LOOP LENGTH AND MAGNETIC FIELD ESTIMATES FROM OSCILLATIONS DETECTED DURING AN X-RAY FLARE ON AT MIC U. Mitra-Kraev and L. K. Harra
Mullard Space Science Laboratory, University College London, Holmbury St. Mary, Dorking, Surrey RH5 6NT, UK
arXiv:astro-ph/0410656v1 27 Oct 2004
Abstract We analyse oscillations observed in the X-ray light curve of the late-type star AT Mic. The oscillations occurred during ?are maximum. We interpret these oscillations as density perturbations in the ?are loop. Applying various models derived for the Sun, the loop length and the magnetic ?eld of the ?are can be estimated. We ?nd a period of 740 s, and that the models give similar results (within a factor of 2) for the loop length (? 5.4 1010 cm) and the magnetic ?eld (? 100 G). For the ?rst time, an oscillation of a stellar X-ray ?are has been observed and results thus obtained for otherwise unobservable physical parameters. Key words: Stars: coronae – Stars: ?are – Stars: individual: AT Mic – Stars: late-type – Stars: magnetic ?elds – Stars: oscillations – X-rays: stars
1. Introduction Oscillations in the solar corona have been observed for many years. Wavelength regimes ranging from hard X-ray right down to radio have been investigated to search for evidence of waves. Periods have been found ranging from 0.02 to 1000 s. Table 1 in Aschwanden et al. (1999) provides an excellent summary of the di?erent periods that have been found, and an explanation for their existence. Most of these waves have been explained by MHD oscillations in coronal loops. Roberts (2000) has provided an excellent review of waves and oscillations in the corona. Many of the observations of waves have been determined from variations in intensity brightness. Recently, however, a huge step forward has been achieved in solar coronal physics due to the high spatial resolution available with the Transition Region and Coronal Explorer (TRACE). The ?rst spatial displacement oscillations have been observed in coronal loops (Aschwanden et al. 1999). It was suggested that these oscillations were triggered by a disturbance from the core ?are site. Various MHD waves were investigated and it was found that a fast kink mode wave provides the best agreement with the observed period of 280±30s. One of the most exciting aspects of observing waves in this fashion is that it potentially provides us with the
capability of determining the magnetic ?eld in the corona. It is notoriously di?cult to measure the magnetic ?eld in the corona. Techniques using the near-infrared emission lines have been successful, but have poor spatial resolution. Most frequently, indirect methods are used such as the extrapolations of the coronal magnetic ?eld from the photospheric magnetic ?eld that can be measured using Zeeman splitting. Nakariakov & Ofman (2001) made use of the ?are-related spatial oscillations sometimes observed when a ?are occurs, to determine the magnetic ?eld. They assume that the oscillation is due to a global standing kink wave, and hence the magnetic ?eld is related to the period of the oscillation, the density of the loop, and the length of the loop. In the case of Nakariakov & Ofman (2001) they found the magnetic ?eld to be 13±9G. The errors on this are large due to errors on the determination of density, loop length and period. Schrijver & Brown (2000) suggest that the oscillations are due to a sudden displacement of the magnetic ?eld at the surface which causes an oscillatory relaxation of the ?eld. Recent work by Nakariakov et al. (2004 suggests that the oscillations are second standing harmonics of acoustic waves tied to the loop length. The model gives a relationship between the oscillation period, the loop length and the average plasma temperature, all of which can be independently observed on the Sun.
Waves are observed across the electromagnetic spectrum on the Sun. However, observations on other stars are rare. One reason for this is that on the Sun, we have the spatial resolution to zoom in on small regions. For example, the transverse TRACE loops that have been observed are away from the main ?are site, and hence have lower emission measure. Analysing an X-ray light-curve, for example, would be unlikely to provide evidence of waves if the emission was averaged over the whole Sun. Oscillations have been observed for optical stellar ?ares (e.g., Mathioudakis et al. 2003 and Mullan et al. 1992). Mathioudakis et found a period of 220 s in the decay phase of a white-light ?are on the RS CVn binary II Peg. Using this they determine a magnetic ?eld of 1200 G. Mullan et al. (1992) again found oscillations in X-ray active red dwarfs. They compared their observations to the possibility that they were observing p-modes. On the Sun p-modes give rise to 5-minute oscillations on the surface. They concluded that p-modes could not produce the amplitudes they observed
Proc. 13th Cool Stars Workshop, Hamburg, 5–9 July 2004, F. Favata et al. eds.
