Conservation of the Flux of Energy in Extra-Galactic Jets

TL;DR

This paper investigates energy flux conservation in turbulent extragalactic jets across different intergalactic medium profiles, deriving analytical and numerical models for jet velocity and trajectory, with applications to radio galaxy observations.

Contribution

It introduces a method to derive jet motion laws from energy flux conservation considering various IGM density profiles, including relativistic effects and radiative losses.

Findings

Derived analytical velocity expressions for different IGM profiles.

Numerically determined jet trajectories in various media.

Applied models to radio galaxy 3C31's synchrotron emission and magnetic field evolution.

Abstract

The conservation of the energy flux in turbulent jets that propagate in the intergalactic medium (IGM) allows us to deduce the law of motion in the classical and relativistic cases. Four types of IGM are considered: constant density, hyperbolic decrease of density, inverse power law decrease of density and a Lane--Emden ($n=5$) profile. The conservation of the relativistic flux for the energy allows us to derive, to the first order, an analytical expression for the velocity. It also allows us to numerically determine the trajectory for the four types of medium. In the case of a Lane--Emden ($n=5$) profile, the back-reaction due to the radiative losses for the trajectory is evaluated both in the classical and the relativistic case. Astrophysical applications are made to the centerline intensity of the synchrotron emission and to the evolution of the magnetic field in the case of the radio-galaxy 3C31.

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

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

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

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

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

$${\left(\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)\right)}^3{\left(x\left(t\right)\right)}^2-{v_0}^3{x_0}^2=0.$$

$$x(t)=\frac{1}{3}{3}^{2/5}\sqrt[5]{{x_0}^2{\left(5t{v}_0+3x_0\right)}^3}.$$

$$x(t)\sim \frac{1}{3}{3}^{2/5}{5}^{3/5}\sqrt[5]{{v_0}^3{x_0}^2}{t}^{3/5}.$$

$$v(t)=\frac{3^{2/5}{x_0}^2{\left(5t{v}_0+3x_0\right)}^2{v}_0}{{\left({x_0}^2{\left(5t{v}_0+3x_0\right)}^3\right)}^{4/5}}$$

$$v(t)\sim \frac{1}{5}\frac{3^{2/5}\sqrt[5]{125}{x_0}^2{v_0}^3{\left(t^{-1}\right)}^{2/5}}{{\left({v_0}^3{x_0}^2\right)}^{4/5}}.$$

$$v(x)=\frac{{x_0}^{2/3}{v}_0}{x^{2/3}}.$$

$$t_{tr}=\frac{3\sqrt[3]{{x_{\max }}^2{x}_0}{x}_{\max }-3{x_0}^2}{5x_0{v}_0}.$$

$$\dot{m}(x)=\rho v(x)\pi (x\tan (\frac{\alpha }{2}){)}^2$$

$$\dot{m}(x)=0.0237n{x}^{4/3}{\left(\tan \left(\alpha /2\right)\right)}^2{x_0}^{2/3}{\beta }_0\frac{M_{\odot }}{yr}$$

$$\rho ={\rho }_0(\frac{x_0}{x})$$

$$x_{0}x\left(t\right){\left(\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)\right)}^3-{v_0}^3{x_0}^2=0$$

$$x(t)=\frac{1}{3}\sqrt[4]{3}\sqrt[4]{x_0{\left(4t{v}_0+3x_0\right)}^3}.$$

$$x(t)\sim \frac{2}{3}\sqrt[4]{3}\sqrt{2}\sqrt[4]{{v_0}^3{x}_0}{t}^{3/4}.$$

$$v(t)=\frac{\sqrt[4]{3}{x}_0{\left(4t{v}_0+3x_0\right)}^2{v}_0}{{\left(x_0{\left(4t{v}_0+3x_0\right)}^3\right)}^{3/4}}$$

$$v(t)\sim \frac{1}{4}\frac{\sqrt[4]{3}\sqrt[4]{64}{x}_0{v_0}^3\sqrt[4]{t^{-1}}}{{\left({v_0}^3{x}_0\right)}^{3/4}}.$$

$$t_{tr}=\frac{3\sqrt[3]{x_{\max }{x_0}^2}{x}_{\max }-3{x_0}^2}{4x_0{v}_0}$$

$$\rho ={\rho }_0(\frac{x_0}{x}{)}^{\delta }$$

$$1/2{\left(\frac{x_0}{x}\right)}^{\delta }{\left(\frac{\mathrm{d}}{\mathrm{d}t}x\left(t\right)\right)}^3{x}^2-1/2{v_0}^3{x_0}^2=0.$$

$$v(x)=\frac{\sqrt[3]{{x_0}^2{\left({\left(\frac{x_0}{x}\right)}^{\delta }\right)}^{2}x}{v}_0}{{\left(\frac{x_0}{x}\right)}^{\delta }x}$$

$$\dot{m}(x)={\rho }_0\pi x{\left(\tan \left(\alpha /2\right)\right)}^2\sqrt[3]{{x_0}^2{\left({\left(\frac{x_0}{x}\right)}^{\delta }\right)}^{2}x}{v}_0$$

$$\dot{m}(x)=0.02375nx{\left(\tan \left(\alpha /2\right)\right)}^2\sqrt[3]{{x_0}^2{\left({\left(1.0\frac{x_0}{x}\right)}^{\delta }\right)}^{2}x}{\beta }_{0}Msun$$

