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 mM–5= logd– 5 + A From Appendix 4 Table 3, we have, for an A0V star, M = 0.6 BV–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==0.6+ 5log 100 – 5 + 4.5 10.1 for the apparent magnitude. 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 GM∆m GM∆m GM∆m ∆E = – ------------------ –– ------------------ ≈ ------------------ r R R where, in this case, the black hole radius R is much smaller than the Sun’s radius r. 2 Using the Schwarzschild radius R=2GM/c for the black hole, we calculate the luminosity as ∆ E GM ∆ m 1 ∆ m L ==------- --------------------------- -------- =--- -------- c 2 ∆ t ()2GM ⁄c2∆ t 2 ∆ t 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 ∆m2 L 9 –13 -------- ==------ 8 . 7 × 10 kg/sec = 1.4×10 solar mass/year ∆tc 2 12 Therefore, the lifetime of the sun is 7.3×10 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. 2/3 Problem 2: In the present day universe the size scale R grows according to R ∝ t . Show that in 1/2 the very early universe, the scale R grew according to R ∝ t . 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 2 2GM 8 π G ------- ==------------- ----------- ρ R 2 dt R 3 NOTE: In clasa and for homework, you worked in the present day universe, which is dominated by matter, so that ρ=A/R3 for some constant A. This leads directly to R12⁄ dR∝ dt so Rt∝23⁄ In the very early universe, radiation dominated and ρ=A/R4 for some constant A: dR 2 8πGA ------- = ----------- ------ dt 3R 2 R 8πGA 12⁄ t ∫RRd =--------------- ∫dt 0 3 0 1 8 π GA 12⁄ --- R 2 = --------------- t 2 3 1/2 and this directly imples that R ∝ t . 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 1/2 early times we have T ∝ 1/t1/2. Putting these two statements together says R ∝ t . 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 tempera- ture, 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: Horizontal branch 0 Giant branch 4 Sun Main sequence 10,000 K 6000 K __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 __B___18. In the neighborhood of our solar system, the stellar density is ≈0.1 stars per cubic par- sec. 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 esti- mate for its distance might be A. 100 kpc B. 500 kpc C. 10 Mpc D. 100 Mpc E. 1000 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. -0.2 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 ____E_________ HII Region ____A or G_____ Planetary Nebula ____C__________ Spiral Galaxy ____F__________ Supernova Remnant ____B__________ Young star cluster ____D or A______ __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 __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 wave- length near A. 1 nm B. 1 µm C.