THE YOUNG BINARY DQ Tau: A HUNT FOR X-RAY EMISSION FROM COLLIDING MAGNETOSPHERES

The observation of DQ Tau was obtained on 2010 January 11–12 with the ACIS camera (Garmire et al. 2003) on board Chandra (Weisskopf et al. 2002) as a combined Guest Observer and Guaranteed Time Observation (ObsID No. 10992, Co-PIs: K. Getman and G. Garmire). The observation was set up based on the expectation of an X-ray flare close to the orbital phase of Φ = 0.98, which corresponds to the peak of the 2008 April large submillimeter flare (Salter et al. 2008, 2010). The total exposure time of the Chandra observation is ∼59 ks, with no data losses or background flaring due to solar activity. The start and end times of the observation are 2010 January 11 at 14:02:43 UT (JD 2455208.09) and 2010 January 12 at 06:31:17 UT (JD 2455208.77), respectively. This time period covers the orbital segment of Φ = 0.95–0.99 assuming the orbital parameters, such as time of periastron passage JD₀ = 2449582.54 and orbital period P = 15.8043 days, determined by Mathieu et al. (1997). To mitigate photon pileup effects during an anticipated X-ray flare, the observation was performed with a 1/8 sub-array of a single ACIS-I3 chip. The aim point of the observation was $04^{\rm {h}}46^{\rm {m}}54\mbox{$.\!\!^{\mathrm s}$}2$, +16°59'353 (J2000), and the satellite roll angle was 2841.

Data reduction follows procedures similar to those described in detail by Broos et al. (2010) and Townsley et al. (2003, Appendix B). Briefly, using the tool acis_process_events from the CIAO 4.2 software package, the latest calibration information (CALDB 4.2.0) on time-dependent gain and a custom bad pixel mask are applied, background event candidates are identified, and the data are corrected for CCD charge transfer inefficiency. Using the acis_detect_afterglow tool, additional afterglow events not detected with the standard Chandra X-ray Center (CXC) pipeline are flagged. The event list is cleaned by "grade" (only ASCA grades 0, 2, 3, 4, 6 are accepted), "status," "good-time interval," and energy filters. The slight point-spread function (PSF) broadening from the CXC software position randomizations is removed.

Using the ACIS Extract (AE) software package³ (Broos et al. 2010), DQ Tau photons are extracted within a polygonal contour enclosing ∼99% of the local PSF. The background is measured locally in a source-free region (Figure 1(a)). More than 6200 DQ Tau net counts in the full (0.5–8 keV) band are detected. The AE package was also used to construct source and background spectra, compute redistribution matrix files (RMFs) and auxiliary response files (ARFs), construct light curves and time–energy diagrams, perform a Kolmogorov–Smirnov variability test, compute photometric properties, correct spectra and light curves for light pileup (Section 2.2), and perform automated spectral grouping and fitting of time-resolved data (Section 3.3).

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With a fast rise phase (e-folding timescale τ_rise = 26.3 ks) and a slow decay phase (τ_decay = 40.9 ks), a single long X-ray flare spanned the entire ∼16.5 hr Chandra exposure (Figure 1(b)). During the observing period the count rate in the full band (and hard band) increased by a factor of two (and three), peaking at about t = 27 ks (Φ ∼ 0.97) after the start of the Chandra observation. The evolution of the median energy of X-ray events in the full band (adaptively smoothed) shows a slower rise and a faster decay than the light curve, reaching a ∼5 ks wide maximum-value plateau at t ∼ 23 ks, i.e., 4–5 ks before the count rate peak (Figures 1(b) and (c)). The X-ray count rate and median energy serve as photometric surrogates for emission measure and plasma temperature (Getman et al. 2010). There is thus an indication for a time delay between the temperature peak and the emission measure peak (see Section 3.3 for an explanation of the effect).

Photon pileup occurs when two or more photons are incident on a single detector region (for Chandra-ACIS this is typically a 3 × 3 pixel region) during a single CCD frame. Two or more photons are thus detected as a single event. Some of these pileup events mimic cosmic rays and are rejected by the on-board processing, while others are telemetered as valid events but with spuriously high energies.⁴ Despite the use of the 1/8 sub-array ACIS mode with the reduced CCD frame of 0.5 s (6.4 times shorter than the nominal frame), the Chandra data of DQ Tau suffer minor pileup. Assuming a single-temperature plasma with kT ∼ 2 keV and an average X-ray column density of N_H ∼ 2 × 10²¹ cm⁻² (Section 3.1), PIMMS⁵ gives an estimate on the pileup fraction (the ratio of the number of frames with two or more events to the number of frames with one or more events) of only 3% at the peak of the Chandra flare. Nevertheless, below we adopt two independent approaches to correct the data for pileup: modeling pileup within the context of spectral fitting using the pileup model of Davis (2001) and reconstructing a pileup free spectrum using an experimental Monte Carlo approach of Broos et al. (2011). Both methods are applied to the forty-five 1000 count overlapping time segments defined in Section 3.3.

In the first approach, for each of the 45 time segments, the pileup model of Davis is applied along with fixed two-temperature soft component and thawed one-temperature hot (flare) component plasma models subject to constant absorption, as defined and discussed in Sections 3.1 and 3.3. According to "The Chandra ABC Guide to Pileup," the following parameters are left frozen at their default values: the maximum number of photons to pileup in a single frame max_ph = 5, the grade correction for single photon detection g₀ = 1.0, and the number of independent 3 × 3 pixel pileup islands in the source extraction region nregions = 1. The frame time parameter frtime is set to that of the Chandra observation's frame time of EXPTIME = 0.5 s divided by the fractional exposure FRACEXPO = 0.995. The grade migration parameter (α), and the fraction of events in the source extraction region to which pileup is applied (psffrac), are varied within the ranges of α = [0.5–1.0] and psffrac = [0.90 − 0.95], respectively.

In the second approach, an experimental Monte Carlo forward-modeling method to reconstruct unpiled DQ Tau spectra from piled-up ACIS data is applied. The essential features of this pileup reconstruction method are briefly described here. A thorough description of the method is given by Broos et al. (2011). A non-physical "nuisance model" with many free parameters feeds an input spectrum to the MARX mirror-detector simulator⁶ which produces photons that have the correct Chandra-ACIS PSF. A physical model of the ACIS CCD produces a piled simulated spectrum. The nuisance model is iteratively adjusted until the piled simulated spectrum is similar to the observed spectrum. The simulation is then run one more time, with pileup disabled (i.e., exactly one photon arrives per frame). The resulting event list is claimed to be similar to what ACIS itself would have produced if pileup were not present, and thus one could attempt to fit it with a physical model. For each of the 45 time segments, reconstructed unpiled spectra of DQ Tau are then fit with fixed two-temperature soft and thawed one-temperature hot plasma component models subject to constant absorption, as defined and discussed in Sections 3.1 and 3.3.

