|Annu. Rev. Astron. Astrophys. 1999. 37:
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4.2. A Relativistic Distance Determination
We note that the detection of a known line from either of the condensations would allow a precise determination of the distance. The Doppler factors, namely, the ratios of observed to emitted frequency (0) for the approaching and receding condensations are given by
In these last two equations = (1 - 2)1/2 is the Lorentz factor. Since we know cos , a determination of either a / 0 or r / 0 will allow the determination of and thus the determination of and of the distance from Equation (4). In the case of cosmologically distant objects, the Equations 1, 2, 4, and 5 are valid replacing the distance D by the angular size distance Da (Peebles 1993), and the rest frequency 0 by 0 / (1 + z), with z being the observed redshift of the central source. The angular size distance is given by Da = (cz / H0)[1 - (1 + q0)z / 2 + ...], where H0 is Hubble's constant and q0 is the dimensionless acceleration (or deceleration) parameter. Then, the observations of proper motions and frequency shifts in extragalactic relativistic ejecta pairs could potentially be used to test between different cosmological models.