The formation of an eccentric gap in a gas disc by a planet in an eccentric orbit

We investigate the effect of a planet on an eccentric orbit on a two-dimensional low-mass gaseous disc. At a planet eccentricity above the planet's Hill radius divided by its semimajor axis, we find that the disc morphology differs from that exhibited by a disc containing a planet in a circular orbit. An eccentric gap is created with eccentricity that can exceed the planet's eccentricity and precesses with respect to the planet's orbit. We find that a more massive planet is required to open a gap when the planet is on an eccentric orbit. We attribute this behaviour to spiral density waves excited at corotation resonances by the eccentric planet. These act to increase the disc's eccentricity and exert a torque opposite in sign to that exerted by the Lindblad resonances. The reduced torque makes it more difficult for waves driven by the planet to overcome viscous inflow in the disc.     

Energy exchange during collision of a pair of disc-sphere galaxies

We have considered the collision of a pair of disc-sphere galaxies under the impulsive approximation to study the variation of the energy change of the disc due to the passage of a spherical galaxy. Various collision parameters like the impact parameter the orientation of the disc galaxy and the relative velocity of the perturber were considered. It is shown that for small impact parameters and high relative velocities, a slightly inclined position of the disc galaxy produces maximum energy exchange. The energy exchange is observed to be asymmetric with respect to the direction of motion of the perturber.     

Gravitational interaction between a planet and an optically thin disc

We investigate the gravitational interaction between a planet and an optically thin protoplanetary disc, performing local three-dimensional hydrodynamical simulations. In the present study, we take account of radiative energy transfer in optically thin discs. Before the stage of planetary accretion, dust opacity is expected to decrease significantly because of grain growth and planetesimal formation. Thus, it would be reasonable to consider optically thin discs in the disc–planet interaction. Furthermore, we focus on small planets that can neither capture disc gas nor open a disc gap. The one-sided torque exerted on a planet by an optically thin disc is examined for various values of the disc optical thickness (<1). In optically thin discs, the temperature behind the density waves is lower than the unperturbed value because of radiative cooling. Heating due to shock dissipation is less effective than radiative cooling. Because of radiative cooling, the density distribution around the planet is not axisymmetric, which exerts an additional torque on the planet. The torque enhancement becomes maximum when the cooling time is comparable with the Keplerian period. The enhancement is significant for low-mass planets. For planets with  3 M_⊕  , the additional one-sided torque can be 40 per cent of the torque in the isothermal case. The radiative cooling is expected to change the differential torque and the migration speed of planets, too.     

Energy, angular momentum and wave action associated with density waves in a rotating magnetized gas disc

Both fast and slow magnetohydrodynamic (MHD) density waves propagating in a thin rotating magnetized gas disc are investigated. In the tight-winding or WKBJ regime, the radial variation of MHD density-wave amplitude during wave propagation is governed by the conservation of wave action surface density which travels at a relevant radial group speed C _g. The wave energy surface density and the wave angular momentum surface density are related to by = and = m respectively, where is the angular frequency in an inertial frame of reference and the integer m , proportional to the azimuthal wavenumber, corresponds to the number of spiral arms. Consequently, both wave energy and angular momentum are conserved for spiral MHD density waves. For both fast and slow MHD density waves, net wave energy and angular momentum are carried outward or inward for trailing or leading spirals, respectively. The wave angular momentum flux contains separate contributions from gravity torque, advective transport and magnetic torque. While the gravity torque plays an important role, the latter two can be of comparable magnitudes to the former. Similar to the role of gravity torque, the part of MHD wave angular momentum flux by magnetic torque (in the case of either fast or slow MHD density waves) propagates outward or inward for trailing or leading spirals, respectively. From the perspective of global energetics in a magnetized gas sheet in rotation, trailing spiral structures of MHD density waves are preferred over leading ones. With proper qualifications, the generation and maintenance as well as transport properties of MHD density waves in magnetized spiral galaxies are discussed.     

The shape of spiral arms

The response by a gaseous disc to perturbations is investigated. Spectral analysis of the resulting pattern indicates that information about the nature of the perturber is maintained only if the perturbation is strong.Paper presented at the IAU Third Asian-Pacific Regional Meeting, held in Kyoto, Japan, between 30 September–6 October, 1984.     