For all 45 time segments, flare plasma temperature and emission measure inferred by the two methods are consistent within 1σ. Both methods indicate very light pileup with ≲4% difference between the observed count rate and that expected if pileup effects were not present (filled versus open circles in Figure 1). Further in the flare analysis (Section 3) we adopt the results from the Broos et al. method.

We use archived, publicly available XMM-Newton (Jansen et al. 2001) observations of DQ Tau. As part of the program aimed at studying X-ray properties of EX Lupi-type outburst (EXOR) stars, the EXOR star DR Tau was observed by XMM-Newton on 2007 February 13 (ObsID no. 0406570701, PI: G. Stringfellow). DQ Tau is 33 northwest of DR Tau, within the field of view of the XMM-Newton EPIC (Strüder et al. 2001; Turner et al. 2001) detectors (Figure 2(a)). Both EPIC-MOS and EPIC-PN cameras were configured in full window mode with medium and thin filters for MOS 1,2 and PN cameras, respectively. The total exposure time for both EPIC-MOS 1,2 detectors is 12.6 ks, and it is 11 ks for the EPIC-PN. The observation start and end times, 04:20:12 UT (JD 2454144.68) and 07:50:28 UT (JD 2454144.83), respectively, correspond to a time period covering the Φ = 0.66–0.67 orbital segment of DQ Tau.

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Our data reduction follows procedures described in the Common and EPIC-related Science Analysis Software (SAS) Threads.⁷ Briefly, the data were reprocessed with the SAS version 9.0.0. The emproc and epproc meta tasks were run to obtain calibrated and concatenated event lists for EPIC-MOS and EPIC-PN detectors, respectively. Short intervals of flaring particle background of 100 s (400 s) long around the observation time of t = 6 ks for MOS (PN) detectors were identified and removed. The event lists were further cleaned with the PATTERN, FLAG, and energy filters.⁸

The DQ Tau photons were extracted within a 335 radius circular region (∼90% encircled PSF). For the MOS 1,2 images, background was measured on the same CCD away from the source; for the PN image, a background region was chosen from a source-free region on a neighboring CCD at the same distance to a corresponding readout node (Figure 2(a)). More than 4100 DQ Tau EPIC net counts in the (0.2–10) keV band were detected. The SAS tasks evselect and epatplot suggest that the EPIC image of DQ Tau is not affected by pileup.

Following the SAS threads, the SAS tasks evselect, backscale, rmfgen, arfgen, epiclccorr and a number of tools from HEASOFT version 6.8 were used to construct source and background spectra, compute RMFs and ARFs, and construct light curves. During the XMM observation, DQ Tau showed no significant variations in its X-ray light curve (Figure 2(b)).

The time-integrated Chandra spectrum is modeled here with a three-temperature APEC plasma emission model (Smith et al. 2001). The APEC model is also employed in the time-resolved flare spectroscopy (Section 3.3). However, for a better modeling of individual spectral features, comparison between the Chandra and XMM-Newton data in Section 3.2 is performed using the VAPEC model (Smith et al. 2001).

The fit to the time-integrated Chandra spectrum is performed using χ² statistics. The light Chandra data pileup (Section 2.2) is ignored here. In the fit, elemental abundances are frozen at 0.3 times solar, which has been previously suggested as typical values for young stellar objects (Imanishi et al. 2001; Feigelson et al. 2002). Solar abundances are taken from Anders & Grevesse (1989). X-ray absorption is modeled using the atomic cross-sections of Morrison & McCammon (1983). The fit is performed with the XSPEC spectral fitting package version 12.5.1n (Arnaud 1996).

The X-ray spectral fitting of the DQ Tau data is ambiguous. At least two qualitatively different spectral model families give similarly good values of reduced χ²_ν ∼ 0.9⁹ (see Figures 3(a) and (b) with analogous VAPEC fits). The best-fit parameters for the first model family are around kT₁ = 0.7 keV, kT₂ = 1.9 keV, kT₃ = 4.3 keV, N_H = 1.3 × 10²¹ cm⁻² (corresponding to a visual extinction of A_V ∼ 0.8 mag using the gas-to-dust relationship of Vuong et al. 2003), and for the second family are around kT₁ = 0.3 keV, kT₂ = 0.9 keV, kT₃ = 3.5 keV, N_H = 3.2 × 10²¹ cm⁻² (A_V ∼ 2 mag). The second solution gives softer temperatures and higher absorption. The high-temperature model component (kT₃) is assumed here to be fully associated with the X-ray emission from the flare. A summary of the low-temperature model components (kT₁ and kT₂) for both solutions is given in Table 1. Both seem physically reasonable. For example, in the first solution, the temperatures (kT₁ and kT₂) and the ratio of emission measures (EM₂/EM₁) are consistent with those for COUP PMS stars (Table 1 in Getman et al. 2010). In the second solution, kT₁ ∼ 0.3 keV might represent a soft excess emission due to accretion (e.g., Telleschi et al. 2007b); DQ Tau is known to experience pulsed accretion flows near periastron (Basri et al. 1997). For both solutions, the absorption-corrected total-band (0.5–8 keV) luminosities of the low-temperature component (kT₁ and kT₂) lie within the locus of XEST Class II stars on the X-ray luminosity versus mass diagram (Telleschi et al. 2007a). Finally, the X-ray column densities from both solutions are consistent with the range of DQ Tau visual extinction estimates given in the literature, from A_V = 0.5 mag (Cohen & Kuhi 1979) to A_V = 2.1 mag (Strom et al. 1989). Thus, in the following analysis of the Chandra flare we will consider both solutions, which we will refer to as low-temperature plasma component Model 1 and Model 2.

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Compared to the emission near periastron (Φ = 0.95–0.99) observed in 2010 January by Chandra, the emission away from periastron (Φ = 0.66–0.67) observed in 2007 February by XMM-Newton is rather constant (Figure 2(b)) and, as shown below, softer. The time-integrated Chandra-ACIS and XMM-Newton-EPIC¹⁰ spectra are compared in Figure 3. Both spectra are fit with a two- or three-temperature VAPEC plasma emission model with individual elemental abundances set to values typical for PMS and for extremely active zero-age main-sequence (MS) stars, as specified in Güdel et al. (2007b).¹¹ The XMM-Newton spectrum can be successfully fit by the low-temperature plasma component Model 1 with no need for a hotter component, but with a flux 1.5 times higher than that of the low-temperature plasma emission in the Chandra spectrum (panels (c) versus (a) in Figure 3). Alternatively, the XMM-Newton spectrum can be fit with the low-temperature plasma component Model 2 with a flux 1.7 times lower than that of the low-temperature emission in the Chandra spectrum, and an additional moderately hot component (kT₃ = 1.9 keV) with a flux 4.5 times lower than that of the Chandra hot component (kT₃ = 3.5 keV) (panels (d) versus (b)). Differences in plasma emission between the two epochs, spaced roughly 3 years apart, could be simply attributed to evolution of magnetic active regions on the stellar surface. The 1.5-fold flux difference between the XMM-Newton spectrum and the Model 1 low-temperature component emission in the Chandra spectrum has an alternate, less likely, explanation—the known suppression of time-integrated X-ray emission in accreting versus non-accreting stars (e.g., Flaccomio et al. 2003; Preibisch et al. 2005); phase-variable mass flows from the circumbinary disk with accretion rates 10–200 times higher near periastron than those away from periastron are proposed to take place in the DQ Tau system (Mathieu et al. 1997; Basri et al. 1997).

