# Harvard Astronomy 201b

## CHAPTER: Excitation Processes: Collisions

In Book Chapter on March 7, 2013 at 3:18 pm

(updated for 2013)

Collisional coupling means that the gas can be treated in the fluid approximation, i.e. we can treat the system on a macrophysical level.

Collisions are of key importance in the ISM:

• cause most of the excitation
• can cause recombinations (electron + ion)

Three types of collisions

1. Coulomb force-dominated ($r^{-1}$ potential): electron-ion, electron-electron, ion-ion
2. Ion-neutral: induced dipole in neutral atom leads to $r^{-4}$ potential; e.g. electron-neutral scattering
3. neutral-neutral: van der Waals forces -> $r^{-6}$ potential; very low cross-section

We will discuss (3) and (2) below; for ion-electron and ion-ion collisions, see Draine Ch. 2.

In general, we will parametrize the interaction rate between two bodies A and B as follows:

${\frac{\rm{reaction~rate}}{\rm{volume}}} = <\sigma v>_{AB} n_a n_B$

In this equation, $<\sigma v>_{AB}$ is the collision rate coefficient in $\rm{cm}^3 \rm{s}^{-1}. <\sigma v>_{AB}= \int_0^\infty \sigma_{AB}(v) f_v~dv$, where $\sigma_{AB} (v)$ is the velocity-dependent cross section and $f_v~dv$ is the particle velocity distribution, i.e. the probability that the relative speed between A and B is v. For the Maxwellian velocity distribution,

$f_v~dv = 4 \pi \left(\frac{\mu'}{2\pi k T}\right)^{3/2} e^{-\mu' v^2/2kT} v^2~dv$,

where $\mu'=m_A m_B/(m_A+m_B)$ is the reduced mass. The center of mass energy is $E=1/2 \mu' v^2$, and the distribution can just as well be written in terms of the energy distribution of particles, $f_E dE$. Since $f_E dE = f_v dv$, we can rewrite the collision rate coefficient in terms of energy as

$\sigma_{AB}=\left(\frac{8kT}{\pi\mu'}\right)^{1/2} \int_0^\infty \sigma_{AB}(E) \left(\frac{E}{kT}\right) e^{-E/kT} \frac{dE}{kT}$.

These collision coefficients can occasionally be calculated analytically (via classical or quantum mechanics), and can in other situations be measured in the lab. The collision coefficients often depend on temperature. For practical purposes, many databases tabulate collision rates for different molecules and temperatures (e.g., the LAMBDA databsase).

For more details, see Draine, Chapter 2. In particular, he discusses 3-body collisions relevant at high densities.

## CHAPTER: Chemistry

In Book Chapter on February 13, 2013 at 10:04 pm

See Draine Table 1.4 for elemental abundances for the proto-solar environment. H:He:C = $1:0.1:3 \times 10^{-4}$ by number. $1:0.4:3.5 \times 10^{-3}$ by mass However, these ratios vary by position in the galaxy, especially for heavier elements (which depend on stellar processing). For example, the abundance of heavy elements (Z > Carbon) is twice as low at the sun’s position than in the Galactic center. Even though metals account for 1% of the mass, they dominate most of the important chemistry, ionization, and heating/cooling processes. They are essential for star formation, as they allow molecular clouds to cool and collapse. Generally, it is easier (i.e. requires less energy) to dissociate a molecule than to ionize something. The lower the electronic state you are trying to ionize, the more energy is needed. The Lyman Limit is the minimum photon energy needed to ionize Hydrogen from the ground state (13.6 eV, 912 Angstrom).

## CHAPTER: Hydrogen Slang

In Book Chapter on February 12, 2013 at 10:02 pm

Lyman limit: the minimum energy needed to remove an electron from a Hydrogen atom. A “Lyman limit photon” is a photon with at least this energy.

$E = 13.6 {\rm eV} = 1~ {\rm Rydberg} = hcR_{\rm H}$,

where $R_{\rm H}=1.097 \times 10^{7} {\rm m}^{-1}$ is the Rydberg constant, which has units of $1/\lambda$. This energy corresponds to the Lyman limit wavelength as follows:

$E = h\nu = hc/\lambda \Rightarrow \lambda=912 \AA$.

Lyman series: transitions to and from the n=1 energy level of the Bohr atom. The first line in this series was discovered in 1906 using UV studies of electrically excited hydrogen gas.

Balmer series: transitions to and from the n=2 energy level. Discovered in 1885; since these are optical transitions, they were more easily observed than the UV Lyman series transitions.

There are also other named series corresponding to higher n. Examples include Paschen (n=3), Brackett (n=4), and Pfund (n=5). The wavelength of a given transition can be computed via the Rydberg equation

$\frac{1}{\lambda}=R_{\rm H} \big(\frac{1}{n_f^2}-\frac{1}{n_i^2}\big)$.

Note that the Lyman (or Balmer, Paschen, etc.) limit can be computed by inserting $n_i=\infty$.

Lyman continuum corresponds to the region of the spectrum near the Lyman limit, where the spacing between energy levels becomes comparable to spectral line widths and so individual lines are no longer distinguishable.