A planet A of mass m is in a circular orbit of radius "a" around a star ( M_star >> m ). Another planet B with the same mass m is in an elliptical orbit around the same star with its length of the semi-major axis equal to "a" . We have just learned that, in this case, the periods of revolution T and the mechanical energies of the two planets are the same. But, are the angular momenta of the planets with respect to the center of the star the same? Is the angular momentum of planet A equal to or different from the angular momentum of planet B, if so which one is greater ?
Don't they both have the same period? Don't they both traverse the same angle 360 degrees in the same time? don't they have the same anglar momentum?
To determine whether the angular momenta of the two planets are the same or different, we need to consider the concept of angular momentum and how it relates to the properties of their respective orbits.
The angular momentum of an object moving in a circular orbit is defined as the product of its moment of inertia (mass times square of the distance from the center of rotation) and its angular velocity. Mathematically, this can be expressed as:
L = I * ω
Here, L represents angular momentum, I is the moment of inertia, and ω is the angular velocity.
Since planet A is in a circular orbit, its mass (m) and orbital radius (a) determine its moment of inertia. The angular velocity of planet A is ω = 2π/T, where T is the period of revolution.
For planet B, which is in an elliptical orbit with length of the semi-major axis equal to a, the moment of inertia is also determined by its mass (m) and the corresponding distance from its axis of rotation. The semi-major axis does not necessarily coincide with the radius of the elliptical orbit, so the moment of inertia differs from that of planet A. However, the angular velocity ω = 2π/T remains the same because both planets have the same period of revolution.
Therefore, based on the definition of angular momentum, and assuming that the moments of inertia of the two planets are different, we can conclude that the angular momenta of planet A and planet B are not the same.
To determine which planet has a greater angular momentum, we need to compare the moments of inertia of the two planets. Since planet A is in a circular orbit, its moment of inertia is likely to be smaller compared to planet B, which is in an elliptical orbit. As a result, planet B is expected to have a greater angular momentum than planet A.
In summary, the angular momenta of planet A and planet B are different, and the angular momentum of planet B is likely to be greater than that of planet A.