Coronal X-Ray Brightness and Photospheric Magnetic Field: A Study ...

THE ASTROPHYSICAL JOURNAL, 539 : 995È1001, 2000 August 20 ( 2000. The American Astronomical Society. All rights reserved. Printed in U.S.A.

CORONAL X-RAY BRIGHTNESS AND PHOTOSPHERIC MAGNETIC FIELD : A STUDY IN CORRELATIONS RICHARD WOLFSON,1 COLIN B. ROALD, AND P. A. STURROCK Center for Space Science and Astrophysics, Stanford University, Stanford, CA 94305, and Department of Physics, Middlebury College ; wolfson=middlebury.edu

AND MARK A. WEBER Department of Physics, Montana State University Received 1999 May 21 ; accepted 2000 March 22

ABSTRACT We have examined correlations between coronal X-ray emission from the Y ohkoh Soft X-Ray Telescope (SXT) and photospheric magnetic Ðeld measurements from the Michelson Doppler Imager (MDI) on SOHO. Our data sets span a 521 day period beginning 1996 July 25, and we have averaged the data temporally into one bin per day and spatially into nine latitude bins, each spanning 15¡. We Ðnd strong correlations between SXT and MDI data at all but extreme latitudes. Phase shifting one data set relative to the other shows that the correlation always peaks at zero shift, indicating that coronal X-ray emission is always most strongly related to the photospheric Ðeld at the same time (essentially, the same longitude). However, higher order peaks occur for phase shifts of the order of 1 solar rotation, and a detailed analysis shows that the exact phasing of these higher order peaks is consistent with di†erential rotation of persistent magnetic structures in the photosphere. Cross-correlation between SXT and MDI data from di†erent latitude bins shows that the high-latitude coronal X-ray emission is most strongly correlated with the photospheric Ðeld at [30¡ and ]30¡. Although this correlation is probably due to projection e†ects, a less likely interpretation is that the coronal magnetic Ðeld, on average, spreads from the photosphere to higher latitudes in the corona. Finally, we compute actual X-ray energy Ñuxes from the SXT data and show that the correlation between X-ray Ñux and photospheric magnetic Ðeld is in reasonable quantitative agreement with a simple model for coronal heating based on the reconnection of magnetic elements in the chromospheric network. Subject headings : MHD È Sun : activity È Sun : corona È Sun : magnetic Ðelds È Sun : rotation È Sun : X-rays, gamma rays 1.

INTRODUCTION

low energy but so frequently as to be essentially continuous, as in ParkerÏs nanoÑare model (Parker 1988 ; but see also Hudson 1991 for calculations suggesting that nanoÑares alone may not suffice for coronal heating). Either way, we expect the coronal heating rate to be a monotonic function of magnetic Ðeld strength, although the details of that monotonic relation may depend on the particular heating model (for model calculations predicting direct relationships between magnetic Ðeld and coronal X-ray Ñux see Sturrock 1996, 1999 ; Sturrock, Roald, & Wolfson 1999 ; Roald, Sturrock, & Wolfson 2000). In this work we explore the connection between coronal heating and the solar magnetic Ðeld, considering speciÐcally the relation between coronal X-ray brightness and the lineof-sight component of the photospheric magnetic Ðeld. The analysis covers a 521 day period beginning 1996 July 25 (the start of Carrington rotation 1912), for which we have both photospheric magnetic Ðeld data from the Michelson Doppler Imager (MDI) on the Solar and Heliospheric Observatory (SOHO) and coronal X-ray data from the Soft X-Ray Telescope (SXT) on Y ohkoh (Tsuneta et al. 1991).

