**2. Theory of relationships**

In this approach, relations between various parameters of the radio sources would be derived based on established laws and theories under certain assumptions. The theories of relationships are outlined.

#### **2.1 Theory of temporal evolution in radio sources**

The standard relationship for an ideal temporal evolution model of extragalactic radio sources can be studied using the relationship between the two key parameters known as the observed linear size (D) and the spectral luminosity (Pv) in a general power-law function as [22]:

$$D \approx D\_0 P\_v^{\pm q} \tag{1}$$

where D0 (a constant) represents D at z � 0 while the slope (q) is the temporal evolution parameter.

Meanwhile, from the Friedmann-Robertson-Walker universe, the radio angular size–redshift ð Þ θ–z relation is described [23, 24] as

$$\theta = \frac{D(\mathbf{1} + \mathbf{z})^2}{d\_L},\tag{2}$$

where D can be represented using the unit of kpc and dL is the luminosity distance expressed [24, 25] as

$$d\_L = \frac{2c}{H\_0 \Omega\_0^2} \left\{ \Omega\_0 x + (\Omega\_0 - 2) \left[ (\Omega\_0 x + 1)^{\frac{1}{2}} - 1 \right] \right\},\tag{3}$$

and c is the speed of light. Moreover, the radio luminosity (P) of a radio source at redshift (z) can be defined as a function of the spectral flux density (*Sv*) at observing frequency (*ν*) according to the [26] as:

$$P = P\_0 (\mathbf{1} + z)^{\beta} \tag{4}$$

where α stands for spectral index. Thus, from [27] work, the relationship between P and z can be approximated to a simple power-law function of the form:

$$P = P\_0(\mathbf{1} + z)^{\beta} \tag{5}$$

where β is the slope of the P–z which is supposed to be constant over all values of z for a given sample of sources. In this scenario, a significant correlation for all values of z in the line with β >0 is expected in a P–z data of different samples of extragalactic radio sources, due to luminosity selection effects in flux density limited samples [27].

Besides, researchers [23, 24, 28] suggest that linear sizes of extragalactic radio sources evolve with cosmological epoch in the form:

$$D \approx D\_0 (\mathbf{1} + \mathbf{z})^{-k} \tag{6}$$

where D0 is the normalized linear size which depends on the assumed cosmology and k represents the evolution parameter which could be a function of both cosmological evolution and luminosity selection effects.

Alternatively, with different values of Ω0, the observed θ–z data of extragalactic radio sources deviate significantly from the standard Friedmann models [29], as a result of an entanglement of two effects, namely linear size evolution [29] and luminosity selection effect [30]. Hence, the linear size evolution of extragalactic radio sources can therefore be expressed as a function of both redshift and luminosity in the form of [24, 30, 31]

$$D\_{(\mathcal{P}, \mathfrak{z})} \approx P^{\pm q} (\mathfrak{1} + \mathfrak{z})^{-m} \tag{7}$$

where m is the residual cosmological evolution parameter defined [24, 27] as:

$$\mathbf{m} = \mathbf{k} \pm \mathbf{q} \clubsuit \tag{8}$$

When the effect of luminosity is controlled and this depends on the product of temporal evolution (q) and luminosity selection effect (β).

It is certain from above relation that if luminosity selection effect is above cosmological evolution, then, when the effect of luminosity is corrected, m < 0, so that Eq. (8) becomes

$$
\mathbf{k} - \mathbf{q}\boldsymbol{\beta} < \mathbf{0} \tag{9}
$$

and *q*> *<sup>k</sup> <sup>β</sup>*. For such sources, a significant positive D–P correlation is envisaged, suggesting a linear relationship between the two parameters. On the other hand, if there is a residual linear size evolution at certain values of k, q < 0 and m > 0, then *q*< *<sup>k</sup> <sup>β</sup>*, indicating a significant D–P anti-correlation and also implies that the luminosity decreases with the expansion of the sources. Hence, the values of k are expected to vary in a given sample. In this scenario, the temporal evolution model for an assumed cosmology can be constrained using the D–z plane.

