Page 1 Astrophysics II, University of Turku 16 January 2009 Lecture 7 : Compton Scattering 7.1 INTRODUCTION Thomsonscattering,orthescatteringofaphotonbyanelec- tron at rest, strictly only applies at low photon energy, i.e. when h?«mec 2 . If the photon energy is comparable to or greater than the electron energy, non-classical e?ects must be taken into account, and the process is called Compton scattering. A further interesting situation develops when the electron is moving — in this case energy can be transferred to the photon, and the process is called inverse Compton scat- tering.Thislastprocessisanimportantmechanisminhigh energy astrophysics. 7.2 THOMSON SCATTERING In Thomson scattering, we have dsT dO = 1 2 r 2 0 (1+cos 2 ?) (7.1) where sT is the Thomson cross-section, O the solid angle, ? is the angle of scattering, and r0 is the classical electron radius, r0 = e 2 mec 2 . (7.2) In Thomson scattering the incident photon and scatter photon have the same wavelength or energy, so this scatter- ing is also called coherent or elastic. If we now move to photons of energy h? > ~ mec 2 , the scattering is modi?ed by the appearence of quantum e?ects, through a change in the kinematics of the collision, and an alteration of the cross-section. 7.3 COMPTON SCATTERING Todothekinematicsofthecollisioncorrectlyathighphoton energy, momentum and energy must be conserved. Lettheincidentphotonhaveenergyh? andmomentum h?/c, the scattered photon have energy h? ' and momentum h? ' /c, and the electron (initially at rest) acquires energy E and momentum pe. The scattering angle is ?. Problem7.1 Showthattheenergyofthescatteredphoton is given by h? ' = h? 1+ h? mec 2 (1-cos?) (7.3) In terms of wavelength, this reduces to ? ' -? =?C(1-cos?) (7.4) Figure 7.1. Compton scattering of an incident photon of energy h? and momentum p to energy h? ' and momentum p ' . The elec- tron is initially at rest and acquires energy E and momentum pe. where ? is the incident photon wavelength, ? ' is the scat- tered photon wavelength and ?C is the Compton wave- length and is given by ?C = h mec =0.02426 ° A. (7.5) Problem 7.2 Prove Eqn 7.4. The compton wavelength can be regarded as a wave- length change ?? in the incident photon. Note that for ?»?C the change is negligible and we get back the Thom- son scattering. In full treatment of the problem yields the Klein- Nishina formula for the scattering cross-section: ds dO = r 2 0 2 ? ' ? 2 ? ? ' + ? ' ? -sin 2 ? (7.6) which can be shown to yield the following formulae for the total cross-section (where x = h? mc 2 ), s˜sT 1-2x+ 26x 2 5 +... , for x« 1, (7.7) and s = 3 8 sT 1 x ln(2x)+ 1 2 , for x»1 (7.8) for the non-relativistic and extremely relativistic cases. The main e?ect is thus to reduce the cross-section at high pho- ton energies, i.e. the scattering of the photons becomes less e?cient. 7.4 INVERSE COMPTON: SCATTERING FROM MOVING ELECTRONS An important case arises when the electrons are no longer considered to be at rest. In inverse scattering, energy is transferred from the electrons to the photons, i.e. it is the oppositeofComptonscattering,inwhichthephotonstrans- fer energy to the electrons. Inverse Compton scattering can Page 2 Astrophysics II, University of Turku 16 January 2009 Lecture 7 : Compton Scattering 7.1 INTRODUCTION Thomsonscattering,orthescatteringofaphotonbyanelec- tron at rest, strictly only applies at low photon energy, i.e. when h?«mec 2 . If the photon energy is comparable to or greater than the electron energy, non-classical e?ects must be taken into account, and the process is called Compton scattering. A further interesting situation develops when the electron is moving — in this case energy can be transferred to the photon, and the process is called inverse Compton scat- tering.Thislastprocessisanimportantmechanisminhigh energy astrophysics. 