3D AMR Simulations of Point-Symmetric Nebulae

Erik-Jan Rijkhorst, Vincent Icke, and Garrelt Mellema, Sterrewacht Leiden, The Netherlandshttp://www.strw.LeidenUniv.nl/AstroHydro3D/Blowing Up Warped Disks in 3D

Planetary nebulae come in a wide variety of shapes, from spherical to highly bipolar. Some nebulae even have a multi-polar or point-symmetric shape [1]. Through two-dimensional hydrodynamic simulations Icke [2] showed that these seemingly enigmatic forms can be easily reproduced by a two-wind model in which the confining disk is warped, as is expected to occur in irradiated disks. Here, we present the extension of such a wind-disk interaction to three dimensions.

Point-Symmetric NebulaeShaping point-symmetric nebulae from the interaction between a warped disk and a fast, tenuous stellar wind proceeds by the following mechanism. As the wind impinges on the inner rim of the disk, a three-dimensional bow shock develops around it. This shock roughly has the shape of two ‘cones’ connected by their tips at the disk’s center. The opening angle of the shock depends inversely on the Mach number of the wind.

Imagining a two-dimensional cut through this developing bow shock, one sees that one branch flies off into space creating a lobe jutting out from the nebula, whereas the other slams into the concave side of the disk, scooping up disk material and thereby producing a set of smaller, unstable lobes. Ultimately, when the density of the disk is not too high, the wind breaks through the concave part of the disk, producing another pair of lobes. A 2D simulation of such an interaction is shown below.

Due to the radiative cooling the swept up shell is momentum driven and highly compressed. Therefore it is thin which is a necessary ingredient for the bow shock to produce the lobes.

Since the true bow shock is three-dimensional in structure and the disk is warped, rotating the two-dimensional cut shows that the concave side of the disk turns into the convex side and vice versa emphasizing the necessity of fully three-dimensional simulations of this interaction to truly understand the emerging point-symmetric structure.

Numerical ImplementationWe used the three-dimensional hydrocode Flash [6] to model the interaction between a spherical wind and a warped disk. This parallelized code implements block-structured adaptive mesh refinement (AMR) [7] (see images below) and a PPM type hydrosolver [8].

We added to the code the proper initial conditions for the wind-disk interaction as follows. To construct the warped disk, Eq. (1) was combined with a constant ‘wedge angle’ and a proper value for the constant A. The disk was given a constant density . The spherical wind was implemented as an inner boundary condition and given a 1/r2 density profile and a constant velocity. The pressure was calculated from an equation of state with a constant Poisson index .

3D AMR SimulationsWe ran wind-disk simulations in three dimensions on a Cartesian grid with a maximum effective resolution of 5123 computational cells using five levels of refinement. These simulations were carried out on the Dutch National Supercomputer (TERAS) using up to 196 processors in parallel.

Since we found in our two-dimensional calculations that simulations with cooling applied through a cooling curve do not result in a qualitatively different morphological outcome compared to calculations with a low value for the Poisson index, we opted not to use the cooling curve module during our three-dimensional simulations to save on computational time and used a value of =1.1 instead.

We used the following parameters: grid size 33, disk wedge angle 5, disk density 20, disk radius 0.6, disk constant A=1.2, wind speed 40, wind density 1, wind radius 0.1, and environment density 1.

On the right we show isosurfaces (left column) and corresponding color-coded plots of the synthesized H images (middle column) where the latter were produced by projecting the three-dimensional data cube at different angles onto the plane of the sky. We simply integrated the density squared along the line of sight and used this as a rough estimate for the emission. One sees that this leads to a wide variety of point-symmetric shapes.

The right column shows similar images for a simulation with a wind velocity that is 1.5 times as high and a disk density that is twice as high as the corresponding parameters from the simulation in the middle column.

Radiation Driven WarpWhen an accretion disk is subject to external torques it may become unstable to warping [3] and when irradiated by a sufficiently luminous central star, the difference in radiation pressure on slightly tilted annuli at different radii will induce a warp, even if the disk is initially flat [4]. For a PN the luminosity of the central star alone is sufficiently high to induce such a radiation driven warp.

Analytical considerations lead to expressions for growth and precession rates and morphologies of the warp. In a Cartesian coordinate system, the warped disk surface is given by

with local disk tilt angle (R, ), and orientation angle of the line of nodes (R, ) where R and are the non-orthogonal radial and azimuthal coordinates respectively, pointing to the surface of the disk.

In our model calculations we adopt the case of a steady precessing disk with no growth and zero torque at the origin [5] for which we have in the precessing frame that = A R and = sin / , with A a constant depending on the viscosity and accretion rate of the disk and the mass and luminosity of the central star. The image on the left shows such a disk surface when viewed from different angles.

2D simulation of the wind-disk interaction illustrating the mechanism described above. On the left the initial disk shape is shown where the central wind has just been switched on. On the right the fully developed bow shock and ‘punched holes’ are presented. The quantity shown in both plots is the log10 of the mass density.

Same figures as on the left but now with the AMR grid structure superimposed. Five different levels of refinement are visible.

References[1] Sahai, R. & Trauger, J.T. 1998, AJ, 116, 1357[2] Icke, V. 2003, A&A, 405, L11[3] Papaloizou, J.C.B. & Pringle, J.E. 1983, MNRAS, 202, 1181[4] Pringle, J.E 1996, MNRAS, 281, 357[5] Maloney, P.R., Begelman, M.C., & Pringle, J.E. 1996, ApJ, 472, 582[6] http://flash.uchicago.edu[7] Berger, M. & Oliger, J. 1984, J. Comp. Phys., 53, 484[8] Woodward, P. & Colella, P. 1984, J. Comp. Phys., 54, 115

An AstroHydro3D Project

AstroHydro3D is an initiative ofVincent Icke, Garrelt Mellema,Norbert Langer and Rien van deWeijgaert

http://www.strw.LeidenUniv.nl/AstroHydro3D/

AcknowledgementsThe software used in this work was in part developed by the DOE-supported ASCI/Alliance Center for Astrophysical Thermonuclear Flashes at the University of Chicago.