# The proper motion VS redshift relation for superluminal radio sources

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## Abstract

Two models for superluminal radio sources predict sharp lower bounds for the apparent velocities of separation. The light echo model predicts a minimum velocity*v*_{min}=2*c*, and the dipole field model predicts*v*_{min}=4.446*c*. Yahil (1979) has suggested that, if either of these models is correct, then*v*_{min} provides a ‘standard velocity’ which can be used to determine the cosmological parameters*H* and*q*_{0}. This is accomplished by estimating a lower envelope for the proper motion vs redshift relation. Yahil also argued that the procedure could easily be generalized to include a nonzero cosmical constant Λ. We derive the formulas relating the proper motion\(\dot \theta\) to the redshift*z* in a Friedmann universe with a nonzero Λ. We show that the determination of a lower envelope for a given sample of measured points\((z_i ,\dot \theta _i )\) yields an estimate of the angle of inclination*φ*_{ i } for each source in the sample. We formulate the estimation of the lower envelope as a constrained maximum likelihood problem with the constraints specified by the expected value of the largest order statistic for the estimated*φ*_{ i }. We solve this problem numerically using an off-the-shelf nonlinearly constrained nonlinear optimization program from the NAg library. Assuming Λ=0, we apply the estimation procedure to a sample of 27 sources with measured values\((z_i ,\dot \theta _i )\), using both the light echo and the dipole field models. The fits give*H*=103 km s^{−1} Mpc^{−1} for the light echo model and*H*=46 km s^{−1} Mpc^{−1} for the dipole field model. In both cases the fits give*q*_{0}=0.4, but the uncertainty in this result is too large to rule out the possibility that*q*_{0}>0.5. When Λ is allowed to be a free parameter, we obtain*H*=105 km s^{−1} Mpc^{−1} for the light echo model and*H*=47 km s^{−1} Mpc^{−1} for the dipole field model. In both cases the fits give*q*_{0}=−1 and Λ/*H* _{0} ^{2} =6.7, but no significance can be attached to these results because of the paucity of measured data at hight redshifts. For all of the fits, we compute the corresponding estimates of the*φ*_{ i } and compare the cumulative distribution of these values with that expected from a sample of randomly oriented sources. In all cases we find a large excess of sources at low-inclination angles (high apparent velocities). The expected selection effect would produce such an excess, but the excess is large enough to suggest a strong contamination of the sample by relativistic beam sources which would only be seen at low inclination angles.

## Keywords

Proper Motion Lower Envelope Apparent Velocity Motion Versus Expected Selection## Preview

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