Motion in the Universe

Definition

Orbital period (T) = the time taken for an object to complete one full orbit

• Units: seconds (s) in SI, but in practice often expressed in minutes, hours, days or years depending on the orbit
• Examples:
  • International Space Station: T ≈ 90 minutes
  • The Moon around Earth: T ≈ 27.3 days
  • Earth around the Sun: T = 1 year
  • Halley's Comet around the Sun: T ≈ 76 years

Orbital speed equation

• In one complete orbit, the body sweeps out a path whose length is the circumference of the orbit (treating it as a circle):

distance per orbit = 2πr

• where r is the average orbital radius (measured from the centre of the body being orbited to the orbiting body, not from the surface)
• Combine this with speed = distance / time to get the orbital speed equation:

v = 2πr / T

• Where:
  • v = orbital speed (m/s)
  • r = orbital radius (m)
  • T = orbital period (s)

Watch out for the radius

• r is from centre to centre, not from the planet's surface
• For a satellite in low Earth orbit at a height h above the surface, the orbital radius is:

r = R_Earth + h

• where R_Earth ≈ 6400 km. Forgetting to add R_Earth is the single most common mistake in this calculation

Why faster-orbiting bodies are closer in

• Looking at v = 2πr / T together with the fact that gravity falls off with distance, you can see the pattern:
  • Close to the central body, gravity is strong, needs high speed to balance, period is short
  • Far from the central body, gravity is weak, only slow speed needed to balance, period is long
• That is why Mercury has a year of 88 days and Neptune has a year of 165 years; and why low satellites zip around the Earth in 90 minutes while geostationary satellites take 24 hours
• Quantitatively, Kepler's third law says T² ∝ r³, but the qualitative rule "closer = faster" is enough here