A. A. Ubachukwu, PASA, 16 (2), 130
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A CONSTRAINT ON THE STANDARD BEAMING MODEL FOR SUPERLUMINAL SOURCES

A. A. Ubachukwu

Dept. of Physics and Astronomy

University of Nigeria, Nsukka

Nigeria

Abstract: We have used two subsamples of superluminal quasars to test the relativistic beaming model, and to place useful constraints on radio source orientation hypothesis and cosmology. Based on the variation of the observed ratio (R) of the core-to-lobe radio luminosities with proper motion (m ) for the subsample of lobe-selected quasars, we show that the observed R-m data can be explained in terms of a bulk relativistic motion with Lorentz factor g » 4. Also, from the observed proper motion - redshift (m - z) plot for this subsample, we show that g » 4 implies a high density universe with deceleration parameter qo = 0.5. Furthermore, from the observed m - z plot for the two subsamples taken separately, we show that both g and m for the core-selected subsample exceed those of the lobe-selected subsample by a factor of _ 2 for the qo = 0.5 world model. This result is demonstrated to be consistent with an orientation-based unified scheme in which lobe-selected quasars lie, on the average, at an angle which is a factor of _ 2-3 larger than that of their core-selected counterparts.

 

 

1. Introduction

In the relativistic beaming model for extragalactic radio sources (Scheuer and Readhead 1979, Blandford and Königl 1979), an object moving relativistically at close angles to the line of sight will appear to have a transverse velocity much greater than the speed of light. This superluminal motion has been detected in a number of sources using the VLBI and quite a significant fraction of them have measured proper motion (m ) and redshift (z). If there is no evolution of proper motion with redshift, the observed upper envelope m -z function can be used to derive useful constraints for relativistic beaming and radio source orientation hypotheses as well as for cosmology (e.g. Hough and Readhead 1987, Cohen et al. 1988, Vermeulen and Cohen 1994).

In the present paper, we derive the value of the Lorentz factor (g , which is a crucial parameter of relativistic beaming scenario) using the observed variation of the ratio (R) of core-to-lobe radio luminosities as a function of proper motion for a sample of superluminal sources. We also show that the derived Lorentz factor (g ) can be used, at least in principle, to constrain the values of deceleration parameter (qo) which is a powerful discriminant for possible cosmological models.

2. Some Basic Relationships

In the simple relativistic beaming model, superluminal sources are believed to consist of one or more radiating components moving at relativistic velocity V relative to a stationary core so that we can write the apparent velocity of separation in units of the speed of light (c) as

 

 

where Q is the angle to the line of sight and b = V/C is related to the bulk Lorentz factor g through,

 

 

In the Friedmann-Robertson-Walker (FRW) universe parameterized by the Hubble constant Ho and deceleration parameter qo, the apparent velocity is also given in terms of the observed proper motion m as (e.g. Pearson and Zensus 1987),

 

 

 

 

where h is a dimensionless parameter which is the Hubble constant in units of 100 kms-1 Mpc.-1 Also, if we assume a simple continuous jet model, the ratio of the beamed flux density of the core to that of the lobe (which is usually considered isotropic) is given by (e.g. Hough and Readhead 1987),

 

 

where RT = R(q = 90o). In the absence of any large spread in the intrinsic distributions of RT and g , R is usually used as an orientation indicator, and this predicts a direct correlation between R and b app and hence m (see Vermeulen et al. 1993). This implies that sources with high values of R are expected to have large values of m . Considering equations (1) and (4), it can be shown that a linear regression of R against b app (or m , using equation (3)) should yield an intercept given by (see Ubachukwu 1998).

 

which can be used to test specific beaming models.

For the present analysis, we have used the sample of superluminal sources compiled by Vermeulen and Cohen (1994). This heterogeneous sample consists of 66 sources out of which 25 were selected on the basis of their core emission, 13 were lobe-selected and the rest were unclassified. Note that two of the sources 0906 + 430 and 1040 + 123 were classified as both core- and lobe-selected but were treated as lobe-selected in the present analysis. Since we are only interested in the classified sources, the final sample consists of 13 lobe- selected - and 23 core-selected quasars. The lobe-selected quasars are usually used to test specific predictions of the beaming models since their radio axes are believed to have a random orientation in the sky and their extended emissions isotropic.

A linear regression of R against m using the lobe-selected quasars in the current sample yields

 

with correlation coefficient r ~ 0.8. We have considered only the largest observed proper motion in each case. Using Ro = 13.1 obtained above in equation (5) and RT = 0.024 (e.g. Orr and Browne 1982) shows that the observed R-m data are consistent with g » 4, which is in close agreement with g = 5 found by Orr and Browne (1982) for a sample of flat- and steep-spectrum quasars.

3. Implications for Cosmology

Following Pearson and Zensus (1988) we write the upper envelope proper motion-redshift (m - z) function in any FRW model as

 

 

for g 2 > > 1. This equation shows that superluminal motion depends on redshift and in the absence of any large intrinsic spread in g , the observed m -z data for any well-defined sample of superluminal sources can be used to test cosmological models. Thus, for any assumed value of h, we can deduce the value of qo required to provide reasonable fit to the observed m -z data using the value of g obtained in the preceding section.

Fig. 1 shows the m -z plot for the present core-selected and lobe-selected quasars. Two curves (A and B) representing two Friedmann world models with qo = 0.1 (curve A) which appears to be consistent with the density of the universe derived from its luminous matter (e.g. Peebles 1988) and qo = 0.5 (curve B) chosen on the supposition of the inflationary world model (Guth 1981), have been superimposed. We adopted h = 1 and g =4 (obtained in the preceding section). Fig. 1 shows that the m -z data for the lobe-selected quasars used in the present analysis are consistent with a Friedmann universe with qo = 0.5. The present result thus favours a closed universe to an open universe.

