A massive black hole is believed to exist at the center of our galaxy (and most

Question

A massive black hole is believed to exist at the center of our galaxy (and most other spiral galaxies). Since the 1990s, astronomers have been tracking the motions of several dozen stars in rapid motion around the center. Their motions give a clue to the size of this black hole.

(a) One of these stars is believed to be in an approximately circular orbit with a radius of about 1.50 103 AU and a period of approximately 30 yr. Use these numbers to determine the mass of the black hole around which this star is orbiting.



(b) What is the speed of this star?

How does it compare with the speed of the Earth in its orbit?

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How does it compare with the speed of light?

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Explanation / Answer

P2 = (4pie2a3) / (G(M1 + M2))

P is the period of the orbiting object (in seconds), a is the radius of the orbit (in meters), G is a constant, M1 is the mass of the first object (in kilograms), and M2 is the mass of the second object (in kilograms). In cases where M1 is way bigger than M2 (like when M1 is the mass of a massive black hole), M1 + M2 is approximately equal to M1. So we'll just call that sum M1, the mass of the black hole.

We'll first convert years to seconds (depending on how you measure one year, this could vary):

P = 30 years × 31556926 seconds / 1 year = 946707780 seconds = 9.4670778×108 seconds

Now, we'll convert Astronomical Units to meters:

a = 1.5×103 AU × 149597870700 meters / 1 AU = 224396806050000 meters = 2.2439680605×1014 meters

Plug those numbers in, as well as G = 6.673×10-11:

(9.4670778×108)2 = (42(2.2439680605×1014)3) / (6.673×10-11(M1))

59807137.6M1 = 4.46076916×1044

M1 = 7.45858996 × 1036 kilograms

For the second part of this question, you can use the formula for speed of an orbiting object:

v = sqrt((GM1)/a)

v is the speed (in meters per second), G is our constant from above, M1 is the mass of the object being orbited (in kilograms), and a is the orbital radius (in meters).

Plug those numbers in:

v = sqrt((6.673×10-11)(7.45858996×1036))/(2.2439680605×1014))

v = sqrt(2.21799818×1012)

v = 1489294.52 m/s, or 1489.29452 km/s

The Earth's orbital speed is 30 km/s, so the star is much faster (almost 50 times faster).

The speed of light is 299792.458 km/s, so the star is much slower (about 200 times slower).

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