MASERS, INTERSTELLAR AND CIRCUMSTELLAR, THEORY

Moshe Elitzur

The radio radiation detected in some lines of certain astronomical molecules is attributed to the natural occurrence of the maser phenomenon (microwave amplification by stimulated emission of radiation), the same as that produced by artificial means in laser devices. Astronomical maser radiation is produced by population inversion of the pertinent transitions and is usually identified as such when at least one of the following properties is observed:

1. Enormous radiation intensity, as measured by its equivalent temperature.

2. Very narrow linewidths.

3. Abnormal line ratios, indicating deviations from equilibrium.

In at least one astronomical maser source there is also direct evidence for population inversion and amplification of radiation. The mechanism that leads to inversion, which is caused by the cycling of molecules through energy levels, is called the pump. The pump process is initiated by excitations from the ground state, due to either external radiation or collisions. The inversion results from a combination of various factors, specific to the particular maser molecule. The strong maser emission detected in our galaxy occurs around late-type stars, where it is called circumstellar, and in the cores of dense molecular clouds that are regions of active star formation, which is then termed interstellar. Maser radiation probes small-scale structure in these sources and is now used to measure distances by kinematic means, the equivalent of the classic moving cluster method. In recent years, maser emission has also been detected in many nearby galaxies.

MASER EFFECT

Population exchange between the two levels of any transition is governed by both collisional and radiative processes. The latter include spontaneous decays from upper to lower level and absorption of external radiation, accompanied by an excitation from lower to upper level. The frequency of the absorbed photon must match the energy separation of the transition. The inverse process, induced or stimulated emission, is the essence of the maser phenomenon. In it, a downward transition is induced by an incoming photon with a matched frequency. To conserve energy and momentum, the transition is accompanied by the emission of another photon whose properties are identical to those of the initial parent photon. The process effectively acts as negative absorption - increasing, instead of decrease the number of photons in the radiation field. If for any reason the population density of the upper level is larger than that of the lower level, the rate of stimulated emission exceeds absorption and the medium amplifies the propagating radiation rather than attenuating it. When the contribution of stimulated emission is included in the absorption coefficient, the latter becomes negative. The same applies to the optical depth and the standard attenuation term exp(-) becomes an amplification factor exp||. The absolute value of the optical depth is then referred to as the maser gain. A gain of more than 20 leads to amplification in excess of 10⁸ and could explain in part the observed exceptional intensities. Additional intensity enhancement results from the focusing of the radiation into a narrow beam, caused by the fact that amplification is proportional to incoming intensity, so stronger rays are amplified more strongly. Observed brightness is frequently expressed in terms of brightness temperature, the temperature of an equivalent blackbody that would be needed to produce the same intensity. Maser brightness temperatures can be as high as ~ 10¹⁵ K.

The exponential growth of the intensity cannot continue indefinitely because it would eventually lead to infinite energy density for sufficiently long masers. A self-limiting process is built right into the maser effect itself: Induced emission removes particles from the upper level, thus reducing the inversion. Because of the excess of induced emissions over absorptions, the inversion decreases once the population exchange between the maser levels is dominated by the interaction with the maser radiation. The intensity then approaches a limit and the maser saturates. The brightness is highest during saturated operation because every pumping event produces a maser photon with the maximal intrinsic efficiency allowed by the pump.

ENVIRONMENTAL CONSTRAINTS

The prevalence of astronomical masers shows that the interstellar medium is apparently capable of producing the maser effect relatively easily, although special efforts are required to achieve the same end in the laboratory. This is a result of the great differences in densities and geometrical dimensions between the two environments. In thermodynamic equilibrium, the populations per substate n of the two levels of a transition separated by energy gap E obey.

n_upper / n_lower = exp (-E / KT)

where k is the Boltzmann constant and T is the temperature. This equilibrium distribution is established when population exchange is dominated by collisions, which is the case for sufficiently high densities. For typical parameters of molecular rotational transitions, the required densities are in excess of ~ 10⁴ cm^-3. This is an extremely low density in the laboratory, achieved only with state of the art vacuum techniques (pressures of ~ 10^-12-10^-13 torr), but it is quite high by interstellar standards. Equilibrium populations are therefore the rule in terrestrial circumstances but are the exception in interstellar space due to the large difference in relevant densities. Once thermal equilibrium is violated, population inversion is a priori almost as likely as its reverse.

