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Meeting Update
Thoughts on parker wind
The solution for the parker wind is found by solving the equations below
v^2 = c_s^2 \psi |
\xi=r/r_p |
\rho = \rho_p \phi |
\lambda=\frac{GM \mu}{r_p k_B T} |
\psi(\xi,\lambda) = \mbox{solve} \left [ \psi - \ln \psi=-3 -4 \ln \frac{\lambda}{2}+4 \ln \xi + 2 \frac {\lambda}{\xi} \right ] |
\phi(\xi,\lambda) = \exp \left [ -\frac{\lambda}{\xi} \left (\xi - 1 \right ) - \frac{1}{2} \psi(\xi,\lambda) \right ] |
Note however, that the actual radius of the planet (and lambda) does not really matter…
v^2 = c_s^2 \psi |
\xi=r/r_s |
\rho = \rho_s \phi |
\psi(\xi) = \mbox{solve} \left [ \psi - \ln \psi=-3 + 4 \ln \xi + \frac {4}{\xi} \right ] |
\phi(\xi) = \exp \left [ -\frac{2}{\xi}\left ( \xi-1 \right ) + \frac{1}{2} \left ( 1 - \psi(\xi) \right ) \right ] |
All that a higher lambda implies, is a smaller planet radius (compared to the sonic radius) - and sampling more and more of a subsonic atmosphere before having a hard boundary.
Also - per our discussion the other day, it seems that
\frac{R_\Omega}{a} = \frac{v_{esc}}{a\Omega} = \sqrt{\frac{2GM_p}{r_p a^2}\frac{a^3}{G\left(M_p+M_s \right) }} = \sqrt{\frac{2aq}{r_p \left(q+1 \right)}}
So this effect is more pronounced when the orbital sepration is small and when the planet radius and mass ratio is close to - ie Hot Jupiters
- Posted: 9 years ago (Updated: 9 years ago)
- Author: Jonathan
- Categories: (none)
Attachments (1)
- parker.jpg (25.0 KB) - added by Jonathan 9 years ago.
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Comments
So smaller r_p would mean a higher \rho_p since we would be driving the wind from deeper in the atmosphere(?)
Yes - for a fixed
and , a smaller would mean a larger . It would also mean that the velocity at the surface would be more and more subsonic.The parker wind doesn't have a 0 velocity inner boundary… Even at the planet 'surface' the parker wind solution has a non zero velocity. Although for r_p << r_s, or equivalently, for lambda >> 1, this velocity is almost 0. See the dimensionless solution to the parker wind.
Presumably if lambda ~ 1 and if the velocity at r_p were zero, the solution would deviate from the parker solution. And then different lambda's would give different dimensionless curves.
Which I guess is what happens with the Stone and Proga type BC's