Hale COLLAGE 2017 Lecture 20 Radiative Processes in Flares I: Bremsstrahlung

Bin Chen (New Jersey Institute of Technology) 8/3/10

The standard flare model

No. 2, 1997 ELECTRON DENSITIES IN SOLAR FLARES 837

Escaping particles Upward Beams III

n acc=3 109 cm-3 e !III=500 MHz Acceleration Site FREQUENCY RS

Downward Beams !dm,1=1 GHz

Trapped particles MW dm,1 10 -3 ne =10 cm DCIM Chromospheric SXR Evaporation Front SXT 11 -3 dm,2 10 -3 n =10 cm ne =5 10 cm dm,2 e ! =2 GHz Precipitating TIME particles HXR HXR

FIG. 10.ÈDiagram of a Ñare model envisioning magnetic reconnectionAschwandenand chromospheric& Benz 1997 evaporation processes in the context of our electron density measurements. The panel on the right illustrates a dynamic radio spectrum with radio bursts indicated in the frequency-time plane. The acceleration site is located in a low-density region (in the cusp) with a density of neacc B 109 cm~3 from where electron beams are accelerated in upward (type III) and downward (RS bursts) directions. Downward-precipitating electron beams that intercept the chromospheric evaporation front with density jumps over neupflow \ (1È5) ] 1010 cm~3 can be traced as decimetric bursts with almost innite drift rate in the 1È2 GHz range. The SXR-bright Ñare loops completely lled up by evaporated plasma have somewhat higher densities of neSXR B 1011 cm~3. The chromospheric upÑow lls loops subsequently with wider footpoint separation while the reconnection point rises higher. 6 region. The type III and RS bursts identied here occur This result has dramatic consequences for the location of during the main impulsive Ñare phase, and their detailed the acceleration site. The extremely low density ratio in the correlation with HXRs was established in several recent acceleration site found in all Ñares, without exception, studies (e.g., Aschwanden et al. 1993, 1995). It was not leaves no room to place the acceleration site inside the known in earlier studies whether the start frequency of SXR-bright Ñare loop (assuming a lling factor near unity). metric type III bursts would be signicantly displaced to If we were to allow for harmonic plasma emission or for lower frequencies than the plasma frequency of the acceler- nonunity lling factors of the SXR loop, the density ratio ation region, because a minimal propagation distance is between the acceleration site and the SXR loop would be needed for electron beams to become unstable (Kane, Benz, even more extreme. Therefore we see no other possibility & Treumann 1982). However, recent studies of the starting than to conclude that the acceleration site is located outside point of combined upward- and downward-propagating the SXR-bright Ñare loop. The next question is the magnetic beam signatures (see bidirectional type III ] RS burst pairs topology that can accommodate such large density gra- in Aschwanden et al. 1995, 1993; Klassen 1996) demon- dients (D2È5 density scale heights). A possible topology is a strated that the centroid position of the acceleration region cusp-shaped magnetic eld geometry above the SXR-bright is close to the start frequency of strong type III bursts. The Ñare loop, where acceleration is assumed to take place only major difficulty with this scenario is the observed beneath the X- or Y-type magnetic reconnection point (see asymmetry of electron numbers accelerated in the upward/ diagram in Fig. 10; for a detailed physical model, see downward direction, which was inferred to be as low as Tsuneta 1996). The magnetic eld lines that connect the 10~2 to 10~3, comparing the electrons detected in inter- cusp with the footpoints can have arbitrarily lower densities planetary space with those required to satisfy the chromo- than the encompassed closed eld lines that have been lled spheric thick-target HXR emission (Lin 1974). It is not clear by evaporated plasma. Of course, the high density gradient whether this asymmetry can be explained by the dominance between the acceleration site and the lled SXR-bright Ñare of closed magnetic eld lines above acceleration regions. In loops can only be maintained in a dynamical process in the following discussion, we associate the start frequency lIII which the reconnection point proceeds to higher altitudes of type III bursts with the electron density neacc in the accel- (Kopp & Pneuman 1976) before chromospheric evapo- eration region. The range of type III start frequencies (220È ration has lled up the cusp volume. This race of the recon- 910 MHz) measured here is found to be nearly identical nection point with the evaporation front in the upward with that (270È950 MHz) of 30 bidirectional III ] RS burst direction may come to a halt in long-duration Ñares, where pairs analyzed in Aschwanden et al. (1995). Comparing the cusp volume becomes clearly lled up (Tsuneta et al. these densities in the acceleration region with those in the 1992; Forbes & Acton 1995). SXR-bright Ñare loop, we nd very low ratios of This Ñare scenario, in which acceleration takes place in a neacc/neSXR \ 0.007È0.127 (with a median of 0.027), i.e., the low-density region above the much denser SXR-bright Ñare density in the acceleration region is 1È2 orders of magnitude loop, would predict a physical separation between non- lower than in the SXR-bright Ñare loop. thermal and thermal electrons. While the SXR-bright Ñare 1) Magnetic reconnection and energy release 2) Particle acceleration and heating 3) Chromospheric evaporation and e- magnetic loop heating reconnection Previous lectures

Following lectures: e- How to diagnose the accelerated particles? • What? • Where? How?

