# 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.

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### 1 Answer

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