Evolution of Radio Source Components and the Quasar/Galaxy Unification Scheme

Abstract

In this work, a theoretical model is developed for explanation of temporal evolution of extragalactic radio sources via beaming, orientation effects and asymmetries. Equation of the form D≈P±q1+z−m is used to account for the D ∼ P/z relation. Also, D≈D01+z−1+z1+z2 accounted properly for Ω0=1 cosmology than the Ω0=0 counterpart in linear size versus redshift of radio sources. Similarly, D=Dc1∓lnPPc1/2 model explained redshift-luminosity relationship of extragalactic radio sources. The results from the regression analyses are q = +0.003 (r = 0.04) for sources with z < 1 and q = −1.59 (r = −0.6) for all z≥1 sources. A critical linear size, Dc of 316kpc which matches the maximum theoretical linear size, Dmax of 0.15D0 at a critical redshift zc∼1 and a critical luminosity Pc=26.33WHz−1 are obtained. The indication of all these results is that the linear size of radio sources evolves up to a certain limit in D–P plane and thereafter decreases with increasing luminosity as predicted in this work.

1. Introduction

1.1 The active galactic nucleus (AGN)

The active galactic nucleus (AGN) is the existence of energetic phenomena in the nuclei, or core regions, of galaxies that cannot be clearly and directly explained to the interactions between the stars and interstellar medium. The source of radiation of AGN is definitely believed to emanate from the gravitational potential of gas from a supermassive black hole accreting mass at the core of the host galaxy. This emitted radiation by AGN is throughout the electromagnetic spectrum. The energy radiation in AGN is non-thermal unlike spectra of stars rather it is primarily as a result of the process of synchrotron radiation. In this scenario, power-law spectra, ${\mathrm{S}}_{\mathrm{\nu }}\sim {\mathrm{\nu }}^{‐\mathrm{\alpha }}$ and high degree of linear polarization assigned to AGN object stand as evidence of synchrotron theory. AGNs have typical luminosity in the range of $10^{33}$ to $10^{40}{\mathrm{WHz}}^{-1}$ [1].

Generally, an AGN possesses peculiar properties like, intense bright and point-like nucleus, radio cores with compact flat spectrum, highly ionized gas relativistically beaming out, variable fluxes observed on a wide timescale range from minutes upwards, extremely high luminosity from the range of $10^6-10^{14}$ solar luminosity, narrow, broad, and sometimes without lines of emission, extended radio jets and lobes.

1.2 Classification of AGN

The naming of AGN into classes and subclasses is mostly based on the exhibition of their properties or morphologies. Some AGN classes may be similar as a result of their evolution; some may be due to observed variability of luminosities [2]. Some AGN are classified based on their viewing angle as seen by the observer, which depends mostly on the obscuration of torus [3]. This brings about an unification scheme in AGNs due to the relativistic beaming and orientation effects.

Optically, AGN may be classified as type I or type II based on the optical spectral line emission. The classification of AGN using their response to radio-loudness is preferable because of the clearer observation obtained from the radio band compared to any bands’ counterpart in the electromagnetic window. In this scenario, AGN can be classified as radio-loud and radio-quiet depending on their radio brightness. When they have ratios of radio (5GHz) to optical (B-band) flux ${\mathrm{F}}_5/{\mathrm{F}}_{\mathrm{B}}\ge 10$, they are called radio-loud, otherwise known as radio-quiet if ${\mathrm{F}}_5/{\mathrm{F}}_{\mathrm{B}}\le 10$ [2, 4]. Since this work is entirely based on the observed radio properties of various AGN samples plus the clarification and consideration of radio window observation, the classification of AGNs based on radio-loudness is represented in Figure 1.

Figure 1.

Classification of AGNs based on radio-loudness.

1.3 Radio-loud AGNs

Radio-loud sources emit more energies in the radio band than in the optical waveband, and hence possess ${\mathrm{F}}_5/{\mathrm{F}}_{\mathrm{B}}$ fluxes ≥ 10 [2]. There are about 15–20% of radio-loud AGN. They are mostly elliptical galaxies in accordance with Hubble turning-fork proposition (population II)—meaning that they contain mostly of old stars with little interstellar gas. These objects are further classified into two: high luminosity objects and low luminosity objects.

