tokens: source, no absorption occurs. An H i absorption line is centered at a frequency, blueshifted or redshifted, according to the H i cloud velocity along the line-of-sight (LOS or radial velocity, V r ). A more realistic scenario is depicted in Figure 1 with a number of H i clouds and continuum sources, represented as discrete clouds. The H i clouds in front of the object of interest (continuum source, red object) will absorb radiation. If the background LOS is adjacent, the distribution of H i and continuum clouds would be nearly the same. Figure 1. Geometry for the case of many H i clouds (yellow) and continuum sources (green) along the line-of-sight toward the object of interest (red). The equation of radiative transfer is , where is the specific intensity, ds is distance along the LOS, is the emission coefficient, and is the linear absorption coefficient. In local thermodynamic equilibrium, one has , the Planck function. For low frequency (), simplifies to the Rayleigh–Jeans formula where T B is the brightness temperature. The solution of the equation of radiative transfer in plane parallel geometry, ignoring scattering, is , with . This is converted to brightness temperature ( T B ) units using . The discrete form of this result using an arbitrary large set of optically thin H i clouds and continuum emission regions isHere the index m sums over the emission regions and the index n sums over the emitting and absorbing H i clouds. is the continuum brightness temperature of cloud m. Here, and are the total optical depth, from each continuum or H i region to the observer, is the optical depth of a single H i region along the LOS, and for all clouds in front of m . The “on” spectrum is the spectrum toward the source and the “off” spectrum is the spectrum toward the background. Subtracting the “off” spectrum from the “on” spectrum, the H i absorption spectrum is given as is the H i optical depth, and is plotted as the absorption spectrum. 3.2. Kinematic Distances and Resolving the Kinematic Distance Ambiguity The radial velocity of the object of interest, or limits on it, can be deduced from the H i absorption spectrum. Then the kinematic distance to the object of interest can be determined, assuming objects move with circular velocity V ( R ) in the Galaxy (Kwee et al. 1954 ). The radial velocity of the object with respect to the Sun is V r : or , where l is the Galactic longitude. Since the rotation curve provides V ( R ) for any R , , R can be found from this equation. The relation between distance, d , to the object from the Sun, in the direction of Galactic longitude, l , and the Galactocentric distance R is given by , which has the solution of . There is a unique solution for d if . For there are two possible distances. This is the kinematic distance ambiguity. For the purpose of this discussion we assume Galactic longitudes of 0°–90°, so that radial velocity is positive for objects inside the solar circle and negative for those
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