Astronomy (Schroeder) Name: fall, 2006 Homework 10 Due Monday, November 6

In this exercise you will construct an H-R diagram for a sample of the nearest and brightest stars, then use the diagram to answer some questions about the sizes of stars. Please draw the diagram on the attached graph paper, on which the temperature and luminosity axes are labeled appropriately. (Note that both axes on the graph are calibrated in a nonlinear way, with units unequally spaced from one end to the other. This kind of “logarithmic” scale is very useful when plotting data that vary over a wide range of values.) A list of stars to include on your diagram is attached. This list consists of two tables of the nearest stars and the brightest stars. (The first table is taken from the web site http://www.chara.gsu.edu/RECONS/TOP100.htm; the second is mostly from Wikipedia.) Plot each of these stars on your H-R diagram, using dots of one color for the brightest stars and dots of another color for the nearest stars. The (absolute) luminosities of the stars are listed in the tables, in units of the sun’s luminosity. (I’ve calculated these from the absolute magnitude values, which are also listed.) To determine the temperatures of the stars, use the following table which relates temperature to spectral type:

Spectral type Temperature (K) 05 44,500 B0 30,000 B5 15,400 A0 9500 A5 8200 F0 7200 F5 6400 G0 6000 G5 5770 K0 5250 K5 4350 M0 3850 M5 3240 For intermediate spectral types, do an approximate interpolation from the table. For instance, a G8 star would be somewhat cooler than a G5 star, but hotter than a K0 star. Don’t worry too much about the precision of your interpolation. After plotting all the stars, label your diagram to classify the various stars as “main sequence,” “giants,” and “white dwarfs.” You may wish to distinguish “giants” from “supergiants.” Consult your textbook or a similar source if you are unsure of how to classify the stars. Then answer each of the questions on the last page. Nearest Stars Apparent Parallax Distance Absolute Luminosity Spectral Magnitude (arc sec) (LY) Magnitude (suns) Type Sun -26.72 0.000015 4.85 1 G2 Proxima Centauri 11.09 0.769 4.24 15.53 0.00005 M5.5 alpha Centauri A 0.01 0.747 4.36 4.38 1.54 G2 alpha Centauri B 1.34 0.747 4.36 5.71 0.45 K0 Barnard's Star 9.53 0.547 5.96 13.22 0.00045 M4.0 Wolf 359 13.44 0.419 7.78 16.55 0.00002 M6.0 Lalande 21185 7.47 0.393 8.29 10.44 0.0058 M2.0 Sirius -1.43 0.380 8.58 1.47 22.49 A1 Sirius B 8.44 0.380 8.58 11.34 0.0025 (B1) BL Ceti 12.54 0.374 8.72 15.4 0.00006 M5.5 UV Ceti 12.99 0.374 8.72 15.85 0.00004 M6.0 Ross 154 10.43 0.337 9.68 13.07 0.00052 M3.5 Ross 248 12.29 0.316 10.32 14.79 0.00011 M5.5 epsilon Eridani 3.73 0.310 10.52 6.19 0.29 K2 Lacaille 9352 7.34 0.304 10.74 9.75 0.011 M1.5 Ross 128 11.13 0.299 10.91 13.51 0.00034 M4.0 EZ Aquarii A 13.33 0.290 11.26 15.64 0.00005 M5.0 Procyon 0.38 0.286 11.40 2.66 7.52 F5 Procyon B 10.7 0.286 11.40 12.98 0.00056 (A6) 61 Cygni A 5.21 0.286 11.40 7.49 0.088 K5.0 61 Cygni B 6.03 0.286 11.40 8.31 0.041 K7.0 GJ 725 A 8.9 0.283 11.52 11.16 0.0030 M3.0 GJ 725 B 9.69 0.283 11.52 11.95 0.00144 M3.5 GX Andromedae 8.08 0.281 11.62 10.32 0.0065 M1.5 GQ Andromedae 11.06 0.281 11.62 13.3 0.00042 M3.5 epsilon Indi A 4.69 0.276 11.82 6.89 0.15 K5 DX Cancri 14.78 0.276 11.82 16.98 0.00001 M6.5 tau Ceti 3.49 0.274 11.88 5.68 0.47 G8

Brightest Stars Sun -26.72 0.000015 4.85 1 G2 Sirius A -1.47 0.380 8.58 1.43 23 A1 Canopus -0.72 0.011 310 -5.61 15000 F0 Arcturus -0.04 0.088 37 -0.31 116 K2 alpha Centauri A -0.01 0.741 4.4 4.34 2 G2 Vega 0.03 0.130 25 0.61 50 A0 Rigel 0.12 0.004 770 -6.75 44000 B8 Procyon A 0.24 0.286 11.4 2.52 9 F5 Achernar 0.5 0.023 140 -2.66 1000 B3 Betelgeuse 0.58 0.008 430 -5.02 8800 M2 Hadar (beta Cent) 0.6 0.006 530 -5.46 13000 B1 Capella A 0.71 0.078 42 0.16 75 G8 Altair 0.77 0.192 17 2.18 12 A7 Aldebaran 0.85 0.050 65 -0.65 158 K5 Capella B 0.96 0.078 42 0.41 60 G0 Spica 1.04 0.013 260 -3.47 2100 B1 Antares 1.09 0.005 600 -5.23 11000 M1 Pollux 1.15 0.096 34 1.06 33 K0 Fomalhaut 1.16 0.130 25 1.74 18 A3 Deneb 1.25 0.001 3000 -8.57 230000 A2 beta Crucis 1.3 0.009 350 -3.85 3000 B0.5 Alpha Centauri B 1.33 0.741 4.4 5.68 0 K0 Regulus 1.35 0.042 77 -0.52 140 B7

1. Consider the two stars Spica (the brightest star in the constellation Virgo) and Sirius B. How do the luminosities of these two stars compare? (That is, which is brighter, and how many times brighter is it? To answer this question, divide the luminosity of the brighter star by the luminosity of the fainter star.) Recall that the Stefan radiation law says

Luminosity = (constant) (Surface area) (Temperature)4. × × Using this law, determine how the surface areas of these two stars compare. Then use the formula 4πr2 for the surface area of a sphere to determine how the radii of the two stars compare. (Again, “how do they compare” means which is more, and by what factor.)

2. Repeat the previous problem for the two stars Betelgeuse and Lalande 21185.

3. Notice that Aldebaran and Regulus have the same luminosity, but different temper- atures. How do their temperatures compare? How do their surface areas compare? How do their radii compare?

4. Given the luminosity and temperature of any star, Stefan’s law allows us to calculate its surface area and radius. Without doing any further calculations, discuss in general how the size of a star depends on its position on the H-R diagram. Where on the diagram are the smallest stars? Where are the largest stars?