The Conservation of Angular Momentum

When a planet is in a circular orbit around a star, it experiences a centripetal force that is provided by the gravitational force between the planet and the star. This gravitational force depends on the mass of the star, the mass of the planet, and the distance between them. According to Newton's law of universal gravitation, this force can be expressed as:

F = G * (M_planet * M_star) / R^2

where F is the gravitational force, G is the gravitational constant, M_planet is the mass of the planet, M_star is the mass of the star, and R is the distance between them.

The angular momentum of the planet in its circular orbit is given by:

L = M_planet * v * R

where L is the angular momentum, v is the velocity of the planet in its orbit, and R is the radius of the orbit.

The Conservation of Angular Momentum

According to the conservation of angular momentum, the angular momentum of the planet remains constant as long as no external torques act on it. In this case, the only force acting on the planet is the gravitational force from the star, which provides the centripetal force necessary for the circular motion. Since the gravitational force is always directed towards the center of the orbit, it does not exert any torque on the planet.

Therefore, the angular momentum of the planet remains constant even if the mass of the star changes. This means that the product of the mass of the planet, the velocity of the planet, and the radius of the orbit remains constant:

M_planet * v * R = constant

The Change in Orbital Radius

As the star loses mass and its mass decreases from M0 to M, the gravitational force between the planet and the star also decreases. In order for the planet to remain in a circular orbit, the velocity of the planet must increase to compensate for the decrease in the gravitational force. This increase in velocity is necessary to maintain the angular momentum constant.

Since the angular momentum is given by the product of the mass of the planet, the velocity of the planet, and the radius of the orbit, and the mass of the planet remains constant, the increase in velocity must be accompanied by a decrease in the radius of the orbit. This is because the product of the velocity and the radius must remain constant in order to keep the angular momentum constant.

Therefore, as the star loses mass and the planet's motion remains circular, the radius of the orbit decreases. This decrease in radius corresponds to the decrease in the distance between the planet and the star.

Summary

- The conservation of angular momentum states that the product of the mass of the planet, the velocity of the planet, and the radius of the orbit remains constant.
- As the star loses mass, the gravitational force between the planet and the star decreases.
- In order for the planet to remain in a circular orbit, the velocity of the planet must increase to compensate for the decrease in the gravitational force.
- This increase in velocity is accompanied by a decrease in the radius of the orbit, as the product of the velocity and the radius must remain constant to keep the angular momentum constant.