Classical and relativistic evolution of an extra-galactic jet with back-reaction

TL;DR

This paper derives analytical and numerical models for the evolution of extragalactic jets in a specific medium, considering classical and relativistic effects and the impact of radiative losses on jet trajectories.

Contribution

It provides first-order analytical expressions for jet velocity and trajectory in a Lane-Emden medium, including relativistic effects and back-reaction from radiative losses.

Findings

Analytical velocity expressions for classical and relativistic jets.

Numerical trajectories accounting for radiative back-reaction.

Comparison of classical and relativistic jet evolution.

Abstract

We consider a turbulent jet which is moving in a Lane--Emden ($n=5$) medium. The conserved quantity is the energy flux, which allows finding, to first order, an analytical expression for the velocity and an approximate trajectory. The conservation of the relativistic flux for the energy allows deriving, to first order, an analytical expression for the velocity, and numerically determining the trajectory. The back-reaction due to the radiative losses for the trajectory is evaluated both in the classical and the relativistic case.

Tables

Table 1. Table 1: Parameters for a classical extra-galactic jet.

parameter	value
$x_0$ (pc)	100
$v_0$ ($\frac{km}{s}$)	10000
$b$ (pc)	10000

Table 2. Table 2: Parameters for a relativistic extra-galactic jet.

parameter	value
$x_0$ (pc)	100
${\beta }_0$	0.9
$b$ (pc)	10000

Equations

$$L\le h_{100}210^{25}\frac{W}{Hzstr}\text{FR-I},$$

$$J=constant\times \rho b_j^2{\overline{v}}_{x,max}^2,$$

$$constant=2\pi {\int }_0^{\infty }{f}^2\xi d\xi ,$$

$$f(\xi )=\frac{{\overline{v}}_x}{{\overline{v}}_{x,max}}with\xi =\frac{x}{b_{1/2}},$$

$$\overline{v}_{x}=\frac{\nu^{(t)}}{x}\frac{2C_{3}^{2}}{\bigl{[}1+\frac{1}{4}(C_{3}\frac{r}{x})^{2}\bigr{]}^{2}}\quad,$$

$$C_3=\sqrt{\frac{3}{16\pi }}\sqrt{\frac{J}{\rho }}\frac{1}{{\nu }^{(t)}}.$$

$$\frac{\overline{v}_{x}(b_{1/2},x)}{\overline{v}_{x,max}(x)}=\frac{1}{2}=\frac{1}{\bigl{[}1+\frac{1}{4}(C_{3}\frac{b_{1/2}}{x})^{2}\bigr{]}^{2}}\quad.$$

$$b_{1/2}=0.0848x,$$

$$C_3=15.17,$$

$$\frac{\overline{v}_{x}(r)}{\overline{v}_{x,max}(r)}=\frac{1}{\bigl{[}1+0.414(\frac{r}{b_{1/2}})^{2}\bigr{]}^{2}}\quad.$$

$$b_j=2\times 4.6b_{1/2}.$$

$$\frac{\alpha }{2}=\arctan \frac{4.6b_{1/2}}{x}.$$

$$\frac{\alpha }{2}=\frac{c_s}{v_j}=\frac{1}{M},$$

$$b_{1/2}=\frac{1}{4.6}\tan (\frac{\alpha }{2})x.$$

$${\nu }^{(t)}=\sqrt{\frac{3}{16\pi }}\sqrt{\frac{J}{\rho }}\frac{1}{C_3}=\sqrt{\frac{3}{16\pi }}\sqrt{constant}b{v}_{x,max}\frac{1}{C_3}.$$

$${\int }_0^{\infty }{f}^2\xi d\xi =0.402,$$

$$constant=2.528.$$

$${\nu }^{(t)}=2.9210^{-9}{b}_1{v}_{100}\frac{\mathrm{p}{\mathrm{c}}^2}{\mathrm{year}}when{C}_3=135.61.$$

