In the context of prominence seismology, a similar approach was proposed by Díaz et al. (2010) to obtain information about the density structuring along prominence threads using the piece-wise longitudinally structured thread model by Díaz et al. (2002) (see Figure 41). 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, , and the non-dimensional length of the thread, ,

Here we follow the notation of Díaz et al. (2010), who use for the thread length, rather than that of Soler et al. (2010a), who denote this length by . 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, . Note, however, that the non-dimensional frequencies of the fundamental mode and its first overtone can be cast as so that the dependence on the length of the tube and the thread Alfvén speed can be removed by considering the period ratio,Equation (44) can be used for diagnostic purposes, once reliable measurements of multiple mode periods are obtained. The curves in Figure 66 display the solution to the inverse problem in the (, ) parameter space for several values of the period ratio. Given the period ratio from an observation, it only depends on in first approximation. Once has been obtained, one can estimate the value of the magnetic field length , since the thread length, , 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. (2010) pointed out two main difficulties in this respect. From a theoretical point of view, the overtone with period 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 , 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., H. 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. (2007), who reported on the presence of two periods, and in their observations of a prominence region. Díaz et al. (2010) used the period ratio from these observations to infer the value for the length of the thread ratio . 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 .

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,

Once the length of the tube is known, an estimate for the prominence Alfvén speed can be inferred from Equation (45). In the example shown by Díaz et al. (2010), the high density contrast limit was used to infer the value .
Living Rev. Solar Phys. 9, (2012), 2
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