It is also believed that D should not increase with z for all z. Although, considering the fact that radio sources are not just rigid rods and the θ–z plane depends on the assumed cosmology, which is also, characterized by the value of Ω0, in principle, this

may not always be the case. Hence, the P–D track of all inclusive radio source samples could be defined [24, 29] as:

$$P = P\_{\text{max}} \exp\left[\pm \left(\frac{D}{D\_c} - 1\right)^2\right] \tag{10}$$

where P max is the maximum luminosity and Dc is the critical linear size at which the P max is emitted by a radio source. Rearranging Eq. (10) gives

$$D = D\_c \left[ \mathbf{1} \mp \left\{ \ln \left( \frac{P}{Pc} \right) \right\}^{1/2} \right] \tag{11}$$

Eq. (11) comprises of two separable components corresponding to k <0 ð Þ � and k> 0 ð Þ þ *:* The D–P relation (c.f. Eq. (11)) is shown in **Figure 7**.

However, using Eq. (3) in (2), the linear size can be defined as:

$$D = \frac{D\_0 \left\{ \Omega\_0 \mathbf{z} + (\mathbf{\Omega}\_0 - \mathbf{\mathcal{I}}) \left[ (\mathbf{\Omega}\_0 \mathbf{z} + \mathbf{1})^{1/2} - \mathbf{1} \right] \right\}}{\left(\mathbf{1} + \mathbf{z}\right)^2},\tag{12}$$

where *<sup>D</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup>*c<sup>θ</sup> H*0Ω<sup>2</sup> 0 is the intrinsic linear size and a constant. Hence, for inflationary universe, Ω<sup>0</sup> ¼ 1, [32], the linear size of a radio source depends on z in the form [27]:

$$D \approx \frac{D\_0 \left\{ \left( 1 + z \right) - \sqrt{1 + z} \right\}}{\left( 1 + z \right)^2} \tag{13}$$

Eq. (13) indicates two components of z: z < 1 and z > 1, where D increases with increasing z in the first so that k > 0, while the revised is the case in the last component for all D and k. On the other hand, assuming Ω<sup>0</sup> ¼ 0, on supposition of a low-density universe, for which dL is given as [33]:

$$d\_L = \frac{2c}{H\_0} \left[ (\mathbf{1} + \mathbf{z})^2 - \mathbf{1} \right],\tag{14}$$

Eq. (2) yields,

$$\theta = \frac{DH\_0(\mathbf{1} + z)^2}{2c\left[\left(\mathbf{1} + z\right)^2 - \mathbf{1}\right]}.\tag{15}$$

Hence, D can be expressed in relation with z as

$$D = D\_0 \left[ 1 - \frac{1}{\left(1 + z\right)^2} \right] \tag{16}$$

where the intrinsic radio size, *<sup>D</sup>*<sup>0</sup> <sup>¼</sup> *<sup>θ</sup>*2*<sup>c</sup> H*<sup>0</sup> . The variation of D with z for both cosmological models is shown in **Figure 3**.

It is obvious from **Figure 3** that there is increase in D as z increases up to a certain maximum point known as a critical value of DC ≈0*:*146 D0 (kpc) at a redshift

*Evolution of Radio Source Components and the Quasar/Galaxy Unification Scheme DOI: http://dx.doi.org/10.5772/intechopen.106244*

**Figure 3.** *Variation of D with z for* Ω<sup>0</sup> ¼ 1 *(a) and* Ω<sup>0</sup> ¼ 0 *(b) cosmologies.*

maximum called critical zc ≈1, and decreases after for Ω<sup>0</sup> ¼ 1. This implies that for parameters, z < 1, k > 0 and q > 0, there is a positive temporal evolution, while for z >1, k < 0 and q < 0 a negative temporal evolution is envisaged which is in good agreement with the predictions of Eq. (13).

