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Update 7/20

Looked again at the Parker wind differential equation:

If we assume spherically symmetric outflow in an isothermal corona, then by conservation of momentum, continuity, and the ideal gas law, we have

\rho u \frac{d u}{d r} = -\frac{d p}{d r} -\rho \frac{G M}{r^2}$ $\frac{1}{r^2} \frac{d (r^2 \rho u)}{d r} = 0$ $p = \frac{2 \rho T}{m}$

Continuity shows that r^2 \rho u = C, where C is a constant. If we substitute for p and rho into the momentum equation, we get a differential equation in u and r:

\frac{1}{u} \frac{d u}{d r}(u^2 - \frac{2 T}{m_p}) = \frac{4 T}{m_p r} - \frac{G M}{r^2}$

Define r_c = \frac{G M m_p}{4 T} so that u_c^2 = \frac{2 T}{m_p} (i.e., sonic radius); substitute and rearrange to find

\frac{d u^2}{u_c^2} - \frac{d u^2}{u^2} = \frac{4 d r}{r} - \frac{4 r_c d r}{r^2}$

Integrate both sides to find the Parker wind equation,

\psi - \log(\psi) = 4 \log(\xi) + \frac{4}{\xi} + C$, where $\psi = (\frac{u}{u_c})^2$ and $\xi = \frac{r}{r_c}$

Modified the domain on a couple of simulations to better resolve the planet/capture sonic radius. Also plotted contour of sonic radius.

Value	Mdot (g/s)
lambda=5	5.6x10¹¹	(unchanged)
M_p = 0.25M_J	> 1.67x10⁹	(not well resolved)	3.7x10⁸
T_p = 5x10⁴ K	Negative throughout	(not resolved)	3.4x10⁸

For lambda=5, may not reach quasi-steady state.

T_p = 5x10⁴ K:

M_p = 0.25 M_J:

Sonic radii:

Continuing to work my way through Zel'dovich.

Posted: 8 years ago (Updated: 8 years ago)
Author: adebrech
Categories: (none)

Attachments (8)

aniso_ambtemp=50_SonicRadius.png (57.9 KB) - added by adebrech 8 years ago.
aniso_lambda=5,bigger_SonicRadius.png (57.2 KB) - added by adebrech 8 years ago.
aniso_lambda=5,bigger_SonicRadius.gif (2.0 MB) - added by adebrech 8 years ago.
aniso_pmass=.25,small_SonicRadius.png (51.0 KB) - added by adebrech 8 years ago.
aniso_pmass=.25,small.gif (953.7 KB) - added by adebrech 8 years ago.
aniso_ptemp=5e4,small_SonicRadius.png (43.7 KB) - added by adebrech 8 years ago.
aniso_ptemp=5e4,small.gif (515.4 KB) - added by adebrech 8 years ago.
aniso_SonicRadius.png (69.6 KB) - added by adebrech 8 years ago.

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Update 7/20

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