Changes between Version 18 and Version 19 of PlanetaryAtmospheres


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Timestamp:
08/31/15 11:37:20 (9 years ago)
Author:
Jonathan
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  • PlanetaryAtmospheres

    v18 v19  
    33== Defining the parameter space for stellar-planetary wind interactions ==
    44
    5 Ignoring magnetic fields, and assuming circular orbits, we can define the problem using these 9 primary variables
     5Assuming circular orbits, we can define the problem using these 11 primary variables
    66
    77|| $M_p$ || Mass of planet ||
    … …  
    99|| $T_p$ || Temperature at planet surface ||
    1010|| $\rho_p$ || Density at planet surface ||
     11|| $B_p$ || Magnetic field strength of planet ||
    1112|| $M_s$ || Mass of star ||
    1213|| $R_s$ || Radius of star ||
    1314|| $T_s$ || Temperature at stellar surface ||
    1415|| $\rho_s$ || Density at stellar surface ||
     16|| $B_s$ || Magnetic field of star ||
    1517|| $a$ || orbital separation ||
    1618
    17 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
    1819
     20Now the Euler equations with 2 body potential are as follows.
     21
     22$\frac{\partial \rho (\mathbf{r})}{\partial t} + \nabla \cdot \rho \mathbf{v(\mathbf{r})} = 0$
     23
     24$ \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 )$
     25
     26$\frac{\partial p(\mathbf{r})}{\partial t} + \mathbf{v(\mathbf{r})} \cdot \nabla p(\mathbf{r}) = - \gamma p(\mathbf{r}) \nabla \cdot \mathbf{v(\mathbf{r})}$
     27
     28$\frac{\partial \mathbf{B(\mathbf{r})}}{\partial t} = \mathbf{\nabla \times v(\mathbf{r}) \times B(\mathbf{r})}$
     29
     30If 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 velocity^2^, and the shape of the potential scales with the length scale. 
     31
     32This implies that $q$ must be constant, $a$ must scale with the length scale, and that $\Omega$ must scale with the inverse time scale.
     33
     34Time 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 9 primary variables to the following eight dimensionless variables that define the interaction
    1935
    2036|| $q=\frac{M_p}{M_s}$ || mass ratio ||
    … …  
    2440|| $\lambda_p=\frac{G M_p m_H}{R_p k_b T_p}$ || characterizes planetary wind ||
    2541|| $\lambda_s=\frac{G M_s m_H}{R_s k_b T_s}$ || characterizes stellar wind ||
     42|| $\beta_p=\frac{B_p m_H}{2 \rho_p k_b T_p}$ || beta of planet ||
     43|| $\beta_s=\frac{B_s m_H}{2 \rho_s k_b T_s}$ || beta of star ||
    2644
    27 Now instead of those 6, we may want to define the following 5 length scales, and density ratio at the bow shock
     45
     46Now 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
    2847
    2948|| $\xi_H=\frac{r_H}{a}$ || Ratio of Hill radius to orbital radius ||
    30 || $\xi_{bow}=\frac{r_{bow}}{a}$ || Ratio of bow shock radius to orbital radius ||
     49|| $\xi_{bow}=\frac{r_{bow}}{a}$ || Ratio bow shock radius to orbital radius ||
    3150|| $\xi_p=\frac{R_p}{a}$ || Ratio of planetary radius to orbital radius ||
    3251|| $\xi_{Mp}=\frac{\lambda_p}{2} \xi_p$ || Ratio of sonic radius of planetary wind to planetary orbital radius ||
    33 || $\chi_{bow}$ || Density ratio at bow shock. ||
     52|| $\chi_{bow} = \frac{\rho_p \left ( r_{bow} \right )}{\rho_s \left ( a-r_{bow} \right )}$ || Density ratio at bow shock. ||
    3453|| $\xi_{Ms}=\frac{\lambda_p}{2} \xi_s$ || Ratio of sonic radius of stellar wind to planetary orbital radius ||
     54|| $\sigma_{p,bow} = \frac{B_p^2 R_p^6}{r_{bow} ^6 \rho_p \left( r_{bow} \right ) \left ( c_p^2+v_p \left(r_{bow} \right)^2 \right ) }$ || Ratio of planetary magnetic pressure to ram pressure (plus thermal) at bow shock ||
