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
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Lecture by: C.P. Dullemond
Literature: Murray & Dermott „Solar System Dynamics“
Example: Pluto is in 3:2 mean motion resonance with Neptune. Every 3 orbits of Neptune around the Sun, Pluto completes 2 orbits.
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
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 planets
are said to be in (p+q):p mean motion resonance.
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
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
= mm resonance
After Murray & Dermott
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:
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.
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
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:
Locations of the p,q-resonances:
Locations of the p,1-resonances:
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.
Jupiter and Saturn
slowly migrate toward
their mutual 2:1 mean
due to interactions with
Once they get in reso-
nance, they rapidly
shake up the entire
outer solar system,
sending many comets
to Earth: The „Late
that caused the craters
on the Moon.
„Nice“ stands for the city in France where model was designed.
Gomes, Levison, Tsiganis, Morbidelli (2005)
Masset & Snellgrove 2001
The Grand Tack
model of Walsch
et al employs this
migration to allow
Jupiter to migrate
inward for a while
and then get „saved“
before plunging into
the Sun by being
by Saturn. The pair
outward again. This
might explain the
emptiness of the
Walsch, Morbidelli, Raymond, O‘Brien, Mandell (2011)