A hydrodynamical model of the circumstellar bubble created by two massive stars

Numerical models of the wind-blown bubble of massive stars usually only account for the wind of a single star. However, since massive stars are usually formed in clusters, it would be more realistic to follow the evolution of a bubble created by several stars. We develope a two-dimensional (2D) model of the circumstellar bubble created by two massive stars, a 40 solar mass star and a 25 solar mass star, and follow its evolution. The stars are separated by approximately 16 pc and surrounded by a cold medium with a density of 20 particles per cubic cm. We use the MPI-AMRVAC hydrodynamics code to solve the conservation equations of hydrodynamics on a 2D cylindrical grid using time-dependent models for the wind parameters of the two stars. At the end of the stellar evolution (4.5 and 7.0 million years for the 40 and 25 solar mass stars, respectively), we simulate the supernova explosion of each star. Each star initially creates its own bubble. However, as the bubbles expand they merge, creating a combined, aspherical bubble. The combined bubble evolves over time, influenced by the stellar winds and supernova explosions. The evolution of a wind-blown bubble created by two stars deviates from that of the bubbles around single stars. In particular, once one of the stars has exploded, the bubble is too large for the wind of the remaining star to maintain and the outer shell starts to disintegrate. The lack of thermal pressure inside the bubble also changes the behavior of circumstellar features close to the remaining star. The supernovae are contained inside the bubble, which reflects part of the energy back into the circumstellar medium.Comment: Accepted for publication in A&A. Six .avi files to be published online (uploaded to ArXiv DC and available as ancillary files) (updated after language corrections

Differential calculus on a Lie algebroid and Poisson manifolds

A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present in this paper the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds.Comment: 46 page

A direct proof of Malus' theorem using the symplectic structure of the set of oriented straight lines

We present a direct proof of Malus' theorem in geometrical Optics founded on the symplectic structure of the set of all oriented straight lines in an Euclidean affine space. Nous pr\'esentens une preuve directe du th\'eor\`eme de Malus de l'optique g\'eom\'etrique bas\'ee sur la structure symplectique de l'ensemble des droites orient\'ees d'un espace affine euclidien

Constraints on gamma-ray burst and supernova progenitors through circumstellar absorption lines. (II): Post-LBV Wolf-Rayet stars

Van Marle et al. (2005) showed that circumstellar absorption lines in early Type Ib/c supernova and gamma-ray burst afterglow spectra may reveal the progenitor evolution of the exploding Wolf-Rayet star. While the quoted paper deals with Wolf-Rayet stars which evolved through a red supergiant stage, we investigate here the initially more massive Wolf-Rayet stars which are thought to evolve through a Luminous Blue Variable (LBV) stage. We perform hydrodynamic simulations of the evolution of the circumstellar medium around a 60 Msol star, from the main sequence through the LBV and Wolf-Rayet stages, up to core collapse. We then compute the column density of the circumstellar matter as a function of radial velocity, time and angle. This allows a comparison with the number and blue-shifts, of absorption components in the spectra of LBVs, Wolf-Rayet stars, Type Ib/c supernovae and gamma-ray burst afterglows. Our simulation for the post-LBV stage shows the formation of various absorption components, which are, however, rather short lived; they dissipate on time scales shorter than 50,000yr. As the LBV stage is thought to occur at the beginning of core helium burning, the remaining Wolf-Rayet life time is expected to be one order of magnitude larger. When interpreting the absorption components in the afterglow spectrum of GRB-021004 as circumstellar, it can be concluded that the progenitor of this source did most likely not evolve through an LBV stage. However, a close binary with late common-envelope phase (Case C) may produce a circumstellar medium that closely resembles the LBV to Wolf-Rayet evolution, but with a much shorter Wolf-Rayet period.Comment: accepted for publication by A&

Radiative cooling in numerical astrophysics: the need for adaptive mesh refinement

Energy loss through optically thin radiative cooling plays an important part in the evolution of astrophysical gas dynamics and should therefore be considered a necessary element in any numerical simulation. Although the addition of this physical process to the equations of hydrodynamics is straightforward, it does create numerical challenges that have to be overcome in order to ensure the physical correctness of the simulation. First, the cooling has to be treated (semi-)implicitly, owing to the discrepancies between the cooling timescale and the typical timesteps of the simulation. Secondly, because of its dependence on a tabulated cooling curve, the introduction of radiative cooling creates the necessity for an interpolation scheme. In particular, we will argue that the addition of radiative cooling to a numerical simulation creates the need for extremely high resolution, which can only be fully met through the use of adaptive mesh refinement.Comment: 11 figures. Accepted for publication in Computers & Fluid

