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 $(q+1):q$ mean motion resonance and undergoing type-I migration in a disc. We present a detailed analysis of the resonance valid for any value of $q$. 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, $(q+1)/q.$ 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 $e_{\rm i}$ of the inner planet reaches under the combined effects of migration and eccentricity damping. We also show that, for a ratio $q$ of inner to outer masses below unity, $e_{\rm i}$ has to pass through a value $e_{\rm i,res}$ 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 $e_{\rm i, res} \simeq 0.6$, as found by previous authors. A necessary condition for onset of an inclination-type resonance is that the asymptotic equilibrium value of $e_{\rm i}$ is larger than $e_{\rm i,res}$. We find that, for $q \le 1$, the system cannot enter an inclination-type resonance if the ratio of eccentricity to semimajor axis damping timescales $t_e/t_a$ is smaller than 0.2. This result still holds if only the eccentricity of the outer planet is damped and $q \lesssim 1$. As the disc/planet interaction is characterized by $t_e/t_a \sim 10^{-2}$, 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

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 $Gy$ time scales or longer, we require that the tidal parameter associated with the planets has to be such that $Q' > \sim 10^{2-3}.$ At the same time, if we start with two subsystems, with the inner three planets comprising the inner one, $Q'$ associated with the planets has to be on the order (and not significantly exceeding) $10^{2-3}$ 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

The response of accretion disks to bending waves: angular momentum transport and resonances

We investigate the linear tidal perturbation of a viscous Keplerian disk by a companion star orbiting in a plane inclined to the disk. We consider m=1 perturbations with odd symmetry with respect to the z=0 midplane. Since the response of a viscous disk is not in phase with the perturbing potential, a tidal torque is exerted on the disk, resulting in a decrease of its angular momentum. This tidal torque is found to be comparable to the horizontal viscous stress acting on the background flow when the perturbed velocities in the disk are on the order of the sound speed. If these velocities remain subsonic, the tidal torque can exceed the horizontal viscous stress only if the viscous parameter \alpha which couples to the vertical shear is larger than that coupled to the horizontal shear. In protostellar disks, bending waves are found to propagate deep into the disk inner parts. If the waves are reflected at the center, resonances occur when the frequency of the tidal waves is equal to that of some free normal global bending mode of the disk. If such resonances exist, tidal interactions may then be important even when the binary separation is large. Out of resonance, the torque associated with the secular perturbation is generally much larger than that associated with the finite frequency perturbations. As long as the waves are damped before they reach the center, the torque associated with the finite frequency perturbations does not depend on the viscosity. These calculations are relevant to disks around young stars and maybe also to disks in X-ray binary systems. (abridged

Bending Instabilities in Magnetized Accretion Discs

We study the global bending modes of a thin annular disc subject to both an internally generated magnetic field and a magnetic field due to a dipole embedded in the central star with axis aligned with the disc rotation axis. When there is a significant inner region of the disc corotating with the star, we find spectra of unstable bending modes. These may lead to elevation of the disc above the original symmetry plane facilitating accretion along the magnetospheric field lines. The resulting non-axisymmetric disc configuration may result in the creation of hot spots on the stellar surface and the periodic photometric variations observed in many classical T Tauri stars (CTTS). Time-dependent behaviour may occur including the shadowing of the central source in magnetic accretors even when the dipole and rotation axes are aligned.Comment: Accepted by MNRAS. 18 pages, 11 figures. LaTeX2e in the MN style. PostScript and HTML files are also available from http://www-star.qmw.ac.uk/~va/ or by e-mail: [email protected]

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 $I$, 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 $H/r$ and mass of dust in the cavity $M_{\rm dust}$. This model is applied to the high inclination ($I=85^{\circ}$) 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 $I \sim 85^{\circ}$ can be obtained for $H/r=10^{-3}$ and $M_{\rm dust} \ge 10^{-12}$ M$_{\odot}$. For higher $H/r$, $M_{\rm dust}$ would have to be close to this lower value and $I$ somewhat less than $85^{\circ}$. 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