SWAYING THREADS OF A SOLAR FILAMENT

The Hα filtergrams resolve many thin threads in the filament. These threads are more clearly seen in the sharpened Hα images. The Hα sequences with cadences of 3.9 s and 18.5 s show that some filament threads sway back and forth in the plane of sky. Ten such swaying threads were selected for investigation. Three of them are marked as solid white lines in panels (c) and (d) of Figure 1. For each thread, two or three perpendicular cuts were made to measure the properties of possible waves (see panel (c) of same figure). The Hα time-slice diagram of each cut clearly shows the swaying signature, see panels (a)–(d) of Figure 2.

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Filament threads undergo a variety of changes that make them hard to follow for more than 5–10 minutes at a time. Their contrast changes and makes them appear and disappear within this timescale. They all move continuously (Zirker et al. 1990; Lin et al. 2003) presumably an effect of motion of their anchoring points in the photosphere. They also often mingle with neighboring threads and observers lose track of them. However, the swaying motion is a separate, characteristic feature of the threads, which may be monitored and studied within the limited time frame.

The temporal variations of the positions of the threads are fitted by sine curves, see panels (c)–(f) of Figure 2, from which the period (P) and the amplitude (A) of the wave are derived. Assuming the wave is propagating along a such thread, the phase difference between the two fitted curves represents the time interval when the wave passes through the two cuts. We derive the time interval (T) from the maximum cross-correlation of the two curves. Taking one of the two curves as a reference, we shift the other curve according to the derived time interval. The deviations of corresponding points between the reference curve and the shifted curve give the uncertainty in the measurement of time interval. Given the distance (L) between the two cuts, one obtains the phase velocities of the waves from V_ps = $\frac{L}{T}$. The uncertainties of the phase velocity can be calculated from the uncertainties in the time interval, since the distance (L) between two cuts is a constant.

Table 1 lists the measured wave properties of the 10 selected Hα swaying threads. These are short-period waves (mean period ∼ 3.6 minutes) and the mean velocities of the swaying motions ($\frac{2\pi A}{P}$) are about 2 km s⁻¹.

The 40 minute Hα Dopplergrams with the cadence of 18.5 s allow us to look for the oscillations in the line-of-sight direction. Thread Nos. 1–3 are selected from this sequence. As two examples, Figure 3 shows the time-slice diagrams in both Hα and the Doppler for thread Nos. 2 and 3.

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Because the position of a slice (marked as some white solid lines in panel (c) of Figure 1) is defined from one image and is fixed, when plotting a time-slice diagram, the filament thread may move out of such a fixed slice "window" due to the swaying effect. Therefore, we reconstruct the short sections of the time-slice diagrams, where the swaying threads are seen in Hα, by adjusting the slice positions slightly (see panels (a) and (c) of Figure 3). The corresponding short section of the Doppler time-slice diagrams is reconstructed accordingly.

For thread Nos. 2 and 3, the temporal variations of their mean Doppler velocities averaged over each slice indicate that the threads oscillate in the line of sight (see panels (b) and (d) of Figure 3). From the power spectrum, the waves are short period (∼5 minutes). Compared with thread Nos. 2 and 3, the Doppler signals of thread No. 1, which is not shown here, are relatively weak.

The amplitude of the derived line-of-sight velocity ranges typically between 1 and 2 km s⁻¹. For threads Nos. 2 and 3, they are 0.8 km s⁻¹ and 1.9 km s⁻¹, respectively. The quantitative study of J. E. Wiik & O. Engvold (2009, in preparation) also demonstrates that in cases when the optical thickness of the artificial filament τ≈ 1 the velocity fluctuation caused by the fine structure of the chromosphere below is suppressed to yield an uncertainty σ≈ 0.5 km s⁻¹. Applying this intrinsic error (0.5 km s⁻¹) caused by the chromosphere below and the "cloud model" correction, the two line-of-sight velocities raise to 1.6 ± 0.5 km s⁻¹ and 3.8 ± 0.5 km s⁻¹.

