Probing solar flare accelerated electron distributions with prospective X-ray polarimetry missions
Natasha L. S. Jeffrey, Pascal Saint-Hilaire, Eduard P. Kontar
aa r X i v : . [ a s t r o - ph . S R ] A ug Astronomy & Astrophysicsmanuscript no. ms c (cid:13)
ESO 2020August 19, 2020
Probing solar flare accelerated electrondistributions with prospective X-ray polarimetrymissions
Natasha L. S. Je ff rey , Pascal Saint-Hilaire , and Eduard P. Kontar Department of Mathematics, Physics & Electrical Engineering, Northumbria University, New-castle upon Tyne, UK, NE1 8ST, e-mail: [email protected] Space Science Laboratory, University of California, Berkeley, USA School of Physics & Astronomy, University of Glasgow, Glasgow, UK, G12 8QQReceived August 19, 2020 / Accepted
ABSTRACT
Solar flare electron acceleration is an extremely e ffi cient process, but the method of accelerationis not well constrained. Two of the essential diagnostics: electron anisotropy (velocity angle tothe guiding magnetic field) and the high energy cuto ff (highest energy electrons produced by theacceleration conditions: mechanism, spatial extent, time), are important quantities that can helpto constrain electron acceleration at the Sun but both are poorly determined. Here, using electronand X-ray transport simulations that account for both collisional and non-collisional transportprocesses such as turbulent scattering, and X-ray albedo, we show that X-ray polarization can beused to constrain the anisotropy of the accelerated electron distribution and the most energeticaccelerated electrons together. Moreover, we show that prospective missions, e.g. CubeSat mis-sions without imaging information, can be used alongside such simulations to determine theseparameters. We conclude that a fuller understanding of flare acceleration processes will comefrom missions capable of both X-ray flux and polarization spectral measurements together. Al-though imaging polarimetry is highly desired, we demonstrate that spectro-polarimeters withoutimaging can also provide strong constraints on electron anisotropy and the high energy cuto ff . Key words.
Sun: flares – Sun: X-rays, gamma rays – Sun: atmosphere – polarization – scattering– acceleration of particles Article number, page 1 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization
1. Introduction
Solar flares are the observational product of magnetic energy release in the Sun’s atmosphere andthe conversion of this magnetic energy into kinetic energies. Processes of energy release and trans-fer acting in flares are initiated by magnetic reconnection in the corona (e.g., Parker 1957; Sweet1958; Priest & Forbes 2000). Many studies show that a substantial fraction of the magnetic en-ergy goes into the acceleration of energetic keV and sometimes MeV electrons (e.g., Emslie et al.2012; Aschwanden et al. 2015, 2017; Warmuth & Mann 2016). Although it is established that so-lar flares are exceptionally e ffi cient particle accelerators and hence, a relatively close astrophysicallaboratory for studying particle acceleration, the exact mechanisms and even the exact locations ofenergy release and / or acceleration are not well-constrained (e.g. Benz 2017). The magnetic energymay be dissipated to particles by plasma turbulence (e.g. Larosa & Moore 1993; Petrosian 2012;Vlahos et al. 2016; Kontar et al. 2017), but it is possible that other mechanisms such as shock ac-celeration in reconnection outflows contribute as well (e.g. Mann et al. 2009; Chen et al. 2015).Further, the acceleration of seemingly distinct populations of electrons at the Sun, and those de-tected in-situ in the heliosphere, as well as the connection between flare-accelerated electrons andMeV to GeV ions, is poorly understood.X-ray bremsstrahlung is the prime diagnostic of flare-accelerated electrons at the Sun (e.g.Kontar et al. 2011). However, one reason that the properties of the acceleration region remain ex-clusive is because bremsstrahlung emission is density weighted, and so flare-accelerated electronsproduce their strongest emission away from the primary sites(s) of acceleration in the corona, in thechromosphere where the density is high. In a standard model, electrons accelerated in the coronastream towards the dense layers of the chromosphere where they lose energy via collisions andproduce hard X-ray (HXR) footpoints (e.g. Holman et al. 2011). Hence, in order to determine theproperties of accelerated electrons and the environment in which they are accelerated, we requirestate-of-the art transport models and diagnostic tools that realistically account for collisions in boththe corona and chromosphere, and constrain non-collisional transport e ff ects such as e.g. turbulentscattering (Kontar et al. 2014). In the last few years, there has been significant advancement in ourunderstanding of electron transport at the Sun. Instead of modelling electron transport with verylittle interaction with the coronal plasma, we now understand the importance of accounting fordi ff usive processes in energy and pitch-angle in the corona (e.g. Je ff rey et al. 2014; Kontar et al.2015). For example, the application of a full collisional model, accounting for energy di ff usion,led to a solution of the ‘low-energy cuto ff ’ problem (e.g. Kontar et al. 2019), whereby the energyassociated with flare-accelerated electrons could not be constrained from the X-ray flux spectrum.However, many of the vital properties required to constrain the acceleration process(es) still remainelusive, since they are di ffi cult to determine from a single X-ray flux spectrum alone.Since the birth of X-ray astronomy, X-ray polarization in solar physics has been under-studied.Observations and studies exist but many results have large uncertainties (e.g. Kontar et al. 2011).This is mainly because the polarimeters were unsuitable for flare observations; they were sec- Article number, page 2 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization ondary add-on missions (such as the polarimeter on board the Reuven Ramaty High Energy SolarSpectroscopic Imager (RHESSI) Lin et al. (2002); McConnell et al. (2002)) or not optimised forsolar observations (e.g. they were astrophysical missions studying gamma ray bursts). However,the X-ray polarization spectrum is an observable that can provide a direct link to several key prop-erties of energetic electrons, including the anisotropy, usually an unknown quantity that is vitalfor constraining both acceleration and transport properties in the corona. Many studies have exten-sively modelled both spatially integrated and spatially resolved solar flare X-ray polarization, i.e.Bai & Ramaty (1978); Leach & Petrosian (1983); Emslie et al. (2008); Je ff rey & Kontar (2011).Here, the aim is not to reiterative the main results of these past studies but to show how prospectivemissions (such as relatively cheap CubeSat missions) can be used alongside electron and X-raytransport simulations to determine vital acceleration parameters, even without imaging informa-tion.Section 2 describes the electron and X-ray transport models used in this study, while Section3 briefly describes some proposed X-ray spectro-polarimeters. In Section 4, we show simulationexamples of spatially integrated X-ray polarization and demonstrate how acceleration parameterscan be determined from spatially integrated X-ray flux and polarization spectra together. We brieflysummarise the study in Section 5.