Shibata et al. 1995 • When? Outline • Introduction • Radiation from energetic particles • Bremsstrahlung à this lecture • Gyrosynchrotron • Other radiative processes (time permitting) • Coherent emission • Inverse Compton • Nuclear processes • Suggested reading: Ch. 5 of Tandberg- Hanssen & Emslie Thermal and non-thermal radiation

• Refer to the distribution function of source particles �(�) (# of particles per unit energy per unit volume) • Radiation from a Maxwellian particle distribution is referred to as thermal radiation • Radiation from a non-Maxwellian particle distribution is referred to as nonthermal radiation • In flare physics, the nonthermal population we consider usually has much larger energies than that of the thermal “background” • Example of a nonthermal distribution function: � � �� = �� ���, where � is the normalization factor to make ∫ � � �� = � à “power law” *See Lecture 17 (by Prof. Longcope) for details on the distribution function 8/3/10

8/3/10

Radio'Emission' Radio'Emission' .'Thermal' bremsstrahlung'(eAp)' .' Gyrosynchrotron'radia5on'(eAB)' .'Thermal' bremsstrahlung'(eAp)' .'Plasma'radia5on'( wAw)' .' Gyrosynchrotron'radia5on'(eAB)' .'Plasma'radia5on'( wAw)'

Radiation from an accelerated charge LarmorLarmor'formulae:'radia5on'from'an'accelerated'charge'''formulae:'radia5on'from'an'accelerated'charge''

dP q2 2q2 dP = q2 a 2 sin2 θ P = a22q2 Larmor formula: dΩ =4πc 3 a 2 sin2 θ P3c=3 a 2 dΩ 4πc 3 3c 3 Rela5vis5c'Larmor'formulae'' Rela5vis5c'Larmor'formulae'' Relativistic Larmor€ formula: € € € θ" θ"

Cases'relevant'to'radio'and'HXR/gammaAray'emission:''

Accelera2on#experienced#in#the#Coulomb#field:'bremsstrahlung' Cases'relevant'to'radio'and'HXR/gammaAray'emission:'' Radio and HXR/gammyAccelera2on#experience#in#a#magne2c#field:#-ray emission in flares: gyromagne5c'radia5on' • Acceleration experienced in the Coulomb field: Accelera2on#experienced#in#the#Coulomb#fieldbremsstrahlung:'bremsstrahlung' • Acceleration experienced in a magnetic field: Accelera2on#experience#in#a#magne2c#field:#gyromagnetic radiationgyromagne5c'radia5on'

10

10 8/3/10

ElectronAion'Bremsstrahlung''

e- ν"

+Ze

Electron-ion bremsstrahlung

.'At'radio'wavelengths,' thermal'bremsstrahlung'or'thermal'freeAfree' radia5on'is'relevant'to'the'quiet'Sun,'flares,'and'CMEs' .'Nonthermal'bremsstrahlung' is'relevant'to'HXR/gammaAray'bursts' • e-i bremsstrahlung is relevant to quiet Sun, flares, and CMEs

11 Bremsstrahlung

• Each electron-ion interaction generates a single pulse of radiation

-e

Ze

Strong interaction Weak interaction 4/2/2017 4 Free–Free Radiation‣ Essential Radio Astronomy

Figure 4.4: The actual power spectrum of the electromagnetic pulse generated by one electron–ion interaction is nearly flat up to frequency , where is the electron speed and is the impact parameter, and declines at higher frequencies. The approximation for all and at higher frequencies (dashed line) is quite good at radio frequencies . Power spectrum of a single interaction The typical electron speed in a K HII region is and the minimum impact parameter is cm, so Hz, much higher than radio frequencies. In the approximation that the power spectrum is flat out to and zero at higher frequencies (dashed line in Figure 4.4), the average energy per unit frequency emitted during a single interaction is approximately Weak interaction

(4.25) which simplifies to Single pulse duration �~�/� Emitted energy

(4.26)

4.3.2 Radio Radiation From an HII Region

http://www.cv.nrao.edu/~sransom/web/Ch4.html 11/23 A note on the impact parameter

• Maximum value �

/ v Debye Length � = = �/�, Where � is the thermal

speed and � = is the plasma frequency

Why this length scale sets �? v � ≈ , where � is the observing frequency

• Minimum value �

v � ≈ , given by maximum possible momentum change of the electron ∆� = 2�

ℏ v� = , from uncertainty principle 4/2/20174/2/2017 4 4Free–Free Free–Free Radiation Radiation‣ Essential‣ Essential Radio Radio Astronomy Astronomy The strengthThe strength and and spectrum spectrum of of radio radio emission emission fromfrom anan H HIIII region region depends depends on theon thedistributions distributions of electron of electron velocities velocities and collision and collisionimpact impact 4/2/2017 parametersparameters (Figure (Figure 4.24.2). The). The distribution distribution ofof dependsdepends4 Free–Freeon on the the electron Radiation electron ‣temperature Essential temperature Radio Astronomy. The . distributionThe distribution of depends of depends on the electron on the electron The strengthnumbernumber densityand density spectrum (cm (cm of) radio and) and the emission the ion ion number number from density an HII region (cm (cm depends). ). on the distributions of electron velocities and collision impact parameters (Figure 4.2). The distribution of depends on the electron temperature . The distribution of depends on the electron number densityIn LTE,In L TE,the (cm theaverage average) and kinetic thekinetic ion ener ener numbergiesgies ofof density electronselectrons and and(cm ions ions ).are are equal. equal. The The electrons electrons are much are much less massive, less massive, so their sospeeds their are speeds much are much higher and the ions can be considered nearly stationary during an interaction (Figure 4.5). higher and the ions can be considered nearly stationary during an interaction (Figure 4.5). In LTE, the average kinetic energies of electrons and ions are equal. The electrons are much less massive, so their speeds are much higher and the ions can be considered nearly stationary during an interaction (Figure 4.5).

