The Astrophysical Journal, 659: L173–L176, 2007 April 20 䉷 2007. The American Astronomical Society. All rights reserved. Printed in U.S.A.
FORWARD MODELING OF HOT LOOP OSCILLATIONS OBSERVED BY SUMER AND SXT Y. Taroyan and R. Erde´lyi Solar Physics and Space Plasma Research Centre, Department of Applied Mathematics, University of Sheffield, Sheffield, S3 7RH, UK;
[email protected]
T. J. Wang Department of Physics, Montana State University, Bozeman, MT 59717-3840
and S. J. Bradshaw Space and Atmospheric Physics, Blackett Laboratory, Imperial College London, Prince Consort Road, London SW7 2BZ, UK Received 2006 December 15; accepted 2007 March 5; published 2007 March 23
ABSTRACT An example of hot active region loop oscillations observed by SUMER and SXT is presented. The hypothesis that a fundamental mode standing slow sausage (acoustic) wave is initiated by a footpoint microflare is tested and confirmed using a forward modeling approach. The oscillation is set up immediately after the heating pulse. The duration, temporal behavior, and total heat input of the microflare are estimated using the oscillation parameters. The rapid energy release is followed by cooling. The time-distance profile of the heating rate along the loop is recovered using the intensity and Doppler-shift time series. Hot loop oscillations are mainly observed in the Doppler shift. The absence of intensity oscillations in this and similar cases is explained. It is also found that the intensity oscillation, unlike the Doppler shift oscillation, undergoes half a period phase variation when the background intensity passes through its maximum, thus making it more difficult to detect. Subject headings: hydrodynamics — line: profiles — Sun: atmosphere — Sun: oscillations Online material: color figures observed with a cadence of 90 s using the 4 # 300 slit during 2000 September 16–20 (Wang et al. 2006, 2007). Five spectral lines, Fe xix l1118 (6.3 MK), Ca xv l1098 and l555#2 (3.5 MK), Ca xiii l1134 (2.2 MK), and Si iii l1113 (0.06 MK) ˚ wide window for each line. After were transmitted, with a 2.2 A processing the raw data following standard procedures (decompression and corrections of flat-field, detector distortions, dead time, and gain effects), a single Gaussian was fit to each line profile to obtain a Doppler shift time series at each spatial pixel. A large number of flarelike brightenings were revealed in the hot flare line Fe xix, and more than 20% of them are found to be associated with Doppler shift oscillations. We present a typical example in this Letter. Figure 1 shows a hot coronal loop seen in the SXT AlMgMn filter. The SUMER slit was located on the loop top. Applying a method described by Wang et al. (2003a), we can determine the geometric parameters of the loop based on a circular model. We obtain the loop length to be 82 Mm, the azimuth angle of the loop baseline to the east-west direction to be 33⬚, and the inclination angle of the loop plane to the vertical to be 30⬚. Assuming that the northern footpoint of the loop is farther than the southern one to the observer, the loop plane should be inclined away from the observer to the vertical in order to match the observed loop shape. With this assumption, we will expect the initial blueshifts as observed if the energy release was located near the northern footpoint and produced a plasma injection along the loop. A brightening seen at the northern footpoint indicates a possible microflare (Fig. 1). This brightening appeared at about 18:28 UT. It was accompanied by a flow as suggested by the blue wing enhancement of the Fe xix line profile. The loop oscillation seems to be set up after 18:35 UT when the line peak position started to move. Therefore, the timing supports the idea that the oscillation could be triggered by a microflare heating at the loop footpoint. Figure 2 shows evolution of the intensities in Fe xix, Ca xv (l1098), and Ca xiii lines along the
1. INTRODUCTION
Wang et al. (2002) have detected oscillations in hot loops (16 MK) using SUMER (SOHO) measurements. The periods, damping times, and quarter-period phase shifts favor the interpretation in terms of a fundamental mode standing slow (acoustic) wave. Recently, Wang et al. (2007) applied the measured periods to infer the loop magnetic field strength. Much work has been dedicated to the study of the influence of various loop parameters on the periods and damping times of such waves. These include thermal conduction and radiation (Ofman & Wang 2002), gravitational stratification (Mendoza-Bricen˜o et al. 2004), compressive viscosity (Sigalotti et al. 2007), and thermal stratification (McEwan et al. 2006). Less attention has been paid to the excitation of these waves. Observations usually indicate that the oscillations are set up immediately following a pulse of hot plasma inflow at one of the loop footpoints (Wang et al. 2005). Taroyan et al. (2005) have established the temporal profile of the footpoint heating pulse, which could lead to such a rapid excitation of the fundamental mode. There are still several questions that remain unanswered, such as the absence of oscillations in the intensity (in the great majority of the cases), the energies required for the excitation of standing waves, and the seismological potential of the Doppler shift oscillations in determining the time-distance profile of the heating rate. The present Letter is aimed at tackling these problems and filling the gaps between theory and observations. The forward modeling approach allows us to reproduce the evolution of loop plasma for a particular example from the microflare heating phase to the cooling phase. 2. OBSERVATIONS
Spectral observations of active region coronae were obtained by SUMER in sit-and-stare mode. Two neighboring active regions (AR 9169 and AR 9167) on the northeast limb were L173
L174
TAROYAN ET AL.
