ASTR 120 · Lecture 03
Kepler extracted empirical laws from Tycho's data; Newton supplied the force law; spacecraft navigation now uses the same mechanics.
Definition: Half the longest diameter of an ellipse; for an orbit it sets the orbit's characteristic size and energy.
Why it matters and connects: Kepler's third law relates semi-major axis to period, making it measurable through repeated positions or timing.
Boundary or common confusion: It is not the instantaneous star–planet distance except for a circular orbit.
Definition: The time required for a body to complete one orbit relative to a specified reference direction.
Why it matters and connects: Period ratios reveal resonances, and period combined with semi-major axis tests the central mass through Keplerian dynamics.
Boundary or common confusion: Sidereal and synodic periods differ; the relevant definition must match the observation.
Definition: A configuration in which two orbital periods are close to a ratio of small integers, causing gravitational kicks to repeat at preferred phases.
Why it matters and connects: Repeated geometry can protect or destabilize orbits, structure rings and asteroid populations, and maintain moon eccentricities.
Boundary or common confusion: A near-integer period ratio alone does not prove resonant locking; the resonant angle must librate rather than circulate.
Definition: A deviation from ideal two-body motion caused by additional masses, nonspherical gravity, drag, radiation, or relativity.
Why it matters and connects: Small accumulated perturbations reveal unseen masses, shape orbital architecture, and must be modeled for spacecraft navigation.
Boundary or common confusion: A perturbation is not necessarily negligible; weak forces acting repeatedly can dominate long-term evolution.
Definition: The differential gravitational acceleration across an extended body.
Why it matters and connects: Tides distort worlds, exchange spin and orbital angular momentum, and can dissipate orbital energy as interior heat.
Boundary or common confusion: Tidal force is a gradient in gravity, not simply strong gravity; dissipation depends on material response and orbital state.
Definition: The region around an orbiting body where its gravity can dominate satellite motion against the tidal influence of the central object.
Why it matters and connects: It provides a first estimate of where moons or rings can remain bound and how planet mass and orbital distance affect satellite systems.
Boundary or common confusion: Stable satellite orbits occupy only part of the Hill sphere and are altered by eccentricity, inclination, and other perturbers.
Keplerian motion, perturbations, and resonances explain why orbits are structured rather than random.
Periods: timing measurements reveal distance through Keplerian scaling.
Ratios: near-integer period ratios can maintain repeated gravitational kicks.
Eccentricity: small departures from circular motion can power tides and geology.
A resonance is not just a numerical coincidence. It means bodies return to nearly the same relative positions often enough for small gravitational tugs to accumulate rather than average away.
Use the diagram to connect orbit size, period, and timing. The key evidence is not the drawn distance alone; it is the repeated alignment implied by measured periods.
Io, Europa, and Ganymede are close to a 1:2:4 chain using the local resonance data table.
Kepler's law predicts the dominant Sun- or planet-centered trend. Resonances, tides, oblateness, and mutual perturbations explain the interesting departures from the simple two-body model.
Given two moons with periods of 1.37 days and 2.74 days, decide whether the 2:1 ratio is dynamically meaningful. What additional evidence would you want: eccentricity, long-term timing, heat flow, or surface geology?
The resonance table is not only an orbital catalog. It predicts where to look for tidal heating: moons locked in resonant chains with nonzero eccentricity should be compared against heat flow, volcanism, fractures, or plume evidence.