The Hertzsprung-Russell Diagram

Introduction

As you may have learned in class, the Hertzsprung-Russell Diagram, or the "HR diagram'', is one of the most important tools used by astronomers: it helps us determine both the ages of star clusters and their distances. In your Astronomy 110 textbooks the type of HR diagram that you will normally encounter plots the Luminosity of a star (in solar luminosity units, L _Sun) versus its temperature (or spectral type). An example is shown here:

The positions of the various main types of stars are labeled in this HR diagram. The Sun has a temperature of 5,800 K, and a luminosity of 1 L _Sun. The Sun is a main sequence "G'' star. All stars cooler than the Sun are plotted to the right of the Sun in this diagram. Cool main sequence stars (with spectral types of K and M) are plotted to the lower right of the Sun. Hotter main sequence stars (O, B, A, and F stars) are plotted to the upper left of the Sun's position. As the Sun runs out of hydrogen fuel in its center, it will become a red giant star-a star that is cooler than the Sun, but 100× more luminous. Red giants are plotted to the upper right of the Sun's position. As the Sun runs out of all of its fuel, it sheds its atmosphere and ends its days as a white dwarf. White dwarfs are hotter, and much less luminous than the Sun, so they are plotted to the lower left of the Sun's position in the HR diagram. The HR diagrams for clusters can be very different depending on their ages. In the following examples, we show the HR diagram of a hypothetical cluster of stars at a variety of different ages. When the star cluster is very young, only the hottest stars have made it to the main sequence. In the HR diagram below, the G, K, and M stars (stars that have temperatures below 6,000 K) are still not on the main sequence, while those stars hotter than 7,000 K (O, B, A, and F stars) are already fusing hydrogen into helium at their cores:

In the next HR diagram, we see a much older cluster of stars (100 million years = 100 Myr). In this older cluster, some of the hottest and most massive stars (the O and B stars) have evolved into red supergiants. The position of the "main sequence turn off'' allows us to estimate the age of a cluster.

In the final HR diagram we have a much older cluster (10 billion years old = 10 Gyr), now stars with one solar mass are becoming red giants, and we say the main sequence turn-off is at spectral type G (T = 5,500 K).

Some white dwarfs (produced by evolved A and F stars) now exist in the cluster. Thus, the HR diagram for a cluster of stars is useful for determining its age.

