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?

How does it compare with the speed of light?

Explanation / Answer

use kepler's law of planetory motion

T^2 = 4*pi^2*a^3/(G*M)

T = 30 *365*24*60*60 = 946080000

Distance a = 1.5*10^3 AU = 2.24*10^14 m

then mass M = 4*3.142^2*2.24^3*10^42/(6.67*10^-11*946080000^2) = 7.43*10^36kg

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B) apply v = sqrt((G*M/a)) = sqrt((6.67*10^-11*7.43*10^36/2.24*10^14)) = 0.723*10^8 = 14.8741*10^5 m/s = 1487.41 km/s

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the earth's orbital speed is 30 km/s which almost all 50 times faster than eatrhs speed

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the speed of the light is 3*10^5 km/s

whgich is about 200 times slower than the speed of light

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