The H-R Diagram

Introduction

In the early part of this century, two astronomers, one Danish and one American, invented a diagram showing the basic characteristics of stars. The color-magnitude diagram, often called the Hertzsprung-Russell (HR) diagram in their honor, has proved to be the Rosetta Stone of stellar astronomy. The purpose of this homework is to give you some familiarity with the diagram. In addition, you will be asked to investigate the types of biases in mesurement used to construct this diagram. Biases are especially important in understanding astronomical data. Unlike laboratory sciences, astronomical experiments must be conducted under the conditions the Universe gives us. Since the astronomer has no direct control over the experiment, it is imperative that he or she understand the prejudices introduced into the data by the human perspective.

However, biases of measurement are found in many other fields of science. An example of a biased study would be to find the weight vs. age relation for all Americans by weighing only members of health clubs. Most active health club members tend to be lighter than the average, and so the average derived would be lower than the true average weight of Americans.

This exercise asks you to make comparisons between two different samples of stars. The bright star table was selected on the basis of apparent brightness NOT the luminosity of the stars. The near star table is all stars within 5 parsecs (about 15-16 lightyears) from the Sun.

Procedure

1. The first thing an astronomer does when faced with a pile of data is gaze at it, contemplates it, and wait for inspiration. Look over the two lists of stars and compare them to the star we know best, the Sun:

1. Characterize the average properties of stars in the "Near Star List" as compared to the Sun. (mention Luminosity, color, and temperature.) "Nearby stars tend to be...."
2. Characterize the average properties of stars in the "Bright Star List" as compared to the Sun. (mention Luminosity, color, and temperature.) "Apparently bright stars tend to be...."

Plot log(L/LSun) versus Temperature for both the lists of nearest stars and brightest stars on the attached graph. Use contrasting colors or symbols so that the two samples of stars are clearly distinguished. The plot you have made will be an HR diagram. Label the region of the diagram where the main-sequence stars, Red Giant stars and White Dwarf stars are found.

2. In a brief sentence or two, comment on the differences in location on your plot of these two groups of stars.

3. Capella has approximately the same temperature as the Sun, yet it is 140 times as bright as the Sun. We know that the luminosity of a star is

L/LSun=(R/RSun)2*(T/TSun)4.

Knowing this, can you propose an explanation for the higher luminosity of Capella?

4. Using this formula,

(R/RSun)2=(L/LSun)(T/TSun)-4

and the lists, find the radius of the hot main-sequence star Vega, the very hot main-sequence star Hadar and the cool main-sequence star Ross 614-A as ratios of the radius of the Sun.

```
RVega/RSun  =

RRoss 614-A/RSun  =

```
Note: In order to find L/LSun from the lists, you need to know about logarithms. Here is a quick reminder:

log(L/LSun)=x

means that

L/LSun=10x

Let's use a real number to work this out. Suppose that x=2, so that

log(L/LSun)=2

Then

L/LSun=102

and therefore

L/LSun=100

So the star is 100 times as luminous as the Sun.

5. From the above calculation, do you see any trend between the size of a main-sequence star and its temperature? If so, what is it?

6. For each list of stars, the nearest stars and the brightest stars, count the number of stars that fall into each of these temperature ranges: 3000 or less; 3001 to 5000; 5001 to 7000, 7001 to 10,000; greater than 10,000. Make a bar graph for each set using the second diagram. Again, use contrasting colors for the two samples so they can be easily distinguished from one another.

Note: some star systems have more than one star in them; count each star in the system individually. For example the 40 Erid system is made of three stars, each of different temperature.

7. Comment briefly on the differences between two samples of stars on the histogram you have drawn.

8. We are not able to catalog all the stars in our Galaxy. However, if we assume that the Sun is situated in a typical piece of Galaxy, we should be able to assemble a sample of stars which accurately reflects the population of the whole Galaxy (e.g. when pollsters want to find out the President's approval rating, they don't ask ALL Americans, but rather they assemble a sample of ~1000 typical Americans and assume that this sample accurately reflects the whole population.)

Which list (the bright star list or the near star list)would best be a representative of the total population of the Galaxy? Explain why!

9. Besides giving us insight into the soul and disposition of stars, the HR diagram can be used to more pragmatic ends. You can use the HR Diagram you constructed to find the distances to stars via the method of spectroscopic parallax. Listed below are the apparent magnitudes and spectral types of six main sequence stars.

## Spectroscopic parallax distance determination

Star Apparent
Magnitude (m)
Spectral
Class
Absolute
Magnitude (M)
m - M Distance
Sirius -1.4 A1
Spica 1.0 B1
Barnard's Star 9.5 M4 V
61 Cygni A 5.2 K5 V
CN Leonis 3.5 M6 V
Tau Cet 3.5 G8

Use the spectral types and the HR diagram to estimate their absolute magnitudes. The difference between the apparent and absolute magnitudes is called the distance modulus. Calculate the distance modulus (m-M) for these six stars. The distance to a star (in parsecs) is given by:

D=10(m-M+5)/5

Calculate the distance to each of these stars. What assumptions are you making as you derive your distance estimate? How accurate do you think your distance estimate is?

