2,305 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
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
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&
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
A torque formula for non-isothermal Type I planetary migration - II. Effects of diffusion
We study the effects of diffusion on the non-linear corotation torque, or
horseshoe drag, in the two-dimensional limit, focusing on low-mass planets for
which the width of the horseshoe region is much smaller than the scale height
of the disc. In the absence of diffusion, the non-linear corotation torque
saturates, leaving only the Lindblad torque. Diffusion of heat and momentum can
act to sustain the corotation torque. In the limit of very strong diffusion,
the linear corotation torque is recovered. For the case of thermal diffusion,
this limit corresponds to having a locally isothermal equation of state. We
present some simple models that are able to capture the dependence of the
torque on diffusive processes to within 20% of the numerical simulations.Comment: 12 pages, 8 figures, accepted for publication in MNRA
Low-mass planets in nearly inviscid disks: Numerical treatment
Embedded planets disturb the density structure of the ambient disk and
gravitational back-reaction will induce possibly a change in the planet's
orbital elements. The accurate determination of the forces acting on the planet
requires careful numerical analysis. Recently, the validity of the often used
fast orbital advection algorithm (FARGO) has been put into question, and
special numerical resolution and stability requirements have been suggested. In
this paper we study the process of planet-disk interaction for small mass
planets of a few Earth masses, and reanalyze the numerical requirements to
obtain converged and stable results. One focus lies on the applicability of the
FARGO-algorithm. Additionally, we study the difference of two and
three-dimensional simulations, compare global with local setups, as well as
isothermal and adiabatic conditions. We study the influence of the planet on
the disk through two- and three-dimensional hydrodynamical simulations. To
strengthen our conclusions we perform a detailed numerical comparison where
several upwind and Riemann-solver based codes are used with and without the
FARGO-algorithm.
With respect to the wake structure and the torque density acting on the
planet we demonstrate that the FARGO-algorithm yields correct results, and that
at a fraction of the regular cpu-time. We find that the resolution requirements
for achieving convergent results in unshocked regions are rather modest and
depend on the pressure scale height of the disk. By comparing the torque
densities of 2D and 3D simulations we show that a suitable vertical averaging
procedure for the force gives an excellent agreement between the two. We show
that isothermal and adiabatic runs can differ considerably, even for adiabatic
indices very close to unity.Comment: accepted by Astronomy & Astrophysic
No snow-plough mechanism during the rapid hardening of supermassive black hole binaries
We present two-dimensional hydrodynamical simulations of the tidal
interaction between a supermassive black hole binary with moderate mass ratio,
and the fossil gas disc where it is embedded. Our study extends previous
one-dimensional height-integrated disc models, which predicted that the density
of the gas disc between the primary and the secondary black holes should rise
significantly during the ultimate stages of the binary's hardening driven by
the gravitational radiation torque. This snow-plough mechanism, as we call it,
would lead to an increase in the bolometric luminosity of the system prior to
the binary merger, which could be detected in conjunction with the
gravitational wave signal. We argue here that the snow-plough mechanism is
unlikely to occur. In two-dimensions, when the binary's hardening timescale
driven by gravitational radiation becomes shorter than the disc's viscous drift
timescale, fluid elements in the inner disc get funneled to the outer disc
through horseshoe trajectories with respect to the secondary. Mass leakage
across the secondary's gap is thus found to be effective and, as a result, the
predicted accretion disc luminosity will remain at roughly the same level prior
to merger.Comment: 5 pages, 5 figures, accepted for publication in MNRA
Long range outward migration of giant planets, with application to Fomalhaut b
Recent observations of exoplanets by direct imaging, reveal that giant
planets orbit at a few dozens to more than a hundred of AU from their central
star. The question of the origin of these planets challenges the standard
theories of planet formation. We propose a new way of obtaining such far
planets, by outward migration of a pair of planets formed in the 10 AU region.
Two giant planets in mean motion resonance in a common gap in the
protoplanetary disk migrate outwards, if the inner one is significantly more
massive than the outer one. Using hydrodynamical simulations, we show that
their semi major axes can increase by almost one order of magnitude. In a
flared disk, the pair of planets should reach an asymptotic radius. This
mechanism could account for the presence of Fomalhaut b ; then, a second, more
massive planet, should be orbiting Fomalhaut at about 75 AU.Comment: 6 pages, 4 figures, accepted for publication by ApJ Letter
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