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The Astrophysical Journal, 590:547–553, 2003 June 10 # 2003. The American Astronomical Society. All rights reserved. Printed in U.S.A.

CONSTRAINTS ON ACTIVE REGION CORONAL HEATING S. K. Antiochos and J. T. Karpen E. O. Hulburt Center for Space Research, Naval Research Laboratory, Washington, DC 20375; [email protected]

and E. E. DeLuca, L. Golub, and P. Hamilton Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 Received 2002 September 23; accepted 2003 February 26

ABSTRACT We derive constraints on the time variability of coronal heating from observations of the so-called active region moss by the Transition Region and Coronal Explorer (TRACE). The moss is believed to be due to million-degree emission from the transition regions at the footpoints of coronal loops whose maximum temperatures are several million degrees. The two key results from the TRACE observations discussed in this paper are that in the moss regions one generally sees only moss, not million-degree loops, and that the moss emission exhibits only weak intensity variations,
10% over periods of hours. TRACE movies showing these results are presented. We demonstrate, using both analytic and numerical calculations, that the lack of observable million-degree loops in the moss regions places severe constraints on the possible time variability of coronal heating in the loops overlying the moss. In particular, the heating in the hot moss loops cannot be truly flarelike with a sharp cutoff, but instead must be quasi-steady to an excellent approximation. Furthermore, the lack of significant variations in the moss intensity implies that the heating magnitude is only weakly varying. The implications of these conclusions for coronal heating models will be discussed. Subject headings: Sun: corona — Sun: transition region — Sun: UV radiation On-line material: mpg animation

the transition region emission appears as a bright twodimensional surface lying on the chromosphere with no obscuration from overlying hot plasma or contamination from underlying cool material, since their emission generally falls outside a narrow TRACE passband, such as the ˚ band. The coronal emission, however, emanates from 171 A a complex fully three-dimensional system of loops. Hence, it is much easier to isolate the evolution of a single loop by observing its transition region. Furthermore, as we show below, the transition region emission is sensitive to the magnitude of the loop heating, so it is an excellent monitor of changes in this heating. We used this approach in a previous work to place constraints of the temporal evolution of flare heating (Antiochos et al. 2000a). In this paper, we show that the TRACE data place important new constraints on the temporal variability of active region loop heating.

1. INTRODUCTION

The mechanism for heating the Sun’s corona has been one of the most intensely studied problems in solar physics during the past 50 years, and yet it remains one of the least understood (e.g., Bray et al. 1991; Mandrini, Demoulin, & Klimchuk 2000). A major obstacle is that we cannot observe directly the structure and dynamics of the coronal magnetic field, which is widely believed to be responsible for the heating. We can only attempt to infer the action of the field by observing its effects on the plasma. Another major difficulty is that the heating is likely to occur on small spatial and temporal scales. Consequently, we need observations with the highest possible spatial, temporal, and spectral resolution if we hope to derive useful constraints on the heating mechanism. At present, the highest resolution observations are from the Transition Region and Coronal Explorer (TRACE), ˚ formed at which can obtain images in the line of Fe ix 171 A 00 6 10 K with a spatial resolution better than 1 and temporal resolution better than 30 s (Golub et al. 1999). There have been numerous studies using UV–X-ray observations from TRACE and other satellites to derive constraints on the spatial structure of coronal heating (e.g., Kano & Tsuneta 1995; Priest et al. 2000; Lenz et al. 1999; Aschwanden, Schrijver, & Alexander 2001; Schmelz et al. 2001), as well as several studies on the temporal structure (e.g., Schrijver et al. 1999; Reale et al. 2000; Klimchuk & Cargill 2001). These studies have tended to focus on observations of the coronal sections of loops, because that is where the bulk of the plasma resides. However, we have argued that for inferring constraints on the heating, it may well be more effective to observe transition region plasma rather than the bulk coronal plasma (Antiochos et al. 2000a). The key point is that

2. TRACE OBSERVATIONS

One of the most interesting new findings from TRACE is the so-called moss (Berger et al. 1999). The moss is observed in the interior—near the polarity inversion line—of large active regions, and is believed to correspond to the 106 K transition region of hot loops with coronal temperatures
ð3 5Þ  106 K. Comparison of TRACE and Yohkoh images shows that the TRACE moss regions do appear to lie at the base of the multi-million–degree Yohkoh loops, and the relative intensity of the images is in good agreement with what one expects for the ratio of transition region to coronal emission for a hot loop (Martens, Kankelborg, & Berger 2000). ˚ image of active region Figure 1 shows a TRACE 171 A 9169 on 2000 September 23. The corresponding magnetic 547

