Interacting winds

A) Construct a numerical hydrodynamics code to follow the interaction between a fast stellar wind and its slower precursor. These kinds of interactions take place when stars change from low effective temperature (`red') to high effective temperature (`blue'), for instance in the post-AGB phase (for low to intermediate mass stars), and when massive stars change from red supergiants to blue supergiants (as happened with the precursor of SN1987A).

Use the following parameters: $\dot{M}=10^{-7}$ $M_\odot$ yr$^{-1}$, $v=1000$ km s$^{-1}$ for the fast wind, and $\dot{M}=10^{-5}$ $M_\odot$ yr$^{-1}$, $v=10$ km s$^{-1}$ for the slow wind. You may assume that all the gas is ionized by the UV radiation from blue star, and that it initially has the typical temperature of an ionized gas, $10^4$ K. Start the interaction between the two winds at an initial radius of $10^{14}$ m. Choose the outer grid radius such that you can follow the interaction for 500 years or more.

The inner boundary condition should be an inflow condition, the outer boundary an outflow condition. The choice of the position of the inner boundary is up to you.

B) The interaction leads to a so called three shock pattern: an outer shock travelling into the slow wind, an inner shock travelling into the fast wind, and a contact discontinuity separating the two; these three boundaries mean that there are four regions in the flow. Identify the three discontinuities in the results of simulation and describe the structure of the four regions of the flow.

C) Compare the results of the simulation with analytical expectations. A full self-similar model was published by Chevalier & Imamura (1983), whereas Kahn (1983) used a simpler `thin shell' model (which assumes efficient cooling, see Sect. 3). How good is the match between the two analytical models and your numerical model? Use these quantities to compare: the (pattern) speed of the outer shock, temperature of the shocked fast wind, position of the three discontinuities. In Chevalier & Imamura (1983) you will have to use Fig. 5 to derive the value of $b_1$ and $b_2$ for the parameters chosen here.

D) Study the effects of resolution. Use as few as 50 grid cells, and then increase the resolution by using more grid cells, up to 10 or 20 times as much (if CPU time allows). What are the differences?

E) Study the effect of reducing the order of the solution. In the Roe solver routine set the parameters sbpar1 and sbpar2 to zero. This will make the code only first order accurate. Compare the result of the simulation with your previous simulations. Has reducing the order of the solution a similar effect as reducing the resolution, or in other words can one compensate for the lower order solution by running at a higher resolution?