Coursehero >> Iowa >> Iowa State >> PHYSICS 660

Course Hero has millions of student submitted documents similar to the one below including study guides, homework solutions, papers, exam answer keys and textbook solutions.

Synchrotron 6. radiation 6.1 The energy loss rate Charges that move in a homogeneous magnetic field take a helical trajectory along the magnetic field line, for their motion component perpendicular the field is turned into gyration by the Lorentz force v B. So the charged particles are constantly accelerated and react with emission of radiation. For an isotropic distribution of relativistic charged particles in a homogeneous magnetic field this radiation process is called synchrotron radiation and details on its characteristics may also be found in the notes of Astronomy 505. A related radiation process is curvature radiation, which involve charged particles that stream along a curved magnetic field. Curvature radiation is important in pulsars, rapidly rotating neutron stars, that possess a dipole magnetic field that is inclined to the rotation axis. Near the poles the rotation induces a strong electric field along the direction of the local magnetic field, so electrons are accelerated to very high Lorentz factor along the magnetic field lines. Strong curvature radiation must then be expected which would appear to the outside observer as gamma-ray continuum emission. As for the other radiation processes involving accelerated charges the efficacy of the synchrotron process is strongly dependent on the mass of the radiating particle, so electrons radiate more efficiently than ions by many orders of magnitude. The electron energy loss rate for synchrotron radiation can be calculated by transforming Larmors formula from the electron restframe into the laboratory frame, in which the electron is moving relativistically. 4 - E = P = T c U B 2 2 3 2 (6.1.1) where UB = 2B 0 is the energy density of the magnetic field and T is the Thomson cross section. As in the case of inverse Compton scattering the energy loss rate scales quadratically with particle energy. The total energy loss rate of relativistic electrons is given by the sum of those for Coulomb scattering and ionization, relativistic bremsstrahlung, and synchrotron radiation and inverse Compton scattering. Neglecting the weak energy dependence that is carried by the Coulomb logarithm as well as the modifications of the inverse-Compton energy loss rate in the Klein-Nishina limit, the total electron energy loss rate of relativistic electrons has the functional form -E() = A + B + C 2 A, B, C = constants (6.1.2) where the constants A and B include the density of ambient gas, whereas the constant C depends on the energy densities of the magnetic field and the ambient photon field. We will see later that the binomial form of the energy loss rate can induce spectral structures that carry information on the three constants and hence on the ambient gas density etc. 1 Synchrotron radiation is polarized. Detailed calculations of the emission coefficient show that for isotropic charged particles the polarization is linear and oriented along the projection on the sky of the magnetic field. However, variations of the magnetic field direction within the source and Faraday rotation inside and outside of the source region modify the polarization orientation and in most cases reduce the degree of polarization, that can be up to 70% at emission. 6.2 Spectral characteristics Synchrotron radiation is emitted by relativistic electrons and therefore we would expect the emission to be strongly bundled in the instantaneous forward direction of the radiating electron. However, the electron gyrates around the magnetic field direction, so the emission beam rapidly sweeps and projects a circle on the sky. The emission spectrum then consists of a series of harmonics of the electron gyration frequency, that lie indistinguishably close to each other and give the impression of a smooth continuum spectrum. The synchrotron spectrum of a single electron then has a form that depends only on a characteristic frequency c = 3 e B sin 2 4 m 40 Hz B nT 2 160 MHz B nT E GeV 2 (6.2.1) where is the pitch angle between the electron momentum vector and the magnetic field and B is the magnetic field component perpendicular to the line-of-sight. For the spectral power of a single electron we find 3 3 3e B 3e B 1/3 dE = F 1.8 exp - (6.2.2) P () = dt d 4 0 m c c 4 0 m c c c where we have used an analytical approximation to the