1.4 High luminosity objects

They have total radio luminosity, ${\mathrm{P}}_{178}\ge 10^{35}{\mathrm{WHz}}^{-1}$, with a highly ionized accretion disk [5]. These comprised of radio-loud quasars, Compact Steep-Spectrum Sources (CSSs) and Fanaroff-Riley class II radio galaxies. The works are mostly on these classes of AGN.

1.4.1 Fanaroff-Riley class II radio galaxies

These are known for their high luminous intensity with the possession of extended double lobes/jets where one side is Doppler enhanced. They also have smooth jets as a result of highly supersonic flows. The jets are also edge-brightened and terminate in hotspots [5, 6, 7]. They polarize linearly with the electric field vector being perpendicularly to the jets. In a unification scheme, [8] suggested that FR II sources are misaligned counterparts of core-dominated quasars.

1.4.2 Quasars

Quasars are classified further into core-dominated and lobe-dominated quasars. These are core-dominated if the radio emissions emanate mostly from the core, otherwise it is a lobe-dominated one.

The core-dominated ones possess properties like flat radio spectra with spectra index, $\mathrm{\alpha }\le 0.5,\left({\mathrm{s}}_{\mathrm{\nu }}-{\mathrm{\nu }}^{‐\mathrm{\alpha }}\right)$ as a result of synchrotron self-absorption mechanism, cores with extremely brightness, broad emission lines and one-sided jets/lobes. These types of quasars dominated the survey at high frequencies and high redshifts. These classes of quasars show more asymmetry than the lobe-dominated counterpart [9].

On the other hand, lobe-dominated quasars, unlike the core-dominated, have two extended lobes straddling a weak compact core. They are also high luminosity sources with total luminosity, ${\mathrm{P}}_{178}\ge 10^{35}{\mathrm{WHz}}^{-1}$. They are characterized by steep radio spectra ($\mathrm{\alpha }>0.5$). They show spectra with broad emission lines; hence, they are referred to as broad-line region sources (BLRS). They also have higher redshifts when compared with radio galaxies [9].

1.4.3 Compact Steep-Spectrum Sources (CSSs)

The CSS sources are characterized by sharp peaks exhibited in their radio spectra. As their names imply, they are compact bright radio sources with a population up to 30% [10]. They are also called “youth” scenario [10] being believed to be the younger phases of powerful large-scale extragalactic radio sources. They have a small radio size, $\mathrm{D}\le 15\mathrm{kpc}$, with a steep radio spectrum, $\mathrm{\alpha }\ge 0.5$, a very high radio luminosity, $\text{logP}>10^{26}{\mathrm{WHz}}^{-1}$ at frequency, ν = 2.7 GHz [10]. CSS radio sources exhibit brightness temperature up to $10^{10}\mathrm{K}$ [10]. Their radio structure is symmetric with low radio polarity and large Faraday rotation measures.

They are CSS radio galaxies if they have double lobes with weak jets and cores emitting weakly, otherwise CSS quasars if they exhibited brighter cores and jets [11]. Majority of CSS radio jets are one-sided and superluminal [11]. From the theory of orientation-based unification scheme, the morphological difference between the CSS radio galaxies and CSS quasars is that objects seen close to the line of sight of the observer are referred to CSS quasars otherwise radio galaxies [2].

1.5 Low luminosity radio sources

Low luminosity radio sources have total radio luminosity, ${\mathrm{P}}_{178}<10^{26}{\mathrm{WHz}}^{-1}$ with less ionized accretion disks [12]. Examples of these sources include FR I radio galaxies and BL Lacertae objects.

1.5.1 Fanaroff and Riley class I (FR I) radio galaxies

According to [12], FR I radio galaxies are characterized by extended double-lobed with low frequency, ν ∼ 400 MHz. FR I has ${\mathrm{R}}_{\mathrm{FR}}<0.5$ as the source faints away from the nucleus, while FR II with brightness further away from the nucleus has ${\mathrm{R}}_{\mathrm{FR}}\ge 0.5$. ${\mathrm{R}}_{\mathrm{FR}}$ ratio is the ratio of the distance between the regions of highest surface brightness to the lowest brightness contour of the central galaxy.

