Changes between Version 25 and Version 26 of PlanetaryAtmospheres

Timestamp:

09/01/15 17:18:48 (9 years ago)

Author:

Jonathan

Comment:

—

PlanetaryAtmospheres

@@ -32 +32 @@
 This implies that $q$ must be constant, $a$ must scale with the length scale, and that $\Omega$ must scale with the inverse time scale.
 
-Time and length similarities allow us to fix the total mass and separation without loss of generality.  In addition, the actual densities don't matter - just their ratios, so we can also fix the planetary density without loss of generality.  So we can reduce the list of 9 primary variables to the following eight dimensionless variables that define the interaction
+Time and length similarities allow us to fix the total mass and separation without loss of generality.  
+In addition, the actual densities don't matter - just their ratios, so we can also fix the planetary density without loss of generality.  So we can reduce the list of 11 primary variables to the following eight dimensionless variables that define the interaction
 
 || $q=\frac{M_p}{M_s}$ || mass ratio ||
@@ -51 +52 @@
 || $\xi_{Mp}=\frac{\lambda_p}{2} \xi_p$ || Ratio of sonic radius of planetary wind to planetary orbital radius ||
 || $\chi_{bow} = \frac{\rho_p \left ( r_{bow} \right )}{\rho_s \left ( a-r_{bow} \right )}$ || Density ratio at bow shock. ||
-|| $\xi_{Ms}=\frac{\lambda_s}{2} \xi_s$ || Ratio of sonic radius of stellar wind to planetary orbital radius ||
+|| $\xi_{s}=\frac{R_s}{a}$ || Ratio of stellar radius to orbital radius ||
 || $\sigma_{p,bow} = \frac{B_p^2 R_p^6}{r_{bow} ^6 \rho_p \left( r_{bow} \right ) \left ( c_p^2+v_p \left(r_{bow} \right)^2 \right ) }$ || Ratio of planetary magnetic pressure to ram pressure (plus thermal) at bow shock || 
 || $\sigma_{s,bow} = \frac{B_s^2 R_s^6}{\left ( a - r_{bow} \right ) ^6 \rho_s \left( a - r_{bow} \right ) \left ( c_s^2+v_s \left(a - r_{bow} \right)^2 \right ) }$ || Ratio of stellar magnetic pressure to ram pressure (plus thermal) at bow shock ||
@@ -85 +86 @@
 
 
-and numerically solve the following 3 equations for $\xi_s$ and $\chi$, and $\lambda_s$
+and numerically solve the following 2 equations for $\lambda_s$ and $\chi$
 || $\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 \chi_{bow} \xi_{Ms} \left ( 1 + \sigma_p \right ) }{\xi_{Mp} \left ( 1 + \sigma_s \right ) } $ ||
 || $\chi=\chi_{bow} \frac{\phi \left ( \frac{1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) }{\phi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right )}$ ||
@@ -97 +98 @@
 and
 
-$c^2=\frac{G M}{r \lambda}$ so $\frac{c_p^2}{c_s^2} = q \frac{\xi_s lambda_s}{\xi_p \lambda_p} = q \frac{\xi_Ms}{\xi_Mp}$
+$c^2=\frac{G M}{r \lambda}$ so $\frac{c_p^2}{c_s^2} = q \frac{\xi_s \lambda_s}{\xi_p \lambda_p} = q \frac{\xi_Ms}{\xi_Mp}$
 
 
 $\xi_{bow}=\mbox{solve} \left [ \left ( 1 + \psi(\frac{1-\xi_{bow}}{\xi_s}) \right) \left ( 1 + \sigma_s \right ) =  q \frac{\xi_Ms}{\xi_Mp}  \left ( 1 + \psi(\frac{\xi_{bow}}{\xi_p}) \right) \left ( 1 + \sigma_p \right ) \right ] $
 
+
+However - apparently those 8 params are not independent.  Instead of specifying $\xi_{Ms}$ let's specify the temperature of the planet.