The two selected Hα threads are found oscillating both in the plane of sky and in the line of sight, with a similar period. Since the location of the target filament is S26W50, the measured swaying motion and oscillation of velocity in line of sight represent inevitably the projected components of the oscillating motions. As noted earlier, the oscillations of the filament threads are probably polarized and the plane of oscillation may attain various orientation relative to the local reference system. Swaying motions will be most clearly observed when a thread sways in the plane of the sky while Doppler signals will be strongest for oscillations along the line of sight.

Combining the corresponding components derived from the swaying motion of the two threads, which are 1.8 ± 0.2 km s⁻¹ and 2.3 ± 0.2 km s⁻¹, and the line-of-sight oscillatory motion, 1.6 ± 0.5 km s⁻¹ and 3.8 ± 0.5 km s⁻¹, we may derive the full vectors as 2.4 ± 0.5 km s⁻¹ and 4.4 ± 0.5 km s⁻¹ which are directed at angles 42° ± 10° and 59° ± 4°, respectively, relative to the plane of the sky. Since the heliocentric position of the target filament is W50, the corresponding local vertical direction becomes 50° relative to line of sight and we conclude that these two threads oscillate in planes which are reasonably close to vertical. It is, however, not possible based on only two cases to draw any general conclusion about the orientation of planes of oscillation of filament threads.

From Table 1, one notices that the amplitudes of the swaying motions are very small. This calls for very high spatial resolution in observations, which may explain why it has not been noticed in earlier studies. On the other hand, for some threads, the amplitudes of the waves passing through two cuts are notably different. In other words, the amplitudes of the waves are changing while they propagate along these threads. Apparent changes can be due to spatial damping of the waves in addition to the noise in the data. The damping phenomenon appears very common in solar filaments/prominences with a damping time corresponding to from 1 to 4 periods (e.g., Molowny-Horas et al. 1999; Terradas et al. 2002). Various possible theoretical mechanisms have been proposed (e.g., Terradas et al. 2001, 2002; Soler et al. 2008; Arregui et al. 2008b), among which the mechanism of resonant absorption seems to be the most efficient one for the damping of kink waves (e.g., Arregui et al. 2008b; Soler et al. 2009). It is also expected this mechanism to be the dominant one for the spatial damping of propagating kink waves, although a theoretical investigation on this issue is needed. Alternatively, the observed change of the wave amplitude might also be caused by a rotation of the oscillating planes. We will look for such evidence from higher temporal resolution data (cadence less than 2 s) in a following filament seismology study.

Next, in order to perform an interpretation of the observed oscillations in terms of linear MHD waves, we consider a simple model representing a prominence thread. We assume a configuration made of a straight and homogeneous cylinder of length L and radius a, filled with filament-like plasma and embedded in an also homogeneous coronal environment. The magnetic field is taken homogeneous and orientated along the tube axis (see a scheme of the model in Figure 4). In this model, we neglect the longitudinal inhomogeneity of the thread plasma for simplicity and consider β = 0, so magnetic effects are dominant over gas pressure effects. By using cylindrical coordinates, we assume that the density profile, ρ₀, only depends on the radial coordinate, r, as follows:

$$\begin{equation} \rho _0\! \left(r \right) = \left\lbrace \begin{array}{@{}lll} \rho _i, & \hbox{if} & r \le a, \\ \rho _e, & \hbox{if} & r > a, \end{array} \right. \end{equation} \tag{ 1 }$$

where ρ_i and ρ_e are the internal and external densities, respectively.

