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Defining the parameter space for stellar-planetary wind interactions

Assuming circular orbits, we can define the problem using these 11 primary variables

M_p	Mass of planet
R_p	Radius of planet
T_p	Temperature at planet surface
\rho_p	Density at planet surface
B_p	Magnetic field strength of planet
M_s	Mass of star
R_s	Radius of star
T_s	Temperature at stellar surface
\rho_s	Density at stellar surface
B_s	Magnetic field of star
a	orbital separation

Now the Euler equations with 2 body potential are as follows.

\frac{\partial \rho (\mathbf{r})}{\partial t} + \nabla \cdot \rho \mathbf{v(\mathbf{r})} = 0

\rho(\mathbf{r}) \left ( \frac {\partial \mathbf{v(\mathbf{r})}}{\partial t} + \mathbf{v(\mathbf{r}) \cdot \nabla v(\mathbf{r})} \right )=-\nabla p(\mathbf{r}) - \mathbf{B(\mathbf{r}) \times \nabla \times B(\mathbf{r})} - \rho \left ( \mathbf{r} \right ) \nabla \phi \left ( \mu, q, a, \mathbf{r} \right )

\frac{\partial p(\mathbf{r})}{\partial t} + \mathbf{v(\mathbf{r})} \cdot \nabla p(\mathbf{r}) = - \gamma p(\mathbf{r}) \nabla \cdot \mathbf{v(\mathbf{r})}

\frac{\partial \mathbf{B(\mathbf{r})}}{\partial t} = \mathbf{\nabla \times v(\mathbf{r}) \times B(\mathbf{r})}

If we consider the dimensionless form for the equations, we find that the dimensionless equations are unchanged under arbitrary changes in density, length, and time, as long as the initial conditions have the same mach number, Alfven number, and as long as the strength of the potential scales with the velocity², and the shape of the potential scales with the length scale.

This implies that q must be constant, a must scale with the length scale, and that \Omega must scale with the inverse time scale.

Time and length similarities allow 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 11 primary variables to the following eight 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
\beta_p=\frac{B_p m_H}{2 \rho_p k_b T_p}	beta of planet
\beta_s=\frac{B_s m_H}{2 \rho_s k_b T_s}	beta of star

Now instead of those 8, we may want to define the following 5 length scales, and density ratio, and magnetic/kinetic ratios at the bow shock

\xi_H=\frac{r_H}{a}	Ratio of Hill radius to orbital radius
\xi_{bow}=\frac{r_{bow}}{a}	Ratio 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} = \frac{\rho_p \left ( r_{bow} \right )}{\rho_s \left ( a-r_{bow} \right )}	Density ratio at bow shock.
\xi_{s}=\frac{R_s}{a}	Ratio of stellar radius to orbital radius
\sigma_{p,bow} = \frac{B_p^2 R_p^6}{r_{bow} ^6 \rho_p \left( r_{bow} \right ) \left ( c_p^2+v_p \left(r_{bow} \right)^2 \right ) }	Ratio of planetary magnetic pressure to ram pressure (plus thermal) at bow shock
\sigma_{s,bow} = \frac{B_s^2 R_s^6}{\left ( a - r_{bow} \right ) ^6 \rho_s \left( a - r_{bow} \right ) \left ( c_s^2+v_s \left(a - r_{bow} \right)^2 \right ) }	Ratio of stellar magnetic pressure to ram pressure (plus thermal) at bow shock

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} ) +\frac{B_s^2}{ \left (a-r_{bow} \right) ^6} = \rho_p(r_{bow}) v_p(r_{bow})^2 +P_p(r_{bow}) + \frac{B_p^2}{ r_{bow} ^6}\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 2 equations for \lambda_s and \chi

\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 \chi_{bow} \xi_{s} \lambda_s \left ( 1 + \sigma_p \right ) }{\xi_{p} \lambda_p \left ( 1 + \sigma_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 (R_t), the wind bow radius R_w (from ram pressure only), and the magnetic bow radius R_M (from magnetic pressure) which give 6 different possible orderings. They lump them into 4 different types - based on 3 length scales.

I	R_t > R_M > R_w
II	R_t > R_w > R_M
III	R_w > R_M > R_t
III	R_w > R_t > R_M
IV	R_M > R_w > R_t
IV	R_M > R_t > R_w

Assuming that the planetary radius is always the smallest, there are still 4 free parameters to the problem. They did not essentially explore

1. Where the location of the stellar sonic surface is in relation to the other length scales
2. Where the location of the planetary sonic surcface is in relation to the other length scales (they mention that the planetary thermal pressure is assumed to be negligible
3. The density ratio at the shock
4. How magnetized the stellar wind is

They also did not explore the role of eccentricity or of the non-isotropic wind from the planet.

A few comments

1. The bow shock radius will always be larger than the magnetic radius and the ram pressure radius, and perhaps it makes more sense to talk about the bow shock radius and the degree of magnetization then to talk about the 'magnetic' and 'ram pressure' radii as independent constructs.
2. Also, note that while \psi looks like a function of two variables, it is really a function of one variable

\psi(\xi, \lambda) = \psi \left ( \xi_\lambda= \frac{\xi}{\lambda} \right ) = \mbox{solve} \left [ \psi - \ln \psi=-3 +4 \left ( \ln 2\xi_\lambda+ \frac{1}{2\xi_\lambda} \right ) \right ]

And if we set the mass ratio q, and the ratio of the sonic surfaces by setting \xi_{Mp} and \xi_{Ms} , we fix the ratio of sound speeds, and the ratio of velocities coming into the bow shock. But we are also setting the ratio of densities at the bow shock \chi_{bow} as well as the location of the bow shock \xi_{bow}. These leads to an overdetermined system with no way to solve for the actual speeds.

So Instead of setting \xi_{Ms} we can set either \xi_s, or \lambda_s.

Sample params

\xi_p	0.005000
\xi_s	0.200000
\xi_H	0.200000
\xi_{bow}	0.250000
\xi_{Mp}	0.010000
\chi_{bow}	1.000000
\sigma_p	0.100000
\sigma_s	0.100000
Orbital separation	0.100000 AU
Mass of Star	0.997333 solar masses
Mass of Planet	2.795926 Jupiter masses
Radius of Star	4.301941 solar radii
Radius of Planet	1.046271 Jupiter radii
Temperature of Star	431184.397104 Kelvin
Temperature of Planet	143479.010318 Kelvin
Density of Star	3.996761e-07
Density of Planet	1.000000e-08
Orbital period	0.031615 years
Magnetic field of Star	0.000036 Gauss
Magnetic field of Planet	0.084617 Gauss

Matlab Code

See the 3 .m files attached

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 #