Annu. Rev. Astron. Astrophys. 1999. 37:
409-443
Copyright © 1999 by . All rights reserved |

**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

(6) |

(7) |

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 *D*_{a}
(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 *D*_{a} = (*cz /
H*_{0})[1 - (1 + *q*_{0})z / 2 + ...], where
*H*_{0} is Hubble's constant and *q*_{0} 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.