Question

A planet has been discovered orbiting around a distant star. a) From measuring the radius of the orbit and the time it takes for the planet to complete it, would we be able to deduce the mass of the planet, the mass of the star, both, or neither? b) A second planet is discovered orbiting the star four times further away than the first one (r2=4r1). How much more time does it take to orbit the star? (In other words, what is the numerical factor relating T2 and T2?)

Answer 1

a) soln.

let the radius of the orbit of planet be a and its time period be T

Then its velocity, v will be :

   ........ ( i )

let the mass of star be M and that of planet be m.

the centripetal force to the planet is provided by the gravitational force acting on it.

.............. ( ii )

substituting the value of v from eqn, (i) in eqn (ii), we get

.......... ( ii )

now we are given time period and radius of orbit, from the last equation we can calculate M i.e. the mass of star

we can't get the mass of planet as it will be cancelled out in eqn (ii).

b) soln.

from eqn. ( iii ) we have,

  

so if radius is increased 4 times.

time period will increase 4³ = 64 times

i.e. T₂ = 64T₁