Thomson scattering: It is the scattering of electromagnetic radiation by a free non-relativistic charged particle.

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Thomson scattering: It is the scattering of electromagnetic radiation by a free non-relativistic charged particle. The electric and magnetic components of the incident wave accelerate the particle.

1 Thomson scattering: It is the scattering of electromagnetic radiation by a free non-relativistic charged particle. The electric and magnetic components of the incident wave accelerate the particle. As it accelerates, it, in turn, emits radiation and thus, the wave is scattered. Thomson scattering is an important phenomenon in plasma physics and was first explained by the physicist J. J. Thomson Nobel Prize 906 The main cause of the acceleration of the particle will be due to the electric field component of the incident wave. The particle will move in the direction of the oscillating electric field, resulting in electromagnetic dipole radiation. The moving particle radiates most strongly in a direction perpendicular to its motion and that radiation will be polarized along the direction of its motion. Therefore, depending on where an observer is located, the light scattered from a small volume element may appear to be more or less polarized.'' Daniele Dallacasa Radiative Processes and MHD Section -5 : Thomson Scattering, Compton scattering & Inverse Compton

2 Thomson scattering Photon-electron interaction - hv m ec h ν me c for a non relativistic particle, the incoming (low-energy) photons can be expressed as a continuous e-m wave and the magnetic field (Ho Eo ) can be neglected (i.e. studied as a consequence of the E field behaviour). the average incoming Poynting flux is S ceo / 8 π the charge feels a force due to a linearly polarized incoming wave: the energy of the scattered radiation is the same as the incoming wave

3 Thomson scattering () Force from a linarly polarised wave F e ε Eo sin ω o t Dipole moment d d e r e r e Eo ( ) e Eo mω o m ε sin ω o t ε sin ω o t describes a dipole with amplitude a (t) d o 0% polarized ( ) e E o mω o ε taking into account the Larmor's formula, the time averaged emitted P is 4 e Eo dp d dσ 3 sin Θ S 3 dω 3c dω 8πm c where dσ dω e4 4 00% polarized sin Θ r sin Θ o mc Differential cross section for the scattering

Particle acceleration (due to incoming radiation) the emitted power by the scattering particle is: from this equation we define the electron Thomson cross section by integrating over all the angles:

4 Particle acceleration (due to incoming radiation) the emitted power by the scattering particle is: from this equation we define the electron Thomson cross section by integrating over all the angles: Back to the ''geometric problem'' the angular distribution of the radiation is given by and this means that each individual scatter produces polarized radiation Cross section independent from frequency (breaks down at high frequencies when the classical physics needs quantum mechanics)

5 Summary. Scattering symmetric wrt the scattering angle (backwardforward). The amount of scattered radiation depends solely on the incident radiation dw σ T c Urad dt 3. Even if the incident radiation is unpolarized, TS can polarize scattered radiation depending on the scattering angle 4. TS is one of the main processes capable of trapping photons in a given region. Optical depth for TS in a medium is τt σ T n dl e Photons are scattered in random directions, i.e. random walk in which each step is the mean free path λ T σ T ne

6 Example: Spectrum of total intensity and polarized light for a type II AGN

Compton scattering ''Compton scattering or the Compton effect, is the decrease in energy (increase in λ) of a high energy (X or γ ray) photon, as it interacts with a free electron.

7 Compton scattering ''Compton scattering or the Compton effect, is the decrease in energy (increase in λ) of a high energy (X or γ ray) photon, as it interacts with a free electron. More important for astrophysics, Inverse Compton scattering also exists, where The photon gains energy (decreasing in wavelength) upon interaction with a high energy electron. The amount the wavelength increases by is called the Compton shift. Although nuclear Compton scattering also exists, what is meant by Compton scattering usually is the interaction involving free electrons only. A.H. Compton Compton effect was observed by Arthur Holly Compton in 93, for which he was awarded the Nobel Prize in Physics in 97.. The effect is important because it demonstrates that light cannot be explained purely as a wave phenomenon. Thomson scattering, the classical theory of charged particles scattered by an electromagnetic wave, cannot explain any shift in wavelength. Light must behave as if it consists of particles in order to explain the Compton scattering.''

