Classical and relativistic flux of energy conservation in astrophysical jets

0.2 MB·

by  L. Zaninetti

Classical and relativistic flux of energy conservation
in astrophysical jets
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n
a
Lorenzo Zaninetti
J

1 Physics Department , via P.Giuria 1,
1 I-10125 Turin,Italy

] E-mail: [email protected]
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Abstract. The conservation of the energy flux in turbulent jets which propagate in
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h
theintergalacticmedium(IGM)allowsdeducingthelawofmotionintheclassicaland
p
- relativistic cases. Three types of IGM are considered: constant density, hyperbolic
o
and inverse power law decrease of density. An analytical law for the evolution of
r
t the magnetic field along the radio-jets is deduced using a linear relation between the
s
a magnetic pressure and the rest density. Astrophysical applications are made to the
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centerline intensity of synchrotron emission in NGC315 and to the magnetic field of

1 3C273.
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Keywords: Galaxies: jets Relativity
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Classical and relativistic flux of energy conservation in astrophysical jets 2
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]. The experiments of
Reynolds 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 extragalactic
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]; as an example the gradients in pressure
are not considered. The application of the theory of turbulence to extragalactic radio-
jets produces a great number of questions to be solved because we do not observe
the turbulent phenomena but the radio features which 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 in order 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?
In order to answer these questions, we derive the differential equations which model
the classical and relativistic conservation of the energy flux for a turbulent jet in the
presence of different types of medium, see Sections 2 and 3. Section 4 presents classical
and relativistic parametrizations of the radiative losses as well as the evolution of the
magnetic field.
2. Energy conservation
The conservation of the energy flux in a turbulent jet requires the perpendicular section
to the motion along the Cartesian x-axis, A
A(r) = π r2 (1)
where r is the radius of the jet. The section A at position x is
0
α
A(x ) = π(x tan( ))2 (2)
0 0
2
where α is the opening angle and x is the initial position on the x-axis. At position x
0
we have
α
A(x) = π(xtan( ))2 . (3)
2
Classical and relativistic flux of energy conservation in astrophysical jets 3
The conservation of energy flux states that
1 1
ρ(x )v3A(x ) = ρ(x)v(x)3A(x) (4)
2 0 0 0 2
where v(x) is the velocity at position x and v (x ) is the velocity at position x , see
0 0 0
Formula A28 in [10].
The selected physical units are pc for length and yr for time; with these units, the
initial velocity v is expressed in pc yr−1, 1 yr = 365.25 days. When the initial velocity
0
is expressed in kms−1, the multiplicative factor 1.02 10−6 should be applied in order
×
to have the velocity expressed in pc yr−1.
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
3
d
x(t) (x(t))2 v 3x 2 = 0 . (5)
0 0
dt −
(cid:18) (cid:19)
The analytical solution of the previous differential equation can be found by imposing
x = x at t=0,
0
1
x(t) = 32/5 5 x 2(5tv +3x )3 . (6)
0 0 0
3
q
The asymptotic approximation is
1
x(t) 32/553/5 5 v 3x 2t3/5 . (7)
0 0
∼ 3
The velocity is p
32/5x 2(5tv +3x )2v
0 0 0 0
v(t) = (8)
x 2(5tv +3x )3 4/5
0 0 0
and its asymptotic approximation
(cid:0) (cid:1)
1 32/5√5 125x 2v 3(t−1)2/5
0 0
v(t) . (9)
∼ 5 (v 3x 2)4/5
0 0
The velocity as a function of the distance is
x 2/3v
0 0
v(x) = . (10)
x2/3
A first comparison can be made with the laboratory data on turbulent jets of [11]
where the velocity of the turbulent jet at the nozzle diameter, D =1, is v = 2.53 m
j 0
s−1 and at D =50 the centerline velocity is v = 0.314 m s−1. The formula (10) with
j
x = 1 and x = 50 gives an averaged velocity of v = 0.186 m s−1 which multiplied by 2
0
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
Classical and relativistic flux of energy conservation in astrophysical jets 4
therefore the centerline velocity is approximately double that of the averaged velocity.
