Question

Astronomical observations of our Milky Way galaxy indicate that it has a mass of about 8.0x10^11 solar masses.

A star orbiting near the galaxy's periphery is 6.0X10^4 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 in yr ?

(b) If its period is 6.0 x 10^7 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. solar masses

Answer 1

(a) From Kepler's third law, we have

T = sqrt ( 4$\pi$²r³ / GM)

T = sqrt ( 4$\pi$² * (6e4 * 3e8 * 365.25 * 24 * 3600)³ / 6.67e-11 * 8e11 * 1.99e30)

T = sqrt (6.8142e31)

T = 8.2548e15 seconds

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(b) Again using the same equation, here T = 6e7 years or 1.89346e15 seconds

therefore,

3.5852e30 = 4$\pi$² * (6e4 * 3e8 * 365.25 * 24 * 3600)³ / 6.67e-11 * M)

M = 3.026e43 Kg

In terms of solar masses,

M = 1.52e13 solar masses.

Please let me know if there is any mistake, I will correct it. Let me know if you have any doubt.