A steady, isothermal, spherically symmetric stellar wind obeys the wind equation (v-…

PHYSICS · Multi-part · Fermi University Physics · Astrophysics & Cosmology · Stellar structure, energy generation and evolution · Astrophysical fluid dynamics and plasmas

Stars from the Inside, Ch The Sun as a test: calibration, sound, neutrinos and a magnetised atmosphere

Problem

A steady, isothermal, spherically symmetric stellar wind obeys the wind equation

\left(v-\frac{c_s^2}{v}\right)\frac{\mathrm dv}{\mathrm dr}=\frac{2c_s^2}{r}-\frac{GM}{r^2},

with constant isothermal sound speed

c_s

, which has a critical point at

r_c=GM/2c_s^2

v(r_c)=c_s

, the radius at which both sides vanish together. ((a)) Write

r=r_c+x

and

v=c_s+u

with

x

u

small, and expand both sides of the wind equation to first order in

x

and

u

. ((b)) Hence show that the velocity gradient at the critical point must satisfy

(\mathrm dv/\mathrm dr)_{r_c}=\pm c_s/r_c

. ((c)) A wind leaves the stellar surface highly subsonic and must reach large

r

supersonic (as the observed solar wind does). Which sign describes this solution, and what would the other sign describe?

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