Moreover, FR I sources are symmetric with smooth and continuous jets which begin as one-sided nearer the core and two-sided at a few kilo parsecs away. FR I sources are located in rich clusters that highly emit x-ray gas. The x-ray gas sweeps back and distorts the FR I radio structure as it moves across the interstellar cluster, hence giving FR I object narrow-angle-tail or wide-angle-tail according to the strength of the ram pressure of the gas [13, 14].

1.5.2 BL Lacertae sources (BL Lacs)

BL Lacs objects are the most violent AGN known. They have properties like very weak or sometimes no radio emission lines, compact radio core, rapid and high peak variable fluxes, superluminal flows. They are also known as blazars just like optically violently variable (OVV) quasars.

1.6 Radio quiet objects

These are objects that emit more in the optical window than they do in the radio band. AGN sources are said to be radio quiet if they emit more of their energy in the optical waveband than in the radio waveband. They have properties like ${\mathrm{F}}_5/{\mathrm{F}}_{\mathrm{B}}$ fluxes ≤10, low luminosity at 6 cm less than $10^{26}{\mathrm{WHz}}^{-1}$, short jets, few relativistic particles and total weakness of the radio sources [15]. Examples of these objects are radio quiet quasars and seyferts (seyfert I and seyfert II). In this work, it is centered more on high luminosity radio-loud objects like FR II radio galaxy and quasar.

1.7 Features observed in AGNs

The observational morphological features of a radio source are radio core, jets, lobes and hotspots, though every source may not exhibit all these features.

1.7.1 Radio core

This is the core engine where the energetic radio emission mechanism in EGRSs is assumed to originate. This core is divided into steep spectrum cores and ultra-compact flat spectrum [16]. The steep spectrum cores are characterized in some radio galaxies by having more extended radio cores of few kilo parsecs in size and steeper spectra ($\mathrm{\alpha }\ge 0.5$), example as found in seyferts and some spiral galaxies. On the other hand, the ultra-compact cores’ counterparts possess properties like 1kpc in sizes and flat radio spectra ($\mathrm{\alpha }<0.5$), signifying synchrotron self-absorption that arises as a result of the re-absorption of some radiate relativistic electrons within the radio source. Quasars objects’ core emissions appear to be more powerful than that of the radio galaxies.

1.7.2 Jets

These are the conduits through which the high-energy particles are transporting from the cores to the other extended radio structures. A typical radio jet is expected to be at least four times as long as its width, in line with the radio core and separable from other features at high resolution [17]. The radio jets which can be one or two-sided exhibit properties like emissions with steep spectra of spectral index, α ̴ 0.5–0.9.

1.7.3 Radio lobes

These are one of the extended structures of radio sources that cover a range of hundred kilo parsec to a few mega parsecs. These are characterized by possession of non-thermal steep spectra of spectral indices, $\mathrm{\alpha }\ge 0.5$, with a high degree of polarization at high frequency. They exhibit morphological features like tail, plumes, bridges and haloes. Tails are structures formed as a result of deflected plasma interacting with external medium, while plumes are extended regions with low luminosity that faints away from the whole source. Bridges occur in the inner lobe regions of radio galaxies, while the haloes are low surface brightness structures containing old aged plasma [18].

1.7.4 Hotspots

Hotspots are the brightest region of the extended radio structure formed at the end of the lobe where kinetic power of the jets is converted into random motion within the relativistic plasma and strengthened magnetic fields [19]. They have a linear size of 1kpc and steep spectra, α ̴ 0.5–1 slightly flatter than that of the surrounding diffuse emission [20]. See Figure 2.

Figure 2.

Unified structure of an typical extragalactic radio source [21].

It has been established that the appearance of EGRSs is substantially modified by relativistic beaming and orientation of the radio axes with the line of sight, leading to asymmetries in the observed radio structures. Similarly, radio sources are known to undergo some form of cosmological as well as temporal evolution. However, the amount of relativistic beaming and the nature of the evolution present in different classes and subclasses of the EGRSs are still a subject of intensive research. In particular, different source samples show a wide range of the amount and nature of temporal evolution as reported in literature. Hence, the aim of this work is to analytically examine the observed radio properties of different samples of EGRSs for radio source structural asymmetry, use relativistic beaming and source orientation model to explain any observed structural asymmetry in the radio sources and finally develop a model that can unambiguously explain the temporal evolution in extragalactic radio sources.