8 Compton scattering photon-electron interaction --

9 Compton scattering photon-electron interaction -- energy conservation: h i m e c h f p e c m e c 4 momentum conservation: p i p f p e pe /c h /c p e p i p f photon momentum p e p i p f p i p f p i p f p i p f cos p e h i h f h i f cos multiply by c then take the square of energy conservation p e c m e c 4 h i h f m e c 4 h i m e c h f m e c h i f compare the two by highlighting pec we get h i f cos h i m e c h f m e c h i f h i f cos h m e c i f f i h cos f i c c me c f i cos me c h cos o cos me c i f h i f h f i Compton wavelength for electrons o h A me c

10 Compton scattering a Java applet showing the geometry of the scattering can be found at

11 Compton scattering a Java applet showing the geometry of the scattering can be found at h e h i h i cos / me c h cos o cos cos [A] me c Final energy of the scattered photon is a function of: e i initial energy of the photon scattering angle The photon always looses energy, unless θ 0, and the scattering is closely elastic when λ λc (i.e. hv mec ) When the photons involved in the collision have large energies, the scattering become less efficient and quantum electrodynamics effects reduce the cross section: the Thomson cross section becomes the Klein-Nishina cross section

12 Cross section: Thomson and Klein Nishina In case hv ~ mec, the probability of interaction decreases, also as a function of the scattering angle; additional constraints come from relativistic quantum mechanics; h x me c if we use T [ 3 x x x 3 x ln x ln x 4 x3 x x x known as the Klein Nishina cross section; if x T it originates from a differential cross section based on the scattering angle d 3 r e [ P h, P h, sin P h, ] d where the probability is defined as P h, h me c cos and the Klein Nishina cross section becomes the Thomson cross section at low photon energies ] 3 ln x 8 x

13 Cross section Klein Nishina The Klein Nishina cross section is relevant at high photon energies (Hard X-Rays and beyond) x

14 Inverse Compton scattering photon-electron interaction -3- Moving, energetic (relativistic), electrons cool down by increasing the energy of seed photons. To make computations simpler, it is possible to consider the centre of momentum frame where the photon energy is much less than~ me c and then~ σ T can be used. In the electron rest frame we have Thomson scattering. Lab system electron rest frame: h νe h νl γ ( β cos θ). Thomson scattering in electron rest frame h νe ' h νe 3. Electron rest frame Lab system h νl ' h ν e ' γ ( + β cos θ ') h νl ' γ h νl i.e. Very efficient energy transfer from electron to photon!

15 Inverse Compton scattering The scattering angle is important for the energy of the outgoing electron: maximum energy gained by the electron if~ θ π and θ ' 0 in the electron rest frame the photon is blue shifted (face on collision) minimum energy gained by the electron if θ 0 and θ ' π in the electron rest frame the photon is red shifted (end on collision) the maximum energy that can be gained is (averaged over all angles): 4 ε' γ ε 3 4 or, in terms of frequency ν' γ ν 3 In case it is possible to measure both the initial and final photon energy/frequency, then γ (or γ ) can be deduced γ 3 ν' 4 ν

16 Inverse Compton scattering emitted energy It is a Lorentz invariant, i.e. the same in the laboratory and electron rest frame E ' h ν ' h ν γ( + β cos θ) E γ( + β cos θ) Let's consider a region where there is a plasma of relativistic electrons and a radiation field: in the electron framework, the photon energy is dε dε ' c σt E 'rad c σ T γ ( + β cos θ) Erad dt dt ' 8π since the emitted energy is a Lorentz invariant. It is then possible to write Erad Uph ( + β cos θ) + β 3 and, finally, the energy coming out the scattering region is dεout c σ T Uph γ + β dt 3 ( )

17 Inverse Compton emission.vs. Synchrotron emission The energy associated with the photons, prior of the scattering is dε in c σ T Uph dt and then the net effect of the Inverse Compton scattering is dεout dt γ dεin dt γ β ( ) dε dt ic ( ) dt ic and if we recall the synchrotron emission we get ( ( ( β ) ] c σ T γ β Uph ( ) dε dt syn / dt )syn UH Uph d ε / dt )ic dε c σ T Uph γ + dε then [ 4 3 c σ T γ β UH

