6.5 Seismology using period ratios of thread oscillations

The widespread use of the concept of period ratios as a seismological tool has been remarkable in the context of coronal loop oscillations (see Andries et al., 2009, for a review). The idea was first put forward by Andries et al. (2005) and Goossens et al. (2006) as a means to infer the coronal density scale height using multiple mode oscillations in coronal loops embedded in a vertically stratified atmosphere. In coronal loop seismology, the ratio of the fundamental mode period to twice that of its first overtone in the longitudinal direction (P1 ∕2P2) mainly depends on the density structuring along magnetic field lines. It can therefore be used as a diagnostic tool for the coronal density scale height.

In the context of prominence seismology, a similar approach was proposed by Díaz et al. (2010Jump To The Next Citation Point) to obtain information about the density structuring along prominence threads using the piece-wise longitudinally structured thread model by Díaz et al. (2002Jump To The Next Citation Point) (see Figure 41View Image). These authors showed that the non-dimensional oscillatory frequencies of the fundamental kink mode and the first overtone are almost independent of the ratio of the thread diameter to its length. Thus, the dimensionless oscillatory frequency depends, basically, on the density ratio of the prominence to the coronal plasma, ρp∕ρc, and the non-dimensional length of the thread, W ∕L,

ωL ---- = f(W ∕L, ρp∕ρc). (41 ) vAp
Here we follow the notation of Díaz et al. (2010Jump To The Next Citation Point), who use 2W for the thread length, rather than that of Soler et al. (2010a), who denote this length by Lp. In order to determine the dimensional frequency when comparing to observations, two additional parameters are needed, namely the Alfvén velocity in the corona or in the prominence (involving some knowledge of the magnetic field strength and density) and the length of the magnetic tube, 2L. Note, however, that the non-dimensional frequencies of the fundamental mode and its first overtone can be cast as
ω1L ----= f1(W ∕L,ρp ∕ρc), (42 ) vAp ω2L- vAp = f2(W ∕L,ρp ∕ρc), (43 )
so that the dependence on the length of the tube and the thread Alfvén speed can be removed by considering the period ratio,
-P1- = F (W ∕L,ρp∕ ρc). (44 ) 2P2
View Image

Figure 66: Plot of the solution lines satisfying P1∕2P2 = constant in the parameter space. The upper line corresponds to P1∕2P2 = 1.25 and the lower one to P1∕2P2 = 3, with each line showing an increment in P1∕2P2 of 0.25 from the previous one (from Díaz et al., 2010Jump To The Next Citation Point).

Equation (44View Equation) can be used for diagnostic purposes, once reliable measurements of multiple mode periods are obtained. The curves in Figure 66View Image display the solution to the inverse problem in the (ρp∕ρc, W ∕L) parameter space for several values of the period ratio. Given the period ratio from an observation, it only depends on W ∕L in first approximation. Once W ∕L has been obtained, one can estimate the value of the magnetic field length 2L, since the thread length, 2W, can be determined quite accurately from the observations.

The use of the period ratio technique needs the unambiguous detection of two periodicities in the same oscillating prominence thread. Díaz et al. (2010Jump To The Next Citation Point) pointed out two main difficulties in this respect. From a theoretical point of view, the overtone with period P 2 is an antisymmetric mode in the longitudinal direction, with a node in the center of the thread and two maxima located outside it. Only for sufficiently long threads, with W ∕L ∼ 0.1, the anti-nodes of the overtone are located inside the thread and could hence be measured in the part of the tube visible in, e.g., Hα. From an observational point of view, no conclusive measurement of the first overtone period has been reported so far in the literature, although there seem to be hints of its presence in some observations by, e.g., Lin et al. (2007Jump To The Next Citation Point), who reported on the presence of two periods, P1 = 16 min and P2 = 3.6 min in their observations of a prominence region. Díaz et al. (2010Jump To The Next Citation Point) used the period ratio from these observations to infer the value for the length of the thread ratio W ∕L = 0.12. Although it is difficult to estimate the length of the particular thread under consideration, assuming a value of 13,000 km, as for other threads analyzed by Lin et al. (2007), results in a magnetic tube length L ∼ 130,000 km.

This new seismological information can be now used to obtain further information about the physical conditions in the oscillating thread. Using analytical approximations for the dimensionless frequency of the first overtone, the following expression for the prominence Alfvén speed as a function of the length of the thread is obtained,

∘-------------- πL W ( W ) vAp = --- 2 --- 1 − --- . (45 ) P1 L L
Once the length of the tube is known, an estimate for the prominence Alfvén speed can be inferred from Equation (45View Equation). In the example shown by Díaz et al. (2010), the high density contrast limit was used to infer the value −1 vAp ∼ 160 km s.
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