Transition region contribution to AIA observations in the context of coronal heating
DDraft version December 4, 2020
Typeset using L A TEX twocolumn style in AASTeX62
Transition region contribution to AIA observations in the context of coronal heating
S. J. Schonfeld and J. A. Klimchuk Institute for Scientific Research, Boston College, Newton, MA 02459, [email protected] NASA Goddard Space Flight Center, Heliophysics Science Division, Greenbelt, MD 20771 (Dated: December 4, 2020)
ABSTRACTWe investigate the ratio of coronal and transition region intensity in coronal loops observed by theAtmospheric Imaging Assembly (AIA) on the Solar Dynamics Observatory (SDO). Using Enthalpy-based Thermal Evolution of Loops (EBTEL) hydrodynamic simulations, we model loops with multiplelengths and energy fluxes heated randomly by events drawn from power-law distributions with differentslopes and minimum delays between events to investigate how each of these parameters influencesobservable loop properties. We generate AIA intensities from the corona and transition region foreach realization. The variations within and between models generated with these different parametersillustrate the sensitivity of narrowband imaging to the details of coronal heating. We then analyzethe transition region and coronal emission from a number of observed active regions and find broadagreement with the trends in the models. In both models and observations, the transition regionbrightness is significant, often greater than the coronal brightness in all six “coronal” AIA channels.We also identify an inverse relationship, consistent with heating theories, between the slope of thedifferential emission measure (DEM) coolward of the peak temperature and the observed ratio ofcoronal to transition region intensity. These results highlight the use of narrowband observations andthe importance of properly considering the transition region in investigations of coronal heating. INTRODUCTIONA consensus understanding of how exactly the plasmaof the Sun’s corona is heated to MK temperatures has re-mained elusive for decades (for more details see reviewsby: Zirker 1993; Walsh & Ireland 2003; Klimchuk 2006,2015; Parnell & De Moortel 2012; Viall et al. 2020).Many physical mechanisms have been proposed to causethis heating (for lists of many such mechanisms see:Mandrini et al. 2000; Cranmer & Winebarger 2019), butthe observations needed to distinguish them are funda-mentally challenging. The basic difficulty is that, for allmechanisms, the heating is highly time dependent with asmall (generally subresolution) spatial scale perpendic-ular to the magnetic field. In this context, it is conve-nient to consider magnetic strands, bundles of magneticflux with approximately uniform plasma properties overtheir cross section (Klimchuk 2006). These propertiesevolve in time in a manner that depends strongly on thedetails of the heating in the strand. The optically thinnature of coronal plasma emission in the extreme ultra-violet (EUV) and X-ray results in confusion between themany overlapping strands along a line of sight (e.g.; Viall& Klimchuk 2011). This makes it impossible to studythe dynamics of a single heating event in isolation. Instead, coronal heating must be studied by deter-mining how the bulk, optically thin plasma responds toheating on observable scales (Hinode Review Team et al.2019). By simulating the observable response of plasmato heating on unobservably small scales it is possible toconstrain the properties of the heating with availableinstrumentation. This is commonly done by simulatingthe evolution of plasma within individual closed mag-netic strands (e.g.; Barnes et al. 2016a,b) and then gen-erating the emission due to collections of these strands(Cargill 1994; Patsourakos & Klimchuk 2008; Warrenet al. 2002; Cargill & Klimchuk 2004; Warren et al. 2010;Bradshaw & Klimchuk 2011; Reep et al. 2013; Viall &Klimchuk 2013; Lionello et al. 2016; Marsh et al. 2018)in observable instrument channels.These coronal models must necessarily consider thecoupled system with the transition region that moder-ates the connection between the hot, tenuous corona andthe cool, dense chromosphere. In observational terms,the transition region has commonly been defined basedon the temperature regime it occupies, ∼ – 10 K.A more appropriate and physically motivated definitionis given by considering models of individual magneticstrands and defining the interface between the coronaand transition region to be the location where ther- a r X i v : . [ a s t r o - ph . S R ] D ec mal conduction changes from being a loss term above(causing cooling) to a gain term below (causing heating;Vesecky et al. 1979). This is the approach taken in theEnthalpy-based Thermal Evolution of Loops (EBTEL)model originally defined in Klimchuk et al. (2008). Theadvantage of this definition is that it more faithfully rep-resents the range of possible states available to coronalloop transition regions. In particular, this acknowledgesthat in hot loops, temperatures commonly associatedwith the corona (above 10 K) can occur in the transi-tion region close to the loop footpoints where the densityand temperature gradients are large. It also allows forthe transition region of an individual loop to evolve dy-namically in time in response to the heating and coolingof the loop as a whole (Johnston et al. 2017a,b, 2019).Despite being a small fraction of the volume of a loop(both because it is confined to near the footpoints andbecause the cross-sectional area of a loop typically in-creases substantially between the high- β photosphereand the low- β corona, e.g.; Guarrasi et al. 2014) thehigher densities in the transition region mean that itemits brightly in the EUV. Therefore, the origin of ob-served EUV emission from lines that emit in the few MKrange is not a priori clear. This emission could originatefrom relatively cooler coronal loops or from the transi-tion regions of much hotter loops. This uncertainty isthe motivation for the present study, to determine howmuch coronal and transition region emission is expectedfrom loop models in the various Atmospheric ImagingAssembly (AIA; Lemen et al. 2012) channels and howthis compares with observations. In Section 2 we brieflydescribe the EBTEL model and the results of varyingloop and heating parameters on the modeled AIA emis-sion. In Section 3 we develop a simple procedure toestimate the transition region contribution in AIA ob-servations and apply it to a number of active regions.We summarize our findings and comment on the impli-cations of these results in Section 4. EBTEL HYDRODYNAMIC SIMULATIONSEBTEL (“enthalpy-based thermal evolution of loops”;Klimchuk et al. 2008; Cargill et al. 2012a,b) modelsthe time evolution of the coronal-averaged temperature,pressure, and density in a single magnetic strand in 0D.It is able to accurately describe subsonic plasma evo-lution under gentle and impulsive heating and can ap-proximately treat complex phenomena