Disc Thermodynamics Research Articles

Modelling planet-induced gaps and rings in ALMA discs: the role of in-plane radiative diffusion

ABSTRACT ALMA observations of protoplanetary discs in dust continuum emission reveal a variety of annular structures. Attributing the existence of such features to embedded planets is a popular scenario, supported by studies using hydrodynamical models. Recent work has shown that radiative cooling greatly influences the capability of planet-driven spiral density waves to transport angular momentum, ultimately deciding the number, position, and depth of rings and gaps that a planet can carve in a disc. However, radiation transport has only been treated via local thermal relaxation, not taking into account radiative diffusion along the disc plane. We compare the previous state-of-the-art models of planet–disc interaction with local cooling prescriptions to our new models that include cooling in the vertical direction and radiative diffusion in the plane of the disc, and show that the response of the disc to the induced spiral waves can differ significantly when comparing these two treatments of the disc thermodynamics. We follow up with synthetic emission maps of ALMA systems, and show that our new models reproduce the observations found in the literature better than models with local cooling. We conclude that appropriate treatment of radiation transport is key to constraining the parameter space when interpreting ALMA observations using the planet–disc interaction scenario.

Calibrated gas accretion and orbital migration of protoplanets in 1D disc models

Context. Orbital migration and gas accretion are two interdependent key processes that govern the evolution of planets in protoplanetary discs. The final planetary properties such as masses and orbital periods strongly depend on the treatment of those two processes. Aims. Our aim is to develop a simple prescription for migration and accretion in 1D disc models, calibrated with results of 3D hydro-dynamic simulations. Our focus lies on non-self-gravitating discs, but we also discuss to what degree our prescription could be applied when the discs are self-gravitating. Methods. We studied migration using torque densities. Our model for the torque density is based on existing fitting formulas, which we subsequently modify to prevent premature gap-opening. At higher planetary masses, we also apply torque densities from hydrody-namic simulations directly to our 1D model. These torque densities allow us to model the orbital evolution of an initially low-mass planet that undergoes runaway-accretion to become a massive planet. The two-way exchange of angular momentum between disc and planet is included. This leads to a self-consistent treatment of gap formation that only relies on directly accessible disc parameters. We present a formula for Bondi and Hill gas accretion in the disc-limited regime. This formula is self-consistent in the sense that mass is removed from the disc in the location from where it is accreted. The prescription is appropriate when the planet is smaller than, comparable to, or larger than the disc scale height. Results. We find that the resulting evolution in mass and semi-major axis in the 1D framework is in good agreement with those from 3D hydrodynamical simulations for a range of parameters. Conclusions. Our prescription is valuable for simultaneously modelling migration and accretion in 1D models, which allows a planet’s evolution to be followed over the entire lifetime of a disc. It is applicable also in situations where the surface density is significantly disturbed by multiple gap-opening planets or processes like infall. We conclude that it is appropriate and beneficial to apply torque densities from hydrodynamic simulations in 1D models, at least in the parameter space we study here. More work is needed in order to determine whether our approach is also applicable in an even wider parameter space and in situations with more complex disc thermodynamics, or when the disc is self-gravitating.

Integrated simulation platform for a locomotive braking system

We used a locomotive and a rail car to examine the working principle of a locomotive braking system. We established models of the locomotive braking system, the locomotive dynamics, and the brake disc thermodynamics and, based on the correlation parameters for each subsystem, built an integrated simulation platform for the locomotive braking system based on Simulink and AMESim. Using this platform, we simulated the characteristics of the pneumatic braking unit, the locomotive braking, and temperature increase in the brake disc under emergency braking conditions. We compared our simulation results with experimental data and the results showed that the integrated simulation platform for the locomotive braking system could successfully be used to study locomotive braking control on the vehicle level. We provide a design optimization method for the development of a braking system, the setting of the anti-skid criterion, and early warning of an increase in the brake disc temperature.

