Astronomy 311: Lecture 7 - Resonance• Solar System consists of 8-9 planets plus their roughly 60 satellites (mostof which were discovered after 20 00.• 10,000 catalgued asteroid orbits, 500 reliable orbits for comets.• Perhaps some 2 × 108objects with radii ≈ 10km beyond Pluto but stillunder the gravitational influence of the Sun.• Most of the time, Solar System dynamics, that is how this collection of ob-jects move under the influence o f gravity, can be determined by ”repeated”application ofF =GM1M2R2.• But subtle gravitational influences that determine Solar System dynamicsare often determined by resonances.• A resonance occurs when there is a simple numerical relationship betweenfrequencies or periods (period is usually 1/frequency).• Could be the rotational and orbital periods of one body: spin-orbit coupling.• Orbit-Orbit coupling• Many possibilities.• Evolutionary forces in Solar System often driven by dissipative forces (for cesthat produce heat) which are connected to the existence of resonances.• An example of Spin-Orbit resonance is the Moon: synchronously rotateswith the Earth.• Mercury’s orbital period is 87.9 days. Its rotation period is 58.6 days.• Is there a spin-orbit resonance there?• Jupiter/Saturn have a 5:2 ”near resonance” which affects their motion onan approx 900 year time scale.• Neptune:Pluto have a 3:2 orbit, orbit resonance: this maximizes their sep-aration at conjunction.• Many of the smaller Solar System objects orbiting beyond Neptune haveorbital periods very close to Pluto’s and ar e in a 3:2 resonance with Neptune:plutinos.1• Some planets also involved in long term resonances associated with theprecession of the planetary orbits in space.• Orbit-Orbit resonance amongst Jupiter’s satellites: Io, Ganymede and Eu-ropa.• Io in a 2:1 resonance with Europa, Europa in a 2:1 resonance with G anymede:so all three involved in a ”joint resonance”.• Whats the orbital period of an asteroid o rbiting 3.28AU from the Sun?Compare this to the orbital period of Jupiter.• So every ot her orbit such an asteroid would receive a stronger tug fromJupiter which would change its orbit.• There are no asteroids orbiting at 3.28AU. Gaps in asteroid belts be-cause of this are called Kirkwood gaps and correspond to orbital periods1/3, 2/5, 3/7, 1/2, 3/5 that of Jupiter.• Lagrangian Points– Kepler and Newton solved the two body problem: two bodies orbitingaround each other.– Three body problem is not solvable in closed form: need to numericallyintegrate.– But in certain simplified cases, Lagrange solved the 3 body problem:Primary body (greatest mass ie Sun), a Secondary (intermediate mass,Earth) and a thord small object (Moon) there ar e 5 points where thethird body’s motion was predictable:– L1-L5 Lagrangian points: L4 and L5 are stable equilbrium