Issue |
A&A
Volume 517, July 2010
|
|
---|---|---|
Article Number | A31 | |
Number of page(s) | 7 | |
Section | Planets and planetary systems | |
DOI | https://doi.org/10.1051/0004-6361/201014072 | |
Published online | 28 July 2010 |
Formation of the resonant system HD 60532
Zs. Sándor1 - W. Kley2
1 - Max Planck Research Group at the Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany
2 -
Institut für Astronomie und Astrophysik, Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany
Received 15 January 2010 / Accepted 20 March 2010
Abstract
Context. Among multi-planet planetary systems there are a
large fraction of resonant systems. Studying the dynamics and formation
of these systems can provide valuable informations on processes taking
place in protoplanetary disks where the planets are thought have been
formed. The recently discovered resonant system HD 60532 is the
only confirmed case, in which the central star hosts a pair of giant
planets in 3:1 mean motion resonance.
Aims. We intend to provide a physical scenario for the formation
of HD 60532, which is consistent with the orbital solutions
derived from the radial velocity measurements. Observations indicate
that the system is in an antisymmetric configuration, while previous
theroretical investigations indicate an asymmetric equilibrium state.
The paper aims at answering this discrepancy as well.
Methods. We performed two-dimensional hydrodynamical simulations
of thin disks with an embedded pair of massive planets. Additionally,
migration and resonant capture are studied by gravitational N-body simulations that apply properly parametrized non-conservative forces.
Results. Our simulations suggest that the capture into the 3:1
mean motion resonance takes place only for higher planetary masses,
thus favouring orbital solutions having relatively smaller inclination
.
The system formed by numerical simulations qualitatively show the same
behaviour as HD 60532. We also find that the presence of an inner
disk (between the inner planet and the star) plays a very important
role in determining the final configurations of resonant planetary
systems. Its damping effect on the inner planet's eccentricity is
responsible for the observed antisymmetric state of HD 60532.
Key words: planets and satellites: formation - celestial mechanics - hydrodynamics - methods: numerical
1 Introduction
A significant fraction of multi-planet planetary systems contain a pair of giant planets engaged in a mean motion resonance (MMR). These planets are mainly in the low order 2:1 MMR (as GL 876, HD 128311, and HD 73526), but in a few systems higher order resonances have been suggested as well; two planets around 55 Cancri may be in 3:1 MMR, or HD 202206 may host a pair of planets in 5:1 MMR. In the recently discovered system, HD 45364, the giant planets are revolving in 3:2 MMR Correia et al. (2009).
The observed orbital solutions and the formation of the majority of resonant systems have been studied thoroughly by many authors. It has been shown, for instance, by Kley et al. (2004) that a sufficiently slow migration process of two giant planets embedded in an ambient protoplanetary accretion disk ends with either a 3:1 or 2:1 resonant configuration depending on the speed of migration. The formation of resonant systems in 2:1 MMR has been modelled exhaustively by hydrodynamical and N-body simulations as well. The system GJ 876 has been investigated by Lee & Peale (2002), Kley et al. (2005), Crida et al. (2008), and the systems HD 128311 and HD 73526 by Sándor & Kley (2006) and Sándor et al. (2007), respectively. Most recently, the formation of the system HD 45364 with planets in the 3:2 MMR has been investigated by Rein et al. (2010). Analytical studies related to the stationary solutions of the 2:1 and 3:1 MMR have been done by Beaugé et al. (2003) and Beaugé et al. (2006), for instance.
Regarding the 3:1 MMR case, the presence of the planet 55 Cancri-c has already been questioned (Naef et al. 2004). Additionally, recent orbital fits indicate that the planets 55 Cancri-c and 55 Cancri-d are not in a resonant configuration (Fischer et al. 2008). Thus, until the recent discovery of the two planets in a 3:1 MMR around the F-type star HD 60532 by Desort et al. (2008), there has been a serious lack of knowledge about the observed behaviour of 3:1 resonant systems. The results of the dynamical study performed by Desort et al. (2008) did not prove without any doubt that the giant planets are in fact in a 3:1 MMR.