Magnitudes and Color Index

While the HR diagrams presented in your class lectures or textbook allow us to provide a very nice description of the evolution of stars and star clusters, astronomers do not actually directly measure either the temperatures or luminosities of stars. Remember that luminosity is a measure the total amount of energy that a star emits. For the Sun it is 10²⁶ Watts. But how much energy appears to be coming from an object depends on how far away that object is. Thus, to determine a star's luminosity requires you to know its distance. For example, the two brightest stars in the constellation Orion (see the "Constellation Highight'' for February in the back of this lab book), the red supergiant Betelgeuse and the blue supergiant Rigel, appear to have about the same brightness. But Rigel is six more times luminous than Betelgeuse-Rigel just happens to be further away, so it appears to have the same brightness even though it is pumping out much more energy than Betelgeuse. The "Dog star'' Sirius, located to the southeast of Orion, is the brightest star in the sky and appears to be about 5 times brighter than either Betelgeuse or Rigel. But in fact, Sirius is a nearby star, and actually only emits 22× the luminosity of the Sun, or about 1/2000^th the luminosity of Rigel! Therefore, without a distance, it is impossible to determine a star's luminosity-and remember that it is very difficult to measure the distance to a star. We can, however, measure the relative luminosity of two (or more) stars if they are at the same distance: for example if they are both in a cluster of stars. If two stars are at the same distance, then the difference in their apparent brightness is a measurement of the true differences in their luminosities. To measure the apparent brightness of a star, astronomers use the ancient unit of "magnitude''. This system was first developed by the Greek astronomer Hipparcos (ca. 190 to 120 BC). Hipparcos called the brightest stars "stars of the first magnitude''. The next brightest were called "stars of the second magnitude''. His system progressed all the way down to "stars of the sixth magnitude'', the faintest stars you can see with the naked eye from a dark location. Astronomers adopted this system and made it more rigorous by defining a five magnitude difference to be exactly equal to a factor of 100 in brightness. That is, a first magnitude star is 100X brighter than a sixth magnitude star. If you are good with mathematics, you will find that a difference of one magnitude turns out to be a factor of 2.5 (2.5 × 2.5 × 2.5 × 2.5 × 2.5 = 100, we say that the fifth root of 100 = 100^1/5 = 2.5). Besides this peculiar step size, it is also important to note that the magnitude system is upside down: usually when we talk about something being bigger, faster, or heavier, the quantity being measured increases with size (a car going 100 mph is going faster than one going 50 mph, etc.). In the magnitude system, the brighter the object, the smaller its magnitude! For example, Rigel has an apparent magnitude of 0.2, while the star Sirius (which appears to be 4.5 times brighter than Rigel) has a magnitude of -1.43. Even though they are a bit screwy, and cause much confusion among Astronomy 110 students, astronomers use magnitudes because of their long history and tradition. So, when astronomers measure the brightness of a star, they measure its apparent magnitude. How bright that star appears to be on the magnitude scale. Usually, astronomers will measure the brightness of a star in a variety of different color filters to allow them to determine its temperature. This technique, called "multi-wavelength photometry'', is simply the measurement of how much light is detected on Earth at a specific set of wavelengths from a star of interest. Most astronomers use a system of five filters, one each for the ultraviolet region (the "U filter''), the blue region (the "B filter''), the visual ("V'', or green) region, the yellow-red region ("R''), and the near-infrared region ("I''). Generally, when doing real research, astronomers measure the apparent magnitude of a star in more than one filter. [Note: because the name of the filter can some times get confused with spectral types, filter names will be italicized to eliminate any possible confusion.] To determine the temperature of a star, measurements of the apparent brightness in at least two filters is necessary. The difference between these two measurements is called the "color index''. For example, the apparent magnitude in the B filter minus the apparent magnitude in the V filter, (B - V), is one example of a color index (it is also the main color index used by astronomers to measure the temperature of stars, but any two of the standard filters can be used to construct a color index). Let us take Polaris (the "North Star'') as an example. Its apparent B magnitude is 2.59, and its apparent V magnitude is 2.00, so the color index for Polaris is (B - V) = 2.59 - 2.00 = 0.59. In Table 12.1, we list the (B - V) color index for main sequence stars. We see that Polaris has the colors of a G star. In Table 12.1, we see that O and B stars have negative (B - V) color indices. We say that O and B stars are "Blue'', because they emit more light in the B filter than in the V filter. We say that K and M stars are very red, as they emit much more V light than B light (and even more light in the R and I filters!). A-stars emit the same amount of light at B and V, while F and G stars emit slightly more light at V than at B . With this type of information, we can now figure out the spectral types, and hence temperatures of stars by using photometry.

Table 13.1: The (B - V) Color Index for Main Sequence Stars

Spectra Type	(B - V)	Spectral Type	(B - V)
O and B Stars	-0.40 to -0.06	G Stars	0.59 to 0.76
A Stars	0.00 to 0.20	K Stars	0.82 to 1.32
F Stars	0.31 to 0.54	M Stars	1.41 to 2.00

The Color-Magnitude HR Diagram

To construct HR diagrams of star clusters, astronomers measure the apparent brightness of stars in two different color filters, and then plot the data into a "Color-Magnitude'' diagram, plotting the apparent V magnitude versus the color index (B - V) as shown below. This following figure shows a color-magnitude diagram for a globular cluster. You might remember from class (or will soon be told!) that globular clusters are old, and that the low mass stars are evolving off the main sequence and becoming red giants. The main sequence turnoff for this globular cluster is at a color index of about (B - V) = 0.4, the color of F stars. An F star has a mass of about 1.5 M _Sun, thus stars with masses near 1.5 M _Sun are evolving off the main sequence to become red giants, so this globular cluster is about 7 billion years old.