Figure 4.5: The number of electrons with speeds to passing by a stationary ion and having impact parameters in the range to during the time interval equals the number of electrons with speeds to in the cylindrical shell shown here. Figure 4.5: The number of electrons with speeds to passing by a stationary ion and having impact parameters in the range to duringThe number the time of electrons interval passing equals any the ion number per unit oftime electrons with impact with parameter speeds to to inand the speed cylindrical range to shell shown is here. 4/2/2017 4 Free–Free Radiation‣ Essential Radio Astronomy Substituting the results for (Equation 4.26) and (Equation 4.28) into Equation 4.29 gives Figure 4.5:The The number number of electrons of electrons passing with any speeds ion per to unit time passingwith impact by a parameter stationary ion to and having and speed impact range parameters to in the is range to during the time interval equals the numberBremsstrahlung emissivity of electrons with speeds to in the cylindrical shell shown here.(4.27) (4.30) The wherenumber of is electrons the normalized passing ( any ion per•) speed # of electrons passing by any ion within unit timedistribution with impactof the electrons. parameter The number to and of speed( such�, � collisionsrange+ �� ) toper unit volume is (4.27) per unit time is and (�, � + ��) (4.31) where is the normalized ( ) speed distribution of the electrons. The number of such collisions per unit volume per unitEquation time is 4.31 exposes a problem: the integral (4.27) • # of collisions per unit volume per unit time (4.28) where The is spectralthe normalized power at frequency ( emitted) speedisotropically distribution per unit ofvolume the electrons. is , where The number is the emission coefficient of such defined collisions by per unit volume per unit timeEquation is 2.26. Thus (4.32) (4.28) • Spectral power emitted at � The spectraldiverges power logarithmically at frequency. There mustemitted be finite isotropically physical limits per unit volumeand is(to be determined), where onis thethe rangeemission of the coefficient impact parameter defined that by Equation 2.26. Thus prevent this divergence: (4.29) (4.28)

The spectral power at frequency emitted isotropically per unit volume is , where is the emission coefficient defined by Equation http://www2.26.cv.nrao.edu/~sransom/web/Ch4.html. Thus (4.33) 12/23(4.29)

The distribution of electron speeds in LTE is the nonrelativistic Maxwellian distribution (see Appendix B.8 for its http://www.cv.nrao.edu/~sransom/web/Ch4.htmlderivation): 12/23 (4.29)

http://www.cv.nrao.edu/~sransom/web/Ch4.html 12/23

Figure 4.6: The nonrelativistic Maxwellian distribution of particle speeds in LTE (Equation 4.34), where is the rms speed of particles with mass at temperature . (4.34) http://www.cv.nrao.edu/~sransom/web/Ch4.html 13/23 4/2/2017 4 Free–Free Radiation‣ Essential Radio Astronomy

Equation 4.34 can be used to evaluate the integral over the electron speeds in Equation 4.33:

(4.35) Thermal bremsstrahlung Substituting so gives • Distribution function of the electron �(�) is Maxwellian (4.36)

(4.37) 4/2/2017 4 Free–Free Radiation‣ Essential Radio Astronomy

(4.38)

In conclusion, the free–free emission coefficient can be written as Equation 4.34 can be used to evaluate the integral over the electron speeds in Equation 4.33:

(4.35) Emission coefficient: (4.39)

Substituting so gives

The remaining problem is to estimate the minimum and maximum impact parameters and . These estimates don’(4.36)t have to be very precise because only their logarithms appear in Equation 4.39. (4.37) To estimate the minimum impact parameter , notice that the net impulse (change in momentum) during a single electron–ion interaction (4.38)

In conclusion, the free–free emission coefficient can be written as

(4.40) (4.39)

The remaining problem is to estimate the minimum and maximum impact parameters and . These estimates don’t have to http://www.cv.nrao.edu/~sransom/web/Ch4.html be very precise because only their logarithms appear in Equation 4.39. 14/23

To estimate the minimum impact parameter , notice that the net impulse (change in momentum) during a single electron–ion interaction

(4.40)

http://www.cv.nrao.edu/~sransom/web/Ch4.html 14/23 Thermal bremsstrahlung 4/2/2017 4 Free–Free Radiation‣ Essential Radio Astronomy

4/2/2017 • Absorption coefficient:4 Free–Free Radiation‣ Essential Radio Astronomy 4/2/2017 4 Free–Free Radiation‣ Essential Radio Astronomy (4.51) (4.51) (4.51) in the Rayleigh–Jeans limit. Thus in the Rayleigh–Jeans limit. Thus• Let’s first consider radio wavelengths. In the in the Rayleigh–Jeans limit. Thus Rayleigh-Jeans regime � � = , so (4.52) (4.52) (4.52)

The limit The(Equation limit 4.48) (Equation is inversely 4.48proportional) is inversely to frequency proportional so the absorption to frequency coefficient so the is absorption not exactly coefficient proportional is to not exactly. A proportional to . A The limitgood numerical (Equation good approximation numerical4.48) is inversely is approximation proportional. is to frequency so. the absorption coefficient is not exactly proportional to . A good numerical approximation is . The total opacityThe of total an HopacityII region is of the an integral HII region of is along the integral the line of sight, along as illustrated the line in of Figure sight, 4.7 as: illustrated in Figure 4.7: The total opacity of an HII region is the integral of along the line of sight, as illustrated in Figure 4.7:

Figure 4.7: Astronomers often approximate HII regions by uniform cylinders whose axis is the line of sight because this gross Figure oversimplification4.7: Astronomers finesses often theapproximate radiative-transfer HII regi problem.ons by uniform It is for cylinders good reason whose that axis astronomers is the line often of sightfeature because in jokes this beginning gross oversimplification Figure finesses 4.7:the radiative-transferAstronomers often problem. approximate It is for HgoodII regi reasonons bythat uniformastronomers cylinders often featurewhose inaxis jokes is beginningthe line of sight because this gross http://www.cv.nrao.edu/~sransom/web/Ch4.htmloversimplification finesses the radiative-transfer problem. It is for good reason that astronomers often feature18/23 in jokes beginning http://www.cv.nrao.edu/~sransom/web/Ch4.html 18/23 http://www.cv.nrao.edu/~sransom/web/Ch4.html 18/23 Thermal bremsstrahlung opacity 4/2/2017 4 Free–Free Radiation‣ Essential Radio Astronomy “Consider4/2/2017 a spherical cow….” 4 Free–Free Radiation‣ Essential Radio Astronomy “Consider a spherical cow….”• Considering a spherical cow…

