Tutorial: Working with color magnitude diagrams to determine
distances to and ages of open clusters.
Evolutionary Tracks of Stars and the Ages of Open Clusters
Shown above are the evolutionary tracks (sometimes called life tracks)
for stars from 1 to 15 solar masses. Note that this graph shows the
change in the temperatures (spectral types) and luminosities (which
could also be expressed as absolute magnitude) as these stars finish up
their main-sequence lifetimes. This graph and others like it do NOT
show how the stars move about the sky, where they are located on the
celestial sphere, nor where they are located in the Galaxy. [To
understand this, think about making a graph that shows the change in
your height and weight over your lifetime. That’s all that graph would
do—it would not indicate how you moved about during the day, what
city you lived in, or how photogenic you are.]
At right are the color-magnitude diagrams (CMDs) for 3 open star
clusters: the Pleiades, the Hyades, and NGC 2682 (No. 2682 in the New
General Catalog). Use the information provided in these figures to
answer the questions on the next page.
2/6/2016
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1.
Give an overall qualitative description of how the temperature and luminosity of a star that has 15 times as
much mass as the Sun changes in temperature and luminosity as it leaves the main sequence.
2.
Give an overall qualitative description of how the temperature and luminosity of a star that has the same mass
of the Sun changes as it leaves the main sequence.
3.
Fill in the following table:
Cluster
MS Turn-off
B – V (approx)
MS Turn-off
Temperature
Approx. age of the
cluster
Approximate mass of
stars at turn-off
Pleiades
Hyades
NGC 2682
4.
Why do CMDs look the way they do? We see a “time-frozen” picture for each of the clusters plotted in the
right-hand-side figures. The Pleiades are the youngest of these 3 clusters, probably a few 10’s of millions of
years old, as it appears that all of the stars—even the most massive ones—are still on the main sequence. The
turn-off for the Hyades is between B –V = 0.0 and 0.5, and so these stars are probably around 1 billion years
old; the cluster has had enough time that some stars have evolved clear to the white dwarf stage. NGC2682 is
by far the oldest. The turn-off is around B – V = 0.6. This cluster has stars that are close to 1 solar mass
leaving the main sequence, meaning that these stars are around 10 billion years old. Give the life-tracks for
stars of different masses in the left-hand-side figure above, do these CMDs make sense? Explain.
The Hyades and the Pleiades Open Clusters
Here is a negative image of how the open clusters the
Hyades and the Pleiades look in the night sky, showing
their relative positions and apparent sizes in the
constellation of Taurus. By just looking at them with the
naked eye, there is no way to tell which cluster is farther
away, and certainly no way to tell how much farther away
that cluster is. But, by using the apparent magnitudes of the
stars in each of these clusters (especially those stars that are
on the main sequence), astronomers can determine their
relative distances. If we knew the actual distance to one of
the clusters, say by an independent method, then we would
know the actual distance to the second cluster.
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We start by observing each cluster through different filters: one centered on blue wavelengths (B), and the other
centered on visible (V, roughly yellow) wavelengths. We then compare the magnitudes registered through each filter (B
– V, or the color index), and plot the results. Within each plot, we are comparing each star to other stars in the same
cluster. For open clusters, we assume that all of the stars in each cluster were born at roughly the same time (they are
all the same age) and from the same giant molecular cloud (they all have roughly the same composition). Thus, the only
difference among the stars in any given cluster is the mass of each star.
In order to compare one cluster to another, we must decide which stars are suitable for comparison. The stars on the
main sequence are the most reliable because we can assume that at a given mass, stars are roughly at the same
luminosity and temperature. We pick B – V = 1.0 (temperature approximately 4500 K).
Relative distances to each cluster
Here we want to just get a rough idea of the relative distances to each of these clusters. We start at B – V = 1.0, go
vertically until we meet the “bottom” of the main sequence at that color (think about drawing a line underneath all of the
points representing the stars on the main sequence) and then read off the corresponding “V” magnitude (this is the
apparent magnitude measured through the V filter). We note that for the Hyades, V = 10, and for the Pleiades, V = 12.
These two magnitudes are approximate. Because of our novice status in this work, our estimated uncertainties give us a
range of + or – 0.5 magnitudes for each V magnitude (range for the Hyades is 9.5 < 10 < 10.5; for the Pleiades, 11.5 <
12 < 12.5).
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Right away, we know that the Pleiades is farther away than the Hyades, because stars at a given temperature are
dimmer. Since we logically assumed these stars should be the same luminosity, the reason that they are dimmer is that
they are farther away.
What is the relative distances between these two clusters? Compare the V magnitudes of the Pleiades versus the Hyades
at B – V = 1.0: 12 – 10 = 2 magnitudes. Since each difference of 1 magnitude corresponds to a luminosity (intensity)
ratio of 2.512, the stars in the Pleiades are about 6.3 times dimmer (2.512 2). The apparent brightness – luminosity –
distance formula tells us that the Pleiades must be about 2.5 times farther away (just reverse the logic in this case, and
round off).
Additional examples for understanding: let’s assume that instead of 2 magnitudes difference, there were actually 3
magnitudes difference. This means that the stars in the more distant cluster would be 2.512 3 = 15.85 times dimmer.
This would correspond to a relative distance of almost 4 times farther away (square root of 15.85). Let’s say there are 5
magnitudes difference between two clusters as measured at a B – V = 1.0. Then, this means that the more distant
cluster is 2.5125 = 100 times dimmer, or 10 times farther away.
Absolute distances to each cluster
Through a number of different methods, astronomers have determined that at B – V = 1.0, the absolute magnitude of the
main sequence stars of the Hyades is 6.6. With an apparent magnitude (as determined here) of 10, this places the
Hyades at a distance of about 48 parsecs.
d  10
 mM 5 


