369 research outputs found
On the horseshoe drag of a low-mass planet. I - Migration in isothermal disks
We investigate the unsaturated horseshoe drag exerted on a low-mass planet by
an isothermal gaseous disk. In the globally isothermal case, we use a formal-
ism, based on the use of a Bernoulli invariant, that takes into account
pressure effects, and that extends the torque estimate to a region wider than
the horse- shoe region. We find a result that is strictly identical to the
standard horseshoe drag. This shows that the horseshoe drag accounts for the
torque of the whole corotation region, and not only of the horseshoe region,
thereby deserving to be called corotation torque. We find that evanescent waves
launched downstream of the horseshoe U-turns by the perturbations of vortensity
exert a feed-back on the upstream region, that render the horseshoe region
asymmetric. This asymmetry scales with the vortensity gradient and with the
disk's aspect ratio. It does not depend on the planetary mass, and it does not
have any impact on the horseshoe drag. Since the horseshoe drag has a steep
dependence on the width of the horseshoe region, we provide an adequate
definition of the width that needs to be used in horseshoe drag estimates. We
then consider the case of locally isothermal disks, in which the tempera- ture
is constant in time but depends on the distance to the star. The horseshoe drag
appears to be different from the case of a globally isothermal disk. The
difference, which is due to the driving of vortensity in the vicinity of the
planet, is intimately linked to the topology of the flow. We provide a
descriptive inter- pretation of these effects, as well as a crude estimate of
the dependency of the excess on the temperature gradient.Comment: Accepted for publication in Ap
On the horseshoe drag of a low-mass planet. II Migration in adiabatic disks
We evaluate the horseshoe drag exerted on a low-mass planet embedded in a
gaseous disk, assuming the disk's flow in the coorbital region to be adiabatic.
We restrict this analysis to the case of a planet on a circular orbit, and we
assume a steady flow in the corotating frame. We also assume that the
corotational flow upstream of the U-turns is unperturbed, so that we discard
saturation effects. In addition to the classical expression for the horseshoe
drag in barotropic disks, which features the vortensity gradient across
corotation, we find an additional term which scales with the entropy gradient,
and whose amplitude depends on the perturbed pressure at the stagnation point
of the horseshoe separatrices. This additional torque is exerted by evanescent
waves launched at the horseshoe separatrices, as a consequence of an asymmetry
of the horseshoe region. It has a steep dependence on the potential's softening
length, suggesting that the effect can be extremely strong in the three
dimensional case. We describe the main properties of the coorbital region (the
production of vortensity during the U-turns, the appearance of vorticity sheets
at the downstream separatrices, and the pressure response), and we give torque
expressions suitable to this regime of migration. Side results include a weak,
negative feed back on migration, due to the dependence of the location of the
stagnation point on the migration rate, and a mild enhancement of the
vortensity related torque at large entropy gradient.Comment: Accepted for publication in Ap
FARGO: a fast eulerian transport algorithm for differentially rotating disks
We present an efficient and simple modification of the standard transport
algorithm used in explicit eulerian fixed polar grid codes, aimed at getting
rid of the average azimuthal velocity when applying the Courant condition. This
results in a much larger timestep then the usual procedure, and it is
particularly well-suited to the description of a Keplerian disk where one is
traditionally limited by the very demanding Courant condition on the fast
orbital motion at the inner boundary. In this modified algorithm, the timestep
is limited by the perturbed velocity and by the shear arising from the
differential rotation. FARGO stands for ``Fast Advection in Rotating Gaseous
Objects''. The speed-up resulting from the use of the FARGO algorithm is
problem dependent. In the example presented here, which shows the evolution of
a Jupiter sized protoplanet embedded in a minimum mass protoplanetary nebula,
the FARGO algorithm is about an order of magnitude faster than a traditional
transport scheme, with a much smaller numerical diffusivity.Comment: 9 pages, 6 figures. Accepted for publication in Astron. Astrophys.