(4.53)(4.53) • At low frequencies, � ≫ 1, optically thick: At frequenciesAt frequencies low enough low enough that that , the H, theII region HII region becomes becomes opaque, opaque, its its spectrum spectrum approaches that that of ofa blackbodya blackbody with with brightness brightness temperaturetemperature approaching approaching the electron the electron temperature temperature ( ( K), K), and and its its flux flux densitydensity obeys obeys the the Rayleigh–Jeans Rayleigh–Jeans approximation approximation � ≈ � � = ∝ � . At very .high At very frequencies, high frequencies, , the H, IIthe region HII region is nearly is nearly transparent, transparent, and and • At high frequencies, � ≪ 1, optically thin:

(4.54) (4.54)

On a log-log plot, the overall spectrum of a uniform HII region looks like Figure 4.8, with the spectral break corresponding to the On a log-logfrequency plot, theat which overall spectrum. of a uniform HII region looks like Figure 4.8, with the spectral break corresponding to the frequency at which .

Figure 4.8: The radio spectrum of an HII region. It is a blackbody at low frequencies, with slope 2 if a uniform cylinder as shown in Figure 4.7 and otherwise. At some frequency the optical depth , and at much higher frequencies the spectral slope

Figure http://www4.8:.cv.nrao.edu/~sransom/web/Ch4.html The radio spectrum of an HII region. It is a blackbody at low frequencies, with slope 2 if a uniform cylinder as shown in19/23 Figure 4.7 and otherwise. At some frequency the optical depth , and at much higher frequencies the spectral slope

http://www.cv.nrao.edu/~sransom/web/Ch4.html 19/23 4/2/2017 4 Free–Free Radiation‣ Essential Radio Astronomy “Consider a spherical cow….”

(4.53)

At frequencies low enough that , the HII region becomes opaque, its spectrum approaches that of a blackbody with brightness temperature approaching the electron temperature ( K), and its flux density obeys the Rayleigh–Jeans approximation . At very high frequencies, , the HII region is nearly transparent, and

(4.54) Thermal bremsstrahlung spectrum in radio On a log-log plot, the overall spectrum of a uniform HII region looks like Figure 4.8, with the spectral break corresponding to the frequency at which .

Figure 4.8: The radio spectrum of an HII region. It is a blackbody at low frequencies, with slope 2 if a uniform cylinder as shown in Figure 4.7 and otherwise. At some frequency the optical depth , and at much higher frequencies the spectral slope

http://www.cv.nrao.edu/~sransom/web/Ch4.html 19/23 Emission measure

4/2/2017 • Optical depth is essential in thermal 4 Free–Free Radiation‣ Essential Radio Astronomy “Consider a spherical cow….” bremsstrahlung

(4.53)

At frequencies low enough that , the HII region becomes opaque, its spectrum approaches that of a blackbody with brightness temperature approaching the electron• Assuming constant temperature ( � K), and its flux density obeys the Rayleigh–Jeans approximation . At very high frequencies, , the HII region is nearly transparent,. and. � ≈ � � (��)

Column emission measure ��� (4.54)

On a log-log plot, the overall spectrum of a uniform HII region looks like Figure 4.8, with the spectral break corresponding to the frequency at which .

Figure 4.8: The radio spectrum of an HII region. It is a blackbody at low frequencies, with slope 2 if a uniform cylinder as shown in Figure 4.7 and otherwise. At some frequency the optical depth , and at much higher frequencies the spectral slope

http://www.cv.nrao.edu/~sransom/web/Ch4.html 19/23 Recap from Lecture 7: Radiative Transfer Equation

• Brightness temperature � = � � = ��

• Effective temperature � = ��

• Using our definitions of brightness temperature and effective temperature, the transfer equation can be rewritten

�� = −� + � ��

• Optically thick source, � ≫ 1, � ≈ � • Optically thin source, � ≪ 1, � ≈ �� Radio Spectral Diagnostics 83 Radio Spectral Diagnostics 77 � spectrum Thermal B C A Bremsstrahlung • Optically thin regime

� = �(1 − � ) ≈ �� . . ? ≈ � � �� ? • Optically thick ?