Wave excitation in three-dimensional discs by external potential

We study the excitation of density and bending waves and the associated angular momentum transfer in gaseous discs with finite thickness by a rotating external potential. The disc is assumed to be isothermal in the vertical direction and has no self-gravity. The disc perturbations are decomposed into different modes, each characterized by the azimuthal index m and the vertical index n , which specifies the nodal number of the density perturbation along the disc normal direction. The   n = 0  modes correspond to the two-dimensional density waves previously studied by Goldreich & Tremaine and others. In a three-dimensional disc, waves can be excited at both Lindblad resonances (LRs; for modes with   n = 0, 1, 2, …  ) and vertical resonances (VRs; for the   n ≥ 1  modes only). The torque on the disc is positive for waves excited at outer Lindblad/vertical resonances and negative at inner Lindblad/vertical resonances. While the   n = 0  modes are evanescent around corotation, the   n ≥ 1  modes can propagate into the corotation region where they are damped and deposit their angular momenta. We have derived analytical expressions for the amplitudes of different wave modes excited at LRs and/or VRs and the resulting torques on the disc. It is found that for   n ≥ 1  , angular momentum transfer through VRs is much more efficient than LRs. This implies that in some situations (e.g. a circumstellar disc perturbed by a planet in an inclined orbit), VRs may be an important channel of angular momentum transfer between the disc and the external potential. We have also derived new formulae for the angular momentum deposition at corotation and studied wave excitations at disc boundaries.     

Magnetic torques between accretion discs and stars

I show in this paper that two types of magnetic torques can appear in the interaction between an accretion disc and a magnetic accretor. There is the well-known torque resulting from the difference in angular velocity between the accretion disc and the star, but in addition there is a torque coming from the interaction between the stellar magnetic field and the disc's own magnetic field. The latter form of magnetic torque decreases in strength more slowly with increasing radius, and will therefore dominate at large radii. The direction of the disc field is not determined by the difference in angular velocity between the star and the disc as in the Ghosh &38; Lamb model, but rather is a free parameter. The magnetic torque may therefore either spin up or spin down the star, and the torque changes sign if the magnetic field in the disc reverses. I suggest that this mechanism can explain the torque reversals that have been observed in some disc-fed X-ray pulsars.     

Non-linear dynamics of the corotation torque

The excitation of spiral waves by an external perturbation in a disc deposits angular momentum in the vicinity of the corotation resonance (the radius where the speed of a rotating pattern matches the local rotation rate). We use matched asymptotic expansions to derive a reduced model that captures non-linear dynamics of the resulting torque and fluid motions. The model is similar to that derived for forced Rossby wave critical layers in geophysical fluid dynamics. Using the model we explore the saturation of the corotation torque, which occurs when the background potential (specific) vorticity is redistributed by the disturbance. We also consider the effects of dissipation. If there is a radial transport of potential vorticity, the corotation torque does not saturate. The main application is to the creation, growth and migration of protoplanets within discs like the primordial solar nebula. The disturbance also nucleates vortices in the vicinity of corotation, which may spark further epochs of planet formation.     

Dynamical friction for accelerated motion in a gaseous medium

Dynamical friction arises from the interaction of a perturber and the gravitational wake it excites in the ambient medium. This interaction is usually derived assuming that the perturber has a constant velocity. In realistic situations, motion is accelerated as for instance by dynamical friction itself. Here, we study the effect of acceleration on the dynamical friction force. We characterize the density enhancement associated with a constantly accelerating perturber with rectilinear motion in an infinite homogeneous gaseous medium and show that dynamical friction is not a local force and that its amplitude may depend on the perturber's initial velocity. The force on an accelerating perturber is maximal between Mach 1 and Mach 2, where it is smaller than the corresponding uniform motion friction. In the limit where the perturber's size is much smaller than the distance needed to change the Mach number by unity through acceleration, a subsonic perturber feels a force similar to uniform motion friction only if its past history does not include supersonic episodes. Once an accelerating perturber reaches large supersonic speeds, accelerated motion friction is marginally stronger than uniform motion friction. The force on a decelerating supersonic perturber is weaker than uniform motion friction as the velocity decreases to a few times the sound speed. Dynamical friction on a decelerating subsonic perturber with an initial Mach number larger than 2 is much larger than uniform motion friction and tends to a finite value as the velocity vanishes in contrast to uniform motion friction.     

