Exercise: Draw HR Diagram of Star Cluster
Ver. 1.10

Exercise: Draw HR Diagram of Star Cluster
Guide for Teachers
INDEX

1. Purpose
2. Contents
3. Outline
4. Background
5. Preparation and Class Plan
6. Comments
7. Example for Answers to the Worksheets
Appendix

1. Purpose
Draw HR diagrams of stars cluster using PC, from FITS-formatted raw frames taken by Hubble
Space Telescope, Subaru Telescope (in Hawaii), and Kiso Schmidt Telescope (in Japan). Then dofollowing activities:
 Estimate an age based on HR diagram of globular star cluster.
 Find out the main sequence in HR diagram of open star cluster.
 Discuss about stellar evolution, by drawing HR diagrams of globular and open star clusters.

2. Contents
 Draw HR diagram (in fact, it is color-magnitude diagram; CM diagram) of star cluster fromarchival data of Hubble Space Telescope (HST) and Kiso Observatory.
 Discuss about stellar evolution on the CM diagram.
 Compare the drawn CM diagram with theoretical curves, and consider age of the star cluster.
 Draw CM diagrams of several star clusters, and compare the ages. [optional]
Things to be prepared:- Image processing software ‘Makali`i’- B and V band FITS images of globular and open clusters- Whole-area images of the globular and open clusters- MS Excel(98,2000,2002)- Excel macro for drawing the CM diagram(c) PAOFITS Teaching Material [Guide for Teachers]
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Exercise: Draw HR Diagram of Star Cluster
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3. Outline
1. Flow of work, explanation of data, and explanation of software
2. Photometry
3. Arrangement of the data, drawing HR (CM) diagram
4. Check and discussion

4. Background4.1. HR Diagram
HR diagram, Heltzsprung -Russell diagram, designed by E. Heltzsprung and H. N. Russell, is adiagram in which the star data are plotted, with its horizontal axis meaning spectral type, and its
vertical axis meaning absolute magnitude.
4.2. Color-Magnitude Diagram of Star Cluster
You can think that stars in the cluster are at nearly the same distance from the Sun. Therefore,
you can draw a plot that has the same meaning as the HR diagram, using apparent magnitude
instead of absolute magnitude for the vertical axis.
It is not easy that you draw an HR diagram after getting spectrum of each star in the cluster.
Thus, astronomers usually draw a color-magnitude diagram (CM diagram) using color index(for example, B-V index) instead of spectral type for the horizontal axis.
4.3. Color Index and Spectral Type4.3.1. Color index
A ratio of brightness in two different wavelengths for a star depends on its surface temperature(that is, spectral type).
Fig.1. shows the difference of temperature of black-body spectrum and difference of ‘color’.
Vertical axis is shown in log scale. Blue, yellow, and red curves indicate black-body spectrawith temperatures of 12000, 6000, and 3000 K.
Shown in Fig. 1, consider a ratio of brightness that is taken through two filters that transmit(c) PAOFITS Teaching Material [Guide for Teachers]
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Exercise: Draw HR Diagram of Star Cluster
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different wavelengths. The
black-body spectrum has
shorter peak-wavelength ofemissionforhighertemperature. Therefore, forthe black-body spectra withdifferent
temperatures,
when you measure the ratioof brightness in two
different wavelengths, it isbrighteratshorter
wavelength than at longer
wavelength for the higher
temperature.

Vertical axis is shown in
log-scale

12000 [K]6000 [K]
3000 [K]Fig1. Difference of temperature of black-body spectrum and differenceof 'color'. Vertical axis is shown in log scale. Blue, yellow, and red
curves indicate black-body spectra with temperatures of 12000,
6000, and 3000 K.

