Lecture #2b-- Thermodynamic and diluted thermodynamic ionization


1) Derivation of the Boltzmann factor
The probability ratio of a subsystem being excited to states that differ in energy by delta Q is given by the ratio of the number of ways the rest of the universe can either have or not have that extra delta Q. The latter is controlled entirely by kT = dQ/d ln N, so the probability ratio is exp(-beta), where beta=Delta Q / kT.
2) Derivation of the Planck Function
The expected excitation of a given photon frequency state is the sum over all n of n*P, normalized by dividing by the sum of P, where P is the relative probability of excitation by n, i.e., P is the Boltzmann factor as a function of n. Here beta = h*nu/kT are the energy steps, so the expectation value is
= sum[n*exp(-n*beta)]/sum[exp(-n*beta)] = 1/(exp(beta)-1).
The Planck function weights these expectation values by the density of states at that nu, which is the surface area of a sphere in three dimensional momentum/energy space, so scales as nu^2, and it also weights the excitation number by the energy of the mode, bringing in another power of nu. The constant comes out 2h/c^2, so B(nu,T) = 2hnu^3/c^2 / (exp(beta)-1).
3) Ionization in thermodynamic equilibrium: the Saha equation
The ionization rate is proportional to B(nu,T), and the recombination rate is proportional to n_e. This results in the probability of a proton being ionized, compared to being neutral, is p_(H II)/p_(H I) = exp(-13.7)*2/lambda^3 *1/n_e, where lambda is the thermal de Broglie wavelength.
4) Ionization in a dilute optically thin Radiation field
Like thermodynamic equilibrium, with the ionizaton rate suppressed by the diluation factor of the radiation field (which is in turn set by the fractional coverage of the star in the sky). The dilution factor simply multiplies the Boltzmann factor above.
The optically thick environment in an H II region will not be treated yet.
5) Energy balance
The presence of ionized hydrogen allows for recombination of free electrons, which can produce copious Lyman continuum emission, as well as Lyman, Balmer and Paschen line emission. This removes energy from the gas and cools it. The cooling per particle for given free electron density is roughly proportional to T for T < 10,000 K for hydrogen, and for T < 100,000 K for metals. The hydrogen cooling peaks at perhaps 20,000 K, because any higher and the electrons are so "hot" they resist forming neutal hydroen long enough to lose substantial energy. This causes a "thermal instability" above 20,000 K for hydrogen, and above 200,000 K if metals are present, which makes it hard to get gas up to those temperatures, but easy to go well beyond once these levels are attained.