The Color-Magnitude Diagram for the Pleiades

In today's lab, you and your lab partners will construct a color magnitude diagram for the Pleiades star cluster. The Pleiades, sometimes known as the "Seven Sisters'' (see the constellation highlight for January at the back of this lab manual), is a star cluster located in the wintertime constellation of Taurus, and can be seen with the naked eye. A wide-angle photograph of the Pleiades is shown below. Many people confuse the Pleiades with the Little Dipper because the brightest stars form a small dipper-like shape.

As you will find out, the Pleiades is a relatively young group of stars. We will be using photographs of the Pleiades taken using two different color filters to construct a Color-Magnitude diagram. If you look closely at the photograph of the Pleiades, you will notice that the brighter stars are larger in size than the fainter stars. Note: you are not seeing the actual disks of the stars in these photographs. Brighter stars appear bigger on photographs because more light from them is detected by the photograph. As the light from the stars accumulates, it spreads out. Think of a pile of sand. As you add sand to a pile, it develops a conical, pyramid shape. The addition of more sand to the pile raises the height of the sand pile, but the base of the sand pile has to spread more to support this height. The same thing happens on a photograph. The more light there is, the larger the spread in the image of the star. In reality, all of the stars in the sky are much to far away to be seen as little disks (like those we see for the planets in our solar system) when viewed/imaged through any existing telescope. We would need to have a space-based telescope with a mirror 1.5 miles across to actually be able to see the stars in the Pleiades as little, resolved disks! [However, there are some special techniques astronomers have developed to actually measure the diameters of stars. Ask your TA about them if you are curious.]

Thus, we can use the sizes of the stars on a photograph to figure out how bright they are, we simply have to measure their diameters! A special tool, called a "dynameter'', is used to measure sizes of circles. You will be given a clear plastic dynameter in class. A replica of this dynameter is shown here:

As demonstrated, a dynameter allows you to measure the diameter of a star image by simply sliding the dynameter along until the edges of the star just touch the lines. In the example above, the star image is 2.8 mm in diameter. On the following two pages are digitized scans of two photographs of the Pleiades taken through B and V filters. These photographs were digitized to allow us to put in an X-Y scale so that you can keep track of which star is which in the two different photographs. You should be able to compare the digitized photographs with the actual photo shown above and see that most of the brighter stars are on all three images.

Figure 13.1: Do not use this figure--it has the wrong scale.

Figure 13.2: Do not use this figure--it has the wrong scale.

Procedure

The first task for this lab is to collect your data. What you need to do for this lab is to measure the diameters of ten of the 63 stars on both digitized photographs. At the end of this lab there is a data table that has the final data for 53 of the 63 stars. It is missing the information for ten of the stars (#'s 7, 8, 13, 18, 30, 39, 53, 55, 61, and 63). You must collect the data for these ten stars.

Task #1: First, identify the stars with the missing data on both of the digitized photographs (use their X,Y positions to do this). Then measure their diameters of these ten stars on both photographs using the dynameter. Write the V and B diameters into the appropriate spaces within the data table. [Note: You will probably not be able to measure the diameters to the same precision as shown for the other stars in the data table. Those diameters were measured using a computer. Do the best you can--make several measurements of each star and average the results.] (15 points)   

Converting Diameters to Magnitudes

Obviously, the diameter you measure of a star on a photograph has no obvious link to its actual magnitude. For example, we could blow the photograph up, or shrink it down. The diameters of the stars would change, but the relative change in size between stars of different brightnesses would stay the same. To turn diameters into magnitudes requires us to "calibrate'' the two photographs. For example, the brightest star in the Pleiades, "Alcyone'' (star #35), has a V magnitude of 2.92, and has a V diameter of 4.4 mm. We have used this star to calibrate our data. Once you have completed measuring the diameters of the stars, you must convert those diameters (in millimeters) into V magnitudes and (B - V) color index. To do so, requires you to use the following two equations:

V(mag) = -2.95×(V mm) + 15.9 (Eq. # 1)
  
and
  
(B - V) = -1.0×(B mm - V mm) + 0.1 (Eq. #2)

