A star has a planet in orbit with an average orbital radius of 7.32x10^11 m.  Using the fact that the period of the planet's orbit is 735 Earth days, calculate the mass of the star.

To do this question we must realize that when any one object is "in orbit" about another object there must be a balance between the centripetal force holding the object in orbit and the centrifugal force (inertia) that wants to cause the object to continue in a straight line.

In the case of planetary orbits the two forces in balance are

Universal Force of Gravity = Centrifugal Force of circular motion

GMsMp/R^2  = MpV^2/R  where

G is universal gravitational constant 6.67x10^-11 Nm^2/kg^2

Ms is mass of the star, Mp is mass of planet, and R is orbital radius.

Cancelling the Mp and one of the R on each side of the equality allows us to solve this equation for Ms:

Ms = V^2R/G

V is the speed with which the planet is orbitting the star.  If the orbit is circular we can use the definition of speed to figure out an expression for V

V = total distance/total time

The distance to complete one orbit is given by 2PiR and the time is the length of the orbital period in seconds

T=735daysX86,400s/day=6.35x10^7 seconds

V = 2Pi x R/T

Ms = 4(Pi^2)(R^2)R/[(T^2)G] = 4Pi(R^3)/[(T^2)G]

Ms =

4Pi^2(7.32x10^11m)^3/[(6.35x10^7s)^2x6.67x10^-11Nm^2/kg^2]

Ms = 2.41x10^37 kg

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