Changes between Version 1 and Version 2 of u/adebrech/Matlab

Timestamp:

07/20/16 11:56:26 (10 years ago)

Author:

adebrech

Comment:

—

u/adebrech/Matlab

@@ -1 +1 @@
 = Parker Wind =
-Solving
 
-psi − ln(psi) = 4 ln(xi) + 4/xi − 3, where psi = (v/v,,s,,)^2^ and xi = r/r,,s,,.
+== Differential Equation ==
+
+The Parker solution to the solar wind assumes a spherically symmetric radial outflow (with velocity u(r)). Then by conservation of momentum,
+
+{{{
+#!latex
+$\rho u \frac{d u}{d r} = -\frac{d p}{d r} -\rho \frac{G M}{r^2}$
+}}}
+
+Continuity (conservation of particle #) gives
+
+{{{
+#!latex
+$\frac{1}{r^2} \frac{d (r^2 \rho u)}{d r} = 0$
+}}}
+
+If we assume an isothermal corona, the ideal gas law is
+
+{{{
+#!latex
+$p = \frac{2 \rho T}{m}$
+}}}
+
+Continuity shows that $r^2 \rho u = C$, 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:
+
+{{{
+#!latex
+$\frac{1}{u} \frac{d u}{d r}(u^2 - \frac{2 T}{m_p}) = \frac{4 T}{m_p r} - \frac{G M}{r^2}$
+}}}
+
+Define $r_c = \frac{G M m_p}{4 T}$ so that $u_c^2 = \frac{2 T}{m_p}$ (i.e., sonic radius); substitute and rearrange to find
+
+{{{
+#!latex
+$\frac{d u^2}{u_c^2} - \frac{d u^2}{u^2} = \frac{4 d r}{r} - \frac{4 r_c d r}{r^2}$
+}}}
+
+Integrate both sides to find the Parker wind equation,
+
+{{{
+#!latex
+$\psi - \log(\psi) = 4 \log(\xi) + \frac{4}{\xi} + C$, where $\psi = (\frac{u}{u_c})^2$ and $\xi = \frac{r}{r_c}$
+}}}
+
+== Plotting ==
+Solving the Parker wind equation:
+
+{{{
+#!latex
+$\psi - \log(\psi) = 4 \log(\xi) + \frac{4}{\xi} - 3$
+}}}
 
 There are 4 types of solutions - combinations of sub/supersonic at the surface and v -> 0/v,,f,, 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.