Show that, for a fixed total angular momentum, the "synchronized" spins situation is the lowest energy state for an orbiting system. (Hint: Consider the sum of the orbital and rotation energies.)

Orbital energy is the energy associated with the movement of an object in orbit around another object. In physics, particularly in celestial mechanics, this is a vital concept. The total orbital energy is the sum of kinetic energy (due to motion) and potential energy (due to gravity). To mathematically express this, we use: \[ E_{\text{orbital}} = \frac{1}{2} m v^2 - \frac{G M_{\text{sun}} m}{r} \] where:

• \( m \) is the mass of the object in orbit
• \( v \) is the object's orbital velocity
• \( G \) is the gravitational constant
• \( M_{\text{sun}} \) is the mass of the central body (e.g., the sun)
• \( r \) is the radius or distance from the center of the central body

In the context of angular momentum, the orbital energy depends on the angular momentum’s magnitude \( L \). This interdependence helps us understand how energy is distributed in an orbiting system.

Spin energy is the energy due to the object's rotation around its own axis. While orbital energy is concerned with path around another object, spin energy focuses on internal rotation. This type of energy becomes significant when analyzing systems like rotating planets or stars. The formula to calculate spin energy is: \[ E_{\text{spin}} = \frac{1}{2} I \theta^2 \] where:

• \( I \) is the moment of inertia
• \( \theta \) is the angular velocity of the spinning object

The moment of inertia \( I \) depends on the mass distribution of the object. For a solid sphere, \( I \) would be different than a hollow one. Spin energy ties into the problem since the total angular momentum depends on both orbital and spin contributions.

Synchronized spins occur when the spin angular momentum \( \textbf{S} \) is aligned or parallel to the orbital angular momentum \( \textbf{L} \). This alignment yields the lowest energy state. Imagine two gyroscopes; when their spins are synchronized, it's easier to balance them on a spinning plate. Similarly, synchronized spins in a system minimize energy. The energy relations in synchronized spins can be expressed as: \[ E = E_{\text{orbital}} + E_{\text{spin}} \] Since both types of energy are functions of the angular momenta, aligning \( \textbf{S} \) and \( \textbf{L} \) minimizes each individual component contribution. This is due to the combined vector magnitude not exceeding the individual magnitudes, strengthening the system's stability. Therefore, synchronized spins lead to a condition where the total energy of the system is minimized, confirming that it is the lowest energy state for the orbiting system.