Demonstrate that the equation of hydrostatic equilibrium can be written as a function of the optical depth ?: dP / d? = g / ? This form of the equation is useful for describing models of stellar atmospheres.
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The equation of hydrostatic equilibrium is given by: $$\frac{dP}{dr} = -\rho g$$ where $P$ is the pressure, $r$ is the radial distance, $\rho$ is the density, and $g$ is the gravitational acceleration. Show more…
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Show that the equation for hydrostatic equilibrium, Eq. ( 6 ), can also be written in terms of the optical depth $\tau,$ as \[\frac{d P}{d \tau}=\frac{g}{\bar{K}}\] This form of the equation is often useful in building model stellar atmospheres. $$\frac{d P}{d r}=-G \frac{M_{r} \rho}{r^{2}}=-\rho g$$
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