2. Electron and X-ray transport
To determine how the properties of flare-accelerated electrons are changed in a hot and collisionalflaring coronal plasma, we use the kinetic transport simulation first discussed in Je ff rey et al. (2014)and Kontar et al. (2015). We model the evolution of an electron flux F ( z , E , µ ) [electron erg − s − cm − ] in space z [cm], energy E [erg], and pitch-angle µ to a guiding magnetic field, using theFokker-Planck equation of the form (e.g., Lifshitz & Pitaevskii 1981; Karney 1986): µ ∂ F ∂ z = Γ m e ∂∂ E " G ( u [ E ]) ∂ F ∂ E + G ( u [ E ]) E Ek B T − ! F + Γ m e E ∂∂µ " (1 − µ ) (cid:18) erf( u [ E ]) − G ( u [ E ]) (cid:19) ∂ F ∂µ + S ( E , z , µ ) , (1)where Γ = π e ln Λ n / m e = Kn / m e , and e [esu] is the electron charge, n is the plasmanumber density [cm − ] (a hydrogen plasma is assumed), m e is the electron rest mass [g], and ln Λ is the Coulomb logarithm. The variable u ( E ) = √ E / k B T , where k B [erg K − ] is the Boltzmannconstant and T [K] is the background plasma temperature. S ( E , z , µ ) plays the role of the electronflux source function. The first term on the right-hand side of Equation 1 describes energy evolution Article number, page 3 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization due to collisions (advective and di ff usive terms), while the second term describes the pitch-angleevolution due to collisions.The functions erf( u ) (the error function) and G ( u ) are given by,erf( u ) ≡ (2 / √ π ) u Z exp( − t ) dt (2)and G ( u ) = erf( u ) − u erf ′ ( u )2 u . (3)Further information regarding these functions and Equation 1 can be found in Je ff rey (2014). Theerror function and G(u) control the lower-energy ( E ≈ k B T ) electron interactions ensuring that theybecome indistinguishable from the background thermal plasma.Equation (1) is a time-independent equation useful for studying solar flares where the electrontransport time from the corona to the lower atmosphere is usually shorter than the observationaltime (i.e. most X-ray spectral observations have integration times of tens of seconds to minutes),but temporal information can be extracted (Je ff rey et al. 2019).Equation (1) models electron-electron energy losses, the dominant electron energy loss mech-anism in the flaring plasma, and both electron-electron and electron-proton interactions for colli-sional pitch-angle scattering . Equation (1) can be easily generalised to model any particle-particlecollisions.The z coordinate traces the dominant magnetic field direction. For simplicity here, in eachsimulation we use a homogenous coronal number density and temperature. At the boundary withthe chromosphere, the number density is set at n = × cm − but the number density rises tophotospheric densities of n ≈ cm − over ≈ ′′ using the exponential density function shownJe ff rey et al. (2019). At the chromospheric boundary, the temperature is set at T ≈ E >> k B T .The simulation ends when all electrons reach the chromosphere and E = Here, we also want to study how non-collisional transport e ff ects such as turbulent scatteringchange the electron distribution and the resulting X-ray polarization. Other non-collisional e ff ectscan change the electron properties, such as beam-driven Langmuir wave turbulence (Hannah et al.2009), electron re-acceleration (Brown et al. 2009) and / or beam-driven return current (Knight & Sturrock For this the pitch-angle term in Equation (1) is multiplied by 2, compared to Je ff rey et al. (2014) andJe ff rey et al. (2019). Article number, page 4 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization Fig. 1.