These equations might seem confusing to you because of the negative number in front of the diameters. But if you remember, the brighter the star, the smaller its magnitude. Brighter stars appear bigger, so bigger diameters mean smaller magnitudes! That is why there is a negative sign. Using the example of Alcyone, its V diameter is 4.4 mm and it has a B diameter of 4.7 mm. Putting the V diameter into equation #1 gives: V (mag) = -2.95×(4.4 mm) + 15.9 = -13.0 + 15.9 = 2.9. So, the V magnitude of Alcyone is correct: V = 2.9, and we have calibrated the photograph. Its color index can be found using Eq. #2: (B - V) = - 1.0×(4.7 - 4.4) + 0.1 = -1.0×(0.4) + 0.1 = -0.20. Alcyone is a B star!   
Task #2: Convert all of the B and V diameters into V magnitudes and (B - V) color index, entering them into the proper column in your data table. Use any of the other stars in the table to see how it is done. Make sure all students in your group have complete tables with all of the data entered. (15 points)

Constructing a Color-Magnitude Diagram

The collection of the data is now complete. In this lab you are getting exactly the same kind of experience in "reducing data'' that real astronomers do. Aren't you glad you didn't have to measure the diameters of all 63 stars? Obtaining and reducing data can be very tedious, tiring, or even boring. But it is an essential part of the scientific process. Because of the possibility of mis-measurement of the star diameters, a real astronomer doing this lab would probably measure all of the star diameters at least three times to insure that they had not made any errors. Today, we will assume you did everything exactly right, but we will provide a check shortly. Now we want to finally get to the goal of the lab: constructing a Color-Magnitude diagram. In this portion of the lab, we will be plotting the V magnitudes vs. the (B - V) color index. On the following page is a blank grid that has V magnitude on the Y axis, and the (B - V) color index on the X axis. Now we want to plot your data onto this blank Color-Magnitude diagram to closely examine what kind of stars are in the Pleiades.   
Task #3: For each star in your table, plot its position where the (B - V) color index is the X coordinate, and the V magnitude is the Y coordinate. Note that some stars will have very similar magnitudes and colors because they are the same types of star. When this happens, simply plot them as close together as possible, making sure they are slightly separated for clarity. All students must complete their own Color-Magnitude diagram. (15 points)

Figure 13.3: The Color-Magnitude Diagram for the Pleiades

Error checking: All of your stars should fit within the boundaries of the Color-Magnitude diagram! If not, go back and remeasure the problem star(s) to see if you have made an error in the B or V diameter or in the calculations.

Results

If you have done everything correctly, you should now have a Color-Magnitude diagram in which your plotted stars trace out the main sequence for the Pleiades. Use your Color-Magnitude diagram to answer the following questions:

1. Are there more B stars in the Pleiades, or more K stars? (5 points)  
2. Given that the Sun is a main sequence G star, draw an "X'' to mark the spot where the Sun would be in your Color-Magnitude diagram for the Pleiades (5 points)  
3. The faintest stars that the human eye can see on a clear, dark night is V = 6.0. If the Sun was located in the Pleiades, could you see it with the naked eye? (5 points)  
4. Are there any red giants or supergiants in the Pleiades? What does this tell you about the age of the Pleiades? (5 points)  

Summary

Please summarize the important concepts of this lab. (35 points)

• Describe how an HR diagram is constructed.
• If you have plotted your HR Diagram for the Pleiades correctly, you will notice that the faint, red stars seem to have a spread when compared to the brighter, bluer stars. Why do you think this occurs? How might you change your observing or measuring procedure to fix this problem? [Hint: is it harder or easier to measure big diameters vs. small diameters?]
• Why are HR diagrams important to astronomers?

Use complete sentences, and proofread your lab before handing it in.

Possible Quiz Questions

1. What is a magnitude? Which star is brighter, a star with V = -2.0, or one with V = 7.0?  
2. In an HR Diagram, what are the two quantities that are plotted?  
3. What are the properties of a white dwarf?  
4. What are the properties of a red giant?  
5. What is a Color Index, and what does it tell you about a star?