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˚ image of active region 9169 on 2000 September Fig. 1.—TRACE 171 A 23. Also shown are close-ups of two regions that have been selected for analysis. This figure is also available as an mpeg animation in the electronic edition of the Astrophysical Journal.

field observations from SOHO-MDI (Scherrer et al. 1995) are presented in Figure 2. We note that the TRACE image is dominated by moss that overlies the large region of negative polarity near the polarity-inversion line. To verify that this moss does, indeed, correspond to the 1 MK transition

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region of high-temperature loops, in Figure 3 we show two SXT images (Acton et al. 1992). We note that AR 9169 was near disk center at this time. We also note that the TRACE moss region clearly corresponds to a bright Yohkoh loop region, in particular, the transition region at the negative polarity footpoint of hot loops that cross the polarityinversion line of Figure 2. As can be seen in Figure 1, the moss resembles a honeycomb-like pattern lying on the chromosphere. When observed with high time cadence,
30 s, this honeycomb pattern appears to be constantly moving on a small scale, while still maintaining its large-scale structure. We have constructed a movie from the TRACE images for the active region of Figure 1 and included it in the electronic version of this paper, movie 1. Figure 1 is the first image in movie 1. The time series of images spans 5 hr with an average cadence of 30 s between images. All images have been corrected for solar rotation. Movie 1 clearly shows that the moss is time variable on a small scale, even though the global structure is unchanged. Possible origins of the moss variability are motions of the magnetic field and/or motions of the sites of coronal heating. We expect coronal flux tubes to be stirred continuously by photospheric flows and jostled by spicules, but also, if the heating is due to some turbulent-like process, we expect it to jump around from flux tube to flux tube. It has been proposed by Martens et al. (2000) that at least part of the time variability is due to displacement and/or obscuration by moving spicules. Obscuration by spicules is clearly evident in moss regions near the limb, and seems a likely explanation for rapid variability, timescales on the order of tens of seconds. However, movie 1 indicates that moss

Fig. 2.—SOHO-MDI magnetograph of 2000 September 23 along with an overlay showing the TRACE field of view

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Fig. 3.—Two YOHKOH-SXT images of 2000 September 23, a full-Sun frame and a short-exposure partial-frame image of AR 9169

patches also appear to move and deform on timescales of minutes. We argue in this paper that all the observed motions of the moss are due to dynamics of the magnetic field and/or spicule obscuration, and not due to variability of the heating. Two key features that we will show imply severe constraints on coronal heating are clearly evident in movie 1. ˚ loops can be seen inside the moss First, almost no 171 A region, even over the dark polarity-inversion line, throughout the 5 hr duration of the TRACE observations of this active region. This result is generally true for moss except when it occurs in emerging flux regions and significant magnetic field evolution is present (Seaton et al. 2001). For ˚ loops are observed developed active regions, TRACE 171 A at locations surrounding the moss, but are rarely observed in the moss regions themselves. If the heating in the moss flux tubes were to turn on and off, however, we would expect these flux tubes to eventually cool down to 106 K and ˚ loops. In fact, theoretical argubecome visible as 171 A ments presented below show that such cooling loops should ˚ images, and TRACE observastrongly dominate the 171 A tions of flares, indeed, are dominated by cooling loops. Flares are well known to be due to impulsive heating that migrates across flux tubes, giving rise to the usual spreading flare ribbons and growing loop arcades. Because such cooling loops are not observed in moss regions, we conclude that the observed motions of the moss emission must be due primarily to motions of the field rather than to migration of the heating across flux tubes. The second important feature of the time series is that, when averaged over some small but finite area to remove effects due to the moss motions, the intensity itself undergoes little change. Figure 4 shows the time history of the average emission per pixel from areas 1 and 2 indicated in Figure 1. Area 1 corresponds to a 20  20 pixel square (1000  1000 ), while area 2 corresponds to a 25  25 pixel square. It is evident from the movie that any one pixel in these squares can show strong variation as moss emission moves in and out of that pixel, but after removing this

motion, the emission is quite steady, only 10% or so variation during the full 5 hr. In Figure 4, we also show for each area the correlation between the image at a particular time and the first image. It is evident that the emission stays fairly well correlated throughout the time history, indicating that, although the moss does undergo rearrangement, its structure and intensity is fairly constant. We present below theoretical and numerical arguments showing that the lack ˚ loops in the moss regions and the weak variation of of 171 A the moss emission severely limit the possible temporal variation of coronal heating in these active region loops. 3. LOOP MODELS