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 (${\mathrm{P}}_{\mathrm{v}}$) in a general power-law function as [22]:

$$D\approx D_0{P}_v^{\pm q}$$E1

where ${\mathrm{D}}_0$ (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 $\left(\mathrm{\theta }–\mathrm{z}\right)$ relation is described [23, 24] as

$$\theta =\frac{D{\left(1+z\right)}^2}{d_L},$$E2

where D can be represented using the unit of kpc and ${\mathrm{d}}_{\mathrm{L}}$ is the luminosity distance expressed [24, 25] as

$$d_L=\frac{2c}{H_0{\mathrm{\Omega }}_0^2}\left\{{\mathrm{\Omega }}_{0}z+\left({\mathrm{\Omega }}_0-2\right)\left[{\left({\mathrm{\Omega }}_{0}z+1\right)}^{\frac{1}{2}}-1\right]\right\},$$E3

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 ($S_v$) at observing frequency ($\nu$) according to the [26] as:

$$P=P_0{\left(1+z\right)}^{\beta }$$E4

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:

$$P=P_0{\left(1+z\right)}^{\beta }$$E5

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 $\mathrm{\beta }>0$ 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:

$$D\approx D_0{\left(1+z\right)}^{-k}$$E6

where ${\mathrm{D}}_0$ 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 ${\mathrm{\Omega }}_0$_, the observed $\mathrm{\theta }–\mathrm{z}$ 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]

$$D_{\left(P,z\right)}\approx P^{\pm q}{\left(1+z\right)}^{-m}$$E7

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

$$\mathrm{m}=\mathrm{k}\pm \mathrm{q\beta }$$E8

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

$$\mathrm{k}-\mathrm{q\beta }<0$$E9

and $q>\frac{k}{\beta }$. 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 $q<\frac{k}{\beta }$, 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 $\mathrm{\theta }–\mathrm{z}$ plane depends on the assumed cosmology, which is also, characterized by the value of ${\mathrm{\Omega }}_0$, 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:

$$P=P_{\max }exp\left[\pm {\left(\frac{D}{D_c}-1\right)}^2\right]$$E10

where ${\mathrm{P}}_{\max }$ is the maximum luminosity and ${\mathrm{D}}_{\mathrm{c}}$ is the critical linear size at which the ${\mathrm{P}}_{\max }$ is emitted by a radio source. Rearranging Eq. (10) gives

$$D=D_c\left[1\mp {\left\{ln\left(\frac{P}{\mathit{Pc}}\right)\right\}}^{1/2}\right]$$E11

Eq. (11) comprises of two separable components corresponding to $\mathrm{k}<0\left(-\right)$ and $\mathrm{k}>0\left(+\right).$ 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:

$$D=\frac{D_0\left\{{\mathrm{\Omega }}_{0}z+\left({\mathrm{\Omega }}_0-2\right)\left[{\left({\mathrm{\Omega }}_{0}z+1\right)}^{1/2}-1\right]\right\}}{{\left(1+z\right)}^2},$$E12

where $D_0=\frac{2\mathit{c\theta }}{H_0{\mathrm{\Omega }}_0^2}$ is the intrinsic linear size and a constant. Hence, for inflationary universe, ${\mathrm{\Omega }}_0=1$, [32], the linear size of a radio source depends on z in the form [27]:

$$D\approx \frac{D_0\left\{\left(1+z\right)-\sqrt{1+z}\right\}}{{\left(1+z\right)}^2}$$E13

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 ${\mathrm{\Omega }}_0=0$, on supposition of a low-density universe, for which d_L is given as [33]:

$$d_L=\frac{2c}{H_0}\left[{\left(1+z\right)}^2-1\right],$$E14

Eq. (2) yields,

$$\theta =\frac{{\mathit{DH}}_0{\left(1+z\right)}^2}{2c\left[{\left(1+z\right)}^2-1\right]}.$$E15

Hence, D can be expressed in relation with z as

$$D=D_0\left[1-\frac{1}{{\left(1+z\right)}^2}\right]$$E16

where the intrinsic radio size, $D_0=\frac{\theta 2c}{H_0}$. The variation of D with z for both cosmological models is shown in Figure 3.

Figure 3.

Variation of D with z for ${\mathrm{\Omega }}_0=1$ (a) and ${\mathrm{\Omega }}_0=0$ (b) cosmologies.

