Synchrotron Radiation, continued Rybicki & Lightman Chapter 6 Also Course notes for “Essential Radio Astronomy” at NRAO, Condon & Ransom .

Synchrotron Radiation,

continued

Rybicki & Lightman Chapter 6Also Course notes for “Essential Radio Astronomy” at

NRAO, Condon & Ransom

http://www.cv.nrao.edu/course/astr534/ERA.shtml

Synchrotron Theory: Summary of Results

1. Synchrotron = relativisitic electrons & B-field

Very spiky E(t) because of beaming

€

ω larmor ≈ 2π ×14 rad s-1

For an electron having ~ 104 the width of the pulse is

€

Δt pulse ≈1

γ 2ω larmour

≈1

104( )

2× 2π ×14

≈10−10 s

B ~5x10-

6G

The time between pulses is ~

€

ν larmour

≈103 s

E(t)

t

2. Spectrum of single energy electron:

Critical frequency

E= electronenergy

x= ν / νc

Each electron of energy E contributesto the spectrum at x=1

sP ωω)(

p = spectral index of particle energiesand s = spectral index of observed radiation

dECEdEEN p)(

2

1

ps

Ergs/s

3. Spectral index related to energy index of electron energy distribution

(Optically thin synchrotron)

4. For optically thin synchrotron, the slope of the spectrum must always be greater than -1/3 because the low-frequency spectrum is a superposition of single spectra, and F(x) ~ ν-1/3

s > -1/3

Single electron spectrum

Summary:

For optically thin emission

For optically thick

Low-frequency cut-off

Thick

Thin

2/)1( pI ννν

2/5 ννν SI

2/)1( pν

Synchrotron Radio Sources

Map of sky at 408 MHz (20 cm). Sources in Milky Way are pulsars, Sne; Diffuse radio spectrumGalactic B-field + cosmic rays

Milky Way magnetic field ~ 5 microGauss, along spiral arms measured via Zeeman splitting of OH masers pulsar dispersion measures polarization of starlight by dust aligned in B-field

c.f. Earth’s Magnetic field: 500,000 microGauss

Spectrum of Cosmic Rays in ISM of the Milky Way has p~2.4

Spectrum of synchrotron radiation s~0.7

Han 2010

Milky WayInterstellar Cosmic RayEnergy spectrum

Casadel & Bindli 2004ApJ 612, 262

energy spectrumhas p~2.4

Synchrotron has s~0.7

M51 Polarization derived fromSynchrotron (6 cm).

Beck 2000

Coherent structure, B-field along arms

Milky Way B-field: TheoryMilky Way B-field: TheoryVertical Field in Center B ~1mG

Horizontal Field in Disc B ~ 3G

Supernova explosion.Supernova explosion.

ScaleScaleEddy turnover timeEddy turnover time

ScaleScaleRotation timeRotation time

V

GG 22101088yearsyearsLLGG 4410102020 mm

00 101077yearsyearsll00 3310101818mm

V

Cowley 2011

6. Dynamo amplification of primordial seed magnetic field E. Parker: Galactic Dynamo

Differentital rotation & convection or SN explosions --> loopsLoops align with existing B-fieldNet result is amplification

Zirker, The MagneticUniverse

7. Minimum Energy and Equipartition

Synchrotron spectrum spectral index electron energy index, but not B-field

B-fields often estimated by assuming “equipartition”

Recall:

€

UB =B2

8πEnergy density of magnetic field

What can we say about the minimum energy in relativistic particles and magnetic fields that is required to produce a synchroton source of a given luminosity?

What is UE?

Assume power-law electron energy distribution

€

N(E) ≈ KE − p

between energies Emin and E max

which produces synchrotron radiation between frequencies νmin and νmax

€

Ue = EN(E)dEE (min)

E(max)

∫ Energy density of electrons

€

L = Lν dνν (min)

ν (max)

∫ Synchrotron radiation luminosity

Substitute

€

N(E) ≈ KE − pand

€

−(dE /dt) ∝ B2E 2Energy per electron from synch.