7.2 THOMSON SCATTERING In Thomson scattering, we have dsT dO = 1 2 r 2 0 (1+cos 2 ?) (7.1) where sT is the Thomson cross-section, O the solid angle, ? is the angle of scattering, and r0 is the classical electron radius, r0 = e 2 mec 2 . (7.2) In Thomson scattering the incident photon and scatter photon have the same wavelength or energy, so this scatter- ing is also called coherent or elastic. If we now move to photons of energy h? > ~ mec 2 , the scattering is modi?ed by the appearence of quantum e?ects, through a change in the kinematics of the collision, and an alteration of the cross-section. 7.3 COMPTON SCATTERING Todothekinematicsofthecollisioncorrectlyathighphoton energy, momentum and energy must be conserved. Lettheincidentphotonhaveenergyh? andmomentum h?/c, the scattered photon have energy h? ' and momentum h? ' /c, and the electron (initially at rest) acquires energy E and momentum pe. The scattering angle is ?. Problem7.1 Showthattheenergyofthescatteredphoton is given by h? ' = h? 1+ h? mec 2 (1-cos?) (7.3) In terms of wavelength, this reduces to ? ' -? =?C(1-cos?) (7.4) Figure 7.1. Compton scattering of an incident photon of energy h? and momentum p to energy h? ' and momentum p ' . The elec- tron is initially at rest and acquires energy E and momentum pe. where ? is the incident photon wavelength, ? ' is the scat- tered photon wavelength and ?C is the Compton wave- length and is given by ?C = h mec =0.02426 ° A. (7.5) Problem 7.2 Prove Eqn 7.4. The compton wavelength can be regarded as a wave- length change ?? in the incident photon. Note that for ?»?C the change is negligible and we get back the Thom- son scattering. In full treatment of the problem yields the Klein- Nishina formula for the scattering cross-section: ds dO = r 2 0 2 ? ' ? 2 ? ? ' + ? ' ? -sin 2 ? (7.6) which can be shown to yield the following formulae for the total cross-section (where x = h? mc 2 ), s˜sT 1-2x+ 26x 2 5 +... , for x« 1, (7.7) and s = 3 8 sT 1 x ln(2x)+ 1 2 , for x»1 (7.8) for the non-relativistic and extremely relativistic cases. The main e?ect is thus to reduce the cross-section at high pho- ton energies, i.e. the scattering of the photons becomes less e?cient. 7.4 INVERSE COMPTON: SCATTERING FROM MOVING ELECTRONS An important case arises when the electrons are no longer considered to be at rest. In inverse scattering, energy is transferred from the electrons to the photons, i.e. it is the oppositeofComptonscattering,inwhichthephotonstrans- fer energy to the electrons. Inverse Compton scattering can 2 produce substantial ?uxes of photons in the optical to X- ray region. Analysis shows that the mean frequency of the photons after the collision increases by a factor ? 2 , so that high frequency radio photons in collisions with relativistic electrons for which ? is of order 10 3 to 10 4 can beboosted in the UV and X-ray regions. There is a practical limit to the amount of boosting possible beyond the Thomson limit (h? ˜ ?mc 2 ), which can be seen from the conservation of energy h? ' =?mc 2 +h?. (7.9) Scattered photon energies are thus limited to ?mc 2 . The power emitted in the case of an isotropic distribu- tion of photons is PComp = 4 3 scU rad ? 2 ß 2 (7.10) whereU rad istheradiationenergydensityofthephoton?eld (before scattering). Note how similiar this is the power due to synchrotron emission P Synch = 4 3 scUB? 2 ß 2 (7.11) where UB is the energy density of the magnetic ?eld. Thus P Synch PComp = UB U rad . (7.12) The losses due to synchrotron and Compton processes are in the ratio of the magnetic ?eld energy density to the photon ?eld energy density, and is independent of ?. The scattered photons may be produced in the source throughsynchrotronradiation,andifthesearebosstedthen theresultantphotonsarecalledSynchrotronSelfComp- ton. 7.5 COMPTONISATION If the spectrum of a source is primarily determined by Compton processes it is termedComptonised. In this case the plasma must be thin enough that other processes, such as bremsstrahlung, do not dominate the spectrum instead. The hotter the gas, the more chance of Comptonisation. Some examples of astrophysical sources in which comp- tonisation is important are: • hot gas near binary X-ray sources • hot plasma in clusters of galaxies • hot plasma near center of active galactic nuclei • primordial gas cooling after the Big Bang 7.5.1 Non-relativistic Comptonisation We consider non-relativistic electrons and photons with en- ergy h?