Planet Formation

Topic:

Resonances

Lecture by: C.P. Dullemond

Literature: Murray & Dermott „Solar System Dynamics“

What is a resonance?

• A resonance is when two characteristic frequencies of a system match up

• Typically such a match-up (and even an almost-match-up) has dynamical consequences (causing instability in an otherwise stable system or stability in an otherwise unstable system)

• In planetary systems numerous possible resonances are possible:– Mean motion resonances– Spin-orbit resonances– Secular resonances– ...

Mean motion resonances

Example: Pluto is in 3:2 mean motion resonance with Neptune. Every 3 orbits of Neptune around the Sun, Pluto completes 2 orbits.

Resonant angle

λ1(t)λ2(t)

λ1(t) and λ2(t) are the true anomalies of theplanets 1 and 2. For circular orbits they aresimply: λ1(t)=Ω1t and λ2(t)=Ω2t. For non-circular orbits they are of course non-linearwith time.

Define now an angle θp,q(t) as follows:

This is called the resonant angle, or resonant argument.

If there exists two natural numbers p and q for which the function θp,q(t)remains bound within a range of 2π for all time t, then the two planetsare said to be in (p+q):p mean motion resonance.

Simple example: q=0, Ω1=Ω2

To get a better „feel“ for the concept of resonant angle, let us have a look at special cases. Let us look at the angles for which q=0:

This is the easiest to visualize: It is simply p times the angle betweenthe two position vectors of the planets.

Now let us assume that planets 1 and 2 are on exactly the same circulat orbit, so that they have exactly the same orbital frequency. Butlet them start with a different true anomaly, or in other words: start withan angle α.

So the resonant angle is constant. These two planets are in 1:1 resonance.

Simple example: q=0, Ω1≅Ω2

Now consider the case where planet 1 is slightly inward of planet 2 (bothstill on circular orbits). In this case the planets slowly approach each other.When they come close, they gravitationally „fly by“ each other, putting each other on different orbits. This is very similar to the horseshow librationwe‘ve seen before. Typically these two new orbits are still very close, and the planets will eventually encounter each other again, and the story will repeat itself. In a corotating frame with average rotation frequency it will then look something like this:

This is in fact exactly what happenswith the moons Epimetheus and Janusof Saturn.

time

θp,q(t)

0

2π

After Murray & Dermott

= mm resonance

Simple example: q=0, Ω1≅Ω2

In other words: the gravitational interaction between the two planets(or moons in the case of Epimetheus and Janus) can cause the angleθp,q(t) (in this case θ1,0(t)) to „bounce“ between two limits.

Without the gravitational forces, if Ω1≅Ω2 we would get instead:

time

θp,q(t)

0

2π

4π

Not bound,so noresonance.

Without gravitational forces, we only get a 1:1 resonance if Ω1===Ω2. This isnever exactly the case! So gravity plays a key role in resonances.

Resonance width & LibrationFor circular orbits, the width of a resonance is the maximum differencein semi-major axis (or in other words, maximum difference in Ω1 and Ω2) for which the gravitational forces between the two planets or moons can still keep them in resonance (i.e. keep the function θp,q(t) bound).

In multi-body problems this means that resonances can in factoverlap.

For massless particles the width is 0. The larger the mass of the planetscompared to the star (or moons compared to the planet) the larger thewidth of their resonance.

The oscillating motion of the resonant angle θp,q(t) is called libration. For thecase of small amplitude libration, the angle θp,q(t) obeys a pendulum equation, which for very small amplitudes is like a harmonic oscillator:

Location of p,q-resonancesLocations of the p,q-resonances:

p=1,2,3,4q=0,1,2,3,4,5

Special case: Lindblad resonances (q=1)Locations of the p,1-resonances:

p=1,2,....,10q=1

Special case: Lindblad resonances (q=1)Lindblad resonances play an important role if a planet is in resonancewith a gas flow. Remember this movie from earlier in the lecture?

If the yellow test particle is a fluid element of a protoplanetary disk,then if it is in p+1:p (=Lindblad) resonance with the planet, it will „hop“ right „onto“ the planet and get a next kick. If not, it will „hop over“ the planet and not get a kick. Gas that is on a Lindblad resonance will thusget strongly perturbed: This is another way to explain the spiral wavescausing planet migration. Hence the name „Lindblad torque“. Since gas is a „fluid“ (and not a collisionless system of particles), theq≠1 resonances cannot play a role for gas disks. Only Lindblad.

Pros and cons of resonances

• Resonances can pump eccentricity efficiently. This can lead to:– Dynamic „heating“ of planetesimals by an embryo– Planet migration in the case of Lindblad resonances in

a gas disk– Instabilities in a multi-planetary system

• Resonances can also:– Lock two planets to each other, preventing instabilities– Modify the nature of planetary migration in disks

„Nice“ Model of Late Heavy Bombardment

Gomes, Levison, Tsiganis, Morbidelli (2005)

t=100 Myr t=879 Myr

t=882 Myr t=1100 Myr

Jupiter and Saturnslowly migrate towardtheir mutual 2:1 meanmotion resonancedue to interactions withthe planetesimals.

Once they get in reso-nance, they rapidly shake up the entireouter solar system, sending many cometsto Earth: The „LateHeavy Bombardment“that caused the craterson the Moon.

„Nice“ stands for the city in France where model was designed.

Migration as a way to push planets into resonance

• If planets are not in resonance, it is not easy to put them into resonance (in a perfect 3 body problem it is not even possible).

• But if the system is embedded in a protoplanetary disk, then each planet migrates at it‘s own pace.

• This can lead two planets to move toward each other‘s orbit.

• They can then get „locked in resonance“• Once they are locked, they migrate together, and

this two-planet migration behaves very differently from one-planet migration.

Pushing planets into resonance

G. Bryden

Outward migration of a locked pair

Masset & Snellgrove 2001

„Grand Tack“ Scenario

Walsch, Morbidelli, Raymond, O‘Brien, Mandell (2011)

The Grand Tackmodel of Walschet al employs thispairwise outwardmigration to allowJupiter to migrateinward for a whileand then get „saved“before plunging intothe Sun by beingresonantly capturedby Saturn. The pairthen migrates outward again. Thismight explain the emptiness of theasteroid belt.