A particle of mass m 𝑚 is under the influence of the gravitational field of a body of mass M ( >> m ) 𝑀 ( >> 𝑚 ) . The particle is moving in a circular orbit of radius r 0 𝑟 0 with time period T 0 𝑇 0 around the mass M 𝑀 . Then, the particle is subjected to an additional central force, corresponding to the potential energy V c ( r ) = m α / r 3 𝑉 𝑐 ( 𝑟 ) = 𝑚 𝛼 / 𝑟 3 , where α 𝛼 is a positive constant of suitable dimensions and r 𝑟 is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius r 0 𝑟 0 in the combined gravitational potential due to M 𝑀 and V e ( r ) 𝑉 𝑒 ( 𝑟 ) , but with a new time period T 1 𝑇 1 , then ( T 2 1 − T 2 0 ) / T 2 1 ( 𝑇 1 2 − 𝑇 0 2 ) / 𝑇 1 2 is given by [G is the gravitational constant.]
A particle of mass m 𝑚 is under the influence of the gravitational field of a body of mass M ( >> m ) 𝑀 ( >> 𝑚 ) . The particle is moving in a circular orbit of radius r 0 𝑟 0 with time period T 0 𝑇 0 around the mass M 𝑀 . Then, the particle is subjected to an additional central force, corresponding to the potential energy V c ( r ) = m α / r 3 𝑉 𝑐 ( 𝑟 ) = 𝑚 𝛼 / 𝑟 3 , where α 𝛼 is a positive constant of suitable dimensions and r 𝑟 is the distance from the center of the orbit. If the particle moves in the same circular orbit of radius r 0 𝑟 0 in the combined gravitational potential due to M 𝑀 and V e ( r ) 𝑉 𝑒 ( 𝑟 ) , but with a new time period T 1 𝑇 1 , then ( T 2 1 − T 2 0 ) / T 2 1 ( 𝑇 1 2 − 𝑇 0 2 ) / 𝑇 1 2 is given by [G is the gravitational constant.]