     55|| $\sigma_{s,bow} = \frac{B_s^2 R_s^6}{\left ( a - r_{bow} \right ) ^6 \rho_s \left( a - r_{bow} \right ) \left ( c_s^2+v_s \left(a - r_{bow} \right)^2 \right ) }$ || Ratio of stellar magnetic pressure to ram pressure (plus thermal) at bow shock ||
    3556
    3657Using the following relations,
    … …  
    4061|| $c_s=\frac{k_B T_s}{m_H}$ || stellar sound speed ||
    4162|| $c_p=\frac{k_B T_p}{m_H}$ || planetary sound speed ||
    42 || $r_{bow}=\mbox{solve} \left [ \rho_s(a-r_{bow}) v_s(a-r_{bow})^2 + P_s(a-r_{bow}) = \rho_p(r_{bow}) v_p(r_{bow})^2 +P_p(r_{bow}) \right ]$ || bow shock standoff distance
     63|| $r_{bow}=\mbox{solve} \left [ \rho_s(a-r_{bow}) v_s(a-r_{bow})^2 + P_s(a-r_{bow} ) +\frac{B_s^2}{ \left (a-r_{bow} \right) ^6} = \rho_p(r_{bow}) v_p(r_{bow})^2 +P_p(r_{bow}) + \frac{B_p^2}{ r_{bow}  ^6}\right ]$ || bow shock standoff distance
    4364
    4465and the dimensionless solution to the Parker Wind
    … …  
    6485
    6586and numerically solve the following 3 equations for $\xi_s$ and $\chi$, and $\lambda_s$
    66 || $\frac{1+\psi\left ( \frac{ 1 - \xi_{bow}}{\xi_s}, \lambda_s \right )}{1+\psi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right )} = \frac{q \xi_{Ms}}{\xi_{Mp}}\frac{\phi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right )}{\phi \left ( \frac{1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) }$ ||
     87|| $\frac{\phi \left ( \frac{1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) \left ( 1+\psi\left ( \frac{  1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) \right )}{\phi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right ) \left ( 1+\psi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right ) \right )} = \frac{q \xi_{Ms} \left ( 1 + \sigma_p \right ) }{\xi_{Mp} \left ( 1 + \sigma_s \right ) } $ ||
    6788|| $\chi=\chi_{bow} \frac{\phi \left ( \frac{1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) }{\phi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right )}$ ||
    6889|| $\lambda_s=\frac{2 \xi_{Ms}}{\xi_s} $ ||
    … …  
    7091
    7192
    72 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.
     93Matsakos et al, compare the ordering of the Hill radius, the bow radius (from ram pressure only), and the bow radius (from magnetic pressure) which give 6 different possible orderings.  They lump them into 4 different types - based on 3 length scales.
    7394
    74 || I || $\xi_H > \xi_\beta > \xi_{bow}$ ||
    75 || II || $\xi_H > \xi_{bow} > \xi_{beta}$ ||
    76 || III || $\xi_{bow} > \xi_\beta > \xi_H$ ||
    77 || III || $\xi_{bow} > \xi_H > \xi_\beta$ ||
    78 || IV || $\xi_\beta > \xi_{bow} > \xi_H$ ||
    79 || IV || $\xi_\beta > \xi_H > \xi_{bow}$ ||
     95|| I || $R_t > R_B > R_w$ ||
     96|| II || $R_t > R_w > R_B$ ||
     97|| III || $R_w > R_B > R_t$ ||
     98|| III || $R_w > R_t > R_B$ ||
     99|| IV || $R_B > R_w > R_t$ ||
     100|| IV || $R_B > R_t > R_w$ ||
     101
     102Assuming that the planetary radius is always the smallest, there are still 4 free parameters to the problem.  They did not essentially explore
     1031. Where the location of the stellar sonic surface is in relation to the other length scales
     1041. 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
     1051. The density ratio at the shock
     1061. How magnetized the stellar wind is
    80107
    81108
    82 
    83 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.
     109Also, 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.
    84110
    85111