On corotation torques, horseshoe drag and the possibility of sustained stalled or outward protoplanetary migration

We study the torque on low-mass protoplanets on fixed circular orbits, embedded in a protoplanetary disc in the isothermal limit. We consider a wide range of surface density distributions including cases where the surface density increases smoothly outwards. We perform both linear disc response calculations and non-linear numerical simulations. We consider a large range of viscosities, including the inviscid limit, as well as a range of protoplanet mass ratios, with special emphasis on the co-orbital region and the corotation torque acting between disc and protoplanet.
For low-mass protoplanets and large viscosity, the corotation torque behaves as expected from linear theory. However, when the viscosity becomes small enough to enable horseshoe turns to occur, the linear corotation torque exists only temporarily after insertion of a planet into the disc, being replaced by the horseshoe drag first discussed by Ward. This happens after a time that is equal to the horseshoe libration period reduced by a factor amounting to about twice the disc aspect ratio. This torque scales with the radial gradient of specific vorticity, as does the linear torque, but we find it to be many times larger. If the viscosity is large enough for viscous diffusion across the co-orbital region to occur within a libration period, we find that the horseshoe drag may be sustained. If not, the corotation torque saturates leaving only the linear Lindblad torques. As the magnitude of the non-linear co-orbital torque (horseshoe drag) is always found to be larger than the linear torque, we find that the sign of the total torque may change even for mildly positive surface density gradients. In combination with a kinematic viscosity large enough to keep the torque from saturating, strong sustained deviations from linear theory and outward or stalled migration may occur in such cases.     

Numerical simulations of type I planetary migration in non-turbulent magnetized discs

Using 2D magnetohydrodynamic (MHD) numerical simulations performed with two different finite-difference Eulerian codes, we analyse the effect that a toroidal magnetic field has on low-mass planet migration in non-turbulent protoplanetary discs. The presence of the magnetic field modifies the waves that can propagate in the disc. In agreement with a recent linear analysis, we find that two magnetic resonances develop on both sides of the planet orbit, which contribute to a significant global torque. In order to measure the torque exerted by the disc on the planet, we perform simulations in which the latter is either fixed on a circular orbit or allowed to migrate. For a     planet, when the ratio β between the square of the sound speed and that of the Alfven speed at the location of the planet is equal to 2, we find inward migration when the magnetic field   B _φ  is uniform in the disc, reduced migration when   B _φ  decreases as   r ⁻¹  and outward migration when   B _φ  decreases as   r ⁻²  . These results are in agreement with predictions from the linear analysis. Taken as a whole, our results confirm that even a subthermal stable field can stop inward migration of an earth-like planet.     

Self-regulated gravitational accretion in protostellar discs

We present a numerical model for the evolution of a protostellar disc that has formed self-consistently from the collapse of a molecular cloud core. The global evolution of the disc is followed for several million years after its formation. The capture of a wide range of spatial and temporal scales is made possible by use of the thin-disc approximation. We focus on the role of gravitational torques in transporting mass inward and angular momentum outward during different evolutionary phases of a protostellar disc with disc-to-star mass ratio of order 0.1. In the early phase, when the infall of matter from the surrounding envelope is substantial, mass is transported inward by the gravitational torques from spiral arms that are a manifestation of the envelope-induced gravitational instability in the disc. In the late phase, when the gas reservoir of the envelope is depleted, the distinct spiral structure is replaced by ongoing irregular non-axisymmetric density perturbations. The amplitude of these density perturbations decreases with time, though this process is moderated by swing amplification aided by the existence of the disc's sharp outer edge. Our global modelling of the protostellar disc reveals that there is typically a residual non-zero gravitational torque from these density perturbations, i.e. their effects do not exactly cancel out in each region. In particular, the net gravitational torque in the inner disc tends to be negative during first several million years of the evolution, while the outer disc has a net positive gravitational torque. Our global model of a self-consistently formed disc shows that it is also self-regulated in the late phase, so that it is near the Toomre stability limit, with a near-uniform Toomre parameter Q ≈ 1.5–2.0. Since the disc also has near-Keplerian rotation, and comparatively weak temperature variation, it maintains a near-power-law surface density profile proportional to r ^−3/2.     

