Old site ​https://clover.pas.rochester.edu/trac/astrobear/blog/crl618Figures

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Paper Section 4, COMPARISON TO MODEL SIMULATIONS

4.1. Models

Here we describe the methodology and implementation of our CRL618 models in detail. We have carried out two Eulerian-grid numerical simulations to follow the formation of nebular lobes via the propagation of a bullet or a jet into a stratified and ridged ambient medium. Fluid dynamics is followed in 3 dimensions using the equations of radiation hydrodynamics. The effects of optically thin cooling have been included using the cooling tables of Dalgarno & McCray (1972). The hydrodynamic equations are solved with the adaptive mesh refinement (AMR) numerical code AstroBEAR2.0 [FOOTNOTE: https://clover.pas.rochester.edu/trac/astrobear/wiki]. In particular, the Euler equations with cooling source terms are solved using a second-order MUSCL Hancock shock capturing scheme and Marquina flux functions (Cunningham et al. 2009). While AstroBEAR is able to solve the equations of magnetohydrodynamics (MHD) and to compute several microphysical processes, such as gas self-gravity and heat conduction, we do not consider these processes in the present study. The computational domain is a rectangle with dimensions 0<x<24000 AU, -4000<y,z<4000 AU. We use a coarse grid with 200x100x100 cells along with two adaptive refinement levels which increases the grid resolution by a factor of 4; the simulations attain a maximum resolution of 20 AU. Typical simulation flow times are of order 400 yr. We use BlueHive [FOOTNOTE: http://www.circ.rochester.edu/wiki/index.php/BlueHive_Cluster] and Blue Gene/P [FOOTNOTE: http://www.circ.rochester.edu/wiki/index.php/Blue_Gene/P] —IBM's parallel cluster and supercomputer, respectively— which are maintained by the Center for Integrated Research Computing of the University of Rochester. We ran simulations for about 2days, using 256 processors.

4.1.1. Initial conditions

We set the ambient medium with a static velocity (V=0) field and a density profile which decreases with distance form the origin (0,0,0) and has a series of periodic spherical ridges spaced by ~333 AU. In detail,

n_amb(r)= 300 / (r/500AU)² * .5 + SIGMA_i exp(-[6d0*{(r/500AU)-5/3-i}]²) particles / cm³,

where r is the radial distance form the origin and i is an integer number. Our choice for such structure is based on Fig. 1 of NF+07, where it is clear that the AGB “spherical halo” containing the lobes of CRL 618 has a stratified structure with semi-periodic ridges of enhanced density separated by ≈1″ (section 3). Observational studies of AGB winds suggests that they expand isotropically with mass-loss rates and velocities of order 10^-5 M_sun/yr and 20km/s, respectively (see e.g. Hrivnak et al. 1989; Bujarrabal et al. 2001). Our model AGB circumstellar medium is static, however, and we do not expect this difference to affect the results of our model because (i) we explore short, ~200yr, nebular expansion times, (ii) the axial velocity of the bullet/jet is more than an order of magnitude faster (300km/s; below) that the expected AGB wind terminal velocities.

BULLET MODEL

We base our bullet model in that of Dennis et al. (2008). At time=0 yr, a spherical bullet is placed in the grid at coordinates (r_b, 0, 0), where r_b=500AU and it is the bullet's, and the jet's, radius, with an axial velocity of 300km/s and a density profile given by

n_b (r) = min( 100*n_amb(r) , r_b²/r² ) particles / cm³.

JET MODEL

For the jet model, we continually inject gas at the grid cells which are located at the bottom face of our computational domain, within r<r_b. The injected gas has a constant axial velocity of 300km/s and a constant density of

n_j = max(n_b(r)) particles / cm³.

4.1.2. Evolution.

We present the results of our CRL 618 numerical simulations using 2d maps of density and velocity field (Figure 3, columns 1 and 3, respectively), and 1d plots of density (Figure 3, columns 2 and 4, top panel) and velocity (Figure 3, columns 2 and 4, bottom panel). All the images in Figure 3 were done using the computational data located at the middle plane of the computational doman, thus no integration of density or velocity along the line of sight was carried out.

The 2d maps in columns 1 and 3 of Figure 3 show the following structures. The bullet/jet (left/right) are the densest whitest central features. Below the bullet, or to the right of the jet (without loss of generality), we see the lobe that is formed by the bullet/jet, which is separated from the ambient medium by a contact discontinuity. Beyond the lobe we see the ambient medium which is stratified and shows a series of concentric spherical shells given by the initial conditions (above). The vertical short red, or blue, lines at the top of the maps show the positions at which the profiles in Figure 3 (columns 2 and 4) have been taken.

The bullet and the jet propagate at 300km/s, away from the ambient medium's densest region (upwards), forming an elongated lobe. We see that the expansion of the lobes is predominantly axial; the major to minor lobes' axes ratio is always > 2. We find that the lobes formed by the bullet and the jet are quite similar (compare left vs. right panels in columns 1 and 3, Figure 3), except for the material located within a bullet/jet radii from the axis. As the bullet (jet) penetrates the ambient's shells, these form regularly separated vertebrae-looking features along the lobe. The axial velocity of these features decreases with radial distance from the axis.

We see that the bullet shrinks as it propagates though the ambient medium (compare columns 1 and 3, Figure 3). Similarly, the head of the jet seems to adopt a conical up-ward pointing geometry as it propagates. Such effect is due to ablation of the material located at the working surface of the bullet/jet as it interacts with the ambient medium. Such effect is consisten with the models of Dennis et al. (2008).

The density profiles in Figure 3 (columns 2 and 4, top panel) show the following.

the evolution of the bullet and the jet quantitatively, as a function of time and position inside the lobes.

Fig. 3. Models of a bullet and jet with outflow speeds of 300 km s–1 propagating through an AGB wind with periodic density ridges spaced by 333 AU after 100 y (left half) and 200y (right half). The grey-scale panels show the density distribution and the streamlines in the midplane for bullets (left half) and jets (right half). The associated densities and radial velocities along lines in the midplane on or displaced from the symmetry axes are plotted to the right of each of the grey-scale images. The offsets are labeled and indicted by vertical tics along the top of each grey-scale image.
Fig 4. Predicted position-velocity diagrams of bullets (left panels) and jets (right panels) at times t = 100y and 200y integrated over a slit. Slit widths are indicated on magnified 3-D model projections of the fingertips (insets). The symmetry axes of the P-V diagrams are inclined by 30o to the plane of the sky. The grey scales are proportional to the surface brightness computed from the square of the density. Note that the bow shocks and their trailing wings—but not the lateral edges of the fingers—are visible in HST images through emission-line filters (cf. Fig. 5). Thus observed P-V diagrams are composed of (1) a tilted line connecting the nucleus and bow shock and (2) a second line left of the bow shock.
Fig 5. Bullet and jet model simulations (midplane) at 100y compared to the eastern fingers of CRL618. [NII] emission is primarily intrinsic whereas other filters become dominated by scattered stellar light to the left of the fingertips.

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Bruce's wish: that Martin/Adam explain why the speeds of the gas in the body of the lobes follow a Hubble-like pattern since Bruce makes the case in section 5 that this is where the H_2 emission is likely to arise.