Left: Di ff erent injected electron anisotropy at the loop apex using S ( µ ) (Equation (11)). Small ∆ µ produces beamed distributions while large ∆ µ produces isotropic distributions. Right: Turbulent scatteringmean free path λ s versus electron energy E using Equation (7) and using λ s , = × , × and 2 × cm. Turbulent scattering quickly isotropises higher energy electrons. The mean free path λ s is also comparedwith the collisional mean free path (using λ = v / Γ ; grey dashed and dotted lines) for three di ff erent densitiesof n = × cm − , n = × cm − and n = × cm − . (e.g. Schlickeiser 1989) where theturbulent scattering di ff usion coe ffi cient D T µµ is related to the turbulent scattering mean free path λ s [cm] and electron velocity v [cm s − ] using D T µµ = v λ s (cid:16) − µ (cid:17) . (4)Using the standard quasilinear theory for slab turbulence (Jokipii (1966); Kennel & Petschek(1966); Skilling (1975); Lee (1982); Kontar et al. (2014)), λ s can be related to the level of turbulentmagnetic field fluctuations δ BB resonant with electrons of velocity v by λ s = vπ Ω ce * δ B B +! − = c √ Em π eB * δ B B +! − , (5)where Ω ce = eB / mc is the electron gyrofrequency [Hz], B is the magnetic field strength [G]and c is the speed of light [cm s − ]. Using for example, E =
25 keV and B =
300 G, 2 × − ≤ δ BB ≤ × − corresponds to 2 × [cm] ≤ λ s ≤ × [cm]. We use this model for turbulent scattering since the details of scattering in the flaring corona are not well-constrained, i.e. there are many models but few observations. Article number, page 5 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization In simulations where we investigate the role of non-collisional turbulent scattering, the govern-ing Fokker-Planck equation becomes µ ∂ F ∂ z = Γ m e ∂∂ E " G ( u [ E ]) ∂ F ∂ E + G ( u [ E ]) E Ek B T − ! F + Γ m e E ∂∂µ " (1 − µ ) (cid:18) erf( u [ E ]) − G ( u [ E ]) (cid:19) ∂ F ∂µ + λ s ( E ) ∂∂µ " (1 − µ ) ∂ F ∂µ + S ( E , z , µ ) . (6)For the majority of coronal flare conditions and electron energies, non-collisional turbulent scat-tering operates on timescales shorter than collisional scattering and can produce greater isotropyand trapping amongst higher energies electrons (see Figure 1, right panel). By combining X-rayimaging spectroscopy and radio observations of the gyrosynchrotron radiation, Musset et al. (2018)find empirically that the scattering mean free path is λ s = λ s , [cm] E ! , (7)where λ s , = × cm. In Section 4, we use λ s , = × cm and λ s , = × cm.In this model higher energy electrons have a smaller turbulent mean free path than lower energyelectrons. The model of Musset et al. (2018) is suitable for the purposes of the paper and while itis based upon the observation of a single flare and has large uncertainties, it clearly shows that themean free path of higher energy electrons (from microwave observations) is smaller than the meanfree path of lower energy electrons (from X-ray observations).Following Je ff rey et al. (2014), and re-writting Equation (6) as a Kolmogorov forward equa-tion (Kolmogorov 1931), Equation (6) can be converted to a set of time-independent stochasticdi ff erential equations (SDEs) (e.g., Gardiner 1986; Strauss & E ff enberger 2017) that describe theevolution of z , E , and µ in Itˆo calculus: z j + = z j + µ j ∆ s ; (8) E j + = E j − Γ m e E j (cid:18) erf( u j ) − u j erf ′ ( u j ) (cid:19) ∆ s + q Γ m e G ( u j ) ∆ s W E ; (9) Article number, page 6 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization µ j + = µ j − Γ m e (cid:18) erf( u j ) − G ( u j ) (cid:19) E j µ j + µ j λ s ( E j ) ∆ s + vuuuuuut (1 − µ j ) Γ m e (cid:18) erf( u j ) − G ( u j ) (cid:19) E j + (cid:16) − µ j (cid:17) λ s ( E j ) ∆ s W µ . (10) ∆ s [cm] is the step size along the particle path, and W µ , W E are random numbers drawn fromGaussian distributions with zero mean and a unit variance representing the corresponding Wienerprocesses (e.g. Gardiner 1986). A simulation step size of ∆ s = cm is used in all simulations,and E , µ and z are updated at each step j . A step size of ∆ s = cm is approximately two ordersof magnitude smaller than the thermal collisional length in a dense ( n = cm − ) plasma with T ≥
10 MK (or the collisional length of an electron with an energy of 1 keV or greater, in a coldplasma). The derivation of Equation (6) and a detailed description of the simulations can be foundin Je ff rey et al. (2014).Equation (6) (and Equations (9) and (10)) diverge as E →
0, and as discussed in Je ff rey et al.(2014), the deterministic equation E j + = (cid:20) E / j + Γ m e √ π k B T ∆ s (cid:21) / must be used for low energieswhere E j ≤ E low using E low = (cid:20) Γ m e √ π k B T ∆ s (cid:21) / – see Je ff rey et al. (2014), following Lemons et al.(2009). For such low energy thermal electrons, µ j + can be drawn from an isotropic distribution µ ∈ [ − , + T and a chosen EM = n V that is dominant at lowerX-ray energies between ≈ −
25 keV, and where V is the volume of this source. Although it ispossible that the thermal component can produce a small detectable polarization of a few percent(Emslie & Brown 1980), we assume that the coronal Maxwellian source is isotropic and hence,produces completely unpolarized X-ray emission in all the simulations shown here. The initial electron anisotropy is chosen using S ( µ ) ∝
12 exp − (1 − µ ) ∆ µ ! +
12 exp − (1 + µ ) ∆ µ ! (11)where ∆ µ controls the electron directivity. As ∆ µ → µ = −
1) and half along the other ( µ = + ∆ µ → ∞ , the electron distribution becomes isotropic (see the left panel of Figure 1).In many of the simulation runs shown in Section 4, the electron directivity is beamed. Since thedirectivity is unknown, a beamed distribution is used so that di ff erences in the studied parametersare clearly seen and understood. Article number, page 7 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization For most of the simulations shown here, we input sensible flaring parameters: a simple powerlaw distribution in energy ( E − δ ) with spectral index of δ =
5, a low energy cuto ff of E c =
