259 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
Horseshoe drag in three-dimensional globally isothermal disks
We study the horseshoe dynamics of a low-mass planet in a three-dimensional, globally isothermal, inviscid disk. We find, as reported in previous work, that the boundaries of the horseshoe region (separatrix sheets) have cylindrical symmetry about the disk´s rotation axis. We interpret this feature as arising from the fact that the whole separatrix sheets have a unique value of Bernoulli´s constant, and that this constant does not depend on altitude, but only on the cylindrical radius, in barotropic disks. We next derive an expression for the torque exerted by the horseshoe region on the planet, or horseshoe drag. Potential vorticity is not materially conserved as in two-dimensional flows, but it obeys a slightly more general conservation law (Ertel´s theorem) that allows an expression for the horseshoe drag identical to the expression in a two-dimensional disk to be obtained. Our results are illustrated and validated by three-dimensional numerical simulations. The horseshoe region is found to be slightly narrower than previously extrapolated from two-dimensional analyses with a suitable softening length of the potential. We discuss the implications of our results for the saturation of the corotation torque, and the possible connection to the flow at the Bondi scale, which the present analysis does not resolve.Fil: Masset, F. S.. Universidad Nacional Autónoma de México; MéxicoFil: BenÃtez Llambay, Pablo. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Centro CientÃfico Tecnológico Conicet - Córdoba. Instituto de AstronomÃa Teórica y Experimental. Universidad Nacional de Córdoba. Observatorio Astronómico de Córdoba. Instituto de AstronomÃa Teórica y Experimental; Argentin
On type-I migration near opacity transitions. A generalized Lindblad torque formula for planetary population synthesis
We give an expression for the Lindblad torque acting on a low-mass planet
embedded in a protoplanetary disk that is valid even at locations where the
surface density or temperature profile cannot be approximated by a power law,
such as an opacity transition. At such locations, the Lindblad torque is known
to suffer strong deviation from its standard value, with potentially important
implications for type I migration, but the full treatment of the tidal
interaction is cumbersome and not well suited to models of planetary population
synthesis. The expression that we propose retains the simplicity of the
standard Lindblad torque formula and gives results that accurately reproduce
those of numerical simulations, even at locations where the disk temperature
undergoes abrupt changes. Our study is conducted by means of customized
numerical simulations in the low-mass regime, in locally isothermal disks, and
compared to linear torque estimates obtained by summing fully analytic torque
estimates at each Lindblad resonance. The functional dependence of our modified
Lindblad torque expression is suggested by an estimate of the shift of the
Lindblad resonances that mostly contribute to the torque, in a disk with sharp
gradients of temperature or surface density, while the numerical coefficients
of the new terms are adjusted to seek agreement with numerics. As side results,
we find that the vortensity related corotation torque undergoes a boost at an
opacity transition that can counteract migration, and we find evidence from
numerical simulations that the linear corotation torque has a non-negligible
dependency upon the temperature gradient, in a locally isothermal disk.Comment: Appeared in special issue of "Celestial Mechanics and Dynamical
Astronomy" on Extrasolar Planetary System
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
Secular interactions between inclined planets and a gaseous disk
In a planetary system, a secular particle resonance occurs at a location
where the precession rate of a test particle (e.g. an asteroid) matches the
frequency of one of the precessional modes of the planetary system. We
investigate the secular interactions of a system of mutually inclined planets
with a gaseous protostellar disk that may contain a secular nodal particle
resonance. We determine the normal modes of some mutually inclined planet-disk
systems. The planets and disk interact gravitationally, and the disk is
internally subject to the effects of gas pressure, self-gravity, and turbulent
viscosity. The behavior of the disk at a secular resonance is radically
different from that of a particle, owing mainly to the effects of gas pressure.
The resonance is typically broadened by gas pressure to the extent that global
effects, including large-scale warps, dominate. The standard resonant torque
formula is invalid in this regime. Secular interactions cause a decay of the
inclination at a rate that depends on the disk properties, including its mass,
turbulent viscosity, and sound speed. For a Jupiter-mass planet embedded within
a minimum-mass solar nebula having typical parameters, dissipation within the
disk is sufficient to stabilize the system against tilt growth caused by
mean-motion resonances.Comment: 30 pages, 6 figures, to be published in The Astrophysical Journa
Evolution of the eccentricity and inclination of low-mass planets subjected to thermal forces: a numerical study
By means of three dimensional, high resolution hydrodynamical simulations we
study the orbital evolution of weakly eccentric or inclined low-mass
protoplanets embedded in gaseous discs subject to thermal diffusion. We
consider both non-luminous planets, and planets that also experience the
radiative feedback from their own luminosity. We compare our results to
previous analytical work, and find that thermal forces (the contribution to the
disc's force arising from thermal effects) match those predicted by linear
theory within %. When the planet's luminosity exceeds a threshold
found to be within % of that predicted by linear theory, its eccentricity
and inclination grow exponentially, whereas these quantities undergo a strong
damping below this threshold. In this regime of low luminosity indeed, thermal
diffusion cools the surroundings of the planet and allows gas to accumulate in
its vicinity. It is the dynamics of this gas excess that contributes to damp
eccentricity and inclination. The damping rates obtained can be up to
times larger than those due to the resonant interaction with the disc, where
is the disc's aspect ratio. This suggests that models that incorporate
planet-disc interactions using well-known formulae based on resonant
wave-launching to describe the evolution of eccentricity and inclination
underestimate the damping action of the disc on the eccentricity and
inclination of low-mass planets by an order of magnitude.Comment: Accepted for publication in MNRA
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