The case for the stars is thesame. Consider that you
measure brightness of star through filters that transmit different wavelengths. Surface
temperature of star ranges from around 3000 to several tens of thousands degrees in Kelvin. For
example, in the optical wavelength range (roughly from 400 to 800 nm), a star with higher
temperature is brighter in shorter wavelengths than in longer wavelengths. When brightness isexpressed in magnitude scale, the ratio of brightness corresponds to difference in magnitude.
The difference in magnitude is called ‘color index’.
In this workbook, we use B-band filter that transmits blue lights for the shorter-wavelength
filter, and V-band filter that transmits green lights for the longer-wavelength filter. Brightness in
B-band image compared to that in V-band one is larger for a higher-temperature star. The ratioof B- and V-bands images is derived by the difference in magnitude, B-V. In this workbook, we
use B-V for the color index (note: B is for blue, V is for visual).
4.3.2. Relation between Color Index and Spectral TypeSpectral type of a star is determined by appearance of absorption lines in spectrum as describedin table 1. These features are determined by stellar surface temperature. Table 2 shows therelation between the spectral type and the color index (B-V).

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Exercise: Draw HR Diagram of Star Cluster
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Table 1. Characteristics of Spectral Types.
Sp. type

Eff. temp. [K]Characteristics

O

45000- 32000

B

30000-10000

A

9700-7500

F

7300-6000

G

6000-4700

K

4700-4000

M
R, N
S

4000-3500
3000
3000

Ionized helium; highly-ionized oxygen, nitrogen, and carbon are
seen.
Hydrogen absorption lines are stronger. Neutral helium lines arestrongest.
Hydrogen absorption lines are strongest. Ionized metallic lines are
getting stronger.
Hydrogen absorption lines are getting weaker. Ionized calcium
absorption lines (H and K lines), and metallic absorption are gettingstronger.
Strong absorption lines of ionized calcium (H and K lines) andmetallic absorption lines are seen. Hydrogen absorption lines do notstand out.
Strong and broad absorption lines of ionized calcium lines (H and K
lines) are seen. Metallic absorption lines are blended each other.
Titanium oxide absorption band stands out.
Carbon and cyanogen absorption band stands out.
Zirconium oxide absorption band stands out.

Table 2. Relation between spectral type and (B-V) color index.
(for main-sequence stars)
Spectral type

Color index (B-V)

O5
B0
B5
A0
A5
F0
F5
G0
G5
K0
K5
M0
M5

-0.3
-0.3
-0.16
0.00
+0.15
+0.33
+0.45
+0.60
+0.68
+0.81
+1.15
+1.4
+1.6

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Exercise: Draw HR Diagram of Star Cluster
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4.4. CM Diagram and Stellar Evolution, Star Cluster's Evolution
All stars in a cluster are considered
to have approximately the same age.
Based on this fact, we can read theage of the cluster from the HR or
CM diagram of the cluster.

Main Sequence

turn off point

A band from the upper left to thelower right in the CM diagram isturn-off point
called main sequence, and stars inthe band are called main-sequencestars.
In the core of a main-sequence star,hydrogen atoms are turning into Fig.2. An example of the CM diagram (Arp et al. 1962, ApJ,
Vol.135, p.311).
helium atoms through nuclear
fusion. The main-sequence stars arestable stars that shine through this nuclear fusion energy.
The more massive a star is, the higher its temperature is, and the brighter the star is. The more
massive star evolves faster, so its life time is shorter. Then, stars at upper left of the main
sequence are more massive and live shorter lives.
Turn-Off Point

