Optics Definitions

Below are some important definitions for optics terms:

Given a gas composed of spherical particles of mass $m$ and radius $s$ and number density (particles per volume) $n$, inside a rectangular slab of infinite size in the x and y directions but with finite height $H$ in the z direction

We define the cross-section per particle $\sigma$ as:

$\sigma = \pi s^2$

We define the column density as number density integrated along some path:

$N = \int_0^H n \mathrm{d} x$

We define the opacity $\kappa$ as the cross-section per unit mass of gas:

$\kappa = \frac {\sigma}{m}$

We define the mass density (mass per volume) $\rho$ as:

$\rho = nm$

We define the optical depth $\tau$ as:

$\tau = \int_0^H n \sigma \mathrm{d} x = n \sigma H$