The final confirmation of the 3:1 MMR between the giant planets in HD 60532 is given in a recent paper by Laskar & Correia (2009), in which two new orbital fits are provided, slightly improving on the previous fit of Desort et al. (2008) with
.
Through a detailed stability analysis of different orbit integrations, Laskar & Correia (2009) suggest that an inclination of
is the most likely configuration for a co-planar model. By this assumption, the planetary masses are increased by a factor of
in comparison
to Desort et al. (2008). In a recent paper, Libert & Tsiganis (2009) analyse the excitation of mutual inclination for planetary systems driven into resonance by planet-disk interaction.
They use a damped N-body
evolution and find that for stronger eccentricity damping, the mutual
inclination is less excited. However, they could not place any
constraints on the observed inclination of the system HD 60532.
The very large difference in planetary masses between the two orbital solutions having
or
also
raises the important question of which orbital solutions are preferred
by formation based on the planetary migration scenario. In the present
paper we aim at answering this question by performing fully
hydrodynamical evolution of planets embedded in the disk. Through this
procedure, we obtain realistic migration and eccentricity damping rates
that will allow us to determine the most probable final state of the
system.
The paper is organized as follows. First, we numerically integrate the orbits of giant planets using the two sets of orbital solutions given by Laskar & Correia (2009) as initial conditions. Then by considering different planetary masses, we investigate the possible capture into the 3:1 MMR between the giant planets. After having formed a resonant system, we compare the results of our simulations to the orbital behaviour of giant planets obtained from numerical integrations of Sect. 2. Finally, by performing gravitational three-body numerical integration with dissipative forces for migration, we study how an inner disk influences the behaviour of the system toward its stationary solutions. We demonstrate that the presence of the inner disk determines the final resonant configuration of the system.
2 Orbital behaviour of the system HD 60532
In this section we present the orbital behaviour of the giant planets around the star HD 60532. To integrate the equations of the three-body problem, we used a Bulirsch-Stoer numerical integrator. The initial conditions for our simulations were calculated from the orbital solutions given by Laskar & Correia (2009), see Fits I and II in Table 1.Table 1: Orbital solutions provided by Laskar & Correia (2009).
A resonant system can be characterized by studying the behaviour of the resonant angles. According to Murray & Dermott (1999), in the planar case of the 3:1 MMR, there are three different resonant angles:
,
,
and
,
where
denotes the mean longitude of the ith planet, and
its periastron. Here index ``1'' stands for the inner and index ``2''
for the outer planet. By introducing the corotational angle
and arbitrarily choosing a
,
all the remaining resonant angles can be expressed with
and
.
If one of
(i=1,2,3) librates around a constant value, the system is in a 3:1 MMR. If
also librates, the system is said to be in apsidal corotation.
By studying Fig. 1,
one can follow the behaviour of the eccentricities and resonant angles
as functions of time for Fit II. The eccentricities show quite
large oscillations, the inner planet's eccentricity varies between
,
while the outer planet's eccentricity is between
.
All resonant angles
are librating,
around
,
,
,
and
around
.
This means that the system is in 3:1 MMR and also in apsidal
corotation, and furthermore the orbits of the giant planets are
anti-aligned. In Fig. 1 we display the behaviour of the resonant angle
,
(as used by Laskar & Correia 2009) and
.
By using Fit I, we found the same dynamical behaviour of the giant planets as Laskar & Correia (2009), therefore we do not display these results here.
![]() |
Figure 1:
Dynamical behaviour of the system HD 60532, where the initial
conditions of the numerical integration are calculated from Fit II
of Laskar & Correia (2009).