Just after the birth of a cluster, stars with various masses shine as main-sequence stars. However,as time goes by, massive stars at upper left of the main sequence near the ends of their lives, andthey turn into red giant stars. In the CM diagram, they move toward right.
The CM diagram of a star cluster with some age has a ‘turn’ in the main sequence. At that point,
the star is just about to leave the main-sequence stage for the red-giant stage. The point is called
turn-off point.
Red-Giant Branch
A star shines almost stably with nuclear energy, burning hydrogen into helium. However, whenabout 10 % of hydrogen in the core has become helium, the core shrinks, and outer layers
expand. That makes the star a red giant. In the CM diagram, evolution from main sequence to
red giant appears a nearly horizontal shift (after entering the red-giant stage, evolutionary path(c) PAOFITS Teaching Material [Guide for Teachers]
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Exercise: Draw HR Diagram of Star Cluster
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goes upward). As the cluster ages, more massive, brighter, and short-lived stars (stars that once
plotted at upper left part of the main sequence in the CM diagram) move to the right, the red
giant branch develops in the CM diagram which stretches to the upper right.
Horizontal Branch
As hydrogen in the core runs out, the core shrinks, and the star becomes red giant. When thecore shrinks even more, temperature goes up farther and helium begins to burn (nuclear fusionof helium). Massive stars start the process of burning helium on the way of the evolution to thered giant. On the other hand, less massive stars like the Sun (less than several times the mass of
the Sun), the helium burning starts all at once, then the burning continues steadily. At earlier ofthis evolutionary stage, the star does not change its brightness much, and moves leftward in the
CM diagram, that is, the temperature is getting higher. Then again, it moves to the upper right.
We see horizontal branch stars, which stays leftward temporarily when they burn their helium.
Isochrone
In the CM diagram of a cluster, you see a curve that depends on the age, shaped by stars having
their own lives depending on the mass. Evolutionary path of a star in the CM diagram can bederived theoretically with a parameter of the stellar mass (in fact, abundance of elements
heavier than hydrogen in the outer layers of a star changes the path). For stars of various masses
with the same age, connecting positions on the evolutionary paths makes the theoretical CM
diagram of a cluster for that age. This curve is called isochrone.
Estimate of Age of Cluster
As mentioned, as time passes after the birth of a cluster, the main-sequence stars change into
red giant in order of stellar mass. In the CM diagram of a cluster with a given age, position for astar that ends its main-sequence life time corresponds to the turn off point. Therefore, if you
draw a CM diagram and find out the turn-off point, you can consider main-sequence life timeof the star at that point as the age of the cluster.

4.5. Photometric Values and Star’s Magnitude
Brightness of stars is often expressed using magnitude system. On the other hand, when you
make photometry for a star in a FITS data, you will get total count in the region where starlights spread (see later). Therefore, it is necessary to convert the count value measured in the
FITS data to the magnitude scale.
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Exercise: Draw HR Diagram of Star Cluster
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Relation between star’s brightness and magnitude
Brightness of an object is expressed in magnitude. There are relations between the brightness
and magnitude as: brighter object has smaller value of magnitude.
 ratio of brightness is expressed as difference in magnitude.
 a brightness ratio of 100 corresponds to a magnitude difference of 5.
For example, a ratio for brightness B1 and B2 is expressed as a difference of magnitudes
between m1 and m2 as a following equation:
m1 - m2 = -2.5 log (B1/B2).
The count value in the FITS data is proportional to the brightness (though we should take into
account many things such as condition of observations, inhomogeneity in sensitivity of camera,
here we think we have data with all necessary corrections). Therefore, above equation can be
rewritten using the count values (put C1 and C2, respectively) as follows:
m1 - m2 = -2.5 log (C1/C2).
In this work, you will derive the color index using counts in B-band and V-band images (put CB
and CV, respectively). Thus:
B-V = mB - mV = -2.5 log (CB/CV).
In analyzing the real data, you should also adjust the filter transmission curve to the standard
filter system. Usually, astronomers observe the standard stars the magnitudes of which aremeasured in the standard system, then calibrate the brightness of stars. In this workbook, we
prepared images that are converted into the standard system. So, you do not need suchcalibration.
Magnitude (apparent magnitude) and absolute magnitude
Brightness we see of an object varies depending on distance. The nearer the object is, thebrighter we see. Therefore, brightness observed from a site (usually, consider Earth) is called
apparent brightness (apparent magnitude). On the other hand, astronomers define absolutemagnitude as an index of intrinsic brightness, independent of distance. For stars, it is defined as(c) PAOFITS Teaching Material [Guide for Teachers]
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Exercise: Draw HR Diagram of Star Cluster
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an apparent magnitude at a distance of 10 pc.
Put apparent magnitude of a star at a distance of d pc from the Sun, m, and put absolutemagnitude, M, and if we can ignore interstellar absorptions, then we have a relation:m - M = 5 log d - 5.

