ON THE ORIGIN OF FANAROFF–RILEY CLASSIFICATION OF RADIO GALAXIES: DECELERATION OF SUPERSONIC RADIO LOBES

Fanaroff & Riley (1974) discovered that the radio galaxies whose linear size l is l ⩾ 10 kpc exhibit a change in morphology from edge-darkened to edge-brightened at a monochromatic power ∼10^24.5 W Hz⁻¹ at a rest-frame frequency of 1.4 GHz. The Fanaroff–Riley type I radio galaxies (FRIs) have the edge-darkened morphology and the subsonic advance speed of their lobes (hereafter subsonic lobes), while the Fanaroff–Riley type II radio galaxies (FRIIs) possess the edge-brightened morphology and the supersonic advance speed of their lobes (hereafter supersonic lobes). Owen & White (1991) and Ledlow & Owen (1996) found a striking separation between the FRIs and FRIIs in the radio-optical luminosity plane. At a given optical luminosity of host galaxies, FRIIs are located on the side of brighter radio luminosity, while FRIs tend to fall on the side of fainter radio luminosity.

The origin of the "FRI/FRII dichotomy," which is an outstanding issue in the astrophysics of extragalactic radio sources, has long been debated. Two promising interpretations for the dichotomy have been proposed. One is that the dividing line can be understood by "the intrinsic differences of the jet's kinetic power" between FRIs and FRIIs (e.g., Meier et al. 1997; Ghisellini & Celotti 2001; Marchesini et al. 2004). The other is that "the deceleration process of the jets" can determine the dividing line, supposing that FRIs are initially supersonic (e.g., Falle 1991; Bicknell 1994; Kaiser & Alexander 1997; Nakamura et al. 2008). The deceleration scenario is also supported by the following two observational facts. One is the observations of relativistic sub-kpc jets in nearby FRIs such as M87 (Biretta et al. 1995) and relativistic sub-pc ones in BL Lac objects (e.g., Gabuzda et al. 1994), whose parent populations are FRIs in an active galactic nucleus (AGN) unified model (e.g., Urry & Padovani 1995). On the other hand, the subsonic lobes at l>10 kpc are observed in FRIs. The second is the existence of a new class of double radio sources in which the two lobes have clearly different FR morphologies at both sides called "HYbrid MOrphology Radio Sources (HYMORS)" (e.g., Gopal-Krishna & Wiita 2000). However, the deceleration process has been unclear.

In this Letter, we especially focus on the deceleration process of lobes to explore the origin of "FRI/FRII dichotomy." How does the supersonic flow decelerate in the first few kpc? As a plausible deceleration process, the entrainment through mixing in a turbulent layer between the jets and the ambient medium has been considered (e.g., De Young 1993a, 1993b; Bicknell 1994; Perucho & Marti 2007; Rossi et al. 2008). De Young (1993a, 1993b) suggested that the deceleration of the advancing lobes due to the entrainment is important for the special cases, i.e., both relatively low kinetic power of jets and sufficiently dense ambient medium. In this Letter, we elucidate another process of deceleration of a radio lobe's advance speed. As a possible deceleration process, Cioffi & Blondin (1992) showed the importance of the jet's head growth by using hydrodynamic simulations. Interestingly, recent observation has suggested that the cross-sectional area of reverse shocked region of jets (hot spots) grows faster in the host galaxy than outside the galaxy. By comparing the dynamical model of radio lobes and hot spots including their head growth, Kawakatu et al. (2008; hereafter KNK08) suggested that this trend that hot spot radius changes with distance can be explained by the strong deceleration of radio lobes.

The goal of this Letter is to elucidate the condition in which initially supersonic lobes become subsonic lobes, which can be closely linked to the origin of "FRI/FRII dichotomy," by considering that the deceleration of radio lobes and hot spots controlled by the growth of radio lobes' head. In Section 2, we provide a simple treatment for the dynamical evolution of radio lobes, in order to derive the dividing line between FRIs and FRIIs. In Section 3, we show the ratio of the jet power and the constant number density of the ambient matter is maximal at the core radius. Then, it is newly predicted that the division between FRIs and FRIIs is determined by the maximal ratio of the jet power L_j to the number density of the ambient matter at the core radius of the host galaxy $\bar{n}_{\rm a}$. In Section 4, we suggest an evolutionary sequence of variously sized radio galaxies based on our findings. A summary is given in Section 5.