On the other hand, there is increase in D up to a critical point Dc≈0*:*8D0 after which it remains constant for Ω<sup>0</sup> ¼ 0. It is now clear that in the two cosmological models, there is indication of Dc at which luminosity is maximum, suggesting that for any assumed cosmological model, the Dc value obtained from D–P turnover can still be found using the D–z plane of the same data. Hence, in any assumed cosmological model, there will be an expected range of the Dc value of 0*:*15D0 to 0*:*8D0 bounded by the two cosmologies. In this scenario, temporal evolution in the current sample of EGRSs would be modeled in terms of the current inflationary one with Ω<sup>0</sup> ¼ 1.

#### *2.1.1 Relativistic beaming based on orientation and radio source asymmetries*

The standard relation for explaining an ideal relativistic beaming and orientation effects for extragalactic sources is often carried using a key parameter known as coredominance, (R) defined [34] as:

$$R = \frac{P\_C}{P\_E} = \frac{P\_{\text{SGHz}}^C}{P\_{1.4GHz}^E} \left( \imath \nearrow \right)^{-a\_E} (1+z)^{-a\_E} \tag{17}$$

where PC represents core-luminosity at 5 GHz, PE is the extended/lobe luminosity at 1.4 GHz and *α<sup>E</sup>* is the lobe spectral index. However, if relativistic beaming effect at small orientation angle, then R can be expressed in terms of the jet speed (β) and inclination angle (*ϕ*) in the form [35]:

$$R = \frac{P\_C}{P\_E} = \frac{R\_T}{2} \left[ (\mathbf{1} - \beta \cos \phi)^{-n+a} + (\mathbf{1} + \beta \cos \phi)^{-n+a} \right] \tag{18}$$

where *RT* is the value of R at *<sup>ϕ</sup>* <sup>¼</sup> <sup>90</sup>° and n is a parameter that depends on the assumed flow model of the radio jet. For radiating plasma with continuous jet n = 2, otherwise n = 3 if the jet consists of blobs.

An obvious outcome of relativistic beaming and orientation effects in AGNs is the wide range of asymmetry observed in their radio structures. The radio source

asymmetry can be explained using the arm-length ratio (Q), defined as the ratio of the distance, from the central engine, of a plasma element emitting radio waves on the approaching jet side (dapp) to that on the receding jet side (drec), [36, 37] given as:

$$Q = \frac{d\_{app}}{d\_{rec}} = \frac{1 + \beta \cos \phi}{1 - \beta \cos \phi} \tag{19}$$

On the other hand, [38, 39] suggested that,

$$x = \frac{Q - 1}{Q + 1} \tag{20}$$

where, *x* represents the index of the asymmetry.

This *x* parameter which believed to have better relationship with orientation when compared to Q can further be defined in terms of the viewing angle as [40, 41]:

$$
\infty = \beta \cos \phi \tag{21}
$$

The relativistic beaming in AGN at small angles to the line-of-sight is fundamentally characterized by beaming enhancement factor (δ) expressed [40, 42] as:

$$\delta = \chi^{-1} (1 - \beta \cos \phi)^{-1} = \tag{22}$$

where, *γ* is the bulk Lorentz factor of the jet [8, 42, 43] defined as:

$$\chi = \frac{1}{\left(1 - \rho^2\right)^{\frac{1}{2}}} \tag{23}$$

Also, assuming <sup>ϕ</sup> <sup>¼</sup> 00 in Eq. (18) and analyzing further, [44, 45] gives

$$R\_{\text{max}} \approx R\_T \gamma^2 \left(2\gamma^2 - 1\right) \approx 2R\_T \gamma^4 \tag{24}$$

While at α ¼ 0 and β � 1, for high luminosity radio-loud AGN sources emitting at small angle to the line-of-sight of the observer, (to a first approximation) Eq. (18) reduces to

$$\cos \phi\_m = \mathbf{1} - \left(\frac{2Rm}{R\_T}\right)^{-\frac{1}{u}} \tag{25}$$

where Rm is the mean value of the R-distribution and ϕ<sup>m</sup> is the mean observation angle.