such as saturatedheat flux and nonthermal electron beam heating. A sig-nificant feature of EBTEL is its speed; it can computethe evolution of a single magnetic strand for one dayof physical time in seconds, orders of magnitude fasterthan comparable 1D models. Despite the simplicity of the model, EBTEL’s results are very similar to the spa-tial average determined along the length of a 1D sim-ulation (Klimchuk et al. 2008; Cargill et al. 2012a,b).In addition to computing the average coronal proper-ties, EBTEL also determines the coronal and transitionregion Differential Emission Measures (DEMs) at eachtime step. Here we use the EBTEL++ implementationdescribed in (Barnes et al. 2016a) and available online athttps://github.com/rice-solar-physics/ebtelPlusPlus.One of the simplifications necessary in the formulationof EBTEL is an assumed ratio of the radiative losses inthe transition region and corona. In the model, this isrepresented by the semiconstant c = R tr /R c where R tr and R c are the total radiative losses from the transitionregion and corona, respectively. This ratio depends onthe fixed length of the strand, the dynamic plasma tem-perature (which influences the plasma scale height), andthe coronal density ( n ) relative to the static equilibriumdensity for a loop with the same temperature ( n eq ). Atlow coronal densities ( n < n eq ) conduction dominatesthe coronal losses and the relative transition region emis-sion is particularly strong (Barnes et al. 2016a). Athigh densities ( n > n eq ) coronal losses are dominatedby radiation, and therefore the relative emission fromthe transition region is reduced (Cargill et al. 2012a).Ignoring corrections for gravitational stratification anddetails of the radiative loss function, which are includedin EBTEL, this ratio smoothly varies with density be-tween the limits c = R tr R c = n ≤ n eq . n (cid:29) n eq (1)which have been chosen to produce results consistentwith HYDrodynamic and RADiative emission (HY-DRAD) 1D loop models (Bradshaw & Mason 2003a,b;Bradshaw et al. 2004; Bradshaw & Cargill 2013) for awide range of coronal heating scenarios. It is importantto note that while these prescriptions have a control-ling influence on the total transition region and coronalemission, they do not directly impact the intensity ofthe individual channels investigated in this study. Thisis due to the nonuniform temperature response of theAIA channels (discussed in Section 2.4) which resultsin their preferential sensitivity to plasma of particulartemperatures. In a given (real or simulated) observa-tion, a particular channel may measure emission fromthe transition region, corona, or a combination of thetwo, independent of c .EBTEL defines two other constants related to thetemperature profile of a 1D strand. One relates the av-erage coronal temperature in the strand, ¯ T , to the apex Figure 1.
Power-law distributions of heating event delaytimes. The maximum of each distribution is 10,800 s (3 hr). temperature, T a : c = ¯ TT a = 0 . T , to the apex temperature: c = T T a = 0 . . (3)These values were chosen based on hydrostatic 1D mod-els computed with HYDRAD, but they are found tobe reasonable representations when subsonic flows arepresent. Equation 3 is particularly important for thecurrent investigation since we are interested in the dis-tinction between the transition region and corona. Thismeans that the calculated coronal temperature deter-mines the maximum temperature of the transition re-gion, which is assumed to cover all temperatures be-tween T and chromospheric temperatures. We stressthat T is a physically motivated temperature that cor-rectly demarcates the region of steep temperature anddensity gradients at the base of a coronal loop.2.1. Power-law distribution of heating events
In these simulations we heat the plasma with a com-bination of a constant background heating (1% of thetotal energy input) and symmetrical triangular impul-sive heating events. Each heating event has a durationof t e = 100s and a total energy input per unit volume (cid:15) e proportional to the delay time to the next event givenby (cid:15) e = 0 .
99 t d (cid:18) FL (cid:19) = 0 . max t e (4)where t d is the random delay until the next event,Q max is the maximum volumetric heating rate duringthe event, and F and L are the energy flux and strand half length (in centimeters) given in table 1. The fac-tor of 0.99 accounts for the 1% constant backgroundheating. The result of this scaling is that each heat-ing cycle has the same time-averaged volumetric heat-ing rate, which is prescribed assuming that the depositedenergy is evenly distributed over the length of the loop.The individual heating events are randomly drawn frompower-law distributions of heating event delay time (t d )shown in Figure 1. These power laws are defined bytheir exponent ( α , the slope when visualized in log-logspace) and minimum and maximum time delay betweenevents. For all models, the maximum delay time is fixedat three hours (10,800 s), that is, each modeled mag-netic strand experiences an impulsive heating event atleast once every three hours.This numerical scheme represents a physical systemthat is driven with a constant energy buildup rate thatreleases some fraction of this energy when a criticalthreshold value is reached. This is consistent with,for example, critical stress reconnection heating drivenby random-walk footpoint motion (Parker 1988; L´opezFuentes & Klimchuk 2015). In this mechanism, thestress in the magnetic field builds with time until a crit-ical level defined in terms of the angle between adjacentmagnetic strands is reached, at which point they recon-nect and release a fraction of the energy stored in thefield. The more energy that is released, the longer itwill take for the magnetic field to return to the criti-cal stressed state and reconnect again. Note, however,that the prescribed heating scheme used here does notassume any particular physical mechanism and is consis-tent with any heating scenario that builds to a thresh-old level. It also yields similar although not identical(due to the fact that the effects of a heating event aredependent on the physical state of the loop when heat-ing begins) average conditions to systems with constantdriving that build to a random stressed state before re-laxing impulsively to some constant minimum energystate (Cargill 2014). Similarly, it will emulate any sys-tem with a power-law distribution of heating event am-plitudes and delay times.2.2. Modeled parameters
We perform a parameter space exploration over rel-evant physical properties of coronal heating. This in-volves computing EBTEL hydrodynamic models forcombinations of four parameters each in two differentstates for a total of 16 different conditions. These pa-rameters are: the length of the magnetic strand, thetime-averaged energy flux into the base of the strand(related to the time-averaged volumetric heating rate:Q = F / L), the minimum event delay time, and the
Table 1.
