Changes between Version 5 and Version 6 of u/zchen/isothermalwind

Timestamp:

02/18/14 13:12:55 (11 years ago)

Author:

Zhuo Chen

Comment:

—

u/zchen/isothermalwind

@@ -25 +25 @@
 Using De l'Hopital's rule at $r=r_c$
 
-$\frac{1}{u}\left.\frac{du}{dt}\right|_{r=r_c}=(-\frac{2a^2}{r_c^2}+\frac{2GM}{r_c^3})/(\frac{2udu}{dr})=\frac{a^2}{r_c^2}\left.\frac{udu}{dr}\right|_{r=r_c}$
+$\frac{1}{u}\left.\frac{du}{dr}\right|_{r=r_c}=(-\frac{2a^2}{r_c^2}+\frac{2GM}{r_c^3})/(\frac{2udu}{dr})=\frac{a^2}{r_c^2}\left.\frac{udu}{dr}\right|_{r=r_c}$
 
 This also shows that numerator and denominator converge to 0 at the same rate. The mathematical appropriation of the isothermal wind is justified.
 
-$\left.\frac{du}{dt}\right|_{r=r_c}=\frac{\pm 2a^2}{GM}$
+$\left.\frac{du}{dr}\right|_{r=r_c}=\frac{\pm 2a^2}{GM}$
 
 We can also solve (4)