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Primary variables

M_p R_p T_p \rho_p M_s R_s T_s \rho_s a

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}}
• 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

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 #