EBTEL Model parametersParameter Symbol Low Value High ValueStrand half length [Mm] L 20 80Energy flux [erg cm − s − ] F 5 × × Minimum delay [s] t min
100 1000Power-law slope α − . − . Note —Parameters of models changed for the different simulations.The parameters held constant in all runs are given in table 2. power-law slope of the distribution of delay times. Theparameters explored here represent typical (and by nomeans extreme) ranges for coronal active regions, whereknown. These parameters are listed in table 1 and de-scribed below. 2.2.1.
Strand length
We simulate strands with half lengths (footpoint toapex) of 20 and 80 Mm, sizes typical of observable loopsin coronal active regions (e.g. those examined in Section3). 2.2.2.
Energy flux
The total energy losses from the corona in active re-gions (i.e. the heating necessary for consistency with ob-servations) are ∼ erg cm − s − (Withbroe & Noyes1977) and we heat our models with half and twice thisvalue to simulate weakly and strongly heated regions.2.2.3. Minimum delay between heating events “Time lag” analysis of active regions using AIA obser-vations suggests that the characteristic delay betweensuccessive heating events is similar to the plasma cool-ing timescale (Viall & Klimchuk 2017), which dependsstrongly on the loop length, but is on the order of athousand seconds. On the other hand, theoretical con-siderations of reconnection-based heating suggest delayson the order of a hundred seconds (Klimchuk 2015). Wetherefore test distributions with minimum delay timesof 100 and 1000 s.2.2.4.
Power-law slopes of event delays
Many observational studies suggest that flares occurwith a power-law distribution (e.g. see discussion in;Parnell & De Moortel 2012), and power-law distribu-tions of nanoflares can explain the observed range inDEM slopes coolward of the emission measure peakfound in active regions (Cargill 2014). Many theoret-ical models have also suggested that nanoflares occurwith a power-law distribution in energy, from a simple
Table 2.
EBTEL fixed model parametersKeyword (description)
Valuetotal time (seconds) tau (initial time step, seconds) . (maximum time step, seconds) (electron-ion equilibrium) Trueuse c1 loss correction Trueuse c1 grav correction Trueuse power law radiative losses Trueuse flux limiting (for conductive cooling)
Falseuse adaptive solver (for dynamic tau)
Trueadaptive solver error 1 × − adaptive solver safety 0 . (c1 during conductive cooling) . (c1 during radiative cooling ) 0 . . (relative to solar) . (1 = electron, 0 = ion) . Note —Relevant EBTEL parameters held constantfor all simulations. More detailed descriptions ofthese keywords are provided through the EBTEL++github repository at https://rice-solar-physics.github.io/ebtelPlusPlus/configuration/ cellular automaton (L´opez Fuentes & Klimchuk 2015)to full three-dimensional magnetohydrodynamic (MHD)simulations (Knizhnik et al. 2018). These models andobservational considerations typically find nanoflare en-ergy distributions with power laws of − . (cid:46) α (cid:46) − . α ≈ −
1. Consequently, our models test heatingevent power laws with α = − α = − .
5. Due tothe proportionality between the delay time and eventenergy (Section 2.1), the energy input from the powerlaws with α = − . α = − Model results
Due to EBTEL’s speed, we are able to simulate a largeamount of solar time in relatively little computationaltime for this study. Each EBTEL model is run for 10 sof solar time and 1000 models with random realizationsof impulsive heating are run for each set of parameters Figure 2.
Time evolution of coronal parameters in EBTEL models of individual magnetic strands.
Left : a strand with ahalf length of L = 20 Mm, average energy flux of F = 5 × erg cm − s − , minimum delay between events of t min = 100s, and a power-law distribution of event sizes with a slope of α = − . Right : a strand with a half length of L = 80 Mm,energy flux of F = 2 × erg cm − s − , minimum delay between events of t min = 1000 s, and a power-law distribution ofevent sizes with a slope of α = −
1. The top panels indicate the volumetric heating rate, the middle panels indicate the coronalelectron temperature, and the bottom panels indicate the coronal electron density. The gray vertical lines mark the end of theequilibration period after which the runs are averaged. to provide a robust average and standard deviation. Intotal, 1 . × s of coronal loop evolution are simulated.Those EBTEL parameters that remain constant acrossall simulations are listed in table 2.The evolution of two of these models is shown in Fig-ure 2. For each of these models, the plasma under-goes many heating and cooling cycles in a single run.Some notable (and expected) features of these simula-tions include: the typically smaller, more frequent heat-ing events in the model with the shorter minimum delayand steeper distribution of event sizes; the more con-sistent plasma temperature and density resulting fromthese more consistent heating events; the more rapidcooling in the shorter strand; and the higher plasmatemperatures in the more strongly heated strand withlarger heating events.While the plasma in these models is evolving on in-dividual magnetic strands, the observable signatures ofthis heating are due to the combination of many hun-dreds or thousands of such strands evolving within asingle resolution element. In addition, because each ofthese strands is evolving in isolation (due to the ex-tremely high ratio of parallel to perpendicular heat con-duction along the magnetic field (van den Oord 1994)), the time average of the evolution of a single strand isequivalent to the average of a snapshot of many strandsat different phases of their heating and cooling cycles.Because of this equivalence, we not only average all 1000runs with each set of parameters together, we also aver-age each run over the duration of its evolution, exceptfor the first 10 s that are discarded to ensure that theinitial conditions of each run have no impact on the re-sults.The average density and temperature for each set ofmodeled parameters is given in Figure 3. We can be-gin to understand the trends by examining the modelswith the highest frequency of heating events, which mostclosely resemble steady heating. These are the caseswith the shortest minimum delay times (t min = 100 s)and steepest distributions ( α = − . T ∝ L / Q / ∝ ( LF ) / (5)where Q = F / L is the volumetric heating rate. Thedensity of that same loop scales as: n ∝ L − / F / (6) Table 3.
