## FANDOM

159 Pages

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 $\rho \propto \left(R-r\right)^3$.