Electrodynamics of Radiation Processes

Transcription

1 Electrodynamics of Radiation Processes 7. Emission from relativistic particles (contd) & Bremsstrahlung Chapter 4: Rybicki&Lightman Sections 4.8, 4.9 Chapter 5: Rybicki&Lightman Sections 5., 5. Four-Vector Notation an event x µ =[ct, x] 4 velocity u µ =[ c, v] 4 momentum p µ =["/c, p] wave 4 vector k µ =[!/c, k)] 4 current density J µ =[ c, J] 4 potential A µ =[ /c, A] 4 force F µ =[ v F/c, F] =/ p v /c " = mc Lorentz factor energy dω = sinθ dθ dφ Emission from Relativistic Particles Emission from relativistic particles The total emitted power is a Lorentz invariant: The power can be evaluated in any frame by just computing in that particular frame & squaring it. It is convenient to express P in terms of the 3-vector acceleration d x /dt rather than the 4-vector acceleration d x µ /dτ. It can be shown that if K is an instantaneous rest frame of a particle, then Angular Distribution of Power in the instantaneous rest frame of the particle, the for convenience amount of energy dw that is emitted into the solid angle dω = sinθ dθ dφ about the direction at angle θ to the x axis. So that x x since energy & momentum form a 4-vector, the transformation of the energy of the radiation is: Thus, from the aberration of light: differentiating since

2 Angular Distribution of Power The power emitted in the rest frame P is found simply by dividing dw by the time interval dt. In frame K there are two possible choices for time interval to divide dw:. dt = γdt this is the time interval during which emission occurs in K With this choice we obtain the emitted power in frame K: P e ( µ) 3. dt A = γ(-βµ)dt this is the time interval of the radiation as received by a stationary observer in K. the extra factor is doppler shift due to the moving source Angular Distribution In observer frame will be highly peaked in forward direction for β ~. given: µ = cos K taylor approx. It follows by expansion that: K' Isotropic (or nearly) in particle frame binomial theorem 4 ( µ) 4 sharply peaked near θ~0 with an angular scale of order Δθ~/γ. With this choice we obtain the received power in frame K: P r 4 ( µ) 4 recieved power is the natural choice - it is the power actually measured by an observer, and it has the required inverse properties. Even if the radiation is isotropic in the particle s frame the angular distribution in the observer s frame will be highly peaked in the forward direction for highly relativistic velocities (β ~ ). Apply to an Emitting Particle in the instantaneous rest frame of the particle the angular distribution is given by the Poynting vector multiplied by the solid angle Θ angle between acceleration and direction of emission Using and Specific cases. acceleration velocity Using, from the aberation of light: definition of projection of a vector onto a plane angular distribution of recieved power use to relate Θ to the angles in frame K Difficult in general, so we consider 3 special cases Specific cases (contd) Specific cases (contd). acceleration velocity a k =0 3. extreme relativistic limit when γ >> the quantity ( - βµ) becomes small in the forward direction, and the radiation strongly peaks in this direction for parallel-acceleration the received radiation pattern is: for perpendicular acceleration: both of these expressions depend on θ solely in combination with γθ and therefore peaking for angles θ ~ /γ.

3 Phase Space Volume Consider a group of particles occupying a slight spread in position and momentum at a particular position. In a co-moving frame, they occupy spatial volume, d 3 x = dx dy dz and a momentum element d 3 p = dp x dp y dp z The group occupies an element of phase space, dv = d 3 p d 3 x Any observer not comoving with the particles will determine the same phase space, in the frame dv = d 3 pd 3 x Phase-space element is Lorentz invariant Equation of Radiative Transfer Radiation density, u ν = energy/volume/dω/dν u v = I v c = Lorentz invariant = I v d dv = energy/volume c also energy/volume = f h p dpd p= h c Lorentz invariant phase-space density of photons, f = dn/dv I 3 f h p dpd = I c d d I 3 / f S 3 because S ν appears in RTE as the difference I ν - S ν S ν transforms as I ν Transformation of a moving, absorbing medium To find the transformation of absorption coefficient, consider material in a frame K streaming with velocity v between two planes parallel to x-axis. Let K be the rest frame of the material. The optical depth τ along the ray must be an invariant since e -τ gives the fraction of photons passing through, simple counting. = Lorentz invariant Bremsstrahlung (or free-free emission) Radiation due to the acceleration of a charge in the Coulomb field of another (different) charge Transformation of sinθ can be found by noting that νsinθ is proportional to the y-component of the 4-momentum k y, and both k y and l are perpendicular to v in both frames, and thus transform the same, l/k y is Lorentz invariant and thus so is να ν. For the emission coefficient, j ν = α ν S ν = Lorentz invariant Bremsstrahlung Bremsstrahlung (free-free) Radiation Virgo clusters is filled with hot gas radiating via bremsstrahlung strongly at kev (several billion degrees). optically thin Bremsstrahlung originates from the acceleration of a charge (e.g., an electron) in coulomb collisions with other charges (e.g., ions or nuclei). The most common situation is the emission from a hot gas as the electrons collide with the nuclei due to their random thermal motions. HII regions: ionised by young hot stars and filled with hot gas, denser but cooler than a galaxy cluster. Virgo Cluster of galaxies, in X-ray Quantifying this process will allow us to obtain physical insights into what is being seen in this image. e.g., mass, temperature, density... 3