The upper panel shows the evolution of the eccentricities (inner planet
red curve, outer planet green curve), middle panel the behaviour of the
resonant angle |
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Regarding the behaviour of the resonant angles and the possible apsidal
corotation, a comprehensive analytical study of the 3:1 MMR has been
done by Beaugé et al. (2003). The authors find that the resonant angle
and
both librate around
if the eccentricity of the inner planet is below a certain limit:
.
Above this limit (i.e. e1>0.13),
there appears asymmetric libration of the resonant angles, whose mean
values depend on the eccentricity of the inner planet. This result has
also been confirmed by Kley et al. (2004), where the mean value of the corotation angle was found to be
.
Returning to the behaviour of the system HD 60532 (obtained from
radial velocity measurements), although the inner planet's eccentricity
is higher than the limit e1=0.13, the corotation angle does not yield any asymmetric libration; on the contrary, it librates around
,
see the bottom panel of Fig. 1. The reason for this behaviour might be that, for some reason, the system could not reach the stationary solution found by Beaugé et al. (2003) during the planetary migration. A possible explanation of this phenomenon will be presented in Sect. 5.
In what follows, we attempt to reproduce the dynamical behaviour of the system shown in Fig. 1 by a migratory formation scenario using hydrodynamical and gravitational 3-body simulations with properly parametrized non-conservative forces for migration.
3 Numerical setup of hydrodynamical simulations
To study the formation of the resonant system HD 60532 through migration of the giant planets, we performed a series of full hydrodynamical simulations of a protoplanetary accretion disk containing two embedded giant planets. For the hydrodynamical simulations we used the locally isothermal version of the FARGO public hydrodynamical code (Masset 2000) and the code RH2D (Kley 1999) both well-suited to investigating planet-disk interactions.For our simulations, we used a flat accretion disk with an aspect ratio H/r=0.05 extending between
and
in dimensionless units. The disk has an initial surface density
profile
,
where the dimensionless surface density value at one distance unit is
.
This value corresponds to 1% of the stellar mass
.
During our simulations, the accretion of the gas in the protoplanetary disk was driven by an alpha-type viscosity with
.
A torque cut-off around the planet of
was also applied. The computational domain (bounded by
and
)
was covered by 256 radial and 500 azimuthal gridcells. The radial
spacing is logarithmic, while the azimuthal is equidistant, resulting
in nearly quadratic gridcells (meaning that at each radius
,
approximately).
The planets were initially placed at r=1.0 and 2.5 distance units from the star. To avoid the transient effects and obtain a quasi steady state initial setup, we first integrated the disk-planets-star system keeping the orbits of giant planets fixed on circular orbits until 500 orbital periods of the inner planet. The surface density distribution of the accretion disk obtained after the above integration time is shown in Fig. 2, where the planetary masses correspond to the higher masses of Fit II, and we used a logarithmic scale for the surface density values for better visibility.
![]() |
Figure 2: The surface density distribution of the protoplanetary accretion disk after 500 periods of the inner giant planet. The disk model is described in Sect. 3, the planetary masses correspond to Fit II. Light colours are for higher, while dark colour shades are for low values of the surface density. For a better visualization we used a logarithmic scale. |
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There is an essential difference between our hydrodynamical setup and the one used by Kley et al. (2004) (Model C in the cited paper). In the latter model, the initial surface density profile tends to zero with
,
because the inner disk was assumed to be cleared by the accretion of
the gas to the star. In such a case, the outer planet feels the
negative torques of only the outer disk and migrates inward until it
captures the inner planet into the 3:1 MMR, which revolves in a
practically gas-free environment. On the other hand, as demonstrated
recently by Sándor et al. (2007) and Crida et al. (2008),
the presence of the inner disk plays a very important role in damping
the eccentricity of the inner planet during the migration process,
hence in shaping the final behaviour of the system. Therefore in our
disk model we did not allow the inner disk to be emptied, so the
planets (mainly the inner one) can feel its effect.
