual stars. This is useful for astronomers because color-magnitude diagrams of clusters then can be used to check theories of how stars evolve, i.e. change over time.
For example, consider a very young cluster. We know that the more massive stars go through their lifecycles faster than the lower mass stars, including the time they take to contract from the interstellar gas cloud into stars burning hydrogen into helium at their cores. This young cluster then should have more massive stars on the zero-age main sequence or ZAMS (the starting point where the stars begin burning hydrogen in their cores) than low mass stars, which haven't made it to the main sequence yet. Those stars are still contracting and heating up (e.g. protostars), so they are bigger and cooler than the cluster's main sequence stars. This corresponds to being to the right of the main sequence, but below the turn-off point. As the cluster ages, the low mass stars reach the main sequence but in the meantime the most massive stars have burned up all the hydrogen in their cores and begin to evolve to the right of the main sequence and into supergiants. The point on the main sequence where stars begin to leave and evolve into giants is called the turn-off point. As more time goes by, more stars peel off the main sequence and move toward the giant region to the upper right corner of the color-magnitude diagram. Armed with this idea, we can find a cluster which we think is young (due to the presence of hot blue stars or some other indicator) or old (due to lack of blue stars), and see if the color-magnitude diagram looks like we think it should to test our theories. Conversely, you can try and tell the age of a cluster by looking at the color-magnitude diagram and comparing it to your theoretical predictions.
This lab is computer-based. It is written in Java and should be accessible from any computer with web access and a Java-capable browser. On the Angell Hall computers, you should use Safari, since other browsers give undesirable results.
Point the browser to the following URL:
The program will run and will begin loading a set of color-magnitude diagrams for eight clusters in our galaxy. Once the images load (this could take a few minutes), take a moment to look through the various plots. (You can change from one cluster to another using the menu at the top right.) Notice that these are quite a bit messier than the main sequence we plotted in the Pleiades lab, because they are snapshots of real clusters at a single point in their lifecycle. Also notice that the diagram plots apparent magnitude, V, versus the (B-V) color index on the bottom x-axis. However, along the top x-axis is a quantity labeled (B-V)o, which is the color index corrected for interstellar reddening (we'll get to that later). A color index is a measure of the color of a star. The smaller the value of (B-V), the bluer the color of the star.
Once you've looked through the various diagrams, go back to the first one. Now click somewhere on the plot. You should see a blue crosshair, which you can drag around to measure values at various points in the color-magnitude diagram. The values are printed in the lower right. Drag the crosshair around to see how they change.
Finally, turn the ZAMS on with the menu in the upper right. This will produce a red gridded overlay; identify the ZAMS line and compare the axes of this with those of the color-magnitude diagram underneath. Moving the slider bars will allow you to slide the ZAMS around; clicking the arrows will allow finer adjustments. Use the menu to turn the ZAMS on and off.
Finding the age of a cluster of stars can actually be fairly accurate for clusters where there is a well-defined turn-off point. The turn-off point is the spot on the main sequence where stars are just starting to move up into the giant region. Below the turn-off, stars are still burning hydrogen in their cores, happily living on the main sequence. Above the turn-off point, stars have exhausted their core hydrogen. For a younger cluster, the turn-off point is closer to the blue (high mass, bright) end of the main sequence. For older clusters, the turn-off point is closer to the red end of the main sequence.
The turn-off point is probably also one of the first things you noticed about the cluster color-magnitude diagrams. To be sure that the turn-off is where you think it is, use the ZAMS overlay. First, match up the upper x-axis of the overlay and the upper x-axis of diagram, at (B-V)o = 0.0, by using the horizontal slider. Then slide the overlay up/down with the vertical slider until you get what you consider to be the best match between the star data points and the ZAMS line. When fitting clusters with a lot of scatter, try to match the narrower parts of scatter to the curve, and generally try to keep the ZAMS to the lower left of the scatter since objects not on the main sequence are probably above and right of the ZAMS. When you've got a match, you will be able to see where the star data 'peels off' from the ZAMS -- this is the turn-off point. Use the crosshair to measure the value of (B-V)o, and record this in Table 1.
