Changes between Version 13 and Version 14 of PlanetaryAtmospheres

Timestamp:

08/27/15 16:15:33 (10 years ago)

Author:

Jonathan

Comment:

—

PlanetaryAtmospheres

@@ -1 +1 @@
 [[BackLinksMenu()]]
 [[PageOutline]]
+== Defining the parameter space for stellar-planetary wind interactions ==
 
-Primary variables
+Ignoring MHD for the moment, and assuming circular orbits, we can define the problem using these 9 primary variables
 
-$M_p$
-$R_p$
-$T_p$
-$\rho_p$
-$M_s$
-$R_s$
-$T_s$
-$\rho_s$
-$a$
+|| $M_p$ || Mass of planet ||
+|| $R_p$ || Radius of planet ||
+|| $T_p$ || Temperature at planet surface ||
+|| $\rho_p$ || Density at planet surface ||
+|| $M_s$ || Mass of star ||
+|| $R_s$ || Radius of star ||
+|| $T_s$ || Temperature at stellar surface ||
+|| $\rho_s$ || Density at stellar surface ||
+|| $a$ || orbital separation ||
+
+Time and length symmetry allows 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 six dimensionless variables that define the interaction
 
 
-Equations - assuming circular orbits and isothermal parker winds
+|| $q=\frac{M_p}{M_s}$ || mass ratio ||
+|| $\chi = \rho_p / \rho_s $ || ratio of densities at surfaces ||
+|| $\xi_p=R_p/a$  || dimensionless planetary radius ||
+|| $\xi_s=R_s/a$ || dimensionless stellar radius ||
+|| $\lambda_p=\frac{G M_p m_H}{R_p k_b T_p}$ || characterizes planetary wind ||
+|| $\lambda_s=\frac{G M_s m_H}{R_s k_b T_s}$ || characterizes stellar wind ||
 
-* $\chi = \rho_p / \rho_s $
-* $\lambda_p=\frac{G M_p m_H}{R_p k_b T_p}$
-* $\lambda_s=\frac{G M_s m_H}{R_s k_b T_s}$
-* $\Omega = \sqrt{\frac{G \left (M_s+M_p \right )}{a^3}}$
-* $r_H=a \left (\frac{M_p}{3M_s} \right )^{1/3}$
-* $v_{esc}=\sqrt{\frac{2 G M_p}{R_p}}$
-* $r_\Omega=\frac{v_{esc}}{2 \Omega}$
-* $q=\frac{M_p}{M_s}$
-* $c_s=\frac{k_B T_s}{m_H}$
-* $c_p=\frac{k_B T_p}{m_H}$
-* $r_{BH} = \frac{2 G M_p}{v_s(a)^2+(a\Omega)^2+c_s^2}$
+Now instead of those 6, we may want to define the following 5 length scales, and density ratio at the bow shock
+
+|| $\xi_H=\frac{r_H}{a}$ || Ratio of Hill radius to orbital radius ||
+|| $\xi_{bow}=\frac{r_{bow}}{a}$ || Ratio of bow shock radius to orbital radius ||
+|| $\xi_p=\frac{R_p}{a}$ || Ratio of planetary radius to orbital radius ||
+|| $\xi_M=\frac{\lambda_p}{2} \xi_p$ || Ratio of sonic radius to planetary orbital radius ||
+|| $\chi_{bow}$ || Density ratio at bow shock. ||
+|| $\xi_{BH}=\frac{r_{BH}}{R_a}$ || Ratio of bondi-hoyle radius to orbital radius ||
+
+Using the following relations, 
+
+||  $\Omega = \sqrt{\frac{G \left (M_s+M_p \right )}{a^3}}$ || orbital angular velocity || 
+|| $r_H=a \left (\frac{M_p}{3M_s} \right )^{1/3} = a \left ( \frac{q}{3} \right ) ^ {1/3}$ || Hill radius ||
+|| $c_s=\frac{k_B T_s}{m_H}$ || stellar sound speed ||
+|| $c_p=\frac{k_B T_p}{m_H}$ || planetary sound speed ||
+|| $r_{BH} = \frac{2 G M_p}{v_s(a)^2+(a\Omega)^2+c_s^2}$ || Bondi-Hoyle radius ||
+|| $r_{bow}=\mbox{solve} \left [ \rho_s(a-r_{bow}) v_s(a-r_{bow})^2 + P_s(a-r_{bow}) = \rho_p(r_{bow}) v_p(r_{bow})^2 +P_p(r_{bow}) \right ]$ || bow shock standoff distance ||
+
+and the dimensionless solution to the Parker Wind
 
