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### 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
- ...

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) and λ2(t) are the true anomalies of the

planets 1 and 2. For circular orbits they are

simply: λ1(t)=Ω1t and λ2(t)=Ω2t. For non-

circular orbits they are of course non-linear

with time.

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

λ2(t)

λ1(t)

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 planets

are 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 between

the 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. But

let them start with a different true anomaly, or in other words: start with

an 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 (both

still 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 libration

we‘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 happens

with the moons Epimetheus and Janus

of Saturn.

θp,q(t)

= mm resonance

2π

0

time

After Murray & Dermott

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 angle

θp,q(t) (in this case θ1,0(t)) to „bounce“ between two limits.

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

4π

Not bound,

so no

resonance.

θp,q(t)

2π

0

time

Without gravitational forces, we only get a 1:1 resonance if Ω1===Ω2. This is

never exactly the case! So gravity plays a key role in resonances.

Resonance width & Libration

For circular orbits, the width of a resonance is the maximum difference

in 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 fact

overlap.

For massless particles the width is 0. The larger the mass of the planets

compared to the star (or moons compared to the planet) the larger the

width of their resonance.

The oscillating motion of the resonant angle θp,q(t) is called libration. For the

case of small amplitude libration, the angle θp,q(t) obeys a pendulum

equation, which for very small amplitudes is like a harmonic oscillator:

Special case: Lindblad resonances (q=1)

Lindblad resonances play an important role if a planet is in resonance

with 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 thus

get strongly perturbed: This is another way to explain the spiral waves

causing planet migration. Hence the name „Lindblad torque“.

Since gas is a „fluid“ (and not a collisionless system of particles), the

q≠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

t=100 Myr

t=879 Myr

Jupiter and Saturn

slowly migrate toward

their mutual 2:1 mean

motion resonance

due to interactions with

the planetesimals.

Once they get in reso-

nance, they rapidly

shake up the entire

outer solar system,

sending many comets

to Earth: The „Late

Heavy Bombardment“

that caused the craters

on the Moon.

t=882 Myr

t=1100 Myr

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

Gomes, Levison, Tsiganis, Morbidelli (2005)

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

The Grand Tack

model of Walsch

et al employs this

pairwise outward

migration to allow

Jupiter to migrate

inward for a while

and then get „saved“

before plunging into

the Sun by being

resonantly captured

by Saturn. The pair

then migrates

outward again. This

might explain the

emptiness of the

asteroid belt.

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

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