Featured Exam Problems
Geostationary Satellite Altitude
A small satellite orbits the Earth so it stays directly above a fixed point on the equator. Using \(5.97 \times 10^{24}\text{ kg}\) for Earth's mass and \(6.37 \times 10^6\text{ m}\) for its radius, how far is the satellite from the surface of the Earth? Assume its mass is \(300\text{ kg}\) and an orbital period of \(86164\text{ s}\).
Ecliptic Longitude & Inclination
Two exoplanets, A and B, have circular orbits around the same central star. Their ascending nodes are located at the same ecliptic longitude, and both planets start there at \(t = 0\). How long will it take for the planets to share an equal ecliptic longitude again, given inclinations \(i_1 = 30^\circ\), \(i_2 = 70^\circ\) and periods \(T_1 = 2\text{ yr}\), \(T_2 = 1\text{ yr}\)?
Kirkwood Gaps & Resonance
When plotting the distribution of asteroids in the asteroid belt by their semi-major axis, we see several regions, known as "Kirkwood gaps," where asteroids' orbits are unstable due to an orbital resonance with Jupiter (\(a = 5.20\text{ AU}\)). What orbital resonance is responsible for the gap marked with the arrow?
