this point make it very hard for the CFD code to function: the extremely small chemical tim e-step and a complex dynamic scenario result in time-steps of the order hundredths of a second rather than several tens of seconds as can be achieved in much cooler and less dense parts of the flow. This is not practical with the current com putational facilities so the models must be started at later times. Alternatively, by doing dynamics-only runs (no chemistry) for earlier times and understanding th a t the physics is only very loosely coupled to the chemistry, it is possible to use later models to make inferences about earlier times. Furtherm ore, the physics calculated in these runs can be supplied as controlling param eters in a purely chemical model.
Using power-law assumptions (equations (2.32) and (2.34))and models as in earlier chapters, we can calculate an initial tem perature and density for any tim e based on a single boundary condition. Evans et al. (1996) measure a density shortly after maximum light of the order of lO^^-lO^^ cm~^ and a tem perature of around 4000 K. It was found impossible to make the model run with a step size greater than thousandths of second with these initial conditions so it was necessary to s ta rt modelling later in the flow. A density of 4.0x10® cm “ ^ calculated for some 20 days after maximum light allowed the model to progress at an acceptable rate (time-steps frequently of the order 1-3 seconds). W hilst this may seem to be long after visual maximum, dust formation does not occur until approxim ately 40 days later which, given th a t the chemistry reaches equilibrium within a m atter of seconds, leaves adequate time for the model to display any quasi-static behaviour of interest. Whilst the initial density is lowered to reflect this s ta rt time, the tem perature is maintained at 4000 Kelvin as a result of experience with CFD models: it is found th a t the shell tem perature does not fall significantly over 20 days. Furtherm ore, Evans et al. (1996) state th a t a tem perature of 4500 K is measured in a ‘relatively cool neutral zone’ when CO is observed; this implies the likelihood of much higher tem peratures elsewhere and more generally, a considerable uncertainty in tem peratures throughout the ejecta. We shall also show in these models th a t in order to develop even very localised regions in which CO can form, the bulk of the ejecta must itself be cool.
Evans et al. (1996) quote a decline rate for the visual m agnitude of ihy = 0.044 m agnitudes d ay“ U From equation (3.18) we find an ejecta velocity of 5.5 X10^ cm s~^. This is consistent with as yet unpublished optical spectra obtained by Delisle and Beauchamp, 1994, showing the Na I D lines at a velocity of -500 k m s“ U We use this as the maximum
160 CHAPTER 5. A FLUID DYNAMIC MODEL
homologous spherical expansion.
At 30 days post-outburst (~20 days post maximum light) the shell radius Rej = 1.30 X 10^4 cm and a shell thickness of O.lRej is assumed (Rawlings and Williams (1989)). A one dimensional model grid of 513 points was chosen after experimentation showed th a t this provided sufficient spatial resolution to show structure within the shell whilst keeping acceptable run times. A scale of 1.95x10^^ cm pixel"^ perm its ten days of evolution to fit within the grid.
It may seem a good idea to set the frame of reference of the model to be th a t of the expanding shell thus ensuring th a t it (the point of interest) remains within the confines of the grid a t all times. Whilst this seems a better and more practical way in which to define the problem, difficulties were encountered th a t have not yet been resolved; small shocks were seen to form at the inside edge of the ejecta shell th a t are neither expected nor observed in models where the frame of reference is th a t of the nova. The reasons for this disparity between physically identical models have not yet been identified and so all models thus far have been set in the white dwarf frame of reference.
Figure 5.1 plots the tem perature, density, radial velocity^, and gas number density a t the s ta rt of the model. The param eters of the 1®*^, i.e. left-most, pixel are considered boundary conditions and are not perm itted to change throughout the execution of the model. In this way we define a constant ‘wind’ entering from the left hand side. This seems a fair approximation; given th a t the material at all grid points behind but not encompassed by the shell is assumed to be entrained by the passage of the ejecta, it is assigned a velocity equal to th a t of the inside edge of the shell. If this were not made continuous by virtue of the continuous wind boundary condition, a low pressure region would form a t the edge of the grid which does not seem to be physically realistic.
The plot of the initial velocity profile (figure 5.1) clearly shows a non-constant profile within the ejecta shell where v oc r. Note th a t this figure covers only the part of the grid in which the shell resides and the radial axis units are in centimetres relative to the left hand edge of the grid which itself is located some 10^^ cm from the white dwarf. This is consistent with a spherical, homologous expansion and the model code itself also assumes spherical symmetry.. The initial conditions assume th a t the extremely tenuous ‘ISM ’, defined at all grid positions not occupied by the shell, is swept up by the shell whilst an
4 n these one dim ensional m odels there is only a. velocity in the x, or radial direction; the ^-com ponent is set to zero throughout
.5.5. WORKING MODELS 161 r ( x I O c m ) 1. 0 Velocity Density 1 0 0 Grid Pi xe l s
F ig u r e 5.1: Initial s t a t e of a sim ple, single shell, C F D e j e c ta m odel. T h e u p p e r axis show s t h e real s p a c e d im e n s io n s of t h e grid w ith t h e left h a n d edge being a t a ra d iu s of lOd^ cm from t h e w h ite d w arf.
162 CHAPTER 5. A FLUID DYNAMIC MODEL
equally tenuous tail follows the principal shell. In practice, it is found th a t the velocity to which this trailing medium is set makes little difference to the evolution of the ejecta shell itself, which is the focus of our attention, except at the inside edge of the shell.
The chemical make-up of the ejecta shell is initially set to be as follows:
S p e c ie s: H H2 0 C
n(i) cm"3 4.04 x 10^ 5.27 x 1 0^ 4.04 x 10^ 8 . 8 x 10^
These abundances are approximately as for the models of Appendix A. All other species are set to a zero-level abundance of 1 0 0 cm “ ^ as are all species at grid-points not located
within the ejecta shell. It is not possible within this model code to set any abundance to zero (for numerical reasons) therefore some appropriate zero-level abundance m ust be chosen. In this case we do wish to have some tenuous medium to represent circum- binary m aterial or the ISM, but no assumptions are made about its composition. W ith all abundances set to a very low level we provide such a medium but it is not sufficiently rich in any particular species for it to alter the chemistry of the ejecta shell. The tem perature is set to 500 K which is ju st below the lowest tem perature for which the model will com pute chemistry. As a result, the medium not contained within the ejectum shell will remain both dynamically and chemically inert until the shell interacts with it. In this way the model is made more efficient for not spending time calculating chemistry outside of the region of interest and the boundary conditions are retained. This satisfies the need for a relatively simple model scenario but is not intended to dismiss the possibility th a t circum-binary m aterial does, in fact, play a rôle in the chemistry of the ejecta. Any contribution from the circum-binary medium would necessarily be atomic rather than molecular as m aterial transfered from this medium to the ejecta shell would have first to traverse a shock region where the extrem e tem peratures can be expected to dissociate all molecules.
It will be seen in figure 5.1 th a t the tem perature profile is extremely simple. The model quickly relaxes away from this crude approximation to something more physically realistic. It is not claimed th a t this initial setting is physically realistic.