Having obtained almost cleared gaps after 500 orbital revolutions
of the inner planet, the planets were released and their orbital
evolution was followed until 4000 periods of the inner planet. For the
planetary masses we used the values published by Laskar & Correia (2009) (see Fits I and II), and also an intermediate value calculated for
.
4 Resonant capture for different masses of the giant planets
In this section we show the results of our hydrodynamical simulations obtained by using different planetary masses.First we investigated the behaviour of the giant planets having low masses, corresponding to the Fit I, where
.
In this case
,
corresponding to
q1=0.0007 and
q2=0.00165 dimensionless mass units (the stellar mass
is unity). We used index ``1'' for the inner and index ``2''
for the outer planet. After releasing them, however, the planets did
not show a convergent migration, and no resonant capture took place.
This was also true for the intermediate value of the inclination
,
where the planetary masses are a factor of two higher than in Fit I.
Finally, we used planetary masses corresponding to Fit II, where
,
and the planetary masses are
and
or, in dimensionless mass units,
q1=0.0021 and q2=0.005. (These
values are quite close to the planetary masses used by Kley et al.
2004.) In this case the giant planets also begin to migrate, but
contrary to the previous two cases, the outer planet migrates faster
than the inner one resulting in a resonant capture.
Before describing the dynamical behaviour of the system after the
capture into the resonance, we should comment on why this phenomenon
does not occur in the cases of lower mass planets. We recall that for a
resonant capture the planets should exhibit convergent migration.
Consequently, if the planets migrate inward with the same speed, or if
the inner planet migrates even faster than the outer one, no resonant
capture will occur. For example, if the inner planet does not open a
sufficiently empty gap, it may migrate faster than a usual type II
migration. It is also true that the inner planet's inward migration is
mainly governed by the torques coming from the middle part of the disk,
which is between the planets. (The outer disk does not really exert
torques to the inner planet because it is separated by the wide gap
opened by the outer planet.) On the other hand, the inward migration of
the inner planet can be stopped and even reversed by a sufficiently
massive inner disk (Crida et al. 2008; Sándor et al. 2007).
From our hydro simulations one can see that only the massive inner
planet of Fit II can open a sufficiently deep gap for a slower
(type II) migration, and could also create a massive inner disk as
pushing the disk's material toward the star. The planets with masses
from Fit I and the intermediate case (with
)
open a shallower gap, the middle disk is more massive, and the inner
disk is less massive for planets having masses from Fit II. This
can be checked in Fig. 3,
where we display the azimuthally averaged surface density profiles of
the disk for the different planetary masses. Thus, as can be seen in
the upper panel of Fig. 4,
the migration of the inner planet (of Fit II) is slow in the
beginning and is stopped after a certain time, and right before the
resonant capture (at
)
its inward migration is even slightly reversed, which is not seen in
the figure. The action of the inner disk means that we might expect a
slow outward migration of the inner planet (Crida et al. 2008).
However, this does not play any role in the resonant capture in our
case, as it is the outer planet that migrates towards the inner one.
![]() |
Figure 3: The azimuthally averaged surface density profile of the protoplanetary disk after 500 periods of the inner giant planet for different planetary masses. Dashed line corresponds to planetary masses of Fit I, dotted line for the intermediate values, which are twice the masses of Fit I, and solid line for masses of Fit II. |
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![]() |
Figure 4:
Dynamical behaviour of the two giant planets embedded in the
surrounding protoplanetary disk: semi-major axes and eccentricities.
During the first 500 orbital periods (of the inner planet), the
planets are kept fixed to obtain a steady state in the disk. The upper panel shows the behaviour of the semi-major axes, while the bottom panel
shows the time evolution of the eccentricities (red line corresponds to
the inner, green line to the outer giant planet). The capture into the
3:1 MMR occurs around 2300 periods of the inner planet. The
planetary masses (
|
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If the effect of the inner disk is not taken into account, the inward migrating outer planet always captures the inner planet into an MMR. In this case, the results of Kley et al. (2004) clearly demonstrate that the final system is either in 3:1 or in 2:1 MMR mainly depending on the speed of the outer planet's migration. A fast migration of the outer planet may result in its crossing the 3:1 MMR without capture and ending the migration in the more robust 2:1 MMR.
