Radiation processes and mechanisms in astrophysics I. R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 2009

Transcription

1 Radiation processes and mechanisms in astrophysics I R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 009

2 Light of the night sky We learn of the universe around us from EM radiation, neutrinos, meteorites, inter-planetary probes and in the years ahead gravitational radiation.

3 Our view of the universe: Galaxies backlit by the CMB Today t = 13.6 Gyr Reionization t = Gyr Recombination t = 380,000 yr

4 The theory of radiation!

5 Theory of radiation - 1 N atoms in thermodynamic equilibrium at temperature T Level N = number density in the upper excited state Level 1 N 1 = number density in the ground state The mechanism that populates and de-populates the levels include interaction with radiation, collisions between atoms. Transition rates: R 1 = exp(-e /kt) R 1 = exp(-e 1 /kt) This is Boltzmann statistics! When the rates are balanced: N 1 R 1 = N R 1 N 1 exp(-e /kt) = N exp(-e 1 /kt) N /N 1 = exp(- (E - E 1 )/kt) = exp(-δe/kt) T is the excitation temperature that describes the net balance owing to the interaction with ambient radiation and collisions. T may be T K, T r or somewhere in-between.

6 Theory of radiation - The interaction between an atomic system with energy levels and radiation is described by three processes: Spontaneous emission coefficient : A 1 (n) is the transition probability per number per unit time per unit frequency band for spontaneous emission. Absorption : B 1 (n) J is the transition probability per unit time per unit frequency band for absorption. J is the intensity of radiation per unit frequency band. Stimulated emission : B 1 (n) J is the transition probability per unit time per unit frequency band for stimulated emission. J is the intensity of radiation per unit frequency band. Einstein s coefficients!

7 Theory of radiation - 3 If the N atoms distributed between the levels 1 & are in statistical equilibrium with the radiation intensity J. N 1 B 1 (n) J = N A 1 (n) + N B 1 (n) J This is a detailed balance. Solving for J J N N A 1 1 B B B B B 1 1 A 1 B 1 E exp kt 1 Do we expect relationships between the Einstein s coefficients?

8 Theory of radiation - 4 Stimulated emission Absorption Stimulated emission and absorption are processes that may be viewed as time reversed. Time reversal symmetry holds for all electromagnetic processes. Therefore, probability for stimulated emission for a given excited particle equals the probability for absorption by some other particle in the lower state. B 1 (n) = B 1 (n)

9 Theory of radiation - 5 Spontaneous emission has no classical explanation. It may be viewed as stimulated emission stimulated by a virtual photon field. In this viewpoint, all emission processes are induced. A 1 (n) = B 1 (n) intensity of the virtual photon field density of states in phase space per unit freq hn c/4π 4p 3 h dp d 8v 3 c A 1 h c 3 B 1

10 Theory of radiation - 6 The relationships between A 1 B 1 and B 1 - also called Einstein s relations - are a reflection of atomic properties and are independent of temperature and whether or not the atoms are in thermodynamic equilibrium. When there is stimulated emission there will also be spontaneous emission. Spontaneous and stimulated emission and absorption are processes present in all radiation mechanisms. They are the processes that establish equilibrium and in the case of thermodynamic equilibrium lead to J 3 h / c h exp kt 1 Planck spectrum

11 Theory of radiation - 7 Kirchhoff s law: Emission coefficient: j The energy emitted per unit time per unit solid angle and per unit volume de j. dv. d. dt Absorption coefficient: The fractional loss of intensity in a beam as it travels unit distance di. I. ds j 3 h / c h exp kt 1 B ( T ) If a gas cloud at temperature T has a blackbody radiator at temperature T behind it: The emission equals the loss! (corrected, of course, for stimulated emission)!

12 Accelerated charges radiate!

13 (Polarized) radiation from an accelerated charge - 1 Charged particle Initially at rest Accelerated to Dv in time Dt E r 1 4 o q r E 1 4 o q r r sin t c 1 4 o q sin rc

14 (Polarized) radiation from an accelerated charge - E 1 4 o q sin rc Radiation E field has a direction n( n) where n is towards apparent location of the charge at the retarded time. Energy radiated per unit solid angle = Poynting flux P( ) ce Total power (integrate over 4p solid angle): o q sin 3 16 c o P q 6 o c 3

15 Charged particles scatter radiation! Non-relativistic particles scattering photons with energy << particle rest mass

16 The interaction of low energy photons with charged particles - 1 Consider an EM wave E.cos(ωt) incident on a free electron EM power in the incident wave S = (e o ce )/ A charge q that is in the path of the EM wave experiences acceleration: And radiates EM power: (per unit solid angle q q E In direction q) P( ) 3 16 c m sin This is a scattering of EM radiation by a charged particle! o qe cos(t) m The differential cross-section is: ( ) And integral cross-section is: P( ) S q 4m oc 8 q ( ) d 3 4omc T sin Thomson Scattering Cross-section

