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Stellar atmosphere (or stellar envelope) is the tenuous outer layer of a star. The mass of this layer is negligible compared with that of the star, so one can assume that gravity  g is constant throughout. The condition for mechanical equilibrium is

 \frac{dP}{dr} = -g \rho

Where  P is the pressure and  \rho is the density. If the atmosphere is adiabatic  P = K \rho ^{\gamma} where  K is a constant and  \gamma is the adiabatic constant. Solving for the density

 \frac{dP}{dr} = \gamma K \rho^{\gamma-1} \frac{d\rho}{dr} = -g \rho

 \gamma K \rho^{\gamma-2} \frac{d\rho}{dr} = -g

  \frac{d}{dr} \left(\rho^{\gamma-1}\right) = -\frac{\gamma-1}{\gamma} \frac{g}{K}

 \rho^{\gamma-1} = \frac{\gamma-1}{\gamma} \frac{g}{K} \left(R - r\right)

 \rho = \left[ \frac{\gamma-1}{\gamma} \frac{g}{K} \left(R - r\right) \right]^{\frac{1}{\gamma-1}}

Where  R is the radius of the star (the point where the density vanishes). For a non radiative atmosphere  \gamma = \frac{5}{3} and  \rho \propto \left(R-r\right)^{3/2} . For radiative atmospheres  \gamma = \frac{4}{3} and  \rho \propto \left(R-r\right)^3 .

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