� = �(1 − � ) ≈ T

From Problem Set #2 Figure 4.2. A “Universal Spectrum" for free-free brightness temperature. The solid line shows the spectrum for a 106 K corona and 104 K chromosphere. The dashed line showsFigure the 4.6. spectrum“Universal Spectra" for gyrosynchrotron emission, for a homogeneous source with for the coronal contribution alone. thermal (left panel) or power-law (right panel) electron energy distributions. The solid lines show the x-mode spectra, while the dashed lines schematically show the o-mode spectra. Thermal spectra are distinguished by their flat optically thick slope, and very steep optically thin slope. ture and density than assuming a single-temperature corona. The fact that the optically thick part of the spectrum gives directly the electron temperature as a function of frequency, and hence height, can be used to determine7. the Exotic LOS Mechanisms variation. In fact, Grebinskij et al. (2000; see also Chapter 6) show that precise measurement of the spectral slope and the degree of polarization areIn sufficient addition to to these standard mechanisms, others have been proposed to ex- determine the longitudinal component of B. Figure 4.3 demonstratesplain certain that this kinds of observed emission. Here we briefly mention the Electron simple technique works amazingly well, at least for model data.Cyclotron The relatively Maser (ECM) mechanism and Transition Radiation. low degree of polarization, and the need for smooth and accurateECM brightness (Holman et al. 1980; Melrose & Dulk 1982) is expected to operate in temperature spectra, make this a challenging but rewarding observationalconvergent ap- magnetic fields in the legs of loops, where downward moving elec- plication for FASR. trons escape the magnetic trap. The remaining particles form an anisotropic 5. Gyroresonance Diagnostics pitch angle distribution, which is unstable to ECM emission. The coherent emis- sion occurs in clusters of short duration ( 10 ms), narrowband ( 10MHz), The free-free emission diagnostics can be used everywhere in the solar at- 12 ª ª mosphere that the magnetic field is not too strong (B 100 G).high However, brightness in ( 10 K) bursts called spike bursts. Too little is known about ∑ the spatial and spectralª characteristics of ECM emission to develop spectral diagnostics (in fact, there is uncertainty whether spike bursts are due to ECM or other wave instabilities—see Chapter 10), and it may be that any such di- agnostics would relate only to the detailed microphysics in the source region. Accounting for the generation and escape of the radiation may yield constraints on the surrounding plasma, however. Bremsstrahlung for X-rays

Single pulse duration �~�/�

-e

Ze

Strong interaction

• Higher electron energy à larger v • Closer encounters à smaller b Introducing the Cross Section

• The cross section � is defined as if the radiation all comes from the impact within an area around a target ion • # of electrons that encounter a single target in �� (assume all e have the same speed and emit a photon at the same wavelength): ����� • # of photons produced per unit volume per unit time: � � � � � • Photon flux at Earth (cm-2 s-1 per unit energy) if the incident electron population remains roughly unchanged, or a “thin target” scenario: �����/(4�� ) � = 1 ��

∫ ≈ ≈ ���/(4�� )�� Differential Cross Section

In fact, � depends on: • Incident electron energy � • Outgoing photon energy �, • Outgoing photon direction Ω

We need a differential cross section: � �/����, written as � �, �, Ω 30 contrast, for emissions in the HXR range ( 10–100 keV), non-thermal bremsstrahlung 30 ⇠ contrast, for emissions in the HXR range ( 10–100 keV), non-thermal bremsstrahlung radiation is the dominant process under most circumstances,⇠ for which the source radiation is the dominant process under most circumstances, for which the source 4/2/2017 electrons are accelerated by4 theFree–Free flare Radiation energy‣ Essential release Radio Astronomy to an average energy E¯ E¯ e i Substituting the results for (Equation electrons4.26) areand accelerated (Equation by the4.28 flare) into energy Equation release4.29 togives an average energy E¯ E¯ often having a power-law energy distribution. e i often having a power-law energy distribution. For a uniform plasma with a temperature T and an ion density ni distributed in a (4.30) For a uniform plasma with a temperature T and an ion density ni distributed in a volume V , the thermal X-ray flux Fthermal(✏) received by an X-ray detector (photons volume V , the thermal X-ray flux Fthermal(✏) received by an X-ray detector (photons(4.31) 2 1 1 cm s keV )atadistanceR from the source can be obtained by integrating over 2 1 1 Equation 4.31 exposes a problem: the integralcm s keV )atadistanceR from the source can be obtained by integrating over all particles times the di↵erential bremsstrahlung cross-section (✏,E) all particles times the di↵erential bremsstrahlung cross-section (✏,E)

niV 1 (4.32) F (✏)= f (En)viV(E)1(✏,E)dE (1.13) thermal F 2 (✏)=e e f (E)v (E)(✏,E)dE (1.13) 4⇡Rthermal✏ 4⇡R2 e e Z Z✏ diverges logarithmically. There mustBremsstrahlung cross section be finite physical limits and (to be determined) on the range of the impact parameter that 3 1 prevent this divergence:where fe(E)(electronscmwhere f (E)(electronscmkeV )isthedi3 keV↵erential1)isthedi electron↵erential density electron distribution density distributionat at • Bremsstrahlung from weak interactionse electron energyelectron of E, ( energy✏,E)isthephysicalparameterdescribingthee of E, (✏,E)isthephysicalparameterdescribingthee↵ective area↵ective area that governs thethat probability governs the of bremsstrahlung probability of bremsstrahlung radiation, which radiation, is discussed which in is detaildiscussed in detail(4.33) in Koch & Motzin(1959Koch). & A Motz widely(1959 used). Aapproximate widely used expression approximate is theexpression non-relativistic is the non-relativistic The distribution of electron• For close encounters, speeds in LTE is the nonrelativistic� �, � Maxwellian, Ω is much more distribution (see Appendix B.8 for its solid-angle-averagedcomplicated (quantum physics). In the nonsolid-angle-averaged Bethe-Heitler (NRBH) Bethe-Heitler cross-section (NRBH) cross-section-relativistic derivation): case, a direction-integrated cross section 2 1/2 2 0Z 11+(1/2 ✏/E) 2 1 NRBH0Z (✏,E1+(1)= ✏/Eln) 2 1 cm keV , (1.14) (✏,E)= ln ✏E 1 (1cm ✏keV/E)1/2, (1.14) NRBH ✏E 1 (1 ✏/E)1/2 25 2 2 where 0 =7.90 10 cm keV and (Z�is the= 0 mean for � > square�) atomic number of the 25 2 ⇥ 2 where 0 =7.90 10 cm keV and Z is the mean square atomic number of the Known as the target⇥ plasmaBethe ( 1.4-Heitler in the solarcross section corona). The bremsstrahlung cross-section is zero at target plasma ( 1.4 in the solar⇡ corona). The bremsstrahlung cross-section is zero at ⇡✏ >Esince an electron cannot emit a photon more energetic than the electron itself. *See Koch & Motz 1959 for full relativistic, angle and polarization dependent cross section ✏ >Esince an electron cannot emit a photon more energetic than the electron itself. For a thermal plasma with a temperature T , the electron energy distribution has T Figure 4.6: The nonrelativisticFor aMaxwellian thermalaMaxwellian-BoltzmannformdescribedbyEquation plasmadistribution with of aparticle temperature speeds in L,TE the (Equation electron4.34 energy), where1.6 distribution. The resulting has X-ray is the flux rms speed of particles with mass at temperature . aMaxwellian-BoltzmannformdescribedbyEquation1.6. The resulting X-ray flux2 Fthermal(✏)isproportionaltothetotalvolumeemissionmeasure⇠V = neV . Usually the(4.34) 2 Fthermal(✏)isproportionaltothetotalvolumeemissionmeasureplasma in the source region is not isothermal but⇠ hasV = an distributioneV . Usually over the temperature, http://www.cv.nrao.edu/~sransom/web/Ch4.html 13/23 plasma in the source region is not isothermal but has a distribution over temperature, Thin target bremsstrahlung

