Planet Wind/Stellar Wind Cooling Times

So we have been seeing lots of cool instabilities in the simulations of the planet wind/stellar wind interactions. So far all these sims have been run as isothermal with

This means we expect cooling to play and important role for both the shocked planetary wind (pw) and shocked stellar wind (sw). I wanted to check these assumptions and so I did a brief calculation.

We are taking about Hot Jupiter's here so the orbital radius is small

Stellar Wind

At that radius the stellar wind is still acceleration and is likely subsonic (M<1). That means no shock. If I assume its a solar type star with a corona the parameters are likely

and using

gives me a number density of

To get a cooling time scale I just use

and I approximate the cooling rate

Putting in #s I get

To see if the wind actually cools I need a dynamical time to compare it to which I chose to be the time crossing time of the shocked flow past the planet (I use Jupiter's radius for the planet and assume the interaction happens at 10 R_j)

This gives me

Thus

and the stellar wind will not cool.

Planetary Wind

For the planetary wind shock things look a bit better but not too much.

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