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					Synchrotron Radiation,
                     continued

               Rybicki & Lightman Chapter 6
Also 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
       B ~5x10-6G                  larmor  2 14 rad s-1

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


              t 
                              1                     1
                                                            1010 s
                         g 2 larmour       10   2 14
                pulse                         4 2




  
                                   g
 The time between pulses is ~                103 s
                                 larmour



                  

E(t)




                                              t
2. Spectrum of single energy electron:

Critical frequency                       x=  / c



                                           E= electron
                                           energy
Each electron of energy E contributes
to the spectrum at x=1
3. Spectral index related to energy index of electron energy distribution

              p = spectral index of particle energies
          and s = spectral index of observed radiation


    NdE
    ECE
    ()
     dE () 
         
        P  Ergs/s
            s           
                        p



                                    p1
                                 s
                                     2



                     (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) ~ -1/3

Single electron spectrum




                                                             s > -1/3
                              
Summary:
                                       
                                      )
                               
                               
                                      p2
                                      (1/
For optically thin emission   I 
For optically thick
                           
                           
                          I S
                              
                              52
                               /


 Low-frequency cut-off




                                  ( )/2
                                   p1

                 Thick

                               Thin
Synchrotron Radio Sources




  Map of sky at 408 MHz (20 cm).
  Sources in Milky Way are pulsars, Sne;
  Diffuse radio spectrumGalactic 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
Milky Way
Interstellar
 Cosmic Ray
Energy spectrum




  energy spectrum
  has p~2.4

  Synchrotron has
      s~0.7


                    Casadel & Bindli 2004
                    ApJ 612, 262
M51 Polarization derived from
Synchrotron (6 cm).

Beck 2000

Coherent structure, B-field along arms
                                                          Cowley 2011

Milky Way B-field: Theory
                       Vertical Field in Center B ~1mG


                                                         Horizontal
                                                         Field in Disc B
                                                         ~ 3G
                         V
                                               V




Supernova explosion.                    LG  4 ´ 10 20 m       Scale
                                                               Rotation time
l0  3 ´ 1018 m    Scale                t G  2 ´ 10 8 years
                   Eddy turnover time
t 0  10 7 years
6. Dynamo  amplification of primordial seed magnetic field
      E. Parker: Galactic Dynamo

Differentital rotation & convection or SN explosions --> loops
Loops align with existing B-field
Net result is amplification




                                                                 Zirker,
                                                                 The Magnetic
                                                                 Universe
7. Minimum Energy and Equipartition


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

   B-fields often estimated by assuming “equipartition”



                    B2
   Recall:     UB                  Energy density of magnetic field
                    8


  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?


                                                                    p
Assume power-law electron energy distribution   N(E)  KE
between energies Emin and E max

which produces synchrotron radiation between frequencies min and max

                     E(m ax)      
              Ue       EN(E )dE               Energy density of electrons
                     E (m in)


                        (m ax)
                L          L d          Synchrotron radiation luminosity
                      (m in)




   
                                               p
     Substitute   N(E)  KE                               and


          (dE /dt)  B E                  2        2
                                                          Energy per electron from synch.



                       K
                              E(m ax)

                                E      1 p
                                               dE        E   2 p
                                                                    E(m ax)


                  Ue
                             E(m in)
                                E(m ax)                            E(m in)
                                                                      E(m ax)
                  L                                            3 p
                         KB   2
                                     E 2 p
                                               dE       B2 E
                                                                      E(m in)
                               E(m in)
     Approximate each electron emits at energy E, and



     So   E max  B 1/ 2       and         E min  B 1/ 2   and



                       B   B1 p / 2  B3 / 2
                                     2 p
                             1/ 2
                 Ue
                    
               L  2 B 1/ 2 3 p B 2 B 3 / 2 p / 2
                      B

     So
                                     3 / 2
                     Ue  B
     
Need total energy density in particles: electrons plus ions

Let             U ions
             
                Ue


Don’t usually know what is, but ~ 40 for cosmic rays near Earth

So total energy        U  U e  U ion  U B
                        (1 )U e  U B
                              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 4
                               1                         “Equipartition”
                field energy   3


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



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




                                                                  Synchrotron
                            Synchrotron
                                                                  Self-Compton
   Synchrotron Lifetimes, for Crab Nebula

                     Photon         Electron   Electron
                    frequency       Energy     lifetime
                    (Hz)            U, (eV)    (Yr)
  Radio (0.5        5x108           3.0x108    109,000
  GHz)
  Optical           5x1014          3.0x1011   109
  (6000A)
  X-ray (4 keV)     1x1018          1.4x1013   2.4
  Gamma Ray         1x1022          1.4x1015   0.024 = 9 days



  5.16 1
t 2     electron decay time,sec.
   B g
                                                Timescales
          
for  =       ,B in teslas                      << age of Crab
                                                Pulsar is
          2
                                                Replenishing energy
Guess what this is an image of ?
Extragalactic radio sources: Very isotropic distribution on the sky

                                                              6cm radio sources



                                                                      right ascension




Milky Way




                                                          North Galactic Pole
Blowup of
North
Pole
                    VLA


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

Extended lobes:
steep spectrum
  s = 0.7-1.2
                    www.jb.man.ac.uk/atlas/dragns.html


DRAGNS: Double-Lobed Radio-loud Active Galactic Nuclei
Cen A, Full moon and CSIRO radio
Observatory

Radio lobes are ~ million light years
across




APOD April 13, 2011
   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 steepens
At 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)

				
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