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Stellar Wind

Starting from equation:

(v^2-a^2)\frac{1}{v}\frac{dv}{dr}=\frac{2a^2}{r}-\frac{A}{r^2}

It is:

\frac{dv^2}{2dr}-a^2\frac{dlnv}{dr}=\frac{2a^2}{r}-\frac{A}{r^2}

Integrate with respect to r:

v^2-2a^2lnv=2a^2lnr+\frac{A}{r}+C (1)

Where the sonic point implies:

A=2a^2 r_c

Substitute above relation in to equation (1) we can get:

\frac{v^2}{a^2}-ln{\frac{v^2}{a^2}}=4ln{\frac{r}{r_c}}+4\frac{r_c}{r}+C

For wind start subsonic and accelerate to supersonic, apply boundary condition that:

v=a,r=r_c

Then C can be determined to be -3

(\frac{v^2}{a^2}-1)-ln{\frac{v^2}{a^2}}=4ln{\frac{r}{r_c}}+4(\frac{r_c}{r}-1) (2)

Given a certain radius and time dependent A(t), we can find the velocity at the radius by solving (2)

In the limit that r\rightarrow\infty,v^2\gg a^2

v\frac{dv}{dr}=\frac{2a^2}{r}

v^2=4a^2ln\frac{r}{r_c}+C

v will grow indefinitely but very slowly. This implies that if wind is to fall back, right hand side should change sign, or it will just keep growing. So, gravity (and radiation driving force) together should give a non-constantly accelerating wind solution—it should decelerate at some radius beyond the sonic radius.

Circumstances might be: Optically thick AGB? Or binary star located at radius larger than sonic point s.t. deepen the potential well significantly.

astrobear

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Stellar Wind

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