17 The interaction of low energy photons with charged particles - This picture is good for the case of low energy photons where hn << mc The Thomson scattering cross-section for electrons has a value Classical electron radius: 6.65 x 10^{-9} m (radius at which electrostatic potential equals rest energy).8 x 10^{-15} m T r e 8 q 3 4 omc q 4 o mc Scattering of photons off electrons: P(scatt within ) = n s T mean free path = 1/(n s T ) N(x) = No exp(-x / mfp) cross-section s T density n

18 The interaction of low energy photons with charged particles - 3 Continuing with the picture of Thomson scattering that is good for the case of low energy photons where hn << mc Photon frequency is not changed on scattering Angular distribution of scattered photons is independent of frequency Scattering cross sections are independent of frequency Scattering by protons is (m p /m e ) less than by electrons Scattering has front-bank symmetry Scattered radiation is linearly polarized

19 What if the photon energy is comparable to the particle rest mass?

20 Interaction of high energy photons with charged particles - 1 If the photon energy hf 1 becomes comparable to or exceeds the rest mass of the charged particle we treat the photon as a particle instead of a EM wave. This is then a collision that is solved in the rest frame of the charged particle by writing conservation equations for energy and momentum along x and y directions: Energy conservation: Momentum conservation: Compton wavelength (equivalent to 511 kev)

21 Interaction of high energy photons with charged particles - The incoming photon, scattered photon and scattered electron are in one plane (just as for reflection and refraction in optics) The scattered photon energy will always be less than or equal to the incident photon energy (some energy goes in the electron after the collision) The decrease in photon energy is minimum on forward scattering; maximum in back scatter. For photons with smaller wavelength (greater energy) the fractional energy change is greater. For protons (more mass) the fractional energy change is less. As the photon energy increases, the collision probability collision crosssection decreases and collision probability decreases. (Klein-Nishina cross section). Very high energy gamma rays have longer mean-free-path through electron-proton gas.

22 What if the charged particles move with relativistic speeds?

23 Interaction of photons with energetic charged particles - 1 Thus far we have considered photons with increasing energy colliding with charged particles at rest. What about the scattering of photons off electrons moving with relativistic speeds? If the electrons are relativistic with Lorentz factor : (1) In its rest frame the electron sees the photon to have an energy hn 1 = g hn o () The scattered photon reduces in energy somewhat in the rest frame of the electron: hn 1 -> hn (This is Thomson/Compton scattering) (3) The energy of the scattered photon is seen in the real world to have again increased energy by factor g: hn f = g hn The scattering increases the photon energy by factor g Hot or relativistic electrons can cause net transfer of energy to the scattered photons Inverse-Compton scattering 1 1 c

24 Interaction of photons with energetic charged particles - As a relativistic electron moves through an isotropic sea of radiation The electron sees most of the photons coming head-on These photons are scattered in all directions with back-front symmetry But we would see the scattered radiation mostly emerging along the electron path And these would have increased energy photons are up-scattered in energy by a factor that is the square of the Lorentz factor.

25 Examples from astrophysics

26 The cosmic microwave background photons propagate to us from recombination. The photons scatter off electrons in the intergalactic and intra-cluster gas along the path. Today t = 13.6 Gyr Reionization t = Gyr Recombination t = 380,000 yr

27 Hot gas in clusters of galaxies. CMB photons are up-scattered in frequency by the hot electrons This is the Sunyaev-Zeldovich Effect:.

28 Thomson scattering of randomly polarized radiation: If incident radiation is isotropic: scattered radiation in randomly polarized If incident radiation has a dipole anisotropy: scattered radiation is randomly polarized If incident radiation has a quadrupole anisotropy: Scattered radiation is linearly polarized Example: scattering of sunlight in the atmosphere

29 Observed CMB polarization Because of Thomson scattering of the Quadrupole Anisotropy in the background CMB radiation by electrons in the ionized IGM 3 K CMB brightness temperature 18 micro K observed quad anisotropy micro K observed TE pol anisotropy Implies an optical depth n e T dl of 0.09 between us and epoch of recombination.

30 A radio galaxy has lobes of plasma containing relativistic electrons The electrons scatter photons at the peak of the 3K blackbody form cosmic microwave background ev CMB photons are upscattered by 1 GeV electrons (Lorentz factor ~,000) to 4 kev. The lobes of relativistic plasma are visible in inverse-compton X-rays.