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 ${\mathrm{D}}_{\mathrm{C}}\approx 0.146{\mathrm{D}}_0$ (kpc) at a redshift maximum called critical ${\mathrm{z}}_{\mathrm{c}}\approx 1$, and decreases after for ${\mathrm{\Omega }}_0=1$. 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 $\mathrm{Dc}\approx 0.8{\mathrm{D}}_0$ after which it remains constant for ${\mathrm{\Omega }}_0=0$. 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 $0.15{\mathrm{D}}_0$ to $0.8{\mathrm{D}}_0$ 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 ${\mathrm{\Omega }}_0=1$.

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 core-dominance, (R) defined [34] as:

$$R=\frac{P_C}{P_E}=\frac{P_{5\mathit{GHz}}^C}{P_{1.4\mathit{GHz}}^E}{\left(\raisebox{1ex}{$1.4$}\!\left/ \!\raisebox{-1ex}{$5$}\right.\right)}^{-{\alpha }_E}{\left(1+z\right)}^{-{\alpha }_E}$$E17

where ${\mathrm{P}}_{\mathrm{C}}$ represents core-luminosity at 5 GHz, ${\mathrm{P}}_{\mathrm{E}}$ is the extended/lobe luminosity at 1.4 GHz and ${\alpha }_E$ 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 ($\varphi$) in the form [35]:

$$R=\frac{P_C}{P_E}=\frac{R_T}{2}\left[{\left(1-\beta cos\varphi \right)}^{-n+\alpha }+{\left(1+\beta cos\varphi \right)}^{-n+\alpha }\right]$$E18

where $R_T$ is the value of R at $\varphi ={90}^{°}$ 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 (${\mathrm{d}}_{\mathrm{app}}$) to that on the receding jet side (${\mathrm{d}}_{\mathrm{rec}}$), [36, 37] given as:

$$Q=\frac{d_{\mathit{app}}}{d_{\mathit{rec}}}=\frac{1+\beta cos\varphi }{1-\beta cos\varphi }$$E19

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

$$x=\frac{Q-1}{Q+1}$$E20

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]:

$$x=\beta cos\varphi$$E21

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:

$$\mathrm{\delta }={\mathrm{\gamma }}^{-1}{\left(1-\mathrm{\beta }cos\varphi \right)}^{-1}=$$E22

where, $\gamma$ is the bulk Lorentz factor of the jet [8, 42, 43] defined as:

$$\gamma =\frac{1}{{\left(1-{\beta }^2\right)}^{\frac{1}{2}}}$$E23

Also, assuming $\mathrm{\varphi }=0^0$ in Eq. (18) and analyzing further, [44, 45] gives

$$R_{\max }\approx R_T{\gamma }^2\left(2{\gamma }^2-1\right)\approx 2R_T{\gamma }^4$$E24

While at $\mathrm{\alpha }=0$ and $\mathrm{\beta }\sim 1$, 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

$$cos{\varphi }_m=1-{\left(\frac{2\mathit{Rm}}{R_T}\right)}^{-\frac{1}{n}}$$E25

where ${\mathrm{R}}_{\mathrm{m}}$ is the mean value of the R-distribution and ${\mathrm{\varphi }}_{\mathrm{m}}$ 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:

$$x=\frac{\mathit{\delta \gamma }-1}{\mathit{\delta \gamma }}$$E26

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:

$$x=x_{\mathit{max}}–\mathit{\lambda D},$$E27

where $x_{\mathit{max}}$ represents the maximum x for a sample at $\mathrm{D}\sim 0\left({\mathrm{\varphi }}_{\mathrm{c}}\ne 0\right)$ to the line-of-sight [44] and λ is the slope. Thus, if $\mathrm{\beta }\sim 1$ for the relativistic jets, analysis of Eq. (20) down to Eq. (28) for optimum beaming gives [40, 45]:

$${\mathrm{\varphi }}_c\approx {sin}^{-1}\left(\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$\gamma $}\right.\right)\approx {cos}^{-1}{x}_m,$$E28

Hence, if relativistic beaming at small viewing angles is responsible for the observed structural asymmetry, the critical viewing angle ${\mathrm{\varphi }}_c$ 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.

Figure 4.