shape function F that itself is an integral over a modified Bessel function. maximum The of emission is emitted around 0.3 c . Equation (6.2.2) makes clear that even for monoenergetic electrons the energy spectrum cannot be emitted harder than 0.33 in the spectral index. If a harder spectrum is observed, it is either not synchrotron radiation or it has been modified by absorption. To each emission process, characterized by an emission coefficient, there is an associated absorption process with a corresponding absorption coefficient. Absorption is usually frequency-dependent and thus modifies the radiation spectrum. In Astronomy 505 we have discussed how the emission and absorption coefficients can be calculated using the spectral power per particle, P (), and how the two coefficients can be used to determine the properties of the radiation field. Here we want to calculate the emision coefficient for isotropic electrons with power-law density spectrum N () = N0 -s . jsy = 1 dE = dt d dV d 4 2 d N0 -s P () 1 1.8 3 e3 B N0 = 16 2 0 m c The integral can be solved to yield j with 0 1/3 1 d -s-2/3 exp - 0 1-s 2 0 2 (6.2.3) 1.8 3 e3 B N0 g(s) 16 2 0 m c g(s) = 2 2 s+ 2-s 3 (6.2.4) 2 3 s 1 exp - - 2 3 We found a clear relation between the electron spectral index and the synchrotron spectral index, = (s - 1)/2, that is identical to the relation for for inverse Compton scattering (note that we calculated the photon number spectrum j/ there). On account of the close relation between electron energy and synchrotron frequency, Eq.(6.2.1), structure in the electron spectrum like a change in spectral index will be observable as a corresponding change in the synchrotron spectrum. Inverse Compton scattering and synchrotron radiation will be produced by the same relativistic electrons, and usually synchrotron emission accounts for emission at low to moderate frequencies, whereas inverse Compton scattering provides high-energy photons. Let us consider as an example electrons with 10 GeV kinetic energy in the interstellar medium where the magnetic field strength is of the order 1 nT. Equation (6.2.1) tells us that the electrons will predominantly synchrotron-radiate around 10 GHz or 4 10-5 eV in photon energy. Inverse Compton scattering on the other hand can involve the microwave background at 10-3 eV, far-infrared radiation from cold gas clouds, or ambient starlight around 1 eV. The scattering would boost the photon energy by the electron Lorentz factor squared, that is a factor 10 9 , so considering all three sources of target photons we would obtain gamma rays in the MeV GeV band. Even if we chose wide spectra, say power laws, for the radiating electrons, we would observe the synchrotron emission at low frequencies and the inverse-Compton component at high photon energies. In both cases the characteristic frequency of emission scales with the square of the electron Lorentz factor times the frequency of the target photons and the non-relativistic gyrofrequency, respectively. In most astronomical objects the magnetic field strength is so low that the non-relativistic gyrofrequency is lower by many orders of magnitude than the frequency of any relevant target photon field, so the synchrotron emission is produced at much lower frequencies than is inverse Compton emission. An exception to this rule a stars with a very strong magnetic field like neutron stars. Let us now calculate the synchrotron absorption coefficient. For relativistic electrons and small frequencies h E we found in Astronomy 505 that the absorption coefficient can be written as an integral over the spectral power per electron, P . = - 1 8 m 2 d 2 P 3 N () 2 (6.2.5) Using a power-law electron spectrum as in the calculation of the emission coefficient we can work out the derivative in the integral. = N0 (2 + s) 8 m 2 d -s-1 P = s 1+ 2 s j (s = s + 1) -2- 2 2 m (6.2.6) The absorption coefficient and therefore the optical depth will be high at low frequencies and low at high frequencies. At high frequencies the source will thus likely be optically thin, at least against synchrotron absorption, and the spectrum of escaping radiation follows the emission coefficient (6.2.4). At low frequencies, at which the system is optically thick, the observed spectrum is given by the source function g(s) m j = (6.2.7) S = 2.5 s (1 + 2 ) g(s + 1) 0 and thus independent of the form of the electron spectrum. 4

Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more. Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents: