Cluster Color Magnitude Diagrams Printable Version (PDF) Data Table in Word
In this exercise you will be finding the ages and distances of star clusters by analyzing their color magnitude diagrams.
We think that stars in star clusters all formed at nearly the same time. The
stars don’t all live the same length of time, however. This is because more massive stars use their
hydrogen fuel more rapidly, so “THE BIG
DIE YOUNG”. As the stars run out of
hydrogen, they leave the main sequence and become giants and supergiants. You
may want to review stellar evolution and reread section 20.5 of Chaisson
Macmillan concerning star cluster color magnitude diagrams before proceeding.
You will be assigned 2
star cluster color magnitude diagrams to analyze. There is a quiz in WebCT that generates random
assignments. Links to the star clusters
are at the end of this write up. Feel free to compare your results with the
ones that other students get. It will probably help if you print out your color
magnitude diagrams before reading on.
Colormagnitude diagrams are like
HertzsprungRussell diagrams. But instead of spectral type or temperature, the
horizontal axis is “color”, the difference in magnitude of the star when measured in two different
wavelength regions. “Color” indicates temperature, but is easier to measure
than spectral types.
On color magnitude diagrams, the main sequence goes from hot
massive stars at the upper left, to cool low mass stars at the lower right. As
stars use up their core hydrogen, they turn
off the main sequence toward the giant region on the right. The very brightest stars still on the main
sequence are just at the end of their lifetimes. So the main sequence lifetime
of a star at the hot, bright end of the main sequence, the main sequence turn off, equals the age of the entire cluster.
You will be calculating the age of your star clusters by finding the main
sequence, then finding the brightness, mass and
age of star at the main sequence
turn off.
Can you find the main sequence on your cluster color
magnitude diagrams? It goes diagonally
up toward the left. Check out the color
magnitude diagram above and the worked example at the end
Magnitudes are on the vertical scale and color on the
horizontal. The magnitudes are generally apparent
visual magnitude (usually indicated by V).
If there are two vertical scales, one will be absolute magnitude (M_{v})
and the other will be apparent magnitude.V. If the absolute magnitude is given,
use it. Otherwise use the horizontal
scale to find the absolute magnitude.
The horizontal scale, called color or color index indicates
temperature. The actual measurements might be
BV the magnitude at
wavelength 4400Å minus the magnitude at 5500Å
VI the magnitude at
wavelength 5500 Å minus the magnitude at 9700Å
RI the magnitude at
wavelength 7100 Å minus the magnitude at 9700Å
These colors are different,
they produce slightly different main sequence curves, as shown below.
These
main sequences probably look different from the ones for your clusters because
the relative sizes of the horizontal and vertical scales are different here
than on your plots. The magnitude level
in your diagram probably will be different too, since these diagrams are in
absolute magnitude, while your cluster data has apparent magnitude. It is
necessary to use the numbers that are on these main sequences.
First find the main sequence turn off for your cluster, just read off the apparent
magnitude and the color at that point (and record them in the data table). Then use the main sequence plot above and
read off the absolute magnitude at the given color.
Now use the magnitude at the main sequence turn off to
compute the following.
A) Distance to cluster
B) Mass of a star at the main sequence turn off
C) Age of the cluster
The formulae are below.
A) Distance to Cluster
_{}
Where
m
is the apparent magnitude
M
is the absolute magnitude that corresponds to the apparent magnitude. Use the
star cluster color magnitude diagram to get an apparent magnitude (m) and a
color (x coordinate) on the main sequence. Use the theoretical main sequence to
get the absolute magnitude for the same value of the color.
D
is the distance in parsecs from Sun to star cluster
When
using this type of formula, you may need to find the value (mM)/5 first, then
use the 10x key and then multiply the answer by 10. Check yourself, if m=12 and
M=3, then the distance is 630.95 pc
B) Mass of star at Main Sequence turn off
For stars on the main
sequence, the massluminosity relationship is
_{}
The relationship between
luminosity and magnitude is
_{}
Using 4.85 for the Sun’s
magnitude produces,
_{}
Substituting for _{} produces
_{}
Or, more simply the mass of a
star on the main sequence is given by
_{}
Use this to find the mass of
the star at the main sequence turn off in solar masses.
(Example, if the absolute
magnitude were 3, the mass would be 1.531solar masses)
C) Age of the Cluster
_{}
A star at the main sequence
turn off has age equal to its main sequence lifetime and also equal to the age
of the cluster. So this value is the age of the cluster.
(Example
The 1.531 solar mass star the main sequence lifetime is 2.79x10^{9}
years.)
A worked example follows. For your homework, fill in the
table. If you are at all uncertain ask
for help and turn in your intermediate results. There is no need to look up the
“right” answer from the literature. The
idea is to get a sensible answer, not particularly to match a value.
Possible Complications
Some of the color magnitude diagrams have (BV) for the bottom axis and (BV)_{o}
along the top. The scale to use is the TOP one. The difference is that gas
between Earth and the star has scattered and absorbed some of the star’s light.
The blue light is affected more than is the red light, so the overall effect is
to make the star look redder and fainter. The value of (BV)o has been
corrected for this reddening and that is what you should use.
Some of the color magnitude diagrams show both apparent and
absloute magnitude. They would show up as two different sets of labels on the
right and left vertical axes. If
the scales don’t already say which magnitude is which, the absolute is probably
is called Mv and is on the right hand side vertical axis. These same diagrams
have the apparent magnitude, V, on the
left hand vertical axis. The absolute magnitude value will be lower than the
apparent.
Some of the magnitude
and color values are negative. Be careful to read the values correctly.
What if you cannot
decide on the main sequence turn off?
Find the main sequence, not near
the turn off, and sketch a line through
it where ever you are sure that you can see it. Pick a point on your sketched
line and read off the color (x axis value).
Now look at the theoretical main sequence and find the absolute magnitude
that goes with the same color.
Calculate (apparent
magnitude of point on main sequence on star cluster color magnitude diagram –
absolute magnitude on theoretical main sequence at the same color)
This difference is the
distance modulus. Normally it will
be positive.
Plot points from the
theoretical main sequence onto the star cluster color magnitude diagram. The x
coordinate from the theoretical main sequence is the same as on the star
cluster diagram. The y coordinate is
absolute magnitude from the theoretical diagram plus the distance modulus. Plot points going toward brighter and
hotter stars.
The theoretical main sequence should run through the main
sequence that you sketched until you reach the main sequence turn off. Then the
star cluster data will diverge from the theoretical with star cluster points on
the right hand side of the diagram. The main sequence turn off is where the two
plots diverge. (If the two plots never match at all, even at faint stars, you
are doing something wrong.)
I suggest that you print out the cluster color magnitude diagrams and draw on
them. If you are turning work via using email, I need only this table or equivalent data. A word version of the table
is available at the top of the write up from
the web version of this write up.( NOT the pdf version). If you are
turning things in by mail or in person, I suggest that you turn in the cm
diagrams and your work. For partial credit on computations, show your work and
turn it in.
Cluster Name 