«mec 2 . From Eqn 7.4 one can show that the rela- tive change in the photon energy ?E/E is given by ?E E = h? mec 2 (1-cos?) (7.13) Problem 7.3 Verify Eqn 7.13 In the electron frame, the scattering is Thomson, and therefore symmetric around the incident direction, so that ?E E = h? mec 2 , for h?«mec 2 . (7.14) This is the average energy increase of the electron for low photon input energies, h?«mec 2 . Now consider high energy photons. In this case the power produced per scattering is given by Eqn. 7.10. For non-relativistic electrons, ?˜ 1, so for electron velocity v, PComp = 4 3 sc v c 2 . (7.15) Thenumberofscatteredphotonspersecondisthenum- berofphotonsencounteredpersecondbyanelectron,which is given by the photon number density, N phot , the photon velocity c and the Thomson cross-section sT N phot csT. (7.16) The photon number density is just N phot = U rad h? (7.17) so we have sTN phot c = sTU rad c h? . (7.18) Comparison of Eqns. 7.15 and 7.18 readily shows that the energy gain of the photons per collision must be ?E E = 4 3 v c 2 for h?»mec 2 . (7.19) Let’s summarise what we have so far: • For h?«mec 2 , the electrons gain energy. (Eqn 7.14). • For h?»mec 2 , the photons gain energy. (Eqn 7.19). 7.5.2 Thermal electrons To make things practical, let’s consider a thermal distribu- tion of electrons, with temperature Te. We have 3 2 kTe = 1 2 mev 2 (7.20) where v is the typical electron velocity. Eqn 7.19 can thus be written ?E E = 4kTe mec 2 for h?«kTe. (7.21) If we now combine the results of Eqns 7.21 and 7.14 we can derive a simple equation for the energy gain/loss for both the high and low frequency regimes ?E E = 1 mec 2 (4kTe-h?). (7.22) Therefore, for • h? = 4kTe, there is no energy exchange • h? > 4kTe, electrons gain energy • h? < 4kTe, photons gain energy Theusualcaseofinterestish? < 4kTe,i.e. theelectrons are “hotter” than the photons. Page 3 Astrophysics II, University of Turku 16 January 2009 Lecture 7 : Compton Scattering 7.1 INTRODUCTION Thomsonscattering,orthescatteringofaphotonbyanelec- tron at rest, strictly only applies at low photon energy, i.e. when h?«mec 2 . If the photon energy is comparable to or greater than the electron energy, non-classical e?ects must be taken into account, and the process is called Compton scattering. A further interesting situation develops when the electron is moving — in this case energy can be transferred to the photon, and the process is called inverse Compton scat- tering.Thislastprocessisanimportantmechanisminhigh energy astrophysics. 7.2 THOMSON SCATTERING In Thomson scattering, we have dsT dO = 1 2 r 2 0 (1+cos 2 ?) (7.1) where sT is the Thomson cross-section, O the solid angle, ? is the angle of scattering, and r0 is the classical electron radius, r0 = e 2 mec 2 . (7.2) In Thomson scattering the incident photon and scatter photon have the same wavelength or energy, so this scatter- ing is also called coherent or elastic. If we now move to photons of energy h? > ~ mec 2 , the scattering is modi?ed by the appearence of quantum e?ects, through a change in the kinematics of the collision, and an alteration of the cross-section. 7.3 COMPTON SCATTERING Todothekinematicsofthecollisioncorrectlyathighphoton energy, momentum and energy must be conserved. Lettheincidentphotonhaveenergyh? andmomentum h?