In the case of the large planetary masses, around 2300 periods of the inner planet, the planets enter into the 3:1 MMR. The evolution of the semi-major axes and eccentricities during the migration and after the resonant capture is shown in Fig. 4. After the resonant capture, the eccentricities first increase and then tend to oscillate (around their mean values). However, the oscillations of the eccentricities remain bounded approximately in the range given by the numerical integration shown in Fig. 1. The inner disk certainly does contribute to the apparently very effective damping of the inner planet's eccentricity, as shown in the formation scenarios of the resonant systems HD 73526 and GJ 876 by Sándor et al. (2007) and Crida et al. (2008).
According to our simulation, the resonant angles
(i=1,2,3) and the corotation angle
show libration,
librates around
,
and the remaining angles around
,
see Fig. 5. The dynamical behaviour of our modelled system is qualitatively the same as the observed one, displayed in Fig. 1, even though in our case the libration amplitudes are somewhat smaller.
![]() |
Figure 5:
Dynamical behaviour of the two giant planets embedded in the
surrounding protoplanetary disk for Fit II. The evolution of the
resonant angle |
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Interestingly though, the behaviour of the resonant angles does not match the theoretical results
of Beaugé et al. (2003) and the numerical simulations done by Kley et al. (2004).
Although the eccentricity of the inner planet exceeds the limit of
,
the libration of
and
is found to be around
.
As already mentioned, this might be the consequence of the system not
yet reaching a stationary solution. In what follows, we study this
unexpected result through numerical integration of the gravitational
three-body problem using properly parametrized non-conservative forces
for migration.
5 Toward the stationary solutions of the 3:1 MMR in HD 60532
In the previous section we presented a reasonably well-working formation scenario for the system HD 60532. The eccentricities of the giant planets and resonant angles (including the corotation angle) of the formed system behave very similarly to the results of numerical integration using initial conditions from Fit II. In this section, we intend to explain why the observed and the formed systems do not fit the previous studies of Beaugé et al. (2003) and Kley et al. (2004). To do so, we performed two series of simulations; in the first one, only the outer planet is forced to migrate inward, while the inner planet does not feel any additional damping forces. In the second one, the inner planet feels the (accelerating) effect of the inner disk as well.The additional non-conservative force responsible for migration and
eccentrity damping can be parametrized by the migration rate
and the eccentricity damping rate
,
or by the corresponding e-folding times
and
of the semimajor axis and eccentricity of the outer planet (see Lee & Peale 2002; Beaugé et al. 2006, for two possible approaches). In our simulations, we used the force corrections as suggested by Lee & Peale (2002). The relations between the damping rates and e-folding times are
,
and similarly for the eccentricities
.
One can define the ratio between the
e-folding times
(or
), which according to Lee & Peale (2002) determines the final state of the system in the case of a sufficiently
slow migration.
If the inner planet is influenced by an inner disk, it may be forced to
migrate outward and its eccentricity is also damped. This effect can
also be modelled by using a (repelling) non-conservative force
parametrized by
(having a positive sign for outward migration), and
for the eccentricity damping. A detailed study of the resulting
dynamical effect of these parametrizations has been presented recently
by Crida et al. (2008).
In the first series of simulations, only the outer planet migrated on different timescales ,
while its eccentricity was damped on timescales
such that their corresponding ratios
were always between
,
which is typically found in hydrodynamical simulations (Kley et al. 2004).