• With a differential cross section � �, � • Taking into account electron distribution �(�) • The directional integrated thin-target bremsstrahlung flux � � (photons cm-2 s-1 per unit energy) becomes

�� � � = � � � � � �, � �� 4��

Where � = ∫ ��� is the column density of the target Thick target bremsstrahlung

• Incident electrons are completely stopped, or thermalized in the source à requires high density. v Usually occurs when energetic electrons precipitating onto the chromosphere • Much quicker energy loss from electrons à lots of X-ray photons emitted à Produces intense X-ray emission • Usually dominates the hard X-ray (~10 keV – 300 keV) spectrum Thick target bremsstrahlung

• Electrons change their energy in time (quickly) � � � � = � � � � � �, � �� 4�� Time and space dependent • Instead we need to have � � � = � � � � � �, � �� 4�� where

() � �, � = � � � � �, � � � � � �� () is the number of photons at energy � emitted per unit energy by an electron of initial energy � Thick target bremsstrahlung

• We need something to describe � � • Q: what is the main mechanism for electron energy loss? • Energy loss mainly due to e-e Coulomb collisions. We need another cross section to describe dE/dt à the Rutherford cross section:

� = ≈ 10cm × , where � = 2�� ln Λ keV

So = −� � � � � � Thick target bremsstrahlung

• Photon flux

δ −2 Ffl � flux&of&NT&eLs& = � � = � � � � 1�E� �, � �� �� L 4��� [&#&cmL2&sL1&]& δ − c NonLthermal&e s& F(E0)& δ&>&2& • Comparing to the thin-target caseELδ$ collision&cross&secMon& −17 2 −2 � � σ e =10 cm × EkeV E & � � = �c � �electrons&� � �, � ��E0& 4�� column&depth:& s z& N(s) = n (s!)ds! [cm−2 ] • Effective column density ∫ e stopping&column:&&N&σe&=&1& 1 � (�, � ) = � � �, � �� E 2 N c 1.4 1017 cm−2 E 2 �� �, � c = 4 = × c,keV 6πe Λ ∝ � stop&cutLoff&

Similar to the stopping column �! ne& From Lecture 10 by 1010& 1012&cmL3& Prof. Longcope Thin target vs. Thick target

• The effective column depth � �, � is independent of the target

• � ≪ � �, � : incident electron distribution is nearly unchanged à thin target

• � ≥ � �, � : substantial change in incident electron distribution à we must use the thick target expression From X-ray spectrum to electron distribution

• � � is what we observe (after taking out instrument response)

�� Thin target � � = � � � � � �, � �� 4��

� Thick target � � = � � � � �� �, � �� �� 4��� • Obtaining � � becomes an inversion problem • Many approaches (Brown 1971 and after), but difficult to obtain an accurate � � due to the “smoothing” effect of the integral (e.g., Craig & Brown 1985) If we pretend to know the form of � � … • Thermal: : � � = �/ exp(−�/��) Maxwellian /()/ • Nonthermal: Power law: � � = ��� () Kappa: � � ∝ 1 + The Astrophysical Journal,799:129(14pp),2015February1 (/) Oka et al.

From Oka et al. 2015 Figure 1. Schematic illustration of previously proposed spectral models of the above-the-looptop source. The distributions represent a plasma population inthe above-the-looptop source alone. The vertical and horizontal axes represent differential density F (E) and electron energy E,respectively,inlogarithmicscale.The models are shown, from left to right, in order of increasing magnitude of non-thermal, power-law component with a fixed spectral slope. Panel (d) may be regarded as the case of a saturated non-thermal tail. We consider that such a distribution can be represented by the kappa distribution. Here, the core distributions (dashed curves) are adjusted as described in the text. Panel (e) is an oversimplified version of non-thermal distribution without a core.

Figure 2. Illustration of the artifact caused by introducing the lower-energy cutoff Ec.Thehypotheticaldata(thickblackhistogram)iscomparedtoa(a)kappa model, (b) thermal+power-law model with a relatively high temperature core, and (c) thermal+power-law model with an extremely high temperature core. For a better modeling, the thermal+power-law model needs to have a very large temperature TM,largerthanTκ .Themodelcurvesarenotfromanactualfit.