18 Synchrotron & Inverse Compton scattering radiative losses The same relativistic electrons radiate via synchrotron and inverse Compton; their contributions add up: ( ) dε dt sy + ic C s ε [ H + 8 π Urad ] The increased cooling rate implies that the electron radiative lifetime is consequently reduced t sy +ic ε H + 8 π Urad (H / 8 π + Urad ) 3 [ev cm ] ε [GeV ] yr Therefore... hard times for relativistic electrons! following example: radiative lifetime for a relativistic electrons in a radio lobe. Heq, Urad, strong dependence on z! HCMB 3.8( + z) μg

19 The Local Radiation Field () Evaluation of Urad: as taken in a number of selected locations, i.e. the lobes of a radio source, far from any galaxy

The Local Radiation Field () Evaluation of Urad: as taken in a number of selected locations in the Milky Way X-rays radio Spatial variation of the total radiation field throughout the Galaxy.

20 The Local Radiation Field () Evaluation of Urad: as taken in a number of selected locations in the Milky Way X-rays radio Spatial variation of the total radiation field throughout the Galaxy. Black line : total radiation field for GC. Magenta line : total radiation field for R0, z5 kpc, Blue line : total radiation field for R 4, z0 kpc, Red line : total radiation field for R, z0 kpc, Green line : total radiation field for R0, z0 kpc. This diagram will be discussed a bit further when considering the ISM

21 Synchrotron Self-Compton Catastrophe Let's consider a spherical cloud with radius R filled with magnetized relativistic plasma located at a distance d from the observer. The low-energy synchrotron photons may be up-scattered by the relativistic electrons via IC: L s 4 π d S (ν)d ν 4 π d Smax ν c f (α) Urad R 4πc R 3 the energy density of the radiation field can then be derived 3L s Urad 4πR c finally the comparison between the IC and Synchrotron losses gives: L IC Ls Urad H /8 π 6L s RHc Smax νc θ Hc f (α) ( )( ) T Bmax 0 K 5 νc GHz f (α) For brightness temperatures above 0 K, the radiation field will undergo to a dramatic amplification; IC would become the most efficient and dominant cooling process (X-Rays) and a source would radiate its energy in a very short time [Compton catastrophe]. Brightness temperature limit in radio source is T B 0 K

22 Comptonization () We want to evaluate if there is a substantial change in the spectrum of the seed photons in case of multiple scattering with hot (but non-relativistic) electrons in thermal equilibrium. Let's consider the process from the point of view of the photons There are interactions where energy is transferred from photons to electrons (C scattering) Δhν hν h ν phot me c There are interactions where energy is transferred from electrons to photons (IC scattering) (positive since we are considering the process from the point of view of the photons) () 4 v Δε ε el 3 c 4 3c 3kT me 4kT me c The net energy transfer is: ΔE E (4kT h ν) me c Thermal equilibrium (/)mev (3/)kT

23 Comptonization -There is not a net energy transfer when 4kT h ν If 4kT h ν photons loose energy and electrons (gas) heats up 4kT h ν photons gain energy and electrons (gas) cools down Let's consider a gas (plasma) region, D in diameter, where Δε 4kT ε me c The opacity to the scattering is where the mean free path is λ mfp ne σ T τe ne σ T D D λ mfp 4kT hν D

24 Comptonization: -3- The total energy gain is ( ) N The total number of collisions is: Δhν hν λ mfp 4kT D me c tot τ e N 4kT me c N dt the mean photon energy as a function of time becomes (4kT / me c )N t h ν(t) h νo e h νo e where y is known as ''Compton y-parameter'' y kt me c N (4kT / me c )N me c kt ne σ T D h νo e 4y italics