High-frequency heating models compared with loop equilibrium scaling lawsL [Mm] F [erg cm − s − ] T [MK] ¯ T theory [MK] n [10 cm − ] n theory [10 cm − ]80 2 × . ± .
08 5 .
21 3 . ± .
04 3 . × . ± .
05 3 .
45 1 . ± .
02 1 . × . ± .
12 3 .
45 6 . ± .
22 6 . × . ± .
07 2 .
26 2 . ± .
08 2 . Note —Temperature and density scaling of EBTEL models with t min = 100 s and α = − . T ) and density ( n ) are compared with the theoretical temperature ( ¯ T theory) anddensity ( n theory) determined for the last three models by applying the scaling laws in referenceto the first model. Figure 3.
Average coronal plasma density ( top ) and tem-perature ( bottom ) for the 16 tested combinations of thestrand parameters. Each simulation is labeled and also in-dicated by the combination of location (left or right panel),color (blue or orange), pattern (solid or stripped), and shad-ing (filled or empty). The black error lines at the top of eachbar indicate the standard deviation as determined by con-sidering the time average of each of the 1000 model runs asa single sample. assuming a radiative loss function with power-law slope β = − . T a and n a , whichhave the same scaling but slightly different constants ofproportionality. We fit the four high-frequency heatingmodels (t min = 100 s and α = − .
5) with linear re-gressions between the modeled and theoretical values todetermine that the constants of proportionality in equa-tions 5 and 6 are 0 .
013 and 0 . β rather thana single value for all temperatures. Models with lowerevent frequency (longer minimum delay and shallowerdistributions) have lower average temperatures and den-sities than the corresponding higher frequency runs. Atfirst this might seem surprising, since high energy eventsthat occur less often produce higher peak temperatures,such as seen in Figure 2. However, the strands coolquickly at these high temperatures and spend the ma-jority of their time in a much cooler state, also character-ized by lower density. This dominates the time averages.2.4. Predicting AIA intensities
Two of the standard products of the EBTEL simula-tions are the time dependent DEMs of the corona andtransition region. These are the plasma density squaredas a function of temperature integrated through theirrespective portion of the modeled atmosphere. EBTELassumes the coronal DEM at any given time is narrowlyand uniformly distributed around the average coronaltemperature in the strand ( ¯ T ). The transition regionDEM is spread between T = 0 . T a = 0 .
67 ¯ T and chro-mospheric temperatures and has a form determined bythe energy balance between thermal conduction, radi-ation, and enthalpy. Using the DEMs, we can simu-late the expected EUV intensity from each componentof the atmosphere. We use the temperature responsefunctions of the “coronal” AIA channels shown in Fig-ure 4 to compute the average coronal and transitionregion intensities of the simulated strands. These re-sponse functions are generated using the IDL routine aia get response.pro version 8 that utilizes version Figure 4.
Temperature response functions of the six “coro-nal” AIA imaging channels. Note that the temperature re-sponse of the 171 ˚A, 193 ˚A, and 211 ˚A channels is concen-trated near a single temperature (quasi-isothermal), whilethe 94 ˚A, 131 ˚A, and 335 ˚A channels have significant re-sponse at two or more temperatures.
C/TR .The results are shown in Figure 5 for all six “coronal”AIA channels and all 16 models. Note that the modelswith strand half lengths of 20 Mm and 80 Mm, corre-sponding to semicircular apex heights of ≈
13 and ≈ Figure 5.
Ratios of coronal to transition region emission insix “coronal” AIA channels for the 16 tested combinations ofthe strand parameters. Each simulation is labeled and alsoindicated by the combination of location (left or right panel,with different scales), color (blue or orange), pattern (solidor stripped), and shading (filled or empty). The black errorlines at the top of each bar indicate the standard deviationin the ratio as determined by considering the time averageof each of the 1000 model runs as a single sample. Note thatthe ratios for the 20 Mm strands are much larger than theratios for the 80 Mm strands.