On spin-up/spin-down torque reversals in disc accreting pulsars

The recent BATSE observations of the spin-up and spin-down of accreting pulsars have shown that the standard formulation for the accretion torque as proposed by Ghosh &38; Lamb may need to be revised. The observations indicate alternate spin-up and spin-down phases driven by torques of similar magnitude and typically larger than the mean torque. The variations of the torque in systems such as Cen X-3 are difficult to explain in terms of changes of the mass accretion rate. The implication is that the torque does not depend on the accretion rate as in the GL model. In this paper we argue that the observed changes in the spin rate can result from stochastic transitions between two magnetospheric states. In particular, we show that intermediate magnetospheric systems are not admissible, because of a disc-induced magnetospheric instability which exists in a star–disc magnetic interaction system. This explains why torque reversal occurs in disc accreting pulsars with similar magnitudes.     

Dynamical friction force exerted on spherical bodies

We present a rigorous calculation of the dynamical friction force exerted on a spherical massive perturber moving through an infinite homogenous system of field stars. By calculating the shape and mass of the polarization cloud induced by the perturber in the background system, which decelerates the motion of the perturber, we recover Chandrasekhar's drag force law with a modified Coulomb logarithm. As concrete examples, we calculate the drag force exerted on a Plummer sphere or a sphere with the density distribution of a Hernquist profile. It is shown that the shape of the perturber only affects the exact form of the Coulomb logarithm. The latter converges on small scales because encounters of the test and field stars with impact parameters less than the size of the massive perturber become inefficient. We confirm in this way the earlier results based on the impulse approximation of small angle scatterings.     

Orbital migrations in planetesimal discs: N-body simulations and the resonant dynamical friction

We have performed N -body numerical simulations of the exchange of angular momentum between a massive planet and a 3D Keplerian disc of planetesimals. Our interest is directed at the study of the classical analytical expressions of the lineal theory of density waves, as representative of the dynamical friction in discs 'dominated by the planet' and the orbital migration of the planets with regard to this effect. By means of a numerical integration of the equations of motion, we have carried out a set of numerical experiments with a large number of particles  ( N ≥10 000)  , and planets with the mass of Jupiter, Saturn and one core mass of the giant planets in the Solar system  ( M _c=10 M_⊕)  . The torque, measured in a phase in which a 'steady forcing' is clearly measurable, yields inward migration in a minimum-mass solar disc  (Σ∼10 g cm^-2  ), with a characteristic drift time of ∼ a few 10⁶ yr. The planets predate the disc, but the orbital decay rate is not sufficient to allow accretion in a time-scale relevant to the formation of giant planets. We found reductions of the measured torque on the planet, with respect to the linear theory, by a factor of 0.38 for M _c, 0.04 for Saturn and 0.01 for Jupiter, due to the increase in the perturbation on the disc. The behaviour of planets whose mass is larger than M _c is similar to the one of type II migrators in gaseous discs. Our results suggest that, in a minimum mass, solar planetesimals disc, type I migrations occur for masses smaller than M _c, whereas for this mass value it could be a transition zone between the two types of migration.     

On the rotation periods of the components of the triple system TYC 9300-0891-1AB/TYC 9300-0529-1 in the Octans Association

Stellar rotation depends on different parameters such as age, mass, initial chemical composition, initial angular momentum, and environment characteristics. The range of values of these parameters causes the dispersion in the rotation period distributions observed in young stellar clusters/associations. We focus our investigation on the effects of different circumstellar environments on stellar rotation. More specifically, we consider the effects of a perturber stellar companion on the accretion-disc lifetime at early evolution stages.We are searching in stellar Associations for visual triple systems where all stellar parameters are similar, with the only exceptions of the unknown initial rotation period, and of the circum-stellar environment, in the sense that one of the two about equal-mass components has a close-by third ‘perturber’ component.In the present study we analyze the 35-Myr old visual triple system TYC 9300-0891-1AB + TYC 9300-0529-1 in the young Octans stellar association consisting of three equal-mass K0V components. We collected from the literature all information that allowed us to infer that the three components are actually physically bound forming a triple system and are members of the Octans Association. We collected broad-band photometric timeseries in two observation seasons. We discovered that all the components are variable, magnetically active, and from periodogram analysis we found the unresolved components TYC 9300-0891-1AB to have a rotation period P = 1.383 d and TYC9300-0529-1 a rotation period P = 1.634 d.TYC 9300-0891-1A, TYC 9300-0891-1B, and TYC 9300-0529-1 have same masses, ages, and initial chemical compositions. The relatively small 16% rotation period difference measured by us indicates that all components had similar initial rotation periods and disc lifetimes, and the separation of 157 AU between the component A and the ‘perturber’ component B (or vice-versa) has been sufficiently large to prevent any significant perturbation/shortening of the accretion-disc lifetime.     

