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.

 

Color-magnitude diagrams are like Hertzsprung-Russell 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 (Mv) 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 

B-V  the magnitude at wavelength 4400Å minus the magnitude at 5500Å

V-I   the magnitude at wavelength 5500 Å minus the magnitude at 9700Å

R-I   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 (m-M)/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 mass-luminosity 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 2.79x109 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 (B-V)  for the bottom axis and (B-V)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 (B-V)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 Turn-Off, m  (or V on some diagrams)

 

 

Type of Color Index

B-V, V-R, or V-I?

 

 

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

(m-M)

 

 

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”.

EXAMPLE  

Horizontal Branch

 

0.5

 

-0.5

 

1.5

 

0

 

V-I

 

Stars just coming onto the Main Sequence would be here, but the few stars here are probably observational scatter.

 

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 V-I 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 B-V or V-I and tell the value

0.4

Absolute Magnitude at Main Sequence Turn Off

2.5

Distance Modulus

(m-M)

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 109 years

Apparent magnitude of Tip of Giant Branch **

18.6

Apparent magnitude of Horizontal Branch**

21.3

V-I =+ 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,  V-I = 0.4. That is the corresponding point on the theoretical diagram. The theoretical main sequence crosses V-I = 0.4 at   magnitude 2.5.

 

So the absolute magnitude, M, at the main sequence turn off is 2.5. The Distance Modulus, (m-M), is 

24.5-2.5=22.  The distance modulus is not always an integer. Usually it is positive.

 

To find the distance, D, in parsecs  use

Substitute for (m-M),

Use the 10x key or the caret (^) key to find 104.4.

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 109 years.  Star cluster ages range from a few million years
(several times 106) to a few billion (several times 109).

 

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

Eridanus

21

NGC 2477

2

Fornax 1

22

NGC 2548

3

Fornax 2

23

Hyades

4

Fornax 3

24

NGC 3114

5

Fornax 4

25

NGC 3532

6

Hyades

26

NGC 5316

7

IC2602

27

NGC 5897

8

M3

28

NGC 6067

9

M37

29

NGC 6087

10

M44

30

NGC 6139

11

M5

31

NGC 6535

12

M53

32

NGC 6626

13

M67

33

NGC 752

14

M80

34

NGC457

15

Mel 111

35

NGC6405

16

Mel 20

36

NGC6494

17

NGC 129

37

NGC884

18

NGC 1817

38

Pal 3

19

NGC 2298

39

Pal 4

20

NGC 2420

40

Pleiades