Astronomy 9/11: Newtonian Gravity and Orbits — Quick Reference

Gravitational law and momentum

• Universal gravitation: $F = \frac{G\, m<em>1\, m</em>2}{r^2}$
• Momentum is conserved in interactions; gravity governs orbital motion rather than a fixed anchor.
• Gravity is a weak force unless masses are large; stronger with more mass and closer distance (inverse-square law).
• Two-body systems share motion about their center of mass; the center of mass is the balance point they orbit around.

Kepler's laws and Newton's synthesis

• Kepler's laws describe planetary motion accurately; Newtonian gravity explains why and enables precise predictions.
• Newtonian gravity allows solving orbital paths; conic sections (ellipses, parabolas, hyperbolas) are possible solutions.
• Newton’s framework lets us predict phenomena like solar eclipses with numerical accuracy.

Two-body dynamics and center of mass

• For two bodies, both bodies orbit their common center of mass; if one mass greatly dominates, the smaller body orbits a center very close to the larger mass.

Kepler's third law and Newton version

• Newtonian Kepler’s third law: $P^2 = \frac{4\pi^2}{G\,(M+m)}\, a^3$
• For solar system-like cases (M ≫ m): $P^2 \approx \frac{4\pi^2}{G\,M_\odot}\, a^3$
• In units of years and AU with M in solar masses: approximately $P^2 \approx \frac{a^3}{M}$ (massive central body effects encoded in the constant; in practice, M ≈ 1 for the Sun).

Period and central mass intuition

• For fixed semi-major axis, increasing central mass decreases the orbital period: $P \propto \frac{1}{\sqrt{M}}$.
• Example: Earth around the Sun at 1 AU has P ≈ 1 year.
• If the central mass were 9× larger, P ≈ 1/3 year for the same orbit.

Tides and gravitational effects

• Tides arise from differential gravity: the near side of Earth feels a stronger pull than the far side, creating bulges.
• Strongest tides occur when the Moon is overhead; solar gravity and rotation modulate the pattern (sun-tide and ocean/land interactions).
• The Moon’s own tides are minimal due to lack of large oceans on the Moon.

Measuring Jupiter's mass with Galilean moons (outline)

• Key formula for a moon of radius $a$ and period $P$ (central mass dominates):
  $M_J \approx \frac{4\pi^2}{G}\, \frac{a^3}{P^2}$
• Assumption: Moon mass is negligible relative to Jupiter; use observations of $a$ and $P$ to estimate $M_J$.
• Observational procedure (Stellarium):
  • View Jupiter and Galilean moons (Io, Europa, Ganymede, Callisto) with FOV around 0.5°.
  • For Callisto (or others), track one full orbit to estimate $P$.
  • Measure orbital radius $a$ (distance from Jupiter) in meters or angular units then convert to meters.
  • Convert $P$ to seconds and $a$ to meters; compute $M_J$ via the formula above.
  • Use multiple moons to cross-check results.
• Practical notes:
  • If angle tools are limited, estimate separations using on-screen angular grid (arcminutes).
  • The four Galilean moons offer multiple independent measurements to improve accuracy.

Practical angular measurements on the sky

• Distances on the sky are angles, not meters; you need the distance to Jupiter to convert angular separations to linear distances.
• In Stellarium, you can use grid/arc-minute cues to estimate separations between Jupiter and its moons when precise tools aren’t available.
• For problems, approximate orbital radii in angular units, then convert to meters once you have the distance to the planet.