The transit time, t , necessary to travel a distance of x can be derived from Eq. (6)
tr max
3√3 x 2x x 3x 2
max 0 max 0
t = − . (11)
tr
5x v
0 0
An astrophysical test can be performed on a typical distance of 15 kpc relative to
the jets in 3C31, see Figure 2 in [12]. On inserting x = 15000pc= 15 kpc, x = 100 pc,
0
and v = 10000 km s−1 we obtain a transit time of t = 2.488107 yr.
0 tr
The rate of mass flow at the point x, m˙ (x), is
α
m˙ (x) = ρv(x)π(xtan( ))2 (12)
2
and the astrophysical version is
M
m˙ (x) = 0.0237nx4/3(tan(α/2))2x02/3β0 ⊙ (13)
yr
where x and x are expressed in pc, n is the number density of protons expressed in
0
particles cm−3, M is the solar mass and β = v0. The previous formula indicates that
0 c
⊙
the rate of transfer of particles is not constant along the jet but increases x4/3.
∝
2.2. An hyperbolic profile of the density
Now the density is assumed to decrease as
x
0
ρ = ρ ( ) (14)
0
x
where ρ = 0 is the density at x = x . The differential equation that models the energy
0 0
flux is
3
d
x x(t) x(t) v 3x 2 = 0 (15)
0 0 0
dt −
(cid:18) (cid:19)
and its analytical solution is
1
x(t) = √4 3 4 x (4tv +3x )3 . (16)
0 0 0
3
q
The asymptotic approximation is
2
x(t) √4 3√2 4 v 3x t3/4 . (17)
0 0
∼ 3
The analytical solution for thpe velocity is
√4 3x (4tv +3x )2v
0 0 0 0
v(t) = (18)
x (4tv +3x )3 3/4
0 0 0
and its asymptotic approximation is
(cid:0) (cid:1)
1 √4 3√4 64x v 3√4 t−1
0 0
v(t) . (19)
∼ 4 (v 3x )3/4
0 0
The transit time can be derived from Eq. (16)
3√3 x x 2x 3x 2
max 0 max 0
t = − (20)
tr
4x v
0 0
and with x = 15000 pc= 15 kpc, x = 100 pc, and v = 10000 km s−1 as in Section 2.1,
0 0
we have t = 5.848106 yr.
tr
Classical and relativistic flux of energy conservation in astrophysical jets 5
Figure 1. Classical velocity as a function of the distance from the nucleus when x0
=100 pc and v0 = 10000km s−1: δ = 0 (full line), δ = 1 (dashes), δ = 1.2 (dot-dash-
dot-dash) and δ =1.6 (dotted).
2.3. An inverse power law profile of the density
Here, the density is assumed to decrease as
x
ρ = ρ ( 0)δ (21)
0
x
where ρ is the density at x = x . The differential equation which models the energy
0 0
flux is
2
1 x δ d 1
0 x(t) x2 v 2x 2 = 0 . (22)
0 0
2 x dt − 2
(cid:18) (cid:19)
(cid:16) (cid:17)
There is no analytical solution, and we simply express the velocity as a function of the
position, x,
x v 1
0 0
v(x) = (23)
x x0 δ
x
q
see Figure 1 (cid:0) (cid:1)
The rate of mass flow at the point x is
x δ
m˙ (x) = ρ 0 πx(tan(α/2))2x v (24)
0 0 0
x
r
(cid:16) (cid:17)
and the astrophysical version is
x δ M
m˙ (x) = 0.0237n0 1.0 0 x(tan(α/2))2x0β0 ⊙ (25)
x yr
r
(cid:16) (cid:17)
where n is the number density of protons expressed in particles cm−3 at x . The
0 0
previous formula indicates that the rate of transfer of particles scales x1−12δ and
∝
therefore at δ = 2 is constant.
Classical and relativistic flux of energy conservation in astrophysical jets 6
3. Relativistic turbulent jets
The conservationof theenergy fluxinspecial relativity (SR)inthe presence ofavelocity
v along one direction states that
1
A(x) (e +p )v = cost (26)
1 v2 0 0
− c2
where A(x) is the considered area in the direction perpendicular to the motion, c is the
speed of light, e = c2ρ is the energy density in the rest frame of the moving fluid, and
0
p is the pressure in the rest frame of the moving fluid, see formula A31 in [10]. In
0
accordance with the current models of classical turbulent jets, we insert p = 0 and the
0
conservation law for relativistic energy flux is
1
ρc2v A(x) = cost . (27)
1 v2
− c2
Our physical units are pc for length and yr for time, and in these units, the speed of
light is c = 0.306 pc yr−1. A discussion of the mass–energy equivalence principle in
fluids can be found in [?].
