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:

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