Thus, it can be shown from Eq. (23) that the asymmetry parameter *x* can be expressed in terms of the beaming enhancement factor as:

$$\boldsymbol{\omega} = \frac{\delta \boldsymbol{\gamma} - \mathbf{1}}{\delta \boldsymbol{\gamma}} \tag{26}$$

Eq. (26) implies that there is an association of relativistic beaming and radio source asymmetry. In asymmetric sources with Q > 1, *x*–D anti-correlation is expected if

geometric projection at small viewing angles is responsible for the observed asymmetry. Following [40, 41], we assume a linear *x*–D relation of the form:

$$
\infty = \varkappa\_{\text{max}} - \lambda D,\tag{27}
$$

where *x max* represents the maximum *x* for a sample at D � 0ð Þ ϕ<sup>c</sup> 6¼ 0 to the lineof-sight [44] and λ is the slope. Thus, if β � 1 for the relativistic jets, analysis of Eq. (20) down to Eq. (28) for optimum beaming gives [40, 45]:

$$
\Phi\_c \approx \sin^{-1}(\mathbb{Y}\_r) \approx \cos^{-1} \mathbb{x}\_m,\tag{28}
$$

Hence, if relativistic beaming at small viewing angles is responsible for the observed structural asymmetry, the critical viewing angle ϕ*<sup>c</sup>* as well as the Lorentz factor γ can be obtained using the *x*–D data.

#### **2.2 Statistical analyses and results**

#### *2.2.1 The source samples*

The data used in the present analysis were drawn from a well-defined source sample of [46] compilation which contains required information on the two objects of interest–ESS quasars and ESS radio galaxies, [34] compilation of 542 extragalactic radio sources and the deep VLA sample of FSRQs compiled by [47]. In these samples, there is wide dispersion in the distributions of the observed parameters.

#### *2.2.2 Distributions of observed radio parameters*

The distributions of the linear size, D, of the extended steep-spectrum sources (ESSs) in logarithm scales are represented in **Figure 4a**. The graph shows D-values of 1896.3, 17.1 and 1879 kpc for the maximum, minimum and range respectively for ESS quasars, while D-values of 5853.30, 29.80 and 5823.5 kpc were obtained for maximum, minimum and range respectively for ESS galaxies. The entire sample yields D-values of 5853.3, 17.1 and 5836.2 kpc for maximum, minimum and range respectively. Further analyses yield median D-values of 148.90, 323.59 and 201.90 kpc for ESS quasars, ESS galaxies and entire sample respectively. The mean D-values of 144.54 � 7.52 kpc and 288.40 � 33.66 kpc were obtained respectively for ESS quasars and radio galaxies.

The distributions of the redshift, z of the ESSs in logarithm scales are represented in **Figure 4b**. The graph shows z-values of 2.88, 0.05 and 2.83 for the maximum, minimum and range respectively for ESS quasars, while z-values of 3.22, 0.006 and 3.21 were obtained for maximum, minimum and range respectively for ESS galaxies. The entire sample yields z-values of 3.22, 0.006 and 3.21 for maximum, minimum and range respectively. Further analyses yield median z-values of 1.89, 1.17 and 1.62 for ESS quasars, ESS galaxies and entire sample respectively. The mean z-values of 1.95 � 0.01 and 1.30 � 0.02 were obtained respectively for ESS quasars and radio galaxies.

The distributions of the radio luminosity, P of the ESSs in logarithm scales are represented in **Figure 4c**. The graph shows P-values of 27.90, 25.71 and 2.18 WHz�<sup>1</sup> for the maximum, minimum and range respectively for ESS quasars, while P-values of 28.01, 24.97 and 3.04 WHz�<sup>1</sup> were obtained for maximum, minimum and range respectively for ESS galaxies. The entire sample yields P-values of 28.01, 24.97 and

**Figure 4.** *Distribution of (a) D, (b) z and (c) P respectively for ESS quasars (lines) and ESS galaxies (plane).*

3.04 WHz�<sup>1</sup> for maximum, minimum and range respectively. Further analyses yield median P-values of 27.04, 26.39 and 26.84 WHz�<sup>1</sup> respectively for ESS quasars, ESS galaxies and the entire data. The mean P-values of 27.01 � 0.01 WHz�<sup>1</sup> and 26.39 � 0.02 WHz�<sup>1</sup> were obtained respectively for ESS quasars and radio galaxies **Figure 4**.