In order to gain some physical insight, let us calculate the expected signatures of time-varying heating using simple scaling law theory. Consider two loops of similar geometry, and let one have a maximum temperature TT ¼ 106 K so ˚ loop, while the other has that it appears as a TRACE 171 A an initial maximum temperature of TY
4  106 K—a Yohkoh loop. Let us first determine the relative emission measure at 106 K of the TRACE loop, EMT , to that of the Yohkoh loop, EMY . If both loops are in static equilibrium, then the ratio of the Yohkoh loop coronal density, NY , to that of the TRACE loop, NT , is given by the usual scaling laws (e.g., Rosner, Tucker, & Vaiana 1978; Craig, McClymont, & Underwood 1978; Vesecky, Antiochos, & Underwood 1979),  2 NY TY ¼ : ð1Þ NT TT Hence, the hotter loop has a substantially denser corona and, consequently, much larger coronal emission measure than the cooler loop. If the geometry of the two loops is similar, then the result above implies that the ratio of their coronal emission measures varies as the fourth power of the ratio of their coronal temperatures, ðTY =TT Þ4 . But TRACE does not observe the coronal section of the hot loop. At 171

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Fig. 4.—Top: Average counts per second per pixel as a function of time for subregion 1 (left) and 2 (right). Bottom: Average correlation coefficient (solid line) and standard deviation (broken line) for subregion 1 (left) and 2 (right).

˚ , TRACE observes only the 106 K transition region plasma A at the footpoints of the Yohkoh loop. In order to determine the emission measure of this plasma, we need to know how the emission measure varies with temperature along the Yohkoh loop; in other words, we need the differential emission measure, DEMðTÞ. This can be determined by using the fact that radiation must be balanced by conduction over most of the temperature range in a loop, which yields the well known result that DEMðTÞ / T (Vesecky et al. 1979). In the Yohkoh loop, therefore, the emission measure at some transition region temperature T is down by a factor T=TY compared to the coronal emission measure. Using this result with equation (1) above, we deduce that the ratio of the emission measure at 106 K of the Yohkoh loop, EMY , to that of the TRACE loop, EMT , is given by EMY =EMT ¼ ðTY =TT Þ3 :

ð2Þ

Hence, TRACE will observe a Yohkoh loop as a bright patch of footpoint emission—the so-called moss (Martens et al. 2000). Let us now determine what TRACE would observe if the Yohkoh loops were undergoing cycles of heating and cooling. During the heating phase TRACE would again observe only a patch of emission at the loop base, because the 106 K plasma will occur only at the transition region during chromospheric evaporation (Antiochos & Sturrock 1978; Antiochos et al. 2000a). During the cooling phase, however, ˚ passthe whole loop must become visible in TRACE’s 171 A band when the maximum coronal temperature decreases to 106 K. The observed intensity at this time depends on the loop density, which in turn depends on whether substantial draining of coronal material occurs during the cooling. If no draining occurs then we can use equation (1) above, implying that the ratio of the emissions measures of the cooling

Yohkoh loop to a steady TRACE loop is given by  4 EMY TY ¼ : EMT TT

ð3Þ

Hence, if the Yohkoh loop had an initial maximum temperature TY ¼ 4  106 K, the cooling Yohkoh loop would appear over 2 orders of magnitude brighter than the steadystate TRACE loop. We expect some draining to occur, however, decreasing the emission measure observed for the cooling loop. From numerical studies of cooling flare loops, we found empirically that the density decreases as the square root of the temperature, N 2 / T (Cargill, Mariska, & Antiochos 1995). Assuming that this relation holds in general, we derive for the ratio of the emissions measures  3 EMY TY ¼ ; ð4Þ EMT TT implying that cooling and draining Yohkoh loops would appear over an order of magnitude brighter than nearby, steady TRACE loops. We also must consider the time required for the Yohkoh ˚ passband. Taking loop to cool through the TRACE 171 A 3 10 as a typical density N
10 cm and using a standard value of 6
1023 for the radiative loss function at 106 K (Cook et al. 1989), we derive a radiative cooling time
¼ P=ðN 2 Þ
1000 s, which is much longer than the exposure times for the images in movie 1. Hence, if cooling loops were present in the TRACE moss regions, they should clearly stand out and dominate the images. In fact, this is exactly what is seen in flares. In the very initial impulsive phase when all the flare plasma is well above 106 K, TRACE observes only ‘‘ moss ’’—the very bright flare ribbons—but as soon as flare loops cool down to the TRACE passband,