Importance of magnetic fields in highly eccentric discs with applications to tidal disruption events

ABSTRACT Whether tidal disruption events (TDEs) circularize or accrete directly as a highly eccentric disc is the subject of current research and appears to depend sensitively on the disc thermodynamics. In a previous paper, we applied the theory of eccentric discs to TDE discs using an α-prescription for the disc stress, which leads to solutions that exhibit extreme, potentially unphysical, behaviour. In this paper, we further explore the dynamical vertical structure of highly eccentric discs using alternative stress models that are better motivated by the behaviour of magnetic fields in eccentric discs. We find that the presence of a coherent magnetic field has a stabilizing effect on the dynamics and can significantly alter the behaviour of highly eccentric radiation-dominated discs. We conclude that magnetic fields are important for the evolution of TDE discs.

Dynamical structure of highly eccentric discs with applications to tidal disruption events

ABSTRACT Whether tidal disruption events circularize or accrete directly as highly eccentric discs is the subject of current research and appears to depend sensitively on the disc thermodynamics. One aspect of this problem that has not received much attention is that a highly eccentric disc must have a strong, non-hydrostatic variation of the disc scale height around each orbit. As a complement to numerical simulations carried out by other groups, we investigate the dynamical structure of TDE discs using the non-linear theory of eccentric accretion discs. In particular, we study the variation of physical quantities around each elliptical orbit, taking into account the dynamical vertical structure, as well as viscous dissipation and radiative cooling. The solutions include a structure similar to the nozzle-like structure seen in simulations. We find evidence for the existence of the thermal instability in highly eccentric discs dominated by radiation pressure. For thermally stable solutions many of our models indicate a failure of the α-prescription for turbulent stresses. We discuss the consequences of our results for the structure of eccentric TDE discs.

Influence of grain growth on the thermal structure of protoplanetary discs

The thermal structure of a protoplanetary disc is regulated by the opacity that dust grains provide. However, previous works have often considered simplified prescriptions for the dust opacity in hydrodynamical disc simulations, for example, by considering only a single particle size. In the present work, we perform 2D hydrodynamical simulations of protoplanetary discs where the opacity is self-consistently calculated for the dust population, taking into account the particle size, composition, and abundance. We first compared simulations utilizing single grain sizes to two different multi-grain size distributions at different levels of turbulence strengths, parameterized through the α-viscosity, and different gas surface densities. Assuming a single dust size leads to inaccurate calculations of the thermal structure of discs, because the grain size dominating the opacity increases with orbital radius. Overall the two grain size distributions, one limited by fragmentation only and the other determined from a more complete fragmentation-coagulation equilibrium, give comparable results for the thermal structure. We find that both grain size distributions give less steep opacity gradients that result in less steep aspect ratio gradients, in comparison to discs with only micrometer-sized dust. Moreover, in the discs with a grain size distribution, the innermost (<5 AU) outward migration region is removed and planets embedded in such discs experience lower migration rates. We also investigated the dependency of the water iceline position on the alpha-viscosity (α), the initial gas surface density (Σg,0) at 1 AU and the dust-to-gas ratio (fDG) and find rice ∝ α0.61Σg,00.8fDG0.37 independently of the distribution used in the disc. The inclusion of the feedback loop between grain growth, opacities, and disc thermodynamics allows for more self-consistent simulations of accretion discs and planet formation.

On the vortex evolution in non-isothermal protoplanetary discs

ABSTRACT It is believed that large-scale horseshoe-like brightness asymmetries found in dozens of transitional protoplanetary discs are caused by anticyclonic vortices. These vortices can play a key role in planet formation, as mm-sized dust – the building blocks of planets – can be accumulated inside them. Anticyclonic vortices are formed by the Rossby wave instability, which can be excited at the gap edges opened by a giant planet or at sharp viscosity transitions of accretionally inactive regions. It is known that vortices are prone to stretching and subsequent dissolution due to disc self-gravity for canonical disc masses in the isothermal approximation. To improve the hydrodynamic model of protoplanetary discs, we include the disc thermodynamics in our model. In this paper, we present our results on the evolution of the vortices formed at the outer edge of an accretionally inactive region (dead zone) assuming an ideal equation of state and taking PdV work, disc cooling in the β-approximation, and disc self-gravity into account. Thermodynamics affects the offset and the mode number (referring to the number of small vortices at the early phase) of the RWI excitation, as well as the strength, shape, and lifetime of the large-scale vortex formed through merging of the initial small vortices. We found that the inclusion of gas thermodynamics results in stronger, however decreased lifetime vortices. Our results suggest that a hypothetical vortex-aided planet formation scenario favours effectively cooling discs.