/c, the scattered photon have energy h? ' and momentum h? ' /c, and the electron (initially at rest) acquires energy E and momentum pe. The scattering angle is ?. Problem7.1 Showthattheenergyofthescatteredphoton is given by h? ' = h? 1+ h? mec 2 (1-cos?) (7.3) In terms of wavelength, this reduces to ? ' -? =?C(1-cos?) (7.4) Figure 7.1. Compton scattering of an incident photon of energy h? and momentum p to energy h? ' and momentum p ' . The elec- tron is initially at rest and acquires energy E and momentum pe. where ? is the incident photon wavelength, ? ' is the scat- tered photon wavelength and ?C is the Compton wave- length and is given by ?C = h mec =0.02426 ° A. (7.5) Problem 7.2 Prove Eqn 7.4. The compton wavelength can be regarded as a wave- length change ?? in the incident photon. Note that for ?»?C the change is negligible and we get back the Thom- son scattering. In full treatment of the problem yields the Klein- Nishina formula for the scattering cross-section: ds dO = r 2 0 2 ? ' ? 2 ? ? ' + ? ' ? -sin 2 ? (7.6) which can be shown to yield the following formulae for the total cross-section (where x = h? mc 2 ), s˜sT 1-2x+ 26x 2 5 +... , for x« 1, (7.7) and s = 3 8 sT 1 x ln(2x)+ 1 2 , for x»1 (7.8) for the non-relativistic and extremely relativistic cases. The main e?ect is thus to reduce the cross-section at high pho- ton energies, i.e. the scattering of the photons becomes less e?cient. 7.4 INVERSE COMPTON: SCATTERING FROM MOVING ELECTRONS An important case arises when the electrons are no longer considered to be at rest. In inverse scattering, energy is transferred from the electrons to the photons, i.e. it is the oppositeofComptonscattering,inwhichthephotonstrans- fer energy to the electrons. Inverse Compton scattering can 2 produce substantial ?uxes of photons in the optical to X- ray region. Analysis shows that the mean frequency of the photons after the collision increases by a factor ? 2 , so that high frequency radio photons in collisions with relativistic electrons for which ? is of order 10 3 to 10 4 can beboosted in the UV and X-ray regions. There is a practical limit to the amount of boosting possible beyond the Thomson limit (h? ˜ ?mc 2 ), which can be seen from the conservation of energy h? ' =?mc 2 +h?. (7.9) Scattered photon energies are thus limited to ?mc 2 . The power emitted in the case of an isotropic distribu- tion of photons is PComp = 4 3 scU rad ? 2 ß 2 (7.10) whereU rad istheradiationenergydensityofthephoton?eld (before scattering). Note how similiar this is the power due to synchrotron emission P Synch = 4 3 scUB? 2 ß 2 (7.11) where UB is the energy density of the magnetic ?eld. Thus P Synch PComp = UB U rad . (7.12) The losses due to synchrotron and Compton processes are in the ratio of the magnetic ?eld energy density to the photon ?eld energy density, and is independent of ?. The scattered photons may be produced in the source throughsynchrotronradiation,andifthesearebosstedthen theresultantphotonsarecalledSynchrotronSelfComp- ton. 7.5 COMPTONISATION If the spectrum of a source is primarily determined by Compton processes it is termedComptonised. In this case the plasma must be thin enough that other processes, such as bremsstrahlung, do not dominate the spectrum instead. The hotter the gas, the more chance of Comptonisation. Some examples of astrophysical sources in which comp- tonisation is important are: • hot gas near binary X-ray sources • hot plasma in clusters of galaxies • hot plasma near center of active galactic nuclei • primordial gas cooling after the Big Bang 7.5.1 Non-relativistic Comptonisation We consider non-relativistic electrons and photons with en- ergy h?«mec 2 . From Eqn 7.4 one can show that the rela- tive change in the photon energy ?E/E is given by ?E E = h? mec 2 (1-cos?) (7.13) Problem 7.3 Verify Eqn 7.13 In the electron frame, the scattering is Thomson, and therefore symmetric around the incident direction, so that ?E E = h? mec 2 , for h?«mec 2 . (7.14) This is the average energy increase of the electron for low photon input energies, h?«mec 2 . Now consider high energy photons. In this case the power produced per scattering is given by Eqn. 7.10. For non-relativistic electrons, ?