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they completely dominate the TRACE images from then on (e.g., Antiochos et al. 2000a). In contrast, we do not see a single loop appear in the moss regions in movie 1. Even more stringent constraints on the heating variability can be obtained from the observed emission variability (Figs. 2a and 2b). The expected dependence of moss emission measure on loop heating rate can easily be derived from the scaling laws. For a uniform heating per unit volume, E, we find that EMY / E 6=7 ;

ð5Þ

which implies that the variation in the average heating rate can be only
12% for the observed 10% variation in moss intensity. Strictly speaking, equation (5) above is valid only for quasi-static loops, but even for the case of flare heating, ˚ is sensitive to we found that the emission measure at 171 A the magnitude of the impulsive heating (Antiochos et al. 2000a). We conclude, therefore, that at least for the active region of Figure 1, the average coronal heating rate must be nearly constant for several hours. 4. NUMERICAL SIMULATIONS

Although the arguments presented above appear compelling, they are not rigorous because the scaling laws are only approximations that are unlikely to apply to loops undergoing rapid evolution. In particular, the amount of draining that will occur during cooling is not clearly known. Therefore, to test the conclusions above, we have simulated the cooling of a hot Yohkoh loop using our one-dimensional adaptive-mesh-refinement code ARGOS, which calculates the detailed dynamical evolution of loop plasma, including fully resolved dynamic transition regions. The code has been used successfully to simulate a variety of highly dynamic loops, and is described in detail in Antiochos et al. (1999). We assumed for the loop geometry a simple semicircle with total length 30,000 km from one chromospheric footpoint to the other and with constant cross-sectional area. A major advantage of using the numerical code is that we can consider completely general forms for both the spatial and temporal structure of the heating. There is mounting evidence that the heating in coronal loops is concentrated near the chromosphere (Aschwanden et al. 2001). Therefore, we assumed a form similar to that in our earlier studies (Antiochos, MacNeice, & Spicer 2000b; Karpen et al. 2001)—spatially localized near each chromospheric footpoint with a scale height of 10,000 km, plus a small spatially uniform background heating, E ¼ f 4:8  103 exp½ðs  s1 Þ= þ 1:5  104 ;

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linearly ramped the heating down to zero over 100 s. The subsequent evolution of the temperature and density at the loop apex are shown in Figure 5. The initial temperature and density are 4:7  106 K and 4:8  1010 cm3, corresponding to a typical, hot Yohkoh loop. It is evident that once the heating turns off, both the temperature and density drop almost linearly with time. In roughly 1 hr, the apex temperature cools down to 106 K , as expected, and the apex density drains down to
1010 cm3. Unlike our previous flare results, considerable draining takes place in this spatially localized heating case. This emphasizes the importance of performing the detailed numerical simulations, rather than relying only on simple scaling laws. From the results presented in Figure 5 we calculated the ˚ passband emission that TRACE would observe in the 171 A using the TRACE response function provided by Handy et al. (1999). Figure 5 also shows the evolution of the integrated intensity from the whole loop, and the intensity from a 5000 km region centered about the apex. As expected, the ˚ apex intensity rises sharply once the temperature 171 A there reaches 106 K and persists for approximately 1000 s. At maximum, the emission from the 5000 km region at the loop apex is almost equal to the total emission from the moss points. By comparing the integrated and apex intensity curves, we note that the bulk of the emission is due to the moss footpoints until t
37; 000 s. After this time, the

ð6Þ

where E is in units of ergs cm3 s1, s is distance along the loop, s1 is the position of the top of the chromosphere, and  ¼ 109 cm is the scale height for the energy deposition. Because real coronal loops are not symmetric, we set the constant f ¼ 1 at the left footpoint and f ¼ 0:75 on the right. This means that the initial equilibrium is not static, but exhibits a small flow from left to right, of order 2 km s1. Note that although the details of the loop evolution will depend on the assumed spatial form of the heating, we do not expect the basic results below, nor any of our conclusions, to be sensitive to the spatial form of the heating. We ran the code for 34,000 s with the heating fixed in order to establish a true equilibrium initial state, and then

Fig. 5.—Top: Evolution of apex temperature and density from a ˚ emission simulation of a cooling loop. Bottom: Evolution of the Fe ix 171 A spatially integrated over an apex region and over the whole loop for the same simulation.