Apparent
Magnitude at Main Sequence TurnOff, m
(or V on some diagrams) 


Type of
Color Index BV, VR,
or VI? 


Numerical
Value of Color Index at Main Sequence Turn Off , from the star cluster data 


Absolute
Magnitude at Main Sequence Turn Off , M 


Distance
Modulus (mM) 


Distance
to cluster in parsecs 


Mass of
Star at Main Sequence Turn Off in solar masses 


Age of
Cluster in years 


Apparent
magnitude of Tip of Giant Branch ** 


Apparent
magnitude of Horizontal Branch** 


**
NOT every cluster shows horizontal branch or tip of the giant branch. If the
feature is lacking, WRITE “not present”.
Horizontal Branch 0.5 0.5 1.5 0 Stars just coming onto the Main Sequence would be
here, but the few stars here are probably observational scatter.
VI
The
Fornax 1 cluster is shown. The main sequence, the giant branch and the
horizontal branch are clearly visible. They are labeled on the second version.
This cluster shows no stars that are still coming onto the main sequence, but
the area where they would be is marked.
Fornax 1 has stars above the main sequence cut off. They are circled
on the second figure. They are usually called
“blue stragglers”. We think that blue
stragglers form when a star in a binary system gains mass from its companion.
The blue straggler “became” a higher mass star. But it started at this main
sequence position long after than the rest of the cluster formed. So blue
stragglers are still on the main sequence and are not used to determine the
cluster age.
To
find the distance and the age of the cluster, we need to look at the main
sequence.
In
this case, the “color” is VI and the turn off is near
Cluster Name 
Fornax 1 
Apparent Magnitude at Main Sequence
Turn Off 
24.5 
Color Index at Main Sequence.
Specify whether it is BV or VI and tell the value 
0.4 
Absolute Magnitude at Main Sequence
Turn Off 
2.5 
Distance Modulus (mM) 
22.0 
Distance to Cluster in pc 
251,189 pc 
Mass of Star at Main Sequence Turn
Off in solar masses 
1.7179 solar masses 
Age of Cluster in years 
1.972 x 10^{9} years 
Apparent magnitude of Tip of Giant
Branch ** 
18.6 
Apparent magnitude of Horizontal
Branch** 
21.3 
VI
=+ 0.4 magnitudes (the x coordinate).
The left hand scale is apparent magnitude and there is no absolute
magnitude given
To
find the absolute magnitude for the main sequence look at the theoretical main sequence and
read off the magnitude at the main sequence turn off, VI = 0.4. That is the corresponding point on
the theoretical diagram. The theoretical main sequence crosses VI = 0.4
at magnitude 2.5.
So
the absolute magnitude, M, at the main sequence turn off is 2.5. The Distance Modulus, (mM), is
24.52.5=22. The
distance modulus is not always an integer. Usually it is positive.
To
find the distance, D, in parsecs use
_{}
Substitute
for (mM),
_{}
_{}
Use the 10^{x} key or the caret (^) key to
find 10^{4.4}.
_{}
Try it yourself to check.
The
mass of a main sequence star is found next. The relationship between luminosity
and magnitude is
_{}
Substituting the Fornax 1
value for the magnitude at main sequence turn off, 2.5, into the equation, we
find
_{}
The main sequence lifetime is
found using:
_{}
So substituting 1.7179 solar
masses for the mass of the star _{}
The unit, solar masses,
cancels with the mass of the Sun leaving
_{}
So the age of the Fornax 1
cluster is 1.97 x 10^{9} years.
Star cluster ages range from a few million years
(several times 10^{6}) to a few billion (several times 10^{9}).
To find the magnitude at the
tip of the giant branch and the magnitude of the horizontal branch, just read
from the diagram. Estimate to the nearest 0.1 magnitude.
The table to the right shows
how a column would look for the Fornax 1 cluster.
Cluster Number 
Name 
Cluster Number 
Name 
1 
21 

2 
22 

3 
23 

4 
24 

5 
25 

6 
26 

7 
27 

8 
28 

9 
29 

10 
30 

11 
31 

12 
32 

13 
33 

14 
34 

15 
35 

16 
36 

17 
37 

18 
38 

19 
39 

20 
40 