T he M odel

We present a tim e dependent, chemical model code th a t integrates the ordinary differential equations describing a given chemistry over time. The physical scenario is one of a simple parcel of gas within an outflow modelled within a co-moving frame of reference. Reac­ tion rates are computed for each time-step with photorates param eterised as a function of photospheric tem perature and computed using realistic fields in advance. Nova param e­ ters such as the nova speed (rhy), initial number density, kinetic tem perature and initial chemical abundances may all be supplied to the code by means of a separate description file.

Two key physical param eters, tem perature and density, are calculated a t each time- step by means of simple power laws. The geometry of the shell and the shape of the cooling function is defined by the density and tem perature power law index supplied as input to the model. The meaning of these indices is described in this section. This choice is one th a t works fast and allows the behaviour of a chemistry under changing conditions to be observed. This may not model the environment as closely as a fluid dynamic approach, but neither does it artificially impose constraints on the chemistry. Our interest is in the chem istry and its behaviour in a changing environment.

In order to explore different shell scenarios let us s ta rt by considering a shell of constant thickness, Sr, radius r, mass M and volume V.

V ~ Airr^Sr p = M / V ,2 P — P s rp + vts ^ _rp-\-vt ^ (2.29)

2.6. THE MODEL 71

where and ts are the density, radius and time respectively a t the point the model sta rts. To is the nova radius a t the time of outburst.

Now consider a homologous spherical expansion where the shell of outer radius r, mass M and thickness Sr = r ( l — n) where n is a constant.

V = -7T [r^ - (r - Sr)^] = 4 7rr^ (l — n) (2.30) M P = P = P = Thus we have a general expression:

P = P 4 7rr^ (l - n) r r J ro + vt, To + vt ro + vt, (2.31) (2.32) ro + vt

where y=2 describes a shell of constant thickness and 7 = 3 a spherical, homologous expan­

sion. The density calculated in this way then appears in the ordinary differential equations describing the chemistry which tracks abundance values as relative abundances; in this way the integrator does not “see” the abundance changes due to the physics th a t would force it to reduce its step size considerably and increase the run-tim e correspondingly. Note th a t the use of relative abundance in this context assumes th a t the medium is com­ posed predominantly of one key species, say H or H2; this probably holds tru e in the case

of nova ejecta.

The tem perature of the ejecta is a highly uncertain param eter, although Beck et al. (1990) and others have attem pted to model the tem perature structure of a shell. We use a simple approach in this work, starting with the notion th a t a t early times, prior to visual maximum, the leading edge of the ejecta and the photosphere almost coincide. In this way, given the boundary condition th a t at r = ro, T = Tphot,Oi the tem perature falls monotonically as:

L = = c (2.33)

T = Tphot,0 L' ej J

At later times however, after visual maximum, the photosphere shrinks whilst the shell continues to expand. The bolometric luminosity is roughly constant however and we find

72 CHAPTER 2. MODELLING TECHNIQUES

th a t, in considering the ejecta to be a thin black shell at a radius r^j in therm al equilibrium, the effective ejecta tem perature is obtained by:

T e / / ~ ' (2.34)

SO in general it holds th a t Tefr oc P where —0.5 ^ a ^ —1.0. In leaving a as a free

param eter we concede th a t the ejecta is, in fact, optically thin so to know the exact tem perature one would have to solve the radiative transfer equations for all heating and cooling processes within the ejecta. The models of Ferland et al, (1979) suggest the electron tem perature does not fall as rapidly ais would be described by a = 0.5. As such, this remains a free param eter.

It is found th a t this model is very easy to prepare and to use; J o makes much of the work involved in adding and deleting reactions quite painless and the use of d a ta files to describe the model make it possible to run grids of models with shell scripts. The function charged with calculating rate coefficients after each step of the integrator takes care of all reactions including special cases. Where d ata concerning particular reactions has been found in the literature, this is encoded separately rather than trying to fit the Arrhenius formula to the rate coefficient curve. Provided th a t the rate file is closed (i.e. all species have both form ation and destruction channels) and well defined with good rate coefficients, the model rarely crashes. Should something be wrong in the chemistry it is usually made clear when either the model crashes or the integrator is forced to keep using a miniscule step size'^. In practice it is found th a t very small initial time-steps are required with the integrator often falling back to steps of the order 10"® seconds. This is indicative of the extremely rapid time-scale over which chemistry occurs in the early nova and represents the period when abundances in the model are ‘relaxing’ to equilibrium values. This is graphically illustrated in figure 3.2 where equilibrium appears to be established after 14 hours (left hand side of the plot). In fact, this equilibrium is established well before, ju st seconds of modelled time after the start.

C om putation takes a m atter of a few minutes even on a low end PC system although most were performed on DEC Alpha machines with 500 MHz processors producing results in under a m inute for most model runs.

A typical m odel wiU step forward w ith steps of the order a second prior to progressing w ith step s of the order of days once the chem istry has relaxed to an equilibrium. If the step size rem ains < 1 second continuously it is indicative o f a problem .