Vol. 659
Fig. 1.—Oscillating soft X-ray loop (outlined with crosses) fitted with a circular model (dark curve). The SUMER spectrometer slit position is indicated as two vertical lines. [See the electronic edition of the Journal for a color version of this figure.]
Fig. 3.—(a) Doppler shift time series in the Fe xix line at a fixed slit position. (b) Average time profile of Doppler shifts along the cut shown in (a). The thick solid curve is the best fit of the oscillations to a damped sine function. [See the electronic edition of the Journal for a color version of this figure.]
SUMER slit observed at the loop top. The brightening first peaked in the Fe xix line at about 18:40 UT, then peaked in the Ca xv and Ca xiii lines about 30 and 50 minutes later. This indicates a cooling of the hot loop down to lower temperatures. Figure 3 shows time series of Doppler shift in the Fe xix line along the slit, indicating a damped oscillation visible for about
4 periods. We measure the oscillation parameters averaged along the slit in a region shown in Figure 3a. The function V(t) p V0 ⫹ Vm sin (qt ⫹ f)e⫺lt
(1)
was then fit to the oscillation (Fig. 3b, solid line), where V0 is the background Doppler shift, Vm is the maximum Doppler shift amplitude, and q, f, and l are the frequency, phase, and decay rate of the oscillations, respectively. We obtain the period, P p 8.1 Ⳳ 0.3 minutes, the decay time, td p 1/l p 7.2 Ⳳ 1.8 minutes, and the Doppler shift amplitude, Vm p 30 Ⳳ 11 km s⫺1. 3. MODELING RESULTS AND DISCUSSION
The loop is represented by a one-dimensional (1D) model. The hydrodynamic equations are solved using HYDRAD (Bradshaw & Mason 2003). Initially, the heating is concentrated near the footpoints (Fig. 4), and the loop has an initial maximum apex temperature of about 6 MK (Fig. 5). The loop is thermally and gravitationally stratified with footpoints anchored in a cool and dense chromosphere (Fig. 5). The inclination angle of the loop plane to the vertical is 30⬚. A heating pulse is applied at the northern footpoint. The precise
Fig. 2.—(a) Time series of the Fe xix line intensity (left line) along the slit. The overlaid contours represent the Ca xv (l1098, middle line) and Ca xiii (right line) intensity time series. The contour levels are 70%, 80%, and 90% of the peak intensity. (b) Average time profile of line-integrated intensity along the cut shown in (a). A 2 pixel smoothing has been applied to the flux curves of the Ca xv (middle line) and Ca xiii (right line) lines in order to remove the noise. [See the electronic edition of the Journal for a color version of this figure.]
Fig. 4.—Heating rate (in logarithmic scale) as a function of time and coordinate along the loop. The brightening at the lower footpoint corresponds to the microflare that initiates the fundamental mode standing wave. [See the electronic edition of the Journal for a color version of this figure.]
No. 2, 2007
FORWARD MODELING OF HOT LOOP OSCILLATIONS
L175
Fig. 6.—Top: Simulated time profiles of line intensities along the cut shown in the middle panel of Fig. 5. The left, middle, and right lines correspond to Fe xix, Ca xv (l1098), and Ca xiii intensities, respectively. Bottom: Corresponding Doppler shift. [See the electronic edition of the Journal for a color version of this figure.]