20 keVand an acceleration rate of ˙ N = × electrons s − and in space, we input a Gaussian at the loopapex with a standard deviation 1 ′′ . The electron flux spectrum is calculated and it is converted to a photon flux spectrum using the fullangle-dependent polarization bremsstrahlung cross section as described in Emslie et al. (2008);Je ff rey & Kontar (2011) and using the cross-section shown in Gluckstern & Hull (1953); Haug(1972) given by σ I ( E , ǫ, Θ ) = σ ⊥ ( E , ǫ, Θ ) + σ k ( E , ǫ, Θ ) , (12) σ Q ( E , ǫ, Θ ) = ( σ ⊥ ( E , ǫ, Θ ) − σ k ( E , ǫ, Θ )) cos 2 Θ , (13)and σ U ( E , ǫ, Θ ) = ( σ ⊥ ( E , ǫ, Θ ) − σ k ( E , ǫ, Θ )) sin 2 Θ , (14)where ǫ is the X-ray photon energy and σ ⊥ ( E , ǫ, Θ ) and σ k ( E , ǫ, Θ ) are the perpendicular and par-allel components of the bremsstrahlung cross-section. Subscripts I , Q , U denote the cross sectionused for the total X-ray flux ( I ) and linear polarization components ( Q , U ) respectively, andcos Θ = cos θ cos β + sin θ sin β cos Φ , (15)relates θ the photon emission angle measured from the local solar vertical, β the electron pitch-angle (e.g. the angle between the electron velocity and the magnetic field) and Φ the electronazimuthal angle measured in the plane perpendicular to the local solar vertical.Using the above cross sections the resulting photon flux I and each specific linear polarizationstate Q and U can be written as: I ( ǫ, θ ) ∝ Z ∞ E = ǫ Z π Φ= Z πβ = F ( E , β ) σ I ( E , ǫ, Θ ) sin β d β d Φ dE (16) Q ( ǫ, θ ) ∝ Z ∞ E = ǫ Z π Φ= Z πβ = F ( E , β ) σ Q ( E , ǫ, Θ ) sin β d β d Φ dE , (17) Article number, page 8 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization U ( ǫ, θ ) ∝ Z ∞ E = ǫ Z π Φ= Z πβ = F ( E , β ) σ U ( E , ǫ, Θ ) sin β d β d Φ dE . (18) Depending on the directivity, some fraction of the emitted X-rays are transported to the photo-sphere. Here Compton backscattering will change the properties of these X-rays before their es-cape towards the observer. This is known as the X-ray albedo component (e.g. Tomblin 1972;Santangelo et al. 1973; Bai & Ramaty 1978) and all X-ray observables including the flux and polar-ization spectra are altered by this albedo component. Hence, we employ the code of Je ff rey & Kontar(2011) to create the X-ray albedo component for all simulation runs. A full discussion regardingthis code can be found in Je ff rey & Kontar (2011).Once the X-ray albedo component has been included, two polarization observables: the degreeof polarization (DOP) and polarization angle Ψ can be calculated using DOP = p Q + U I , (19)and Ψ =
12 arctan − U − Q ! . (20)The negatives in Equation 20 ensure that Ψ = ◦ corresponds to a dominant polarization di-rection parallel to the local solar radial direction (negative DOP) and Ψ = ◦ corresponds to adominant polarization direction perpendicular to the local solar radial direction (positive DOP).We note that high energy electrons (MeV) can produce spatially integrated Ψ = ◦ and thus posi-tive DOP, due to electrons with higher energies scattering through larger angles, providing a usefuldiagnostic for the presence of MeV electrons. Moreover, Emslie et al. (2008) showed that values ofspatially integrated Ψ other than 0 ◦ or 90 ◦ are possible when the tilt of the flare loop moves awayfrom the local vertical direction (loop tilt τ ). Finally, we note that in the case of spatially resolvedobservations, the angle of polarization Ψ can have values other than Ψ = ◦ or Ψ = ◦ due to theCompton scattered albedo component as shown in Je ff rey & Kontar (2011), providing a detailedprobe of electron directivity in the chromosphere.However, here we will only study spatially integrated X-ray data for relatively low electronenergies <
300 keV and for flare loops aligned along the local solar vertical. Hence
Ψ = ◦ andDOP is negative for all simulations. Therefore, since Ψ and the sign of DOP do not change, we willonly show the DOP results (and as a percentage only). Article number, page 9 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization Fig. 2.
The resulting spatially integrated DOP and flux spectra for three injected electron distributions witheither beamed, ∆ µ = . ∆ µ = . δ = E c =
20 keV (vertical grey dotted line), E H =
100 keV and ˙ N = × e s − , and corona plasma properties of: n = × cm − and T =
20 MK, plotted for a flare located at aheliocentric angle of θ = ◦ . All spectra include an albedo component and a coronal background thermalcomponent with EM = n V = . × cm − , using a chosen V = cm . For example error calculations,we use an e ff ective area of 5 cm and a time bin of 120 s. No turbulent scattering is present.
3. Prospective X-ray polarization instrumentation
Instrumentation to specifically detect solar flare HXR polarization are currently being researchedand developed and three such instruments are summarised here. Currently, the most advanced con-cept is the Gamma-Ray Imager / Polarimeter for Solar flares (GRIPS; Duncan et al. (2016)). GRIPSis a balloon-borne telescope designed to study solar-flare particle acceleration and transport. It hasalready flown once in Antarctica in January 2016, and will be re-proposed for flight during the nextSolar Maximum. GRIPS can do imaging spectro-polarimetry of solar flares in the ∼
150 keV to ∼ ∼ . ′′ . GRIPS’skey technological improvements over the current solar state of the art at HXR / gamma-ray energies,RHESSI, include 3D position-sensitive germanium detectors (3D-GeDs) and a single-grid modula-tion collimator, the multi-pitch rotating modulator (MPRM). Focusing optics or Compton imagingtechniques are not adequate for separating magnetic loop footpoint emissions in flares over theGRIPS energy band, and indirect imaging methods must be employed. The GRIPS MPRM covers13 spatial scales from 12 . ′′ to 162 ′′ . For comparison, RHESSI could only image gamma-ray emis-sions at two spatial scales (35 ′′ and 183 ′′ ). For photons that Compton scatter, usually &
150 keV, the
Article number, page 10 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization energy deposition sites can be tracked, providing polarization measurements as well as enhancedbackground reduction through Compton imaging. The nominal GRIPS balloon payload has a min-imum detectable polarization (MDP) signal of ∼
3% in the 150-650 keV band for 2002-July-23X-flare, while a spacecraft version will likely be closer to ∼ ∼ ∼ ≥
20 keV, 3-sigma) for a similar flare. Sapphire modulesare designed to be stackable, with the decrease (improvement) in MDP for N modules roughlybehaving as 1 / √ N .Other proposed missions include the Japanese PhoENiX mission (Physics of Energetic andNon-thermal plasmas in the X ( = magnetic reconnection) region; Narukage (2019)) which willhave an X-ray spectro-polarimeter onboard measuring over an energy range of 60-300 keV (and20-300 keV for spectroscopy).
