Changes between Version 2 and Version 3 of u/adebrech/Matlab
- Timestamp:
- 07/27/16 10:35:52 (9 years ago)
Legend:
- Unmodified
- Added
- Removed
- Modified
-
u/adebrech/Matlab
v2 v3 80 80 [[Image(ParkerWrongApprox.jpg)]] 81 81 82 == Numerical Integration of DE == 83 84 Write 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 93 Calculate 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 82 97 = Change in Bow Shock with Magnetic Field = 83 98 If 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,,.