Astronomy 161: An Introduction to Solar System Astronomy Prof. Richard Pogge, MTWThF 2:30

# Lecture 21: Dance of the Planets

## Key Ideas

Every object in the Solar System feels gravitational pulls from all of the other objects in the Solar System.

The Three-Body Problem

• Lagrange Points
• Examples: Earth-Moon and Sun-Jupiter systems
Gravitational Interactions
• Long-range Perturbations (discovery of Neptune)
• Close Encounters (slingshot effect)

Resonances

• Orbital (Mean-Motion) Resonances

## Beyond Kepler

Newton's formulation of Kepler's Laws of Planetary Motion is only strictly true for idealized systems with only 2 massive bodies.

But, the Solar System is a many-body system:

• 8 Planets
• Multiple Moon Systems around 6 planets
• Millions of small Asteroids and Icy Bodies
• Lots more smaller debris (meteors & comets)
How do we address this many-body problem?

## The Three-Body Problem

Let's start relatively simple:

What is the orbit of a small body in the combined gravitaional field of two larger objects orbiting each other?

Some examples:

• Spacecraft in the Earth-Moon system
• An asteroid orbiting the Sun near Jupiter
A formal solution of the problem was proposed by French mathematician and physicist Joseph-Louis Lagrange (1736-1813)
• Solved the "Restricted 3-Body Problem", in which the third body has a negligible mass compared to two more massive bodies in circular orbits around their common center of mass.
• Found 5 "islands of equilibrium" that orbit in lock-step with the two main bodies.
These are known as the Lagrange Points, labelled L1 through L5.

Earth-Moon Lagrange Points

Earth-Moon Lagrange Points

L4 and L5 are stable:
Can have objects trapped in stable "tadpole" orbits.

L1 through L3 are unstable:
Objects at these points are easily nudged out of their orbits and drift away (no long-term orbits without special circumstances, like firing engines for "station-keeping maneuvers").
[21.1]

Jupiter Trojan Asteroids

An example of objects trapped in stable orbits at the L4 and L5 Lagrange points are the Trojan Asteroids of Jupiter. These are two families of asteroids that that follow and lead Jupiter around the Sun as part of the Sun-Jupiter system.

Jupiter Trojan Asteroids.

## Gravitational Interactions

To a first approximation, the orbits of most objects around the Sun are simple 2-body Keplerian orbits:
• The Sun is ~1000x more massive than all of the planets, moons, and small bodies combined.
• The Solar System (today) is mostly empty, with large distances between big objects.
Since the Gravitational Force gets weaker as the inverse square of the distance between objects:
• Gravitational pulls from the other bodies are typically small compared to the Sun's gravity.
• But, they can accumulate over time...
• or, they can be strong if there are chance close encounters.

## Long-Range Perturbations

Long-range interactions between two massive bodies orbiting the Sun.

The extra object-to-object gravitational forces accelerate the bodies relative to their Keplerian orbits

• The orbits are perturbed from simple Keplieran paths
• Effect is strongest with the two objects are lined up at opposition (closest approach to each other).
• Less massive objects are accelerated more.
• The perturbations can accumulate over time, though generally they mostly cancel out.
Any systematic deviation of a body's orbit from a simple Keplerian path is usually a sign that the body is being perturbed by the gravity of another, nearby object.

## The Discrepant Orbit of Uranus

William Herschel accidentally discovered the planet Uranus in 1781 while sweeping the sky with his telescope in his backyard in Bath, England.

Uranus is the 7th planet, orbiting beyond Saturn.

Subsequent measurements of the orbit of Uranus started showing systematic discrepancies between the predicted and actual positions of Uranus in the sky. By the 1840s, these discrepancies had become as large as 1 arcminute!

• Uranus was accelerating in its orbit because of the combined gravity of other massive objects in the Solar System.
The problem was, if you added up the perturbations from the known planets, it wasn't enough to explain the discrepancy seen.
• Extra acceleration was from the gravity of an 8th, as yet unknown massive planet orbiting somewhere beyond Saturn!

## The Discovery of Neptune

Two theorists, Urban LeVerrier in France and John Couch Adams in England, predicted that the deviations were due to the gravitational influence of another, unknown massive planet beyond Uranus.

Using the deviant motions of Uranus, they independently calculated where this unknown 8th planet should be.

• Adams was ignored by English astronomers.
• Leverrier convinced Galle in Berlin to search.

On Sept 23 1846, Galle found Neptune only 52 arcminutes from where Leverrier predicted it would be!

