854 research outputs found
On the formation of the Kepler-10 planetary system
In this paper, we investigate the conditions required for the 3 and 17 Earth
mass solid planets in the Kepler-10 system to have formed through collisions
and mergers within an initial population of embryos. By performing a large
number of N-body simulations, we show that the total mass of the initial
population had to be significantly larger than the masses of the two planets,
and that the two planets must have built-up farther away than their present
location, at a distance of at least a few au from the central star. The planets
had to grow fast enough so that they would detach themselves from the
population of remaining, less massive, cores and migrate in to their present
location. By the time the other cores migrated in, the disc's inner edge would
have moved out so that these cores cannot be detected today. We also compute
the critical core mass beyond which a massive gaseous envelope would be
accreted and show that it is larger than 17 Earth masses if the planetesimal
accretion rate onto the core is larger than 10^{-6} Earth mass per year. For a
planetesimal accretion rate between 10^{-6} and 10^{-5} Earth mass per year,
the 17 Earth mass core would not be expected to have accreted more than about 1
Earth mass of gas. The results presented in this paper suggest that a planetary
system like Kepler-10 may not be unusual, although it has probably formed in a
rather massive disc.Comment: 12 pages, accepted for publication in MNRA
The effects of disc warping on the inclination of planetary orbits
The interaction between a planet located in the inner region of a disc and
the warped outer region is studied. We consider the stage of evolution after
the planet has cleared-out a gap, so that the planetary orbit evolves only
under the gravitational potential from the disc. We develop a secular analysis
and compute the evolution of the orbital elements by solving Lagrange's
equations valid to second order in the eccentricity. We also perform numerical
simulations with the full disc potential. In general, the interaction between
the disc and the planet leads to the precession of the orbit. The orbital plane
therefore becomes tilted relative to the disc's inner parts, with no change in
the eccentricity. When the inclination approaches 90 degrees, there is an
instability and the eccentricity increases. In this case, both the inclination
and the eccentricity develop large variations, with the orbit becoming
retrograde. As the eccentricity reaches high values, we would expect tidal
capture on a short orbit of the planet by the star to occur. This instability
happens when the disc is severely warped, or if there is a significant amount
of mass in a ring inclined by at least 45 degrees relative to the initial
orbital plane. The inclination of the orbit does not depend on the semimajor
axis nor on the planet's mass. However, for a significant inclination to be
generated on a timescale of at most a few Myr, the planet should be beyond the
snow line. The process described here would therefore produce two distinct
populations of inclined planets: one with objects beyond the snow line with at
most moderate eccentricities, and another with objects on short circularized
orbits.Comment: 25 pages, 4 figures, accepted for publication in MNRA
First-order mean motion resonances in two-planet systems: general analysis and observed systems
This paper focuses on two-planet systems in a first-order mean
motion resonance and undergoing type-I migration in a disc. We present a
detailed analysis of the resonance valid for any value of . Expressions for
the equilibrium eccentricities, mean motions and departure from exact resonance
are derived in the case of smooth convergent migration. We show that this
departure, not assumed to be small, is such that period ratio normally exceeds,
but can also be less than, Departure from exact resonance as a
function of time for systems starting in resonance and undergoing divergent
migration is also calculated. We discuss observed systems in which two low mass
planets are close to a first-order resonance. We argue that the data are
consistent with only a small fraction of the systems having been captured in
resonance. Furthermore, when capture does happen, it is not in general during
smooth convergent migration through the disc but after the planets reach the
disc inner parts. We show that although resonances may be disrupted when the
inner planet enters a central cavity, this alone cannot explain the spread of
observed separations. Disruption is found to result in either the system moving
interior to the resonance by a few percent, or attaining another resonance. We
postulate two populations of low mass planets: a small one for which extensive
smooth migration has occurred, and a larger one that formed approximately
in-situ with very limited migration.Comment: Accepted for publication in MNRA
Evolution of eccentricity and orbital inclination of migrating planets in 2:1 mean motion resonance
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, has to
pass through a value of order 0.3 for the system to enter an
inclination-type resonance. Numerically, we confirm that such a resonance may
also be excited at another, larger, value , as found
by previous authors. A necessary condition for onset of an inclination-type
resonance is that the asymptotic equilibrium value of is larger
than . We find that, for , the system cannot enter an
inclination-type resonance if the ratio of eccentricity to semimajor axis
damping timescales is smaller than 0.2. This result still holds if
only the eccentricity of the outer planet is damped and . As the
disc/planet interaction is characterized by , 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.Comment: 22 pages, 10 figures, accepted for publication in MNRA
Eccentricity pumping of a planet on an inclined orbit by a disc
In this paper, we show that the eccentricity of a planet on an inclined orbit
with respect to a disc can be pumped up to high values by the gravitational
potential of the disc, even when the orbit of the planet crosses the disc
plane. This process is an extension of the Kozai effect. If the orbit of the
planet is well inside the disc inner cavity, the process is formally identical
to the classical Kozai effect. If the planet's orbit crosses the disc but most
of the disc mass is beyond the orbit, the eccentricity of the planet grows when
the initial angle between the orbit and the disc is larger than some critical
value which may be significantly smaller than the classical value of 39
degrees. Both the eccentricity and the inclination angle then vary periodically
with time. When the period of the oscillations of the eccentricity is smaller
than the disc lifetime, the planet may be left on an eccentric orbit as the
disc dissipates.Comment: 13 pages, 4 figures, accepted for publication in MNRA
Linear Analysis of the Hall Effect in Protostellar Disks
The effects of Hall electromotive forces (HEMFs) on the linear stability of
protostellar disks are examined. Earlier work on this topic focused on axial
field and perturbation wavenumbers. Here we treat the problem more generally.