Magnitude used in HR (CM) diagram of star cluster
Usually, the vertical axis of the HR diagram indicates absolute magnitude. On the other hand,for objects the distance to them can be considered nearly the same, such as stars in a star cluster
or a galaxy, we have a diagram with the same meaning using apparent magnitude instead ofabsolute magnitude for the vertical axis.
4.6. Processes of FITS Data
FITS data’s pixels have values of count that correspond to brightness. Brightness of a star ismeasured by summing up values of count with a star’s region.
FITS data analyzing software ‘Makali`i’ is software that can see count in the pixel at any
position (x, y), integrate counts within any region, and operate counts between frames easily.
In the photometry mode of Makali`i, when you click a star you want to photometry with the
cursor on its center, center of star’s image is calculated, and Makali`i returns total countintegrated in a circle area with a given radius. Besides, an average value at an annular region
around the area of the star is subtracted as the background level. The radius of the aperture as
the star area is given as FWHM * 1.5 + 0.5 pixels for auto mode (FWHM: full width at half
maximal brightness). As the background area, inner radius of the annulus is given
FWHM * 3 - 0.5 pixels and width of the annulus is 2 pixels automatically (under the automode). Image data taken by CCD camera contains noise and inhomogeneity in sensitivity due
to characteristics of the camera. This workbook provides data with proper corrections for that.

5. Preparation and Class Plan
Preparation
1. Download and install Makali`i.
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Exercise: Draw HR Diagram of Star Cluster
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2.
3.
4.
5.

Download FITS data.
Download whole-area image of the cluster.
Teacher should work through once before the class day, from a point of student’s view.
If students should have time to get acquainted with handling Makali`i, have some time.

Class plan (example)
1st period: Introduction; explain the work, see image data carefully (or start the photometry)
2nd period: Do photometry earnestly, output to CSV file
3rd period: Convert count values measured from data to magnitudes and colors, draw CM
diagram (using Excel macro). Discuss based on the CM diagram.

