Changes between Version 2 and Version 3 of u/adebrech/Matlab


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Timestamp:
07/27/16 10:35:52 (9 years ago)
Author:
adebrech
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  • u/adebrech/Matlab

    v2 v3  
    8080[[Image(ParkerWrongApprox.jpg)]]
    8181
     82== Numerical Integration of DE ==
     83
     84Write the differential equation for $\frac{d u}{d r}$ as a parameterized system:
     85
     86{{{
     87#!latex
     88$\frac{d u}{d s} = -2uc^2(r-\frac{GM}{2c^2})$
     89
     90$\frac{d r}{d s} = -r^2(u^2-c^2)$
     91}}}
     92
     93Calculate the Jacobian and evaluate at the critical point to linearize the system, then move a small distance along the stable eigenvector (i.e. tangent to the stable manifold, the transonic solution of interest). Integrate from these points out.
     94
     95$Jacobian = \begin{bmatrix} 0 & 2c^3 \\ \frac{GM}{2c^3} & 0 \end{bmatrix}$
     96
    8297= Change in Bow Shock with Magnetic Field =
    8398If sigma,,*,, and sigma,,p,, are equal, the bow shock radius is unchanged with or without magnetic field - ratio of radius to orbital separation, chi,,bow,, = 0.240468. With sigma,,*,, = 1, sigma,,p,, = 0.1, chi,,bow,, = 0.148204; sigma,,*,, = 0.5, sigma,,p,, = 0.1, chi,,bow,, = 0.187300; sigma,,*,, = 0.1, sigma,,p,, = 0.5, chi,,bow,, = 0.302483; sigma,,*,, = 0.1, sigma,,p,, = 1, chi,,bow,, = 0.363674; and with sigma,,*,, = 0.5 and sigma,,p,, = 1, chi,,bow,, = 0.297793 ≈ chi,,Coriolis,,.