Both axisymmetric and nonaxisymmetric cases are investigated. Though seldom
explicitly included in calculations, HEMFs appear to be important whenever
Ohmic dissipation is. They allow for the appearance of electron whistler waves,
and since these have right-handed polarization, a helicity factor is also
introduced into the stability problem. This factor is the product of the
components of the angular velocity and magnetic field along the perturbation
wavenumber, and it is destabilizing when negative. Unless the field and angular
velocity are exactly aligned, it is always possible to find destabilizing
wavenumbers. HEMFs can destabilize any differential rotation law, even those
with angular velocity increasing outward. Regardless of the sign of the angular
velocity gradient, the maximum growth rate is always given in magnitude by the
local Oort A value of the disk, as in the standard magnetorotational
instability. The role of Hall EMFs may prove crucial to understanding how
turbulence is maintained in the ``low state'' of eruptive disk systems.Comment: 34 pages, 10 figures, AAS LaTEx, v.4.0. Submitted to Ap
Discs and Planetary Formation
The formation, structure and evolution of protoplanetary discs is considered.
The formation of giant planets within the environment of these models is also
discussed.Comment: 22 pages, LaTeX (including 6 figures), uses paspconf.sty, epsf.sty
and rotate.sty, to be published in Proceedings of the EC Summer School on
'Astrophysical Discs', eds J. A. Sellwood and J. Goodman, ASP Conf. Serie
Tidally-induced warps in protostellar discs
We review results on the dynamics of warped gaseous discs. We consider tidal
perturbation of a Keplerian disc by a companion star orbiting in a plane
inclined to the disc. The perturbation induces the precession of the disc, and
thus of any jet it could drive. In some conditions the precession rate is
uniform, and as a result the disc settles into a warp mode. The tidal torque
also leads to the truncation of the disc, to the evolution of the inclination
angle (not necessarily towards alignment of the disc and orbital planes) and to
a transport of angular momentum in the disc. We note that the spectral energy
distribution of such a warped disc is different from that of a flat disc. We
conclude by listing observational effects of warps in protostellar discs.Comment: 10 pages, LaTeX (including 1 figure), uses paspconf.sty and epsf.sty,
to be published in Proceedings of the EC Summer School on 'Astrophysical
Discs', eds J. A. Sellwood and J. Goodman, ASP Conf. Serie
The TRAPPIST-1 system: Orbital evolution, tidal dissipation, formation and habitability
We study the dynamical evolution of the TRAPPIST-1 system under the influence
of orbital circularization through tidal interaction with the central star. We
find that systems with parameters close to the observed one evolve into a state
where consecutive planets are linked by first order resonances and consecutive
triples, apart from planets c, d and e, by connected three body Laplace
resonances. The system expands with period ratios increasing and mean
eccentricities decreasing with time. This evolution is largely driven by tides
acting on the innermost planets which then influence the outer ones. In order
that deviations from commensurability become significant only on time
scales or longer, we require that the tidal parameter associated with the
planets has to be such that At the same time, if we start
with two subsystems, with the inner three planets comprising the inner one,
associated with the planets has to be on the order (and not significantly
exceeding) for the two subsystems to interact and end up in the
observed configuration. This scenario is also supported by modelling of the
evolution through disk migration which indicates that the whole system cannot
have migrated inwards together. Also in order to avoid large departures from
commensurabilities, the system cannot have stalled at a disk inner edge for
significant time periods. We discuss the habitability consequences of the tidal
dissipation implied by our modelling, concluding that planets d, e and f are
potentially in habitable zones.Comment: 27 pages, 15 figures, accepted for publication in MNRA
A circumbinary disc model for the variability of the eclipsing binary CoRoT 223992193
We calculate the flux received from a binary system obscured by a
circumbinary disc. The disc is modelled using two dimensional hydrodynamical
simulations, and the vertical structure is derived by assuming it is
isothermal. The gravitational torque from the binary creates a cavity in the
disc's inner parts. If the line of sight along which the system is observed has
a high inclination , it intersects the disc and some absorption is produced.
As the system is not axisymmetric, the resulting light curve displays
variability. We calculate the absorption and produce light curves for different
values of the dust disc aspect ratio and mass of dust in the cavity
. This model is applied to the high inclination ()
eclipsing binary CoRoT 223992193, which shows 5-10% residual photometric
variability after the eclipses and a spot model are subtracted. We find that
such variations for can be obtained for and
M. For higher , would
have to be close to this lower value and somewhat less than .
Our results show that such variability in a system where the stars are at least
90% visible at all phases can be obtained only if absorption is produced by
dust located inside the cavity. If absorption is dominated by the parts of the
disc located close to or beyond the edge of the cavity, the stars are
significantly obscured.Comment: 17 pages, 6 figures, accepted for publication in MNRA
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