Coronal heating is a major unsolved problem in solar physics. Even though the full solar luminosity radiates from the SunÏs photosphere, the second law of thermodynamics precludes thermal transfer of any of this vast energy Ñux from the relatively cool (D6 kK) photosphere to the hot (D2 MK) corona. Therefore, the energy that heats the corona must be supplied by nonthermal means, e.g., as mechanical or electromagnetic energy. The high electrical conductivity of the corona and the dominance of magnetic pressure in the low corona make magnetic energy a particularly attractive source for coronal heating, and for this reason most theories of coronal heating invoke solar magnetic Ðelds (see Ulmschneider, Priest, & Rosner 1991 for a review of coronal heating mechanisms). Most of these theories fall into two broad classes : those involving magnetosonic or Alfven waves and those in which photospheric footpoint motions result in slow buildup of magnetic energy. In wave models, waves generated at the coronal base or below carry energy into the corona, where the wave energy is dissipated. In the magnetic energy buildup models, conversion of magnetic to thermal energy occurs when regions of high current density develop. Ohmic dissipation and/or tearing-mode reconnection then result in heating and, in the case of reconnection, nonthermal high-energy particles. This process may be relatively infrequent and highly energetic, as in large solar Ñares, or it may occur at

2.

PREPARATION OF THE DATA

We obtained coronal X-ray brightness data from Y ohkoh SXT single-frame desaturated (SFD) composite images. Data were averaged into nine bins. Each bin covered 1¡ in longitude and was centered on the central meridian. Bins covered 15¡ in latitude and were centered every 15¡ from [60¡ to ]60¡ (see Weber et al. 1999 for more details on the

1 On leave from Middlebury College.

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data reduction). Within each latitude bin, all the data from a given day, typically 20 images, were also averaged. Data were taken with both the thin aluminum (Al) and aluminum-magnesium (AlMg) Ðlters. In most of our work on correlations between X-ray brightness and photospheric magnetic Ðeld, we used the Al Ðlter data numbers (DN) as a proxy for the actual X-ray energy Ñux. We generally analyze the logarithm of the DN, always taking the logarithm after doing the averaging. In ° 4 we discuss how we used both Al and AlMg Ðlter data in an attempt to convert the data numbers to actual energy Ñux. Errors were estimated from the statistical variance in the data ; the actual variances so calculated resulted in relative uncertainties on the order of 6%È9%, varying somewhat with latitude bin. Relative uncertainties for individual days may be higher or lower ; our values are the means for the entire data set. The MDI data were taken from synoptic data sets prepared by the MDI team (T. Hoeksema 1999, private communication) ; each set covers one Carrington rotation and contains 360 ] 180 measurements of the line-of-sight photospheric Ðeld, one for each 1¡ of longitude and one for each of 180 evenly spaced increments in sin (latitude). We averaged the MDI data into the same latitude bins as the SXT data. As with the SXT data, all MDI data for a given day were also averaged at each latitude bin ; this amounted to combining typically 13 of the 1¡ longitude strips. In the MDI averaging, we formed both the average line-of-sight magnetic Ðeld, SBT, and the average Ðeld magnitude, S o B o T. In statistical comparisons of MDI and SXT data, we removed MDI data for those days for which SXT data were missing. There were 41 of those days for the Al Ðlter, distributed apparently randomly either as isolated days or in groups of at most a few days. MDI errors were estimated using the fact that pixels in individual MDI magnetograms have noise levels from 5 to 15 G depending on the exposure time (T. Hoeksema 1999, private communication). In the substantial averaging that went into producing the synoptic maps and then the nine bins corresponding to the SXT data, that noise level is reduced to about 0.2 G. There are also systematic errors and uncertainties in the absolute calibration relative to other magnetographs, but we do not consider these in our analysis. 3.