$$\frac{d^2}{d{x}^2}Y(x)+2\frac{\frac{d}{dx}Y(x)}{x}+(Y(x){)}^n=0,$$

$$\rho ={\rho }_{c}Y(x{)}_n^n,$$

$$P=K{\rho }^{1+\frac{1}{n}},$$

$$T=K^{\prime }Y(x),$$

$$Y(x)=\frac{1}{(1+\frac{x^2}{3}{)}^{1/2}},$$

$$\rho (x)={\rho }_c\frac{1}{(1+\frac{x^2}{3}{)}^{5/2}}.$$

$$\rho (r;b)={\rho }_c\frac{1}{(1+\frac{r^2}{3b^2}{)}^{5/2}}.$$

$$\delta=\frac{\big{|}x-x_{app}|}{x}\times 100\quad,$$

$$A(r)=\pi r^2$$

$$A(x_0)=\pi (x_0\tan (\frac{\alpha }{2}){)}^2$$

$$A(x)=\pi (x\tan (\frac{\alpha }{2}){)}^2.$$

$$\frac{1}{2}\rho (x_0)v_0^{3}A(x_0)=\frac{1}{2}\rho (x)v(x{)}^{3}A(x)$$

Full text

\doinum

10.3390/—— \pubvolumexx

\historyReceived: xx / Accepted: xx / Published: xx \Title Classical and relativistic evolution of an extra-galactic jet with back-reaction

\AuthorLorenzo Zaninetti 1,

\corres zaninetti@ph.unito.it

\PACS 98.54.Gr 98.62.Nx 98.70.Dk

1 Introduction

The study of extra-galactic jets started with the observations of NGC 4486 (M87), where ‘a curious straight ray lies in a sharp gap in the nebulosity …’, see Curtis (1918) and Figure 1.

At the moment of writing, the extra-galactic radio sources are classified on the basis of the position of the brightest radio emitting regions with respect to the channel, see Fanaroff and Riley (1974); Kembhavi and Narlikar (1999) for details. FR-I, after Fanaroff & Riley, have hot spots that are more distant from the nucleus (a typical example is Cygnus A) and luminosity, $L$, at 178 Mhz of

[TABLE]

where $h_{100}=H_{0}/100$ and $H_{0}$ is the Hubble constant. FR-II radio galaxies have emission uniformly distributed along the channel (typical example 3C449) and luminosities greater than the above value or, in other words, the more powerful radio galaxies are classified as FR-II. A list of the properties, length in kpc and power in Watt, of extra-galactic radio jets can be found in Bridle and Perley (1984); Liu and Zhang (2002). In the following we will study jets with small openings, such as that of M87.

The problem of the velocity of extra-galactic radio jets has been analysed in two ways:

The velocity of the jet is constant over many kpc and takes the value $v$. Due to the fact that is thought that this velocity is nearly relativistic, it is parametrized as $\beta=\frac{v}{c}$, where $c$ is the velocity of light. As an example, Hardcastle and Sakelliou (2004) analysed some wide-angle tail radio galaxies and found a terminal velocity of $\beta=0.3$. 2. 2.

The velocity of the jet decreases with an imposed law, see Laing and Bridle (2002), or is evaluated by a numerical code, see Nawaz et al. (2014, 2016). In this case, the relativistic parameter $\beta$ decreases along the trajectory.

Recently the problem of the decrease of the velocity along a turbulent jet has been solved, imposing the conservation of the flux of momentum, see Zaninetti (2015), or imposing the conservation of the energy flux, see Zaninetti (2016). The approach using the conservation of the flux of energy is attractive because it has the same dimension of the luminosity. Further on, the jets are radiating away in the various observational bands, such as radio, optical, infrared, etc., and we briefly recall that the extra-galactic radio source covers a range in observed luminosity from $10^{19}\frac{W}{Hz}$ to $10^{28}\frac{W}{Hz}$, see Kellermann and Richards (1998).

Therefore the flux of energy available at the beginning of the jet will progressively decrease due to the radiative losses. This paper, in Section 2, introduces the Lane–Emden ($n=5$) density profile and consequently derives an approximate trajectory to first order as well as a numerical trajectory to second order in the presence of losses. In Section 3 we present a series solution for the relativistic trajectory to first order and a numerical solution to second order. Section 4 models the intensity of the radio-jet in 3C31.