Determination of Large Asteroid Masses by the Dynamical Method

Masses of 19 asteroids have been determined from the analysis of their gravitational effect on the motion of perturbed bodies. The following asteroids were selected as perturbed bodies: (1) those which had single close encounters with the perturbing asteroid; (2) those whose mean motion was in a 1 : 1 commensurability with that of the perturber and which had close or moderate recurrent encounters with the perturber. The perturber mass was determined from observations of several tens of perturbed asteroids that were selected from these two groups. The selection criterion was the error of the mass determined from observations of only one asteroid. Positional observations of the asteroids on the interval 1900–2002 were used. The masses were determined with errors by an order-half an order of magnitude smaller than the masses found. The results are compared with those of other authors.     

Iron Kα line profiles and the inner boundary condition of accretion flows

Recent X-ray observations have shown evidence for exceptionally broad and skewed iron Kα emission lines from several accreting black hole systems. The lines are assumed to be due to fluorescence of the accretion disc illuminated by a surrounding corona and require a steep emissivity profile increasing into the innermost radius. This appears to question both standard accretion disc theory and the zero-torque assumption for the inner boundary condition, both of which predict a much less extreme profile. Instead it argues that a torque may be present due to magnetic coupling with matter in the plunging region or even to the spinning black hole itself. Discussion so far has centred on the torque acting on the disc. However, the crucial determinant of the iron line profile is the radial variation of the power radiated in the corona. Here we study the effects of different inner boundary conditions on the coronal emissivity and on the profiles of the observable Fe Kα lines. We argue that in the extreme case where a prominent highly redshifted component of the iron line is detected, requiring a steep emissivity profile in the innermost part and a flatter one outside, energy from the gas plunging into the black hole is being fed directly to the corona.     

Accretion disc–stellar magnetosphere interaction: field line inflation and the effect on the spin-down torque

We calculate the structure of a force-free magnetosphere which is assumed to corotate with a central star and which interacts with an embedded differentially rotating accretion disc. The magnetic and rotation axes are aligned, and the stellar field is assumed to be a dipole. We concentrate on the case when the amount of field line twisting through the disc–magnetosphere interaction is large , and consider different outer boundary conditions. In general the field line twisting produces field line inflation (e.g. Bardou & Heyvaerts), and in some cases with large twisting many field lines can become open. We calculate the spin-down torque acting between the star and the disc, and we find that it decreases significantly for cases with large field line twisting. This suggests that the oscillating torques observed for some accreting neutron stars could be caused by the magnetosphere varying between states with low and high field line inflation. Calculations of the spin evolution of T Tauri stars may also have to be revised in the light of the significant effect that field line twisting has on the magnetic torque resulting from star–disc interactions.     

Storm fronts over galaxy discs: models of how waves generate extraplanar gas and its anomalous kinematics

The existence of partially ionized, diffuse gas and dust clouds at kiloparsec scale distances above the central planes of edge-on, galaxy discs was an unexpected discovery about 20 years ago. Subsequent observations showed that this extended or extraplanar diffuse interstellar gas (EDIG) has rotation velocities approximately 10–20 per cent lower than those in the central plane, and has been hard to account for. Here, we present results of hydrodynamic models, with radiative cooling and heating from star formation. We find that in models with star formation generated stochastically across the disc, an extraplanar gas layer is generated as long as the star formation is sufficiently strong. However, this gas rotates at nearly the same speed as the midplane gas. We then studied a range of models with imposed spiral or bar waves in the disc. EDIG layers were also generated in these models, but primarily over the wave regions, not over the entire disc. Because of this partial coverage, the EDIG clouds move radially, as well as vertically, with the result that observed kinematic anomalies are reproduced. The implication is that the kinematic anomalies are the result of three-dimensional motions when the cylindrical symmetry of the disc is broken. Thus, the kinematic anomalies are the result of bars or strong waves, and more face-on galaxies with such waves should have an asymmetric EDIG component. The models also indicate that the EDIG can contain a significant fraction of cool gas, and that some star formation can be triggered at considerable heights above the disc mid-plane. We expect all of these effects to be more prominent in young, forming discs, to play a role in rapidly smoothing disc asymmetries and in working to self-regulate disc structure.