As in G08a, flare spectral modeling is performed here on multiple adjacent overlapping time segments. Each time segment is specified by moving a rectangle ("boxcar") kernel of variable width through the Chandra event time series to encompass 1000 X-ray counts. Start time of each segment is specified to be offset from the start time of the previous segment by 0.1 times the duration of the previous segment. Forty-five segments are defined. The typical width of the smoothing kernel is 7 ks at the peak of the flare and 10 ks at the base of the flare (Table 2). Among the 45 overlapping segments, sets of five to six independent segments can be identified. Examining dozens of overlapping time segments along the decay phase of a flare light curve, rather than just a handful of independent intervals, allow higher time resolution which often results in the discovery of a more detailed, often more complex flare behavior (Appendix A in G08a).

Both piled and unpiled spectra (Section 2.2) from each segment are fit in XSPEC with thawed one-temperature APEC hot component (flare) and fixed two-temperature APEC cooler component (Table 1) models subject to constant WABS absorption. As in Section 3.1, elemental abundances are set to 0.3 times solar, and fits are performed using χ² statistics. Examples of the spectral modeling are shown in Figure 4, and the unpiled spectral results (plasma temperature, emission measure, corrected-for-absorption X-ray luminosities) for both Model 1 and Model 2 scenarios are reported in Table 2.

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For Model 1 and Model 2, respectively, Figures 5 and 6 give the inferred temporal evolution of the plasma temperature and the emission measure as well as flare evolution in the $\log T - \log \sqrt{{\rm EM}}$ plane. Without pileup correction (gray points), flare temperatures would have been overestimated by 20%–25% and emission measures would have been underestimated by 10%. During the first 15–20 ks, the temperature profile shows a plateau with a relatively constant plasma temperature around 50–60 MK, followed by a drop to 40–50 MK. This might be an indication of a precursor flare event. Afterward, the temperature profile is typical of many big flares from the COUP sample of flares: starting with a 5 ks period where the temperature rapidly rises to 60–90 MK, then in the next 5 ks it rapidly drops to 30–40 MK, and decreases more gradually to 25–30 MK over the following >10 ks.

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A 5–8 ks time delay is seen between the temperature peak and the emission measure peak. Such delays are often observed in solar (e.g., Milkey et al. 1971; Guedel et al. 1996, and the Appendix of this work), stellar (e.g., Reale 2007, and references therein), and flares from PMS stars (e.g., Table 1 in G08a). The delay can be interpreted as an efficient cooling of the plasma by conduction and radiation while the emission measure is still rising due to ongoing evaporation in the loop (Guedel et al. 1996). The effect is applicable to both flares from single and multiple loops (see the Appendix).

The derived slope of the trajectory in the DQ Tau flare's temperature–density diagram ζ = 1.1–1.5 indicates mild or no sustained heating in the framework of a single-loop flaring model (see Section 3.4.1 for more details).

3.4.1. Flare Loop Length

In order to derive the size of the coronal structure associated with the Chandra flare in DQ Tau, we employ the time-dependent hydrodynamic model of Reale et al. (1997, hereafter R97) for a single dominant coronal magnetic loop. Those authors establish a formula for estimating a loop's half-length L (throughout the text the terms "size" or "length" will be used to indicate the loop's half-length) accounting for the possibility of prolonged heating during the decay phase. They find

$$\begin{equation} L = \frac{\tau _{{\rm decay}} \sqrt{T_{pk}}}{3.7 \times 10^{-4} F(\zeta)}, \end{equation} \tag{ 1 }$$

where L is the half-length of the loop (cm), τ_decay is the flare decay e-folding timescale (s), and T_pk is the loop apex flare temperature (K) at the time of the maximum emission measure. F(ζ) is a correction factor for prolonged heating that is a function of the slope ζ of the trajectory in the temperature–density diagram. In practice, F(ζ) and T_pk must be calibrated for each X-ray observatory; the slope ζ is usually measured in the log T − log(EM^1/2) plane, where EM is the evolving emission measure and EM^1/2 is used as a proxy for the plasma density. We reproduce the calibration formulas for F(ζ) and T_pk derived for Chandra-ACIS by Favata et al. (2005):

$$\begin{equation} F(\zeta) = \frac{0.63}{\zeta -0.32} + 1.41 \end{equation} \tag{ 2 }$$

and

$$\begin{equation} T_{pk} = 0.068 \times T_{{\rm obs}}^{1.2}, \end{equation} \tag{ 3 }$$

where T_pk is a temperature at the loop apex (K) and T_obs is an observed "average" loop temperature (K) obtained from our Chandra-ACIS data. The ζ ≃ 1.5 corresponds to a freely decaying loop with no sustained heating, while ζ ∼ 0.32 corresponds to a loop with a long sustained heating.

This model is simplistic in a number of ways. It assumes that the plasma has a uniform density with a unity filling factor confined within a single semicircular loop of uniform cross-section. Furthermore, the model assumes that this geometry remains unaltered during the flare, that energy is efficiently transported along the magnetic field lines of the loop, and that there is continuous energy balance between the loop heating and the thermal conduction and radiative losses. Despite these limitations, the R97 model has been applied to a variety of solar and stellar flares, including flares from PMS stars in the Orion Nebula Cluster (ONC) and Taurus molecular cloud star-forming regions studied by Favata et al. (2005), Getman et al. (2008a), and Franciosini et al. (2007). The model has the advantage over earlier and simpler cooling loop models (e.g., Rosner et al. 1978; Serio et al. 1991), which neglect reheating during the decay phase and thereby tend to overestimate loop sizes. Note, however, the limitations of the single-loop approach and its application to the DQ Tau flare are discussed in Section 6.1.

The decay timescale inferred for the DQ Tau flare is τ_decay = 40.9 ks (Section 2). In the case of the Model 1 (Model 2) the observed "average" loop temperature at the time of the peak of the emission measure (time segment 27 in Table 2) is T_obs = 68.6^+14.0_−9.9 MK (T_obs = 48.9^+6.7_−5.5 MK), and the slope of the trajectory in the temperature–density diagram is ζ = 1.5 ± 0.6 (ζ = 1.1 ± 0.5; see Figures 5 and 6). Applying formulas (1)–(3), we find the size of the DQ Tau flaring coronal structure L = (5.8 ± 0.7) × 10¹¹ cm (L = 5.2 ± 0.6 R_⋆) and L = (4.4 ±  0.6) × 10¹¹ cm (L = 4.0 ±  0.6 R_⋆) for the Model 1 and Model 2, respectively. This size is comparable to independently inferred heights for the mm emitting regions of 3.7–6.8 R_⋆ above the stellar surface (Salter et al. 2010). The loop size of L = 4–5 R_⋆ is also comparable to half the ≳8 R_⋆ component separation near the periastron passage (Mathieu et al. 1997).