2 Conservation of the flux of energy

A turbulent jet is defined as a jet which has the same density as the surrounding intergalactic medium (IGM), see the next subsection for details. The conservation of the flux of energy in a turbulent jet has been explored in Zaninetti (2016) for three types of IGM, with the following radial dependences: constant density profile, hyperbolic and inverse power law density profiles. Here we analyse the case of a Lane–Emden (n=5) density profile, to which a subsection will be dedicated.

2.1 The turbulent jet

Turbulent jets are a subject of laboratory experiments, as an example, see Figure 2.

The theory of turbulent jets emerging from a circular hole can be found in different books with different theories, see Bird et al. (2002), Landau (1987), and Goldstein (1965). The basic assumptions common to the three already cited approaches are

The rate of momentum flow, $J$, represented by

[TABLE]

is constant; here $x$ is the distance from the initial circular hole, $b_{j}(x)$ is the jet’s diameter at distance $x$, $\overline{v}_{x,max}$ is the maximum velocity along the the centreline, $constant$ is

[TABLE]

where

[TABLE]

and $\rho$ is the density of the surrounding medium, see equation (5.6-3) in Bird et al. (2002). 2. 2.

The jet’s density $\rho$ is constant over the expansion and equal to that of the surrounding medium. The pressure is absent in this theory.

Omitting the details of the computation, an expression can be found for the average velocity $\overline{v}_{x}$, see equation (5.6-21) in Bird et al. (2002),

[TABLE]

where $\nu^{(t)}$ is the kinematical eddy viscosity and $C_{3}$ is as follows, see equation (5.6-23) in Bird et al. (2002),

[TABLE]

An important quantity is the radial position, $r=b_{1/2}$, corresponding to an axial velocity one-half of the centreline value, see equation (5.6-24) in Bird et al. (2002),

[TABLE]

The experiments in the range of Reynolds number, $Re$, $10^{4}\leq Re\leq 310^{6}$ (see Reichardt (1942), Reichardt (1951) and Schlichting (1979)) indicate that

[TABLE]

and as a consequence

[TABLE]

and therefore

[TABLE]

The average velocity, $\overline{v}_{x}$, is $\approx~{}1/100$ of the centreline value when $r/b_{1/2}$ = 4.6 and this allows seeing that the diameter of the jet is

[TABLE]

On introducing the opening angle $\alpha$, the following new relation is found:

[TABLE]

The generally accepted relation between the opening angle and Mach number, see equation (A33) in De Young (2002), is

[TABLE]

where $c_{s}$ is the velocity of sound, $v_{j}$ the jet’s velocity, and $M$ the Mach number. The new relation (12) replaces the traditional relation (13). The parameter $b_{1/2}$ can therefore be connected with the jet’s geometry:

[TABLE]

If this approximate theory is accepted, equation (8) gives $\alpha=42.61^{\circ}$: this is the theoretical value that yields the so called Reichardt profiles. The value of $b_{1/2}$ fixes the value of $C_{3}$ and therefore the eddy viscosity is

[TABLE]

In order to continue, the integral that appears in $constant$ should be evaluated, see equation (3). Numerical integration gives

[TABLE]

and therefore

[TABLE]

On introducing typical parameters of jets like $\alpha$=$5^{\circ}$, ${v}_{x,max}$=$v_{100}=v[{\mathrm{km/sec}}]/100$, $b_{j}=b_{1}$, where $b_{1}$ is the momentary diameter in ${\mathrm{pc}}$, it is possible to deduce an astrophysical formula for the kinematical eddy viscosity:

[TABLE]

This paragraph concludes by underlining the fact that in extra-galactic sources it is possible to observe both a small opening angle, $\approx~{}5^{\circ}$ and great opening angles, i.e. $\approx~{}34^{\circ}$ in the outer regions of 3C31 Laing and Bridle (2004).