3.1. Constant density in SR
The conservation of the relativistic energy flux when the density is constant can be
written as a differential equation
d α 2 dx(t) 2 −1
ρc2 x(t) π (x(t))2 tan 1 dt
dt 2 − c2
(cid:18) (cid:19) (cid:0) (cid:1) !
(cid:16) (cid:16) (cid:17)(cid:17)
α 2 v 2 −1
ρc2v πx 2 tan 1 0 = 0 . (28)
0 0
− 2 − c2
(cid:18) (cid:19)
(cid:16) (cid:16) (cid:17)(cid:17)
An analytical solution of the previous differential equation at the moment of writing
does not exist but we can provide a power series solution of the form
x(t) = a +a t+a t2 +a t3 +... (29)
0 1 2 3
see [13, 14]. The coefficients a up to order 4 are
n
a = x
0 0
a = v
1 0
1 v 3(5c6 11c4v 2 +3c2v 4 +3v 6)
0 0 0 0
a = −
2 3 x 2(c2 +v 2)(c4 +2c2v 2 +v 4)
0 0 0 0
1 v 3(5c6 11c4v 2 +3c2v 4 +3v 6)
0 0 0 0
a = − . (30)
3 3 x 2(c2 +v 2)(c4 +2c2v 2 +v 4)
0 0 0 0
In order to find a numerical solution of the above differential equation we isolate
the velocity from Eq. (28)
v(x;x ,β ,c) =
0 0
β 2x2 x2 + x4β 4 2x4β 2 +4β 2x 4 +x4 c
1 0 0 0 0 0
− −
(31)
2 (cid:16) p β x 2 (cid:17)
0 0
Classical and relativistic flux of energy conservation in astrophysical jets 7
where β = v0 and separate the variables
0 c
x β x 2
0 0
2 dx
Zx0 β02x2 x2 + x4β04 2x4β02 +4β02x04 +x4 c
− −
t(cid:16) p (cid:17)
= dt . (32)
Z0
The indefinite integral on the left side of the previous equation has an analytical
expression
AN
I(x;β ,c,x ) = (33)
0 0
AD
where
AN =
iβ x2 ix2 iβ x2 ix2
2β 3x 6√2 4 2 0 +2 4+2 0 2
0 0
s − x02 β0x02s x02 − β0x02 ×
i(β 2 1)
F(1/2x√2 0 − ,i)
× s β0x02
iβ i
β 3x 2x3 0 x4β 4 2x4β 2 +4β 2x 4 +x4
− 0 0 sx02 − β0x02 0 − 0 0 0
q
iβ i
+β x 2x3 0 x4β 4 2x4β 2 +4β 2x 4 +x4
0 0 sx02 − β0x02 0 − 0 0 0
q
iβ i iβ i
+β 5x 2x5 0 2β 3x 2x5 0
0 0 sx02 − β0x02 − 0 0 sx02 − β0x02
iβ i iβ i
+4β 3x 6x 0 +β x 2x5 0 (34)
0 0 sx02 − β0x02 0 0 sx02 − β0x02
and
AD =
iβ i
6cβ 2x 4 0 x4β 4 2x4β 2 +4β 2x 4 +x4 (35)
0 0 sx02 − β0x02 0 − 0 0 0
q
where i = √ 1 and
−
x 1
F(x;m) = dt (36)
√1 t2√1 m2t2
Z0 − −
is the elliptic integral of the first kind, see formula 17.2.7 in [15]. Figure 2 shows the
behaviour of β as function of the distance.
A numerical solution can be found by solving the following non-linear equation
I(x;β ,c,x ) I(x ;β ,c,x ) = t (37)
0 0 0 0 0
−
and Figure 3 presents a typical comparison with the series solution. The relativistic rate
Classical and relativistic flux of energy conservation in astrophysical jets 8
Figure 2. Relativistic β as a function of the distance from the nucleus when x0
=200 pc and β0 =0.9 in the case of constant density.