### *2.2.3 D: P/z correlation*

**Figure 5** represents the scatter plot of linear size, D against the redshift, z. The median value data in seven redshift bins is superimposed on the plot. Critical investigation of the plot shows that on average, the linear size increases with increasing redshift up to a value logDc = 2.5 kpc (Dc = 316.23 kpc) at zc ¼ 1, after which it

*Evolution of Radio Source Components and the Quasar/Galaxy Unification Scheme DOI: http://dx.doi.org/10.5772/intechopen.106244*

**Figure 5.** *Scatterplot of logD (kpc) against z for entire sources (circle) with median values (square) superimposed [24].*


#### **Table 1.**

*D–P/z regression analyses results for both z < 1 and* z≥1*=*P<Pc *and* P ≥Pc *[24].*

decreases with increasing redshift [24]. This is in agreement with the prediction made in **Figure 3(a)**. Hence, the present data obviously proved consistency with the inflationary model of the universe ð Þ Ω<sup>0</sup> ¼ 1 . The median values give significant trends with correlation coefficients of +0.95 and � 0.90 for zc ¼ 1 and zc ≥1 respectively. Results of the regression analyses of the D–P/z data for zc ¼ 1 and zc ≥1 are summarized in **Table 1**.

In modelling the temporal evolution of the sample, **Figure 6** represents the scatter plot of projected linear size (D) and the radio luminosity (P). Similarly, the median value data in eight uniform luminosity bins are superimposed on the plot. There is an obvious trend indicating that the linear size first increases

with increasing luminosity up to a certain value and thereafter decreases with increasing luminosity. The median value data showed very significant trends. This suggests that the turnover occurs at a critical point of luminosity, logPc ¼ 26*:*33 WHz�<sup>1</sup> and logDc = 2.51 kpc (316.23 kpc) [24]. A summary of the results of these regression analyses of the D–P and D–z data for sources with P ≤ Pc and P > Pc is presented in **Table 1**.

The results in **Table 1** did not show any obvious trend in D–P relation for sources with P< Pc. However, for sources with P≥Pc, there is a fairly strong correlation. Therefore, the weak trend found in the region below z = 1 and P< Pc is believed to be due to the effects of luminosity selection. In this scenario, the low redshift samples, z < 1 have more impacts on average luminosity-redshift plane than the high redshift, z≥ 1 counterparts in any flux density limited samples.

#### **Figure 6.**

*Scatterplot of logD(kpc) against logP (*WHz‐<sup>1</sup>*) for both sources with* P≤Pc *(circle),* P≥Pc *(square) and median (triangle) values superimposed [24].*

#### *2.2.4 Luminosity selection effect on temporal evolution model*

**Figure 7** represents the P–z scatter plot of the sample. There is a turnover at critical point logPc ¼ 26*:*33 W/Hz and z � 0.05 in the P–z plane. This point of P is presumably the value of luminosity in **Figure 6** that will correspond to Dc at zc � 1. However, the critical luminosity, Pc found in the P–z plane is inconsistent with sources at z = 1. Hence, the luminosity-redshift relation of the current data did not assume Ω<sup>0</sup> ¼ 1 model but Ω<sup>0</sup> ¼ 0 cosmological model (low-density universe) [24].

In **Figure 6**, the effect of Eqs. (10) and (11) in the light of the temporal evolution model was considered. Hence, the data are grouped into two; P < Pc and P≥Pc. The results of the regression analyses are shown in **Table 1**. These opine that the linear size, D of radio sources increases up to critical radio luminosity, logPc <sup>¼</sup> <sup>26</sup>*:*33WHz�<sup>1</sup> and decreases thereafter. This is in agreement with Eq. (10), hence suggesting that eqn. (10) is approximately correct to zero order. This is applicable when critical linear size Dc is used. Dc ¼ 316kpc is obtained from the D - z plane at the turning point. This Dc corresponds to theoretical Dc ¼ 0*:*14D0 at zc ¼ 1 for Ω<sup>0</sup> ¼ 1 in **Figure 3a** and logPc = 26.33 WHz�<sup>1</sup> in **Figure 7**. The indication of this is that D0 (� 2100 kpc in the