CORRELATION ANALYSIS

To study the SXT/MDI correlations, we Ðrst prepared scatter plots and calculated correlation coefficients for the logarithms of the SXT versus MDI data. We use logarithms both because the SXT data span several orders of magnitude and because we will be looking for power-law relations between the coronal and photospheric data. Scatter plots and correlations were made for each of the nine latitude bins centered at [60¡, [45¡, . . . , 45¡, 60¡. We used the SXT Al Ðlter data numbers and considered both the Ðeld B and the magnitude o B o . There was essentially no correlation between SXT DN and the average signed Ðeld B, hardly surprising given that the Ðeld for each bin is an average over a large area 15¡ in latitude and 1 day or about 13¡ in longitude. In what follows we therefore consider only correlations with o B o . Figure 1 shows scatter plots for three of the latitude bins, centered at [30¡, the equator, and ]60¡. Other plots in the range [30¡È]30¡ latitude look very similar to those shown in Figure 1 for [30¡ and the equator, while the plot for [60¡ is similar to that for ]60¡. Plots for [45¡ and ]45¡ show somewhat more correlation

FIG. 1.ÈScatter plots of SXT data number (DN) vs. the magnitude o B o of the photospheric magnetic Ðeld as measured by MDI, shown for three of the nine latitude bins considered. Other plots in the range [30¡È]30¡ look similar to those shown here for [30¡ and the equator, while the [45¡ and ]45¡ show somewhat more correlation in their scatter plots than do the [60¡ and ]60¡ data. Each point in these scatter plots is marked with a cross whose extent in the horizontal and vertical directions represents, respectively, estimated errors in magnetic Ðeld and SXT DN.

than those at the extreme latitudes. The SXT-MDI relations visually evident in Figure 1 and similar scatter plots can be summarized quantitatively in the correlation coefficients, which we show plotted as a function of latitude in Figure 2. We also list the correlations and the slope of the best-Ðt line (i.e., the power-law index relating DN and o B o ) in Table 1 ; we return to the question of the power-law index in ° 4. The two data sets are evidently strongly correlated at lower latitudes, but the correlation falls o† rapidly toward the poles.

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CORONA-PHOTOSPHERE CORRELATIONS

FIG. 2.ÈCorrelation coefficient between SXT DN and MDI o B o as a function of latitude. The lack of signiÐcant correlation at high latitudes may be due to the absence of active regions.

The plot for 60¡ in Figure 1 suggests a physical reason for this : the higher latitudes lack the strong Ðelds and large X-ray Ñuxes that are associated, presumably, with active regions found at lower latitudes. The lower X-ray emission associated with the quiet corona and/or polar coronal holes evidently shows much less correlation with the underlying photospheric Ðeld. For the lower latitude plots, such as those shown for the equator and [30¡ in Figure 1, most of the data are concentrated in a seemingly uncorrelated mass similar to the entire data set for the 60¡ plot. This suggests that the SXT-MDI correlations are due at least in part to the high-Ðeld, high-emission areas corresponding to active regions. We test this suggestion by computing correlation coefficients for subsets of the data divided according to whether the magnetic Ðeld associated with a given DN- o B o pair is less than or greater than a cuto† value B , here taken c the [60¡ to be 100.5 G (roughly the upper limit on o B o for and ]60¡ data). The results are shown in Figure 3. For the o B o \ B curve, the absence of any correlation coefficient c about 0.4 suggests that the quiet corona is not greater than as strongly correlated with the photospheric magnetic Ðeld, but even here the correlation is signiÐcant at lower latitudes. On the other hand, it is possible that the relatively high noise level in the low-Ðeld data simply masks a correlation that exists even in the quiet corona. Meanwhile the data with o B o [ B for latitudes in the range [30¡ to ]30¡ show very strong ccorrelation, conÐrming our suggestion that the correlations between X-ray Ñux and photospheric magnetic Ðeld in the overall data are due primarily to what

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FIG. 3.ÈCorrelation coefficients for portions of the data sets corresponding to o B o [ B (upper curve) and o B o \ B (lower curve), where the c c cuto† magnetic Ðeld strength is B \ 100.5 G. The coefficients for o B o [ c B are not shown for the ]60¡ and [60¡ latitude bins because these c include, respectively, only 2 and 4 high-Ðeld points, rendering the resulting correlation coefficients meaningless. The number of data points for o B o [ B at other latitudes ranges from 13 to 140, from a total of 480 points in c time series. A long-term quadratic Ðt has been removed from the data each to eliminate solar cycle inÑuences on the correlations.

