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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_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 )
• \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_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}, \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}

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. 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

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 #