**Figure 7.** *Scatterplot of logP (W/Hz) against log (1 + z) for all sample.*

### *Evolution of Radio Source Components and the Quasar/Galaxy Unification Scheme DOI: http://dx.doi.org/10.5772/intechopen.106244*

observed data) is approximately the linear size at the earliest epoch (at z � 0.02 in the observed data of the present sample). The correlation coefficient, r � +0.4, �0.5 and � 0.9 respectively for sources with P < Pc, P ≥ Pc and the median values respectively were obtained. These suggest that there are positive and negative correlations in the D–P track at P < Pc and P≥Pc respectively, indicating that the Dc ¼ 316kpc of the observed samples is consistent with Dc = 0.14D0 just in accordance with the theory for the inflationary model of the universe, Ω<sup>0</sup> ¼ 1 only, at zc ¼ 1 and Pc ¼ 26*:*33WHz�<sup>1</sup> , proving Eq. (11) to be perfectly correct. Hence, temporal evolution in extragalactic radio sources can be explained.

### *2.2.5 The* x *– D relationship*

There is a fairly significant trend in the *x*–D plot, which is obvious at the upper envelope function. This yields: x = 0.35–0.0006 D with a correlation coefficient <sup>r</sup> � � 0.5, chance probability <sup>ρ</sup> � <sup>10</sup>�<sup>10</sup> and critical viewing angle, *<sup>ϕ</sup>*<sup>c</sup> <sup>≈</sup>70o, which corresponds to γ ¼ 1*:*1. The upper envelope function gave correlation coefficient, <sup>r</sup> � 0.9, *<sup>ϕ</sup>*<sup>c</sup> <sup>≈</sup> 48o and <sup>γ</sup> <sup>¼</sup> <sup>1</sup>*:*3. The analyses for separate objects, radio galaxy and quasar sub-samples, of upper envelope functions yield *ϕ*<sup>c</sup> ≈59o; γ = 1.2 and *ϕ*<sup>c</sup> ≈33o; γ ¼ 1*:*8, for radio galaxies and quasars, respectively. These results correspond to angular separation *<sup>ϕ</sup>*sep of � 26o , based on the upper envelope functions [40]. The results are shown in **Table 2** and **Figure 8**.

The results opine that the relativistic beaming and source orientation effects are the major cause of large-scale structural asymmetries observed among powerful


#### **Table 2.**

x*–D regression analyses results [40].*

#### **Figure 8.**

*Scatterplot of the fractional separation difference (x) as against projected linear size (D) for quasars (•) and radio galaxies (Δ) [40].*

extragalactic radio sources, which are more obvious in core-dominated sources with large core-to-lobe luminosity ratio.

#### **2.3 General discussions and conclusions**

#### *2.3.1 Discussions*

The study of the redshift effect dependence of radio size in extragalactic radio sources has been of great interest since inception of the universe. This is obviously important in cosmological studies more especially in testing of the evolution in the extragalactic radio sources. In this scenario, a theoretical model that can best interpret the D–P track is developed. According to [48] the analyzes obtained, the high redshift radio sources show high bending angles, distorted and smaller structures than the low redshift radio sources. According to the researchers, the effects on the radio source evolution depend mostly on intracluster/ or interstellar medium. A plausible model was first developed by [49] for strong linear size evolution for radio galaxies. This model explained that the typical radio sources, both giant galaxies and sub-galactic quasars, evolve at high redshifts, z.

