The sun is about 25,000 light years from the center of the galaxy and travels approximately in a circle with a period of 170,000,000 years. The earth is 8 light minutes from the sun. From these data alone, find the a approximate gravitational mass of the galaxy in units of the sun's mass. You may assume that the gravitational force on the sun may be approximated by assuming that all the mass of the galaxy is at its center.
Text Solution
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Sol. For the motion of the earth around the sun,
= 
,
Where r is the distance from the earth to the sun, v is the velocity of the earth, m and m s are the masses of the earth and the sun respectively.
For the motion of the sun around the center of the galaxy,
= 
,
Where R is the distance from the sun to the center of the galaxy, V is the velocity of the sun and M is the mass of the galaxy.
Hence M = 
= 
.
Using V = 2пR/T, v = 2пr/t, where T and t are the periods of revolution of the sun and the earth respectively, we have
M = 
m s .
With the data given, we obtain
M = 1.53 × 10 11 m s .
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