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The linear MHD wave modes supported by this flux tube model have been investigated in detail (e.g., Spruit 1982; Edwin & Roberts 1983; Cally 1986). Among the possible wave modes, the kink mode is the only one that can produce a significant transverse displacement of the cylinder axis, and so it is the best candidate for an interpretation of the present oscillations. Moreover, the kink mode produces short-period oscillations of the order of minutes for typical filament plasma conditions, which is also compatible with the measured periods. Since the observed threads appear to be very thin and long structures, it seems reasonable to consider the so-called thin tube approximation, i.e., L ≫ a. In such an approximation, the kink mode phase velocity (also called the kink speed), namely, c_k, is given by

$$\begin{equation} c_k = \sqrt{\frac{2 B_0^2}{\mu \!\left(\rho _i + \rho _e \right)}}, \end{equation} \tag{ 2 }$$

where B₀ is the magnetic field strength and μ = 4π × 10⁻⁷NA⁻² is the magnetic permittivity. Equation (2) can be rewritten in terms of the internal Alfvén speed, $v_{\mathrm{A}i}= B_0 / \sqrt{ \mu \rho _i }$, and the density contrast, c = ρ_i/ρ_e, as follows:

$$\begin{equation} c_k = v_{\mathrm{A}i}\sqrt{ \frac{2c}{c + 1} }. \end{equation} \tag{ 3 }$$

Figure 5(a) displays the ratio c²_k/v²_Ai as a function of c. We see that for large density contrast the curve becomes flat and so the ratio c²_k/v²_Ai is then almost independent of the density contrast. For typical coronal and filament densities, one has c ≈ 200. In such a situation, the factor $\sqrt{ \frac{2c}{c + 1} } \approx \sqrt{2}$ in Equation (3) and so the kink speed directly depends on the internal Alfvén speed

$$\begin{equation} c_k \approx \sqrt{2} v_{\mathrm{A}i}. \end{equation} \tag{ 4 }$$

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Now, assuming that thread oscillations observed from the Hα sequences are the result of a propagating kink mode, the measured phase velocities, V_ph, can be related to the kink speed, so c_k = V_ph. Therefore, it is possible to use Equation (4) to give a seismological estimation of the thread Alfvén speed

$$\begin{equation} v_{\mathrm{A}i}\approx V_{\rm ph} / \sqrt{2}. \end{equation} \tag{ 5 }$$

The use of the MHD seismology to obtain information of the plasma physical conditions has been applied by some authors in the context of coronal loop oscillations (e.g., Nakariakov & Ofman 2001; Arregui et al. 2007, 2008a; Goossens et al. 2008) and prominence oscillations (Terradas et al. 2008). Since the density contrast in the case of coronal loops typically has a small value, the factor with the density contrast cannot be dropped from Equation (3) and so it is not possible to give a direct estimation of the Alfvén speed from the observed kink speed. However, the much larger density of filament threads allows us to provide a more accurate determination of the Alfvén speed.

Table 2 contains an estimation of the Alfvén speed of each swaying thread observed in the Hα sequences by taking expression (5) into account. We see that the Alfvén speed varies in a wide range, which suggests that physical properties (i.e., density, magnetic field strength, etc.) significantly changes in different threads of the filament. Once the Alfvén speed is determined, we have a relation between the magnetic field strength and the thread density, i.e., $B_0 = v_{\mathrm{A}i}\sqrt{\mu \rho _i}$. Since the thread density is an unknown parameter, we cannot uniquely determine the magnetic field strength or vice versa. Figure 5(b) shows the dependence B₀ with ρ_i corresponding to four selected threads. The density has been varied in a wide range, 10⁻¹² kg m⁻³ < ρ_i < 10⁻⁹ kg m⁻³, and subsequently the magnetic field strength is in the range 0.1 G ≲ B₀ ≲ 20 G. This range agrees with the usually reported values of the magnetic field strength in prominences and filaments (e.g., Patsourakos & Vial 2002). From Figure 5(b), it can be also observed that the larger the density, the larger the magnetic field strength, which is compatible with the idea that active region filaments have stronger magnetic fields. By assuming a typical value of ρ_i = 5 × 10⁻¹¹ kg m⁻³ for the thread density and considering the results for the 10 studied threads, one obtains a magnetic field strength which varies in the range 0.5 G ≲ B₀ ≲ 5 G, see Table 2. Although we have to be cautious with the values obtained in this very rough estimation, the present results suggest that the magnetic field in the filament is not homogeneous and could largely vary between threads. Also, this could happen for the density, but the results are more sensible to variations of the magnetic field strength than variations of the density.