Understanding the Problem:
We have a planet moving in a circular orbit around a star. Initially, the star has a mass M₀ and the planet orbits at a radius R. The star starts losing its mass adiabatically and eventually reaches a mass M, where M< />₀. We need to determine the new radius of the planet's orbit if its motion remains circular.

Solution:

1. Conservation of Angular Momentum:
The key concept to understand in this problem is the conservation of angular momentum. Angular momentum is conserved when there are no external torques acting on the system. In this case, the only significant force acting on the planet is the gravitational force of the star.

2. Angular Momentum Equation:
The angular momentum of the planet is given by the equation:
L = mvr
where L is the angular momentum, m is the mass of the planet, v is its velocity, and r is the radius of its orbit.

3. Conservation of Angular Momentum:
As the star loses mass, the gravitational force it exerts on the planet decreases. However, since there are no external torques acting on the system, the angular momentum of the planet must remain constant.

4. Relationship between Mass, Velocity, and Radius:
The gravitational force between the star and the planet is given by the equation:
F = G(M₀M)/r²
where G is the gravitational constant.

The centripetal force required to keep the planet in a circular orbit is given by:
F = mv²/r

Setting these two equations equal to each other and solving for v, we get:
v = √(GM₀/r)

5. Relationship between Initial and Final Masses:
Using the above equation, we can express the initial velocity of the planet as:
v₀ = √(GM₀/R)

Similarly, the final velocity of the planet can be expressed as:
v = √(GM/r)

Since angular momentum is conserved, we can equate the initial and final angular momenta:
m₀v₀R = mvR

Simplifying this equation, we get:
(M₀/M) = (R²/r²)

6. Determining the Final Radius:
We are given that the motion of the planet remains circular even after the star loses mass. So, the radius of its orbit at that time will be equal to r.

Using the relationship (M₀/M) = (R²/r²) derived in the previous step, we can rewrite it as:
r² = R²(M/M₀)

Taking the square root on both sides, we get:
r = R√(M/M₀)

Therefore, the radius of the planet's orbit will become R√(M/M₀