Exercise 1.3: Distances to Open Clusters
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:
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.
The Distance to the Pleiades
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!
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:
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