2.2 The Lane–Emden profile

The self gravitating sphere of a polytropic gas is governed by the Lane–Emden differential equation of the second order

[TABLE]

where $n$ is an integer, see Lane (1870); Emden (1907); Chandrasekhar (1967); Binney and Tremaine (2011); Zwillinger (1989). The solution $Y(x)_{n}$ has the density profile

[TABLE]

where $\rho_{c}$ is the density at $x=0$. The pressure $P$ and temperature $T$ scale as

[TABLE]

[TABLE]

where K and K*′* are two constants. For more details, see Hansen and Kawaler (1994).

Analytical solutions exist for $n=0$, 1, and 5. The analytical solution for $n$=5 is

[TABLE]

and the density for $n$=5 is

[TABLE]

The variable $x$ is non-dimensional and we now introduce the new variable $x=r/b$

[TABLE]

2.3 Preliminaries

The chosen physical units are pc for length and yr for time; with these units, the initial velocity $v_{{0}}$ is expressed in pc yr*-1*. When the initial velocity is expressed in km s*-1*, the multiplicative factor $1.02\times 10^{-6}$ should be applied in order to have the velocity expressed in pc yr*-1*. In these units, the speed of light is $c=0.306$  pc  yr*-1*. The goodness of the approximation of a solution is evaluated through the percentage error, $\delta$, which is

[TABLE]

where $x$ is the analytical or numerical solution and $x_{app}$ the approximate solution, see Abramowitz and Stegun (1965).

2.4 Classical solution to first order

The conservation of the energy flux in a straight turbulent jet the concept of the perpendicular section section to the motion along the Cartesian $x$-axis, $A$

[TABLE]

where $r$ is the radius of the jet. The section $A$ at position $x_{0}$ is

[TABLE]

where $\alpha$ is the opening angle and $x_{0}$ is the initial position on the $x$-axis. At position $x$ we have

[TABLE]

The conservation of energy flux states that

[TABLE]

where $v(x)$ is the velocity at position $x$ and $v_{0}(x_{0})$ is the velocity at position $x_{0}$, see Formula A28 in De Young (2002). We now assume that a Lane–Emden ($n=5$) density profile is valid, see equation (22). Then the above conservation law becomes

[TABLE]

where $v(x)$ is the velocity at position $x$, $v_{0}(x_{0})$ is the velocity at position $x_{0}$ and $\alpha$ is the opening angle of the jet. The above equation is a cubic equation which has one real root plus two non-real complex conjugate roots. Here and in the following we take into account only the real root. The real analytical solution for the velocity without losses is

[TABLE]

The asymptotic expansion of above velocity, $v_{a}$, with respect to the variable $x$, which means $x\rightarrow\infty$, is

[TABLE]

The trajectory can be found by the indefinite integral of the inverse of the velocity as given by equation (29):

[TABLE]

where ${~{}_{2}F_{1}(a,b;\,c;\,v)}$ is a regularized hypergeometric function, see Abramowitz and Stegun (1965); von Seggern (1992); Thompson (1997); Olver et al. (2010). The trajectory expressed in terms of $t$ as a function of $x$ is

[TABLE]

The above equation can not be inverted in the usual form, which is $x$ as a function of $t$. The asymptotic trajectory can be found by the indefinite integral of the inverse of the asymptotic velocity as given by equation (30)

[TABLE]

The equation of the asymptotic trajectory is

[TABLE]

and the solution for $x$ of the above equation, the asymptotic trajectory, is

[TABLE]

Figure 3 shows a typical example of the above asymptotic expansion.

2.5 Solution to second order

Let us suppose that the radiative losses are proportional to the flux of energy

[TABLE]

Inserting in the above equation the velocity to first order as given by equation (29) the radiative losses, $Q(x;x_{0},v_{0},b,\epsilon)$, are

[TABLE]

where $\epsilon$ is a constant which fixes the conversion of the flux of energy to other kinds of energies, in this case, the radiative losses. The sum of the radiative losses between $x_{0}$ and $x$ is given by the following integral, $L$,

[TABLE]

The conservation of the flux of energy in the presence of the back-reaction due to the radiative losses is

[TABLE]

The analytical solution for the velocity to second order, $v_{c}(x;b,x_{0},v_{0})$, is

[TABLE]

and Figure 4 shows an example.