and that it must be coronal oscillations, which have, until now, not been observed. In this work we observe for the ?rst time an oscillation in the corona of AT Mic. We measure the periodicity, and hence determine the magnetic ?eld. This is the ?rst observation of an X-ray oscillation during a stellar ?are. 2. Target AT Mic (GJ 799A/B) is an M-type binary dwarf, with both stars of the same spectral type (dM4.5e+dM4.5e). Both components of the binary ?are frequently. The radius of AT Mic given by Lim et al. (1987) is 2.6 1010 cm, using a stellar distance of 8.14 pc (Gliese & Jahreiss 1991). Correcting for the newer value for the distance from HIPPARCOS (10.2±0.5 pc, Perryman et al. 1997), using that the stellar radius is proportional to the stellar distance for a given luminosity and spectral class, we obtain a stellar radius of r? = 3.3 1010 cm = 0.47 r⊙ . The mass of AT Mic given by Lim et al. (1987) is m? = 0.4 m⊙ . 3. Observation
For example, Svestka (1989) shows solar ?are light curves and a ?at top is not seen. An alternative to oscillations for this ?are is that repeated and rapid ?aring is occurring. We do not consider this option in this paper, and assume for our analysis that we are observing a coronal oscillation. The purpose of our analysis is to determine through wavelet analysis the period and amplitude of the oscillation. We determine the magnetic ?eld and loop length of the coronal loop assuming that the oscillation is due to an acoustic wave. As a validity check the value of loop length was compared to the value determined from a radiative cooling model. 4. Results 4.1. Oscillation during flare maximum
For our analysis we used the XMM-Newton observations of AT Mic on 16 October 2000 during revolution 156. Raassen et al. (2003) have analysed this data spectroscopically, obtaining elemental abundances, temperatures, densities and emission measures, while a comparative ?are analysis between X-ray and simultaneously observed ultraviolet emissions can be found in Mitra-Kraev et al. (2004). 4.2. Loop length from oscillation Here, we solely used the 0.2–12 keV X-ray data from the Interpreting these oscillations as the second spatial harpn-European Photon Imaging Camera (EPIC-pn). Figure 1 shows the AT Mic light curve. The observa- monics of an acoustic wave within the ?are (Nakariakov et al. 2004) tion started at 00:42:00 and lasted for 25.1 ks (? 7 h). we can then determine the ?are loop length from There is one large ?are, starting ? 4 h into the obser1 vation, increasing the count-rate from ?are onset to ?are L [Mm] ≈ (1) · P [s] · T [MK]. 6.7 peak by a factor of 1.7, and lasting for 1 h 25 min. It shows a steep rise (rise time τr = 1100 s) and decay (de- Inserting the above obtained period P = 740 s and a cay time τd = 1700 s). There is an extended peak to this ?are temperature T = 24 MK, we obtain a loop length ?are, which shows clear oscillatory behaviour. The am- L = 5.4 1010 cm. Note that the temperature carries a 1σ plitude of the oscillation is around 16%. Applying multi- error of ? 16%, therefore the error of L is at least 8% temperature ?tting, Raassen et al. (2003) obtain a mean (?L > 0.5 1010 cm). It is not straightforward to calculate ?are temperature T = 24 ± 4 MK and a quiescent tem- the corresponding 1σ error of P . With these numbers, the perature Tq = 13 ± 1 MK, and from the O vii line ratio a inferred speed of sound is cs = L/P ≈ 730 km/s. ?are and quiescent electron density of n = 4+5 1010 cm?3 ?3 and nq = 1.9 ± 1.5 1010 cm?3 , respectively. The total 4.3. Loop length from radiative cooling times ?are and quiescent emission measures are EM = (19.5 ± 0.8)1051 cm?3 and EMq = (12.2 ± 0.5)1051 cm?3 . The loop length can also (and independently of any osOne of the reasons why we concentrated on this obser- cillation) be estimated from rising and cooling times obvation is that following the rise phase of the ?are there was tained from the temporal shape of the ?are, applying a a period of approximately 40 minutes when the light curve ?are heating/cooling model (see, e.g., Cargill et al. 1995). stayed at a high intensity level and showed strong oscilla- We follow the approach by Hawley et al. (1995) who intions. The oscillatory activity can be clearly seen by eye vestigated a ?are on AD Leo observed in the extreme ul(see Fig. 1), and shows behaviour suggestive of damping. traviolet. The shape of this ?are is very similar to our This character is much di?erent from solar ?ares where ?are on AT Mic, but roughly 10 times larger (in duration the ?are reaches a peak rapidly, followed by a slow decay. as well as in the increase of the count rate). It also shows
To investigate periodicities during the observation, we applied a continuous wavelet transformation to the 10s-binned-data, using a Morlet wavelet (see, e.g., Torrence & Compo 199 The wavelet coe?cients with the lowest periods are displayed in Fig. 1. The ?are oscillation is picked up with a 5σ signi?cance level. The local maximum with the shortest period and a signi?cance of more than 5σ is simultaneous with the ?are top and identi?es the oscillation, which, at the local maximum, has a period of P = 740 s . The period interval where the wavelet coe?cients have a signi?cance of 5σ or more, is [730,920] s.