Figure 2 illustrates this. Let us consider hypothetical data con- look for a more appropriate (preferably single) functional form taining a super-hot thermal core combined with the saturated of the entire non-thermal distribution with no spectral break. non-thermal component. In Figure 2(a), the hypothetical data To the authors’ knowledge, the kappa distribution is, so far, the (thick black histogram) are actually generated from the kappa only alternative to reasonably describe such a distribution. distribution (thick gray curve). The thermal core of this distribu- The power-law model with no thermal core (Figure 1(e)) is tion is represented by the adjusted Maxwellian described above unphysical as a representation of the entire distribution,5 but it and is shown in red. It has the temperature TM 35 MK. Now, may still be useful if the core temperature was sufficiently low if we try to fit this hypothetical data with the= thermal+power- and if the observation was limited only to the higher energy, law model, a dip appears between the thermal and power-law power-law part of the distribution. In fact, the lower-energy components (Figure 2(b)). In Figure 2,thelower-energycutoff cutoff Ec (at typically 15–20 keV) is generally considered to Ec 40 keV. If we raise Ec,thedipbecomeslargerandleadsto be an upper limit to the actual value of Ec (e.g., Holman et al. = a worse fit. If we lower Ec,thepowerlawdeviatesfromthedata. 2011;Kruckeretal.2010;Krucker&Battaglia2014). However, The dip remains significant even if we raise the temperature TM previous observations (and case studies presented in this paper) up to the kappa temperature Tκ 56 MK (Figure 2(b)). To fill indicate that if the power law were extended to the lower-energy = this dip, we need to raise the core temperature even higher, up to, range below Ec,theelectronfluxwouldbesohighthatit for example, TM 75 MK (Figure 2(c)). Note also that raising contradicts the fact that the above-the-looptop source is not = the core temperature TM has the effect of reducing the density visible in the lower-energy X-rays (Krucker et al. 2010;Oka of the thermal core component. Therefore, the thermal+power- et al. 2013). If the power law cannot reasonably be extended to law model will give us a systematically higher temperature and the lower-energy range, then we would need to consider a hot lower density (or emission measure, EM) than the kappa model, thermal component to fill the gap below Ec (e.g., Holman et al. due to the very sharp, lower-energy cutoff Ec. Of course, this would be less significant for a less-sharp cutoff (for example, FPL(E) √E for E

3 Thermal bremsstrahlung

• Resulted from � � with a Maxwellian distribution • Radio thermal bremsstrahlung do not require high speed electrons • X-ray thermal bremsstrahlung does require high speed electrons � > �. For 3 keV X-ray, �~3.5×10 � à from flaring loops From Krucker & Battaglia 2014 Question: • Why is thermal bremsstrahlung not so important in optical and UV? 8/3/10

8/3/10 Suppose'the'energe5c'electron'distribu5on'func5on'is'a'power'law:'' Nonthermal thin target bremsstrahlung

With'a'change'of'variables'from'• Assume electron distribution is a power law:E to'E/ε we'then'have'''' � � = ��� Suppose'the'energe5c'electron'distribu5on'func5on'is'a'power'law:'' and'the'photon'count'at'1'AU'is'recast'as''''''• Plug in � � for thin target, we have

� � = With'a'change'of'variables'from'E to'E/ε we'then'have''''

� � ∝ �(/) and'the'photon'count'at'1'AU'is'recast'as'''''' This'result'was'obtained'by'Brown'(1971),'although'instead'it'was'• � � usually have a power-law shape in HXRs: formulated'in'terms'of'the'observed'photon'spectrum,'taken'to'be'a' power'law,' � � = �� If thin-target, the electron energy spectrum is This'result'was'obtained'by'Brown'(1971),'although'instead'it'was' from'which'N(E)'was'inferred'through'(/inversion) :''' formulated'in'terms'of'the'� � ∝ � observed'* See J. Brown 1971 for detailsphoton'spectrum,'taken'to'be'a' power'law,'

from'which'N(E)'was'inferred'through'inversion:'''

Now'consider'the'case'where'we'have'a'con5nuous'injec5on'of'fast'electrons' into'the'source'volume'where'they'suffer'energy'losses'via'collisions'on'free' (and'bound)'electrons'and'are'brought'to'a'stop.' Now'consider'the'case'where'we'have'a'con5nuous'injec5on'of'fast'electrons' For'a'fully'ionized'plasma'we'have''into'the'source'volume'where'they'suffer'energy'losses'via'collisions'on'free' (and'bound)'electrons'and'are'brought'to'a'stop.'

For'a'fully'ionized'plasma'we'have'' where'

where' An'electron'injected'with'an'energy'Eo'can'radiate'photons'with'energy'ε'via' bremsstrahlung'un5l'the'electron’s'energy'has'fallen'below'ε.'It’s'photon' An'electron'injected'with'an'energy'E 'can'radiate'photons'with'energy'ε'via' produc5on'rate'is'then'given'by:' o bremsstrahlung'un5l'the'electron’s'energy'has'fallen'below'ε.'It’s'photon' produc5on'rate'is'then'given'by:'

where'' where''

21 21 Nonthermal thick target bremsstrahlung

• Assume electron distribution is a power law: � � = ��� • Plug in � � for thick target, we have

� � ∝ �() Much flatter than thin-target! • Inverting from a power-law photon spectrum

� � ∝ �(/)

• Inferred spectral index is steeper by 2 for thick target than thin target Thick target

Thermal Thin target X-ray emission in flares

In solar corona: low density à very few collisions à energy loss small (dE< 10 cm THICK target

photosphere photosphere Other bremsstrahlung contributions

Bremsstrahlung emissivity The Astrophysical Journal,750:35(16pp),2012May1 Chen & Bastian from an electron beam

• At higher (relativistic) energies, (a) (b) (c)

corrections for the e-i cross Increasing � section must be included. Moreover, the radiation pattern becomes highly beamed Chen & Bastian 2012