The temporal evolution of radio linear size could be interpreted in the light of observed D–z plane for radio source samples. Hence, the D–z correlation helps in constraining cosmological/temporal models. According to [50], the amount of linear size evolution required to interpret the observed θ–z data can be in the range of k = 2.0 and 1.5 to k = 1.2 and 0.75 for Ω ¼ 1 and 0 respectively. A clear investigation from the median values in **Figure 5** shows that the D–z curve of the observed data is in good agreement with that of the theoretical D–z plane of **Figure 3**. In other word, for Ω ¼ 1, there is an increase in the linear size of radio sources up to a certain point known as critical linear size, Dc which corresponds to critical luminosity, Pc at critical redshift, zc � 1 and thereafter decreases. This steep change in P–z relation of extragalactic radio sources may occur around z = 0.3 [27, 33] or z = 1 (see also [46], Fig. 2) [51]. This is due to the luminosity selection effect occurred as a result of Malmquist bias in flux density limited samples [27]. In this effect, the present work adopted samples based on zc ¼ 1 as predicted earlier with the theory. This raise and fall of the radio sources in the D–z plane are consistent with the observational results showing that radio galaxies and quasars exhibit different D–P correlations.

In this scenario, the median values of D–P of the entire data superimposed in **Figure 6** clearly show that the observed D–P data of the sample are consistent with the theoretical curves (c.f. **Figure 3a**). Hence, this suggests that the P–D data of the present sample could be used to constrain the temporal evolution model.

A cursory look in **Table 1** shows that there is steep change in q as predicted above in the subsample around z ≥ 1 with fairly trends with a significant correlation in the median values obtained from the eight appropriate luminosity bins of the entire sample. The results show that the apparent lack of D–P correlation in the samples below z = 1 was due to the luminosity selection effects in the observed data. In other words, in flux density limited samples, the low redshift (z < 1) sources exhibit more dependence on luminosity as a function of redshift than the high redshift (z > 1) source counterparts. Below z = 1, the P–z weak correlation is expected, while Beyond z = 1, the luminosities have a much milder dependence on z. The regression analyses yield q = +0.002 for all samples around z ≤ 1 and q = �1.59 at z = ≥ 1. Hence, at z < 1, there is a positive temporal evolution model while at z ≥ 1, negative temporal evolution is obtained for zc � 1 and Ω<sup>0</sup> ¼ 1 as predicted. This shows that the temporal

*Evolution of Radio Source Components and the Quasar/Galaxy Unification Scheme DOI: http://dx.doi.org/10.5772/intechopen.106244*

evolution model can be well understood using Ω<sup>0</sup> ¼ 1 than Ω<sup>0</sup> ¼ 0. In this scenario, the temporal evolution is positive when z < 1, k > 0 and q > 0, and otherwise for z > 1, k < 0 and q < 0. Thus, the linear size increases as a function of radio luminosity up to a maximum value called critical D-value, Dc ¼ 316kpc in consistence with the maximum theoretical linear size, D max <sup>¼</sup> <sup>0</sup>*:*14D0 at zc � 1 and Pc <sup>¼</sup> <sup>26</sup>*:*33WHz�<sup>1</sup> and thereafter decreases with increasing luminosity, as predicted earlier in this work.

Several authors have argued that the unified scheme will vanish out if the derived positive D–P dependence for radio galaxies (D � <sup>P</sup><sup>q</sup> for q � 0.3) contrasts with the negative D–P dependence in quasars [52]. Ubachukwu and Onuora [50] suggested an inverse correlation of D–P of the form D � <sup>P</sup>�0*:*52. They found out that radio galaxies locate at low redshifts with q � 0.3 [31], while radio quasars located at high redshifts with q � �0.5 [48]. The D–P turnover in extragalactic radio sources occurs around Dc = 100kpc [29]. It was argued [24] that at D � 1Mpc and critical luminosity, Pc � 26WHz�<sup>1</sup> , extragalactic radio sources evolve and thereafter decrease.

In *x*–D regression analyses, there is an apparent lack of *x*–D anti-correlation in the entire samples and radio galaxies subsample. There was a fairly *x*–D anti-correlation in quasars subsample which was obvious in the upper envelope function with r � �0.5, 0.30 and � 0.3 for the entire sample, radio galaxies and quasars respectively. Hence, the results suggest that orientation effects may be more necessary in explaining the properties of the radio quasars population than that of the radio galaxies populations in which the expected *x*–D anti-correlation was not observed and also implies that the two objects are the same, the difference is just the angle at which the observer took in viewing them.
