15.2: Phenomena Which Produce Departures from Local Thermodynamic Equilibrium

a. Principle of Detailed Balancing

Under the assumption of LTE, the material particles of the gas are assumed to be in a state that can be characterized by a single parameter known as the temperature. Under these conditions, the populations of the various energy levels of the atoms of the gas will be given by Maxwell-Boltzmann statistics regardless of the atomic parameters that dictate the likelihood that an electron will make a specific transition. Clearly the level populations are constant in time. Thus the flow into any energy level must be balanced by the flow out of that level. This condition must hold in any time-independent state. However, in thermodynamic equilibrium, not only must the net flow be zero, so must the net flows that arise from individual levels. That is, every absorption must be balanced by an emission. Every process must be matched by its inverse. This concept is known as the principle of detailed balancing.

Consider what would transpire if this were not so. Assume that the values of the atomic parameters governing a specific set of transitions are such that absorptions from level 1 to level 3 of a hypothetical atom having only three levels are vastly more likely than absorptions to level 2 (see figure 15.1). Then a time-independent equilibrium could only be established by transitions from level 1 to level 3 followed by transitions from level 3 to level 2 and then to level 1. There would basically be a cyclical flow of electrons from levels 1 → 3 → 2 → 1. The energy to supply the absorptions would come from either the radiation field or collisions with other particles. To understand the relation of this example to LTE, consider a radiation-less gas where all excitations and de-excitations result from collisions. Then such a cyclical flow would result in energy corresponding to the 1 → 3 transition being systematically transferred to the energy ranges corresponding to the transitions 3 → 2 and 2 → 1. This would lead to a departure of the energy momentum distribution from that required by Maxwell-Boltzmann statistics and hence a departure from LTE. But since we have assumed LTE, this process cannot happen and the upward transitions must balance the downward transitions. Any process that tends to drive the populations away from the values they would have under the principle of detailed balancing will generate a departure from LTE. In the example, we considered the case of a radiationless gas so that the departures had to arise in the velocity distributions of the colliding particles. In the upper reaches of the atmosphere, a larger and larger fraction of the atomic collisions are occurring with photons that are departing further and further from the Planck function representing their thermodynamic equilibrium distribution. These interactions will force the level populations to depart from the values they would have under LTE.

Figure 15.1 shows the conditions that must prevail in the case of detailed balancing (panel a) and for interlocking (panel b).The transition from level(1) to level (3) might be a resonance line and hence quite strong. The conditions that prevail in the atmosphere can then affect the line strengths of the other lines that otherwise might be accurately described by LTE.

b. Interlocking

Consider a set of lines that have the same upper level (see Figure 15.1). Any set of lines that arise from the same upper level is said to be interlocked (see R.Woolley and D.Stibbs¹). Lines that are interlocked are subject to the cyclical processes such as we used in the discussion of detailed balancing and are therefore candidates to generate departures from LTE. Consider a set of lines formed from transitions such as those shown in Figure 15.1. If we assume that the transition from 163 is a resonance line, then it is likely to be formed quite high up in the atmosphere where the departures from LTE are the largest. However, since this line is interlocked with the lines resulting from transitions 3 → 2 and 3 →1, we can expect the departures affecting the resonance line to be reflected in the line strengths of the other lines. In general, the effect of a strong line formed high in the atmosphere under conditions of non-LTE that is interlocked with weaker lines formed deeper in the atmosphere is to fill in those lines, so that they appear even weaker than would otherwise be expected. A specific example involves the red lines of Ca II(λλ8498, λλ8662, λλ8542), which are interlocked with the strong Fraunhofer H and K resonance lines. The red lines tend to appear abnormally weak because of the photons fed into them in the upper atmosphere from the interlocked Fraunhofer H & K lines.

c. Collisional versus Photoionization

We have suggested that it is the relative dominance of the interaction of photons over particles that leads to departures from LTE that are manifest in the lines. Consider how this notion can be quantified. The number of photoionizations from a particular state of excitation that takes place in a given volume per second will depend on the number of available atoms and the number of ionizing photons. We can express this condition as \[N_i R_{i k}=4 \pi \int_{\nu_0}^{\infty} \frac{\kappa_\nu \rho J_\nu}{h \nu} d \nu=4 \pi \int_{\nu_0}^{\infty} \frac{\alpha_\nu J_\nu}{h \nu} d \nu\label{15.1.1}\]

The frequency ν0 corresponds to the energy required to ionize the atomic state under consideration. The integral on the far right-hand side is essentially the number of ionizing photons (modulo \(4\pi\), so that this expression really serves as a definition of \(\mathrm{R_{ik}}\) as the rate coefficient for photoionizations from the ith state to the continuum. In a similar manner, we may describe the number of collisional ionizations by \[N_i C_{i k} \equiv N_i N_e \int_{v_0}^{\infty} \sigma(v) f(v) d v=N_i N_e \Omega_{i k}\label{15.1.2}\]

Here, \(C_{\mathrm{ik}}\) is the rate at which atoms in the ith state are ionized by collisions with particles in the gas. The quantity \(\sigma(\mathrm{v})\) is the collision cross section of the particular atomic state, and it must be determined either empirically or by means of a lengthy quantum mechanical calculation; and \(f(\mathrm{v})\) is the velocity distribution function of the particles.

In the upper reaches of the atmosphere, the energy distribution functions of the constituents of the gas depart from their thermodynamic equilibrium values. The electrons are among the last particles to undergo this departure because their mean free path is always less than that for photons and because the electrons suffer many more collisions per unit time than the ions. Under conditions of thermodynamic equilibrium, the speeds of the electrons will be higher than those of the ions by \((\mathrm{m_h}A/\mathrm{m_e})^½\) as a result of the equipartition of energy. Thus we may generally ignore collisions of ions of atomic weight A with anything other than electrons. Since the electrons are among the last particles to depart from thermodynamic equilibrium, we can assume that the velocity distribution \(f(\mathrm{v})\) is given by Maxwell-Boltzmann statistics. Under this assumption \(\boldsymbol{\Omega}_{\mathrm{ik}}\) will depend on atomic properties and the temperature alone. If we replace \(\mathrm{J_\nu}\) with \(\mathrm{B_\nu(T)}\), then we can estimate the ratio of photoionizations to collisional ionizations \(\mathrm{R_{ik}/C_{ik}}\) under conditions that prevail in the atmospheres of normal stars. Karl Heintz Böhm² has used this procedure along with the semi-classical Thomson cross section for the ion to estimate this ratio. Böhm finds that only for the upper-lying energy levels and at high temperatures and densities will collisional ionizations dominate over photoionizations. Thus, for most lines in most stars we cannot expect electronic collisions to maintain the atomic-level populations that would be expected from LTE. So we are left with little choice but to develop expressions for the energy-level populations based on the notion that the sum of all transitions into and out of a level must be zero. This is the weakest condition that will yield an atmosphere that is time-independent.