1. Consider our Sun - it is in orbit around the center of our Milky Way Galaxy. The velocity of the Sun in its orbit is about 250 km/s. The distance to the center of the galaxy is about 9.1 kpc (kiloparsecs). We can use Kepler's third law to calculate the mass of the galaxy interior to the Sun's orbit. We assume that the orbit is circular so that the semimajor axis is just the radius of the circular orbit = 9.1 kpc. First we need to calculate the number of AU's in 9.1 kpc. (Note that 1 Крс - 1000 рс - 3260 1t yrs and 1 pc - 206,265 AU.) %3D a =r =9.1kpc = (9.1kpc) 1000 pc 206,265AU] 1kpc AU Sun 1pc

Observations indicate that each galaxy contains a supermassive black hole at its center. These black holes can be hundreds of thousands to billions of times more massive than the Sun. Astronomers estimate the size of such black holes using multiple methods. One method, using the orbits of stars around the black hole, is an application of Kepler's third law. The mass of the black hole can be found by using the given equation, where a is the semi-major axis in astronomical units, P is the period in years, and k is a constant with a value of 1 Mo X year²/ AU³. a³ M = k- p² What is the mass of a supermassive black hole if a star orbits it with a semimajor axis of 959 AU and a period of 13.3 years? mass: Another method measures the speed of gas moving past the black hole. In the given equation, v is the velocity of the gas (in kilometers per second), r is the distance of the gas cloud from the black hole (in kilometers), and G is Newton's gravitational constant. In this equation, G = 1.33 ×…

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