Relationships between Galactic Radio Continuum and H Emission

Basic mechanisms of thermal emission

In this section we summarize relevant radiation theory and present the relationship between H and radio frequency radiation from a thermal Hydrogen plasma.

An optically thin population of excited Hydrogen atoms will radiate in the H spectral line a power (erg s ster or W ster)

where is the Einstein coefficient for spontaneous emission and is the number density of Hydrogen in the n=3 excited state. (Note that the rate of collisional transitions between the L = S, P, D angular momentum states of n=3 may be too low to promote a state of detailed balance in low-density regions of the ISM, and that a more complete theory would need to consider these states separately). In a plasma in statistical equilibrium (see Mihalas 1978) the population density is determined by the statistical balance between the various opportunities for radiative and collisional excitation and destruction of the state. The radiative rates depend on ``external'' radiation, such as that from an exciting star, and in ``internal'' radiation from the gas itself. The collisional rates are dominated by electron collisions, and so depend on the electron density and electron temperature.

The population density can be related to the proton and electron densities, and , by

Here, is the statistical weight and the ionization energy from n=3, is the partition function of atomic Hydrogen, and is the departure coefficient reflecting the difference between the actual population of the level and the population that would occur in local thermodynamic equilibrium at the same temperature and the same values of and .

Values of the departure coefficients may be predicted by solving the equations of statistical balance appropriate to various circumstances. In view of the complexity of the coupled equations, a number of convenient approximations are usually made in the analysis of low-density, radiatively excited nebulae (``H II regions''). Perhaps the simplest approximation is that any photon produced in the gas is immediately radiated away. This approximation - called ``Case A'' - is not usually a valid approximation in cosmic plasmas. A slightly more complicated approximation which is often valid is so-called ``Case B'' recombination. In this approximation it is supposed that every Lyman line photon is scattered locally many times, ultimately being converted to Lyman- and a higher-series photon, while all of the higher-series photons are emitted with no further scattering.

When Case B is valid, and the density is so low that collisional excitation and de-excitation can be ignored, it is straightforward to set up and solve a system of statistical balance equations for the ionization equilibrium and bound level population densities in hydrogen. These yield self-consistent values of the departure coefficient and other parameters. The value of depends on the electron temperature and the nature of the ambient radiation field, and might be typically . Under Case B conditions the ratio of the emissivity of any Balmer line to that in H (the Balmer decrement) is also readily predicted. Agreement or otherwise between the predicted and observed ratios then provides a check on the validity or otherwise of Case B conditions. Alternatively, any difference between the predicted and observed ratio allows an estimate of the extinction if Case B is assumed.

The approximations underlying Case B equilibrium are not always satisfied in actual emission regions. A more complete and accurate approach to the spectroscopic diagnosis of H emission from the interstellar medium requires a computer study of a complex set of statistical balance equations. This approach is particularly important in regions with high densities (where collisions are relatively more important) and near dynamical features such as contact discontinuities, ionization fronts and shock waves (where density gradients and time-dependent processes might be important).

A volume of cosmic plasma containing thermal electrons and protons will radiate thermal bremsstrahlung according to (e.g. Osterbrock, 1974, Eq 4.22)

Here, is the Gaunt factor for free-free radiation, for which an approximation is available (e.g. Rohlfs, 1986, Eq. 8.27). The total brightness of thermal radiation along a specified line of sight is the integral of this expression, provided that the line of sight is optically thin. For sufficiently low frequencies and sufficiently high emission measures (the emission measure is the integral along the line of sight) the line of sight will not be optically thin, and the brightness of the radio thermal continuum is then given by the (black body) Planck function. In this case, the shape of the radio continuum spectrum as it varies form the optically thick (low frequency) to the optically thin (high frequency) part provides additional information on physical conditions in the plasma (e.g., Rohlfs 1986, Section 8.4).

The consequence of the theory outlined above is the prediction of a close relationship between the thermal radio continuum flux density and H intensity along a line of sight free from other emission or any absorption processes. The relationship corresponds to

where depends only on physical and atomic constants. Note that in this equation the factor is of order 0.1, but would be uncertain by perhaps as much as an order of magnitude as a result of uncertainties in the effects of collisions and radiative transfer in the determination of the departure coefficient.

© Copyright Astronomical Society of Australia 1997