25 Comptonization -4The spectrum of the radiation is substantially modified when ν f y 4kT i.e. 4 me c N 4kT ne σ T D me c h νf 4kT namely τe [ ln i.e. y ( ) 4kT me c h νo 4kT 4 and the final equilibrium is achieved when ln ν(t ) > νo > ( ) 4kT h νo ] / If this condition is met (thermal equilibrium), the modified photon spectrum must follow the Bose-Einstein distribution u(ν)d ν 8 π h ν3 c 3 (h ν / kt)+μ e dν

26 Comptonization -5- Modified Spectrum At equilibrium, the photons will follow a Bose - Einstein distribution: 8 π h ν3 u(ν)d ν 3 dν (h ν / kt)+μ c e where μ is known as chemical potential (which is 0 for Planck's BB spectrum) and measures the rate at which photons are produced hν in case kt + μ we have u(ν)d ν 8πh ν 3 3 h ν / kt μ e e c i.e. the Wien's law modified by e μ

27 Kompaneets' equation The exact description of the photon scattering (i.e. find the observed spectrum originated by various values of the y-parameter) is better obtained in the phase-space through the Kompaneets' equation: n if n y x u(ν)c 3 8πhν x x and 3 [( 4 x n + n + Recoil effect n x hν kt )] Doppler motion Induced/stimulated emission Emission spectra come out as power-law with index 3+m where Examples: Accretion disks in XRBs (μ Q), AGN,...S-Z effect 3 9 m ± + 4 y

28 Thermal bremsstrahlung spectrum with comptonization (accretion disks)

29 Comptonization: the spectrum is modified as a function of the opacity to this process Remember: opacity is the square of the number of scattering processes!

30 Thermal bremsstrahlung spectrum with comptonization (accretion disks)

31 Comptonization: Astyrophysical examples possible models of comptonization

$Some fraction of these photons is seen directly by the Earth observer. Red and blue photon trajectories correspond to soft and hard (upscattered) photons, respectively.$

32 A suggested geometry of GX Disk and NS soft photons are upscattered off the hotter plasma of the TL located between the accretion disk and NS surface. Some fraction of these photons is seen directly by the Earth observer. Red and blue photon trajectories correspond to soft and hard (upscattered) photons, respectively. In our model, two Comptonization components are considered: the first one (Comptb), with "seed" (disk) photon temperature Ts kev and the CC electron temperature varies from 3 kev to kev. The second Comptonized component (Comptb with Ts ~.5 kev) and are related to the NS surface and its inner part of the TL (boundary) layer, respectively. Seifina, Titarchuk & Filippo Frontera, 03

Sunyaev-Zeldovich effect Rashid Sunyaev Jakov Zeldovich 94 987 - thermal: high energy

the spectrum, second order effect, also known as Ostriker Vishniac effect Where does

33 Sunyaev-Zeldovich effect Rashid Sunyaev Jakov Zeldovich thermal: high energy electrons interact with CMB photons via IC scattering - kinematic: bulk motions modify the spectrum, second order effect, also known as Ostriker Vishniac effect Where does it take place? For more information on SZ effect Carlstrom + 00 ARAA, vol 40, p

35 Sunyaev-Zeldovich effect () Galaxy clusters have hot electrons capable of IC scattering of CMB photons. A CMB photon have a % probability of interacting with a high energy ICM electron. v3+m fixed cross-over freq

36 Sunyaev-Zeldovich effect () From Carlstrom + 00 ARAA, vol 40, p v3+m fixed cross-over freq

satellite. A full sky temperature map is at http://www.

37 Sunyaev-Zeldovich effect: the Planck's view The same galaxy cluster as a function of frequency as seen by the Planck satellite. A full sky temperature map is at

38 Comptonization summary Effective in rarified plasma (no significant competing cooling mechanism) Hot electrons! (must transfer energy to low energy photons) Total number of photons is conserved, i.e. possible BB photon distribution is modified Equilibrium is achieved in saturated sources (i.e. when the energy of the radiation is balanced With that of the electron: hν 4KT

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

Radiation processes and mechanisms in astrophysics I R Subrahmanyan Notes on ATA lectures at UWA, Perth 18 May 009 Light of the night sky We learn of the universe around us from EM radiation, neutrinos,