Changes between Version 12 and Version 13 of PlanetaryAtmospheres

Timestamp:

08/27/15 12:15:45 (9 years ago)

Author:

Jonathan

Comment:

—

PlanetaryAtmospheres

@@ -15 +15 @@
 
 
-Equations
+Equations - assuming circular orbits and isothermal parker winds
+
 * $\chi = \rho_p / \rho_s $
 * $\lambda_p=\frac{G M_p m_H}{R_p k_b T_p}$
 * $\lambda_s=\frac{G M_s m_H}{R_s k_b T_s}$
-* $\omega = \sqrt{\frac{G \left (M_s+M_p \right )}{a^3}}$
+* $\Omega = \sqrt{\frac{G \left (M_s+M_p \right )}{a^3}}$
 * $r_H=a \left (\frac{M_p}{3M_s} \right )^{1/3}$
+* $v_{esc}=\sqrt{\frac{2 G M_p}{R_p}}$
+* $r_\Omega=\frac{v_{esc}}{2 \Omega}$
+* $q=\frac{M_p}{M_s}$
+* $c_s=\frac{k_B T_s}{m_H}$
+* $c_p=\frac{k_B T_p}{m_H}$
+* $r_{BH} = \frac{2 G M_p}{v_s(a)^2+(a\Omega)^2+c_s^2}$
+
+* $\psi(\xi,\lambda) = \mbox{solve} \left [ \psi - \ln \psi=-3 -4 \ln \frac{\lambda}{2}+4 \ln \xi + 2 \frac {\lambda}{\xi} \right ]$
+* $\phi(\xi,\lambda) = \exp \left [ -\frac{\lambda}{\xi} \left (\xi - 1 \right ) - \frac{1}{2} \psi(\xi,\lambda) \right ]$
+* $v_s(r) = \sqrt{\psi(\frac{r}{R_s}, \lambda_s)}c_s$
+* $P_s(r)=\frac{\rho_s T_s k_B}{m_H} \phi \left ( \frac{r}{R_s}, \lambda_s \right )$
+* $\rho_s(r)=\rho_s \phi \left ( \frac{r}{R_s}, \lambda_s \right )$
+* $v_p(r) = \sqrt{\psi(\frac{r}{R_p}, \lambda_p)}c_p$
+* $P_p(r)=\frac{\rho_p T_p k_B}{m_H} \phi \left ( \frac{r}{R_p}, \lambda_p \right )$
+* $\rho_p(r)=\rho_p \phi \left ( \frac{r}{R_p}, \lambda_p \right )$
+
+
+* $r_{bow}=\mbox{solve} \left [ \rho_s(a-r_{bow}) v_s(a-r_{bow})^2 + P_s(a-r_{bow}) = \rho_p(r_{bow}) v_p(r_{bow})^2 +P_p(r_{bow}) \right ]$
+
+
+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
+
+|| $q=\frac{\xi_H^3}{3}$ ||
+|| $\xi_p=\xi_p$ ||
+|| 
+
+