˜ 1, so for electron velocity v, PComp = 4 3 sc v c 2 . (7.15) Thenumberofscatteredphotonspersecondisthenum- berofphotonsencounteredpersecondbyanelectron,which is given by the photon number density, N phot , the photon velocity c and the Thomson cross-section sT N phot csT. (7.16) The photon number density is just N phot = U rad h? (7.17) so we have sTN phot c = sTU rad c h? . (7.18) Comparison of Eqns. 7.15 and 7.18 readily shows that the energy gain of the photons per collision must be ?E E = 4 3 v c 2 for h?»mec 2 . (7.19) Let’s summarise what we have so far: • For h?«mec 2 , the electrons gain energy. (Eqn 7.14). • For h?»mec 2 , the photons gain energy. (Eqn 7.19). 7.5.2 Thermal electrons To make things practical, let’s consider a thermal distribu- tion of electrons, with temperature Te. We have 3 2 kTe = 1 2 mev 2 (7.20) where v is the typical electron velocity. Eqn 7.19 can thus be written ?E E = 4kTe mec 2 for h?«kTe. (7.21) If we now combine the results of Eqns 7.21 and 7.14 we can derive a simple equation for the energy gain/loss for both the high and low frequency regimes ?E E = 1 mec 2 (4kTe-h?). (7.22) Therefore, for • h? = 4kTe, there is no energy exchange • h? > 4kTe, electrons gain energy • h? < 4kTe, photons gain energy Theusualcaseofinterestish? < 4kTe,i.e. theelectrons are “hotter” than the photons. Lecture 7 : Compton Scattering 3 7.6 THE COMPTON PARAMETER Nowconsideraplasmacloudofelectrondensityne andchar- acteristicsizeD.Theopticaldepthte toComptonscattering is just te =nesTD. (7.23) If te»1, the cloud is optically thick, and a large num- ber of scatterings are required for a photon to escape from the cloud. The path taken by the photon to get out of the cloud is termed arandom walk. Figure 7.2 illustrates two possible paths for a photon executing a random walk from the center of an optically thick region until it reaches the edge and escapes. Suppose the photon typically moves a distance l before being scattered. In order to escape from the cloud, the pho- ton must move a distance D where D = v Nl (7.24) where N is the number of scatterings. Problem 7.4 Consider a photon emitted in an in?nite, homogeneous scattering region. Let it travel a distance r1 before the ?rst scatter, a distancer2 before the second scat- ter, and so on. The displacement of the photon R after N scatterings R =r1 +r2 +r3 +...r N (7.25) Show that the mean total displacement D = |R| is of order D = v Nl (7.26) where l is the mean free path. The optical thickness t of the region is of order R/l. Show that the number of scatterings is given by N ˜t 2 . (7.27) The mean free path l is given by l = 1 nesT . (7.28) After one scattering, the initial energy E is increased by ?E, where ?E E = 1+ 4kTe mec 2 (7.29) and so after N scatterings, the energy is E ' where E ' E = 1+ 4kTe mec 2 N . (7.30) We now de?ne y, the called the Compton y-parameter, as y = (Number of scatterings)×(Energy gain/scattering). 7.6.1 Weak Comptonisation Problem 7.5 By assumption, 4kTe « mec 2 . Use this to show that Figure 7.2. Compton scattering of a photon in a mildly optically thick region. The photon begins at the central dot and executes a random walk until it reaches the edge of the cloud and escapes. A shorter and a longer random walk are shown. E ' E =e 4y . (7.31) This is the energy gain for weak Comptonisation, in which the photon energy remains small relative to the electron energy. 7.6.2 Strong Comptonisation In strong Comptonisation, the photon energy is increased until the electron and photon energy distributions approach equilibrium, i.e. there is further no net energy gain by one population to the other. In this case the photons are “heated” to such a temperature that h? =4kTe. (7.32) Lettheopticalthicknessnecessaryforthistohappenbe t. If the medium is optically thick, then a large number of scatterings are required for photons to escape. The number of scatterings N is given by Eqn 7.27, and it thus follows from Eqn 7.31 that 4kTe h?0 = exp h 4 kTe mec 2 t 2 i . (7.33) In this case the medium will be strongly Comptonised, and the photon spectrum will approach an equilibrium form given by Page 4 Astrophysics II, University of Turku 16 January 2009 Lecture 7 : Compton Scattering 7.1 INTRODUCTION Thomsonscattering,orthescatteringofaphotonbyanelec- tron at rest, strictly only applies at low photon energy, i.e. when h?