Final answer:
The correct option is (D) The radius of the orbit will not change.
The orbit of a planet around a star will not change if the star shrinks to half its radius while maintaining its mass. The gravitational force and orbital characteristics depend on the mass of the star and the orbital radius, both of which remain the same in this scenario.
Step-by-step explanation:
If a star of mass M suddenly shrinks to half its radius without any loss of mass, the orbit of a planet of mass m revolving around it will not change. This is because according to Newton's law of universal gravitation, the force between two masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Since neither the masses involved nor the orbital distance of the planet from the center of the star have changed, the gravitational force between the star and the planet remains unchanged.
Furthermore, the orbital characteristics such as the period and the speed depend on this gravitational force and the distance between the bodies involved. Since these factors are unaltered by the star's change in radius, the orbital speed and period would remain the same. Therefore, the radius of the planet's orbit will not change, corresponding to option (D).