In order to obtain the requirement for an initially supersonic lobe to turn into a subsonic lobe, we combine the ram pressure equilibrium between the hot spots and ambient medium with the observed relation between "the hot spot radius" and "the linear size of radio sources" (Figure 1 in KNK08).

We consider here a pair of relativistic jets propagating in an ambient medium ρ_a(l_h) where l_h is the distance from the jet apex. The equation of motion along the jet axis, i.e., the momentum flux of a relativistic jet is balanced to the ram pressure of the ambient medium spread over the effective cross-sectional area of the cocoon head, A_h(l_h),

$$\begin{eqnarray} L_{\rm j}/c&=&\rho _{\rm a}(l_{\rm h})v_{\rm HS}^{2}(l_{\rm h})A_{\rm h} (l_{\rm h}), \end{eqnarray} \tag{ 1 }$$

where L_j and v_HS(l_h) are the total kinetic energy of jets and the hot spot velocity, respectively. Here, we assume that L_j is constant in time during the activity period t ≈ 10^6–8 yr, which is a typical age of FRIs and FRIIs (e.g., Carilli et al. 1991; Parma et al. 1999; O'Dea et al. 2009; Machalski et al. 2009). Hereafter, we do not consider the entrainment effect of the ambient medium. This is justified for higher L_j (e.g., Scheck et al. 2002) but we will discuss this later on. At the hot spots, the flow of shocked matter is spread out by the oblique shocks (e.g., Lind et al. 1989) and the vortex via shocks (e.g., Smith et al. 1985). Thus, the effective "working surface" for the advancing jet is larger than the cross-sectional area of hot spots (Begelman & Cioffi 1989; Kino & Kawakatu 2005; hereafter KK05; Kawakatu & Kino 2006). The evolution of A_h(l_h) could be determined by the density profile of ambient medium and the growth rate of the cross-section of cocoons (e.g., Begelman & Cioffi 1989; Loken et al. 1992; KK05). However, it is difficult to obtain A_h(l_h) analytically because this is determined by the nonlinear effects (Smith et al. 1985; Lind et al. 1989; Scheck et al. 2002; Mizuta et al. 2004; Perucho & Marti 2007). Then, we assume A_h(l_h) as being proportional to r²_HS(l_h) as follows:

$$\begin{equation} A_{\rm h}(l_{\rm h})\equiv f\pi r_{\rm HS}^{2}(l_{\rm h}), \end{equation} \tag{ 2 }$$

where r_HS(l_h) is the hot spot size. Although the parameter f (>1) is determined by the nonlinear effects (e.g., the vortex and oblique shock) at the hot spots (e.g., Mizuta et al. 2004), the two-dimensional relativistic numerical simulations show f = const. (e.g., Scheck et al. 2002; Perucho & Marti 2007). Thus, we here suppose f = const.

For the ambient density profile ρ_a(l_h), we assume a double power-law distribution obtained from X-ray observations (e.g., Trinchieri et al. 1986; Mathews & Brighenti 2003; Allen et al. 2006; Fukazawa et al. 2006) as

$$\begin{equation} \rho _{\rm a}(l_{\rm h})=\left\lbrace \begin{array}{@{}l@{\qquad}ll@{}} \bar{\rho }_{\rm a}(l_{\rm h}/l_{\rm c})^{-\alpha _{\rm inner}} & {\rm for} & l_{\rm h} < l_{\rm c}, \\ \\ \bar{\rho }_{\rm a}(l_{\rm h}/l_{\rm c})^{-\alpha _{\rm outer}} & {\rm for} & l_{\rm h} > l_{\rm c}, \end{array} \right. \end{equation} \tag{ 3 }$$

where $\bar{\rho }_{\rm a}$ is the mass density of the ambient matter at l_c where the slope index of ρ_a(l_h) changes and α ⩾ 0 is the slope index of ρ_a(l_h). Here $\bar{\rho }_{\rm a}=\bar{n}_{\rm a}m_{\rm p}$, where m_p is the proton mass. Based on the observed temperature profile of elliptical galaxies from X-ray observations (e.g., Allen et al. 2001; Fukazawa et al. 2006), we assume the ambient temperature profile T_a(l_h) as follows:

$$\begin{equation} T_{\rm a}(l_{\rm h})=\left\lbrace \begin{array}{@{}l@{\qquad}ll@{}} T_{\rm a,c} & {\rm for}& l_{\rm h} < l_{\rm c}, \\ \\ T_{\rm a,c}(l_{\rm h}/l_{\rm c})^{\beta } & {\rm for} & l_{\rm c} < l_{\rm h} < l_{\rm g}, \\ \\ T_{\rm a,g} & {\rm for} & l_{\rm h} > l_{\rm g}, \end{array} \right. \end{equation} \tag{ 4 }$$

where T_a,g = T_a,c(l_g/l_c)^β with β ⩾ 0 and l_g is the distance where T_a(l_h) = T_a,g.

From the radio observations (Jeyakumar & Saikia 2000; KNK08), the evolution of r_HS(l_h) can be assumed to be a broken power-law distribution as

$$\begin{equation} r_{\rm HS}(l_{\rm h})=\left\lbrace \begin{array}{@{}l@{\qquad}ll@{}} \bar{r}_{\rm HS}(l_{\rm h}/l_{\rm c})^{\gamma _{\rm inner}} & {\rm for} & l_{\rm h} < l_{\rm c}, \\ \\ \bar{r}_{\rm HS}(l_{\rm h}/l_{\rm c})^{\gamma _{\rm outer}} & {\rm for} & l_{\rm h} > l_{\rm c}, \end{array} \right. \end{equation} \tag{ 5 }$$

where $\bar{r}_{\rm HS}$ is the hot spot radius at l_c.

Based on simple treatments for the dynamical evolution of radio sources, we will derive the condition in which initially supersonic lobes turn into subsonic lobes. We suppose that the growth of radio sources changes drastically when the advance speed along the jet axis equals the sound speed of the ambient medium, c_s(l_h) (e.g., Gopal-Krishna & Wiita 1987, 1991). This criterion is simply described as

$$\begin{equation} v_{\rm HS}(l_{\rm h})= c_{\rm s}(l_{\rm h}). \end{equation} \tag{ 6 }$$

Here c_s(l_h) = (5kT_a(l_h)/3m_p)^1/2, where k is the Boltzman constant. Using Equations (2) and (6), Equation (1) can be expressed as follows:

$$\begin{eqnarray} L_{\rm j}/c&=&f\pi r_{\rm HS}^{2}(l_{\rm h})\rho _{\rm a}(l_{\rm h}) c_{\rm s}^{2}(l_{\rm h}). \end{eqnarray} \tag{ 7 }$$

By substituting Equations (3), (4), and (5) into Equation (7), the critical line between the supersonic lobes and subsonic ones is expressed as a function of l_h as follows:

$$\begin{equation} \left(\frac{L_{\rm j}}{\bar{n}_{\rm a}} \right)_{\rm crit} \,{=}\, fcm_{\rm p}\bar{r}^{2}_{\rm HS} \left\lbrace \begin{array}{@{}ll@{}} c^{2}_{\rm s, c}(l_{\rm h}/l_{\rm c})^{\,\,(2\gamma -\alpha)_{\rm inner}} & {\rm for}\; l_{\rm h} < l_{\rm c}, \\ \\ c^{2}_{\rm s, c}(l_{\rm h}/l_{\rm c})^{\,\,(2\gamma -\alpha)_{\rm outer}+\beta } & {\rm for}\; l_{\rm c} < l_{\rm h} < l_{\rm g}, \\ \\ c^{2}_{\rm s, g}(l_{\rm h}/l_{\rm c})^ {\,\,(2\gamma -\alpha)_{\rm outer}} & {\rm for}\; l_{\rm h} > l_{\rm g}, \end{array} \right. \end{equation} \tag{ 8 }$$

