Since, for any line, the pattern of splitting depends on the angle 0, it would seem straightforward to read off both B||, the field component along the line of sight, and the orthogonal component However, our discussion has thus far assumed an idealized spectral line of zero width. In fact, outside of maser environments, each OH line is at least broadened thermally, i. e., by the Doppler shifts resulting from the molecules' random motion. The magnitude of such broadening is, from Appendix E,
Avtherm vs faiA\
Vo c where vs is the sound speed in the cloud. The estimate here is good to within a factor of order unity. The quantity vs is (RTkin/f)1/2, where R is the ideal gas constant and f, the mean molecular weight relative to hydrogen, is about 2. Comparing this line broadening to the magnetic line splitting in equation (6.33), we find
AVmag b B
Apart from masers, the actual B-values of interest range in magnitude from roughly 10 |G in cold, quiescent molecular clouds to 100 |G near HII regions. Hence, the well-separated lines in Figure 6.8 are generally not seen, and a more indirect approach is necessary.
The mechanism of thermal broadening operates identically on all polarization components of the radiation. Imagine viewing the 1665 MHz line, for example, through two circularly polarizing filters of opposite helicity. The two profiles, Ir(v — vo) and Il(v — vo), would be identical in shape, but shifted in frequency by Avmag (see Figure 6.9). This fact suggests that we consider the difference of the two profiles. In general, the electric field vector of any monochromatic radiation can be decomposed into right- and left-handed circularly polarized waves. The intensity difference of these components is known as the Stokes V-parameter, one of three independent scalar quantities that fully characterize the polarization state of the radiation. In the present case, the radiation is not monochromatic, but Ir(v — vo) — Il(v — vo) is still termed the V-spectrum of the source.4
We noted earlier that both Ir(v — vo) and Il(v — vo) are reduced in magnitude from the total intensity I(v — vo) by cos 9. Hence we may write their difference as
Ir(v — Vo) — Il(v — Vo ) = cos 9 [I (v — Vo — AVmag ) — I (v — Vo + AVmag)]
Using equation (6.33) for Av, we have the approximate equality
Equation (6.37) represents a practical means of determining the field component B||. One first measures directly the V-spectrum on the left side of the equation. It is best to evaluate the
4 In practice, one measures all three Stokes parameters by employing polarizing filters. One may then define the fractional polarization of the beam, whether circular or linear, through algebraic combinations of these parameters; see also equation (2.51).
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