«mec 2 . If the photon energy is comparable to or greater than the electron energy, non-classical e?ects must be taken into account, and the process is called Compton scattering. A further interesting situation develops when the electron is moving — in this case energy can be transferred to the photon, and the process is called inverse Compton scat- tering.Thislastprocessisanimportantmechanisminhigh energy astrophysics. 7.2 THOMSON SCATTERING In Thomson scattering, we have dsT dO = 1 2 r 2 0 (1+cos 2 ?) (7.1) where sT is the Thomson cross-section, O the solid angle, ? is the angle of scattering, and r0 is the classical electron radius, r0 = e 2 mec 2 . (7.2) In Thomson scattering the incident photon and scatter photon have the same wavelength or energy, so this scatter- ing is also called coherent or elastic. If we now move to photons of energy h? > ~ mec 2 , the scattering is modi?ed by the appearence of quantum e?ects, through a change in the kinematics of the collision, and an alteration of the cross-section. 7.3 COMPTON SCATTERING Todothekinematicsofthecollisioncorrectlyathighphoton energy, momentum and energy must be conserved. Lettheincidentphotonhaveenergyh? andmomentum h?/c, the scattered photon have energy h? ' and momentum h? ' /c, and the electron (initially at rest) acquires energy E and momentum pe. The scattering angle is ?. Problem7.1 Showthattheenergyofthescatteredphoton is given by h? ' = h? 1+ h? mec 2 (1-cos?) (7.3) In terms of wavelength, this reduces to ? ' -? =?C(1-cos?) (7.4) Figure 7.1. Compton scattering of an incident photon of energy h? and momentum p to energy h? ' and momentum p ' . The elec- tron is initially at rest and acquires energy E and momentum pe. where ? is the incident photon wavelength, ? ' is the scat- tered photon wavelength and ?C is the Compton wave- length and is given by ?C = h mec =0.02426 ° A. (7.5) Problem 7.2 Prove Eqn 7.4. The compton wavelength can be regarded as a wave- length change ?? in the incident photon. Note that for ?»?C the change is negligible and we get back the Thom- son scattering. In full treatment of the problem yields the Klein- Nishina formula for the scattering cross-section: ds dO = r 2 0 2 ? ' ? 2 ? ? ' + ? ' ? -sin 2 ? (7.6) which can be shown to yield the following formulae for the total cross-section (where x = h? mc 2 ), s˜sT 1-2x+ 26x 2 5 +... , for x« 1, (7.7) and s = 3 8 sT 1 x ln(2x)+ 1 2 , for x»1 (7.8) for the non-relativistic and extremely relativistic cases. The main e?ect is thus to reduce the cross-section at high pho- ton energies, i.e. the scattering of the photons becomes less e?cient. 7.4 INVERSE COMPTON: SCATTERING FROM MOVING ELECTRONS An important case arises when the electrons are no longer considered to be at rest. In inverse scattering, energy is transferred from the electrons to the photons, i.e. it is the oppositeofComptonscattering,inwhichthephotonstrans- fer energy to the electrons. Inverse Compton scattering can 2 produce substantial ?uxes of photons in the optical to X- ray region. Analysis shows that the mean frequency of the photons after the collision increases by a factor ? 2 , so that high frequency radio photons in collisions with relativistic electrons for which ? is of order 10 3 to 10 4 can beboosted in the UV and X-ray regions. There is a practical limit to the amount of boosting possible beyond the Thomson limit (h? ˜ ?mc 2 ), which can be seen from the conservation of energy h? ' =?mc 2 +h?. (7.9) Scattered photon energies are thus limited to ?mc 2 . The power emitted in the case of an isotropic distribu- tion of photons is PComp = 4 3 scU rad ? 