We now consider a few of the species that have been especially fruitful in the study of molecular clouds. Table 5.1 lists abundances and important transitions for all the molecules discussed in this chapter, as well as some others of interest. The sixth column gives the energy difference between the upper and lower states. This energy is expressed as an equivalent temperature by using the Boltzmann constant kB. Also given is the Einstein coefficient Aul, i. e., the probability per time of spontaneous decay from the upper to the lower state. Appendix B introduces the Einstein coefficients more systematically. Finally, Table 5.1 lists the critical density, ncrit = Aul/7ul, where 7ul is the rate of collisionally induced downward transitions in the molecule, as measured per unit colliding partner. This quantity, as we have noted, is an estimate of the minimum ambient density at which collisions depopulate the upper state before it can decay through radiation. The utility of ncrit will become evident as we proceed through specific examples.
It is an unfortunate fact that the chief constituent of cold interstellar clouds, the hydrogen molecule, is also among the most difficult to detect. Even the lowest excited energy levels, those corresponding to molecular rotation, are too far above the ground state to be easily populated. In addition, H2 consists of two identical hydrogen atoms and therefore lacks a permanent electric dipole moment. A rotationally excited molecule must radiate through a relatively slow quadrupole transition. To find H2, it is best to look in hotter environments, such as clouds irradiated by a luminous star or shocked by a stellar wind. Here, photons or particle collisions can excite vibrational and electronic states that do decay in a relatively brief time. The molecule was actually first detected in 1970, through rocket observations that found several ultraviolet absorption lines in the direction of the O star £ Persei. These lines arise from the photo-excitation of electronic states in H2 within an intervening diffuse cloud.
Since the hydrogen molecule plays a dominant role in many aspects of star formation, it is worthwhile to examine its transitions in some detail. We begin with the rotational levels. In classical mechanics, the kinetic energy of a dumbbell rotating about an axis through the center of mass and perpendicular to the plane of rotation is given by
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