are presumably active regions with strong surface Ðeld and high coronal X-ray emission. We also show the e†ect of introducing a high-Ðeld cuto† in Table 1, where in the rightmost column we list the least-squares Ðt slope of log (DN) versus log ( o B o ) with points where o B o [ B removed. At c the lower latitudes, where high-Ðeld active regions occur, the resulting slopes are signiÐcantly lower, indicating that the reported slopes for the full data set are substantially inÑuenced by the active regions. The uncertainties in the slopes shown in Table 1 are the standard error estimates from the least-squares regression ; note that they are much smaller for the highly correlated low-latitude data. We also attempted to calculate slope uncertainties from the error estimates on both o B o and SXT DN for the individual points in our time series. We did so through a Monte Carlo simulation in which we multiplied the uncertainty in each point (e.g., one-half the error bar length in Fig. 1) by a di†erent random number from a normal distribution of random numbers with mean zero and standard deviation 1. The result was a new data set with random errors. For each latitude bin we generated 100 such sets, did a least-squares Ðt on each, and calculated the mean and standard deviation of the 100 least-squares slopes. Errors, taken as the standard deviations of the slopes, were roughly a factor of 5 lower than the errors

TABLE 1 CORRELATION COEFFICIENTS AND LINEAR FIT SLOPES Latitude Bin

log (DN)-log ( o B o ) Correlation

Slope

Slope for o B o \ B

[60¡ . . . . . . . . [45¡ . . . . . . . . [30¡ . . . . . . . . [15¡ . . . . . . . . 0¡ . . . . . . . . . . . . 15¡ . . . . . . . . . . . 30¡ . . . . . . . . . . . 45¡ . . . . . . . . . . . 60¡ . . . . . . . . . . .

0.08 0.35 0.84 0.80 0.81 0.82 0.78 0.30 [0.02

0.48 ^ 0.28 1.39 ^ 0.17 1.80 ^ 0.05 1.73 ^ 0.06 1.74 ^ 0.06 1.49 ^ 0.05 1.43 ^ 0.05 0.92 ^ 0.15 [0.10 ^ 0.20

0.50 1.30 1.63 1.51 1.21 1.02 0.92 0.77 [0.11

c

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computed in the least-squares Ðtting process on the original data. We have therefore reported the latter, i.e., the larger errors, in Table 1. Because our Monte Carlo procedure applies normally distributed errors to the actual o B o and DN values while the least-squares slope is Ðt to the logarithm of the data, the means of the slopes from the Monte Carlo simulation are somewhat lower than the slopes computed directly from the logarithms of the data. Our data span a time period of 521 days, long enough for variations associated with the solar cycle to have some inÑuence on the correlations. We therefore Ðtted both the SXT and MDI data with quadratic functions of time normalized to zero mean, subtracted the Ðt, and recomputed the correlation coefficients. This procedure has essentially no e†ect on the correlations for o B o [ B and only a c modest e†ect for o B o \ B . Thus, the correlations we have c computed evidently result from real, short-term correlated variations in photospheric magnetic Ðeld and coronal X-ray emission. An alternate reason why detrending the data might have little e†ect on correlations would be if the correlated data were dominated by one or two large active regions, a situation that would also call into question many of our other conclusions. To demonstrate that this is not the case, we show in Figure 4 the entire MDI data set, folded with the photospheric period appropriate to the central latitude of each bin (i.e., the period calculated from the Komm, Howard, & Harvey 1993 Ðts to di†erential rotation). That the data with high values of log ( o B o ) are widely distributed in time indicates that the correlations are not dominated by just a very small number of active regions. The solar photosphere is known to rotate di†erentially, with the fastest rotation at the equator (Komm et al. 1993). The corona, in contrast, shows evidence of rigid rotation (Weber et al. 1999 ; Wang et al. 1988 ; Hoeksema & Scherrer 1987 ; Parker 1987). We might, therefore, expect that coronal emission would be correlated most strongly with the photospheric Ðeld at a slightly di†erent longitude or, in our data set, a di†erent time, as the magnetic Ðeld linking photosphere and corona gets stretched by the di†erent rotation rates. We have performed correlations of our two data

