Exercise 1.3: Distances to Open Clusters

Introduction

Open clusters are physically related groups of stars held together by mutual gravitational attraction. They are thought to originate from large cosmic gas and dust clouds in the Milky Way, and continue to orbit the galaxy within the disk.  Most open clusters have only a short lifetime as stellar swarms.  As these star clusters drift along their orbits, some of their members can escape.  An average open cluster has spread most of its stars along its orbital path after several 100 million years.  The escaped individual stars will continue to orbit the Galaxy on their own and are called field stars.  All field stars in our galaxy and other galaxies are thought to have their origin in clusters.

Open clusters have been known since prehistoric times:  The Pleiades (M45), the Hyades and the Beehive or Praesepe (M44) are the most prominent examples, but Ptolemy had also mentioned M7 and the Coma Star Cluster (Mel 111) as early as 138 AD.  First thought to be nebulae, it was Galileo who, in 1609, discovered that they are composed of stars while observing M44.  As open clusters are often bright and easily observable with small telescopes, many of them were discovered with the earliest telescopes.

The fact that stars in a cluster all lie at approximately the same distance allows us to determine the distance by making an H-R diagram of the cluster stars.  However, we construct the diagram by plotting apparent brightness (magnitude) rather than the true luminosities along the vertical axis.  Once we plot a sufficient number of stars, the cluster's main sequence appears prominently.  Because we already know the true luminosities of the main sequence stars from the standard H-R diagram, we can calculate the distance to the cluster with the luminosity-distance formula.  The method of obtaining the cluster distance is called main sequence fitting because it relies on "fitting" the main sequence on the cluster diagram to the standard main sequence.  Here are two examples of H-R diagrams of clusters.

Notice how the Main Sequence does not fully continue across the diagram and seems to bend over towards the right.

Main Sequence Fitting

We can use Sun-like stars as standard candles (those objects for which we are likely to know the true luminosity) because we know that they are similar to the Sun and because we can measure the Sun's luminosity quite easily.  However, Sun-like stars are relatively dim, and cannot be easily detected at great distances.  To measure beyond 1000 light-years or so, we need a brighter standard candle.  However, before we can use any main sequence star as a standard candle, we must first have some way of knowing its true luminosity.  The key to understanding this process is to remember that we can use the luminosity-distance formula in two ways:

1. First, we identify a star cluster that is close enough for us to determine its distance by parallax and plot its H-R diagram.  Because we know the distances to the cluster's stars, we can use the luminosity-distance formula to establish their true luminosities from their apparent magnitudes.
2. Then we can look at stars in other clusters that are too far away for parallax measurements and measure their apparent magnitudes.  If we assume that the main sequence stars in the other clusters have the same true luminosities as their counterparts in the nearby cluster, we can calculate their distances with the luminosity-distance formula.

The nearest star cluster with a well populated main sequence, the Hyades Cluster in the constellation Taurus, is crucial to this technique.  We have measured the true distance to the Hyades Cluster with another method which we will not employ here.  Knowing this distance allows us to determine the true luminosities of all its stars with the luminosity-distance formula.  We can find the distances to other star clusters by comparing the apparent brightnesses of their main sequence stars with those in the Hyades Cluster and assuming that all main sequence stars of the same temperature (or color) have the same luminosity.

Another well-known cluster is the Pleiades star cluster, also located in the constellation Taurus.  We're going to determine it's distance by comparing the apparent magnitudes of it's stars to the absolute magnitude of stars close enough to have accurate parallax distances.  The Pleaides stars should trace out a nice main sequence, as will the stars whose absolute magnitudes are known.  However, since the Pleaides are farther than 10 parsecs, that Main Sequence should be significantly fainter.  We can then use the distance modulus formula to compare the difference in brightness between the two Main Sequences and find the distance to the Pleiades!

Distance Modulus

The distance modulus is the difference between the apparent and absolute magnitudes of a star, the quantity (m-M).  This distance modulus is related to the distance by the following formula:

To convert this into the distance to the cluster, we must invert the distance modulus equation.  Apparent magnitude is sometimes referred to as V:

and then solving for the distance gives:

Use this formula to find the distance to each cluster and record the distance (in parsecs) in your notebook.

We're going to use EXCEL to make some simple plots of a few Pleiades stars and some "standard stars".  Rather than using a cluster of known distance, we will use stars whose distances are known by parallax.  These Standard Stars are nearby stars of different distances.  Knowing their distance allows us to know their absolute magnitudes (the magnitude they would have if they were all 10 parsecs away).

You're going to plot the temperature versus magnitude for both these known stars and 10 stars from the Pleiades.  Both groups will trace out a main sequence.  Before we get started, think about these questions:

1. If I gave you the apparent magnitudes of the Standard Stars and asked you to plot those values versus the temperature of those stars, would you see a main sequence?
2. How will the main sequence of the Pleiades stars compare to the main sequence traced out by the standard stars?

Proceed as follows:

2. Click on the Chart option (an icon with blue, yellow, and red bars).
3. Select XY (Scatter).  Click on the TOP option (no lines).  Click on NEXT.
4. Click on tab labeled "Series" then click on Add.
5. Name the first series "Standard Stars".  Use the icon with the red arrow to choose the Standard Star Temperatures for the X-Axis. Do the same to put the Absolute Magnitude in the Y-Axis.
6. Click on Add again.  This time do the same for the "Pleiades Stars" as you did for the Standard Stars.
7. Click on NEXT.  Label the Chart, and the X and Y-axes.
8. Click on NEXT and put the Chart on a new sheet.
9. Now, there's just one problem.  Look at some of the HR diagrams you've seen before.  How are the X and Y-axes oriented?  Are they different than yours?  To fix this, double click on X-axis then select "Scale".  Select "Values in Reverse".  Do the same for the Y-axis.
10. Print out your plot.  Draw a SMOOTH CURVE through the scatter of points for EACH set of data.
11. Pick a few places along your curves and measure the difference in the magnitudes.  Average these and use this to calculate the distance to the Pleiades!

Discussion Questions:

1. Why do you think we used an entire cluster rather than just a single star in the Pleiades?
2. Do you think this method could work to find the distance to other galaxies? Why or why not?  Try this:  the Absolute Magnitude of the sun is +4.8.  What would its apparent magnitude be if it was in the Andromeda Galaxy, 860,000 parsecs away?  Could the Hubble Space telescope detect it?  The HST can detect objects as faint as about m=+30.  Would their be any other problems other than just being able to detect something that faint?