2 ß 2 (7.10) whereU rad istheradiationenergydensityofthephoton?eld (before scattering). Note how similiar this is the power due to synchrotron emission P Synch = 4 3 scUB? 2 ß 2 (7.11) where UB is the energy density of the magnetic ?eld. Thus P Synch PComp = UB U rad . (7.12) The losses due to synchrotron and Compton processes are in the ratio of the magnetic ?eld energy density to the photon ?eld energy density, and is independent of ?. The scattered photons may be produced in the source throughsynchrotronradiation,andifthesearebosstedthen theresultantphotonsarecalledSynchrotronSelfComp- ton. 7.5 COMPTONISATION If the spectrum of a source is primarily determined by Compton processes it is termedComptonised. In this case the plasma must be thin enough that other processes, such as bremsstrahlung, do not dominate the spectrum instead. The hotter the gas, the more chance of Comptonisation. Some examples of astrophysical sources in which comp- tonisation is important are: • hot gas near binary X-ray sources • hot plasma in clusters of galaxies • hot plasma near center of active galactic nuclei • primordial gas cooling after the Big Bang 7.5.1 Non-relativistic Comptonisation We consider non-relativistic electrons and photons with en- ergy h?«mec 2 . From Eqn 7.4 one can show that the rela- tive change in the photon energy ?E/E is given by ?E E = h? mec 2 (1-cos?) (7.13) Problem 7.3 Verify Eqn 7.13 In the electron frame, the scattering is Thomson, and therefore symmetric around the incident direction, so that ?E E = h? mec 2 , for h?«mec 2 . (7.14) This is the average energy increase of the electron for low photon input energies, h?«mec 2 . Now consider high energy photons. In this case the power produced per scattering is given by Eqn. 7.10. For non-relativistic electrons, ?˜ 1, so for electron velocity v, PComp = 4 3 sc v c 2 . (7.15) Thenumberofscatteredphotonspersecondisthenum- berofphotonsencounteredpersecondbyanelectron,which is given by the photon number density, N phot , the photon velocity c and the Thomson cross-section sT N phot csT. (7.16) The photon number density is just N phot = U rad h? (7.17) so we have sTN phot c = sTU rad c h? . (7.18) Comparison of Eqns. 7.15 and 7.18 readily shows that the energy gain of the photons per collision must be ?E E = 4 3 v c 2 for h?»mec 2 . (7.19) Let’s summarise what we have so far: • For h?«mec 2 , the electrons gain energy. (Eqn 7.14). • For h?»mec 2 , the photons gain energy. (Eqn 7.19). 7.5.2 Thermal electrons To make things practical, let’s consider a thermal distribu- tion of electrons, with temperature Te. We have 3 2 kTe = 1 2 mev 2 (7.20) where v is the typical electron velocity. Eqn 7.19 can thus be written ?E E = 4kTe mec 2 for h?«kTe. (7.21) If we now combine the results of Eqns 7.21 and 7.14 we can derive a simple equation for the energy gain/loss for both the high and low frequency regimes ?E E = 1 mec 2 (4kTe-h?). (7.22) Therefore, for • h? = 4kTe, there is no energy exchange • h? > 4kTe, electrons gain energy • h? < 4kTe, photons gain energy Theusualcaseofinterestish? < 4kTe,i.e. theelectrons are “hotter” than the photons. Lecture 7 : Compton Scattering 3 7.6 THE COMPTON PARAMETER Nowconsideraplasmacloudofelectrondensityne andchar- acteristicsizeD.Theopticaldepthte toComptonscattering is just te =nesTD. (7.23) If te»1, the cloud is optically thick, and a large num- ber of scatterings are required for a photon to escape from the cloud. The path taken by the photon to get out of the cloud is termed arandom walk. Figure 7.2 illustrates two possible paths for a photon executing a random walk from the center of an optically thick region until it reaches the edge and escapes. Suppose the photon typically moves a distance l before being scattered. In order to escape from the cloud, the pho- ton must move a distance D where D = v Nl (7.24) where N is the number of scatterings. Problem 7.4 Consider a photon emitted in an in?nite, homogeneous scattering region. Let it travel a distance r1 before the ?rst scatter, a distancer2 before the second scat- ter, and so on. The displacement of the photon R after N scatterings R =r1 +r2 +r3 +...r N (7.25) Show that the mean total displacement D = |R| is of order D = v Nl (7.26) where l is the mean free path. The optical thickness t of the region is of order R/l. Show that the number of scatterings is given by N ˜t 2 . (7.27) The mean free path l is given by l = 1 nesT . (7.28) After one scattering, the initial energy E is increased by ?E, where ?E E = 1+ 4kTe mec 2 (7.29) and so after N scatterings, the energy is E ' where E ' E = 1+ 4kTe mec 2 N . (7.30) We now de?ne y, the called the Compton y-parameter, as y = (Number of scatterings)×(Energy gain/scattering). 