4. Results
Here, we investigate how spatially integrated X-ray DOP changes with anisotropy, high energy cut-o ff and non-collisional turbulent scattering. DOP (and indeed polarization angle) will vary with theflare geometrical properties such as heliocentric angle and the properties of the flare loop such asloop tilt to the local vertical direction, but these properties can always be estimated from flare imag-ing (see Appendix A, Figure A.1). In the following sections, all results are shown for a heliocentricangle of 60 ◦ and a loop tilt of 0 ◦ (the loop apex is parallel to the local vertical direction). Traditionally, the prime diagnostic of X-ray polarization is its direct link with electron directivity.Figure 2 shows the spatially integrated DOP (%) versus X-ray energy for four di ff erent injectedelectron anisotropies from completely beamed to isotropic , for a given set of otherwise identicalcoronal plasma parameters and electron spectral and spatial parameters. The resulting DOP spectracontains an albedo component and a thermal component as described in Section 2. As expected, theDOP at all energies clearly decreases with increasing injected electron isotropy. In Figure 2, we plotthe results for an average coronal number density of n = × cm − and coronal temperature of T =
20 MK. The plasma properties of the corona do change the resulting DOP slightly, particularlyat lower energies below 50 keV, however, properties such as the average coronal number density For comparison, the emitting electron distribution has been forced to be completely isotropic and the DOPspectrum peaking at 2-3% results purely from the backscattered albedo component (e.g. Bai & Ramaty 1978;Je ff rey & Kontar 2011). Article number, page 11 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization Fig. 3.
The resulting spatially integrated X-ray flux and DOP spectra for injected electron distributions withdi ff erent high energy cuto ff s of E H =
100 keV, E H =
200 keV and E H =
300 keV. Each use the followingidentical electron and plasma properties of: δ = E c =
20 keV (vertical grey dotted line), a beamed distri-bution and ˙ N = × e s − , and coronal plasma properties of: n = × cm − and T =
20 MK, plottedfor a flare located at a heliocentric angle of 60 ◦ . All DOP spectra include an albedo component and a coronalthermal component ( EM = . × cm − ). Higher E H produces a clear flattening in the DOP spectra afterthe low energy cuto ff . No turbulent scattering is present. can be estimated from the X-ray flux spectrum and / or EUV spectral observations, and applied tothe DOP spectrum.In Figure 2, we plot the X-ray flux and DOP over an energy range of ≈ −
100 keV, wherethe flare count rates are highest (most flare spectra are steeply decreasing power laws). As anexample in Figures 2-4, we calculate sensible flux and DOP error values using counts = flux × e ff ective area ( A ) × time bin ( ∆ T ) × energy bin ( ∆ E ) and assuming that photons = counts. The erroron the flux is then calculated as flux error = √ counts A ∆ t ∆ E and corresponding DOP error as DOP error = flux error × DOPflux . Here we use A = and ∆ t =
120 s, with ∆ E shown in each figure. We plotthe resulting DOP spectra for a flare located at a heliocentric angle of 60 ◦ (the DOP should growas the heliocentric angle approaches the limb for the majority of flare observations). Another important diagnostic of the acceleration mechanism is the high energy cuto ff (the highestenergy electrons produced by the acceleration process). The high energy cuto ff is also dependent onthe spatial extent of the acceleration region and the acceleration time, making it an important mod- Article number, page 12 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization elling constraint. Although the presence of high (MeV) energy electrons might be determinablefrom microwave observations if present e.g. Melnikov et al. (2002); Gary et al. (2018), we showthat the high energy cuto ff changes the trend in the DOP spectra (and also the sign of the po-larization angle Ψ for MeV energies as discussed in Je ff rey & Kontar (2011)). We show that thepresence of higher energy keV electrons in the distribution can decrease the overall DOP at allobserved energies.In Figure 3, we plot the resulting DOP spectra from three injected electron distributions withdi ff erent high energy cuto ff s of 100 keV, 200 keV and 300 keV. After 20 keV, we can see that thegradient of the DOP spectrum decreases with an increase in the high energy cuto ff E H , producinga flattening and then a decrease in DOP with energy as the high energy cuto ff increases to E H =
300 keV, over a typical observed flare energy range of ≈ −
100 keV.This occurs because the resulting X-ray directivity is also dependent on the bremsstrahlungcross section. For example, an 80 keV X-ray is more likely to be emitted by an electron of energy100 keV than an electron of energy 80 keV (i.e. magnitude of the cross section), with the directivityof the emitted 80 keV X-ray decreased (i.e. the cross section is more isotropic; see Appendix B,Figure B.2). However, solar flare electron energy spectra are steeply decreasing power laws and tocheck that the X-ray emission from high energy electrons can indeed contribute enough emissionto significantly decrease the X-ray directivity and DOP, two electron distributions with E H = E H =
300 keV, and δ = E H .This ultimately means that electron distributions with higher electron energies will producelower energy X-rays with a smaller DOP, even for the same injected beaming. This is an importantdiagnostic tool that can help to constrain the highest energies in the electron distribution, even whenthey are completely undetectable by other means and this diagnostic is unique to X-ray polarization.This also means that the DOP spectrum is always a result of both the electron anisotropy and highenergy cuto ff and both parameters should be determined in tandem. Other transport e ff ects such as turbulent scattering also alter the properties of the electron distri-bution, including the anisotropy and hence the X-ray polarization. An estimation of the plasmaproperties from the X-ray flux spectrum, imaging and EUV observations can determine the e ff ectsdue to collisions, however, constraining the presence and properties of additional non-collisionaltransport e ff ects outside of the acceleration region can be challenging in the majority of flares.Since DOP is sensitive to changes in anisotropy, we test whether it would be possible to use theDOP spectra to separate acceleration isotropy with isotropy produced by additional transport pro-cesses in the corona outside of the acceleration region. Article number, page 13 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization Fig. 4.