6. Supplement
Corresponding to the National Curriculum Guidelines in Japan (Heisei 15 version)
Senior High School
Earth Science I(2) Atmosphere, Ocean, and Universe
Earth Science II(4) Extensive Studies(c) PAOFITS Teaching Material [Guide for Teachers]
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Exercise: Draw HR Diagram of Star Cluster
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7. Example for Answers to the Worksheets
7.1. Globular Cluster Part
Q.1. Referring the below figure, write down names to parts in your CM diagram.
It is possible that picking out the turn-off point varies. It is the position where the main
sequence leaves. Note that it is not the position where the red-giant branch begins. There may
not be other problems.
Q.2. Why do you think are the main-sequence stars rare on the dotted part above the turn-off
point?
Because of change in energy generation mechanism, the main-sequence stars are considered
to move toward the red-giant branch. We see this because we limit the data only for a cluster.
Q.3. Red-giant branch tells us the evolutionary path of stars in the globular cluster. Write anarrow for the path that you guess. Why can you guess so?
It is possible that many students answer that the main-sequence stars evolve as the red-giantstars along the line where plots are concentrated in the CM diagram. However, it is notcorrect. The evolutionary path of each star is actually near the theoretical line (see teacher’s
guide for details).
Q.4. What are horizontal-branch stars?
Hint: They are relatively stable on their ways through their evolution.
Note that we can not say that they are a group of stars that are changing from main sequence
to red giant. They are temporarily shining steadily. We can say that they are at a stage ofhelium fusion (see teacher’s guide for details).
Q.5. How bright is the horizontal branch (in magnitude)? If your classmates investigate otherglobular cluster, compare your estimation with his or hers. What can you conclude from that?
Let students read absolute magnitude (the vertical-line value) of the horizontal branch in the
CM diagram.
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Exercise: Draw HR Diagram of Star Cluster
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Q.6. How bright is the turn-off point from the main sequence to the red giant? Compare resultswith your classmates. Is there any difference among yours?
Let students read absolute magnitude (the vertical-line value) of the turn-off point in the CM
diagram. Depending on their readings, the value of absolute magnitude can vary.
Q.7. The turn-off point from the main sequence to the red giant differs from globular clusters to
globular clusters. What makes this difference?
The absolute magnitude (the vertical-line value) for the point differs. In a younger cluster,
more stars stays in the main sequence, and the absolute magnitude of the turn-off point issmaller (that is, brighter, and upper in the diagram).
Q.8. Fainter stars below the turn-off point, in the lower part of the diagram, spread like skirt of amountain. What makes this distribution?
The stars spread into horizontal direction. Fainter stars have poor quality in pictures. Since
the color index is calculated from two photometry values, deviation from the true value grows
for the color. Besides, since we can not distinguish cluster stars from others, these other stars(we say them background stars) may contaminate the cluster area. The fainter the star is, thebigger the number of stars is, so faint stars spread in the lower area.
Q.9. The position of the turn-off point varies as the globular cluster ages. Consider that (i), (ii),and (iii) in the diagram below are for three clusters. Which one is for the oldest cluster? Why
do you think so?
The oldest one is shown by (iii). Because even less-massive stars that are slow in hydrogenfusion and thus slow in leaving the main sequence turned into red giants. Let students
understand that duration time of the nuclear fusion (we can say it nearly the life time of a star)is shorter for more massive stars, and that stars plotted in the left of the CM diagram havehigher temperatures and larger masses.
Q.10. Estimate the age of the globular cluster by comparing with the theoretical curve ofglobular cluster’s evolution.
An age of about 10 billion years is expected. Recent detailed studies show that the age of thisglobular cluster is 8.3 billion years old. We do not expect such fine value. Being not 1 billion(c) PAOFITS Teaching Material [Guide for Teachers]
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years is a good answer.
Q.11. Which is the closest to your estimation for the age? What can you say when you get the
information of age of globular cluster?
(1) Birth of human beings(2) Birth of Earth(3) Formation of the Galaxy(4) Origin of the universe
The answer is (3). (1) is as early as about several million years ago, (2) is about 4.6 billion
years ago, and (3) is of order of 10 billion years old because it is considered to have happened
1 billion years after the birth of the universe. (4) is considered to be about 14 billion years old.

7.2. Open Cluster Part
Q.1. Find the main sequence from the printed-out CM diagram. Enclose the main sequence byline with a width as narrow as possible.
Though the width expands in the region fainter than 3 in absolute magnitude, upper part canbe encircled by a shape band. At least at the upper part, the main sequence can be enclosed
like an example of school textbook’s HR diagram. Note that the upper top is just moving to the
red-giant branch, so the upper sequence turns to the right a little.
Q.2. What are the characteristics of the main-sequence stars? Explain considering meanings ofabsolute magnitude in the vertical line and color index in the horizontal line.
Confirm the contents in the school textbook.
Q.3. For the Sun, V=4.85 and B-V=0.65.
(1) Plot this on the CM diagram.
(2) What characteristics do stars that reside upper than the Sun's position have? Think it byconsidering the meanings of brightness and color index.
Let students understand that they are brighter than the Sun, and that they are bluer than theSun. They are more massive than the Sun as well.
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Exercise: Draw HR Diagram of Star Cluster
Ver. 1.10(3) In the diagram, stars that reside lower than the Sun do not concentrate but spread, andthe number of stars are not large. What do you think makes this?
This question is the same as Q.8. of globular cluster part.
Q.4. Consider that the shape of the CM diagram is general for all clusters. See the two main
sequences shown by a solid and a dotted curves in the diagram below (separated by a gap in
the vertical direction). What characteristics of the clusters differ to make this difference?
Apparent brightness is different. The shape of the CM diagram does not change. So, to change
the apparent brightness, the distance to the cluster must differ. The B-V indices of cluster stars
are ratios of brightness in B and V. So, B-V does not change because B and V magnitudes
change similarly as the cluster go farther.
Q.5. Compare the main sequence of the HR (CM) diagram shown in your school textbook withthat you found out in Q.1. for NGC 1912.
(1) What do you notice in the upper part of the main sequence? How is the difference like?
Let students notice that upper top part of the main sequence is turning to the right. See Q.1.
(2) Why do you think makes this?
See Q.1.

