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At the Galaxy’s Core. Astronomers have observed a small,
Chapter 13, Problem 37E(choose chapter or problem)
At the Galaxy’s Core. Astronomers have observed a small, massive object at the center of our Milky Way galaxy (see Section 13.8). A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about 200 km/s. (a) Determine the mass of the object at the center of the Milky Way galaxy. Give your answer both in kilograms and in solar masses (one solar mass is the mass of the sun). (b) Observations of stars, as well as theories of the structure of stars, suggest that it is impossible for a single star to have a mass of more than about 50 solar masses. Can this massive object be a single, ordinary star? (c) Many astronomers believe that the massive object at the center of the Milky Way galaxy is a black hole. If so, what must the Schwarzschild radius of this black hole be? Would a black hole of this size fit inside the earth’s orbit around the sun?
Questions & Answers
QUESTION:
At the Galaxy’s Core. Astronomers have observed a small, massive object at the center of our Milky Way galaxy (see Section 13.8). A ring of material orbits this massive object; the ring has a diameter of about 15 light-years and an orbital speed of about 200 km/s. (a) Determine the mass of the object at the center of the Milky Way galaxy. Give your answer both in kilograms and in solar masses (one solar mass is the mass of the sun). (b) Observations of stars, as well as theories of the structure of stars, suggest that it is impossible for a single star to have a mass of more than about 50 solar masses. Can this massive object be a single, ordinary star? (c) Many astronomers believe that the massive object at the center of the Milky Way galaxy is a black hole. If so, what must the Schwarzschild radius of this black hole be? Would a black hole of this size fit inside the earth’s orbit around the sun?
ANSWER:Soultion Step 1 In this problem we must centripetal force with gravitational force.This is because the gravity keeps the particles around the orbit, to form a ring. Let M be the mass of the object at the center of the galaxy and m be the mass of a particles. 2 2 GM/r = mv r where m is canceled GM/r = v where r is canceled GM/r = v 2 The radius is 7.5 light years. Convert velocity from km/s to m/s. GM/r = v 2 11 15 2 = 6.6710 × 10 × M/(7.5 × 9.461 × 10 ) = 200000 = 9.4 × 10 28× M = 4.0 × 10 10 M = 4.26 × 10 kg7