 * $\psi(\xi,\lambda) = \mbox{solve} \left [ \psi - \ln \psi=-3 -4 \ln \frac{\lambda}{2}+4 \ln \xi + 2 \frac {\lambda}{\xi} \right ]$
 * $\phi(\xi,\lambda) = \exp \left [ -\frac{\lambda}{\xi} \left (\xi - 1 \right ) - \frac{1}{2} \psi(\xi,\lambda) \right ]$
+
+which gives us the following solutions for the stellar and planetary winds
 * $v_s(r) = \sqrt{\psi(\frac{r}{R_s}, \lambda_s)}c_s$
 * $P_s(r)=\frac{\rho_s T_s k_B}{m_H} \phi \left ( \frac{r}{R_s}, \lambda_s \right )$
@@ -38 +56 @@
 * $\rho_p(r)=\rho_p \phi \left ( \frac{r}{R_p}, \lambda_p \right )$
 
-
-* $r_{bow}=\mbox{solve} \left [ \rho_s(a-r_{bow}) v_s(a-r_{bow})^2 + P_s(a-r_{bow}) = \rho_p(r_{bow}) v_p(r_{bow})^2 +P_p(r_{bow}) \right ]$
-
-
-Time and length symmetry allows 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 six dimensionless variables that define the interaction
-
-$q$, $\chi$, $\xi_p$, $\xi_s$, $\lambda_p$, $\lambda_s$
-
-where $\xi_p=R_p/a$ and $\xi_s=R_s/a$
-
-
-Now instead of those 6, we may want to define 
-
-|| $\xi_H=\frac{r_H}{a}$ || Ratio of Hill radius to orbital radius ||
-|| $\xi_{bow}=\frac{r_{bow}}{a}$ || Ratio of bow shock radius to orbital radius ||
-|| $\xi_p=\frac{R_p}{a}$ || Ratio of planetary radius to orbital radius ||
-|| $\xi_{\Omega}=\frac{r_\Omega}{a}$ || Ratio of centrifugal radius to orbital radius ||
-|| $\chi_{bow}$ || Density ratio at bow shock. ||
-|| $\xi_{BH}=\frac{r_{BH}}{R_a}$ || Ratio of bondi-hoyle radius to orbital radius ||
-
-This then allows us to calculate
+We can directly calculate
 
 || $q=\frac{\xi_H^3}{3}$ ||
 || $\xi_p=\xi_p$ ||
-|| 
+|| $\lambda_p=\frac{2 \xi_M}{\xi_p} $ ||
 
+and numerically solve the following 3 equations for $\xi_s$, $\lambda_s$, and $\chi$
+
+|| $\xi_{BH}=\frac{2 \xi_s}{\psi\left ( \xi_s^{-1} \right) + \frac{q+1}{q} + \frac{1}{q \lambda_s}}$ ||
+|| $\chi_{bow}=\chi \frac{\phi \left ( \frac{\xi_{bow}}{\xi_p} \right )}{\phi \left ( \frac{1 - \xi_{bow}}{\xi_s} \right ) }$ ||
+|| $\frac{1+\psi\left ( \frac{ 1 - \xi_{bow}}{\xi_s} \right )}{1+\psi \left ( \frac{\xi_{bow}}{\xi_p} \right )} = \frac{q \lambda_s \xi_s \chi_{bow}}{\lambda_p \xi_p \chi}$
+
+
+As a side note, we have 
+|| $\frac{v_{esc}}{c_p}=\sqrt{2 \lambda_p}$ || planetary escape speed ||
+|| $\xi_\Omega=\frac{v_{esc}}{2 a \Omega}=\frac{2q}{ (q + 1)\xi_p}$ || dimensionless radius at which coriolis forces bend planetary wind - not independent ||