There are no analytical results for the trajectory corrected for radiative losses, and a numerical example is shown in Figure 5.

The inclusion of back-reaction allows the evaluation of the jet’s length, which can be derived from the minimum in the corrected velocity to second order as a function of $x$,

[TABLE]

which is

[TABLE]

The solution for $x$ of the above minimum determines the jet’s length, $x_{j}$,

[TABLE]

where

[TABLE]

and

[TABLE]

Figure 6 shows $x_{j}$ numerically.

3 Conservation of relativistic flux of energy

The corrections in special relativity (SR) for stable atomic clocks in satellites of the Global Positioning System (GPS) are applied to satellites which are moving at a velocity of $\approx 3.87\frac{km}{s}$, see Ashby (2003); Ashby and Nelson (2009).

In astrophysics we deal with velocities near that of light and therefore we should introduce relativistic conservation laws. The conservation of the relativistic flux of energy in SR in the presence of a velocity $v$ along one direction states that

[TABLE]

where $A(x)$ is the considered area in the direction perpendicular to the motion, $c$ is the speed of light, $e_{0}=c^{2}\rho$ is the energy density in the rest frame of the moving fluid, and $p_{0}$ is the pressure in the rest frame of the moving fluid, see formula A31 in De Young (2002) and Zaninetti (2016). In accordance with the current models of classical turbulent jets, we insert $p_{0}=0$. Then the conservation law for the relativistic flux of energy is

[TABLE]

In the presence of a Lane–Emden ($n=5$) density profile, as given by equation (22) and $A(x)$ as given by equation (26), the conservation of relativistic flux of energy for a straight jet takes the form

[TABLE]

where $v$ is the velocity at $x$, $v_{0}$ is the velocity at $x_{0}$, $\beta=\frac{v}{c}$ and $\beta_{0}=\frac{v_{0}}{c}$. The solution for $\beta$ to first order is

[TABLE]

where

[TABLE]

The equation for the relativistic trajectory is

[TABLE]

The integral in the above equation does not have an analytical solution and should be integrated numerically. In order to have analytical results, two approximation are now introduced. The first approximation computes a truncated series expansion for the integrand of the integral in equation (51), which transforms the relativistic equation of motion into

[TABLE]

with

[TABLE]

where

[TABLE]

In the above analytical result we have the time as a function of the distance, see Figure 7 where the percentage error at $x=15$ kpc is $\delta=15.91\%$.

The second approximation computes a Padé approximant of order [2/1], see Adachi and Kasai (2012); Aviles et al. (2014); Wei et al. (2014), for the integrand of the integral in equation (51)

[TABLE]

with

[TABLE]

where

[TABLE]

The above equation can be inverted, but the analytical expression for $x=G(t;x_{0},\beta_{0},b)$ as a function of time is complicated and is omitted here. As an example, with the parameters of Table 2, we have

[TABLE]

with

[TABLE]

and

[TABLE]

An example is shown in Figure 8, where the percentage error at $x=15$ kpc is $\delta=4.81\%$.

3.1 Relativistic solution to second order

We now suppose that the radiative losses are proportional to the relativistic flux of energy. The integral of the losses, $L_{r}$, between $x_{0}$ and $x$ is

[TABLE]

The conservation of the relativistic flux of energy in the presence of the back-reaction due to the radiative losses is

[TABLE]

where

[TABLE]

The solution of the above equation, to second order, for $\beta$ is

[TABLE]

where

[TABLE]

The relativistic equation of motion with back-reaction can be solved by numerically integrating the relation in equation (51). Figure 9 gives an example.

4 Astrophysical applications

We now analyse two models for the synchrotron emission along the jet.

4.1 Direct conversion

The flux of observed radiation along the centre of the jet, $I_{c}$, in the classical case is assumed to scale as

[TABLE]

where $L$, the sum of the radiative losses, is given by equation (38).