€

Ue

L∝

K E1− pdEE(min)

E (max)

∫

KB2 E 2− pdEE(min)

E (max)

∫∝ E(min)

E(max)

E2−p

B2

E(min)

E(max)

E 3−p

Approximate each electron emits at energy E, and

So

€

Emax ∝ B−1/ 2and

€

Emin ∝ B−1/ 2and

€

Ue

L∝

B−1/ 2( )

2− p

B2 B−1/ 2( )

3− p =B−1+ p / 2

B2B−3 / 2+ p / 2 = B−3 / 2

So

€

Ue ∝ B−3 / 2

Need total energy density in particles: electrons plus ions

Let

€

η ≡U ions

Ue

Don’t usually know what η is, but η~ 40 for cosmic rays near Earth

So total energy

€

U = Ue +U ion +UB

= (1+η)Ue +UB

€

∝ B−3 / 2

€

∝ B2

So there is a B for which U is minimum

Find minimum U by taking dU/dB and setting = 0

Result: Get minimum energy when

€

particle energy

field energy≈

4

3≈1 “Equipartition”

So given an optically thin synchrotron source of luminosity L, Assume equipartition, and then compute B numerical formulae on Condon & Ramson web site

Physically plausible: B field cannot have U>>U(particles) and still haveCoherent structures

Large extragalactic jets have an enormous amount of particle energy as It is, so putting more energy into particles makes theory more difficult

Crab Nebula

The Crab Nebula, is the remnant of a supernova in 1054 AD, observed as a "guest star" by ancient Chinese astronomers. The nebula is roughly 10 light-years across, and it is at a distance of about 6,000 light years from earth. It is presently expanding at about 1000 km per second. The supernova explosion left behind a rapidly spinning neutron star, or a pulsar is this wind which energizes the nebula, and causes it to emit the radio waves which formed this image.

Radio emission of M1 = Crab Nebula, from NRAO web site

IR

Optical

Radio X-ray(Chandra)

Crab Nebula Spectral Energy Distribution from Radio to TeV gamma rays see Aharonian+ 2004 ApJ 614, 897

SynchrotronSynchrotronSelf-Compton

Photon frequency(Hz)

Electron EnergyU, (eV)

Electron lifetime(Yr)

Radio (0.5 GHz)

5x108 3.0x108 109,000

Optical (6000A)

5x1014 3.0x1011 109

X-ray (4 keV) 1x1018 1.4x1013 2.4

Gamma Ray 1x1022 1.4x1015 0.024 = 9 days

Synchrotron Lifetimes, for Crab Nebula

€

=5.16

B2

1

γ electron decay time, sec.

for α =π

2,B in teslas

Timescales<< age of CrabPulsar is Replenishing energy

Guess what this is an image of?

Extragalactic radio sources: Very isotropic distribution on the sky

6cm radio sources

North Galactic Pole

Milky Way

right ascension

Blowup ofNorthPole

VLA

Core of jets:flat spectrum s=0 to .3

Extended lobes:steep spectrum s = 0.7-1.2

DRAGNS: Double-Lobed Radio-loud Active Galactic Nuclei

www.jb.man.ac.uk/atlas/dragns.html

APOD April 13, 2011

Cen A, Full moon and CSIRO radioObservatory

Radio lobes are ~ million light years across

FR I vs. FR II

On large scales (>15 kpc)

radio sources divide into

Fanaroff-Riley Class I, II

(Fanaroff & Riley 1974 MNRAS 167 31P)

FRI: Low luminosity

edge dark

Ex.:Cen-A

FRII: High luminosity

hot spots on outer edge

Ex. Cygnus A

Lobes are polarized synchrotron emission with well-ordered B-fields Polarization is perpendicular to B

8. Synchrotron spectra steepen with age

Energy radiated by electrons

€

∝ E 2

So high energy electrons lose their energy faster than low energy electrons

Spectrum steepensAt high freqencies:

Typically:

Cores have “flat spectra” s~0.5

Outer lobes have “steep spectra” s ~ 1.5-2

Real spectra can be complex: non-uniform B-fields, geometries

(Kellerman & Owen 1988)