Question

A star has a mass of 3 solar masses (3 times the mass of the Sun). If a planet is 1 AU from this star, should its orbit be more or less than 1 Earth year? Explain why or why not. (show work)

You see a planet that takes 1 year to orbit its star, but the planet averages 10 AU from its star. Explain what this tells you about the star.

Answer 1

Ans- We know that the orbital period of a planet revolving around a star is given as-

T= (4π2r3/GM)1/2 (M is mass of star,r is radius of the orbit,G is the Gravitational constant)

Also, The distance of earth from sun is 1 AU.

Now the orbital period of earth is-

T1= (4π2r3/GM)1/2 (M is mass of sun,r is radius of the Earth orbit,G is the Gravitational constant)

According to the question-

-Mass of star is 3M (3 times the mass of the Sun).

- orbital radius is same as earth.

T2= (4π2r3/3GM)1/2 (M is mass of star,r is radius of the planet orbit,G is the Gravitational constant)

as, T2=1/√3T1 ( T1>T2 ) so the orbital time will be less than 1 Earth year.

b) Orbital time of planet = 1 year.

Radius of the Orbit = 10AU=10r (earth orbital radius r)

now,

(4π2r3/GM)1/2 ={4π2(10r)3/GM2}1/2   (M2 is mass mass of star)

M2=31.63M

Which means that it will be 31.63 heavier than mass of our sun.

Hope you have understood the answer.

Best Wishes!