Download Evolution of eccentricity and inclination of migrating planets in the 2

E VOLUTION OF ECCENTRICITY AND ORBITAL INCLINATION OF
MIGRATING PLANETS IN 2:1 MEAN MOTION RESONANCE
Jean Teyssandier & Caroline Terquem
Institut d’Astrophysique de Paris & University of Oxford
A BSTRACT
We determine, analytically and numerically, the conditions needed for a system of two migrating planets trapped in a 2:1 mean motion resonance to
enter an inclination-type resonance. We provide an expression for the asymptotic equilibrium value that the eccentricity of the inner planet reaches
under the combined effects of migration and eccentricity damping. We also show that, for a ratio of inner to outer masses below unity, the inner
eccentricity has to pass through a value of order 0.2-0.3 for the system to enter an inclination-type resonance. Numerically, we confirm that an
inclination-type resonance may also be excited at another, larger, eccentricity of 0.6, as found by previous authors. We find that, for a mass ratio below
unity, the system cannot enter an inclination-type resonance if the ratio of eccentricity damping timescale to migration timescale is smaller than 0.2.
As the disc/planet interaction is characterized by such ratios to be of the order of 10−2, we conclude that excitation of inclination through the type of
resonance described here is very unlikely to happen in a system of two planets migrating in a disc.
I NTRODUCTION
I LLUSTRATIVE CASES
As of today, 98 extrasolar multiple planet systems
have been detected by radial velocity surveys and
420 by the Kepler mission. A significant fraction
of these systems contain planet pairs in or near a
2:1 mean motion resonance.
Capture in mean motion resonance is thought
to be the result of convergent migration of planets. The evolution of planetary systems in mean
motion resonance has been studied with three–
dimensional simulations [e.g., Thommes and Lissauer, 2003]. It was found that a system of planets in an eccentricity–type resonance may enter
an inclination–type resonance if the eccentricity
of the inner planet becomes large enough. In this
context, high orbital inclinations can be reached
starting from nearly coplanar configurations.
Studies of resonant inclination excitation are important as this mechanism has been proposed
to explain the fact that some extrasolar planets
have an orbit which is inclined with respect to the
stellar equatorial plane.
In Teyssandier and Terquem [2014], we derive
analytically a necessary condition for the onset of
inclination–type resonance. The analysis has to
be done to second order in eccentricities and inclinations. We also perform numerical simulations
to investigate the regime of high eccentricities.
Left column, from top to bottom: Semi–major axes (in AU), eccentricities and inclinations of the two planets, ∆ϖ = ϖi − ϖo and ∆Ω = Ωi − Ωo versus time (in
years). Right column, from top to bottom: resonant angles ϕ1, ϕ2, ϕ6, ϕ7 and ϕ8 †† versus time (in years). All the angles are given in degrees. In the plots
displaying the eccentricities and inclinations, the upper curves represent e i and I i, respectively, whereas the lower curves represent e o and I o, respectively.
R ESONANT M IGRATION
We are interested in the case where convergent
migration of two planets has led to capture into
a 2:1 mean motion resonance. The inner planet
has a mass m i , and the outer one a mass m o.
We define q = m i /m o and α = a i /a o, the constant
semi-major axis ratio. The orbital evolution of
each planet is a combination of disc-planet interactions (in the form of orbital migration and
eccentricity damping) and planet–planet interactions (in the form of resonant and secular interactions). The evolution of semi-major axis and
eccentricity is :
0
time (years)
0
0
time (years)
time (years)
Evolution of a system in 2:1 mean motion resonance with q = 0.7 and t e /t a = 0.25. Shortly
after the beginning of the simulation, the planets are captured into an eccentricity–type resonance: e i grows until it reaches e i,eq = 0.2
and ∆ϖ, ϕ1 and ϕ2 librate about 0◦. After t ≃
8 × 105 years, e i starts decreasing while e o gets
larger, and the value about which ϕ2 librates
switches rather abruptly to 180◦. Throughout
the evolution, e i stays too small to allow for
inclination–type resonance. Accordingly, the
inclinations remain small, the resonant angles
ϕ6, ϕ7 and ϕ8 behave chaotically and ∆Ω = 0
throughout the evolution.
Here t a and t e are the migration and eccentricity
damping timescales. Migration is only applied to
the outer planet. The resonant part is modelled
by a disturbing function which is expanded to
second-order in eccentricity and inclination [Murray and Dermott, 2001]. From these equations we
find that the eccentricities are excited by the resonant migration, but are damped by the planetdisc interactions. The inner planet’s eccentricity
reaches the equilibrium value
(
1
t e /t a
e i,eq =
p
2 1+ q α
time (years)
Evolution of a system in 2:1 mean motion resonance with q = 0.7 and t e /t a = 0.8. Here again,
shortly after the beginning of the simulation,
the planets are captured into an eccentricity–
type resonance: e i grows until it reaches e i,eq =
0.35 and ∆ϖ, ϕ1 and ϕ2 librate about 0◦. After
t ≃ 3.2 × 106 years, e i starts decreasing while
e o gets larger, and the value about which ϕ2
librates switches to 180◦. When e i ≃ 0.3, an
inclination–type resonance starts to develop.
