The distance modulus in the presence of absorption is given by

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1 Problem 4: An A0 main sequence star is observed at a distance of 100 pc through an interstellar dust cloud. Furthermore, it is observed with a color index B-V = 1.5. What is the apparent visual magnitude of the star? The distance modulus in the presence of absorption is given by m M = 5logd 5 + A From Appendix 4 Table 3, we have, for an A0V star, M = 0.6 B V = 0 where the magnitudes and absorption are in the visual (V) band. Therefore, the color excess is 1.5 0=1.5, and so A=3 1.5=4.5. This gives m = log = 10.1 for the apparent magnitude.

2 Problem 3: Imagine that all of the Sun s mass is concentrated in a thin spherical shell at the Sun s radius. Imagine further that the Sun is powered by this mass slowly falling piece by piece into a black hole at the center of the sphere. If 100% of this energy is radiated away from the surface of the Sun, calculate the lifetime of the Sun, given its observed luminosity. Comment on your answer. If a small bit of mass m falls from a radius r onto a radius R, then the energy released is given by E = GM m r GM m R GM m R where, in this case, the black hole radius R is much smaller than the Sun s radius r. Using the Schwarzschild radius R=2GM/c 2 for the black hole, we calculate the luminosity as L E GM = = m t ( 2GM) c = t m c 2 t 2 This is precisely the same relationship used when we studied Cygnus X-1 and also appeared on the second class exam. The rate at which mass is used is therefore m t 2 L 9 13 = c 2 = kg/sec = solar mass/year Therefore, the lifetime of the sun is years. This is nearly a thousand times longer than the age of the solar system. In other words, we cannot exclude this model of powering stars on the basis of lifetime.

3 Problem 2: In the present day universe the size scale R grows according to R t 2/3. Show that in the very early universe, the scale R grew according to R t 1/2. In class, we came up with the following expression (or something like it), based on the concept of escape velocity and the size scale of the universe: dr dt 2 = 2GM = R 8 π G ρ R 3 2 NOTE: In clasa and for homework, you worked in the present day universe, which is dominated by matter, so that ρ=a/r 3 for some constant A. This leads directly to In the very early universe, radiation dominated and ρ=a/r 4 for some constant A: and this directly imples that R t 1/2. R 1 2 dr dt so R t 2 3 R 0 dr πG = A dt 3 R 2 RdR = 8πGA t dt R π GA = t 3 A different way of doing this problem uses the fact that, as demonstrated on a homework problem, T 1/R (at all times) and, as demonstrated in class, for very early times we have T 1/t 1/2. Putting these two statements together says R t 1/2.

4 Part II Answer the following four short problems. Each is worth 10 points. Problem 1: Sketch an HR diagram, plotted as absolute magnitude versus temperature, for the stars in M5, displayed in image E (labeled AAT 70). Include at least two labeled tick marks on each axis. Label the primary features of the diagram. Also,indicate the position on the diagram of a star like our sun. M5 is a globular cluster. These are collections of very old stars, so the brightest are yellow and red giants, as opposed to blue main sequence stars. You can see that clearly by comparing it with image D, a very young star cluster (the Pleiades). The HR diagrams of globular clusters all look the same and reflect the fact that the more massive stars have evolved off the main sequence. Figure 16-10A in your textbook sketches several gobular clusters, and 16-10B in fact displays it for M5 itself. However, the vertical scale must be converted to absolute magnitude (as is done in 16-10A) and the color index horizontal scale must be converted to temperature, using, for example, equation 11-11b. The sun lies on the main sequence, just near the turnoff, at a temperature near 6000K. The minimal acceptable sketch for full credit looks something like the following: 0 Horizontal branch Giant branch 4 Sun Main sequence 10,000 K 6000 K

5 E 23. The object which is farthest from the Earth is shown in A. Image A B. Image B C. Image C D. Image E E. Image F B 24. The red glow in Image A is most likely A. Hydrogen Lyman emission B. Hydrogen Balmer emission C. Hydrogen 21cm emission D. Synchrotron radiation E. Blackbody radiation E 25. The oldest collection of stars is shown in A. Image A B. Image B C. Image C D. Image D E. Image E A 26. New stars are presently being formed in A. Image A B. Image B C. Image C D. Image D E. Image E D 27. Which of the following is located at the center of Image B: A. Hot, young star B. Typical solar mass star C. White dwarf star D. Neutron star E. Black hole A 28. The dark patch in image G is A. An interstellar dust cloud B. A supermassive black hole C. Caused by cold interstellar gas D. Very close to the solar system E. A region where stars have died