The R97 model assumes that loop growth is unrestrained by gravity. This is clearly the case for DQ Tau flare, where the derived half-length of the flaring loop is much shorter than the pressure scale height,¹² H_p, both near the stellar surface (H_p ∼ 7–16 R_⋆) and at a few stellar radii from the surface (H_p ≳ 100 R_⋆).

3.4.2. Cooling Timescales and Flare Loop Thickness

The analysis described in Section 3.4.1 only provides the loop length. The ratio between the loop's cross-sectional radius and the loop length (β = r/L), which indicates the thickness of the loop, remains to be determined. For the cool (T ∼ 1 MK) solar active region coronal loops spatially resolved by the Transition Region and Coronal Explorer (TRACE; Handy et al. 1999), β ranges from 0.01 to 0.06 for loops with a half-length of L < 100 Mm, and β falls in the narrow range 0.007–0.02 for larger loops (Table 3 and Section 4.2 in Aschwanden et al. 2000).¹³ For giant coronal flaring structures found in ONC PMS stars, Favata et al. (2005) and G08a assumed β = 0.1. This assumption might not hold (Section 3.2 in Favata et al. 2005). To check the applicability of the Reale method to big PMS flares, Favata et al. (2005) performed detailed hydrodynamic simulations of the big flare in COUP star 1343. Their loop simulations with β = 0.02 were in good agreement with the data.

Similar to the DQ Tau flare with no (Model 1, ζ = 1.5) or mild (Model 2, ζ = 1.1) sustained heating, the COUP 1343 flare loop is freely decaying with no sustained heating (ζ ≳ 1.5; G08a). Assuming similarity in flare decay cooling processes between the DQ Tau and COUP 1343 flares and using the COUP 1343 flare as a test bed, further comparison of the cooling timescales with the light curve decay timescales for both flares allows estimation of the thickness of the DQ Tau loop.

The dominant cooling mechanisms of a coronal plasma are thermal conduction and radiation with timescales of (e.g., Cargill et al. 1995)

$$\begin{equation} \tau _{{\rm con}} = \frac{3 n_e k_b T L^2}{\kappa _0 T^{7/2}}, \tau _{{\rm rad}} = \frac{3 n_e k_b T}{n_e^2 P(T)}, \end{equation} \tag{ 4 }$$

where k_b is the Boltzmann constant, n_e is the electron density, T is the measured plasma temperature, L is the loop length, κ₀ is the coefficient of thermal conductivity (Spitzer 1965), taken here as κ₀ = 10⁻⁶ erg s⁻¹ cm⁻¹ K^−7/2, and P(T) is the radiative loss function for an optically thin plasma (e.g., Rosner et al. 1978; Mewe et al. 1985). These timescales can be expressed as functions of independent quantities already derived in our previous analyses, such as, the loop length L (Section 3.4.1), temperature T, emission measure EM, X-ray luminosity in the wide 0.01–50 keV band L_X,0.01–50 (Table 2) as well as the quantity of interest here, the loop thickness β. At any given time t,

$$\begin{equation} V = 2 \pi \beta ^2 L^3 \end{equation} \tag{ 5 }$$

$$\begin{equation} n_e(t) \simeq \sqrt{\frac{{\rm EM}(t)}{V}} \end{equation} \tag{ 6 }$$

$$\begin{equation} L_{X,0.01\hbox{--}50}(t) = n_e(t)^2 V P(T), \end{equation} \tag{ 7 }$$

where V is the loop volume, so that

$$\begin{eqnarray} \tau _{{\rm con}}(t) &=& \frac{3 k_b}{\kappa _0 T(t)^{5/2} \beta } \sqrt{\frac{{\rm EM}(t) L}{2 \pi }}, \tau _{{\rm rad}}(t)\nonumber\\ &=& \frac{3 k_b T(t) \beta }{L_{X,0.01\hbox{--}50}(t)} \sqrt{{\rm EM}(t) 2 \pi L^3}. \end{eqnarray} \tag{ 8 }$$

For the COUP 1343 flare, the evolution of its T, EM, and L_X,0.01–50 as derived by G08a is shown in Figure 7(a). Figures 7(b)–(f) compares cooling timescales obtained from Equation (8) for the loop length of 4.8 R_⋆ (Favata et al. 2005; Getman et al. 2008a) and a range of β with the observed light curve decay timescale of τ_decay = 24 ks (G08a). For solar and stellar flares, if sustained heating is mild or absent, it is anticipated that from the end of the heating phase (indicated by the temperature peak) to the time of peak density (the emission measure peak) the plasma cooling is governed by conduction; at the density peak the radiation cooling and conduction times become equal; afterward the radiation cooling dominates (e.g., Reale 2007). During the flare decay the combined conduction and radiation cooling time τ_th (1/τ_th = 1/τ_con + 1/τ_rad) is naturally expected to be somewhat close to the observed flare decay time τ_decay. That is precisely what happens in the evolution of the COUP 1343 cooling timescales with a choice of β = 0.02 (panel (c)). In this case, during the flare decay the τ_th differs from τ_decay by no more than a factor 1.3, with the conduction and radiation times matching at the emission measure peak t = 230 ks and with radiation losses dominating afterward. With a choice of β < 0.02, τ_th is systematically lower than τ_decay (panel (b)), while with β>0.02, τ_th is systematically higher than τ_decay (panels (d)–(f)) with radiation losses becoming negligible at β ∼ 0.1 (panel (f)). For the COUP 1343 flare our cooling timescale analysis thus gives β = 0.02 and is in complete agreement with the results of the detailed simulations of Favata et al. (2005). This validates our use of cooling timescales for the estimation of the loop thickness.

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This method has also been applied to the DQ Tau flare; the results of the modeling with the Model 1 are shown in Figure 8. Panels (b)–(e) compare cooling timescales obtained from Equation (8) for a loop length of 5 R_⋆ (Section 3.4.1) and a range of β with the observed light curve decay timescale of τ_decay = 40.9 ks (Section 2). With a choice of β = 0.03 (Figure 8(c)) the evolution of cooling timescales in DQ Tau follows the correct pattern seen in Figure 7(c) for the COUP 1343 flare. With a choice of β < 0.03, τ_th is systematically lower than τ_decay (Figure 8(b)), while for β>0.03, τ_th is systematically higher than τ_decay (Figures 8(d) and (e)). The results of the cooling timescale analysis are also in agreement with the loop length analysis (Section 3): with a choice of loop length L ≪ 5 R_⋆, τ_th is systematically lower than τ_decay (Figure 8(f)), while for L ≳ 5.5 R_⋆, τ_th is systematically higher than τ_decay (Figures 8(h) and (i)). The model L = 4.5 R_⋆, β = 0.03 is also plausible (Figure 8(g)). For the DQ Tau flare our cooling timescale analysis thus gives β = 0.03 in the case of Model 1 (Figure 8), and β = 0.06 in the case of Model 2 (results are not shown). The Model 2 assumption gives a shorter and thicker flaring loop with the twice the volume of a Model 1 loop.