FIG. 4.ÈData from the entire MDI data set, folded with the di†erential rotation periods appropriate to the center of each latitude bin. The horizontal axis thus represents the phase relative to the photospheric rotation period. Note that the high values are spread over time, indicating that the correlations are not dominated by a very small number of active regions.

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sets, introducing both positive and negative phase lags between the SXT and MDI data. At all latitude bins, however, we Ðnd the strongest correlation between SXT data numbers and the immediately underlying magnetic Ðeld (i.e., zero phase lag). The correlation is sharply peaked, dropping rapidly on a timescale of a few days. Figure 5 shows the correlation as a function of phase lag for two latitude bins. As is evident in Figure 5, we Ðnd peaks in the correlation for phase lags that are a multiple of the solar rotation period, suggesting the presence of long-lived structures either at the photosphere or in the corona or both. In addition to peaks associated with solar rotation, we Ðnd some intermediate peaks that probably represent noise due to spurious correlations arising when a region of strong X-ray emission is phase lagged enough to overlap an active region with which it may be physically unrelated. Focusing on phase shifts spanning a few days on either side of the 27.3 day Carrington period shows that the phase lag where the maximum correlation occurs varies slightly with latitude. Figure 6 shows this variation, along with the di†erential rotation curve for photospheric magnetic features from Komm et al. (1993). In preparing this Ðgure, we have estimated the position of the centroid of each correlation peak by taking into account asymmetries in the curve when plotted on a phase-shift scale running from [20 to [35 or from ]20 to ]35 days. In all cases the estimated centroid location is within 0.5 days of the actual peak location. Figure 6 makes clear that the value of the phase shift at the correlation peak is related to the phenomenon of di†erential rotation. If the correlations are the result of persistent structures both at the photosphere and in the corona, then we would expect peaks at phase lags equal to both the

FIG. 5.ÈCorrelations between phase-shifted time series for the latitude bins centered at (top) [45¡ and (bottom) the equator. Phase shift on the horizontal axis is the time by which the MDI data have been shifted relative to the SXT data. Only times with data points available in both series are used in computing the correlations. The strong central peak occurs at zero phase shift for all latitude bins. Note the two sets of higher order peaks associated with shifts of about 1 and 2 solar rotation periods. Fig. 6 explores the secondary peaks in more detail.

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FIG. 6.ÈPhase shifts associated with the secondary peaks for all latitude bins, from plots like those of Fig. 5. The shifts were estimated from higher resolution plots of correlation vs. phase, for phase shifts in the range [20 to [35 and ]20 to ]35 days. The vertical axis is the absolute value of the phase shift, which may be positive or negative. Diamonds represent positive phase shifts (MDI advanced relative to SXT), while triangles represent negative shifts. The solid curve is the phase shift one would expect if the secondary peaks were due to persistent photospheric features exhibiting the di†erential rotation proÐle of Komm et al. (1993). The Komm et al. proÐle has been corrected to give the synodic period.

