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Homework Solutions: Chapter 10, The Deaths of Stars. [ RQ: 6, 8, 12 Problems: 3 , 5 , 9 ] RQ 6: How can star clusters confirm our theories of stellar evolution? Answer: Star clusters contain stars of a wide variety of masses that all formed at about the same time. Since our models tell us that massive stars should evolve faster than low mass stars, we expect to see differences in the star clusters based on the mass of the stars. One clear test that our models are at least on the right track, is that we find no star clusters with O and B main sequence stars, and a large number of giants and supergiants. All of the star clusters show us that when giants and supergiants are present in the clusters, there are few if any massive stars (O and B stars) on the main sequence. We can take models of many different masses of stars and determine what a cluster should look like after a given time. We can then compare the HR diagram from our model with that of the observations of the actual cluster to see how well they agree. For most part, the agreement is quite good. RQ 8: What causes an aging giant star to produce a planetary nebula? Answer: A planetary nebula is produced as the carbon-oxygen core of a star contracts and its outer envelope expands. The intense radiation from the hotter inner layers ionizes the innermost gas of the envelope and drives it outward from the star. As this gas overtakes mass lost earlier in the stars life as a giant, material piles up to form a dense shell of ionized gas. The gas is continually ionized by the radiation from the hot contracting core. RQ 12: How can we understand the Algol paradox? Answer: Algol is a binary star composed of a five solar mass main sequence star and a one solar mass giant. This presents a paradox because massive stars evolve faster than low mass stars, so the five solar mass stars should leave the main sequence before the one solar mass star. The solution to this paradox is mass transfer. Suppose that the system was originally composed of a five solar mass star and a one solar mass star. After about 200 million years the five solar mass star would run out of hydrogen and begin to expand to become a red giant. If as it expands, the star fills its Roche lobe, material will be drawn off of the five solar mass star and accreted onto the one solar mass star that is still on the main sequence. As the giant continues to expand, more and more mass will be passed from the giant to the star that is still on the main sequence. If four solar masses of material can be drawn off of the giant and deposited on the main sequence star, the main sequence star will end up with a mass of five solar masses and the giant will have a mass of only one solar mass. Thus through mass transfer as the massive star expands to become a giant, the star still on the main sequence can become more massive than the giant. PR 3: About how far apart are the stars in an open cluster? in a globular cluster? (Hint: What share of the cluster’s volume belongs to a single star?) Answer: An open cluster is a collection of 10 to 1000 stars in a region about 25 pc in diameter. i) If there are 10 stars then each star has a volume equal to, 4 3 12.53 V 817.7 pc 3 V 10 Hence the maximum distance rmax between two stars in an open cluster is, 4 3 3 817.7 rmax rmax 195.3 rmax 5.8 pc 3 ii) If there are 1000 stars then each star has a volume equal to, 3 4 3 12.5 V V 8.18 pc 3 1000 Hence the minimum distance rmin between two stars in an open cluster is, 4 3 3 8.18 rmin rmin 1.95 rmin 1.25 pc 3 Finally, 1.25 pc r 5.8 pc A globular cluster is a collection of 10 5 to 10 6 stars in a region about 20 pc in diameter. i) If there are 10 6 stars in a region of 10 pc diameter then each star has a volume equal to, 4 3 103 V 0.0042 pc 3 V 10 6 Hence the minimum distance rmin between two stars in a globular cluster is, 4 3 3 0.0042 rmin rmin 0.001 rmin 0.1 pc 3 ii) If there are 10 5 stars then each star has a volume equal to, 4 3 103 V 0.042 pc 3 V 10 5 Hence the maximum distance rmax between two stars in a globular cluster is, 4 3 3 0.042 rmax rmax 0.01 rmax 0.21 pc 3 Finally, 0.1 pc r 0.21 pc PR 5: If the Ring Nebula is expanding at a velocity of 15 km/sec, typical of planetary nebulae, how old is it? Answer: The Ring Nebula has a diameter of about D=1 light-year. Its radius therefore is R=0.5 ly which is also the distance of the gas shell from its center. The gas must have had enough time to travel the distance of 0.5 ly at a speed of v=15 km/sec. Therefore, the age of the nebula must be at least, R 0.5 300000Km sec 1year 150000 T T year T 10000 years v 15Km sec 15 The Ring Nebula must be at least 10000 years old. PR 9: The Crab Nebula is now 1.35pc in radius and is expanding at 1400 Km sec . About when did the supernova occur? (Hint: 1pc equals 3 1013 Km) Answer: The speed of the expansion is constant and so we can write, dis tan ce 1.35 3 1013 Km T T 2.89 1010 sec velocity 1400 Km sec But, 30days 24hours 60 min utes 60 sec onds 1year 12months month day hour min ute 1year 3.11 10 7 sec Hence, 2.89 1010 sec T T 929 years 3.11 10 7 sec year