where c_s,i(l_h) = (5kT_a,i(l_h)/3m_p)^1/2 with i="c" or "g," (2γ − α)_inner ≡ 2γ_inner − α_inner and (2γ − α)_outer ≡ 2γ_outer − α_outer. To determine the critical line, the slope index of $(L_{\rm j}/\bar{n}_{\rm a})_{\rm crit}$ is important. Since the slope index depends on the values of α and γ, we can classify the following three cases, i.e., (1) (2γ − α)_inner ⩾ 0 and (2γ − α)_outer ⩽ 0, (2) (2γ − α)_inner ⩾ 0 and (2γ − α)_outer>0, and (3) (2γ − α)_inner < 0 and any (2γ − α)_outer. For the slope of ambient medium α, it is possible to constrain as 0 ⩽ α_inner ⩽ α_outer = 1–2 from X-ray observations (e.g., Trinchieri et al. 1986; Mathews & Brighenti 2003; Allen et al. 2006; Fukazawa et al. 2006). As for r_HS(l_h), KNK08 found γ_inner = 1–1.5 and γ_outer = 0.3–0.5 by using about 120 radio sources. From these values of α and γ, the allowed ranges are given by

$$\begin{equation} 0\le (2\gamma -\alpha)_{\rm inner}\le 3 \,\,\,{\rm and}\,\, -1.5\le (2\gamma -\alpha)_{\rm outer}\le 0. \end{equation} \tag{ 9 }$$

Thus, the cases (2) and (3) can be safely ruled out, then hereafter we will consider the case (1). In case (1), there exists the maximal ratio of the jet power and the number density of the ambient medium at l_c, $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ because of (2γ − α)_inner ⩾ 0 and (2γ − α)_outer ⩽ 0. Importantly, the presence of $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ is independent of the values of α, β, and γ for case (1), except for (2γ − α)_inner = 0 and β = 0. The existence of $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ is physically understood in terms of the strong deceleration of radio lobes and hot spots in host galaxies due to the growth of the cross-sectional area of radio lobes (see KNK08). The independent observations also imply the deceleration of the advance speed of hot spots (O'Dea & Baum 1997; Labiano 2008). Note that the maximum of $(L_{\rm j}/\bar{n}_{\rm a})$ disappears only for α_inner = α_outer = 2, β = 0, γ_inner = 1 and γ_outer ⩽ 1 because $(L_{\rm j}/\bar{n}_{\rm a})_{\rm crit}$ is independent of l_h. If this is the case, the division between FRIs and FRIIs is determined by T_a and r_HS at the distance from nuclei where the A_h growth phase starts, i.e., a few pc.

Figure 1 shows the critical lines (the solid lines) for β = 1 [case (a)] and for β = 0 [case (b)], assuming f = 10, (2γ − α)_inner = 3, (2γ − α)_outer = 0, and l_c = 1 kpc. Concerning β, Fukazawa et al. (2006) found that the temperature slightly increases with l_h, i.e., 0 ⩽ β ⩽ 1 (l_c < l_h < l_g), for X-ray luminous elliptical galaxies, while for X-ray faint galaxies shows a flat temperature profile against l_h, i.e., β = 0. For T_a (e.g., Allen et al. 2001; Fukazawa et al. 2006), we assume T_a,c = 2 × 10⁶ K and T_a,g = 1 × 10⁷ K for case (a) and T_a,c = T_a,g = 2 × 10⁶ K for case (b). For case (a), l_g = 5 kpc because of l_g/l_c = T_a,g/T_a,c. As we mentioned, we see that the maximum of $L_{\rm j}/\bar{n}_{\rm a}$ dividing between the supersonic lobes and the subsonic lobes (the shaded region in Figure 1) appears at l_c. From Equation (8), the maximal ratio of the jet power and the number density at l_c can be described as $(L_{\rm j}/\bar{n} _{\rm a})_{\rm max}=fcm_{\rm p}\pi \bar{r}^{2}_{\rm HS}c^{2}_{\rm s,i}$ where i="g" for case (a) and i="c" for case (b). The $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ is expressed as

$$\begin{eqnarray} \bigg(\frac{L_{\rm j}}{\bar{n}_{\rm a}} \bigg)_{\rm max}=&&1\times 10^{45}\bigg(\frac{f}{10}\bigg) \bigg(\frac{c_{\rm s, i}}{10^{-3}c}\bigg)^{2}\nonumber\\ &&\times\, \bigg(\frac{\bar{r}_{\rm HS}}{0.3\;{\rm kpc}}\bigg)^{2} \;{\rm erg}\;{\rm s}^{-1}\;{\rm cm}^{3}, \end{eqnarray} \tag{ 10 }$$