photospheric and coronal rotation periods, if di†erent. The fact that we see just one peak and that it shifts with latitude to follow reasonably closely the standard photospheric differential rotation curve suggests either that the persistent structures are predominantly photospheric orÈa more radical suggestionÈthat the corona shares the photospheric di†erential rotation proÐle (see Weber et al. 1999 for evidence that the corona may exhibit di†erential rotation during the rising phase of the solar cycle). Given the many studies suggesting rigid-body coronal rotation (e.g., Wang et al. 1988), the former explanation is far more likely. Indeed, correlation peaks associated with weaker but persistent coronal structures not partaking in di†erential rotation may be buried within the rather broad peaks that we have identiÐed with persistent photospheric structures (see Weber et al. 1999 for evidence of such persistent structures in the SXT data). We next consider cross-correlations among di†erent latitude bins. Before subtracting the long-term quadratic trend, such analysis shows substantial correlations between corresponding latitude bins in the northern and southern hemispheres. These disappear when the long-term trend is subtracted, indicating that they are due to the general trend in solar activity and its well-known migration toward lower latitudes as the cycle progresses. Correlations that remain after the long-term trend subtraction are shown in Figure 7, where each curve represents a Ðxed latitude for SXT data, with varying latitude for MDI data. (We show only SXT latitudes from [60¡ to the equator ; correlation curves for the northern hemisphere are essentially mirror images.) For lower latitudes (between [30¡ and ]30¡), the correlation is strongly peaked when the MDI latitude is the same as the SXT latitude. However, at higher latitudes ([45¡, ]45¡, [60¡, and ]60¡), the SXT data correlate most strongly with MDI data at [30¡ and ]30¡. These correlations are most likely due to projection e†ects ; for example, coronal emission from 0.4 R radially above the 30¡ photosphere _ will appear at 44¡. However, the Y ohkoh on-disk images do

FIG. 7.ÈCross-correlations between SXT and MDI data for di†erent latitude bins. Each graph is for a Ðxed SXT latitude from [60¡ to the equator ; northern hemisphere graphs are very nearly mirror images of the southern hemisphere graphs shown here. At each SXT latitude, we compute correlations with MDI data for all latitude bins ; these latitudes are given on the horizontal axes. Each dashed vertical line represents the Ðxed latitude of the SXT data for each graph, so the point where the curve intersects the dashed line gives the correlation between MDI and SXT data for the same latitude bin. At high SXT latitudes the peak correlation is not with MDI data at the same latitude but with MDI data at 30¡. This is most likely a projection e†ect, but we cannot rule out the possibility that it suggests a magnetic connection between the low-latitude photosphere and the corona at higher latitudes.

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not give the height of the coronal X-ray emission, and in the event that emission takes place lower in the corona, we cannot rule out the more remote possibility that our correlations suggest a magnetic connection between the photosphere at lower latitudes and the corona at higher latitudes. 4.