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Quantitative evaluation of the relation between the IRAM and Chandra data of DQ Tau can be deduced within the framework of the generalized Neupert effect introduced in Guedel et al. (1996). According to this formalism, the temporal profile of the total thermal energy E(t) of the plasma in the flaring loop can be expressed as the convolution of the kinetic energy influx by non-thermal electrons into the chromosphere (assumed proportional to the observed radio flux) with radiative and conductive cooling (assumed to follow an exponential decay):

$$\begin{equation} E(t) = \alpha \int _{t_0}^{t} F_R(t^{\prime }) e^{-(t-t^{\prime })/\bar{\tau }_{{\rm th}}} dt^{\prime }, \end{equation} \tag{ 9 }$$

where α is an assumed constant conversion factor between the thermal energy flux from electrons injected into the chromosphere and the observed radio flux F_R, $\bar{\tau }_{{\rm th}}$ is the time-averaged cooling time of the thermal plasma in the loop, t₀ is the start time of the radio flare, and t' < t.

In the case of DQ Tau, the captured late decay phase of the mm flare observed with IRAM can be fit to an exponential model:

$$\begin{equation} F_R(t) = F_R(t_1) e^{(t1-t)/\tau _R}, \end{equation} \tag{ 10 }$$

where t₁ = 15.5 ks is the earliest time of the portion of the mm flare decay that was observed and τ_R = 15.9 ks is the decay e-folding timescale during that phase (Figure 9(a)). The expected thermal energy profile is then

$$\begin{equation} E(t) = C_1 + \alpha F_R(t_1) \int _{t_1}^{t} e^{(t1-t^{\prime })/\tau _R} e^{-(t-t^{\prime })/\bar{\tau }_{{\rm th}}} dt^{\prime }, \end{equation} \tag{ 11 }$$

where the averaged plasma cooling time over the rise phase of the Chandra flare is $\bar{\tau }_{{\rm th}} \sim 30$ ks (see Figure 8(c)) and the C₁ is a constant term accounting for the integration over the unseen part of the mm flare. The integral in Equation (11) can be solved analytically; however two constant parameters remain unknown: the expected total thermal energy at the end of the unseen part of the mm flare (C₁) and the rate of the injected energy at the beginning of the captured decay of the mm flare (α × F_R(t₁)).

Equation (11) defines the thermal energy profile using the IRAM data. On the other hand, using Equations (5) and (6) the thermal energy $E(t) = 3 n_e(t) k_b T(t) V = 3 k_b T(t) \sqrt{2 \pi {\rm EM}(t)} \beta L^{3/2}$ can be inferred from our X-ray flare analysis; the resulted profile (using X-ray Model 1) is shown as the thick solid curve in Figure 9(b). Although the energy is subject to significant uncertainty, we are not so much interested in the individual values as in the general evolution of thermal energy that is observed: an early rise phase (t < 19 ks), when the energy is approximately constant around 3 × 10³⁵ erg; a late rise phase, when the energy rapidly rises to ∼5 × 10³⁵ erg within a 6 ks period; and a flare decay phase, when the energy falls to ∼2 × 10³⁵ erg during the remaining 15 ks period. For Model 2 the evolution is similar, but the energies are systematically higher by factors of ∼1.2–1.4.

A family of solutions to Equation (11) exists with constant parameter values around C₁ = 0.5 × 10³⁵ erg and α × F_R(t₁) = 0.8 × 10³² erg s⁻¹, resulting in thermal energy profiles (such as the dashed curve in Figure 9(b)) that resemble the profile derived from the X-ray modeling during the late, but not the early, stage of the X-ray flare rise phase.¹⁴ This resemblance of the thermal energy profiles derived independently from the microwave and X-ray modeling provides a quantitative indication of a Neupert-like effect. The divergence of the dashed and solid curves in Figure 9(b) for the very early rise phase of the X-ray flare (t < 19 ks) could be due to three possible effects that we have not modeled: (1) X-ray emission excess from flaring events in other coronal loops; (2) time dependence of the energy conversion factor α during the early decay phase of the microwave flare; (3) inaccurate estimation of the conduction cooling timescale τ_con, electron density n_e, and thermal energy E during the very early stages of chromospheric evaporation.

Finally, we assume that the second mm flare detected by IRAM is related to a separate and independent flaring event in another coronal loop. The weaker X-ray emission associated with the secondary event has a negligible impact on the observed decay phase of the primary large X-ray flare.

Large flares on the Sun often involve arcades of dozens or hundreds of sequentially reconnected magnetic loops, which are often observed to have different temperatures with the cooler loops lying below the hotter ones (e.g., Reeves & Warren 2002, and references therein). Despite the morphological complexity of such flares, their X-ray light curves often have simple exponential rise and decay shapes.¹⁶ In view of this solar analogy, the concept of the single-loop approach and its application to powerful spatially unresolved stellar flares might be questionable.

R97 have applied their single-loop method to flare decays of 20 solar M- and C-class flares monitored by the Soft X-ray Telescope (SXT; Tsuneta et al. 1991) on board the Yohkoh solar observatory satellite (Ogawara et al. 1991). Loop sizes derived using the method were shown to agree with the length of X-ray structures measured from direct inspection of SXT images (Figure 6 in R97). In the Appendix, we apply the approach of R97 to five solar X-class flares that are clearly associated with arcades of multiple loops seen in EUV TRACE images. In contrast to the general view that single-loop models tend to overestimate flaring loop sizes of complex flare events, for all of our five test bed flares, the equations of R97 yield a single-loop length comparable to or shorter than the lengths of the individual EUV loops measured from the TRACE images.

Here, we are unable to give a definite answer to the question of why the loop lengths of multi-loop X-class flares derived using the R97 approach can be comparable to the observed loop lengths, considering that the application of the single-loop approach to multi-loop flares is a priori incorrect.

The consistency may be by chance. For instance, for all five flares, the inferred slope ζ in the log T – $\log \sqrt{{\rm EM}}$ diagram is found to be close to ∼0.4. Let us ignore for a moment the fact that the physical meaning of the slope ζ for multi-loop flare arcades is generally different from that of a single loop (i.e., different plasmas in different loops heated and cooled sequentially versus a single plasma in a single loop heated and cooled). Within the framework of the single-loop model of R97, ζ ∼ 0.4 is close to the lowest values allowed, and the equations of R97 produce smaller loop lengths at smaller slopes. In fact, in the regime of low ζ (ζ < 0.7), the loop length is steeply decreasing with decreasing ζ, by a factor of 10 when ζ changes from 0.7 to 0.4.