6 B 18. In the neighborhood of our solar system, the stellar density is 0.1 stars per cubic parsec. If we take this to be a measure of the matter density in the universe, the universe would be A. Open B. Closed C. Flat D. Spherical E. We don t know the Hubble constant well enough to tell C 19. The full disk of a typical spiral galaxy covers 15 arcmin on the sky. A reasonable estimate for its distance might be A. 100 kpc B. 500 kpc C. 10 Mpc D. 100 Mpc E Mpc A 20. The goalposts on a football field are separated by about 100 m, but this distance is constantly expanding according to Hubble s Law. After one year, this distance increases by about A. 10 nm B. 10 µm C. 10 mm D. 10 cm E. 10 m C 21. The intrinsic color index B-V of the Sun is approximately A B. 0.2 C. 0.6 D. 1.0 E. 1.4 The remaining questions in Part I all refer to the sheet of color optical images. 22. Indicate the image (A through G) which best typifies the following classes of objects. One point for each answer. Put down only one answer on each line, even if you think there may be more than one possibility. Globular Cluster HII Region Planetary Nebula Spiral Galaxy Supernova Remnant Young star cluster E A or G C F B D or A

7 A 12. Extremely strong hydrogen lines, surface temperature close to 10,000K, and color index B-V=0 are all characteristics of stars with spectral type A. A0 B. B0 C. F0 D. G0 E. K0 C 13. A star forms on the main sequence with twice the mass of our sun. It s lifetime on the main sequence, relative to that for a solar mass star, is A. The same B. Half as long C. 1/4 as long D. 1/9 as long E. 1/16 as long D 14. The evolutionary path of a star takes it horizontally and to the left on an HR diagram. Which of the following is true: A. The radius and temperature are both decreasing B. The radius and temperature are both increasing C. The radius is increasing and the temperature is decreasing D. The radius is decreasing and the temperature is increasing E. The change in size of the star cannot be determined C 15. The Milky Way galaxy has a bulge and disk characterstic of spiral galaxies. These features are most plainly seen in images taken in A. Visible light B. Ultraviolet light C. Infrared light D. 21cm radio observations E. X-rays D 16. Compared with population I stars, population II stars of the same mass A. Are smaller B. Are more blue C. Are more luminous D. Have longer total lifetimes E. Can generally be found in star forming regions E 17. It makes sense for Dallas Cowboys fans to study Astronomy because A. The Cowboys make their opponents see stars B. The Cowboys have a lot of stars on their team C. The Cowboys wear stars on their helmets D. All of the above E. It doesn t make sense for anyone to be a Dallas Cowboys fan

8 A 6. Which of the following is not considered to be an active galaxy: A. M31 B. M87 C. Cygnus A D. Centaurus A E. 3C273 C 7. The final stage in the evolution of our Sun will most likely be a A. Black hole B. Neutron star C. White dwarf D. Planetary nebula E. Red giant C 8. A blackbody radiates with a temperature T=3K. The signal is strongest for a wavelength near A. 1 nm B. 1 µm C. 1 mm D. 1 cm E. 1 m E 9. A patch of sky shows a dark region nearly devoid of stars when viewed in visible light. However, an infrared image shows a small area within the region that is more than ten times as bright as the Sun. You are most likely observing A. A nova B. A pulsar C. A black hole D. A planetary nebula E. A collapsing cloud that will eventually form a star E 10. Star #1 is 100 times more luminous than star #2. Star #1 is also 100 times farther away than star #2. The difference in apparent magnitudes m 1 -m 2 is A. -5 B C. 0 D. 2.5 E. 5 A 11. Which of the following is not known to be associated with astrophysical jets: A. Supernovae B. Star formation C. Active galaxies D. Quasars E. SS433

9 Part I: Multiple choice worth two points each. Give the best choice for each question. D 1. Which of the following were the last to appear in the early universe: A. Protons B. Neutrons C. Electrons D. Hydrogen atoms E. Helium nuclei E 2. Hydrogen Balmer Hα emission is observed at a wavelength of nm in a distant galaxy. The distance to the galaxy is independently found to be 450 Mpc. From this data, we determine the Hubble constant to be A. 50 km/sec Mpc B. 62 km/sec Mpc C. 75 km/sec Mpc D. 87 km/sec Mpc E. 100 km/sec Mpc A 3. The Hubble Space Telescope has made important recent observational contributions to our best value for the Hubble constant. These have mainly been from A. Cepheids measured in galaxies closer that 25 Mpc B. Cepheids measured in galaxies farther than 25 Mpc C. Supernova observations in very distant galaxies D. Verification of the Tully-Fisher relationship E. Establishing quasars as Active Galactic Nuclei C 4. Most of the galaxies in the local group A. Are similar to the Milky Way B. Are spiral galaxies C. Display negative redshift values D. Are within 100 kpc of each other E. Are more massive than the Milky Way B 5. Saturn s orbital distance from the Sun is about 10 times that for the Earth. Saturn s orbital velocity around the Sun (in km/sec for example) is which of the following, relative to the orbital velocity for the Earth: A. 1/10 B. 1/ 10 C. The same D. 10 E. 10

10 Final Exam Astronomy Fall 1996 NAME: Solution Key You have three hours to complete this exam. Part I has 27 multiple choice questions, each worth 2 points, and one one six-point question. Part II has four short problems, each worth 10 points. You are to answer all questions on both parts. You may use your textbook (Zeilik), workbook (Hoff), and class notes and handouts. You may not share these resources with another student during the test. GOOD LUCK! Score Part I: Part II: Total Score:

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