Varshni,Y.P.: 1976, *Astrophys.Space Sci.*, **42**, 369.

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The angular size of a radio source is a particularly sensitive test of cosmological models (Hoyle, 1959; Sciama, 1971; Peebles, 1971), since for Friedmann models a source with linear diameter

A few years ago, Legg (1970), using currently available data, made a study of 32
radio galaxies and 25 quasars which are predominantly double in structure. He
plotted the angular separations of the source components against the redshift. Curves
obtained for the angular size from two world models, namely those of Einstein-de
Sitter, and de Sitter, for a source of a fixed linear size of 290 kpc were found to
provide reasonable upper envelopes to the plotted points. Macdonald and Miley (1971)
studied the structures of 79 quasars with a resolution of up to 3 arcseconds. Miley (1971)
made a compilation of the then available data on 127 quasars and from plots of
`largest angular size' (LAS) versus the redshift *z*, concluded that, on the average,
component size decreases with increasing redshift. A Euclidean world model with
source size of 400 kpc appeared to provide the best envelope. The indication was that
the angular size falls off with redshift faster than one might expect for values of the
deceleration parameter, *q0*, as determined from the apparent magnitude-redshift
diagram for galaxies. Also, there was no evidence for any change in the slope at large
redshifts as is expected from any of the conventional cosmological models. This led
to the enunciation of the third paradox of
Kellermann (1972) :

*`The linear dimensions of radio sources depend on redshift in just such a way to cancel the geometrical effects of the redshift'*.

Recently, Wardle and Miley (1974) have studied the structures of 39 quasars with
the NRAO interferometer at wavelengths of 11.1 and 3.7 cm. The LAS-*z* diagram is
extended to 166 quasars, and the plot is compared to the results from the following
three world models for a source of linear size 500 kpc.

(I) Static Euclidean model :

theta = 17.2/z arcsec (1)(II) Einstein-de Sitter model :

theta = 8.6 (1+z)^2 / ( 1 + z - sqrt(1+z)) arcsec (2)(III) A variant of (II) in which account is taken of the `ram pressure' for the containment of extended radio sources; it involves an additional parameter. The resulting relationship is

theta = 8.6 (1+z)^(6/5) / ( 1 + z - sqrt(1+z)) arcsec (3)None of the three models provide a satisfactory solution. The only conclusion that can be made is that (III) appears to be better than the other two.

In practice there is a basic difficulty in drawing conclusions as to which is the best
curve. Ideally speaking, an envelope function should lie above all the points, which
certainly is not the case here. No analytical method has been proposed which could
be used as a criterion. The behaviour of any curve has to be judged by the eye alone,
which, of course, is subjective and rather unsatisfactory. Also the evidence for the
existence of an envelope is based on very few points. The LAS-*z* envelope is
sometimes used as an argument in favour of the cosmological interpretation of the apparent
redshifts of quasars. It is important to establish whether or not the existence of the
envelope is real or whether it arises because of some statistical or selection effects.
We demonstrate here that the observed behaviour of the envelope of the LAS-*z* plot
is a consequence of statistical effects and is just a reflection of the redshift distribution
diagram, and that it has no intrinsic physical significance.

We consider here 164 quasars out of the 166 in the total list of Miley (1971) and Wardle and Miley (1974). The difference of two is due to the fact that one of the quasars lies outside the scale of Figure 3 of Wardle and Miley (1974), and for another one, we have been unable to find the redshift. We may note here that there seem to be only 164 points in Figure 3 of Wardle and Miley (1974).

**Fig.1.***'Largest angular size-redshift' plot for 164 quasars. The vertical axis represents
the largest angular size in arcseconds. The numbers indicate the
total number of quasars which lie in the respective rectangles. Blank rectangles have no quasars
in them. Curves I, II, and III represent Equations (1), (2), and (3) respectively.*

We have plotted the LAS versus *z* in Figure 1. This is essentially equivalent to
Figure 3 of Wardle and Miley (1974). However, instead of showing the individual
points, we have shown the total number of quasars in compartments having redshift
widths of 0.2 and LAS widths of 20 arcsec. The three theoretical curves (Equations
(1), (2) and (3)) are also shown. Figure 2 is a histogram showing the number of quasars
as a function of the LAS, without reference to the redshift. Figure 3 shows the distribution
of emission-line redshifts for the 164 quasars under consideration.

**Fig.2.***Histogram showing the distribution of the largest angular size (LAS) of quasars.*

It will be noticed from Figure 2 that the majority of quasars (126) have a LAS
<=20 arcsec. These quasars play hardly any role in determining the envelope function.
Also, Figure 2 shows that there is a steep decrease in the number of quasars with
increasing LAS. If we select randomly *n* quasars from these 164, the distribution of the
LAS values of these *n* quasars will be similar to the one shown in Figure 2. If *n* is
large, e.g. about 30, most of them will have a LAS <= 20 arcsec, but we can expect to have
a few quasars in the tail of the distribution up to LAS=150 arcsec. On the other hand,
if *n* is small, e.g. about 7, practically all of them will have a LAS <= 20 arcsec. The extent of
the tail will depend on the number *n*. Next consider the redshift distribution of quasars
shown in Figure 3.

**Fig.3.** *Distribution of emission-line redshifts for 164 quasars.*

It will be noticed that the general trend is that the number of
quasars decreases with increase of redshift. It is obvious from our previous discussion
that the extent of the tail will also decrease with increase of redshift. We can place our
arguments on a quantitative footing by making use of the sampling theory (see, e.g.,
Cochran, 1963). The mean number of objects of a certain type in a sub-population of
size *n* is given by

E = n A / N (4)where

V = n(A/N)(1-A/N)(N-n)/(N-1) (5)In Figure 4 we show the expected mean number of quasars and the standard deviation in the various compartments. The upper quantity in each compartment is the mean number of quasars and the lower quantity is the standard deviation. To avoid non-significant numbers like 0.12, the tail of the distribution was integrated such that the last number was about 1.

**Fig.4.** *Theoretical largest angular size-redshift plot, obtained from statistical
considerations. The upper quantity in each compartment is the expected mean number of quasars
and the lower quantity is the standard deviation.*

It will be noticed that Figure 4 is quite similar to Figure 1. In
other words, the envelope function in the LAS-*z* plot is just a reflection of the redshift
distribution, and in itself has no physical significance. The result of the present
investigation is consistent with our suggestion that there are no redshifts in quasar
spectra (Varshni,
1973,
1974a,
1974b,
1975a,
1975b).

The conclusion of this communication was further confirmed by randomly pairing
LAS and *z* values on a computer. Several runs were made. The resulting plots are
found to be quite similar to Figure 3 of Wardle and Miley (1974).

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The first key to wisdom is assiduous and frequent questioning ... For by doubting we come to inquiry, and by inquiry we arrive at truth.

-Perter Abelard,

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