Contrary to the "by chance" explanation above, loop length numbers can be comparable due to the presence of a single loop (or localized multiple loops ignited simultaneously and undergoing similar heating and cooling processes) that dominates the flare X-ray emission. For instance, an arcade-like structure with a single primary loop dominating the rise and early decay phases of a flare was proposed (based on a detailed modeling) for the complex flare on Proxima Centauri (Reale et al. 2004). At least for the best studied of the five solar X-class flares considered here, the Bastille Day flare, we do no find compelling reasons for the presence of such a dominant X-ray emitting structure (see the Appendix).

Clearly, even if the loop lengths inferred via the single-loop method happen to be comparable to the observed values for such complex flare arcade events, the method would not predict a correct loop geometry and would likely underestimate the emitting volume by at least a factor comparable to the number of instantaneously heated loops in the arcade, N_loop, and would overestimate the average electron density of emitting plasma by ${\sim} \sqrt{N_{{\rm loop}}}$. For instance, during the Bastille Day flare arcade, N_loop ∼ 20 out of the ∼100 observed loops fired near the flare peak time (Aschwanden & Alexander 2001).

For all five solar X-class flares considered here, the inferred slope ζ in the log T – $\log \sqrt{{\rm EM}}$ diagram is found to be close to the value of ∼0.4. Observationally, such a value is among the lowest values seen in solar flares (e.g., Sylwester et al. 1993). Such a shallow slope indicates continued heating during the decay phase of the X-ray emission. Continued heating is also what one would expect for multi-loop two-ribbon flares where the decay phase could be entirely driven by the heating released through sequential reconnection in individual loops (Kopp & Poletto 1984).

The slope ζ ≳ 1 in the log T – $\log \sqrt{{\rm EM}}$ diagram for the DQ Tau flare is significantly larger than the ζ ∼ 0.4 found for complex powerful solar flares (see the Appendix), suggesting heating behavior different from the five solar X-class flares. The possibility that the DQ Tau flare could involve an arcade-like structure with a single loop dominating the rise and early decay phases, as was proposed for the complex flare on Proxima Centauri (Reale et al. 2004), cannot be excluded. On the other hand, within the framework of a model involving explosive chromospheric evaporation into a single loop (e.g., Reale 2007), an evaporation time of only 3–5 ks is needed for a plasma with temperature T_obs ∼ 60–90 MK (Section 3.3) to propagate with an isothermal sound speed $v_s \rm {(cm\, s^{-1})}$$= \sqrt{(5/3 k_b T_{{\rm obs}})/(\mu m_p)} = 1.5 \times 10^4 \sqrt{(}T_{{\rm obs}}\rm {(K)})$ (e.g., Aschwanden et al. 2000) and to fill a DQ Tau flaring loop of a semi-length L = 4–5 R_⋆. The observed rise timescale of the DQ Tau light curve, τ_rise = 26.3 ks, is much longer than 3–5 ks and thus might indicate a complex history of heating during the early phase of the flare, e.g., due to multiple loops.

It is also worth mentioning that the time delay between temperature and emission measure peaks observed in the DQ Tau flare (Section 3.3) cannot be used to argue in favor of a single-loop event. This effect is commonly seen in multi-loop flares and can be explained by invoking the principle of linear superposition (see the Appendix). Likewise, the Neupert-like effect observed in the DQ Tau flare (Section 4) cannot be used to argue in favor of a single-loop event. If the Neupert effect occurred in each loop of a multi-loop system, then an observation that did not spatially resolve the loops would show the Neupert effect. Low-resolution observations of multi-loop solar flares commonly exhibit the Neupert effect (Veronig et al. 2002).

After all, in view of the fact that the observed timescales, temperatures, and X-ray luminosities of PMS stars (DQ Tau and ONC stars) are much higher than those of typical solar flares (Section 5), one could argue that the solar analogy might not be directly applicable. PMS and magnetically active older stars possess stronger, than solar, surface and global magnetic fields and larger volumes for magnetic fields to interact, so the extreme flaring behavior, beyond solar analogy, is expected (e.g., Section 2.2 in Benz & Güdel 2010 and Section 6.2 in this work). But then the single-loop approach developed on solar analogy might not be applicable either.

We conclude that we can neither prove the presence of loop arcades analogous to the solar cases nor refute the presence of single flaring loops longer than any seen on the Sun. The results from the single-loop X-ray modeling presented in Sections 3.4 and 4.1 should not be treated as definitive and should be considered with caution.

The origin of the big flares from the COUP sample themselves is unclear. For some of the 32 most powerful COUP flares whose inferred coronal structures reach several stellar radii, Favata et al. (2005) suggest magnetic loops linking the stellar photosphere with the inner rim of the circumstellar disk. For the extended sample of >200 flares, G08a and G08b find that large (a few to several stellar radii in length based on the single-loop approach of R97) coronal structures are present in both disk-bearing and disk-free stars; however G08a and G08b also find a subclass of super-hot flares with peak plasma temperatures exceeding 100 MK that are preferentially present in highly accreting systems. G08a and G08b further propose that the majority of big flares from the COUP sample can be viewed as enhanced analogs of the rare solar long-decay events (LDEs; see Section 7.3 in G08b and references therein). LDEs are eruptive solar flare events that produce X-ray emitting arches and streamers with altitudes reaching up to several hundred thousand kilometers (L ∼ 0.5–1 R_☉). In panels (b), (d), and (e) of Figure 10, representative solar LDEs (gray diamonds; compiled in G08a) are shown to have systematically higher flare durations and coronal lengths than the more typical solar flares (solid and dotted gray loci). We wish to emphasize that the analogies of the big flares from the COUP sample to solar LDEs ("growing" systems of giant multiple loops) are not based on information about the specific detailed geometry of COUP flares (the geometry is really unknown to us and is only simplistically modeled as a large single loop), but are instead based on the simple fact that the observed flare durations and model-inferred characteristic loop scales for COUP flares are the largest ever reported from PMS stars, provided that these model-inferred loop scales are close to the truth. The most widely accepted model for the origin of LDEs is that the impulsive flare near the solar surface (L ≲ 10⁻² R_☉) blows open the overlying large-scale magnetic field with subsequent reconnection of large-scale magnetic lines through a vertical current sheet. The large-scale magnetic field of PMS stars is likely far stronger than in the Sun and can sustain giant X-ray arches and streamers with sizes L ∼ 1–10 R_⋆. For the DQ Tau flare a different mechanism from that of a solar LDE-like reconnection can be considered.

There are at least three independent supporting lines of evidence suggesting that the DQ Tau flare could be produced through a process of colliding magnetospheres.

