# Radiation Lecture 07

### From AstroBaki

## Contents |

### Increasing Grain Size

We return to the model in which an infinite plane wave passes through an aperture, is focused by an infinite lens, and shines on a wall. In this model, as the slit aperture widens (a increases), then the diffraction pattern narrows. Thus, for larger grain sizes, there is more forward scattering than in other directions. The fitting formula for the power pattern for large grains is:

where . Note that this formula fails for . If *g* = 1, then we have isotropic scattering, and as , *F*(θ) peaks increasingly in the θ = 0 direction (it is increasingly “forward throwing”).

### Forward Scattering

Forward scattering can actually increase the intensity of light in some areas over if there were no scattering at all. If particles are smaller that the wavelength of light, there is isotropic scattering. For larger particles, the power is more concentrated in the perfectly forward and backward directions. This is why, where there is fog, you dim your headlights. The larger scattering particles in the fog actually increases the intensity of light reflected back at you and at oncoming cars.

### Collisional Excitation Cross-sections

The Einstein analog:

So the Rate of Excitations *R*_{ex} is given by:

Suppose we have some distribution of relative velocities given by *f*(*v*)*d**v*, where *f* is the fraction of collisions occurring with relative velocities [*v*_{rel},*v*_{rel} + *d**v*]. Then:

where *q*_{12} is the “collisional rate coefficient” [*c**m*^{3}*s*^{ − 1}]. Then the Rate of de-excitation is given by:

We recognize now that is the rate of excitations of A using B moving at relative velocity *v*. If we have detailed balance, then this has to be the same as the rate of de-excitation .

Where *m*_{r} is the reduced mass . However many *B*_{slow} are created by collisional excitation, the same number are used for the reverse de-excitation. This is **detailed balance**.

Second, under thermal equilibrium, particles have a **Maxwellian** velocity distribution:

(Maxwellian velocity distribution) In thermal equilibrium,

Now, assuming detailed balance and thermal equilibrium,

This is the “Einstein analog”.

For a specific case, *B* = *e*^{ − }, *A* = ion with bound electron.

- Incident electron has kinetic energy >
*h*ν_{21}.

- Coulomb focusing gives cross-section.

We want to know how far away an electron with *v* can be aimed and still hit the *a*_{0} radius cloud around the ion. This is *b*, the **impact parameter**. Our collision cross-section = π*b*^{2}. Our angular momentum is conserved, so

We know that , where is the velocity to the original electron velocity. This is a result of it falling toward the ion. Then:

Generally, the Coulomb focusing factor > 1 because we want to excite, not ionize. , so:

Ω is the “collisional strength”, and generally is 0 below the *v* threshold, goes to 1 at the threshold, and decreases for increasing *v*, with some occasional spikes. Generally, it is of order 1, with some slight temperature dependency.

- 2000 K gas. , so . Then