Figure 11. Polar diagram of the normalized emissivity from a monodirectional electron as a function of the logarithm of the photon energy ϵγ and the angle of the LOS • Additional contributions need to be includedrelative to the beam direction. The electron beam propagates in the +x direction. The electron distribution function is a power law between 10 keV and 100 MeV and δ has an index δ 3. The emissivity has been multiplied by ϵγ in all cases. The concentric circles indicate contours of constant photon energy: 20, 50, 100, 200, 500, • e-e bremsstrahlung à important at and 1000 keV. (a)�= Thin-target> 300 bremsstrahlungkeV with a emissivity. The EIB and EEB contributions have been summed; (b) the same for ICS of monoenergetic EUV photons flatter spectrum (Haug 1975; (0.2Kontar keV) upscattered2007) by the electron beam; (c) same as panel (b), but for SXR photons (2 keV). • e+-e bremsstrahlung (Haug 1985)(A color version of this figure is available in the online journal.) • i-e bremsstrahlung is usually insignificant (Emslie & Brown using the same parameters used to compute the examples of ICS emission in the mildly relativistic regime shown in 1985; Haug 2003) −2 Dashed: δ−beam 10 Dotted: cone−beam Figure 9.Wenotethattheresultingspectraarequalitatively Dash−dotted: pancake similar to those resulting from ICS on these distributions. In Solid: isotropic 10−4 particular, the extreme case of a monodirectional electron beam directed along the LOS results in a significant enhancement to the thin-target emissivity and a substantially flatter spectrum −6 10 than the isotropic case, for which α 3.5 for non-relativistic photon energies (20–80 keV) and α =2.9forγ -ray photon en- = 10−8 ergies (200–800 keV). The cone beam and pancake anisotropy result in more modest enhancements, and their spectra are in- termediate to the isotropic and monodirectional beam (for the 10−10 cone beam, α 2.5 and 2.0 for the 20–80 keV and 200–800 keV ranges, respectively;= for the pancake anisotropy, α 3.2 and = −12 2.5, respectively). We note that for the monodirectional beam, 10 EIB and EEB asymptotically approach equality as the photon energy increases (cf. Dermer & Ramaty 1986), but the EEB 10−14 contribution becomes less prominent in the beam-cone and pan- flux (photons per keV second electron) 1 10 100 1000 cake distributions. Note, too, that the degree of enhancement HXR photon energy (keV) of each of the anisotropic cases relative to the isotropic case is Figure 12. Bremsstrahlung photon spectra at the Sun (photons per keV per less dramatic than for ICS. There is essentially no enhancement second per electron) from anisotropic electron distributions along the LOS. at 10 keV photon energies owing to the fact that the electron Different kinds of anisotropies are plotted—a monodirectional electron beam distribution cuts off at 10 keV, but this changes as the photon (dashed line), a cone-beam distribution with a half-angle width of ∆θb 10◦ energy increases: to an enhancement of perhaps 1.5 orders of (dotted line), a pancake electron distribution with a half-angle width= of magnitude for the monodirectional beam at 100 keV, 1order ∆θp 10◦ (dash-dotted line), and an isotropic electron distribution (solid ! line).= The electron energy distribution is assumed to have a single power-law of magnitude for the cone beam, and a factor of 2forthe form with a spectral index δ 3, extending from 10 keV to 100 MeV (same as pancake distribution. This can be understood as a consequence∼ that in Figure 9). The results= are also normalized to one source electron above 8 3 of the more modest degree of directivity of bremsstrahlung 0.5 MeV,and the ion number density ni is assumed to be 10 cm− .Notethatthe break in the spectra at 10 keV is from the lower energy cutoff of the electron emission compared with ICS. The HXR (20–80 keV) and distribution at 10 keV. ∼ γ -ray (200–800 keV) spectral indices that result from these cases are summarized in Table 1.Weconcludefromthisexer- cise that the effect of electron anisotropies on mildly relativistic (2008)forthecaseofanisotropicelectrondistributionandthin- and ultrarelativistic ICS emission is significantly larger than is target EIB emission. That is, for photon energies well below the the case for thin-target bremsstrahlung emission, all other things cutoff (ϵ Ec) the spectral index is very hard (α 1.5). The being equal. inclusion≪ of EEB may change the spectral index of∼ the photon We now turn our attention to the relative roles of ICS and spectrum by !0.1. thin-target bremsstrahlung in the production of HXR and con- We have also calculated the bremsstrahlung photon spectra, tinuum γ -ray emission. As was discussed by MM10, a com- including the contributions from both EIB and EEB, resulting parison between the relative roles of ICS and bremsstrahlung from the same electron anisotropies considered in Section 3, is somewhat problematic because the HXR photons resulting namely, a monodirectional electron beam, a cone-beam distri- from the two mechanisms are due to electrons from very differ- bution with a half-angle width of ∆θb 10 ,andapancake ent parts of the electron energy distribution. The high-energy = ◦ electron distribution with a half-angle width of ∆θp 10◦. electrons responsible for ICS make essentially no contribu- Figure 12 shows the corresponding results from a single= power- tion to the HXR bremsstrahlung emission. Similarly, the much law electron energy distribution with a spectral index δ 3.0 lower energy electrons responsible for HXR bremsstrahlung = 10 Summary

• Bremsstrahlung emission is one of the most important diagnostics for energetic electrons in flares • Thermal bremsstrahlung: radio, X-ray • Nonthermal bremsstrahlung: X-ray • Thin target à corona • Thick target à chromosphere (sometimes corona) • To obtain � � , we need • Observation of X-ray spectrum � � with high resolution • Application of the correct emission mechanism(s) • Appropriate inversion