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Defining the parameter space for stellar-planetary wind interactions

Ignoring magnetic fields, and assuming circular orbits, we can define the problem using these 9 primary variables

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

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

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_{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_{Ms}=\frac{\lambda_p}{2} \xi_s	Ratio of sonic radius of stellar wind to planetary 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_{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 )
• \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 )

We can directly calculate

q=\frac{\xi_H^3}{3}

\xi_p=\xi_p

\lambda_p=\frac{2 \xi_{Mp}}{\xi_p}

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}

Matsakos et al, compare the ordering of the Hill radius, the bow radius, and the magnetic radius which give 6 different possible orderings. They lump them into 4 different types.

I	\xi_H > \xi_\beta > \xi_{bow}
II	\xi_H > \xi_{bow} > \xi_{beta}
III	\xi_{bow} > \xi_\beta > \xi_H
III	\xi_{bow} > \xi_H > \xi_\beta
IV	\xi_\beta > \xi_{bow} > \xi_H
IV	\xi_\beta > \xi_H > \xi_{bow}

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.

Planetary Atmospheres

Profiles

• Density

• Enclosed Mass

• Pressure - and rho*R*T mismatch

Module supports

• Global simulation in a fixed frame
• Global simulation in a rotating frame
• Local simulation in a rotating frame
• Spatial based Refinement of planet
• Stellar envelope in HSE
• Source code problem.f90​
• Data file problem.data​
• Uses Particles, Ambients, Clumps, and Refinement Objects

Still working on

• Basic testing of hydrostatic equilibrium for planet
• Line transfer for stellar heating
• Second AMR implicit solve may need to be added later (ie Howell and Greenough 2002)

Results

Working on getting stable planetary atmosphere using profile without a core. Turning on characteristic limiting seems to cause numerical artifacts which lead to 'explosion'. See ticket #