The turn-off point's (B-V)o can be turned into an estimated age for the cluster using some theoretical calculations. Luckily, you don't have to do any calculations. Use the graph of cluster age vs. turn-off point color included in this lab by matching your (B-V)o to the value on the x-axis. Follow straight up to the solid line in the graph, and then over to the left to find the age. The labels on the y-axis should be read as the little number (1.5, 2, 3, etc.) times the large number that is labeled every ten spaces (107, 108, etc.). For example, you could have an age of 4 x 107. Record the age in Table 1.
Hopefully you remember distance modulus from the Pleiades lab. If not, or if the concept is unclear, look it up in your textbook and ask your instructor to explain distance modulus again. Briefly, the distance modulus is the quantity (m-M), the difference between the apparent and absolute magnitudes. In the Pleiades lab, you measured the vertical offset between the Pleiades stars (m) and a set of standard stars (M) in the HR diagram. Here you are doing much the same thing by sliding the standard stars (the ZAMS) down until they line up with the cluster stars. To find the distance modulus on these color-magnitude diagrams, keep your ZAMS overlay lined up as before so the ZAMS matches the star data points. Clicking the crosshairs at any point will measure V from the cluster color-magnitude diagram and MV from the ZAMS overlay. With a little math you should be able to get (V-MV). (If you click where MV = 0, that makes the math even easier.) Record V, MV and the distance modulus in Table 2.
To convert this into the distance of the cluster, we must invert the distance modulus equations.
(V-MV) = -5 log( d/10)
d = 10 (V-Mv+5)/5
Record the distance (in parsecs) in Table 2 as well.
There is another thing that we can learn from these color-magnitude diagrams, after the age and distance of the cluster. This last quantity doesn't have much to do with the cluster itself however, but rather it tells us about what it is between us and the cluster. That is, the interstellar dust.
Interstellar dust acts in similar ways to dust in our planet's atmosphere. You have probably noticed how the sun or bright full moon can appear orange or even red at sunset or sunrise. The reason is that at those times you are looking through a lot of dust in the air. Dust tends to absorb and selectively scatter blue light more effectively than red. The red light can pass by the dust grains more easily than the blue, so we end up receiving a higher fraction of the red light that is emitted than of the blue. This applies to dust in our atmosphere (resulting in a redder, dimmer sun at sunset) and also applies to dust between the stars (resulting in much the same thing).
So two things occur due to interstellar dust: starlight reddening (redder light) and extinction (less light).
The color index (B-V) that we observe is redder than the true color index of the stars because of the reddening effect of the dust. We can make an estimate of how much reddening has occurred by using the spectral type to calculate an intrinsic, un-reddened color index. This is labeled (B-V)o. The difference between the observed color index (B-V) and the intrinsic color index (B-V)o is called the color excess, or E(B-V), of the star. This is defined as:
E(B-V) = (B-V) - (B-V)o
If you look at the upper and lower x-axes on the cluster color-magnitude diagrams, you will notice that in each case the (B-V) and (B-V)o scales are offset such that the (B-V)o scale is shifted to the right with respect to the lower scale (to the right corresponds to a lower value of the color index, and thus is bluer). This should make sense to you, in light of the reddening effects of dust. Measure the color excess for each diagram by using the crosshair to measure (B-V)o and (B-V) at some point in the plot. Once again, a little math should give you E(B-V). (And once again, clicking where (B-V)o = 0 will make this trivial.) Record (B-V)o, (B-V) and the color excess in the Table 3.
Stars also appear dimmer due to interstellar dust. This is called extinction. Extinction affects our measurement of a star's brightness and therefore our determination of the star's distance if we use the distance modulous equation. Let's find out how extinction has affected our data.
In Table 4 are actual, carefully-measured distances to these star clusters. Rewrite your distances in the column marked d' (your measured distance) and calculate the ratio d'/d for the third column. Notice the errors.
Now answer the questions contained within the worksheets. (Note: #3 asks you to fill in Graph 1, so don't worry about that until you get there.)
Last Update: 10/31/12 by SAM
Copyright Regents of the University of Michigan.