Extra Credit: White dwarfs are 100× less luminous than the Sun, but are hot, and have a negative color index (B - V) = -0.2. Given that a factor of 100 = 5 magnitudes, is it possible to plot the positions of white dwarfs on your Color-Magnitude diagram for the Pleiades? (3 points)

Table 13.2: Data Table

#	X	Y	V(mm)	B(mm)	V(mag)	(B - V)
01	157.00	832.00	3.10	2.89	6.76	0.31
02	157.61	832.20	2.49	2.00	8.50	0.59
03	178.33	821.70	2.37	1.70	8.91	0.77
04	183.40	830.51	2.32	1.60	9.06	0.82
05	190.53	818.94	2.24	1.52	9.29	0.82
06	190.62	834.99	2.23	1.52	9.32	0.81
07	192.98	865.44
08	197.37	754.50
09	202.78	696.35	2.23	1.46	9.32	0.87
10	203.87	810.57	2.36	1.72	8.94	0.74
11	210.57	789.29	2.32	1.62	9.06	0.80
12	212.22	693.49	2.48	1.97	8.58	0.61
13	233.44	830.40
14	234.34	759.27	2.35	1.57	8.97	0.88
15	235.50	751.74	2.40	1.85	8.82	0.65
16	246.00	807.00	3.26	3.07	6.28	0.29
17	252.95	795.24	2.75	2.35	7.78	0.50
18	254.95	688.02
19	259.60	730.54	2.39	1.74	8.85	0.75
20	260.00	795.00	2.35	1.77	8.97	0.68
21	265.00	792.00	2.24	1.48	9.29	0.86
22	265.00	831.00	2.95	2.65	7.20	0.40
23	266.66	831.82	2.20	1.36	9.41	0.94
24	269.27	731.47	2.18	1.33	9.47	0.95
25	270.00	789.00	2.31	1.62	9.09	0.79
26	274.00	790.00	2.32	1.70	9.06	0.72
27	276.28	836.35	2.50	1.98	8.53	0.62
28	277.19	811.96	2.22	1.55	9.35	0.77
29	283.00	792.00	2.35	1.75	8.97	0.70
30	285.00	774.00
31	288.00	786.00	2.20	1.42	9.41	0.88
32	289.50	852.50	2.18	1.54	9.47	0.74
33	291.00	822.00	4.24	4.46	3.39	-0.12
34	297.00	822.00	3.46	3.38	5.69	0.18
35	298.00	793.00	4.40	4.70	2.92	-0.20
36	299.00	749.00	4.09	4.23	3.83	-0.04
37	304.00	773.00	2.39	1.79	8.85	0.70
38	308.00	777.00	2.31	1.67	9.09	0.74
39	310.00	794.04
40	312.00	748.00	3.35	3.20	6.02	0.25

Table 13.3: Data Table (cont.)

#	X	Y	V(mm)	B(mm)	V(mag)	(B - V)
41	316.46	832.35	2.52	2.01	8.47	0.61
42	317.00	766.00	3.93	4.00	4.31	0.03
43	319.14	731.31	2.38	1.81	8.88	0.67
44	320.29	742.55	2.17	1.46	9.50	0.81
45	322.43	819.50	2.17	1.52	9.50	0.75
46	325.00	756.00	3.62	3.57	5.22	0.15
47	327.00	787.00	2.20	1.47	9.41	0.83
48	327.80	841.25	2.34	1.68	8.99	0.76
49	329.00	771.00	2.87	2.52	7.43	0.45
50	332.00	794.00	2.62	2.14	8.17	0.58
51	335.13	732.56	2.28	1.54	9.17	0.84
52	347.41	654.23	2.15	1.43	9.55	0.82
53	352.00	756.00
54	359.05	685.95	2.35	1.70	8.97	0.75
55	361.00	807.00
56	368.31	692.12	2.35	1.69	8.96	0.76
57	375.90	729.41	2.20	1.50	9.41	0.80
58	375.90	729.41	2.36	1.73	8.94	0.73
59	386.00	813.00	2.37	1.72	8.91	0.75
60	387.50	683.69	2.20	1.54	9.41	0.76
61	397.48	769.11
62	410.49	839.98	2.34	1.62	8.99	0.82
63	420.52	720.04