Radiative Processes in Astrophysics 11. Synchrotron Radiation & Compton Scattering Eline Tolstoy http://www.astro.rug.nl/~etolstoy/astroa07/ Synchrotron Self-Absorption synchrotron emission is accompanied by absorption in which a photon interacts with the a charge in a magnetic field and is absorbed giving up energy to the charge. Stimulated emission (negative absorption) can also occur. These processes can best be studied using the Einstein Coefficients (lecture 2). for a given photon energy h! there are many possible transitions between states differing in energy by h!. This means that the absorption coeff must be summed over all upper states E 2, and lower E 1 : assuming that the emission & absorption are isotropic - which means for synchrotron emission that the magnetic field is tangled and has no net direction, and the particle distribution is also isotropic. now have to write this formula for synchrotron emission
Synchrotron using Einstein coefficients have also made use of continuous nature - moving n(e 1 ) from under summation and replacing with n(e 2 -h!), this is allowed because f 21 acts like "-fn & enforces energy relation E 1 = E 2 - h! Continuing... next want to replace this sum over what are formally discrete states with an integral over continuum states to do this we identify n(e) with the local density of states in phase space make classical assumption h!<< E (which is anyway implicit), then identify #h! with "E and then the integral simplifies we now want to add a power-law distribution for N(E) and the spectrum for a single electron P(!,E) power-law distribution of particles thus, absorption coeff is:
spectrum spectrum for a single electron P(!,E) for a power law distribution of electrons can now predict the observed flux spectrum in the optically thick and thin limits Optically thick vs. thin optically thick: from Ginzburg & Syrovatski ARAA 1969 p a(p) b(p) 1 0.283 0.960 2 0.103 0.700 3 0.0742 0.650 optically thin case the turnover frequency! max is normally too low to be seen in most astrophysical situations, unless the density (or brightness temperature) is very high. e.g., Crab nebula relativistic electrons don t easily absorb photons, especially for weak B however, some sources, e.g., quasars turnover at 1 GHZ
combining both limiting cases, spectrum looks like: the synchrotron spectrum of a source with a power law electron distribution Various causes of spectral turnover
HII regions ordinary electrons at 10 4 K (HII regions for example) are much stronger sources of thermal (radio) emission than synchrotron. electrons must be ultra-relativistic to produce a similar power in the radio - where $ B can be Doppler boosted by the potentially large factor % 2. in situations where particle velocities are relativistic the densities are generally low and so synchrotron emission tends to dominate thermal bremsstrahlung (the emissivity goes with square of density) Molonglo Galactic Plane Survey (843 MHz) SNR - rdaiating strongly synchrotron MGPS (843 MHz) IRAS (60!m) MGPS and IRAS combined
A Case Study: The Crab Nebula assume prolate spheroid (a > b = c) for which Properties Energy: use Miley s expression with z=0, k=1 and #=1 then the field at minimum energy is: and the corresponding minimum energy Cooling time: consider electrons in the Crab nebula synchrotron radiating at 20keV (1eV = 1.602x1012 erg) or!=4.8x1018hz the magnetic field strength is Bmin= 3.86 x 10-4gauss with a Lorentz factor % ~ 4 x 107 the electron energy, E= %mec2 ~ 30 erg and the emitted power is P=1.7 x 10-8 erg s-1. so the cooling time is of order: energy from pulsar: P = 0.0333s & = 4.21 x 10-13 similar to the energy required to power the crab nebula!
Spectral energy distribution s s not affected by age cooling effective E for which E for which injection rate equal to Spectral energy distribution: Interpreting the break in terms of age of nebula: interpret break frequency! b to be point where t 1/2 ~T for the injected electrons setting clearly we are consistent at best, but our approximations are insufficient
SN as particle accelerators given the large amount of kinetic energy associated with SNRs is it possible that SNRs are the source of most of the relativistic particles in our Galaxy? from all sky radio surveys we know the Galaxy is filled with cosmic rays because the radio emission is almost entirely synchrotron from highly relativistic electrons, which are also observed in SNR Bonn 408MHz all sky survey the equipartition condition allows the calculation of B how much cosmic ray energy in SNRs 0.8 0.7 0.6 SN convert a good fraction of their blast energy into CRs. 0.5 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 B RADIO GALAXIES the centre of our Galaxy is a powerful source of synchrotron radio emission although a strong emitter this is very small compared to the centre of active galaxies
perhaps the most powerful objects in the Universe are radio galaxies. they contain large reservoirs of non-thermal energy located 100s of kpc from the nuclei, the power centres of these galaxies 1.5 light-months diameter 6000 light-years long polarized emission blobs move & change in intensity & polarisation It was first detected in 1956 by Burbidge in a jet emitted by M87, as a confirmation of a prediction by Iosif S. Shklovsky in 1953 Centaurus A Synchrotron radiation from jets and black hole. Hot Spot VLA Observations at 6cm lobe magnetic field strength tesla = G 30 000 light-years long
a simple light travel time argument illustrates that the relativistic particles filling these reservoirs must be accelerated in situ the lifetime of a relativistic particle is ultimately dependent upon the rate at which it looses energy (emits radiation) typically and the light travel time for a distance so how is energy transfered in radio galaxies? the particles must be accelerated within the lobes and perhaps all along the way from the nucleus to the lobes. is Next & Final Radiative Process: Compton Scattering
Compton Scattering scattering of radiation by particles when an ultra-relativistic electron with Lorentz factor % encounters a photon, the resulting collision tends to upshift the photon frequency! to &% 2!, with the exact coefficient dependent on the scattering geometry. Generalising Thomson Scattering To generalise to relativistic case need to include quantum effects in 2 ways: 1. since photon carries momentum, the electron will recoil, this causes a small energy correction in the expression for '. equivalent to a wavelength shift: This correction to expressions for thomson scattering can be ignored for radiation at wavelengths longer than %-ray 2. classical electron cross-section no longer valid for large photon energies, which reduces the cross-section for highly energetic photons. Corrections come from the Klein-Nishina formula [given in Eqns (7.4-7.6) in R&L]
Inverse Compton Scattering occurs when an electron in motion carries a large amount of kinetic energy (relativistic motion). conservation of momentum in the electron s rest frame results in the transfer of energy from the electron to the photon, which gains by a factor % 2 in the observer s frame. This process converts low-energy photons to high-energy by a factor &% 2. Since photons can have energy as high as 100keV and still be in the Thomson limit, enormous energies can be produced. If the energy is any higher both quantum effects act to reduce the effectiveness of the process by making ' 1 < ' and reducing the probability of scattering. Photon energies larger then ~%mc 2 cannot be obtained. Inverse Compton power for single scattering Let s be a bit more quantitative, and derive average formulas for the case of an isotropic distribution of electrons. Say we have an initial phase density (lorentz invariant), n(p),for the incoming photons. by carefully working out the Lorentz transformations, and assuming an isotropic distribution of photons, can derive the following expression for total power emitted through inverse compton process. This refers to a single, specific electron energy (%), for a general case, N(%), P tot =! P comp N(%)d%
General energy distribution of electrons 1. power-law energy distribution: (for % 1 < % < % 2 ) 2. thermal distribution of electrons Inverse Compton spectra for single scattering Now that we have derived the power due to inverse compton scattering, the next step is to derive the spectrum. In general this will depend on both the energy distribution of the incoming photons (incoming spectrum) and the energy distribution of the electrons.