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

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

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

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

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

Full text

Keywords: galaxies, jets, relativity

1. Introduction

The analysis of turbulent jets in the laboratory offers the possibility of applying the theory of turbulence to some well-defined experiments, see [1, 2]. Reynolds experiments can be seen in [3]. Analytical results for the theory of turbulent jets can be found in [4, 5, 6, 7]. Recently, the analogy between laboratory jets and extra-galactic radio-jets has been pointed out, see [8, 9]. We briefly recall that the theory of ‘round turbulent jets’ can be defined in terms of the velocity at the nozzle, the diameter of the nozzle, and the viscosity, see Section 5 in [6]. However, in this example, the gradients in pressure are not considered. The application of the theory of turbulence to extra-galactic radio-jets raises many questions because we do not observe the turbulent phenomena, but the radio features that have properties similar to the laboratory’s turbulent jets, i.e., similar opening angles. We now pose the following questions:

• •

Is it possible to apply the conservation of the flux of energy to derive the equation of motion for radio-jets in the cases of constant and variable density of the surrounding medium?

• •

Can we extend the conservation of the flux of energy to the relativistic regime?

• •

Can we model the behaviour of the magnetic field and the intensity of synchrotron emission as functions of the distance from the parent nucleus?

• •

Can we model the back reaction on the equation of motion for turbulent jets due to radiative losses?

To answer these questions, in Sections 2. and 3., we derive the differential equations that model the classical and relativistic conservation of the energy flux for a turbulent jet in the presence of different types of medium. Sections 2.5. and 3.4. present the classical and the relativistic parametrization of the radiative losses for the Lane–Emden ($n=5$) profile. Section 4. introduces two models for the synchrotron emission along the jet.

2. Energy Conservation

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

[TABLE]

where $r$ is the radius of the jet. 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 [10].

The selected physical units are pc for length and yr for time; with these units, the initial velocity $v_{{0}}$ is expressed in pc yr*-1*, 1 yr = 365.25 days. 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*. More details can be found in [11, 12]

2.1. Constant Density

In the case of constant density of the intergalactic medium (IGM) along the $x$-direction, the law of conservation of the energy flux, as given by Eq. (4), can be written as a differential equation

[TABLE]

The analytical solution of the previous differential equation can be found by imposing $x=x_{0}$ at t=0,

[TABLE]

The asymptotic approximation is

[TABLE]

The velocity is

[TABLE]

and its asymptotic approximation

[TABLE]

The velocity as a function of the distance is

[TABLE]

A first comparison can be made with the laboratory data on turbulent jets of [13] where the velocity of the turbulent jet at the nozzle diameter, $D_{j}$=1, is $v_{0}=2.53$ m s*-1* and at $D_{j}$=50 the centerline velocity is $v=0.314$ m s*-1*. The formula (10) with $x_{0}=1$ and $x=50$ gives an averaged velocity of $v=0.186$ m s*-1* which multiplied by 2 gives $v=0.372$ m s*-1*. This multiplication by 2 has been done because the turbulent jet develops a profile of velocity in the direction perpendicular to the jet’s main axis and, therefore, the centerline velocity is approximately double that of the averaged velocity. The transit time, $t_{tr}$, necessary to travel a distance of $x_{max}$ can be derived from Eq. (6)

[TABLE]

An astrophysical test can be performed on a typical distance of 15 kpc relative to the jets in 3C 31, see Figure 2 in [14]. On inserting $x=15000\,$pc$=15$ kpc, $x_{0}=100$ pc, and $v_{0}=10000$ km s*-1* we obtain a transit time of $t_{tr}=2.488\,10^{7}$ yr.

The rate of mass flow at the point $x$, $\dot{m}(x)$, is

[TABLE]

and the astrophysical version is

[TABLE]

where $x$ and $x_{0}$ are expressed in pc, $n$ is the number density of protons expressed in particles cm*-3*, M_{\hbox{\odot}} is the solar mass and $\beta_{0}=\frac{v_{0}}{c}$. The previous formula indicates that the rate of transfer of particles is not constant along the jet but increases $\propto x^{4/3}$.

2.2. A Hyperbolic Profile of the Density

Now the density is assumed to decrease as

[TABLE]

where $\rho_{0}=0$ is the density at $x=x_{0}$. The differential equation that models the energy flux is

[TABLE]

and its analytical solution is

[TABLE]

The asymptotic approximation is

[TABLE]

The analytical solution for the velocity is

[TABLE]

and its asymptotic approximation is

[TABLE]

The transit time can be derived from Eq. (16)

[TABLE]

and with $x=15000$ pc$=15$ kpc, $x_{0}=100$ pc, and $v_{0}=10000$ km s*-1* as in Section 2.1., we have $t_{tr}=5.848\,10^{6}$ yr.

2.3. An Inverse Power Law Profile of the Density

Here, the density is assumed to decrease as

[TABLE]

where $\rho_{0}$ is the density at $x=x_{0}$. The differential equation which models the energy flux is

[TABLE]

There is no analytical solution, and we simply express the velocity as a function of the position, $x$,

[TABLE]

see Figure 2.3.