Download Astronomy 311: Lecture 7 - Resonance • Solar System consists of 8

Astronomy 311: Lecture 7 - Resonance
• Solar System consists of 8-9 planets plus their roughly 60 satellites (most
of which were discovered after 2000.
• 10,000 catalgued asteroid orbits, 500 reliable orbits for comets.
• Perhaps some 2 × 108 objects with radii ≈ 10km beyond Pluto but still
under the gravitational influence of the Sun.
• Most of the time, Solar System dynamics, that is how this collection of objects move under the influence of gravity, can be determined by ”repeated”
application of
GM1 M2
.
F =
R2
• But subtle gravitational influences that determine Solar System dynamics
are often determined by resonances.
• A resonance occurs when there is a simple numerical relationship between
frequencies 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 (forces
that produce heat) which are connected to the existence of resonances.
• An example of Spin-Orbit resonance is the Moon: synchronously rotates
with 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 on
an approx 900 year time scale.
• Neptune:Pluto have a 3:2 orbit, orbit resonance: this maximizes their separation at conjunction.
• Many of the smaller Solar System objects orbiting beyond Neptune have
orbital periods very close to Pluto’s and are in a 3:2 resonance with Neptune:
plutinos.
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• Some planets also involved in long term resonances associated with the
precession of the planetary orbits in space.
• Orbit-Orbit resonance amongst Jupiter’s satellites: Io, Ganymede and Europa.
• Io in a 2:1 resonance with Europa, Europa in a 2:1 resonance with Ganymede:
so all three involved in a ”joint resonance”.
• Whats the orbital period of an asteroid orbiting 3.28AU from the Sun?
Compare this to the orbital period of Jupiter.
• So every other orbit such an asteroid would receive a stronger tug from
Jupiter which would change its orbit.
• There are no asteroids orbiting at 3.28AU. Gaps in asteroid belts because of this are called Kirkwood gaps and correspond to orbital periods
1/3, 2/5, 3/7, 1/2, 3/5 that of Jupiter.
• Lagrangian Points
– Kepler and Newton solved the two body problem: two bodies orbiting
around each other.
– Three body problem is not solvable in closed form: need to numerically
integrate.
– 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 are 5 points where the
third body’s motion was predictable:
– L1-L5 Lagrangian points: L4 and L5 are stable equilbrium points.
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