Changes between Version 17 and Version 18 of PlanetaryAtmospheres

Timestamp:

08/28/15 18:01:49 (9 years ago)

Author:

Jonathan

Comment:

—

PlanetaryAtmospheres

@@ -30 +30 @@
 || $\xi_{bow}=\frac{r_{bow}}{a}$ || 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 ||
+|| $\xi_{Mp}=\frac{\lambda_p}{2} \xi_p$ || Ratio of sonic radius of planetary wind 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 ||
+|| $\xi_{Ms}=\frac{\lambda_p}{2} \xi_s$ || Ratio of sonic radius of stellar wind to planetary orbital radius ||
 
 Using the following relations, 
@@ -40 +40 @@
 || $c_s=\frac{k_B T_s}{m_H}$ || 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 ||
+|| $r_{bow}=\mbox{solve} \left [ \rho_s(a-r_{bow}) v_s(a-r_{bow})^2 + P_s(a-r_{bow}) = \rho_p(r_{bow}) v_p(r_{bow})^2 +P_p(r_{bow}) \right ]$ || bow shock standoff distance 
 
 and the dimensionless solution to the Parker Wind
@@ -60 +59 @@
 || $q=\frac{\xi_H^3}{3}$ ||
 || $\xi_p=\xi_p$ ||
-|| $\lambda_p=\frac{2 \xi_M}{\xi_p} $ ||
+|| $\lambda_p=\frac{2 \xi_{Mp}}{\xi_p} $ ||
 
-and numerically solve the following 3 equations for $\xi_s$, $\lambda_s$, and $\chi$
 
-|| $\xi_{BH}=\frac{2 \xi_s}{\psi\left ( \xi_s^{-1}, \lambda_s \right) + \frac{q+1}{q} + \frac{1}{q \lambda_s}}$ ||
-|| $\chi_{bow}=\chi \frac{\phi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right )}{\phi \left ( \frac{1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) }$ ||
-|| $\frac{1+\psi\left ( \frac{ 1 - \xi_{bow}}{\xi_s}, \lambda_s \right )}{1+\psi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right )} = \frac{q \lambda_s \xi_s \chi_{bow}}{\lambda_p \xi_p \chi}$
+
+and numerically solve the following 3 equations for $\xi_s$ and $\chi$, and $\lambda_s$
+|| $\frac{1+\psi\left ( \frac{ 1 - \xi_{bow}}{\xi_s}, \lambda_s \right )}{1+\psi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right )} = \frac{q \xi_{Ms}}{\xi_{Mp}}\frac{\phi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right )}{\phi \left ( \frac{1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) }$ ||
+|| $\chi=\chi_{bow} \frac{\phi \left ( \frac{1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) }{\phi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right )}$ ||
+|| $\lambda_s=\frac{2 \xi_{Ms}}{\xi_s} $ ||
 
 
@@ -81 +81 @@
 
 
-In general, the planet radius will always be the smallest.  You can probably argue that the Hill radius will in general be larger than the Bondi Hoyle radius, since
-
-$\frac{r_{H}}{r_{BH}}=\frac{v_s(a)^2+(a\Omega)^2+c_s^2}{2 G M_p} a\left ( \frac{q}{3} \right ) ^ {1/3} $
-$ = \frac{v_s(a)^2+G(M_s+M_p)/a+c_s^2}{2 G M_p} a\left ( \frac{q}{3} \right ) ^ {1/3} =\left ( \frac{ \left ( \psi(\frac{1}{\xi_s})+1 \right )}{2\xi_s \lambda_s q}+\frac{1+q}{2q} \right )\left ( \frac{q}{3} \right ) ^ {1/3} $ 
-
-Now $\psi \approx< 5$, $q \approx 1/1000$, $\xi_s \approx 1/100$, so in general $r_H >> r_{BH}$
-
-However, the location of the stellar sonic radius compared to a can be used to constrain the velocity of the stellar wind - and would presumably have more bearing on the dynamics of the bow shock.
-
-
-
-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 ||
+In general, the planet radius will always be the smallest.  But they ignored the location of the sonic surfaces for both the planet and stellar winds as well as the density contrast at the bow shock - which all might be important for determining the morphology of resulting flows.