Supp. Serie
Saturated torque formula for planetary migration in viscous disks with thermal diffusion: recipe for protoplanet population synthesis
We provide torque formulae for low mass planets undergoing type I migration
in gaseous disks. These torque formulae put special emphasis on the horseshoe
drag, which is prone to saturation: the asymptotic value reached by the
horseshoe drag depends on a balance between coorbital dynamics (which tends to
cancel out or saturate the torque) and diffusive processes (which tend to
restore the unperturbed disk profiles, thereby desaturating the torque). We
entertain here the question of this asymptotic value, and we derive torque
formulae which give the total torque as a function of the disk's viscosity and
thermal diffusivity. The horseshoe drag features two components: one which
scales with the vortensity gradient, and one which scales with the entropy
gradient, and which constitutes the most promising candidate for halting inward
type I migration. Our analysis, which is complemented by numerical simulations,
recovers characteristics already noted by numericists, namely that the viscous
timescale across the horseshoe region must be shorter than the libration time
in order to avoid saturation, and that, provided this condition is satisfied,
the entropy related part of the horseshoe drag remains large if the thermal
timescale is shorter than the libration time. Side results include a study of
the Lindblad torque as a function of thermal diffusivity, and a contribution to
the corotation torque arising from vortensity viscously created at the contact
discontinuities that appear at the horseshoe separatrices. For the convenience
of the reader mostly interested in the torque formulae, section 8 is
self-contained.Comment: Affiliation details changed. Fixed equation numbering issue. Biblio
info adde
On the corotation torque in a radiatively inefficient disk
We consider the angular momentum exchange at the corotation resonance between
a two-dimensional gaseous disk and a uniformly rotating external potential,
assuming that the disk flow is adiabatic. We first consider the linear case for
an isolated resonance, for which we give an expression of the corotation torque
that involves the pressure perturbation, and which reduces to the usual
dependence on the vortensity gradient in the limit of a cold disk. Although
this expression requires the solution of the hydrodynamic equations, it
provides some insight into the dynamics of the corotation region. In the
general case, we find an additional dependence on the entropy gradient at
corotation. This dependence is associated to the advection of entropy
perturbations. These are not associated to pressure perturbations. They remain
confined to the corotation region, where they yield a singular contribution to
the corotation torque. In a second part, we check our torque expression by
means of customized two-dimensional hydrodynamical simulations. In a third
part, we contemplate the case of a planet embedded in a Keplerian disk, assumed
to be adiabatic. We find an excess of corotation torque that scales with the
entropy gradient, and we check that the contribution of the entropy
perturbation to the torque is in agreement with the expression obtained from
the linear analysis. We finally discuss some implications of the corotation
torque expression for the migration of low mass planets in the regions of
protoplanetary disks where the flow is radiatively inefficient on the timescale
of the horseshoe U-turns.Comment: 36 pages, 19 figures, accepted for publication in Ap
Reversing type II migration: resonance trapping of a lighter giant protoplanet
We present a mechanism related to the migration of giant protoplanets
embedded in a protoplanetary disc whereby a giant protoplanet is caught up,
before having migrated all the way to the central star, by a lighter outer
giant protoplanet. This outer protoplanet may get captured into the 2:3
resonance with the more massive one, in which case the gaps that the two
planets open in the disc overlap. Two effects arise, namely a squared mass
weighted torque imbalance and an increased mass flow through the overlapping
gaps from the outer disc to the inner disc, which both play in favour of an
outwards migration. Indeed under the conditions presented here, which describe
the evolution of a pair of protoplanets respectively Jupiter and Saturn sized,
the migration is reversed, while the planets semi-major axis ratio is constant
and the eccentricities are confined to small values by the disc material. The
long-term behaviour of the system is briefly discussed, and could account for
the high eccentricities observed for the extrasolar planets with semi-major
axis a>0.2 AU.Comment: 5 pages, 4 figures. Accepted for publication in MNRA
Simulating planet migration in globally evolving disks
Numerical simulations of planet-disk interactions are usually performed with
hydro-codes that -- because they consider only an annulus of the disk, over a
2D grid -- can not take into account the global evolution of the disk. However,
the latter governs planetary migration of type II, so that the accuracy of the
planetary evolution can be questioned.
To develop an algorithm that models the local planet-disk interactions
together with the global viscous evolution of the disk, we surround the usual
2D grid with a 1D grid ranging over the real extension of the disk. The 1D and
2D grids are coupled at their common boundaries via ghost rings, paying
particular attention to the fluxes at the interface, especially the flux of
angular momentum carried by waves. The computation is done in the frame
centered on the center of mass to ensure angular momentum conservation.
The global evolution of the disk and the local planet-disk interactions are
both well described and the feedback of one on the other can be studied with
this algorithm, for a negligible additional computing cost with respect to
usual algorithms.Comment: 12 pages, 11 figures, accepted for publication in A&
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