Differences in coronal brightness between models are notdue to differences in depth.The results from Figure 5 yield the following generaltrends. In interpreting these trends, it is important tokeep two things in mind. First, at any given time duringthe evolution of a strand, the transition region temper-ature extends to more than half of the apex (maximum)temperature in the strand (equation 3). Second, theclassification of heating frequency into high, intermedi-ate, and low is based on the delay between successiveheating events relative to the plasma cooling time. • In all cases, the ratio is much larger in the 20 Mmstrand than the 80 Mm strand. This is due partlyto the 40 Mm coronal depth scaling describedabove. The coronal intensity used in the ratio isover and under represented in the short and longstrands, respectively, compared to the full strandlength simulated with EBTEL. There is an addi-tional real effect. During a low to intermediatefrequency heating and cooling cycle, the transi-tion region emits in a narrow temperature bandcentered on T for the entire time that the apex iscooling from its peak value to approximately 2 T .The corona, on the other hand, emits at this tem-perature only for the short time that it takes thecoronal plasma to cool through the band. Strandsthat start their cooling from a higher peak temper-ature are therefore expected to have a smaller ra-tio of corona to transition region intensity. Longerstrands tend to reach higher temperatures. Withstrong impulsive heating, the temperature rises tothe point at which thermal conduction cooling bal-ances the energy input. This determines the max-imum apex temperature. We can estimate thistemperature from Q = F / L ∝ T / a / L , whichshows that T a ∝ (FL) / . • In the 171 ˚A, 193 ˚A, 211 ˚A. and 335 ˚A chan-nels, R
C/TR is smaller with the larger energy flux,all else equal. This can also be explained bythe argument above. Larger F implies hotter T a ,which means that the transition region radiates forlonger. The 94 ˚A and 131 ˚A channels often displaythe opposite effect, which may be due to their sec-ond, high-temperature peaks. The 193 ˚A channelalso has a second, high-temperature peak, but itsreduced sensitivity compared to the primary peakand its very high temperature mean that it hasa negligible influence on the channel response inthese modeled loops and solar observations out-side of flares. • In general, the channels with higher temperatureresponses (94 ˚A, 211 ˚A, and 335 ˚A) have largerratios than the channels with cooler temperatureresponses (131 ˚A, 171 ˚A, and 193 ˚A). A variationof the above argument applies here. The maxi-mum apex temperature of a strand is of course thesame, regardless of the observing channel. A given T that begins in the transition region at the startof cooling switches to being in the corona whenthe apex cools to approximately 2 T . This hap-pens sooner for larger T , so the transition regionemission turns off more quickly in hotter channels,and R C/TR is larger. Real channels are of coursesensitive to a broad range of temperatures, but thebasic concept applies. • For cases with α = − t min has almost no effect.This is because the energy input is dominated bylarger heating events with longer delay times. • For cases with α = − . t min has a large effect,particularly for the 20 Mm strands. This is due tothe cooling time of a 20 Mm strand being of order1000 s, and therefore these small-event-weighteddistributions are heated in either a high- or low-frequency regime depending on the choice of min-imum cutoff. The effect for the 80 Mm strands isless pronounced since even 1000 s is less than thecooling time. • In the 80 Mm strands, the ratios are largest inthe low-frequency heating scenarios (with the ex-ception of the 94 ˚A channel in strands experi-encing high energy flux). This is consistent withthe findings from Patsourakos & Klimchuk (2008)that found impulsive (nonstatic equilibrium) heat-ing produced larger corona to footpoint ratios inTRACE observations. • The arguments above do not apply to models withhigh-frequency heating, since they experience min-imal cooling. Plasma that begins in the coronastays in the corona, and plasma that begins inthe transition region stays in the transition region.R
C/TR still has a strong temperature sensitivity,but for a different reason. Higher temperaturechannels are better “tuned” to the corona than tothe transition region, so the ratio is larger. A goodexample is the scenario with L = 20 Mm, averageenergy flux of F = 5 × erg cm − s − , mini-mum delay between events of t min = 100 s, and α = − .
5. The time-series for one of these modelsis shown in the left panel of Figure 2 which illus-trates that the temperature is tightly constrainedaround the average of 2 . C/TR in the 211 ˚A and 335 ˚A (whichalso has significant sensitivity at these tempera-tures) channels while yielding the lowest R
C/TR for models with the same energy flux in the otherchannels.Overall, Figure 5 clearly demonstrates that EBTELmodels of the solar atmosphere indicate both that thetransition region contributes significantly to the inten-sity of AIA observations and that this contribution hasstrong dependence on the details of the underlying coro-nal heating. In the channels with strong response to thelowest temperatures, particularly 131 ˚A and 171 ˚A, thisanalysis suggests that the majority of observed emissioncould be due to plasma more accurately attributed tothe transition region than the corona, for a wide rangeof loop lengths. This is also true of the hotter channelsin the long loops. Furthermore, in every channel except335 ˚A, R
C/TR is different by more than a factor of 2 forcertain combinations of minimum delay and event dis-tribution power law for a given loop length and energyflux. While these results are difficult to apply directlyto the interpretation of observational data, as explainedin Section 3, they highlight the importance of consider-ing contributions from the transition region when usingobservations to characterize coronal heating.Before proceeding to consider observations, we notethat Patsourakos & Klimchuk (2008) used EBTEL sim-ulations to investigate the coronal and transition regionemission as observed in the 171 ˚A channel of the Tran-sition Region And Coronal Explorer (TRACE; Handyet al. 1999). Their approach differs from ours in thatthey treated observations near the limb, assuming thatthe line of sight is perpendicular to the plane of thestrand, and spreading the transition region emission over2 Mm vertically from the solar surface. They found in-tensity ratios of about 1 /
600 and 1 /
35 for steady andlow-frequency impulsive heating, respectively, in a 25Mm (half length) strand. These ratios correspond to ∼ .
03 and ∼ . AIA OBSERVATIONSSince the launch of the Solar Dynamics Observatory(SDO; Pesnell et al. 2011) in 2010, the AIA (Lemen et al.2012) has become the default imager for studies of thesolar corona. However, as demonstrated in Section 2.4,a significant portion of the light observed in the AIA channels may originate in the transition region ratherthan the corona. In the following sections, we makesimplifying assumptions about the geometry of observedactive regions to distinguish the observed coronal andtransition region contributions to the six “coronal” AIAchannels.3.1.
Observationally separating the corona andtransition region
On the Sun, a single line of sight typically passesthrough the coronae of one set of strands and the tran-sition regions of an entirely different set of strands, notthe corona and transition region of the same strand, asassumed in the modeling described in Section 2. This isonly a minor concern for understanding coronal heatingif the strands are similar, but that is often not the case.Instead, to compare with the modeled magnetic strands,we must investigate multiple lines of sight containing ob-served coronal and transition region emissions that arephysically linked by the magnetic field. This is possi-ble whenever individual loops, or collections of loops,and their associated footpoint(s) can be identified in animage.An example for active region NOAA 11268 is shown inFigure 6. We select this region because of its widely sep-arated bipole magnetic field structure with easily iden-tifiable loops that clearly terminate in a compact con-centration of strong photospheric magnetic fields. Inaddition, the loop top region we identify as a sample ofthe corona (blue box) has very weak photospheric mag-netic fields along the line of sight, suggesting that therewill be very little contribution from transition regionplasma associated with other structures. The smaller or-ange box identifies the transition region footpoints thatwe associate with these loops. We analyze the averageover five minutes of full cadence (12 s) data in orderto minimize the impact of any particularly short-termvariability within the region. While this average mayincorporate multiple complete heating cycles (e.g. ift min = 100s) we expect no information loss from thisprocedure due to the inherent averaging in the observa-tions caused by the many overlapping and out of phasestrands along a line of sight. This 5 minute averaging isconsistent with the procedure from Warren et al. (2012)discussed in Section 3.2.The average intensities within the boxed regions areused to determine the characteristic coronal and tran-sition region intensities of the prominent loops in thisregion. Because the photospheric magnetic field withinthe blue box resembles that within the quiet-Sun, wesubtract the average intensity of the quiet Sun (iden-tified by the green boxes in the upper and lower left0
Figure 6.
