Changes between Version 1 and Version 2 of u/zchen/isothermalwind

Timestamp:

02/16/14 14:24:26 (11 years ago)

Author:

Zhuo Chen

Comment:

—

u/zchen/isothermalwind

@@ -13 +13 @@
 $u=a$ is a singular point and $\frac{2a^2}{r}-\frac{GM}{r^2}=0$ is critical point. If the wind is to experience constant acceleration, u will exceed a at some radius such that the equation become singular. Therefore the reasonable picture for the constant accelerating wind is that numerator goes to 0 as the denominator goes to 0.
 
-Let $r=r_c$ when $\frac{2a^2}{r}-\frac{GM}{r^2})=0$ therefore $r_c=\frac{GM}{2a^2}$
+Let $r=r_c$ when $\frac{2a^2}{r}-\frac{GM}{r^2}=0$ therefore $r_c=\frac{GM}{2a^2}$
 
-$\frac{2a^2}{r}-\frac{GM}{r^2})<0$ while $u<a$
+$\frac{2a^2}{r}-\frac{GM}{r^2}<0$ while $u<a$
 
-$\frac{2a^2}{r}-\frac{GM}{r^2})>0$ while $u>a$
+$\frac{2a^2}{r}-\frac{GM}{r^2}>0$ while $u>a$
 
-$\frac{2a^2}{r}-\frac{GM}{r^2})=0$ while $u=a$
+$\frac{2a^2}{r}-\frac{GM}{r^2}=0$ while $u=a$
 
-Any other combination will not give constant accelerating wind but wind that experience deceleration in some region. Like supersonic wind 
+Any other combination will not give constant accelerating wind but wind that experience deceleration in some region. Like supersonic wind in $r<r_c$ region will decelerate and subsonic wind in $r>r_c$ region will also decelerate.
+
+Using De l'Hopital's rule at $r=r_c$
+
+$\frac{1}{u}\left.\frac{du}{dt}\right|_{r=r_c}=(-\frac{2a^2}{r_c^2}+\frac{2GM}{r_c^3})/(\frac{2udu}{dr})=\frac{a^2}{r_c^2}\left.\frac{udu}{dr}\right|_{r=r_c}$
+
+This also shows that numerator and denominator converge to 0 at the same rate. The mathematical appropriation of the isothermal wind is justified.
+
+$\left.\frac{du}{dt}\right|_{r=r_c}=\frac{\pm 2a^2}{GM}$
+
+Remark: use the