X-RAY ENERGY FLUX AND THE PHOTOSPHERIC FIELD

The DN versus o B o correlations discussed above are important in establishing links between coronal X-ray emission and the photospheric magnetic Ðeld. However, comparison with theory is facilitated by having the actual physical X-ray energy Ñux. Converting the SXT DN to X-ray Ñux proves a formidable problem, and here we compare the results from several approaches. The conversion from DN to X-ray Ñux requires that we know the temperature of the X-rayÈemitting plasma. With SXT, computing the temperature requires data from two di†erent SXT Ðlters, for which the ratio of X-ray counts then yields a temperature. (The Y ohkoh data analysis routine SXT–TEEM accomplishes this calculation.) Our data, however, are averaged over large areas and full-day time intervals. It is unlikely that the plasma is isothermal over all the emitting regions contributing to this average. Furthermore, data from the di†erent Ðlters may have been taken at di†erent times in the averaging period, rendering temperature calculations suspect. Additionally, our data originate in SXT SFD composite images rather than the SXT single-frame raw (SFR) data. With these caveats in mind, we nevertheless attempted to compute SXT temperatures based on Ðlter ratios and used these as inputs to the routine SXT–ERG–PER–DN. With the data already normalized to 1 s exposures, we could then divide by the area of the SFD pixels (each 4A. 9 ] 4A. 9 ) to get the X-ray Ñux in ergs s~1 cm~2. Data from the Al Ðlter were available for 480 days of the 521 day period under study, and data from the AlMg Ðlter for 443 days. Data were available from both Ðlters simultaneously for 416 days, and thus for these days we could use SXT–TEEM to attempt a temperature calculation based on the ratio of counts in the two Ðlters. Typically, some 350 of the 416 days with data from both Ðlters yielded valid results for the temperature, although this number varied slightly with latitude bin (in the SXT analysis routines, a valid temperature has 5.5 \ log (T ) \ 8). The fact that a substantial fraction of the data were not consistent with temperatures in the valid SXT range conÐrms our earlier concern that temperature calculations based on our highly averaged data should not be taken too seriously. Even when SXT–TEEM produced valid results, the computed temperatures Ñuctuated widely over some 2 orders of magnitude, again suggesting that the temperature calculations are not particularly reliable. Where SXT–TEEM yielded temperatures in its valid range, we used these temperatures as inputs to the routine SXT–ERG–PER–DN to compute conversion factors between the original data (in SXT data number, or DN) and the actual energy. Scaling by the pixel area then gave the energy Ñux in ergs s~1 cm~2. Because we have little faith in the SXT–TEEM temperature calculations for our averaged data, we explored two other approaches to the DN-to-Ñux conversion. Our Ðrst alternative followed the SXT–TEEM analysis but restricted the resulting temperatures to the range 1È3 MK. Statistics of the Ñux-DN relation di†ered insigniÐcantly from those calculated using the full, unrestricted tem-

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perature range. Second, we calculated the Ñux based on Ðxed temperatures. This is the approach adopted by Fisher et al. (1998) in their study of correlations between SXT X-ray Ñux and photospheric magnetic Ðeld in active regions. Their approach was clearly an approximation, one that is further degraded here because, in our case, the corona includes both quiet and active regions. We note in passing a new method described by Acton, Weston, & Bruner (1999), which might enable a more accurate calculation of X-ray Ñux from SXT DN. We are still studying this recent work. Figure 8 is a scatter plot including all our data and showing the X-ray Ñux versus photospheric magnetic Ðeld for the full SXT–TEEM temperature range and for a Ðxed temperature of 2 MK. In both cases the relation is roughly linear, suggesting a power law between X-ray Ñux and magnetic Ðeld. The Ðxed-temperature approach yields a tightly correlated relation for which the power-law index is 1.47. This is well below the value 1.95 suggested by a recent model by Roald et al. (2000) that links coronal heating with reconnection of magnetic elements in the chromospheric network, although it lies midway between the values 1 and 2 suggested in simpler models by Sturrock (1996, 1999). Other Ðxed temperatures yield exactly the same slope because any Ðxed temperature results in a linear relation between SXT DN and X-ray Ñux ; only the vertical position of the points in Figure 8 (i.e., the overall scaling of the Ñux-DN relation) would change with temperature. Methods based on Ðlterratio temperatures yield better Ðts to the model, with slopes of 1.85 when the actual SXT–TEEM temperature is used and 1.79 when the SXT–TEEM results are limited to the range 1 MK \ T \ 3 MK. However, in both these cases the correlation declines substantially. If we once again divide the data according to whether log ( o B o ) is greater or less than 0.5, then we Ðnd, as before, that most of the correlation is due to the high-Ðeld data.

FIG. 8.ÈScatter plot showing the relation between photospheric magnetic Ðeld and X-ray Ñux, the latter calculated using two di†erent approaches to the coronal temperature. Dots represent the Ñux calculation based on a temperature determined from Al and AlMg Ðlter ratios, using SXT–TEEM. Crosses represent the Ñux calculation based on the assumption of a Ðxed temperature of 2 MK. The correlation between X-ray Ñux and magnetic Ðeld is higher with the Ðxed-temperature assumption, but the implied power-law index is higher with the Ðlter-ratio temperatures. For log ( o B o ) [ 0.5, both methods give slopes in the range 1.86È1.88, reasonably close to the value 1.95 suggested by the coronal heating model of Roald et al. (2000). Errors in this scatter plot should be comparable to those in Fig. 1.