In the second series of runs, besides the migration of the outer
planet, the inner planet's eccentricity was also damped, while its
semi-major axis was not influenced. Typical cases of these simulations
can be seen in Figs. 6 and 7. In these particular cases the following migration and eccentricity damping timescales were applied; for the outer planet
,
(or
,
years), and if the inner disk's influence on the inner planet was also taken into account, we used for the inner planet
(as its semi-major axis was not influenced) and
.
![]() |
Figure 6:
Behaviour of the eccentricities (red for the inner, green for the outer planet), the resonant angle |
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![]() |
Figure 7:
Behaviour of the eccentricities (red for the inner, green for the outer planet), the resonant angle |
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Comparing Figs. 6 and 7,
it can be seen immediately that, in the first case when the inner
planet was unaffected, the resonant angles behave according to the
theoretical predictions of Beaugé et al. (2003) (asymmetric libration of the resonant angles), while in the second case the libration of the resonant angles is around
and
,
respectively. The inclusion of the inner disk helps to keep the inner
planet's eccentricity at lower values close to those coming from the
observations. In Fig. 6 the eccentricity of the inner planet tends to 0.5, while in Fig. 7 it will show only small amplitude oscillations around the value 0.3.
Finally, we can conclude that the effect of an inner disk in the three-body simulations (with dissipative forces) alters the final state of the resonant configuration (here a 3:1 MMR), keeping the resonant angles far from their mean libration, thus impeding the system for reaching the equilibrium solutions suggested by Beaugé et al. (2003). In both cases the systems will remain in their final configrations, when the damping effect of the disk is reduced and eventually non existent.
6 Summary
After the discovery of the system HD 60532 (Desort et al. 2008), the thorough dynamical analysis of Laskar & Correia (2009)
has reliably established the first resonant system that contains two
giant planets in a higher order mean motion resonance, here 3:1. Until
now, only one system, 55 Cancri, has been proposed as a candidate
for a system where two giant planets might be in 3:1 MMR. On the other
hand, for 55 Cancri, the existence of the resonant solution could
not be confirmed by Naef et al. (2004), and a new non-resonant solution was later found by Fischer et al. (2008).
Thus according to our present knowledge, HD 60532 is the only
known system containing giant planets in the 3:1 MMR. Recently, Laskar & Correia (2009) has improved the orbital solution found by Desort et al. (2008).
In their new fit that assumes a relatively small inclination between
the common orbital plane of the planets and the tangent plane of the
sky,
,
the giant planets have quite high masses:
and
.
The main path to forming a resonant planetary system is thought to be by convergent migration of planets. For an alternative scenario, formation through a scattering process has been proposed by Raymond et al. (2008). Migration is the the result of dissipative forces originating in an ambient accretion disk, which act along with the mutual gravitational forces between the planets and the central star. Being a dissipative process, a sufficiently long-lasting migration process brings the system close to a stationary solution, which corresponds to a periodic solution of the system. The stationary (minimum energy) solutions in the systems yielding 3:1 MMR have been studied by a Hamiltonian approach by Beaugé et al. (2003). It has been found that, depending on the ratio of the eccentricities, the resonant angles can exhibit antisymmetric (symmetric anti-aligned) or asymmetric libration. These results have also been confirmed by numerical simulations of Kley et al. (2004).
In a first step, we integrated the system HD 60532 numerically
using the (observed) initial conditions calculated from Fits I
and II of Laskar & Correia (2009). We found that all resonant angles librate and that the system is in apsidal corotation. Since
oscillates around
,
observationally the system lies in an antisymmetric configuration.
However, for the observed mean eccentricities, the Hamiltonian approach would suggest an asymmetric configuration.
