Version 2 (modified by 9 years ago) ( diff ) | ,
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Parker Wind
Differential Equation
The Parker solution to the solar wind assumes a spherically symmetric radial outflow (with velocity u(r)). Then by conservation of momentum,
Continuity (conservation of particle #) gives
If we assume an isothermal corona, the ideal gas law is
Continuity shows that
, where C is a constant. If we substitute for p and rho into the momentum equation, we get a differential equation in u and r:Define
so that (i.e., sonic radius); substitute and rearrange to findIntegrate both sides to find the Parker wind equation,
Plotting
Solving the Parker wind equation:
There are 4 types of solutions - combinations of sub/supersonic at the surface and v → 0/vf as xi → infinity. The only physical solution is subsonic at the surface and has a nonzero velocity infinitely far away, and passes through xi = 1, psi = 1.
First attempted to solve equation in Mathematica - solution appears correct up to the sonic radius, but the incorrect solution (v → 0) appears for r > rs. Next attempted to plot approximations for r << rs and r >> rs in Matlab, but as would be expected, the approximations fail near r = rs. Numerically solving the equation in Matlab gives the opposite case of Mathematica - correct solution for r > rs, but not for r < rs.
Looking at Jonathan's code for calculating the Parker wind to find the correct approximation, try to implement it in a different way to test understanding. Works for r > rs, but not r < rs.
Pertinent line is: yy(i)=vpasolve(y - log(y) == -3 + 4*log(xx(i))+4/xx(i),y,(xx(i) > 1) * xx(i) + (xx(i) ⇐ 1)*exp(3-4*log(xx(i))-4/xx(i)))
Interpreted as:
if xx(i) ≥ 1
yy(i)=vpasolve(y - log(y) == -3 + 4*log(xx(i))+4/xx(i));
else
yy(i)=vpasolve(y - log(y) == exp(3-4*log(xx(i))-4/xx(i)));
end
Correct & incorrect plots:
Change in Bow Shock with Magnetic Field
If sigma* and sigmap are equal, the bow shock radius is unchanged with or without magnetic field - ratio of radius to orbital separation, chibow = 0.240468. With sigma* = 1, sigmap = 0.1, chibow = 0.148204; sigma* = 0.5, sigmap = 0.1, chibow = 0.187300; sigma* = 0.1, sigmap = 0.5, chibow = 0.302483; sigma* = 0.1, sigmap = 1, chibow = 0.363674; and with sigma* = 0.5 and sigmap = 1, chibow = 0.297793 ≈ chiCoriolis.
Attachments (4)
- ParkerApprox.jpg (11.8 KB ) - added by 9 years ago.
- parkerwind.jpg (12.3 KB ) - added by 9 years ago.
- ParkerWrong.jpg (12.0 KB ) - added by 9 years ago.
- ParkerWrongApprox.jpg (11.6 KB ) - added by 9 years ago.
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