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

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 ${\mathrm{WHz}}^{-1}$ for the maximum, minimum and range respectively for ESS quasars, while P-values of 28.01, 24.97 and 3.04 ${\mathrm{WHz}}^{-1}$ were obtained for maximum, minimum and range respectively for ESS galaxies. The entire sample yields P-values of 28.01, 24.97 and 3.04 ${\mathrm{WHz}}^{-1}$ for maximum, minimum and range respectively. Further analyses yield median P-values of 27.04, 26.39 and 26.84 ${\mathrm{WHz}}^{-1}$ respectively for ESS quasars, ESS galaxies and the entire data. The mean P-values of 27.01 ± 0.01 ${\mathrm{WHz}}^{-1}$ and 26.39 ± 0.02 ${\mathrm{WHz}}^{-1}$ 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 ${\mathrm{z}}_{\mathrm{c}}=1$, after which it 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 $\left({\mathrm{\Omega }}_0=1\right)$. The median values give significant trends with correlation coefficients of +0.95 and − 0.90 for ${\mathrm{z}}_{\mathrm{c}}=1$ and ${\mathrm{z}}_{\mathrm{c}}\ge 1$ respectively. Results of the regression analyses of the D–P/z data for ${\mathrm{z}}_{\mathrm{c}}=1$ and ${\mathrm{z}}_{\mathrm{c}}\ge 1$ are summarized in Table 1.

Figure 5.

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

Parameters	D–P			D–z
	${\mathrm{D}}_0$	q	r	${\mathrm{D}}_0$	k	r
$\mathrm{z}<1$	2.37	0.003	0.004	2.55	−1.09	−0.12
$\mathrm{z}\ge 1$	2.67	−1.59	−0.6	−1.59	2.67	−0.5
$\mathrm{P}<{\mathrm{P}}_{\mathrm{c}}$	4.91	0.17	0.21	2.51	0.81	−0.05
$\mathrm{P}\ge {\mathrm{P}}_{\mathrm{c}}$	16.48	−0.32	−0.7	2.63	−1.50	−0.6

Table 1.

D–P/z regression analyses results for both z < 1 and $\mathrm{z}\ge 1/\mathrm{P}<{\mathrm{P}}_{\mathrm{c}}$ and $\mathrm{P}\ge {\mathrm{P}}_{\mathrm{c}}$ [24].

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

Figure 6.

Scatterplot of logD(kpc) against logP (${\mathrm{WHz}}^{‐1}$) for both sources with $\mathrm{P}\le {\mathrm{P}}_{\mathrm{c}}$ (circle), $\mathrm{P}\ge {\mathrm{P}}_{\mathrm{c}}$ (square) and median (triangle) values superimposed [24].

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, ${\text{logP}}_{\mathrm{c}}=26.33{\mathrm{WHz}}^{-1}$ 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 $\mathrm{P}<{\mathrm{P}}_{\mathrm{c}}$. However, for sources with $\mathrm{P}\ge {\mathrm{P}}_{\mathrm{c}}$, there is a fairly strong correlation. Therefore, the weak trend found in the region below z = 1 and $\mathrm{P}<{\mathrm{P}}_{\mathrm{c}}$ 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, $\mathrm{z}\ge 1$ counterparts in any flux density limited samples.

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 ${\text{logP}}_{\mathrm{c}}=26.33$ 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 ${\mathrm{D}}_{\mathrm{c}}$ at ${\mathrm{z}}_{\mathrm{c}}\sim 1$. However, the critical luminosity, ${\mathrm{P}}_{\mathrm{c}}$ 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 ${\mathrm{\Omega }}_0=1$ model but ${\mathrm{\Omega }}_0=0$ cosmological model (low-density universe) [24].

Figure 7.