where c_s,i = 1 × 10⁻³c corresponds to T_a,i = 1 × 10⁷ K and $\bar{r}_{\rm HS}=0.3\;{\rm kpc}$ which is derived from Figure 1 in KNK08. Note that $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ for case (b) is factor 5 smaller than that for case (a) because of $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}\propto T_{\rm a}$. The predicted $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ in Equation (10) is about 3 orders of magnitude higher than the ratio below which the entrainment can work efficiently, i.e., $L_{\rm j}/\bar{n}_{\rm a} \le 10^{42}\;{\rm erg}\;{\rm s}^{-1}\;{\rm cm}^{3}$ (De Young 1993a, 1993b; hereafter DY93a, b). Thus, by comparing with the future measurements of L_j and $\bar{n}_{\rm a}$ for the variously sized radio sources, we will be able to reveal which deceleration processes, i.e., the jet's head growth or entrainment, can divide FRIs and FRIIs.

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Our results suggest that the supersonic lobes can be maintained up to ∼1 Mpc when $L_{\rm j}/\bar{n}_{\rm a}$ is larger than its maximal value. On the other hand, in the case of below $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$, the lobes are initially supersonic but the supersonic lobes decelerate and then turn into subsonic lobes in the core radius of the host galaxy, or the lobes are initially subsonic. The division between FRIs and FRIIs can be determined by $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$, because the radio lobes of FRIs and FRIIs can be subsonic and supersonic, respectively. This indicates that FRIs favor the higher $\bar{n}_{\rm a}$ than FRIIs at given L_j. This is consistent with the observational results showing that FRIs are located at the centers of clusters of galaxies while FRIIs are discovered in the fields or at the edge of clusters of galaxies (e.g., Prestage & Peacock 1988; Miller et al. 2002). The division in the Owen–White diagram has been shown to be much less clear based on the latest Sloan Digital Sky Surveys sample (Best 2009). This may indicate that radio luminosity and optical magnitude of the host galaxy are not the fundamental physical quantities for the demarcation between FRIs and FRIIs. In order to judge whether our prediction is reasonable, it will be essential to compare with the observed L_j and $\bar{n}_{\rm a}$ for the variously sized radio galaxies (i.e., CSOs, MSOs, FRIs, and FRIIs).

Lastly, we discuss how $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ depends on the different choice of parameters, i.e., f, T_a, and $\bar{r}_{\rm HS}$. As for parameter f, A_h must be smaller than πl²_h because of the elongated morphology of radio lobes. According to the observed relation between the hot spot size, r_HS and the projected linear size, l_h relation (see Figure 1 in KNK08) f can be 10 < f < 10². For four FRIIs, by directly comparing πr²_HS with A_h, the range of f can be obtained as 10 < f < 10² (e.g., Ito et al. 2008; see Carilli et al. 1991). Thus, we here consider a wide range of f, i.e., 10 < f < 10². As c_s becomes larger, $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ increases at fixed $\bar{r}_{\rm HS}$ and vice versa (see Equation (10)). In order to satisfy v_HS ⩾ c_s, the larger $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ is required as c_s increases because of $v_{\rm HS} \propto L_{\rm j}/ \bar{n}_{\rm a}$ (see Equation (1)). According to the current observations, the ranges of temperature could be narrow, i.e., 2 × 10⁶ K ⩽ T_a,c ⩽ 1 × 10⁷ K and 1 × 10⁷ K ⩽ T_a,g ⩽ 2 × 10⁷ K (e.g., Allen et al. 2001; Fukazawa et al. 2006). For the hot spot radius at l_c, the range of $\bar{r}_{\rm HS}$ could be $0.3\;{\rm kpc} \le \bar{r}_{\rm HS} \le 1\;{\rm kpc}$ (see Figure 1 in KNK08) because of 1 kpc ⩽ l_c ⩽ 10 kpc (e.g., Fukazawa et al. 2006). Considering all uncertainties, the allowed range of $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ can be ≈10⁴⁴–10⁴⁷ erg s⁻¹ cm³, which is more than 2 orders of magnitude higher than the ratio below which the entrainment would be important (e.g., DY93a, b). To summarize, it will be essential to measure ρ_a(l_h), T_a(l_h), and f more accurately, in order to determine a critical line dividing between FRIs and FRIIs.

