= 13.6 eV n-2 , where n, the principal quantum number, can be 1, 2, 3, etc., and where ^ep is the reduced mass, memp/(me + mp). The zero of energy, reached asymptotically for large n, represents the marginally bound state. The ultraviolet spectral lines generated by downward transitions to n =1 form the Lyman series, where n = 2 ^ 1 yields the Lya line at 1216 A, n = 3 ^ 1 yields Ly3 at 1026 A, etc. Similarly, the visible lines created by the jump to n = 2 constitute the Balmer series, with n = 3 ^ 2 being the 6563 A Ha line we have already encountered.

The Formation of Stars. Steven W. Stahler and Francesco Palla Copyright © 2004 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim ISBN: 3-527-40559-3

Figure 2.1 Origin of the 21 cm line. The hydrogen atom has greater energy when the spins of its proton and electron are parallel.

In the nonrelativistic treatment, a state with quantum number n actually consists of n2 levels of identical energy. Each level is labeled not only by n, but also by a quantum number l, corresponding classically to the magnitude of the electron's orbital angular momentum L, and by a third quantum number ml, corresponding to the projection of L along any fixed axis. For a given n, l can take on any integer value from 0 to n — 1, while ml can range from —l to +l. Thus, in the n =1 state normally found in interstellar gas, both l and ml are zero.

The electron also has an intrinsic spin, which can be identified classically with another angular momentum vector S and an associated magnetic moment (. In the electron's own reference frame, the motion of the charged proton creates a magnetic field which torques the spinning electron. The relativistic Dirac theory shows how this spin-orbit interaction causes the n2 levels to have slightly different energies. Such fine splitting of the energy vanishes whenever l = 0. Even in this case, however, a smaller, hyperfine splitting is present due to the fact that the proton itself has an intrinsic spin I and therefore a magnetic moment. In a semi-classical picture, the energy of the atom depends on whether the vectors S and I are parallel or antiparallel (see Figure 2.1). Quantum mechanically, the two states are labeled by a quantum number F, which is 1 and 0 in the upper and lower states, respectively. The energy difference is so small, only 5.9 x 10~5 eV, that the wavelength of the emitted photon lies in the radio regime.

Within a region of HI gas, a hydrogen atom can be excited to the F = 1 state by collision with a neighboring atom. Usually, the same atom is later collisionally deexcited, but there is a small probability that the downward transition occurs through emission of a 21 cm photon. Despite the long interval between emission events for a given atom, 1 x 107 yr on average, an appreciable radio signal can be built up by a sufficiently large number of atoms. Indeed, the HI column density Nhi, i. e., the number of atoms per unit area along any line of sight, is directly proportional to the received intensity of the 21 cm radiation.

Converting a column density into a volume density requires knowing the spatial location of the emitting gas. The actual received 21 cm signal is both shifted and spread out over a finite width in wavelength due to the Doppler effect. That is, any motion of a hydrogen atom toward or away from the observer changes the received wavelength from the intrinsically emitted value. The overall shift in peak intensity mainly stems from the differential rotation of the Galaxy. Since the pattern of rotation is well known, velocity shifts can be correlated with positions within the disk. Analysis of the emission profiles in many directions thus yields the number density nHI throughout large regions.

Occasionally, one finds an extragalactic radio source, such as a quasar, behind a region of atomic hydrogen. If this source emits in the 21 cm regime, some of its photons will be absorbed by the intervening matter. Figure 2.2 shows a representative HI spectrum toward the quasar 3C 161.1 The upper solid curve was obtained by averaging the signal from four lines of sight slightly offset from the point-like source. The resulting profile represents emission from HI gas, only some of which stems from the region of interest. When the telescope is pointed directly at the quasar, the additional signal is nearly constant in wavelength, except for the finite range removed by absorption. Hence, subtraction of the previous average from the on-source signal yields the absorption spectrum, also shown in the figure. In warmer gas, there is more collisionally induced emission of 21 cm radiation. The corresponding dip in the spectrum is therefore less pronounced. We can see, then, how 21 cm absorption profiles are useful probes of the gas temperature.

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