Changes between Version 26 and Version 27 of PlanetaryAtmospheres


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Timestamp:
09/02/15 12:53:16 (9 years ago)
Author:
Jonathan
Comment:

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  • PlanetaryAtmospheres

    v26 v27  
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    8886and numerically solve the following 2 equations for $\lambda_s$ and $\chi$
    89 || $\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_{Ms} \left ( 1 + \sigma_p \right ) }{\xi_{Mp} \left ( 1 + \sigma_s \right ) } $ ||
     87|| $\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 ) } $ ||
    9088|| $\chi=\chi_{bow} \frac{\phi \left ( \frac{1 - \xi_{bow}}{\xi_s}, \lambda_s \right ) }{\phi \left ( \frac{\xi_{bow}}{\xi_p}, \lambda_p \right )}$ ||
    9189|| $\lambda_s=\frac{2 \xi_{Ms}}{\xi_s} $ ||
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    93 
    94 Note
    95 
    96 $r_{bow}=\mbox{solve} \left [ \rho_s c_s^2 \phi(\frac{1-\xi_{bow}}{\xi_s}) \left ( 1 + \psi(\frac{1-\xi_{bow}}{\xi_s}) \right) \left ( 1 + \sigma_s \right ) =  \rho_p c_p^2  \phi(\frac{\xi_{bow}}{\xi_p}) \left ( 1 + \psi(\frac{\xi_{bow}}{\xi_p}) \right) \left ( 1 + \sigma_p \right ) \right ] $
    97 
    98 and
    99 
    100 $c^2=\frac{G M}{r \lambda}$ so $\frac{c_p^2}{c_s^2} = q \frac{\xi_s \lambda_s}{\xi_p \lambda_p} = q \frac{\xi_Ms}{\xi_Mp}$
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    103 $\xi_{bow}=\mbox{solve} \left [ \left ( 1 + \psi(\frac{1-\xi_{bow}}{\xi_s}) \right) \left ( 1 + \sigma_s \right ) =  q \frac{\xi_Ms}{\xi_Mp}  \left ( 1 + \psi(\frac{\xi_{bow}}{\xi_p}) \right) \left ( 1 + \sigma_p \right ) \right ] $
    104 
    105 
    106 However - apparently those 8 params are not independent.  Instead of specifying $\xi_{Ms}$ let's specify the temperature of the planet.
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    1221061. How magnetized the stellar wind is
    123107
     108They also did not explore the role of eccentricity or of the non-isotropic wind from the planet.
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    125 Also, 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.
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     111
     112== A few comments ==
     113
     1141. 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.
     1151. Also, note that while $\psi$ looks like a function of two variables, it is really a function of one variable
     116
     117 $\psi(\xi, \lambda) = \psi \left ( \xi_\lambda= \frac{\xi}{\lambda} \right ) = \mbox{solve} \left [ \psi - \ln \psi=-3 +4  \left ( \ln 2\xi_\lambda+  \frac{1}{2\xi_\lambda} \right ) \right ]$
     118
     119 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.
     120
     121 So Instead of setting $\xi_{Ms}$ we can set either $\xi_s$, or $\lambda_s$.
     122
     123
     124== Sample params ==
     125
     126|| $\xi_p$ || 0.005000 ||
     127|| $\xi_s$ || 0.200000 ||
     128|| $\xi_H$ || 0.200000 ||
     129|| $\xi_{bow}$ || 0.250000 ||
     130|| $\xi_{Mp}$ || 0.010000 ||
     131|| $\chi_{bow}$ || 1.000000 ||
     132|| $\sigma_p$ || 0.100000 ||
     133|| $\sigma_s$ || 0.100000 ||
     134|| Orbital separation || 0.100000 AU ||
     135|| Mass of Star || 0.997333 solar masses ||
     136|| Mass of Planet || 2.795926 Jupiter masses ||
     137|| Radius of Star || 4.301941 solar radii ||
     138|| Radius of Planet || 1.046271 Jupiter radii ||
     139|| Temperature of Star || 431184.397104 Kelvin ||
     140|| Temperature of Planet || 143479.010318 Kelvin ||
     141|| Density of Star || 3.996761e-07 ||
     142|| Density of Planet || 1.000000e-08 ||
     143|| Orbital period || 0.031615 years ||
     144|| Magnetic field of Star || 0.000036 Gauss ||
     145|| Magnetic field of Planet || 0.084617 Gauss ||
     146
     147== Matlab Code ==
     148
     149See the 3 .m files attached
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