The HMI line-of-sight magnetogram and the six “coronal” AIA channel observations of active region NOAA 11268.Each of the AIA images are five minute averages of full cadence data. The green squares indicate the regions designated as quietsun, the blue square indicates the loop tops in the corona, and the orange rectangle indicates the footpoints and transition regionof these same loops. The blue and red contours in the AIA images indicate the extent of the ±
200 G photospheric line-of-sightmagnetic field. corners) from the intensity of the corona (blue box).This has very little impact on the analysis because thequiet sun intensity is small compared to the loop inten-sity in these channels. We make two different assump-tions about the source of the intensity in the orangebox that we call the transition region. First, we as-sume that all of the emission comes from the transitionregion. Second, we acknowledge that some of the in-tensity is due to the overlying corona, and assume thatthe coronal component is identical to that in the bluebox. This is likely an overestimate because we expectcoronal emission to diminish from the polarity inversionline outward, both horizontally and vertically, becausethe heating rate varies directly with the magnetic fieldstrength. Shorter strands tend to be brighter — due totheir increased density, as seen in Figure 3 — and theline of sight intersects more short strands in the bluebox than in the orange box. See Figure 1 in Klimchuk& Bradshaw (2014). The observed R
C/TR using both assumptions aboutthe contribution of the corona to the orange box is plot-ted for each channel in Figure 7. In all cases, the bluebars indicate that the transition region is brighter thanthe corona. When we take into account that there willbe some contribution from the corona in the box iden-tified as the transition region, the ratio increases, andsignificantly in the case of the 211 ˚A and 335 ˚A channels.This is not surprising since, in this small active region,we might expect these two relatively hotter channels tobe the brightest in the corona, as can be seen in theimages. In reality, the true ratios likely fall somewherebetween the blue and orange bars in Figure 7.While we do not anticipate any single EBTEL modelwill agree with the ratios observed in this active region,because it contains contributions from a large numberof magnetic strands of differing length and, presumably,heating properties, it is encouraging to see the same gen-eral trends as those identified in the models. Regardlessof the foreground coronal subtraction, the transition re-1
Figure 7.
Observed R
C/TR in the six “coronal” AIA chan-nels for active region NOAA 11268. The blue bars representa scenario where there is no overlying corona in the transitionregion (orange box) while the orange bars assume that theidentified coronal intensity (blue box) is also present in thetransition region. The true ratios likely fall between thesetwo representations. gion is brighter than the corona in the 131 ˚A and 171 ˚Achannels that are sensitive to lower temperature plasmaand the corona is relatively brighter in the 211 ˚A and335 ˚A channels that are sensitive to hotter plasma. The193 ˚A channel samples intermediate temperatures andexhibits an intermediate ratio. The fact that the 94 ˚Aratio closely resembles that of 171 ˚A and 193 ˚A suggeststhat its emission is dominated by the low temperaturepeak in its temperature response function (Figure 4) andtherefore that there is less plasma near ∼ ∼ The impact of loop geometry
In addition to the single strand/multistrand differ-ence, there are geometrical effects that impact the com-parison of the modeled and observed intensity ratios.The observed loops, or at least their envelope, appearto be considerably more compact (particularly in lati-tude) at their footpoints than at their apexes. Hence,the orange transition region box is smaller than the bluecoronal box. The intensities that are used in the ratiosare the spatial averages over the boxes. The coronalvalue is smaller than would be the case if all the emissionwere confined to a smaller area, i.e., an expanding ver-sus nonexpanding loop. Since the models do not accountfor this effect, the modeled corona-to-transition regionintensity ratios would need to be decreased for a moredirect comparison with the observed ratios. Another ge-ometric difference is that the models assume a coronalpath length of 40 Mm, whereas the line-of-sight depth ofloops within the blue box could be larger or smaller. Fi- nally, the coronal values from the models are the spatialaverages along a strand, whereas the observed coronalintensities are near the loop apexes. Gravitational strat-ification would suggest that the modeled ratios shouldbe decreased somewhat for a more direct comparisonwith the observations, particularly for the 80 Mm loops.Since the modeled ratios are, if anything, too smallcompared to the observations, these corrections wouldmake the discrepancy worse. However, we stress thatthe modeled ratios are highly idealized. The point ofthe present study is not to reproduce the observationsas closely as possible, but rather to (1) demonstrate thatthe transition region makes an important contributionto intensities observed in AIA “coronal” channels and(2) demonstrate that R
C/TR is sensitive to the detailsof the heating and therefore has diagnostic potential.In future work, we will construct more realistic modelsalong the lines of those in e.g., Warren & Winebarger(2006, 2007); Lundquist et al. (2008a,b); Bradshaw &Viall (2016); Nita et al. (2018); Barnes et al. (2019).3.2.
