Question

Consider a high Earth-orbiting satellite of angular momentum per unit mass \(\vec{h}\) and eccentricity \(e\). The mass of the Earth is \(M\) and \(G\) is the universal gravitational constant. The distance between the satellite’s center of mass and the Earth’s center of mass is \(r\), the true anomaly is \(\theta\), and the phase angle is zero. Which of the following statements is/are true?

• A. The trajectory equation is \(r = \frac{|\vec{h}|}{GM(1+e \cos \theta)}\)
• B. The trajectory equation is \(r = \frac{|\vec{h}|^2}{GM(1+e \cos \theta)}\)
• C. \(\vec{h}\) is conserved
• D. The sum of potential energy and kinetic energy of the satellite is conserved

Hint:

For orbital motion under central forces, angular momentum and total energy are conserved, and the trajectory can be described by the vis-viva equation.

Answer 1

The Correct Option is B, C, D

Step 1: Understand the orbital mechanics.
In orbital dynamics, the trajectory of a satellite in a central force field, like the Earth’s gravitational field, can be expressed using the conservation of angular momentum \(\vec{h}\). The orbit equation is based on the conservation of angular momentum and energy, where the trajectory follows the equation: \[ r = \frac{|\vec{h}|}{GM(1 + e \cos \theta)}. \] Step 2: Evaluate each option.
Option (A): This is the correct trajectory equation for elliptical orbits based on the conservation of angular momentum.
Option (B): This is a different form of the orbital equation, also correct, representing the same relationship.
Option (C): \(\vec{h}\) is conserved because there is no external torque acting on the satellite.
Option (D): The total mechanical energy (kinetic + potential) in a gravitational orbit is conserved in the absence of non-conservative forces.