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The power-law index for this subset of the data is in the range 1.86È1.88 for both Ñux calculation methods used for Figure 8. This more substantial agreement with the Roald et al. model is somewhat surprising because the model is based on assumptions about the quiet network, not active regions. It may be that the extreme scatter in the low-Ðeld data masks a relation that holds in both types of solar regions. 5.

DISCUSSION

We have examined correlations between coronal X-ray brightness as measured by the Y ohkoh SXT and photospheric magnetic Ðeld as measured by MDI on SOHO. Not surprisingly, correlations with the signed magnetic Ðeld, as averaged over our 15¡ latitude bins, are essentially nonexistent. However, correlations between SXT DN and the unsigned Ðeld, o B o , are signiÐcant. At lower latitudes correlation coefficients are substantial, around 0.8, while the correlation coefficients decline to near negligible values at the highest latitudes considered. Physically, these correlations hint at a relation between the strength of the photospheric Ðeld and the heating processes that ultimately result in coronal X-ray emission, although they provide no insight into the nature of such a relation. More detailed temporal and spatial analysis of the correlations yields several other implications. Phase shifting one data set relative to the other shows that the maximum correlation always occurs for zero phase shift. However, higher order peaks with substantial correlation occur at phase shifts whose magnitude approximates the solar rotation period. This suggests the presence of persistent structures in the photosphere, the corona, or both. A closer focus on phase shifts near the 27.3 day Carrington rotation period shows that the phase shift at which secondary peaks occur increases with poleward latitude. This e†ect agrees quantitatively with what one would expect if the higher order correlation peaks were due to persistent structures participating in solar di†erential rotationÈmost likely photospheric structures. The lack of higher order peaks at a latitude-independent phase shift suggests the absence of

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persistent structures in rigid rotation. To the extent that the corona rotates rigidly, this implies that coronal structures in our SXT data set are more ephemeral than magnetic regions on the photosphere. Cross-correlations between coronal and photospheric data at di†erent latitudes reveal a modest correlation between corresponding latitude bins in the northern and southern hemisphere, but these disappear when long-term quadratic trends in time, presumably associated with the solar cycle, are subtracted from each data set. The remaining correlations are, at low latitudes, strongest between the coronal X-ray emission and the photospheric Ðeld for the same latitude bin. However, at [45¡ and ]45¡ and higher, the coronal X-ray emission correlates most strongly with the photospheric Ðeld at [30¡ and ]30¡. This crosscorrelation is most likely the result of three-dimensional projection e†ects, but our Y ohkoh data do not allow us to rule out a less probable interpretation, namely, that magnetic Ðeld lines extend from the low-latitude photosphere to higher latitudes in the corona. Finally, approximate calculations of the actual X-ray energy Ñux from SXT data yield relations between X-ray Ñux and photospheric magnetic Ðeld that are in reasonable quantitative agreement with a simple coronal heating model involving emergence and di†usion of magnetic elements at the photosphere. We thank Todd Hoeksema of the Stanford MDI group for supplying the MDI data used in our study and John Emerson of the Middlebury College mathematics department for helpful comments on statistical methodology. The Stanford portion of this work was supported by NASA grants NAG5-6118 and NAS8-37334 ; at Montana, M. Weber was supported by NASA contract NAS8-37334 as well as by a Montana Space Grant Fellowship through the Montana Space Grant Consortium Federal Grant NGT40041. Y ohkoh is a mission of the Japan Institute for Space and Astronautical Sciences with participation by NASA and the UK SERC. The SXT was prepared by the Lockheed Palo Alto Research Laboratory, the National Astronomical Observatory of Japan, and the University of Tokyo.

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