The resulting X-ray flux and DOP spectra for injected beamed electron distributions with E H =
100 keV without turbulent scattering, and with turbulent scattering (using λ s , = × cm or λ s , = × cm) situated in the coronal loop over a distance of [-10 ′′ , + ′′ ] from the loop apex. Each use the followingidentical electron and plasma properties of: δ = E c =
20 keV (vertical grey dotted line), a beamed distribu-tion and ˙ N = × e s − , and coronal plasma properties of: n = × cm − and T =
20 MK, plotted fora flare located at a heliocentric angle of θ = ◦ . All DOP spectra include an albedo component and a coronalthermal component ( EM = . × cm − ). The results suggest that a lack of coronal turbulent scatteringcould be detectable from the DOP spectrum. Parameter Simulation Estimated MethodEM 0 . × cm − (0 . ± . × cm − X-ray flux, f_vthT 25 MK 23 . ± . × cm − (4 . ± . × cm − X-ray flux, f_vth (EM) and VL 24 ′′ ′′ X-ray imaging˙ N × e s − (7 . ± . × e s − X-ray flux, f_thick_warm δ . ± E c . ± E H
113 keV ≈
100 keV X-ray DOP ∆ µ Beamed Beamed ( ∆ µ < .
1) X-ray DOP λ s , No turbulence No turbulence X-ray DOP
Table 1.
A comparison of the determined parameters and those input into the simulation. Here we demonstratethat using the X-ray flux and DOP spectra in tandem can help us to estimate all of the important accelerationparameters together and ultimately help to constrain the acceleration process(s) in flares.
In Figure 4, as one example, we plot the DOP versus energy for three identical beamed electrondistributions with E H =
100 keV, with no coronal turbulence and with turbulence ( λ s , = × cm and λ s , = × cm) situated in the coronal loop over a distance of [-10 ′′ , + ′′ ] from theloop apex. As expected, the presence of turbulence in the corona increases the electron isotropy Article number, page 14 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization Fig. 5.
An example of how the X-ray flux and DOP spectra can be used together to estimate the propertiesof the accelerated electron distribution. Applying the warm target function (f_thick_warm in OSPEX) to theX-ray flux helps to constrain the coronal plasma properties and the following acceleration parameters: lowenergy cuto ff E c , the spectral index δ and the rate of acceleration ˙ N . Ignoring any other transport mechanisms,and once constrained, the resulting X-ray DOP spectra should only be dependent upon the high energy cuto ff E H and acceleration anisotropy ∆ µ , and turbulent scattering (using λ s , ), if present. Top panels: (left) The ‘fluxdata’ (black) and the resulting f_vth + f_thick_warm fits to the X-ray flux spectrum (red and green respectively,this also contains an albedo component not shown), and (right) the ‘DOP data’ (black) and four simulationruns using the constrained properties from X-ray flux spectra and the values of E H , ∆ µ and λ s , shown. Bottompanels: Residuals and resulting reduced χ for each fit. Both the X-ray flux and DOP are fitted between 3 and90 keV (grey dash-dot line). and hence, reduces the DOP at all energies. Even very beamed distributions produce low levels ofDOP when turbulent scattering is present, as shown in Figure 4.Using DOP only, it is di ffi cult to distinguish between an initially isotropic electron distributionand the presence of strong turbulence in the corona outside of the acceleration region, althoughit is suggestive that if strong turbulence is present in the corona then it is also highly likely to bepresent in the coronal acceleration region. Turbulence greatly a ff ects higher energy electrons andisotropises them quickly leading to low DOP at all energies. The results indicate that a lack ofturbulence in the flaring corona can be determined from the DOP spectrum. Figure 4 also showsinteresting di ff erences between strong and milder turbulent scattering (i.e. slightly greater DOPat large energies from stronger turbulent scattering) that will be investigated further for di ff erentturbulent conditions beyond the scope of this study. The above results show that X-ray polarization is dependent on several important electron acceler-ation properties and the coronal plasma properties. Moreover, this shows that the DOP spectrum isa powerful diagnostic tool, particularly when used alongside the X-ray flux spectrum.As a preliminary demonstration of how we could extract vital acceleration parameters fromcombined X-ray flux and DOP spectra together, we simulate a flare at a heliocentric angle of 60 ◦ Article number, page 15 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization with certain accelerated electron and background coronal properties (see Table 1), producing theresulting X-ray flux and DOP spectra shown in Figure 5. To each spectrum, we add noise and weassume a spacecraft background level of 10 − photons s − cm − keV − at all energies. We now treatthe outputs as observational data with unknown properties. Firstly, to the X-ray flux spectrum, weapply the Solar Software (SSW) / OSPEX fitting function routines (as with real flare data). We fit theX-ray flux spectrum with an isothermal function (f_vth in OSPEX) and warm-target fitting function(f_thick_warm in OSPEX), using the steps described in Kontar et al. (2019). In this fit, as is usualwith data, we set E H =
