Changes between Version 21 and Version 22 of PlanetaryAtmospheres

Timestamp:

08/31/15 14:58:18 (9 years ago)

Author:

Jonathan

Comment:

—

PlanetaryAtmospheres

@@ -86 +86 @@
 
 and numerically solve the following 3 equations for $\xi_s$ and $\chi$, and $\lambda_s$
-|| $\frac{\phi \left ( \frac{1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) \left ( 1+\psi\left ( \frac{  1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) \right )}{\phi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right ) \left ( 1+\psi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right ) \right )} = \frac{q \xi_{Ms} \left ( 1 + \sigma_p \right ) }{\xi_{Mp} \left ( 1 + \sigma_s \right ) } $ ||
+|| $\frac{ 1+\psi\left ( \frac{  1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) }{ 1+\psi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right ) } = \frac{q \xi_{s} \left ( 1 + \sigma_p \right ) }{\xi_{p} \left ( 1 + \sigma_s \right ) } $ ||
 || $\chi=\chi_{bow} \frac{\phi \left ( \frac{1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) }{\phi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right )}$ ||
 || $\lambda_s=\frac{2 \xi_{Ms}}{\xi_s} $ ||
+
+
+Note
+
+$r_{bow}=\mbox{solve} \left [ \rho_s c_s^2 \phi(\frac{1-\xi_{bow}}{\xi_s}) \left ( 1 + \psi(\frac{1-\xi_{bow}}{\xi_s}) \right) \left ( 1 + \sigma_s \right ) =  \rho_p c_p^2  \phi(\frac{\xi_{bow}}{\xi_p}) \left ( 1 + \psi(\frac{\xi_{bow}}{\xi_p}) \right) \left ( 1 + \sigma_p \right ) \right ] $
+
+and
+
+$c_s^2=\frac{G M_p}{r_p}$ so $\frac{c_p^2}{c_s^2} = q \frac{\xi_s}{\xi_p}$
+
+
+$\xi_{bow}=\mbox{solve} \left [ \left ( 1 + \psi(\frac{1-\xi_{bow}}{\xi_s}) \right) \left ( 1 + \sigma_s \right ) =  q \frac{\xi_s}{\xi_p}  \left ( 1 + \psi(\frac{\xi_{bow}}{\xi_p}) \right) \left ( 1 + \sigma_p \right ) \right ] $