Annu. Rev. Astron. Astrophys. 1999. 37: 409-443
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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 (nu0) for the approaching and receding condensations are given by

Equation 6 (6)

Equation 7 (7)

In these last two equations gamma = (1 - beta2)1/2 is the Lorentz factor. Since we know beta cos theta, a determination of either nua / nu0 or nur / nu0 will allow the determination of beta and thus the determination of theta 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 nu0 by nu0 / (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.

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