200 keV, a value above the highest energy used in the fitting (90 keV). Fromfitting, we determine the following accelerated electron parameters: ˙ N = (7 . ± . × es − , δ = − . ± . E c = . ± . T = . ± . EM = (0 . ± . × cm − and n = √ EM / V = (4 . ± . × cm − where V is the volume of the hot plasma, assuming a sensible flare volume of V ≈ × cm − and loop length L = ′′ , as shown in Table 1. Once these parameters are constrained, we canfix them in the simulation. Then, the only unknowns are E H , ∆ µ and any coronal loop turbulencedescribed here using λ s , . Using di ff erent simulation runs, we can constrain and determine theseparameters by comparison with the observed flare X-ray DOP spectrum. In Figure 5, four such runsare shown and compared with the DOP spectrum; the runs use di ff erent anisotropies, high energycuto ff s and one with turbulent scattering filling the entire corona loop. The residuals and goodnessof fit χ (calculated as the sum of the residuals divided by the number of fitted energy bins) of eachresulting simulation ‘model’ are also shown as an example of how observations and models can becompared.In Table 1, we show all the determined electron and plasma parameters, the method used toobtain each parameter and compare with the actual parameters that were used in the original ‘data’simulation. Using X-ray flux and X-ray DOP observations together provides us with a fuller un-derstanding of the solar flare acceleration mechanism. Our analysis shows that current modellingis capable of producing estimates of E H and ∆ µ , and determining whether turbulence exists in thecorona. It will be possible to determine the uncertainties on these variables using many simulationruns and a full Monte Carlo parameter space analysis. However, this full uncertainty analysis isbeyond the scope of the current work, since the aim here is to demonstrate how the DOP spectrumcan be used as a powerful diagnostic tool alongside the X-ray flux spectrum if the data becomesavailable.The DOP is also sensitive to other electron acceleration parameters such as e.g. spectral indexand breaks in the spectrum. However, as shown these parameters can be constrained from the X-rayflux spectrum before analysing the DOP spectrum, showing the importance of studying the X-rayflux and DOP spectra in tandem. Article number, page 16 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization
5. Summary
X-ray polarimetry is a vital tool for constraining the solar flare acceleration mechanism especiallywhen used alongside the X-ray flux spectrum. The X-ray DOP spectrum is highly sensitive tocurrently unknown properties such as the accelerated electron pitch-angle distribution, highest en-ergy accelerated electrons and the presence of turbulence in the corona. We have simulation toolsavailable to analyse the X-ray DOP spectrum in detail, if the observations become available andalthough imaging polarimetry is highly desired, our results show that missions without imagingcan also provide strong constraints on electron anisotropy and the high energy cuto ff .In this paper we specifically discussed transport processes that can change the pitch angle distri-bution of electrons such as Coulomb collisions and turbulent scattering due to magnetic fluctuationsbut other transport processes may be present. We used a model with a simple homogenous back-ground plasma in the corona since we cannot study the individual plasma properties of each flare.Spatial variations in parameters such as number density along the loop will cause changes in theDOP spectrum. If such changes can be inferred from the data, then they can be incorporated intothe model before the inference of acceleration properties. Acknowledgements.
The development of the electron and photon transport models are part of an international team grant(“Solar flare acceleration signatures and their connection to solar energetic particles” )from ISSI Bern, Switzerland. NLSJ acknowledges IDL support provided by the UK Science and Technology FacilitiesCouncil (STFC). We thank the anonymous referee for their useful comments that helped to improve the text of the paper.
References
Alaoui, M. & Holman, G. D. 2017, ApJ, 851, 78Aschwanden, M. J., Boerner, P., Ryan, D., et al. 2015, ApJ, 802, 53Aschwanden, M. J., Caspi, A., Cohen, C. M. S., et al. 2017, ApJ, 836, 17Bai, T. & Ramaty, R. 1978, ApJ, 219, 705Benz, A. O. 2017, Living Reviews in Solar Physics, 14, 2Brown, J. C. 1971, Sol. Phys., 18, 489Brown, J. C., Turkmani, R., Kontar, E. P., MacKinnon, A. L., & Vlahos, L. 2009, A&A, 508, 993Chen, B., Bastian, T. S., Shen, C., et al. 2015, Science, 350, 1238Duncan, N., Saint-Hilaire, P., Shih, A. Y., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Con-ference Series, Vol. 9905, Proc. SPIE, 99052QEmslie, A. G. 1980, ApJ, 235, 1055Emslie, A. G., Bradsher, H. L., & McConnell, M. L. 2008, ApJ, 674, 570Emslie, A. G. & Brown, J. C. 1980, ApJ, 237, 1015Emslie, A. G., Dennis, B. R., Shih, A. Y., et al. 2012, ApJ, 759, 71Gardiner, C. W. 1986, Appl. Opt., 25, 3145Gary, D. E., Chen, B., Dennis, B. R., et al. 2018, ApJ, 863, 83Gluckstern, R. L. & Hull, M. H. 1953, Physical Review, 90, 1030Hannah, I. G., Kontar, E. P., & Sirenko, O. K. 2009, ApJ, 707, L45Haug, E. 1972, Sol. Phys., 25, 425Holman, G. D., Aschwanden, M. J., Aurass, H., et al. 2011, Space Sci. Rev., 159, 107Je ff rey, N. L. S. 2014, PhD thesis, University of GlasgowJe ff rey, N. L. S. & Kontar, E. P. 2011, A&A, 536, A93 Article number, page 17 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization Je ff rey, N. L. S., Kontar, E. P., Bian, N. H., & Emslie, A. G. 2014, ApJ, 787, 86Je ff rey, N. L. S., Kontar, E. P., & Fletcher, L. 2019, ApJ, 880, 136Jokipii, J. R. 1966, ApJ, 146, 480Karney, C. 1986, Computer Physics Reports, 4, 183Kennel, C. F. & Petschek, H. E. 1966, J. Geophys. Res., 71, 1Knight, J. W. & Sturrock, P. A. 1977, ApJ, 218, 306Kolmogorov, A. 1931, Mathematische Annalen, 104, 415Kontar, E. P., Bian, N. H., Emslie, A. G., & Vilmer, N. 