Figure 3. Non-linear relativistic solution as given by Eq. (37) (full line) and series
solution as given by Eq. (29) (dashed line) when x0 =100 pc and β0 =0.999.
of mass flow in the case of constant density is
m˙ (x) =
ρ β 2x2 x2 + x4β 4 2x4β 2 +4β 2x 4 +x4 cπx(tan(α/2))2
0 0 0 0 0
− −
(38)
(cid:16) p (cid:17)
2 1 β 2 β 2x2 x2 + x4β 4 2x4β 2 +4β 2x 4 +x4
0 0 0 0 0 0
− − −
r
(cid:0) (cid:1)(cid:16) p (cid:17)
3.2. Inverse power law profile of density in SR
The conservation of the relativistic energy flux in the presence of an inverse power law
density profile as given by Eq. (21) is
d α 2 x δ dx(t) 2 −1
ρ c2 x(t) π (x(t))2 tan 0 dt +1
0
dt 2 x(t) − c2
(cid:18) (cid:19) (cid:18) (cid:19) (cid:0) (cid:1) !
(cid:16) (cid:16) (cid:17)(cid:17)
α 2 v 2 −1
ρ c2v πx 2 tan 0 +1 = 0 . (39)
− 0 0 0 2 − c2
(cid:18) (cid:19)
(cid:16) (cid:16) (cid:17)(cid:17)
Classical and relativistic flux of energy conservation in astrophysical jets 9
Figure 4. Relativistic β for the relativistic energy flux conservation as a function of
the distance from the nucleus when x0 =100 pc and β0 =0.9: δ = 0 (full line), δ = 1
(dashes), δ =1.2 (dot-dash-dot-dash) and δ =1.4 (dotted).
This differential equation does not have an analytical solution. An expression for β as
a function of the distance is
1 1 x δ x δ
β(x) = β 2x2 0 x2 0 +√D (40)
2 β x 2 0 x − x
0 0 (cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
with
D =
2 2 2
x δ x δ x δ
0 β 4x4 2 0 β 2x4 + 0 x4 +4β 2x 4 . (41)
0 0 0 0
x − x x
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17)
The behaviour of β as a function of the distance for different values of δ can be
seen in Figure 4. A power series solution for the above differential equation (39) up to
order three gives
a = x
0 0
a = v
1 0
1 v 2(c2δ δv 2 2c2 +2v 2)
0 0 0
a = − − . (42)
2
2 x (c2 +v 2)
0 0
Figure 5 shows a comparison between the numerical solution of (39) with the series
solution.
The relativistic rate of mass flow in the case of an inverse power law for the density
is
ρ x0 δ β 2x2 x0 δ x2 x0 δ +√D cπx2(tan(α/2))2
0 x 0 x − x
m˙ (x) = (43)
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1) 2
2β x 2 1/4 1 β 2x2 x0 δ x2 x0 δ +√D +1
0 0 − β02x04 0 x − x
r
(cid:16) (cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
where ρ is the density at x and D was defined in Eq. (41).
0 0
Classical and relativistic flux of energy conservation in astrophysical jets 10
Figure 5. Non-linear relativistic solution as given by Eq. (39) (full line) and series
solution as given by Eq. (42) (dashed line) when x0 =100 pc and β0 =0.999.
4. The losses
The previous analysis does not cover the radiative losses. The astrophysical version of
the relativistic energy flux as represented by Eq. (27 ) is
dE nβ R 2 erg
= 1.348 1049 0 100 (44)
dt × 1 β 2 s
0
−
where R is the radius of the jet expressed in units of 100 pc, and n is the number
100
density of protons expressed in particles cm−3. The above luminosity is 4–5 orders
of magnitude too high for the radio sources here considered. In order to explain
this discrepancy, one model assumes that extragalactic jets are much lighter than the
surroundings. The second model assumes that the observed intensity of radiation, I ,
ν
at a given frequency ν is a fraction of the energy flux
dE erg
I = ε (45)
ν
dt s
where ε represents the efficiency of conversion of the relativistic energy flux into
radiation. At the moment of writing there is no exact evaluation of the efficiency of
conversion. We now outline two different models for the radiative losses and a model
for the magnetic field.
4.1. Losses through recursion
In the classical case, with constant density, we can model the radiative losses through
the following recursive equation obtained by modifying Eq. (5)
1 α 1 α 1 α
ρv3 π(x tan( ))2 +ε ρv3π(x tan( ))2 = ρv3π(x tan( ))2 (46)
2 n+1 n+1 2 2 n n 2 2 n n 2
where
x = x +v ∆t . (47)
n+1 n n