Astronomical observations of our Milky Way galaxy indicate that it has a mass of

Question

Astronomical observations of our Milky Way galaxy indicate that it has a mass of about 8. x 1011 solar masses. A star orbiting near the galaxy's periphery is 5.8 x 104 light years from its center. (For your calculations, assume that the galaxy's mass is concentrated near its center.) a) What should the orbital period of that star be? Identify the source of the centripetal force that makes the star orbit the galaxy. What is the relationship between orbital speed and period of rotation? yr (b) If its period is 5.3 x 107 years instead, what is the mass of the galaxy? Such calculations are used to imply the existence of "dark matter" in the universe and have indicated, for example, the existence of very massive black holes at the centers of some galaxies. Check your calculations. solar masses

Explanation / Answer

1 solar mass = 1.989 x 10^30 kg

Mass of our Milky Way = 8 x 10^11 solar mass = 15.91 x 10^41 kg

Distance, r = 5.8 x 10^4 light years = 5.8 x 10^4 x 9.461 x 10^15 = 5.487 x 10^20 m

A) Orbital velocity, vo = (G M/ r)^1/2 = (6.67 x 10^-11 x 15.91 x 10^41 / 5.487 x 10^20)^1/2 = 4.397 x 10^5 m/s

Orbital period, T = 2 pi r/ vo = 7.84 x 10^15 sec = 2.486 x 10^8 years

B) If orbital period, T = 5.3 x 10^7 years = 1.67 x 10^15 sec

Orbital velocity, vo = 2 pi r/T = 2.06 x 10^6 m/s

Mass of galaxy, M = vo^2 r/ G = (2.06 x 10^6)^2 x 5.487 x 10^20/ 6.67 x 10^-11 = 3.51 x 10^43 kg = 1.76 x 10^13 solar masses

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