7.3. Open Cluster and Globular Cluster Part
Q.1. Decide which type is your CM diagram, referring diagrams below. Write names to parts inyour diagram.
The type B is for a globular cluster, and the type A is for an open cluster.
Q.2. Compare results by classmates who investigated open cluster data or globular cluster data.
Exchange comments.
(1) Mention as many points as possible which are similar to and different from each other.
Overlapping is frequent at lower part, fainter than 4 in absolute magnitude. The difference(c) PAOFITS Teaching Material [Guide for Teachers]
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between their plots is clear as going upper part.
(2) Compare absolute magnitude of the turn-off point.
It is expected that for an open cluster it is about 0, and for a globular cluster it is fainter than
4.
(3) Consider that the difference in part upper than the turn-off point is the shift in thehorizontal direction. What characteristic of stars makes this shift?
The horizontal axis means color index. The right side indicates the redder color.
(4) Since the energy generation system changes as the star ages, a star evolves from a
protostar to a main-sequence star, a red-giant star, and a white dwarf. Which do you
think is older, the open cluster or the globular cluster? Why do you conclude that?
See answer to Q.9 of globular cluster part.
Q.3. Compare derived CM diagram and theoretical curve. Estimate age of cluster you
investigated.
(1) Estimation of the age.
The estimation is about 10 billion years old for globular cluster. See Q.10 of globular cluster
part. It is about 1 billion years old for open cluster. In both cases, if the order is correct, it is agood answer.
(2) Why do you guess so?
It is easy if the diagram happens to be fitted with the theoretical curve. If not so, byinterpolating the curves, you can read the value for the age.

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Appendix
Flow of a class (plan 1)[1h] Introduction
Confirmation about the HR diagram
Confirmation about the stellar evolution(mass and speed of evolution)
Evolution of star cluster as a whole is shown in the HR diagram.
About the vertical axis, absolute magnitude, photometry
About the horizontal axis, color index, photometry of two images[2h] Photometry using Makali`i
Explanation about photometry
Photometry works
Saving the results, output to CSV file, saving the FITS data[3h] Drawing Graph
Showing theoretical curve data
Instructing about fitting of coordinates of the diagram
Output to CSV file
Input to Excel sheet, running the macro, printing out the graph[4h] Discussion
Estimation of the age by fitting the theoretical curve
Presentation by some students
Discussion about the difference among students
Flow of a class (plan 2)[Preparation]
Image processing software Makali`i
FITS image of a globular cluster (NGC 362), B#2 & V#2
MS Excel(98,2000,2002)[Purpose]
Draw HR diagrams of stars cluster using PC, from excellent FITS-formatted raw frames taken(c) PAOFITS Teaching Material [Guide for Teachers]
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by Hubble Space Telescope, Subaru Telescope and others. Based on the diagram, derive age ofglobular cluster and discuss about evolution of the Galaxy.
[Procedure]<Photometry>
1. Start the application software Makali`i.
2. Open B and V FITS images.
3. Press Photometry button for the B image. Photometry window pops up. The mode
should be changed from semi-auto to auto. Drag the photometry window below to hide.
Similarly, do for the V image.
4. Then, pick out the same star in both B and V images and click by turns. Click as many as
50 to a few hundred stars.
[note for picking out stars](1) Avoid the case that two stars lie in one-click circle. If so, you will get the mixed
value.
(2) Avoid stars that lie near the edge of frame.
(3) Click by turns the same star in B and V frames.
5. When you want to interrupt the work, be sure that you save in files with new names for B
and V images. By saving so, you can resume your work from the previous point.
6. After finishing photometry for all stars, save the photometry data sheet to CSV-formatted
files for B and V images, respectively.
7. Quit Makali`i. Makali`i asks the saving name: select the overwrite saving mode.
<Numerical processes, drawing graphs>(c) PAOFITS Teaching Material [Guide for Teachers]
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