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Fig. 6.—Evolution of the apex temperature and density, and the ˚ emission from a simulation of a loop with a spatially integrated Fe ix 171 A sinusoidal variation in the heating.

emission is dominated by plasma in the coronal section of the loop. The emission rises as the bulk of the plasma ˚ , but only by a factor of approxibecomes visible in 171 A mately 1.7. Note that if no draining had occurred, then the fact that EMðTÞ / T in the initial static-loop equilibrium would predict an increase by a factor of more than 4. But even with the substantial draining in the simulation, we still find that cooling loops would be bright and clearly visible in the TRACE images. As discussed above, more stringent constraints on the temporal variability can be obtained by considering the variability of the moss emission itself. In order to test the predicted sensitivity of the moss emission to heating rate (eq. [5]), we performed a simulation in which we imposed a sinusoidal variation instead of turning the heating off. The heating rate of equation (6) was multiplied by a factor 1 þ sin½ðt  t0 Þ=2000, where t0 ¼ 34; 000 is the time required to establish a true equilibrium initial state. The ˚ emission for oneresulting evolution of the integrated 171 A half period of the sinusoidal variation is shown in Figure 6. Note that since the heating is always larger than its initial value, the emission plotted in Figure 6 originates solely from the moss footpoints. It is evident that this emission tracks the heating quite well, increasing by a factor of 1.7, in close agreement with the scaling law prediction. We conclude, therefore, that since the observed variation in the moss emission is only
10%, the variation in the coronal-loop heating in the active region of Figure 1 must be well below a factor of 2 during the whole 5 hr duration of the TRACE observations. 5. DISCUSSION

TRACE observations of moss in the interiors of active regions appear to place severe constraints on the possible temporal variability of the coronal heating there. Our

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results rule out the possibility of a truly impulsive, shotlike heating, as would be expected from flarelike reconnection, for example. It is still possible, however, that a nanoflaretype model (Parker 1988) would be compatible with our findings, depending on the parameters of the nanoflare model. However, our results place significant constraints on two key parameters of the models, the time interval between heating events in any one flux tube and the variation in magnitude between events. First, the average time between nanoflares must be less than the cooling time for hot Yohkoh plasma,
1000 s; otherwise TRACE loops would appear in the moss regions. Second, the magnitude of the events cannot vary too greatly ( by a factor of ~2); otherwise, the moss intensity would vary more strongly than we observe. It will be interesting to determine whether these conditions are satisfied by the various nanoflare models that have been proposed. In particular, Katsukawa & Tsuneta (2001) have argued recently for picoflare heating in active region loops based on their observations of Yohkoh intensity fluctuations. It seems likely that their conclusions are compatible with our moss results, because the Yohkoh fluctuations are small, but a quantitative comparison remains to be done. Although our results cannot rule out the possibility of high-frequency impulsive heating, they clearly imply that the observed time variability of the moss must be due primarily to magnetic field motions and obscuration rather than to migration of the heating across flux tubes. If so, we conclude that the heating is somehow insensitive to the continual stirring of the magnetic field by spicular and photospheric motions, remaining on average fairly uniform in space as well as in time. It should be emphasized that this conclusion holds only for the interiors of mature active regions. In fact, it is likely that these are the only locations in the corona where the heating could possibly be steady. The point is that these are the only locations where the coronal field is not changing on a rapid timescale. In quiet regions the so-called magnetic carpet implies that the coronal field is undergoing continuous reconnection and evolution (Title & Schrijver 1998), and the field in newly emerging active regions is clearly highly dynamic. Furthermore, the field at the edges of developed active regions must be continually interacting and reconnecting with the surrounding quiet region field. Therefore, the magnetic field in the corona is likely to be quasi-stationary only in the interiors of large active regions. However, even though the magnetic field is stable there, the coronal plasma is somehow still being heated continuously. Our hope was that the TRACE moss observations would allow us to determine timescales for this heating and thereby gain some insight into its nature, but if anything, our results have made the coronal heating process even more of a mystery. This work has been supported in part by grants from NASA and ONR, and by a NASA contract to Lockheed Martin.

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