The above relation connects the observed intensity of radiation with the rate of energy transfer per unit area. In the relativistic case

[TABLE]

where $L_{r}$ is given by equation (61)

A statistical test for the the goodness of fit is the observational percentage of reliability, $\epsilon_{\mathrm{obs}}$,

[TABLE]

In order to make a comparison with the observed profile of intensity, we chose 3C31, see Figure 8 in Laing and Bridle (2002); Figure 10 shows the theoretical synchrotron intensity as well as the observed one.

4.2 The magnetic field of equipartition

The magnetic field in CGS has an energy density of $\frac{B^{2}}{8\pi}$ where $B$ is the magnetic field. The presence of the magnetic field can be modeled assuming equipartition between the kinetic energy and the magnetic energy

[TABLE]

Inserting the above equation in the classical equation for the conservation of the flux of energy (27), a factor 2 will appear on both sides of the equation, leaving unchanged the result for the deduction of the velocity to first order. The magnetic field as a function of the distance $x$ when the velocity is given by equation (29) and in the presence of a Lane–Emden ($n=5$) profile for the density is

[TABLE]

where $B_{0}$ is the magnetic field at $x=x_{0}$. We assume an inverse power law spectrum for the ultrarelativistic electrons, of the type

[TABLE]

where $K$ is a constant and $p$ the exponent of the inverse power law. The intensity of the synchrotron radiation has a standard expression, as given by formula (1.175) in Lang (1980),

[TABLE]

where $\nu$ is the frequency, $H_{\perp}$ is the magnetic field perpendicular to the electron’s velocity, $l$ is the dimension of the radiating region along the line of sight, and $\alpha_{p}(p)$ is a slowly varying function of $p$ which is of the order of unity. We now analyse the intensity along the centreline of the jet, which means that the radiating length is

[TABLE]

The intensity, assuming a constant $p$, scales as

[TABLE]

where $I_{0}$ is the intensity at $x=x_{0}$ and $B_{0}$ the magnetic field at $x=x_{0}$. We insert Eq. (70) in order to have an analytical expression for the centreline intensity

[TABLE]

The above equation for the intensity is relative to the unit area; in order to have the intensity on the centreline, $I_{c}$, we should make a further division by the area of interest, which scales $\propto\,x^{2}$

[TABLE]

Figure 11 shows the theoretical synchrotron intensity with the variable magnetic field as well as the observed one for 3C31.

5 Conclusions

Classical case The approximate trajectory of a turbulent jet in the presence of a Lane–Emden ($n=5$) medium has been evaluated to first order, see equation (35). The solution for the velocity to first order allows the insertion of the back-reaction, i.e. the radiative losses, in the equation for the flux of energy conservation, see equation (39), and as a consequence the velocity corrected to second order, see equation (40). The trajectory, calculated numerically to second order, is shown in Figure 5. The radiative losses allow evaluating the length at which the advancing velocity of the jet is zero. This length has a complicated analytical expression and was presented numerically, see Figure 6.

Relativistic case In the relativistic case it is possible to derive an analytical expression for $\beta$ to first order, see equation (49), and to second order (taking into account radiative losses), see equation (64). The relativistic trajectory to first order has been evaluated through a series, see equation (52) or a Padé approximant of order [2/1], see equation (58). The relativistic equation of motion to second order (back-reaction) has been evaluated numerically, see Figure 9. In other words, with the introduction of the radiative losses, the length of the classical or relativistic jet becomes finite rather than infinite.

An astrophysical application The radiative losses are represented by equation (37) in the classical case and by (61) in the relativistic case. A division of the two above quantities by the area of interest allows deriving the theoretical rate of energy transfer per unit area, which can be compared with the intensity of radiation along the jet, for example, 3C31, see Figure 10. The spatial behaviour of the magnetic field is introduced under the hypothesis of equipartition between the kinetic and magnetic energy, see equation (70), and this allows closing the standard equation for the synchrotron emissivity, see equation (4.2).