Fig. 5.—Hydrodynamic evolution of the loop: the time-distance plots are for the temperature, velocity, and density. The dark edges in the temperature and density plots represent the cool and dense chromospheric footpoints of the loop. The cut between the horizontal solid lines (s1 p 38 Mm, s2 p 48 Mm) in the velocity diagram indicates the position of the SUMER slit. [See the electronic edition of the Journal for a color version of this figure.]
form of the pulse that leads to rapid formation of a fundamental mode standing wave is given by Taroyan et al. (2005). The corresponding brightening can be seen in Figure 4. The duration of the pulse is 7.5 minutes, which approximately matches the fundamental mode period. Pulses with much shorter or longer durations could result in propagating waves. Such phenomena have already been observed by Berghmans et al. (2001) and Doyle et al. (2006). The hydrodynamic evolution of the loop plasma is plotted in Figure 5. The timedistance plots represent the temperature, velocity, and density. The middle panel of Figure 5 shows that a standing wave is formed rapidly, within a single period, in agreement with the SUMER plot in Figure 3 (assuming that the starting time of the pulse which sets up the standing wave is 18:28 UT). Assuming a loop radius of 1 Mm, the total energy input required to recover the initial Doppler shift is 1.5 # 10 27 ergs; i.e., the brightening seen in Figures 1 and 4 represents a microflare. The wave is subsequently damped mainly due to the action of thermal conduction. The inclusion of viscosity would enhance the damping as shown by Sigalotti et al. (2007). Figure 3 shows that the Fe xix line becomes predominantly redshifted and gradually returns to the pre-event zero level. Figure 2 shows that after an initial temperature increase the loop undergoes cooling that also distorts the standing wave profile and increases the oscillation period. The fact that the microflare phase and the cooling phase are close in time and space suggests that these are manifestations of
different phases of the same process, e.g., reconnection or some other process. In order to recover the intensity time profile seen in Figure 2 and the redshift behavior that follows the initial large amplitude oscillation, the heating rate is decreased to 20% of its initial value at the bottom (northern) footpoint within about 25 minutes. The heating at the top (southern) footpoint is decreased to 10% of the initial value but on a longer timescale so as to gradually change the redshifts into blueshifts. In order to compare the hydrodynamic quantities with SUMER measurements, synthetic spectra are generated. The Fe xix, Ca xv (l1098), and Ca xiii lines are selected. The details of the applied procedure are described by Taroyan et al. (2006). The heliographic position of the loop, its orientation with respect to the east-west direction, and the location of the SUMER slit are taken into account. The evolution of the intensities and the Doppler shift is plotted in Figure 6. The intensity is integrated over the horizontal cut in the middle panel of Figure 5, which represents the location of the SUMER slit. Similar to Figure 2, each line suffers an intensity increase followed by a decrease. The initial increase in the Fe xix line intensity is a consequence of the microflare and the corresponding temperature increase. The subsequent decrease represents the cooling phase. The intensities of the cooler Ca xv and Ca xiii lines consecutively pass through their peaks as the loop cools to lower temperatures. The initial negative blue shift of about 32 km s⫺1 is followed by a damped oscillation that gradually becomes redshifted similar to Figure 3b. The redshift gradually decreases and eventually turns into blueshift. The good agreement between the measured and simulated results indicates that the inferred time-distance profile of the heating rate displayed in Figure 4 has been chosen correctly. Just like in the SUMER data, the oscillations appear only in the Doppler shift. Therefore, the simulations confirm that the interpretation in terms of a fundamental mode acoustic standing wave is correct despite the absence of any oscillations in the intensity. The theoretical explanation for this important feature is the following: the Doppler shift is proportional to the plasma
L176
TAROYAN ET AL.
Fig. 7.—Same as in Fig. 6, but with a cut between s1 p 8 Mm, s2 p 18 Mm along the loop. [See the electronic edition of the Journal for a color version of this figure.]
