Celestial Mechanics
Celestial mechanics is the physics of orbits. This domain tests Kepler's three laws, Newton's law of gravitation, orbital energy and angular momentum, tides, and how the Sun–Earth–Moon system produces observable effects. Expect numerical items on orbital speed, period, and escape velocity alongside conceptual items on why real orbits depart from ideal ellipses.
Kepler and Newton
- First law: orbits are conic sections; a closed orbit is an ellipse with the primary at one focus. Periapsis distance is a(1 − e), apoapsis a(1 + e).
- Second law: the radius vector sweeps equal areas in equal times, which is conservation of angular momentum. Speed is highest at periapsis and lowest at apoapsis.
- Third law: P² = a³ when P is in years and a in AU for orbits dominated by the Sun. The general form is P² = a³/(M₁ + M₂), which is how exoplanet and binary masses are found.
- Newton: F = GMm/r². For a circular orbit v = √(GM/r); escape velocity is √(2GM/r), exactly √2 times the circular speed at the same radius.
Energy, tides, and perturbations
- Bound orbits have negative total energy. Adding energy increases a and lengthens P; the body is slower at every radius but its periapsis speed still exceeds its old one.
- Tidal force scales as M/d³, so it falls off faster than gravity. Tides stretch a body, dissipate energy, and drive tidal locking; inside the Roche limit a moon is pulled apart.
- Real orbits precess because of oblateness (Earth's J₂) and third-body pulls; resonances (e.g., the Laplace resonance of Io, Europa, Ganymede) keep eccentricities pumped and cause tidal heating.
Worked example
A satellite circles Earth at 400 km altitude. Earth's radius is 6.371 × 10⁶ m, so r = 6.771 × 10⁶ m and GM = 3.986 × 10¹⁴ m³/s². Then v = √(3.986 × 10¹⁴ / 6.771 × 10⁶) = √(5.89 × 10⁷) ≈ 7.67 km/s. The period is T = 2πr/v = 2π(6.771 × 10⁶)/7670 ≈ 5.55 × 10³ s, about 92 minutes. Escape speed from that altitude is 1.414 × 7.67 ≈ 10.8 km/s, not the familiar 11.2 km/s surface value - the trap is using surface numbers at altitude.
Traps that cost marks
- The second law gives constant angular momentum, not constant speed.
- P² = a³ fails unless the central mass dominates and the units are AU and years; the Moon does not obey it around the Sun in those units.
- Astronauts are not in "zero gravity"; at 400 km g is still about 8.7 m/s². They are in free fall.
- Geosynchronous (period 23h56m) is not the same as geostationary, which also requires a circular, equatorial, prograde orbit.
- Retrograde motion of Mars is a perspective effect of Earth overtaking it on an inner, faster orbit.