Analyzing the Warren et al. (2012) active regions
For each of the 15 active regions studied in Warrenet al. (2012) (except their Region 13 Box 2) we repeatthe analysis performed in Section 3.1. The regions iden-tified as coronal are those defined and analyzed in theoriginal paper while the transition region boxes are de-termined by eye based on the apparent connectivity ofthe loop features in each region. These active regionsand the associated boxes indicating the corona, transi-tion region, and quiet Sun are shown in Figure 8. Weexpect that the wide range of active region structuresand viewing geometries represented in this sample willminimize any particular geometrical bias introduced byanalyzing a single region. In addition, these regions rep-resent a wide range of physical scales with potentiallydifferent heating properties.We compute R
C/TR in each channel for each activeregion individually. The distributions of these ratios areplotted in Figure 9. Notice that while the ratios are onaverage larger than the ratios found in NOAA 11268,in most cases the transition region is still brighter thanthe corona. Only in the 94 ˚A and 335 ˚A channels isthis not generally the case. Previous analysis of theseactive regions determined that they have DEMs peakingbetween log(T [K]) = 6 . .
6, where the 94 ˚A and 335˚A channels have the highest relative response. It is notsurprising, therefore, that R
C/TR is greatest in thesechannels. While there is some plasma above the DEMpeak, the slopes of the DEMs are quite steep, and thereis very little plasma at log(T [K]) ∼ .
1, the temperatureof the strong secondary peak in the 131 ˚A channel. The2
Figure 8.
AIA 211 ˚A five minute average images of the active regions from Warren et al. (2012). The boxed regions andcontours highlight the same features as in Figure 6. The bottom right panel shows the individual region from Figure 6 forcomparison.
131 ˚A intensity ratios are consequently smaller, althoughstill elevated compared to NOAA 11268.Warren et al. (2012) measured the power-law index ofthe DEM distribution in the range 6 . ≤ log(T) ≤ . α DEM , in alog-log plot. We compare the coronal DEM slopes withR
C/TR . A sample of these relationships is shown for the171 ˚A and 335 ˚A channels in Figure 10. There is a clearanticorrelation in the 171 ˚A channel, in which larger in-tensity ratios correspond to smaller slopes, i.e., flatterDEM distributions. The same trend appears in the 335 ˚A channel, but with much larger scatter. To quantifythe trends, we perform multiple statistical analyses, asreported in table 4. Because the distributions appearapproximately linear, we compute the Pearson correla-tion coefficient. The negative coefficients indicate theinverse relationships while the larger magnitudes of the171 ˚A, 193 ˚A, and 211 ˚A channels indicate tighter cor-relations (less scatter).One disadvantage of the Pearson analysis is that itassumes that the measured quantities are normally dis-tributed, i.e., that the errors in the measurements fol-low a normal distribution. We have no indication that3
Figure 9.
Observed R
C/TR in the Warren et al. (2012)active regions. The green lines indicate the median of allthe active regions, the box indicates the lower and upperquartiles, and the whiskers indicate the extremes. This plotis equivalent to the blue bars in Figure 7
Figure 10.
Correlation between R
C/TR and α DEM identifiedin Warren et al. (2012) for each active region in the 171 ˚Aand 335 ˚A channels. The best-fit linear relationship anderror region for each channel are plotted to guide the eye.Statistics about the relationship between R
C/TR and α DEM for each channel are given in table 4. this is or is not the case. We therefore also performa nonparametric, or rank ordered, statistical analysis,which is valid for any measurement distribution. We usethe weighted t-statistic described in Efron & Petrosian(1992), following the implementation in Porter & Klim-chuk (1995). The probability that α DEM and R
C/TR are random is given by P(tw) in table 4. A small valueindicates a high probability of correlation. The fourth
Table 4.
Correlation between α DEM and R
C/TR
AIA channel r P(tw) χ χ
94 ˚A − .
21 0 . − . − .
06 : 0 . − .
45 0 . − . − .
78 : − . − .
78 0 . − . − .
42 : − . − .
73 0 . − . − .
82 : − . − .
68 0 . − . − .
37 : − . − .
42 0 . − . − .
16 : 0 . Note —r is the Pearson correlation coefficient between α DEM and R
C/TR . P(tw) is the probability of drawingthe observed distribution from a uniform random distri-bution. χ is the most probable exponential in the rela-tionship R C/TR ∝ ( α DEM ) χ . χ is the 90% confidenceinterval of χ . column indicates the most probable χ in the assumed re-lationship R C/TR ∝ ( α DEM ) χ , and the final column givesthe 90% confidence interval of χ . From these analyses,we see that all channels except 94 ˚A and 335 ˚A haverobust inverse correlations. The relationship between α DEM and R
C/TR in the 94 ˚A and 335 ˚A channels islikely random, which could be due to their significantlynonisothermal temperature response functions. Again,the 131 ˚A channel is functionally isothermal in these ob-servations because there is very little plasma above 10MK in these regions.Warren et al. (2012) also measured the slopes of thecoronal DEM with log(T) ≥ .