velocity inside the loop. In a 1D plasma with no source terms, the velocity of a standing wave is expressed in the form
v(t, s) p v 0 sin
, (2ptP ) sin (2ps Pc ) k
(2)
k
where v 0 is the velocity amplitude, s is the coordinate along the loop, L is its length, and c is the sound speed. Each mode is characterized by a period Pk p 2L/(ck) and a harmonic number k p 1, 2, …, where k p 1 corresponds to the fundamental mode. Thus, the fundamental mode has nodes at the footpoints and an antinode with maximum velocity at the apex. The intensity, on the other hand, depends on the density perturbation which, according to the equation of continuity and equation (2), has the form r(t, s) p ⫺r 0 sin
, [2p(t ⫺P P /4)] cos (2ps Pc ) k
k
(3)
k
where r 0 is the amplitude of the density perturbation. Equation (2) shows that there is a quarter-period phase shift between the velocity and density oscillations. It also shows that the density oscillation, unlike the velocity oscillation, has a node at the apex and antinodes at the footpoints of the loop. The slit is located
Vol. 659
around the apex of the loop, which explains the absence of intensity oscillations in both the SUMER data and the simulations. This is confirmed by Figure 7, where the total intensity is integrated along a loop segment away from the apex. Figure 7 displays intensity oscillations in all three selected lines. Such a case of clear intensity oscillations has been observed by SUMER (Wang et al. 2003b). On the other hand, the Doppler shift oscillations become smaller. From an observational perspective, equations (2)–(3) also suggest that the detection of intensity oscillations without accompanying Doppler shift oscillations at the apex could indicate the presence of a first harmonic with k p 2. The Doppler shift oscillations would be in antiphase on different sides of the apex. A first harmonic could be excited by a microflare at the apex, which would drive flows in opposite directions. Such oscillations have been studied by Nakariakov et al. (2004) in a somewhat different context. The lack of evidence for the presence of the first harmonic in the SUMER data supports the idea that small-scale energy releases such as microflares predominantly occur at the footpoints of active region hot loops. Apart from the well-known quarter-period phase shift between the intensity and Doppler shift oscillations, Figure 7 displays another interesting feature: the phase dependence of the intensity oscillation on the background intensity trend. During the initial large-amplitude phase of the oscillation, the Fe xix intensity oscillation is in antiphase with the Ca xv and Ca xiii intensity oscillations. Note that during the same time interval the background Fe xix intensity decreases, whereas the background intensities in Ca xv and Ca xiii increase. As the background intensity in Ca xv passes through its peak, the corresponding oscillation in Ca xv suffers half a period phase shift, becoming inphase with the Fe xix intensity oscillation. Therefore, the phase of the intensity oscillation changes by half a period when the background intensity passes through its maximum, which suggests that, in general, standing acoustic waves should more easily be detectable in the Doppler shift than in the intensity because the former does not display any phase variations in time. This phenomenon will be discussed in more detail in a forthcoming paper. The simulations were run on White Rose Grid (Iceberg node at Sheffield University). Y. T. is grateful to the Leverhulme Trust and to PPARC for financial support. R. E. acknowledges M. Ke´ray for patient encouragement and NSF, Hungary (OTKA, ref. TO43741). T. J. W.’s work is supported by NASA grant NNG06GA37G, and S. J. B. is funded by a PPARC Fellowship. Facilities: SOHO (SUMER), YOHKOH (SXT).
REFERENCES Berghmans, D., McKenzie, D., & Clette, F. 2001, A&A, 369, 291 Bradshaw, S. J., & Mason, H. E. 2003, A&A, 407, 1127 Doyle, J. G., Taroyan, Y., Ishak, B., Madjarska, M. S., & Bradshaw, S. J. 2006, A&A, 452, 1075 McEwan, M. P., Donnelly, G. R., Diaz, A. J., & Roberts, B. 2006, A&A, 460, 893 Mendoza-Bricen˜o, C. A., Erde´lyi, R., & Sigalotti, L. Di G. 2004, ApJ, 605, 493 Nakariakov, V. M., Tsiklauri, D., Kelly, A., Arber, T. D., & Aschwanden, M. J. 2004, A&A, 414, L25 Ofman, L., & Wang, T. J. 2002, ApJ, 580, L85 Sigalotti, L. Di G, Mendoza-Brice˜ no, C. A., & Luna-Cardozo, M. 2007, ApJ, submitted
Taroyan, Y., Bradshaw, S. J., & Doyle, J. G. 2006, A&A, 446, 315 Taroyan, Y., Erde´lyi, R., Bradshaw, S. J., & Doyle, J. G. 2005, A&A, 438, 713 Wang, T. J., Innes, D. E., & Qiu, J. 2007, ApJ, 656, 598 Wang, T. J., Innes, D. E., & Solanki, S. K. 2006, A&A, 455, 1105 Wang, T. J., Solanki, S. K., Curdt, W., Innes, D. E., & Dammasch, I. E. 2002, ApJ, 574, L101 Wang, T. J., Solanki, S. K., Curdt, W., Innes, D. E., Dammasch, I. E., & Kliem, B. 2003a, A&A, 406, 1105 Wang, T. J., Solanki, S. K., Innes, D. E., & Curdt, W. 2005, A&A, 435, 753 Wang, T. J., Solanki, S. K., Innes, D. E., Curdt, W., & Marsch, E. 2003b, A&A, 402, L17