4 Bremsstrahlung free-free emission Produced by collisions between particles in hot ionized plasmas predominantly from collisions between electrons and ions In an electron-ion collision we can take the ion to be unaccelerated Precise results require quantum treatment, but useful approximate results can be obtained from classical calculation of the dipole radiation. Emission from single speed electrons. compute radiation power spectrum from a single collision with given electron velocity & impact parameter.. Integrate over impact parameter to get the emission from a single speed electron component 3. Integrate over a thermal distribution of electron velocities to obtain thermal bremsstrahlung emissivity 4. Consider thermal bremsstrahlung absorption & emission from a plasma with relativistic electron velocities Single speed electrons Assume electrons move rapidly enough so that they don t deviate from a straight line. This is the small angle scattering regime. dipole moment is d= -er second derivative: d = where v is velocity of the electron Taking the fourier transform: remember FT! ˆd(!) Z of e d is! ˆd(!) = v e i!t dt e v Derive in asymptotic limits of large and small frequencies for ωτ >> the exponential oscillates rapidly and the integral is small for ωτ << the exponential is unity An electron of charge e moving past an ion of charge Ze with an impact parameter b collision time, τ=b/v change of vel, Δv Single speed electrons: Δv Since path is almost linear, change in velocity is predominantly normal to path. Thus integrate the component of acceleration normal to path: thus for small angle scatterings, the emission from a single collision: to determine the total spectrum for a medium with an ion density, n i an electron density, n e and a fixed electron speed, v the flux of electrons (per unit area per unit time) incident on one ion is n e v and the area is πbdb per single ion. total emission per unit time per unit volume per unit frequency range dw d! = 8!4 3c 3 ˆd(!) Dipole approx. Lecture 4 minimum value of the impact parameter Total emission the integral requires the full range of impact parameters, however it turns out to be a good approximation using only low frequency (b<<v/ω) form substitute b max is a value of b beyond where b<<v/ω is applicable and the contribution of the integral becomes negligible, so it is of order v/ω A possible value for the lower limit on the integration b min is when the straight-line approximation ceases to be valid, this occurs when Δv~v Quantum option for lower limit another option is quantum in nature, and treats collision process in terms of classical orbits using the classical form of the uncertainty principle when b v min >> bq min then a classical description of scattering is valid and bv min is the appropriate choice for b min. this occurs when where Rydberg energy for H When b v min << bq min or equivalently /mv >> Z Ry then the uncertainty principle plays a big role and the classical calculation cannot strictly be used, but using b q min gives results of the correct order of magnitude. For any regime the exact results are conveniently stated in terms of a correction factor, or Gaunt factor g ff (v,ω). the gaunt factor the gaunt factor is a function of the energy of the electron and of the frequency of the emission, and extensive tables exist in the literature. 4

5 The Gaunt Factor The Gaunt Factor Gaunt factor usually in the range. to.5. u= h kt Consider a more realistic plasma than single speed electrons A plasma in thermal equilibrium means that the electrons have a Maxwell-Boltzmann velocity distribution: have to average the single speed electron expression we just derived over a thermal distribution of speeds M-B means, the probability dp that a particle has a velocity in range d 3 v is assuming velocity distribution is isotropic Now we have to integrate over this distribution. the limits of the integration should account for the fact that at frequency ν the incident velocity must be such that photon discreteness effect because otherwise the photon of energy hν could not be created. perform integral thus, in CGS units (erg s - cm -3 Hz - ), we have for the emission velocity averaged Gaunt factor 5