### 3.3 Determination of coronal magnetic fields

In the low plasma- limit, the expression for the kink
speed (8) reduces to
and contains two unknown parameters, the Alfvén speed and the density ratio . By
observationally measuring and considering the density ratio as a parameter, the Alfvén speed in the
loop can be determined. Assuming a density ratio , we obtain for
the kink speed of , for the event on the 14 July 1998 (see Nakariakov and
Ofman, 2001, for more details).
The Alfvén speed is defined by the magnetic field strength and the density of the medium.
Consequently, by using Equation (30), we can estimate the value of the magnetic field in the loop:
(there is a typo in Equation (6) of Nakariakov and Ofman (2001), corrected, e.g., in Roberts and
Nakariakov (2003)).
A practical formula for the magnetic field determination by the observables is

where the magnetic field is in , the distance between the footpoints is in , the number
density in the loop is in , and the oscillation period is in ; is the effective particle
mass with respect to the proton mass. In the solar corona, because of the presence of heavier elements,
. Applying this formula, Nakariakov and Ofman (2001) estimated the magnetic field in an
oscillating loop observed on the 14 July 1998, as (see Figure 12, where the number density is
measured in ). This error bar can be significantly reduced by improving the determination of the
density in the loop and by better statistics.
A similar estimation for the field strength (about ) was obtained by Roberts et al. (1984) from
the observations of Koutchmy et al. (1983) discussed in Section 3.2. However, in contrast with the TRACE
observations, the lack of the direct observability of the oscillating loop did not make the interpretation of
the oscillations in terms of the kink modes absolutely secure.
Asai et al. (2001) observed microwave quasi-periodic pulsations with a periodicity of , which are
associated with a global kink oscillation. Using Equation (32) and assuming , we find the loop
to have a magnetic field strength of . This value is consistent with a magnetic field
extrapolation (see Asai et al., 2001, which found a magnetic field strength of ). For
an alternative interpretation of this observation in terms of the global fast sausage mode, see
Section 4.