Figure 1. The light curve and wavelet coe?cients of AT Mic. The observation started on 2000-10-16 at 00:42:00 h and lasted for 25’100 s (? 7 h). The upper panel shows the observed light curve, binned up to 100 s. The vertical dashed lines indicate the start and end of the rise and decay phase of the ?are. The lower panel shows the absolute values of a section of the corresponding wavelet coe?cients (see main text) divided by their standard deviation. The 3, 4 and 5 σ signi?cance contour lines are drawn. The two dashed lines mark the border of the cone of in?uence. A local maximum is clearly seen with a period of around 740 s during the ?are peak.
a ?at top with a possible oscillation. The ?are loop energy equation for the spatial average is given by 3 p = Q ? R, ˙ (2) 2 with Q the volumetric ?are heating rate, R the optically thin cooling rate and p the time rate change of the loop ˙ pressure. During the rise phase, strong evaporative heating is dominant (Q ? R), while the decay phase is dominated by radiative cooling and strong condensation (R ? Q). At the loop top, there is an equilibrium (R = Q). The loop length can be derived as 1500 4/7 3/7 · τd · τr · T 1/2 , (3) L= 4/7 (1 ? x1.58 ) d where τr is the rise time, τd the ?are decay time (indicated in Fig. 1 with the vertical dotted lines), T the apex ?are temperature and x2 = cd /cmax , with cmax the peak count d
rate and cd the count rate at the end of the ?are. Inserting these values, the loop length becomes L = 2.3 1010 cm, which is about a factor of 2 smaller that the loop length inferred from the oscillation. 4.4. Determination of the Magnetic field Interpreting the oscillations in terms of global standing kink waves, Nakariakov & Ofman (2001) derive a relation for the magnetic ?eld B= √ L 8π P ρ 1+ ρext , ρ (4)
with ρ the mass density inside, and ρext the mass density outside the loop. Note that there is a correction factor of 0.64 in the equation as presented by Nakariakov & Ofman (2001)
(Mathioudakis et al. 2003). We can simplify Eq. (4) by inserting Eq. (1) and using ρ = mp · n and ρext = mp · nq , with mp the proton mass. Then, the magnetic ?eld is given by √ 8π nq B= . (5) · 105 · T mp n 1 + 6.7 n Inserting the temperature and densities from the spectroscopy results, we obtain a magnetic ?eld of B = 115 ± 56 G. Note that the large error is resulting mainly from the density uncertainty. Using the loop length from the radiative cooling approach (Sect. 4.3) and Eq. 4, the magnetic ?eld is B ≈ 50 G. 4.5. Pressure balance To maintain stable ?are loops, the gas pressure of the evaporated plasma must be smaller than the magnetic pressure B2 . (6) 2nkT ≤ 8π Knowing the ?are density and temperature, we get a lower limit for the magnetic ?eld B > 80±60 G. Again, there is a large error because of the large uncertainty for the density. Shibata & Yokoyama (2002) assume pressure balance and give equations for B(EM, nq , T ) and L(EM, nq , T ) (their Eqs 7a,b). Using these relations, we obtain a magnetic ?eld of B = 70 ± 40 G and a loop length of L = (2.8 ± 1.7) 1010 cm. 5. Discussion and Conclusions We have used three di?erent approaches to determine the loop length of the AT Mic ?are and two di?erent approaches for the magnetic ?eld. The loop length derived in Sect. 4.2 from the ?are oscillation is the largest with L = 5.4 · 1010 cm, while the loop length derived from radiative cooling times (Sect. 4.3) is somewhat smaller by roughly a factor of 2, L = 2.3 1010 cm. The loop length derived from pressure balance (Sect. 4.5) is similar to the latter one, L = (2.8±1.7) 1010 cm. The loop length derived by assuming that the variations seen are due to oscillations is a factor of 2 di?erent to the loop length determined by more usual methods. Considering the assumptions that have to be made, this is good agreement, and gives us con?dence that we are, for the ?rst time, observing a stellar coronal loop oscillating. The magnetic ?eld ranges from 50 G (radiative cooling) to 115±56 G (oscillation), with the value from pressure balance in between with 70±40 G. Again, these three values are all consistent. This was the ?rst time that an oscillation during ?are peak was observed in X-rays in a stellar ?are and ?are loop length and magnetic ?eld derived from it. The values are consistent with other ?are models.
Acknowledgements We acknowledge ?nancial support from the UK Particle Physics and Astronomy Research Council (PPARC). UMK would
also like to thank the European Space Agency (ESA) and the University College London (UCL) Graduate School for ?nancial assistance to attend the Cool Stars 13 conference.
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