• 1.  
  Although the DQ Tau flare duration, morphology, and plasma temperature are typical of those of big flares from the COUP sample (Section 5), the probability of observing a big COUP-like flare is small. Within the COUP observation window (∼1200 ks) on average three big flares per bright PMS star were detected. This number is derived from the analysis of the 161 brightest X-ray PMS stars (G08a); the frequency of big flares for the remaining 1200 fainter COUP PMS stars is even lower. The DQ Tau Chandra observation was designed for an X-ray flare to appear close to the orbital phase of Φ = 0.98, which corresponds to the peak of the 2008 April large submillimeter flare (Salter et al. 2008). The detected Chandra flare indeed peaks within 15 ks of this orbital phase. The probability of detecting a big COUP-like flare within 15 ks of a pre-specified point in time is small (Prob < 15 × 2/(1200/3) = 0.075).
• 2.  
  Due to the lack of systematic monitoring observations in mm-bands, mm flare activity in PMS stars has been rarely reported (e.g., Bower et al. 2003; Salter et al. 2010). The DQ Tau X-rays are accompanied by a relatively unique non-thermal mm activity. The X-rays and mm are likely related through a Neupert-like effect (Section 4). The observed re-appearance of this unique mm activity near periastron is proposed to be associated with processes caused by the interacting magnetospheres (Salter et al. 2008, 2010).
• 3.  
  It is natural to expect large-scale magnetic structures to be involved in a process of colliding magnetospheres. The loop flaring sizes inferred from the X-ray flare analysis (4–5 R_⋆) are comparable to half the separation of the DQ Tau components near periastron (Section 3.4.1). This third item of evidence is completely model dependent and should be considered with caution (Section 6.1).

Based on the concept of interacting magnetospheres, we can further comment on energetics and possible flare loop geometry.

We can show that a crude estimate of the power expected to be dissipated by magnetic reconnection from colliding magnetospheres in DQ Tau is in agreement with the derived peak flare X-ray luminosity (L_X,0.5–8 = (4–5) × 10³⁰ erg s⁻¹; see Table 2) and the modeled rate of the kinetic energy injected into the chromosphere by non-thermal electrons (α × F_R(t₁) = (0.6–0.8) × 10³² erg s⁻¹; Section 4).

For a large-scale dipolar topology and a magnetic field B(L) ≃ B_ph/(L/R_⋆ + 1)³ with photospheric field strengths in the range 1–6 kG consistent with measurements of Zeeman broadening and circular polarization of photospheric lines in PMS stars (e.g., Johns-Krull 2007; Donati et al. 2008), the field strength at distances of 4–5 R_⋆ from the stellar surface is expected to be B(4–5 R_⋆) ≃ 5–50 G. Within the framework of the simple magnetic reconnection model shown in Figure 11, the initially dipolar fields of average strength B = 5–50 G at 4–5 R_⋆ from the stellar surface colliding at a speed of v ∼ 100 km s⁻¹ (Mathieu et al. 1997) reconnect with the rate of energy release E_m/τ_R ∼ 10³⁰–10³² erg s⁻¹, which is in agreement with the numbers given above. It is interesting to note that the analogous procedure applied to cases of star–planet magnetic interaction has difficulty explaining an excess X-ray emission associated with stellar chromospheric hot spots rotating synchronously with close-in giant planets (Lanza 2009).

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The model of interacting magnetospheres in RS CVn-type binaries by Uchida & Sakurai (1985) predicts the existence of X-ray emitting loop structures connecting the stars. The model of star–planet magnetic interaction by Lanza (2009) expects complex topologies, including loops connecting the planet with the stellar surface. While our X-ray modeling method and many related results are limited to the scenario of a single X-ray emitting loop on a single star, other scenarios for the DQ Tau loop geometry are possible: multiple X-ray emitting loops on a single star (Section 6.1), two or more loops appearing on both stars simultaneously, or a single or multiple loops connecting the two stars. For our modeled scenario, where the loop's footprints are anchored on the surface of a single star, potential destruction of the loop by centrifugal forces is not a serious issue. The derived length for the X-ray emitting structure of 4–5 R_⋆ is only comparable to and does not exceed the Keplerian corotation radius R_cor = 4.7 R_⋆.¹⁷

R97 applied their single-loop approach to solar X-ray flares observed with Yohkoh/SXT and showed that the method provided loop scaling sizes comparable to those measured in SXT images. In this section, we further check on the validity of the method by applying it to solar X-class flares clearly associated with multiple loop arcade geometries spatially resolved by TRACE. To better mimic stellar X-ray observations, which cannot resolve flare structures, the analysis of these solar X-class flares is carried out on disk-integrated X-ray data obtained by GOES.

From the TRACE solar flare catalog¹⁸ we choose only limb flares, where the heights of coronal structures derived from high-resolution TRACE images at EUV are most reliable. Out of the 75 X-class flares listed in the catalog, ∼10 are limb flares, and of those at least the following four have been studied in detail: X3.7 and X2.5 flares on 1998 November 22 (Warren 2000); X1.5 flare on 2002 April 21 (Gallagher et al. 2002); and X3.1 flare on 2002 August 24 (Li & Gan 2005; Reznikova et al. 2009). To these four flares we add the well-studied Bastille Day (2000 July 14) near-disk center flare (Aschwanden & Alexander 2001). All these five flare events are associated with arcades of multiple loops (Figure 12). At least for the two events on 2002 April 21 (Gallagher et al. 2002) and 2002 Aug 24 (Li & Gan 2005), "rising" flaring loop systems are reported and are believed to be associated with the formation of new loops as magnetic reconnection progresses upward. The heights of the EUV loops inferred from TRACE images, H, are given in Column 9 of Table 3. We assume that the X-ray emitting loops for the five events have heights comparable to or higher than the observed EUV loops; this assumption is generally expected due to effects of loop cooling and shrinkage and is found for both 1998 Nov 22 flares (Figure 3 in Warren 2000). As a sanity check of our GOES X-ray flare analysis, presented below, the three brightest M-class flares from the original test bed sample of R97 are also analyzed.

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The GOES X-ray data have been analyzed using the GOES software tools within an IDL based data analysis software package for Solar Physics, SolarSoft (Freeland & Handy 1998).¹⁹ Background subtraction for these powerful solar flares is ignored. Plasma temperature and emission measure profiles are derived by applying the filter-ratio method (goes_chianti_tem.pro program from White et al. 2005) to the observed GOES light curves in the (0.5–4 Å) and (1–8 Å) bands.

Light curve timescales, observed peak plasma temperature and emission measure, and slope ζ of the trajectory in the log T – $\log \sqrt{{\rm EM}}$ diagram are shown in Figure 12 and are given in Columns 2–6 of Table 3. Estimates on loop half-length from the rise, L_rise, and decay, L_decay, phases of light curves, using the single-loop approach, are also tabulated (Columns 7 and 8) and compared to the heights of the corresponding EUV loops, H (Column 9). Loop half-length from the rise timescale of the light curve, assuming an explosive chromospheric evaporation, is given as $L_{{\rm rise}}{\rm {(cm)}} = \tau _{{\rm rise}}{\rm {(s)}} \times 1.5 \times 10^{4} \sqrt{T_{\rm obs,pk}\rm {(K)}}$ (Section 6.1). Loop half-length from the decay phase of the light curve is calculated using the R97 method (Section 3.4.1) with the instrument-dependent parameters in the calibration formulas for F(ζ) and T_pk taken for GOES9 from Table A.1 of Reale (2007). Formal statistical errors on derived GOES temperature and emission measure are expected to be ≲4% (White et al. 2005), leading to statistical errors on the slope ζ of ≲3% and errors on the loop half-length derived from the decay phase of the light curve L_decay of ≲30%.

