Circular Satellite Orbits: Motion, Energy, and Launch Requirements

All flashcards

How does circular-orbit period depend on radius?

For a much smaller orbiting mass, $T=2\pi\sqrt{r^3/(GM)}$, where $r$ is measured from the central body's center.

What is the relationship between escape speed and circular orbital speed?

At the same radius, $v_{esc}=\sqrt{2}\,v_{orbit}$.

What does negative total energy signify for a circular orbit?

The system is gravitationally bound; the orbiting object lacks enough energy to reach infinite separation with leftover kinetic energy.

What are the energy values for a circular orbit around a much larger mass?

$K=GMm/(2r)$, $U=-GMm/r$, and $E=-GMm/(2r)$, with potential energy zero at infinite separation.

Why does a real rocket require more energy than the payload's idealized orbital energy increase?

The rocket must also accelerate its hardware and propellant, not just change the payload's mechanical energy.

Quiz questions

1. What conditions make a circular satellite orbit geostationary?Answer: It has Earth's rotation period, lies above the equator, and moves in Earth's rotation direction.
2. Under the circular-orbit model for a satellite much less massive than its central body, how can the central mass be estimated from orbital radius $r$ and period $T$?Answer: $M=\frac{4\pi^2r^3}{GT^2}$
3. When two orbiting bodies have comparable masses, how do they move?Answer: Both orbit their common center of mass.
4. For a circular orbit, what does negative total mechanical energy indicate?Answer: The orbiting object is gravitationally bound to the system.
5. In an ideal comparison from rest at Earth's surface to a circular orbit, what contributes to the change in total mechanical energy?Answer: The increase in gravitational potential energy and the orbital kinetic energy.