An appreciable maser effect requires large gain, which in turn implies a substantial number of molecules along the line-of-sight. This conflicts with the necessary deviation from thermal equilibrium, which requires that the densities be small, and the only way to reconcile these opposite demands is with large dimensions. At a density of ~ 10⁴ cm^-3, an appreciable gain is achieved only for linear dimensions in excess of ~ 10¹⁰ cm - almost as much as the radius of the Sun. One of the methods to overcome this difficulty in the laboratory is to increase the radiation path length by bouncing the laser light between mirrors, effectively increasing the linear dimensions of the system by the many passes of the laser beam in the resonant cavity. This technique is possible only in systems that can maintain a high degree of phase coherence. On the other hand, typical lengths of astronomical masers are at least ~ 10¹³ cm (the same as the radius of the Earth's orbit around the Sun), and the required gains are obtained during simple photon propagation as in a single-pass laser. Interstellar space is therefore a natural environment for maser operation. Using laboratory terminology, astronomical masers are single-pass, lossless, gaseous lasers without feedback.

The observed profiles of interstellar lines usually indicate the presence of highly supersonic motions in the emitting regions. Because maser amplification is achieved by induced emission, the maser photons must seek paths that maintain good coherence in the component of the velocity along the line-of-sight; otherwise the transition frequencies of molecules encountered by the maser photons would be shifted by the Doppler effect, making amplification impossible. Observations show indeed that maser sources are comprised of many emission spots, each with its own well defined velocity. These single features are often shaped like elongated tubes, or cylinders, which need not be well-defined physical entities, but rather directions that developed the required velocity coherence by chance. Because of their high brightness, the spots can be studied individually using techniques of high-resolution interferometry. This provides the opportunity for probing small scale structure in the host environments.

SOURCES AND PUMPING

Maser emission is associated with both the early and late stages in the life of a star. This is a fortunate coincidence because these are generally regarded as the most interesting phases of stellar evolution. Late-type stars display strong maser radiation in transitions of all three "classical" maser molecules - OH, H₂O, and SiO. The masers occur in distinct regions located at different distances from the central star. The SiO masers involve rotational transitions inside excited vibration states, which lie high above the ground state. These levels can maintain substantial populations only close to the star where the excitation rates are high, and that is where SiO masers are located. The H₂O and OH masers, on the other hand, emanate from transitions in the ground vibration states of these molecules and do not require such extreme conditions for pumping. Both are located in shells that are part of an expanding wind that blows away from the star. The H₂O masers require higher temperatures than OH masers and occur at distances of up to ~ 10¹⁵ cm from the central star (by comparison, the radius of Pluto's orbit around the Sun is about 6 x 10¹⁴ cm). The OH maser shell extends further out, to a radius larger by another order of magnitude at least.

Regions of active star formation, located at the cores of many molecular clouds, display the most powerful and spectacular maser emission observed in the Galaxy in both OH and H₂O, and in at least one case also in SiO. The OH masers appear to be surface phenomena on the edges of very compact H₂O regions, ionized spheres around young and very hot stars. The H₂O masers usually trace high velocity flows (velocities of ~ 200 km s^-1) from some centers of activity that presumably erupt at a certain stage of the star formation process.

Pump analysis involves the construction of models capable of producing the observed maser output with parameters that can be checked against other observations. The pumping mechanism is termed collisional or radiative according to the nature of the process that dominates the molecular excitation from the ground state. In general, both types of pumping can produce inversion under the right conditions (although in certain, specific circumstances only one may be capable of inversion) and the nature of the pump in any particular source is determined by the relative strengths of both processes. Radiative pump models can be more easily confronted with observations because they relate two directly observed quantities - the number of photons observed in the maser transition and in the pump bandwidth. Indeed, the masers whose detailed modeling has been most successful are the OH masers in late-type stars that are pumped by infrared radiation resulting from the reemission of the stellar radiation by the dust particles that permeate the stellar wind. Detailed models of the H₂O masers in these sources show that pumping is controlled by collisions. The model predicts correctly the location of the H₂O maser region and its variation with the stellar mass loss rate, but these observational tests are not as direct. Current models of masers in star-forming regions are not as detailed yet, reflecting perhaps the somewhat poorer overall understanding of these sources.

1. Cohen, R.J. (1989). Compact maser sources. Rep. Prog. Phys. 52 881.
2. Elitzur, M. (1982). Physical characteristics of astronomical masers. Rev. Nod. Phys. 54 1225.
3. Genzel, R. (1986). Strong interstellar masers. In Masers, Molecules and Mass Outflows in Star Forming Regions, A.D. Haschick, ed. Haystack Observatory, p. 233.
4. Herman, J. and Habing H.J. (1985). OH/IR stars. Phys. Rep. 124 255.
5. Reid, M.J. and Moran, J.M. (1981). Masers. Ann. Rev. Astron. Ap. 19 231.
6. See also Masers, Interstellar and Circumstellar.