Transcript Resonances

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 almostmatch-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) 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 noncircular orbits they are of course non-linear
with time.
λ2(t)
λ1(t)
Define now an angle θp,q(t) as follows:
q p,q (t) = (p + q)l2 (t) - pl1 (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:
q p,0 (t) = p [ l2 (t) - l1 (t)]
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.
q p,0 (t) = p [Wt + a -Wt ] = pa
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)
2π
= mm resonance
0
After Murray & Dermott
time
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:
d 2q p,q (t)
2
=
w
0 sin (q p,q (t))
2
dt
Location of p,q-resonances
Locations of the p,q-resonances:
p=1,2,3,4
q=0,1,2,3,4,5
Special case: Lindblad resonances (q=1)
Locations of the p,1-resonances:
p=1,2,....,10
q=1
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=882 Myr
t=879 Myr
t=1100 Myr
Gomes, Levison, Tsiganis, Morbidelli (2005)
Jupiter and Saturn
slowly migrate toward
their mutual 2:1 mean
motion resonance
due to interactions with
the planetesimals.
Once they get in resonance, 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.
„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
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)