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

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 $\mathrm{P}\ge \mathrm{Pc}$. 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, $\text{logPc}=26.33{\mathrm{WHz}}^{-1}$ 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. $\mathrm{Dc}=316\mathrm{kpc}$ is obtained from the D - z plane at the turning point. This Dc corresponds to theoretical ${\mathrm{D}}_{\mathrm{c}}=0.14{\mathrm{D}}_0$ at ${\mathrm{z}}_{\mathrm{c}}=1$ for ${\mathrm{\Omega }}_0=1$ in Figure 3a and logPc = 26.33 ${\mathrm{WHz}}^{-1}$ in Figure 7. The indication of this is that ${\mathrm{D}}_0$ (∼ 2100 kpc in the 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 $\mathrm{P}\ge {\mathrm{P}}_{\mathrm{c}}$ respectively, indicating that the ${\mathrm{D}}_{\mathrm{c}}=316\mathrm{kpc}$ of the observed samples is consistent with D_c = 0.14D₀ just in accordance with the theory for the inflationary model of the universe, ${\mathrm{\Omega }}_0=1$ only, at ${\mathrm{z}}_{\mathrm{c}}=1$ and ${\mathrm{P}}_{\mathrm{c}}=26.33{\mathrm{WHz}}^{-1}$, 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 r ∼ − 0.5, chance probability ρ ∼ 10⁻¹⁰ and critical viewing angle, ${\varphi }_{\mathrm{c}}\approx 70^{\mathrm{o}}$, which corresponds to $\mathrm{\gamma }=1.1$. The upper envelope function gave correlation coefficient, r ∼ 0.9, ${\varphi }_{\mathrm{c}}\approx {48}^{\mathrm{o}}$ and $\mathrm{\gamma }=1.3$. The analyses for separate objects, radio galaxy and quasar sub-samples, of upper envelope functions yield ${\varphi }_{\mathrm{c}}\approx {59}^{\mathrm{o}}$; γ = 1.2 and ${\varphi }_{\mathrm{c}}\approx {33}^{\mathrm{o}}$; $\mathrm{\gamma }=1.8$, for radio galaxies and quasars, respectively. These results correspond to angular separation $\left({\varphi }_{\mathrm{sep}}\right)$ of $\sim {26}^{\mathrm{o}}$, based on the upper envelope functions [40]. The results are shown in Table 2 and Figure 8.

Source parameters	Radio galaxies	Quasars	Galaxies + quasars
$x_m$	0.16	0.25	0.20
λ	+0.000056	−0.0002	+0.000002
r	+0.3	−0.3	−0.5
Sig. (%)	5.0	5.0	5.0

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 (∆) [40].

The results opine that the relativistic beaming and source orientation effects are the major cause of large-scale structural asymmetries observed among powerful 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 $\mathrm{\theta }–\mathrm{z}$ data can be in the range of k = 2.0 and 1.5 to k = 1.2 and 0.75 for $\mathrm{\Omega }=1$ 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 $\mathrm{\Omega }=1$, there is an increase in the linear size of radio sources up to a certain point known as critical linear size, ${\mathrm{D}}_{\mathrm{c}}$ which corresponds to critical luminosity, ${\mathrm{P}}_{\mathrm{c}}$ at critical redshift, ${\mathrm{z}}_{\mathrm{c}}\sim 1$ 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 ${\mathrm{z}}_{\mathrm{c}}=1$ 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 ${\mathrm{z}}_{\mathrm{c}}\sim 1$ and ${\mathrm{\Omega }}_0=1$ as predicted. This shows that the temporal evolution model can be well understood using ${\mathrm{\Omega }}_0=1$ than ${\mathrm{\Omega }}_0=0$. 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, $\mathrm{Dc}=316\mathrm{kpc}$ in consistence with the maximum theoretical linear size, ${\mathrm{D}}_{\max }=0.14{\mathrm{D}}_0$ at ${\mathrm{z}}_{\mathrm{c}}\sim 1$ and ${\mathrm{P}}_{\mathrm{c}}=26.33{\mathrm{WHz}}^{-1}$ 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 ($\mathrm{D}\sim {\mathrm{P}}^{\mathrm{q}}$ 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 $\mathrm{D}\sim {\mathrm{P}}^{-0.52}$. 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, ${\mathrm{P}}_{\mathrm{c}}\sim 26{\mathrm{WHz}}^{-1}$, 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.

3. Conclusions

The following major conclusions were drawn from the work. A satisfactory theoretical model was obtained to explain the observed D–P track, with sources at z ≥ 1 showing stronger anti-correlation than those around z < 1. A problem was developed for these objects to match-up with D–z positive correlation as predicted in this paper at z < 1. Finally, the distributions of radio galaxies and quasars on the x–D plane are consistent with the beaming scenario, with quasars being more consistent with relativistic beaming and source orientation model than radio galaxies.