In order to evolve into the large-scale [l_h>l_c(⩽10 kpc)] supersonic lobes, it is necessary to hold the condition which $L_{\rm j}/\bar{n}_{\rm a}$ is larger than $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$. Moreover, the dividing line between the subsonic and supersonic, $(L_{\rm j}/\bar{n}_{\rm a})_{\rm crit}$ depends on l_h. Thus, the evolutionary path of radio sources can be divided by $(L_{\rm j}/\bar{n}_{\rm a})_{\rm crit}$ (see Equation (8)). Observationally, several authors have discovered young and compact radio sources such as compact symmetric objects (CSOs; l_h < 1 kpc) and medium-size symmetric objects (MSOs; l_h = 1–10 kpc) (e.g., Wilkinson et al. 1994; Fanti et al. 1995; Readhead et al. 1996).

On the basis of our findings (see Figure 1), we can predict the fate of compact and young radio galaxies as follows.

• 1.  
  If $L_{\rm j}/\bar{n}_{\rm a} > (L_{\rm j}/\bar{n}_{\rm a})_{\rm max} =10^{44\hbox{--}45}\;{\rm erg}\;{\rm s}^{-1}\;{\rm cm}^{3}$, the evolutionary sequence appears as CSOs → MSOs → FRIIs.
• 2.  
  When $L_{\rm j}/\bar{n}_{\rm a} < (L_{\rm j}/\bar{n}_{\rm a})_{\rm max} =10^{44\hbox{--}45}\;{\rm erg}\;{\rm s}^{-1}\;{\rm cm}^{3}$, the evolutionary track is as follows: CSOs → distorted MSOs → FRIs.
• 3.  
  If $(L_{\rm j}/\bar{n}_{\rm a}) < 10^{36}\;{\rm erg}\;{\rm s}^{-1}\;{\rm cm}^{3}$ at l_h = 1 pc, the evolutionary track appears as the FRI-like CSOs → distorted MSOs → FRIs.

For case (3), the distorted MSOs might correspond to ∼1 kpc low power compact (LPC) radio sources (e.g., Kunert-Bajraszewska et al. 2005; Giroletti et al. 2005) because the radio morphology of LPCs tends to be irregular and some LPCs show FRI-like morphology (e.g., Giroletti et al. 2005). Note that the deceleration of supersonic lobes via the entrainment may be also important for cases (2) and (3).

We briefly comment on the number of radio sources with supersonic lobes per bin of projected size, N(l_h). As seen in Figure 1, the region of the supersonic lobes allowed is larger as the size of the AGN jets is smaller. This might imply that the number of CSOs having supersonic lobes is larger than that of MSOs. However, in order to predict N(l_h) we need to consider the luminosity evolution of radio sources, the distribution of L_j and $\bar{n}_{\rm a}$. This is left as our future work.

We examine the origin of "FRI/FRII dichotomy" by considering the deceleration of expanding radio lobes. We explored the condition of a supersonic lobe turning into a subsonic one, comparing the observed r_HS–l_h relation with the ram pressure confinement along the jet axis. We then found that the dividing line between the supersonic lobes and subsonic ones is determined by the single parameter, i.e., the ratio of the jet power (L_j) and the number density of the ambient medium at the core radius of hosts ($\bar{n}_{\rm a}$). Importantly, there exists the maximal ratio of $(L_{\rm j}/\bar{n}_{\rm a})$ and its value is $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}\approx 10^{44\hbox{--}47}\;{\rm erg}\;{\rm s}^{-1}\;{\rm cm}^{3}$, taking account of considerable uncertainties. This is more than 2 orders of magnitude higher than the ratio below which the entrainment would be important (e.g., DY93a, b). Thus, it will be able to test whether the predicted maximal ratio $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ divides the FRIs from the FRIIs, by comparing with the future observation of L_j and $\bar{n}_{\rm a}$ for the variously sized radio sources.

We thank the anonymous referee for several comments helpful toward improving our work. N.K. is financially supported by the Japan Society for the Promotion of Science (JSPS) through the JSPS Research Fellowship for Young Scientists.