To model the formation of the system HD 60532 and find an
explanation for this special antisymmetric configuration, we assumed
that the system formed through a planet-disk interaction process. We
performed a series of fully hydrodynamical simulations with low,
intermediate, and high planetary masses
,
assuming
,
,
and
,
respectively. Since the eccentricity of the inner planet oscillates around a moderate mean value
,
we assumed that during the migration process an inner disk (between the
inner planet and the central star) was present, providing an efficient
damping mechanism on the inner planet's eccentricity. We found that the
convergent migration through planet-disk interaction, which takes the
planets into the 3:1 MMR only occurred for the highest planetary
masses. The dynamical behaviour of the resulting resonant planetary
system is then indeed very similar to the one obtained from the radial
velocity observations. Through our full hydrodynamicalsimulations, we
support the small inclination
of the system as suggested by Laskar & Correia (2009).
To understand why the system does not appear to be in a miminum energy configuration to the 3:1 MMR, we performed a series of dedicated 3-body simulations with additonal forces emulating the effects of the protoplanetary disk. In particular, we studied situations with and without the influence of an inner disk. We found that it is exactly the effect of the inner disk that distiguishes between an antisymmetric or asymmetric final configuration. Here, its presence is responsible for the system not reaching the minimum energy asymmetric configuration. Nevertheless, we point out that the solutions obtained in the antisymmetric state are indeed stationary solutions, which are stable when the effects of the disk are reduced.
In related earlier works, we had already established that the observed smallness of the mean eccentricities in resonant planetary systems can also be attributed to the effect of an inner disk (Crida et al. 2008; Sándor et al. 2007). We conclude that an inner disk during the migration process has directly observable consequences on the post-formation dynamical behaviour of resonant planetary systems in mean motion resonances.
AcknowledgementsThis work was partly supported by the German Research Foundation (DFG) under grant KL 650/11 within the collaborative research group The formation of planets (FOR 759).
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All Tables
Table 1: Orbital solutions provided by Laskar & Correia (2009).
All Figures
![]() |
Figure 1:
Dynamical behaviour of the system HD 60532, where the initial
conditions of the numerical integration are calculated from Fit II
of Laskar & Correia (2009).
The upper panel shows the evolution of the eccentricities (inner planet
red curve, outer planet green curve), middle panel the behaviour of the
resonant angle |
Open with DEXTER | |
In the text |
![]() |
Figure 2: The surface density distribution of the protoplanetary accretion disk after 500 periods of the inner giant planet. The disk model is described in Sect. 3, the planetary masses correspond to Fit II. Light colours are for higher, while dark colour shades are for low values of the surface density. For a better visualization we used a logarithmic scale. |
Open with DEXTER | |
In the text |
![]() |
Figure 3: The azimuthally averaged surface density profile of the protoplanetary disk after 500 periods of the inner giant planet for different planetary masses. Dashed line corresponds to planetary masses of Fit I, dotted line for the intermediate values, which are twice the masses of Fit I, and solid line for masses of Fit II. |
Open with DEXTER | |
In the text |
![]() |
Figure 4:
Dynamical behaviour of the two giant planets embedded in the
surrounding protoplanetary disk: semi-major axes and eccentricities.
During the first 500 orbital periods (of the inner planet), the
planets are kept fixed to obtain a steady state in the disk. The upper panel shows the behaviour of the semi-major axes, while the bottom panel
shows the time evolution of the eccentricities (red line corresponds to
the inner, green line to the outer giant planet). The capture into the
3:1 MMR occurs around 2300 periods of the inner planet. The
planetary masses (
|
Open with DEXTER | |
In the text |
![]() |
Figure 5:
Dynamical behaviour of the two giant planets embedded in the
surrounding protoplanetary disk for Fit II. The evolution of the
resonant angle |
Open with DEXTER | |
In the text |
![]() |
Figure 6:
Behaviour of the eccentricities (red for the inner, green for the outer planet), the resonant angle |
Open with DEXTER | |
In the text |
![]() |
Figure 7:
Behaviour of the eccentricities (red for the inner, green for the outer planet), the resonant angle |
Open with DEXTER | |
In the text |
Copyright ESO 2010
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