Galaxy Clusters in radio → Non-thermal phenomena

Galaxy Clusters in radio Galaxy Clusters in radio Î Î Non Non - - thermal phenomena thermal phenomena

Luigina Feretti

Istituto di Radioastronomia CNR Bologna, Italy

Tonanzintla, GH2005, 4-5 July 2005

Lecture 1 :

Theoretical background related to radio emission

Synchrotron spectrum

Evolution of synchrotron spectrum with time Age of a radio source (lifetime)

Energy content in a radio source Minimum energy density

Equipartition magnetic field

Limitations of the classical approac Polarization of synchrotron emission

Physical quantities

Flux density : S(ν) = apparent

(monochromatic flux)

Jy = 10^-26 W Hz^-1 m^-2

Surface brightness : B(ν) =

intrinsic Jy/arcsec²

D

4

) ( L

π ν

Σ π

= ν Ω

ν

d D D

4

) ( L d

) (

S

2

Synchrotron Radiation from 1 particle Synchrotron Radiation from 1 particle

2

2 c

v 1

1

− / Particle of energy E = γ m c² γ = Lorentz factor =

Lorentz Force F = q v ^X H

Î Helical path along the field lines

Radiated Power :

Big losses for high γ and small m (Î electrons) Energy loss proportional to the energy itself 1 electron : b= 2.37 x 10^-3 cgs units

2 2

2 2 7

4 4 2

2 3 2

4

H bE

H c E

m q 3

H 2 c

m q 3

2 dt

dE

⊥

⊥

⊥

=

= γ

=

−

θ

⊥

= H sin

H

θ

2 2 9

2 2 15

H E

10 6

H 10

6 dt 1

dE

⊥

−

⊥

−

×

=

γ

×

= .

(erg s^-1 if E in GeV and H in G)

2 18

2

5 3 2

2

10 3

. 6

2 . 4

4 1 4

1

E H H

c m E qH mc

qH

c peak

⊥

⊥

⊥

⊥

×

=

≈

=

=

γ ν

γ π ν π

cgs

H γ

10^-6 G ≥ 1000 1 G ≅ 10⁴ 10 G ≅ 10⁵

By integrating the equation for the energy losses:

2

2 ⊥

= bE H dt

dE

It is obtained that the particle energy decreases with time as:

t E bH E E

0 2 0

1 +

=

The time when E=1/2 E₀ can be defined as the characteristic time t* = particle lifetime

0 2

* 1

E t bH

⊥

=

Similarly,we can define a characteristic energy E*

*

* 1

t E bH

⊥

=

such that a particle with energy E₀ > E* will lose its energy in a time t*

Synchrotron Radiation from an ensemble of particles

Synchrotron Radiation from an ensemble of particles

Particle energy distribution:

N(E)dE = N₀E^-δ dE N(E)dE = Number of particles per unit volume with energy between E and E+dE

incoherent emission, no internal absorption (optically thin), isotropic particle velocity

Î J_syn(ν) = f(δ) N₀H_⊥^(δ+1)/2ν^(-δ+1)/2

J(ν) is the monochromatic power radiated per unit volume (specific emissivity)

Log J

Log ν ν^-α

Typical values in extragalactic radio sources are α = 0.7 – 0.8

2

−1

= δ

α :spectral index J_synch ∝ N₀ H_⊥^α+1 ν^-α

Time evolution of the synchrotron spectrum

We have derived the critical time t* and the critical energy E* for 1 electron

In an ensemble of particles, with power-law energy distribution, the energy losses modify the energy distribution

Particle continuity equation:

) , ) (

, ) (

, ) (

,

( Q E t

T

t E t N

E t N

E E

t t E N

conf

=

⎟ +

⎠

⎜ ⎞

⎝

⎛

∂

∂

∂ + ∂

∂

∂

particle flux escape rate injection rate

Particle production in a single event N(E,0) = N₀E^-δ

Particles with higher energy suffer stronger radiation losses.

After a certain time particles with E > E* have lost their energy thus the energy distribution of the particles goes rapidly to 0 for energies E > E*

As a consequence, the synchrotron spectrum J_synch ∝ ν^-α

undergoes a modification

there is a critical frequency ν*, corresponding to E*

such that :

for ν < ν* :

the spectrum is the same ν > ν* :

the spectrum goes rapidly to zero

(some emission present because the synch. em. is not monocromatic)

ν*

ν* is related to the age of the radiating electrons ν* shifts to lower frequencies with time

Log J

Log ν ν^-α

t ^ν*

ν* ν*

ν* = break frequency

In case of injection of new particles, injection rate Q(E,t) =AE^-δ for ν < ν* : the spectrum is the same

spectral index = α

ν > ν* : the spectrum steepens by 0.5 spectral index = α + 0.5

Log J ν^-α

ν^-α-0.5

Log ν ν*

Summary of synchrotron spectra

ν^{-α Log J}

Log ν Log J

Log ν

Log J

Log ν

ν^-α

ν^-α-0.5 ν^-α

ν* ν*

Standard Aged With Injection ν* is related to ν* is related to the source the time since lifetime last injection

Computation of electron (source) age

Freq. of synch.

radiation in cgs

⊥

=

= π

ν 6 3 x 10 E H

c m E qH

4

1

5 3

2

. *

*

*

We subsitute: b= 2.37 x 10^-3 cgs units

to find the relation between t* and ν*

H

bt E 1

⊥

=

*

2 5

0 5 0 12

H 10 H

x 06 1

t

ν

=

. *

*

We have to take into account the distribution of electron pitch angles and the IC losses

IC power : - ∝ E² as for synchrotron emission

dt dE

The energy density of the CMB radiation field equals the energy density of a magnetic field

H_CMB = 3.25 (1+z)² µG

Taking into account electron pitch angles and IC losses, the final formula for the computation of radio source age is

( )

[ ]

2 CMB 2

5 0

z H 1

H 1590 H

t

.

