Chapter 10, Problem 10.1P

To determine

The equation of hydrostatic equilibrium in the term of an optical depth is,

    $\frac{dP}{d\tau }=\frac{g}{\overline{k}}$.

Answer to Problem 10.1P

The equation of hydrostatic equilibrium in the term of an optical depth is $\frac{dP}{d\tau }=\frac{g}{\overline{k}}$.

Explanation of Solution

Write the expression for an optical depth.

    $d{\tau }_{\lambda }=-k\rho ds$        (1)

Write the expression for hydrostatic equilibrium in terms of acceleration due to gravity.

    $\frac{dP}{dr}=-\rho g$        (2)

Conclusion:

Substitute the value of $-\frac{d{\tau }_{\lambda }}{k_{\lambda }ds}=\rho$ from equation (1) in equation (2)

    $\begin{array}{l}\frac{dP}{dr}=-\left(-\frac{d\tau }{kds}\right)\times g\\ \frac{dP}{d\tau }=\frac{g}{\overline{k}}\end{array}$

Thus, the equation of hydrostatic equilibrium in the term of an optical depth is $\frac{dP}{d\tau }=\frac{g}{\overline{k}}$.