This was possible because between Uranus' discovery in 1781 and the 1840s, Neptune passed through opposition with Uranus, when the perturbation of Uranus by Neptune's gravity is strongest. If their configuration had been conjunction, there would not have had a measurable perturbation. Neptune would have eventually been discovered by accident like Uranus, in fact Galileo saw it while observing Jupiter but thought it was a fixed star! [21.2]

## Close Encounters

Close encounters between objects have much stronger effects:
• They can dramatically alter the orbits of one or both bodies.
• Smaller objects are more strongly affected by close encounters with larger bodies.
Examples:

Short Period Comets
An incoming comet on a large elliptical or near-parabolic orbit with a very long period is driven into a smaller elliptical orbit with a shorter period after a close gravitational encounter with Jupiter.

The dashed path is the orbit the comet would follow if it had no Jupiter encounter.
The solid line show the new, smaller ellitpical comet orbit after a close gravitational encounter with Jupiter,

Gravitational Slingshot
A spacecraft catches up with Jupiter from behind, and is accelerated by Jupiter's strong gravity. This, combined with Jupiter's orbital motion, give the spacecraft a boost in speed, slinging it into the outer solar system.

This method of "gravity assist" is used to send spacecraft into the outer Solar System by robbing Jupiter of a tiny bit of its orbital energy. This is much more energy efficient than having to carry a very heavy payload of fuel to rocket the spacecraft to those speeds.

This was done by all outer Solar System explorers (Voyager 1 & 2, Pioneer 10 & 11, Cassini, and New Horizons).

Above is the Cassini gravity assist trajector to Saturn. Boosts from Venus, Earth, and Jupiter were used to get Cassini to Saturn with minimal expenditure of fuel (and so less spacecraft weight).

The reverse process, where the encounter occurs with the spacecraft getting in front of the planet, can be used to take energy from the spacecraft, dropping it deeper into the inner Solar System. This is done for spacecraft bound for Mercury like MESSENGER.

## Orbital Resonances

Small perturbations at opposition usuall happen at different places along the object's orbit.
• The combined effects tend to average out over long times.
But, if the periods of the object and its perturber are whole-number ratios, you can get regular, periodic pertubations at the same place in the orbit.
• The repeated accelerations add up over time and amplify.
• If there are regular close encounters, the smaller object can get destablized over time and cleared out of its original orbit.
• If the synchronization avoids close encounters, the orbit can be stabilized.
We say that such synchronized pairs of orbits are Orbital Resonances.

The analogy is to consider a child on a swing being pushed another person:

If the pushes of the child come at random times, sometimes pushing with their swing and boosting them, other times pushing against their swing and slowing them down, they average out and the swing doesn't change much.

However, if the pushes are all timed just right so that you push the child with their swing each time, the in-phase pushes build up and the child's swing gets amplified.

Such a well-timed push is called exciting a resonance.
Orbital resonances are a way to amplify small long-range gravitational perturbations.

## Naming Resonances

Resonances are named for the number of orbits completed by each body in the pair.
• The first number is the number of orbits completed by body being most perturbed.
• The second number is the number of orbits completed in the same time by the perturber
Examples:
Pluto is in a 2:3 Resonance with Neptune
Pluto completes 2 orbits for every 3 orbits of Neptune.
Pluto is the smaller object being driven by Neptune.

Asteroid Hilda is in a 3:2 Resonance with Jupiter
Hilda completes 3 orbits for every 2 orbits of Jupiter.
Hilda is the smaller object being driven by Jupiter.

## Resonances in the Solar System

Some examples of important orbital resonances in the Solar System:
Main Asteroid Belt Resonances with Jupiter
Kirkwood Gaps - unstable resonances cleared of asteroids
Asteroid Families - stables resonances populated with asteroids orbiting with the same periods.
Trojan Asteroids - 1:1 resonance with Jupiter

Kuiper Belt Resonances with Neptune
Pluto and the Plutinos in 2:3 resonance orbits
Twotinos - objects in 1:2 resonance orbits

Jupiter and Saturn Systems
Jupiter: 1:2:4 Laplace Resonance of the moons Io, Europa, and Ganymede
Saturn: many resonant moons and resonant gaps in the rings.
We will meet each of these as we begin our exploration of the Solar System in upcoming lectures.

## Dynamical Evolution (again)

The Solar System is not a "static" clockwork that moves exactly the same forever
• The dynamical state of the Solar System changes over time as the planets interact via their mutual gravitation.
Close Encounters and Resonances amplify these changes:
• Close encounters re-arrange orbits, and can even kick small bodies out of the Solar System.
• Some resonances destabilize small-body orbits, depopulating some orbits (Kirkwood Gaps in the Asteroids, Ring Gaps around Saturn)
• Some resonances are stable, and objects swept into them can form distinct dynamical "families" of objects.
All of these effects have helped "shape" the Solar System over its long history.

As we explore the Solar System in more detail later in the course, we will keep on the lookout for signs of this "dynamical evolution", and use it to help read the dynamical history of the Solar System.

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Updated: 2016 March 30