Table 1: Bright Stars
Star M(V) log(L/Lsun) Temp Type Star M(V) log(L/Lsun) Temp Type
Sun 4.8 0.00 5840 G2 Sirius A 1.4 1.34 9620 A1
Canopus -3.1 3.15 7400 F0 Arcturus -0.4 2.04 4590 K2
Alpha
Centauri A
4.3 0.18 5840 G2 Vega 0.5 1.72 9900 A0
Capella -0.6 2.15 5150 G8 Rigel -7.2 4.76 12140 B8
Procyon A 2.6 0.88 6580 F5 Betelgeuse -5.7 4.16 3200 M2
Achemar -2.4 2.84 20500 B3 Hadar -5.3 4.00 25500 B1
Altair 2.2 1.00 8060 A7 Aldebaran -0.8 2.20 4130 K5
Spica -3.4 3.24 25500 B1 Antares -5.2 3.96 3340 M1
Fomalhaut 2.0 1.11 9060 A3 Pollux 1.0 1.52 4900 K0
Deneb -7.2 4.76 9340 A2 Beta Crucis -4.7 3.76 28000 B0
Regulus -0.8 2.20 13260 B7 Acrux -4.0 3.48 28000 B0
Adhara -5.2 3.96 23000 B2 Shaula -3.4 3.24 25500 B1
Bellatrix -4.3 3.60 23000 B2 Castor 1.2 1.42 9620 A1
Gacrux -0.5 2.10 3750 M3 Beta Centauri -5.1 3.94 25500 B1
Alpha Centauri B 5.8 -0.42 4730 K1 Al Na'ir -1.1 2.34 15550 B5
Miaplacidus -0.6 2.14 9300 A0 Elnath -1.6 2.54 12400 B7
Alnilam -6.2 4.38 26950 B0 Mirfak -4.6 3.74 7700 F5
Alnitak -5.9 4.26 33600 O9 Dubhe 0.2 1.82 4900 K0
Alioth 0.4 1.74 9900 A0 Peacock -2.3 2.82 20500 B3
Kaus Australis -0.3 2.02 11000 B9 Theta Scorpii -5.6 4.14 7400 F0
Atria -0.1 1.94 4590 K2 Alkaid -1.7 2.58 20500 B3
Alpha Crucis B -3.3 3.22 20500 B3 Avior -2.1 2.74 4900 K0
Delta Canis Majoris -8.0 5.10 6100 F8 Alhena 0.0 1.90 9900 A0
Menkalinan 0.6 1.66 9340 A2 Polaris -4.6 3.74 6100 F8
Mirzam -4.8 3.82 25500 B1 Delta Vulpeculae 0.6 1.66 9900 A0

Table 2: Nearby Stars
Star M(V) log(L/Lsun) Temp Type Star M(V) log(L/Lsun) Temp Type
Sun 4.8 0.00 5840 G2 *Proxima
Centauri
15.5 -4.29 2670 M5.5
*Alpha
Centauri A
4.3 0.18 5840 G2 *Alpha
Centauri B
5.8 -0.42 4900 K1
Barnard's Star 13.2 -3.39 2800 M4 Wolf 359 (CN Leo) 16.7 -4.76 2670 M6
HD 93735 10.5 -2.30 3200 M2 *L726-8 ( A) 15.5 -4.28 2670 M6
*UV Ceti (B) 16.0 -4.48 2670 M6 *Sirius A 1.4 1.34 9620 A1
*Sirius B 11.2 -2.58 14800 DA Ross 154 13.1 -3.36 2800 M4
Ross 248 14.8 -4.01 2670 M5 Epsilon Eridani 6.1 -0.56 4590 K2
Ross 128 13.5 -3.49 2800 M4 L 789-6 14.5 -3.90 2670 M6
*GX Andromedae 10.4 -2.26 3340 M1 *GQ Andromedae 13.4 -3.45 2670 M4
Epsilon Indi 7.0 -0.90 4130 K3 *61 Cygni A 7.6 -1.12 4130 K3
*61 Cygni B 8.4 -1.45 3870 K5 *Struve 2398 A 11.2 -2.56 3070 M3
*Struve 2398 B 11.9 -2.88 2940 M4 Tau Ceti 5.7 -0.39 5150 G8
*Procyon A 2.6 0.88 6600 F5 *Procyon B 13.0 -3.30 9700 DF
Lacaille 9352 9.6 -1.93 3340 M1 G51-I5 17.0 -4.91 2500 M7
YZ Ceti 14.1 -3.75 2670 M5 BD +051668 11.9 -2.88 2800 M4
Lacaille 8760 8.7 -1.60 3340 K5.5 Kapteyn's Star 10.9 -2.45 3480 M0
*Kruger 60 A 11.9 -2.85 2940 M3.5 *Kruger 60 B 13.3 -3.42 2670 M5
BD -124523 12.1 -2.93 2940 M3.5 Ross 614 A 13.1 -3.35 2800 M4
Wolf 424 A 15.0 -4.09 2670 M5 van Maanen's Star 14.2 -3.78 13000 DB
TZ Arietis 14.0 -3.70 2800 M4 HD 225213 10.3 -2.23 3200 M22
Altair 2.2 1.00 8060 A7 AD Leonis 11.0 -2.50 2940 M3.5
*40 Eridani A 6.0 -0.50 4900 K1 *40 Eridani B 11.1 -2.54 10000 DA
*40 Eridani C 12.8 -3.20 2940 M3.5 *70 Ophiuchi A 5.8 -0.40 4950 K0
*70 Ophiuchi B 7.5 -1.12 3870 K5 EV Lacertae 11.7 -2.78 2800 M4