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 ?
The angular momentum of an object can be calculated as the product of its moment of inertia and its angular velocity. In the case of a planet orbiting a star, the angular momentum can be expressed as:
L = I * ω
where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
For a circular orbit, the moment of inertia of the planet is given by:
I = m * r^2
where m is the mass of the planet and r is the radius of the circular orbit.
Therefore, the angular momentum for planet A can be written as:
L_A = m * r_A^2 * ω_A
For an elliptical orbit, the moment of inertia can be approximated by taking the average of the maximum and minimum distances (r_max and r_min) of the planet from the star:
I = m * ((r_max^2 + r_min^2) / 2)
Again, the angular momentum for planet B can be written as:
L_B = m * ((r_max^2 + r_min^2) / 2) * ω_B
Since the two planets have the same masses and the same mechanical energies, their angular velocities (ω) are the same. This means that the only difference in the angular momentum lies in the value of the moment of inertia.
In the case of planet B, the maximum distance (r_max) equals the length of the semi-major axis (a), and the minimum distance (r_min) equals the length of the semi-minor axis (b). Therefore, the moment of inertia for planet B can be written as:
I_B = m * ((a^2 + b^2) / 2)
Comparing the expressions for angular momentum:
L_A = m * r_A^2 * ω
L_B = m * ((a^2 + b^2) / 2) * ω
Given that r_A = a and b = a for planet B, we can substitute these values into the expressions:
L_A = m * a^2 * ω
L_B = m * ((a^2 + a^2) / 2) * ω
= m * (2a^2 / 2) * ω
= m * a^2 * ω
Therefore, we find that the angular momenta of the two planets, L_A and L_B, are actually equal. In this case, the angular momentum is the same regardless of whether the planet is in a circular or elliptical orbit.
To determine whether the angular momenta of the planets with respect to the center of the star are the same, we need to consider the definition of angular momentum and how it relates to the given information.
Angular momentum (L) is defined as the product of the moment of inertia (I) and the angular velocity (ω) of an object rotating about an axis. In the case of a planet in a circular orbit, the angular momentum is given by L = Iω, where I is the moment of inertia and ω is the angular velocity.
Now, let's analyze planet A which is in a circular orbit. The moment of inertia of a planet depends on its mass (m) and the radius (a) of its orbit. Since the planet orbits in a perfect circle, the moment of inertia can be calculated as I_A = m × a^2. The angular velocity (ω_A) is related to the period of revolution (T) through the equation ω_A = 2π/T.
Combining these equations, we have L_A = I_A × ω_A. Substituting the values for I_A and ω_A, we get L_A = m × a^2 × (2π/T).
Now let's analyze planet B, which is in an elliptical orbit. The length of the semi-major axis is given as "a." In an elliptical orbit, the moment of inertia can still be calculated as I_B = m × a^2. However, the angular velocity is not constant throughout the orbit. The angular velocity varies depending on the position of the planet in its elliptical path.
Therefore, we cannot directly equate the angular momenta of planet A and planet B due to the difference in their angular velocities.
In summary, the angular momentum of planet A, L_A, is given by L_A = m × a^2 × (2π/T), while the angular momentum of planet B, L_B, would involve integrating the angular momentum over the elliptical path, taking into account the varying angular velocities. The angular momentum of planet A will not be the same as the angular momentum of planet B, and it is not possible to determine which one is greater without additional information about planet B's elliptical orbit.