Accordingly, the resonant angles ϕ6, ϕ7 and ϕ8
start librating about 180◦, 0◦ and 180◦, respectively, while ∆Ω librates about 180◦. The inclinations grow quickly.
10
)
2a i e2i
da i
da i
=
−
,
dt
dt res
te
(
)
da o
a o 2a o e2o
da o
=
− −
,
dt ( dt ) res t a
te
ei
de i
de i
=
− ,
dt ( dt )res t e
eo
de o
de o
=
− .
dt
dt res t e
0
0
time (years)
time (years)
Evolution of a system in 2:1 mean motion
resonance with q = 0.7 and t e /t a = 4. Here
again, shortly after the beginning of the simulation, the planets are captured into an
eccentricity–type resonance and e i and e o grow.
In the present case, ϕ1 and ϕ2 librate about 0◦
throughout the simulation. When e i ∼ 0.6, the
system enters an inclination–type resonance
and ∆Ω, ϕ6, ϕ7 and ϕ8 start librating about
180◦, 180◦, 0◦ and 180◦, respectively, while the
inclinations start growing.
†† ϕ1 = 2λo − λi − ϖi, ϕ2 = 2λo − λi − ϖo, ϕ6 = 4λo − 2λi −
2Ωi, ϕ7 = 4λo − 2λi − Ωi − Ωo, ϕ8 = 4λo − 2λi − 2Ωo.
PARAMETER S PACE
τe / τa
(
0
1
0.1
0.1
1
q = mi / mo
no I-res
I-res, φ2 = 180
I-res, φ2 = 0
I-res, φ2 = 180, tres > 3 Myr
I-res, φ2 = 0, tres > 3 Myr
This plot shows the occurrence of inclination–type resonance as a function of q = m i /m o and t e /t a in systems
evolved between 0 and 6 Myr. Crosses represent systems that did not enter an inclination–type resonance,
open and filled symbols represent systems that entered
an inclination–type resonance with (ϕ1, ϕ2) = (0◦, 180◦) or
(ϕ1, ϕ2) = (0◦, 0◦), respectively. Circles and triangles represent systems that entered an inclination–type resonance before or after t = 3 Myr, respectively.
The dashed
(
)
p
lines are the curves t e /t a = 4e2i,res 1 + q α with e i,res =
0.3 (lower line) and e i,res = 0.57 (upper line). In agreement with theoretical expectations, systems in between
those two lines entered an inclination–type resonance
with (ϕ1, ϕ2) = (0◦, 180◦) if evolved long enough, whereas
systems above the upper line entered an inclination–
type resonance with (ϕ1, ϕ2) = (0◦, 0◦).
Hence, it clearly appears that if t e /t a ≲ 0.2, an
inclination–type resonance cannot be achieved. For
mass ratios of unity and larger, t e /t a has to be of order
of unity, at least, for an inclination–type resonance to
occur.
)1/2
.
An inclination–type resonance can be triggered
when the nodes precess at the same rate, and
match the periaspis precession. It imposes a condition on the inner eccentricity. We find that for
most mass ratios, e i,res ≃ 0.3. An inclination–type
resonance will be triggered if e i,res < e i,eq, i.e.,
t e /t a
p > 4e2i,res.
1+ q α
R EFERENCES
C ONCLUSIONS
C.D. Murray and S.F Dermott. Solar System
Dynamics. Cambridge University Press,
2001.
• We have shown that, for a system of two resonant migrating planets to enter an inclination–type resonance,
the eccentricity of the inner planet has to reach a value e i,res ≃ 0.2 − 0.3.
J. Teyssandier and C. Terquem. . Submitted
to MNRAS, 2014.
E. W. Thommes and J. J. Lissauer. . ApJ, 597:
566–580, 2003.
• A necessary condition for the system to enter an inclination–type resonance is that e i,res < e i,eq. This leads to
a condition on the ratio of the eccentricity damping timescale to the migration timescale, t e /t a, as a function
of the mass ratio. For q ≤ 1, we find that the system cannot enter an inclination–type resonance if t e /t a < 0.2.
• For a wide range of planet masses, the eccentricity damping timescale due to planet/disc interaction is
of a few 103 years. As type I and type II migration timescales are on the order of 105 years, it implies
that t e /t a ∼ 10−2, too low for the value needed for e i,res to be reached. We conclude that it is unlikely that
misaligned systems formed via this mechanism.