Stellar irradiated discs and implications on migration of embedded planets

The strength and direction of migration of embedded low mass planets depends on the disc's structure. It has been shown that, in discs with viscous heating and radiative transport, the migration can be directed outwards. In this paper we investigate the influence of a constant dM/dt-flux through the disc, as well as the influence of the disc's metallicity on the disc's thermodynamics. We focus on dM/dt discs, which have a net mass flux through them. Utilizing the resulting disc structure, we determine the regions of outward migration in the disc. We perform numerical hydrosimulations of dM/dt discs with viscous heating, radiative cooling and stellar irradiation in 2D in the r-z-plane. We use the explicit/implicit hydrodynamical code FARGOCA that includes a full tensor viscosity and stellar irradiation, as well as a two temperature solver that includes radiation transport in the flux-limited diffusion approximation. The migration of embedded planets is studied by using torque formulae. For a disc of gas surface density and viscosity, we find that the discs thermal structure depends on the product Sigma_G \nu and the amount of heavy elements, while the migration of planets additionally to the mentioned quantities, depends on the amount of viscosity itself. As a result of this, the disc structure can not be approximated by simple power laws. During the lifetime of the disc, the structure of the disc changes significantly in a non-linear way in the inner parts. In the late stages of the discs evolution, outward migration is only possible if the metallicity of the disc is high. For low metallicity, planets would migrate inwards and could potentially be lost. The presented disc structures and migration maps have important consequences on the formation of planets, as they can give hints on the different formation mechanisms for different types of planets as a function of metallicity.

Stellar irradiated discs and implications on migration of embedded planets

The strength and direction of migration of low mass planets depends on the disc's thermodynamics. In discs where the viscous heating is balanced by radiative transport, the migration can be directed outwards, a process which extends the lifetime of growing planetary embryos. We investigate the influence of opacity and stellar irradiation on the disc thermodynamics. Utilizing the resulting disc structure, we determine the regions of outward migration. We perform two-dimensional numerical simulations of equilibrium discs with viscous heating, radiative cooling and stellar irradiation. We use the hydrodynamical code NIRVANA that includes a full tensor viscosity and stellar irradiation, as well as a two temperature solver that includes radiation transport in the flux-limited diffusion approximation. The migration is studied by using torque formulae. In the constant opacity case, we reproduce the analytical results of a black-body disc: the stellar irradiation dominates in the outer regions -- leading to flaring -- while the viscous heating dominates close to the star. We find that the inner edge of the disc should not be significantly puffed-up by the stellar irradiation. If the opacity depends on the local density and temperature, the structure of the disc is different, and several bumps in the aspect ratio H/r appear, due to transitions between different opacity regimes. The bumps in the disc can shield the outer disc from stellar irradiation. Stellar irradiation is an important factor for determining the disc structure and has dramatic consequences for the migration of embedded planets. Compared to discs with only viscous heating, a stellar irradiated disc features a much smaller region of outward migration for a range of planetary masses. This suggests that the region where the formation of giant planet cores takes place is smaller, which in turn might lead to a shorter growth phase.

Rayleigh adjustment of narrow barriers in protoplanetary discs

Sharp density features in protoplanetary discs, for instance at the edge of a magnetically dead zone, have recently been proposed as effective barriers to slow down or even stop the problematically fast migration of planetary cores into their central star. Density features on a radial scale approaching the disc vertical scale height might not exist, however, since they could be Rayleigh (or more generally Solberg-Hoiland) unstable. Stability must be checked explicitly in one-dimensional viscous accretion disc models because these instabilities are artificially eliminated in the process of reducing the full set of axisymmetric equations. The disc thermodynamics, via the entropy stratification, and its vertical structure also influence stability when sharp density features are present. We propose the concept of Rayleigh adjustment for viscous disc models: any density feature that violates Rayleigh stability (or its generalization) should be diffused radially by hydrodynamical turbulence on a dynamical time-scale, approaching marginal stability in a quasi-continuous manner.