5


 10
 106.65 


5


 10
8.4
5
 101.68  47.86
Using this number and the relative distances determined above, we find the Pleiades are 48 * 2.5 = 120 pc away. The
current values for the distance modulus to the Pleiades range from 5.3 to 5.6 corresponding to distances of 115 – 132 pc
away. That is about as certain as the distance to the Pleiades is at the moment, despite the best attempts by different
teams of astronomers! Our value determined here, especially considering our uncertainties, puts us in great agreement
with the best astronomers!
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Determining the ages of these open clusters
The Hyades star cluster is closer to the Sun (Earth), but is it older or younger than the Pleiades? There are a number of
ways to determine the age of a cluster, the primary one being the age of the stars that are just “turning off” of the main
sequence. Theory tells us that stars have a limited life fusing hydrogen to helium in their cores. The lifetime depends
on the mass of the star—the more massive the star, the shorter the lifetime. Let’s look at this in two different ways: A
table, and a diagram.
Spectral Color Lifetime
Type (B-V) (years)
O
–0.4
< 106
B
–0.2
3 × 107
A
0.2
4 × 108
F
0.5
4 × 109
G
0.7
1 × 1010
K
1.0
6 × 1010
M
1.6
> 1011
These values are all
approximate!
Given the information above
for the Hyades, we find that
this cluster has a turn-off at
about B – V = 0.1. This means
that stars between spectral
types A and B (temperatures of
around 10 – 15,000 K) are
evolving off of the main
sequence.
Age = about 4 × 108
For the Pleiades, we find the
turn-off at about B – V = –0.1.
This means that stars closer to
temperatures of 15 – 20,000 K
(maybe B5 or so) are evolving
off of the main sequence.
Age = about 108 years
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Take a closer look at the following diagrams and note how the “turn-off” of the main-sequence has been determined for
each cluster. Think about taking a “vertical line” and, starting at the y-axis, move the line horizontally until it runs up
against the place where there are the hottest stars—still some on the main sequence, but also some in the sub-giant
region of the H-R diagram. After reviewing the turn-off, rank these clusters from oldest to youngest. Compare your
answers to the key at the bottom of the page.
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TO = 0.0
TO = – 0.05
TO = + 0.15
TO = + 0.03