2014, ApJ, 780, 176Kontar, E. P., Brown, J. C., Emslie, A. G., et al. 2011, Space Sci. Rev., 159, 301Kontar, E. P., Je ff rey, N. L. S., & Emslie, A. G. 2019, ApJ, 871, 225Kontar, E. P., Je ff rey, N. L. S., Emslie, A. G., & Bian, N. H. 2015, ApJ, 809, 35Kontar, E. P., Perez, J. E., Harra, L. K., et al. 2017, Physical Review Letters, 118, 155101Larosa, T. N. & Moore, R. L. 1993, ApJ, 418, 912Leach, J. & Petrosian, V. 1983, ApJ, 269, 715Lee, M. A. 1982, J. Geophys. Res., 87, 5063Lemons, D. S., Winske, D., Daughton, W., & Albright, B. 2009, Journal of Computational Physics, 228, 1391Lifshitz, E. M. & Pitaevskii, L. P. 1981, Physical kinetics (Course of theoretical physics, Oxford: Pergamon Press, 1981)Lin, R. P., Dennis, B. R., Hurford, G. J., et al. 2002, Sol. Phys., 210, 3Mann, G., Warmuth, A., & Aurass, H. 2009, A&A, 494, 669McConnell, M. L., Ryan, J. M., Smith, D. M., Lin, R. P., & Emslie, A. G. 2002, Sol. Phys., 210, 125Melnikov, V. F., Shibasaki, K., & Reznikova, V. E. 2002, ApJ, 580, L185Musset, S., Kontar, E. P., & Vilmer, N. 2018, A&A, 610, A6Narukage, N. 2019, in American Astronomical Society Meeting Abstracts, Vol. 234, American Astronomical Society Meet-ing Abstracts ff rey, N. L. S., Martinez Oliveros, J. C., et al. 2019, in AGU Fall Meeting Abstracts, Vol. 2019, SH31C–3310Santangelo, N., Horstman, H., & Horstman-Moretti, E. 1973, Sol. Phys., 29, 143Schlickeiser, R. 1989, ApJ, 336, 243Skilling, J. 1975, MNRAS, 172, 557Strauss, R. D. T. & E ff enberger, F. 2017, Space Sci. Rev., 212, 151Sweet, P. A. 1958, in IAU Symposium, Vol. 6, Electromagnetic Phenomena in Cosmical Physics, ed. B. Lehnert, 123Tomblin, F. F. 1972, ApJ, 171, 377Vlahos, L., Pisokas, T., Isliker, H., Tsiolis, V., & Anastasiadis, A. 2016, ApJ, 827, L3Warmuth, A. & Mann, G. 2016, A&A, 588, A116Zharkova, V. V. & Gordovskyy, M. 2006, ApJ, 651, 553 Article number, page 18 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization Appendix A: Change in DOP versus energy with heliocentric angle and loop tilt
The spatially integrated DOP changes with both flare location on the solar disk (heliocentric angle)and with loop tilt (how the loop apex is tilted with respect to the local vertical; the polarizationangle also changes with loop tilt Emslie et al. (2008)). However, both the heliocentric angle andloop tilt can be estimated from imaging the flare in di ff erent wavelengths (e.g. EUV). We showsome examples of how DOP changes with heliocentric angle and loop tilt in Figure A.1. Fig. A.1.
Left: Change in spatially integrated X-ray DOP versus energy for four di ff erent heliocentric anglesof 20 ◦ , 40 ◦ , 60 ◦ and 80 ◦ (as shown by the coloured dots). In this example, all electron distributions are beamedwith E H =
100 keV (top panel) and E H =
300 keV (bottom panel). Right: Change in spatially integrated X-ray DOP and flux versus energy for di ff erent loop tilts of τ = ◦ , 25 ◦ , 45 ◦ (the apex of the loop is tilted byan angle relative to the local vertical - see small cartoon). Each (left and right) uses the following identicalelectron properties of: δ = E c =
20 keV (grey dotted line), and ˙ N = × e s − , and corona plasmaproperties of: n = × cm − and T =
20 MK. All spectra include an albedo component and a coronalbackground thermal component (EM = . × cm − ). Article number, page 19 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization Appendix B: Bremsstrahlung cross section and the contribution from highenergy electrons
In subsection 4.2, we determined that a high energy cuto ff leads to lower DOP across all energiesand explained that this is due to the production of a more isotropic X-ray distribution when higherenergy electrons are present. To confirm this, in Figure B.1 we plot the resulting X-ray fluxes fromelectrons of energy E ≤
40 keV (dashed line) and from electrons of energy E >
40 keV (solid line)separately, but from a same emitting electron distribution. This is shown for an electron distributionwith a high energy cuto ff of E H =
100 keV (top panel) and E H =
300 keV (middle panel), and aspectral index of δ =
5. The bottom panel of Figure B.1 shows the X-ray contribution ratio (definedas the X-ray emission from E >
40 keV divided by the X-ray emission from E ≤
40 keV) versusenergy up to 40 keV, for electron distributions with E H =
100 keV and E H =
300 keV respectively.
Fig. B.1.
Resulting X-ray fluxes from electrons of energy E ≤
40 keV (dashed line) and from electrons ofenergy E >
40 keV (solid line) separately, for two di ff erent high energy cuto ff of E H =
100 keV (top) and E H =
300 keV (middle). Bottom: X-ray contribution ratio (the X-ray emission from E >
40 keV divided bythe X-ray emission from E ≤
40 keV) versus energy up to 40 keV (vertical grey dotted line), for electrondistributions with E H =
100 keV and E H =
300 keV. The horizontal black dashed-dot line denotes a ratio of1.
Although, the contribution will vary with other properties such as spectral index δ , we can seethat for both E H =
100 keV and E H =
300 keV, the X-ray contribution from higher energy electrons( E >
40 keV) dominates from above 25 keV ( E H =
300 keV) and above 30 keV ( E H =
100 keV).
Article number, page 20 of 21e ff rey et al.: Probing flare accelerated electrons with X-ray polarization Therefore, it shows that the bulk of the X-rays >
25 keV ( E H =
300 keV) and >
35 keV ( E H = E >
40 keV (Figure B.2).
Fig. B.2.
Total bremsstrahlung cross section ( σ I ) for the emission of a 40 keV X-ray by electrons of di ff erentof energies 40, 50, 100 and 300 keV (left) and for the emission of an 80 keV X-ray by electrons of di ffff