6, hotter than the peak.We compare those slopes with the intensity ratios andfind no significant correlation in any channel.We can offer a partial explanation for the robust in-verse correlation between α DEM and R
C/TR in the 131 ˚A,171 ˚A, 193 ˚A, and 211 ˚A channels. Consider, for exam-ple, the 211 ˚A channel with a peak response at 2 MK.This channel measures the corona of loops with coro-nal temperatures near 2 MK but the transition regionof loops with coronal temperatures near 4 MK. Funda-mentally, R
C/TR in a given channel correlates positivelywith emission measure at the peak of the temperatureresponse function and negatively with emission measureat temperatures greater than about twice the peak ofthe temperature response function. However, the exactexplanation depends on the frequency with which theplasma is heated. • In the case of high-frequency heating, individualstrands evolve very little. A shallow coronal DEMslope (small α DEM ) indicates that nearly as manystrands are held at a quasi-constant coronal tem-4 perature of, say, 2 MK as are held at a quasi-constant coronal temperature of 4 MK. A steepslope (large α ) indicates the dominance of hotstrands. Consequently, R C/TR will be smaller (rel-atively brighter transition region) when the slopeis steep (relatively more hot strands). • For low-frequency heating, the same argument asdiscussed in Section 2.4 applies. Strands experi-encing low-frequency heating that begin their cool-ing from higher initial temperatures have smallerR
C/TR . Because α DEM is calculated over a fixedtemperature range (6 . ≤ log(T) ≤ . . α DEM ) and result in larger R
C/TR . • In the intermediate frequency heating regime,strands cool partially before being reheated. Asteep DEM slope (large α DEM ) indicates that rel-atively more strands begin their cooling at a highermaximum temperature and/or are reheated beforecooling to lower temperatures. The same argu-ments that explain the anticorrelation betweenR
C/TR and α DEM in the low-frequency heatingcase apply here, with an additional, reinforcingeffect. If the coronal segment of a strand nevercools through the peak response of a given chan-nel, that channel will collect even less coronalemission leading to a smaller R
C/TR .These effects are not uniform across all AIA channelsand depend on the shape of the temperature responsefunction, but apply generally to the 131 ˚A, 171 ˚A, 193˚A, and 211 ˚A channels that are quasi-isothermal withpeak response below 2 MK. We also note that no modelin Section 2 has exclusively high-, intermediate-, or low-frequency heating. They all include a mixture of thethree, with the relative proportions being different frommodel to model. The same is likely true in these ob-served active regions. CONCLUSIONUsing the computational efficiency of EBTEL model-ing and active regions studied by Warren et al. (2012) weinvestigated the theoretical and observed contribution ofthe transition region to AIA images. For this analysis,we defined the transition region from a physically mean-ingful perspective as the volume of the solar atmosphereabove the chromosphere that is heated (while the coronais cooled) by thermal conduction, rather than more tra-ditional observational definitions based on plasma tem-perature. With this definition, the transition region is confined to low altitudes, as in the conventional picture.This study involved two major investigations: an explo-ration of the parameter space of relevant coronal heatingvariables, with particular focus on the frequency of im-pulsive heating events, and a study of observed activeregions to provide an observational anchor for the mod-els.The EBTEL models revealed that, consistent withprevious studies (e.g.; Patsourakos & Klimchuk 2008),imaging observations often described as “coronal” areexpected to have significant contribution from transi-tion region plasma. We find that the ratio of coronalto transition region emission is very different for theindividual AIA channels and depends strongly on theheating parameters, demonstrating promising diagnos-tic potential. In general, we find that those scenar-ios with higher frequency heating events lead to highertime-averaged coronal temperatures and densities, butlower maximum temperatures and densities. However,observed intensities depend on the full DEM distribu-tion, including both the coronal and transition regioncontributions, and it is not possible to easily predictthe brightness in a channel based on the time-averagedcoronal temperature and density alone. We also findthat those strands subjected to the highest frequencyheating agree quite well with theoretical expectations forcoronal loops in static equilibrium. Overall, our analy-sis suggests that in shorter strands, the emission fromthe transition region and corona are comparable, whilethe emission from long strands tends to be dominatedby the transition region, particularly in the higher fre-quency heating scenarios.We performed a simple analysis of observed AIA ac-tive regions, comparing the intensity of emission fromcoronal and transition region plasma identified based ontheir morphology and relation to photospheric magneticfields. Analyzing observations of active region NOAA11268, we find an overall consistency with the models.The observations confirm the general trend in the mod-els that the 335 ˚A, 211 ˚A, and sometimes 94 ˚A channels(i.e. those associated with the hotter plasma) have thelargest ratios and the 131 ˚A, 171 ˚A, and 193 ˚A chan-nels have the smallest ratios. The observed ratios de-pend, however, on assumptions about how much overly-ing coronal emission is present above the footpoint tran-sition region emission. These same observational trendspersist when analyzing the 15 active regions from War-ren et al. (2012), although they have generally higherratios. All of these active regions suggest that AIA ob-servations of loops sample a similar level of emissionfrom the corona and transition region.5We also analyzed the relationships between R
C/TR and the slopes of the DEMs determined by Warren et al.(2012). We find that there is a consistent negative re-lationship between the slope of the DEM coolward ofthe temperature peak and R
C/TR in the observed re-gions. This is consistent with theoretical expectationsbased on low, intermediate, or high frequency impulsiveheating.We note that, particularly for the longer 80 Mmstrands, the models suggest that the ratio of coronal totransition region intensity should be significantly smallerthan is observed. One potential explanation for thisis the absorption of transition region emission fromspicules extending from the underlying chromosphere.This has been found to cause up to a factor of 2 de-crease in the observed transition region intensity (DePontieu et al. 2009), which would increase the observedratios compared to model predictions, consistent withour findings.We made no attempt to ascribe a particular heat-ing model to the studied active regions because indi-vidual zero-dimensional EBTEL models are inadequateto properly characterize the complexity of active regionobservations. It is unreasonable to expect the modelof a single magnetic strand to replicate observationsfrom even simple active regions. In addition, there isambiguity due to the somewhat arbitrary choice of ob-servational path length assigned to the coronal emis-sion in the EBTEL models. Both of these uncertainties can be largely resolved by studying this effect in three-dimensional models of active regions where the true ex-tent of the corona can be more accurately estimated.We have begun to construct such models, based on ob-served photospheric magnetograms, using the approachdescribed in Nita et al. (2018).Despite the idealized nature of the modeling and ob-servational analyses presented here, they clearly demon-strate the importance of considering the transition re-gion in active region models, particularly when they areused to study coronal heating. Depending on how theactive region is heated, failing to include the transitionregion could lead to significant underestimation of theAIA emission from the region.Data supplied courtesy of the SDO/HMI and SDO/AIAconsortia. SDO is the first mission launched for NASA’sLiving With a Star (LWS) Program. EBTEL++ is de-veloped and maintained by the Rice University SolarPhysics Research Group. The authors would like tothank Harry Warren for providing data from Warrenet al. (2012). SJS’s research was supported by an ap-pointment to the NASA Postdoctoral Program at theGoddard Space Flight Center, administered by Univer-sities Space Research Association under contract withNASA. This work of JAK was supported by the God-dard Space Flight Center Internal Scientist FundingModel (competitive work package) program.REFERENCES
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