7.6.1 Weak Comptonisation Problem 7.5 By assumption, 4kTe « mec 2 . Use this to show that Figure 7.2. Compton scattering of a photon in a mildly optically thick region. The photon begins at the central dot and executes a random walk until it reaches the edge of the cloud and escapes. A shorter and a longer random walk are shown. E ' E =e 4y . (7.31) This is the energy gain for weak Comptonisation, in which the photon energy remains small relative to the electron energy. 7.6.2 Strong Comptonisation In strong Comptonisation, the photon energy is increased until the electron and photon energy distributions approach equilibrium, i.e. there is further no net energy gain by one population to the other. In this case the photons are “heated” to such a temperature that h? =4kTe. (7.32) Lettheopticalthicknessnecessaryforthistohappenbe t. If the medium is optically thick, then a large number of scatterings are required for photons to escape. The number of scatterings N is given by Eqn 7.27, and it thus follows from Eqn 7.31 that 4kTe h?0 = exp h 4 kTe mec 2 t 2 i . (7.33) In this case the medium will be strongly Comptonised, and the photon spectrum will approach an equilibrium form given by 4 Figure 7.3. Comptonisation of photons for various values of the potential, µ. From top to bottom the curves show the spectrum for µ = 0,0.1,0.3.1.0,3.0 and 10.0. Note that µ = 0 is the pure black-body curve. u? = 8ph? 3 c 3 h exp h? kT +µ i -1 . (7.34) This form, for µ = 0, is just the Planck (or blackbody) spectrum. When the equilibrium state involves two species (photons and electrons), then we can introduce a chemical potential between them, µ. The e?ect on the Planck spec- trum is illustrated in ?gure 7.3 for various potentials µ. For µ» 1, the spectrum at high frequency approaches the Wien spectrum with an extra factor of e -µ , u? = 8ph? 3 c 3 e -µ e - h? kT (7.35) and at low frequency u? ?? 3 . (7.36) Figure 7.4 illustrates strong comptonisation of a bremsstrahlung spectrum in an optically thick, non- relativistic medium. The bremsstrahlung spectrum dom- inates at low frequency and shows a characteristic self- absorption region (I? ? ? 2 ) and a ?at region (I? ? ? 0 ). At higher frequency, photons have been multiply scattered via the Compton process so that a Wien spectrum forms (I? ?? 3 e -h?/kT ). 7.7 KOMPANEETS EQUATION In the previous section we derived forms for the spectrum when the amount of Comptonisation in a non-relativistic mediumisweakandstrong.Whataboutintermediatecases? The equation which describes these cases was derived by Kompaneets in 1949. Its derivation is non-trivial. The equa- tioninvolvestheevolutionofthedistributionofthephotons in phase space, n, ?n ?y = 1 x 2 ? ?x h x 4 n+n 2 + ?n ?x i (7.37) Figure7.4.SaturatedComptonisationofabremsstrahlungspec- trum. See text for details. where x = h? kT (7.38) and the compton y-parameter is generalised to an integral along the photon path y = Z kTe mec 2 sTnedl (7.39) and where the terms in the square brackets represent • the increase/decrease in photon numbers in frequency space, (the ?n ?x term) • cooling of photons/electron recoil, (the n term) • and the cooling due to induced Compton scattering, (the n 2 term) In general, solutions to the Kompaneets equation must be found numerically, although there are some useful limit- ing cases which can be solved analytically.Read More

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