For the three brightest M-class flares from the original sample of R97, comparison of their flare properties derived using Yohkoh/SXT data (R97) with those inferred from our GOES analysis shows that GOES peak plasma temperatures are systematically higher by 30%, slope ζ differences are within 30%, and loop half-length L_decay differences are within 50% (with a formal statistical precision of ≲30% on individual flares). Effects of neglecting background emission in GOES data are expected to be more pronounced for M-class rather than for X-class flares.

The result of our analysis is that for the five X-class flares associated with arcades of multiple loops (Figure 12), loop half-lengths L_decay (or more precisely loop heights, 0.64 × L_decay, assuming semi-circular loops stand vertically on the solar surface) derived from disk-integrated GOES data using the single-loop approach of R97 are shorter or comparable to the heights of the EUV loops, H, measured from a direct inspection of high-resolution TRACE images (Table 3). It is important to note that the application of the single-loop approach to multi-loop flares is a priori incorrect, and the similarity in loop length values can occur by chance (Section 6.1) or because of a special physical phenomenon taking place, namely, the presence of a dominant X-ray emitting structure. The latter does not seem to apply to the Bastille Day flare (see below). Meanwhile, the loop half-lengths derived from the rise phases of the GOES light curves with the assumption of chromospheric evaporation in a single loop, L_rise (or their semi-circular heights, 0.64 × L_rise) are systematically larger by a factor of 4–7 compared to the measured heights of the EUV loops. It is also important to note two other aspects of our analyses. First, for all five flares the inferred slope of the trajectory in the log T – $\log \sqrt{{\rm EM}}$ diagram is ζ ∼ 0.4, indicating that heating is the dominant process in the decay phase of a flare (Section 6.1). Second, the effect of time delay between temperature and emission measure peaks is clearly observed in all five multi-loop X-class flares (Figure 12). This confirms the general expectation that the effect (described in Section 3.3) is applicable to a multi-loop flare case as well as to a single loop. In a multi-loop case the effect can be explained by the principle of linear superposition of flares from individual loops to produce a flare for a complete event (e.g., Aschwanden & Alexander 2001). Below we apply the principle of superposition to the Bastille Day flare.

If individual loops in a flaring loop system are ignited nearly simultaneously and are subject to a similar evolution, which is rather quite a rare condition (or a single loop dominates the flare X-ray emission), then for the complete event, according to the superposition principle, the flare temperature profile (including its peak plasma temperature), the flare light curve shape (specifically the e-folding decay timescale), and the slope ζ on the log T – $\log \sqrt{{\rm EM}}$ diagram should be similar to those of individual loops. In such case, for the complete event the equation of R97 would yield loop length similar to those of individual loops (or a single dominant loop). Is this the case for the five solar X-class flares studied here? At least for the best-studied case, the Bastille Day flare, we are unable to collect strong evidences in support of such a picture.

The plasma temperature profile of the Bastille Day event (Figure 12) has a major peak (near 10:18:20 UT) and two bumps seen along the rise (near 10:10:00 UT) and the decay (near 10:28:20 UT) phases of the profile, respectively. The classic time delay from temperature to emission measure is seen at the major peak (ΔT ∼ 340 s) and decay-phase bump (ΔT ∼ 200 s). According to the TRACE images (Figure 7 in Aschwanden & Alexander 2001) and the movie,²⁰ the rise-phase temperature bump is likely related to the appearance of several EUV loops at the westernmost edge of the arcade (near 10:10:00 UT), the major temperature/emission measure peaks are associated with ∼20 EUV loops (Aschwanden & Alexander 2001) that start appearing at about 10:14:00 UT (and last up to 11:00:00 UT), and the later temperature/emission measure bumps are likely related to numerous loops at the eastern part of the arcade that produce weaker flares appearing around 10:38:00 UT.

Using the principle of superposition, we simulate profiles of the complete event as EM_tot(t) = ∑EM_i(t) and T_tot(t) = ∑(T_i(t) × EM_i(t))/∑EM_i(t) where the summation is over the 20 EUV loops associated with the major peak. Here, we assume that the T_i(t) and EM_i(t) profiles for individual loop events have shapes similar to the profiles of the observed complete event (Figure 12). This simplistic simulation shows that if the loops fire nearly simultaneously (within the time period less than the observed T − EM peak delay of 340 s, or even more precisely less than 2 minute period), then the temperature profile (both the shape and the peak), the shape of the emission measure profile as well as the T − EM peak delay of the complete event look similar to those of individual loop events. Otherwise, for a complete event the simulation predicts broader shapes of the temperature and emission measure profiles and longer T − EM peak delay than those of individual loop events, e.g., the delay becomes twice longer, if the loops fire sequentially (and evenly spaced in time) within the 10 minute time period. Thus, the ignition of 20 loops (that are subject to a similar evolution) within the 2 minute time period might mimic a single dominant loop. Note, however, that in our analysis we make an assumption that the profiles for individual loop events have shapes similar to the observed profiles of the complete event. Many other solutions must exist. For example, using the principle of superposition, Aschwanden & Alexander (2001) show that individual loop events not firing near-simultaneously, with temperature and emission measure profiles much shorter than for the complete event, can produce the observed profiles of the complete event.

One might also argue that EUV images might not provide a true picture of an X-ray flare event. The X-ray band Yohkoh/SXT image of the Bastille Day flare corresponding to the major EUV peak (Figure 3 in Aschwanden & Alexander 2001) shows a single bright feature (spanning 20 × 40 arcsec²) within the more extended weaker emission, which may be evidence that a few localized EUV-like loops²¹ dominate the X-ray emission (Reale et al. 2004). However, since only a single instant X-ray image (Figure 3 in Aschwanden & Alexander 2001) is available, it is unclear how the X-ray emission evolves spatially during the early phase of the flare.

Assuming the presence of a dominant X-ray emitting structure (a single loop or multiple similar loops firing simultaneously) during the Bastille Day flare, we can apply the R97 approach strictly to the initial decay segment of the X-ray light curve (Reale et al. 2004). The slope of the initial decay (prior to the bump at the late decay phase) on the log T – $\log \sqrt{{\rm EM}}$ diagram is ζ ∼ 2, much steeper than the ζ = 0.41 obtained for the entire decay phase of the light curve, leading to an unrealistically large loop size of 200,000 km, which is a factor of 10 larger than the loop lengths measured directly from the EUV images. Considering the above, it seems to us that there is no clear evidence in support of a dominant X-ray emitting structure in the Bastille Day flare.