Multi-dimensional models of circumstellar shells around evolved massive stars

Massive stars shape their surrounding medium through the force of their stellar winds, which collide with the circumstellar medium. Because the characteristics of these stellar winds vary over the course of the evolution of the star, the circumstellar matter becomes a reflection of the stellar evolution and can be used to determine the characteristics of the progenitor star. In particular, whenever a fast wind phase follows a slow wind phase, the fast wind sweeps up its predecessor in a shell, which is observed as a circumstellar nebula. We make 2-D and 3-D numerical simulations of fast stellar winds sweeping up their slow predecessors to investigate whether numerical models of these shells have to be 3-D, or whether 2-D models are sufficient to reproduce the shells correctly. We focus on those situations where a fast Wolf-Rayet (WR) star wind sweeps up the slower wind emitted by its predecessor, being either a red supergiant or a luminous blue variable. As the fast WR wind expands, it creates a dense shell of swept up material that expands outward, driven by the high pressure of the shocked WR wind. These shells are subject to a fair variety of hydrodynamic-radiative instabilities. If the WR wind is expanding into the wind of a luminous blue variable phase, the instabilities will tend to form a fairly small-scale, regular filamentary lattice with thin filaments connecting knotty features. If the WR wind is sweeping up a red supergiant wind, the instabilities will form larger interconnected structures with less regularity. Our results show that 3-D models, when translated to observed morphologies, give realistic results that can be compared directly to observations. The 3-D structure of the nebula will help to distinguish different progenitor scenarios.Comment: Accepted for publication in A&A. All figures in low resolution. v2: language corrections and addition of DOI numbe

Thin shell morphology in the circumstellar medium of massive binaries

We investigate the morphology of the collision front between the stellar winds of binary components in two long-period binary systems, one consisting of a hydrogen rich Wolf-Rayet star (WNL) and an O-star and the other of a Luminous Blue Variable (LBV) and an O-star. Specifically, we follow the development and evolution of instabilities that form in such a shell, if it is sufficiently compressed, due to both the wind interaction and the orbital motion. We use MPI-AMRVAC to time-integrate the equations of hydrodynamics, combined with optically thin radiative cooling, on an adaptive mesh 3D grid. Using parameters for generic binary systems, we simulate the interaction between the winds of the two stars. The WNL+O star binary shows a typical example of an adiabatic wind collision. The resulting shell is thick and smooth, showing no instabilities. On the other hand, the shell created by the collision of the O star wind with the LBV wind, combined with the orbital motion of the binary components, is susceptible to thin shell instabilities, which create a highly structured morphology. We identify the nature of the instabilities as both linear and non-linear thin-shell instabilities, with distinct differences between the leading and the trailing parts of the collision front. We also find that for binaries containing a star with a (relatively) slow wind, the global shape of the shell is determined more by the slow wind velocity and the orbital motion of the binary, than the ram pressure balance between the two winds. The interaction between massive binary winds needs further parametric exploration, to identify the role and dynamical importance of multiple instabilities at the collision front, as shown here for an LBV+O star system.Comment: 10 pages, 13 figures. Accepted for publication in A&

Luminous Blue Variables & Mass Loss near the Eddington Limit

During the course of their evolution, massive stars lose a substantial fraction of their initial mass, both through steady winds and through relatively brief eruptions during their Luminous Blue Variable (LBV) phase. This talk reviews the dynamical driving of this mass loss, contrasting the line-driving of steady winds to the potential role of continuum driving for eruptions during LBV episodes when the star exceeds the Eddington limit. A key theme is to emphasize the inherent limits that self-shadowing places on line-driven mass loss rates, whereas continuum driving can in principle drive mass up to the "photon-tiring" limit, for which the energy to lift the wind becomes equal to the stellar luminosity. We review how the "porosity" of a highly clumped atmosphere can regulate continuum-driven mass loss, but also discuss recent time-dependent simulations of how base mass flux that exceeds the tiring limit can lead to flow stagnation and a complex, time-dependent combination of inflow and outflow regions. A general result is thus that porosity-mediated continuum driving in super-Eddington phases can explain the large, near tiring-limit mass loss inferred for LBV giant eruptions.Comment: Conference proceedings, Massive Stars as Cosmic Engines, IAU Symp 250, ed. F. Bresolin, P. A. Crowther, & J. Puls (Cambridge Univ. Press