*

* + ν

= +

H in µG, ν in GHz, t in Myr

Anisotropic (KP)

( )

[ ]

2 CMB 2

5 0

z 1 3 H

H 2 1060 H

t ^.

.

*

* + ν ⁻

+

=

(Murgia 2001, Slee et al 2001)

Energy content in a radiosource Energy content in a radiosource

The magnetic field cannot be estimated unambiguously from the synchrotron emission

J_synch ∝ N₀ H_⊥^α+1 ν^-α

The radio emission that we detect is due to the relativistic electrons in a magnetic field

The total energy in the radiosource is thus contributed by - electrons

- magnetic field Plus a third component:

- protons (relativistic)

E

= E

+ E

+ E

Energy in magnetic field:

π Φ

= V 8

E H

2 H

Energy in protons :

we take it proportional to that in electrons

E_pr = k E_el

The energy in electrons can be derived from the observed radioemission and can be expressed as a function

of the synchrotron luminosity (observable), the value of the magnetic field, and a constant which depends on the

energy spectrum of radiating electrons (slope δ ) E_el = L_syn H_⊥^-3/2f(δ,ν₁,ν₂)

Finally:

E

= (1+k)

π V 8

H²

8 V L H

H k

1 c

2 syn

2

3

Φ

+ π +

= ( )

E

= E

+ E

+ E

E E_H

E_min E_part

H_eq

E_tot

The expression for E_tot is minimum for:

el

H k E

E (1 ) 4

3 +

= EQUIPARTITION

7 4

syn 7

3 7 3 7 4

tot

1 k V L

E

∝ ( + )

Φ

7 4

syn 7

4 7

4 7

tot

1 k

V L

V

u

= E

∝ ( + )

Φ

7 2

syn 7

2 7

2 7

2 2

1

eq

f 1 k V L

H = ( δ , ν , ν )( + )

Φ

(Pacholczyk 1970)

By writing the synchrotron luminosity as the observed source brightness I₀ at the frequency ν₀, and the

source depth d (to be inferred), applying the K-correction, assuming Φ = 1 (same volume in particles and magnetic field), and expressing the parameters in commonly used units:

7 / 4 7

/ 4 0 7 / ) 4 12 ( 7

/ 4 0 7 / 4 12

min

= 1 . 23 x 10

( 1 + k ) ( 1 + z )

I d

u ν

u_min in erg/cm³ ν₀ in MHz

I₀ in mJy/arcsec² d in kpc

Constant computed for α = 0.7, ν₁ = 10 MHz, ν₂ = 100 GHz

2 1

eq u

7 H 24

/ min ⎟

⎠

⎜ ⎞

⎝

= ⎛ π Usually k = 0 or k = 1

assumed for clusters

Limitations of the classical approach :

The synchrotron luminosity is observed between

the frequencies ν₁ and ν_2, usually taken =10MHz and 100 GHz.

The electron energies corresponding to these frequencies depend on the magnetic field value

Thus the total energy obtained for particles refer

to particles in the energy interval E₁ and E₂ which depends on the value of H.

Moreover, electrons with e.g. γ < 1000, i.e. radiating below 10 MHz, are neglected in this computation

5 3 2

s

m c

E qH 4

1

= π

ν

The minimum energy condition is now obtained by imposing the minimum condition in the formula involving the Energies

E

= (1+k)

The equipartition field obtained with this approach H’

relates to the value H obtained with the classical formula with ν₁=10 MHz and ν₂=100 GHz, as:

) (

.

'

α

−

γ

≈

7 eq 3

2 1

eq

1 1 H

H

valid for α > 0.5 H’ and H in Gauss

Φ H V

π

'²

(Brunetti, Setti and Comastri 1997) (Beck and Krause 2005)

γ_min = 50

α = 1.15

α = 0.65

1.050.95 0.850.75

The synchrotron radiation from a population of relativistic

electrons in a uniform magnetic field is linearly polarized, with the electric vector perpendicular to the magnetic field which has generated the synchrotron emission.

In the optically thin case, for isotropic electron distribution, and electron power-law energy spectrum:

the degree of intrinsic linear polarization is N(E)dE = N₀E^-δ dE

Polarization Polarization

8 0 75

7 0 3

3

P

3 ≈ . − . +

δ +

= δ

The above value is reduced in the more realistic cases where

- the magnetic field is not uniform, since regions where the magnetic field has different orientations give radiation with

different polarization angle orientations, which tend to average (or cancel) each other.

- there is Faraday rotation effect arising both from instrumental limitations (beamwidth – bandwidth) or within the source itself

(Sokoloff et al. 1998, 1999 :

how fractional pol. is affected by magnetic field configurations)