4. Implications for Unified Theories

In the Orr and Browne (1982) unified scheme, core-dominated quasars are believed to be lobe-dominated quasars seen at smaller angles to the line of sight. This immediately predicts that superluminal velocity will be higher and more frequent in core-dominated than in lobe-dominated quasars. This implies that a larger value of Lorentz factor would be required to provide a good fit to the observed m -z data for core-selected quasars than for their lobe-dominated counterparts.

We can recast equation (7) in the form (see Pearson and Zensus 1987),

 

where

 

 

It follows immediately from equation (8) that for given values of qo and h, we can write

 

 

where the subscripts c and L stand for core- and lobe-selected quasars respectively. The last equation thus suggests that the apparent values of the Lorentz factor derived from superluminal data will be different for the two classes of object because of orientation effects (see Padoani and Urry 1992). A good fit to the m -z data for the core-selected quasars is also indicated in Fig. 1 (Curve C) for the qo = 0.5 world model. This curve corresponds to a Lorentz factor g c = 9 obtained by Cohen et al. (1988). It therefore follows that g c/g L » 2.3 for the present sample. Curve B and Curve C (Fig. 1) also show that within the region of overlap in redshift (0.2 £ z £ 2), m c/m L » 2.4. These results are consistent with equation (10) and suggest that the Lorentz factor and proper motion are a factor _ 2 larger for core-selected quasars than for lobe-selected quasars.

Furthermore, in a simple standard beaming model, the boosting factor d is given by (e.g. Vermeulen and Cohen 1994

 

Vermeulen and Cohen (1994) have shown that the optimum value of d can be obtained by n setting dd /dq = 0 in the last equation and this corresponds to

 

 

We can therefore write,

 

 

This clearly demonstrates that equation (10) is consistent with a pure-orientation-based unified scheme in which the core-dominated quasars are their lobe-dominated counterparts seen end-on. This can be quantitatively tested by estimating the values of q c and q L from the observed R distributions for the core-and lobe-selected quasars respectively. These distributions give median values of Rm,c = 12.6 and Rm,L = 0.22. Using g c = 9 and g L = 4 (obtained earlier) respectively in equation (4) and for a typical value of RT = 0.024 (e.g. Orr and Browne 1982), we obtain q m,c » 13o and q m,L » 38o. This yields Sin q L/Sin q c » 2.7 which agrees closely with m c/m L » 2.4 obtained above.

5. Summary

We have examined a simple statistical consequence of the standard model of superluminal sources and its implications for cosmology and unification schemes. Although the sample statistics are necessarily small, the overall results are however very intriguing. The present analysis shows a strong correlation between the core-to-lobe flux ratio (R) and proper motion (m ) for a subsample of lobe-selected quasars used. This is consistent with the scenario in which R and m are good beaming indicators. The observed R-m correlation is

shown to be consistent with a Lorentz factor g » 4. Also, from a plot of proper motion- redshift (m -z) function, it is shown that a value of g L = 4 implies that the data could be understood in terms of a high density universe with deceleration parameter qo = 0.5.

Furthermore, from a comparison of the observed m -z plot for core- and lobe-selected quasars at similar redshifts, it is shown that m and g for core-selected quasars exceed those of lobe-selected quasars by a factor of _ 2. This is also shown to be consistent with a pure orientation-based unified scheme in which the core-selected quasars are on the average at an angle » 13o to the line of sight while their lobe-dominated counterparts are at » 38o to the line of sight. We however need more data especially at high redshifts to confirm these results.

 

Acknowledgement

This work was supported by the University of Nigeria Senate Research Grant (No. 93/121). I also wish to thank the anonymous referee for some useful criticisms.

 

 

 

Fig. 1 Plot of m -z data for the core-selected quasars circles (o) and lobe-selected quasars crosses (x). All sources with m £ 0 have been omitted.

 

 

References

Blandford, R.D. and Königl, A. (1979). Astrophys. J. 232, 34.

Cohen, M.H., Barthel, P.D., Pearson, T.J. and Zensus, J.A. (1988). Astrophys. J. 329, 1.

Guth, A.H. (1981). Phys. Rev. A, 23, 347.

Hough, D.H. and Readhead, A.C.S. (1987). In Superluminal radio sources ed. J.A. Zensus and T.J. Pearson (Cambridge University Press) p. 114.

Hough, D.H. and Readhead, A.C.S. (1989). Astrophys. J. 321, L11.

Pearson, T.J. and Zensus, J.A. (1987). In Superluminal radio sources ed. J.A. Zensus and T.J. Pearson (Cambridge University Press, p. 12.

Padovani, P. and Urry, C.M. (1992). Astrophys. J. 387, 449.

Peebles, J.P.E. (1988). Pub. Stron. Soc. Pac. 100, 670.

Orr, J.L. and Browne, I.W.A. (1982). Mon. Not. R. Astron. Soc. 200, 1067.

Scheuer, P.A.G. and Readhead, A.C.S. (1979). Nat. 277, 182.

Ubachukwu, A.A. (1998). Astrophys & Sp. Sci. in press.

Vermeulen, R.C., Bernstein, R.A., Hough, D.H. and Readhead, A.C.S. (1993). Astrophys. J. 417, 541.

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