3.3 Determination of coronal magnetic fields

In the low plasma- $b$ limit, the expression for the kink speed (8

) reduces to

and contains two unknown parameters, the Alfvén speed $CA0$ and the density ratio $re/r0$ . By observationally measuring $CK$ and considering the density ratio as a parameter, the Alfvén speed in the loop can be determined. Assuming a density ratio $r /r = 0.1 e 0$ , we obtain $C = 756 ± 100 km s- 1 A$ for the kink speed of $-1 1020 ± 132 km s$ , 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 $B 0$ is in $G$ , the distance between the footpoints $d$ is in $m$ , the number density in the loop $n0$ is in $-3 m$ , and the oscillation period $P$ is in $s$ ; $m$ is the effective particle mass with respect to the proton mass. In the solar corona, because of the presence of heavier elements, $m = 1.27$ . Applying this formula, Nakariakov and Ofman (2001

) estimated the magnetic field in an oscillating loop observed on the 14 July 1998, as $13 ± 9 G$ (see Figure 12

, where the number density is measured in $cm -3$ ). This error bar can be significantly reduced by improving the determination of the density in the loop and by better statistics.

Figure 12:

The magnetic field inside a coronal loop as function of plasma density inside the loop, determined by Equation (32

). The external to internal density ratio is 0.1. The solid curve corresponds to the central value of the kink speed $CK = 1030 ± 410 km s- 1$ (for the event of the 4 July 1999), and the dashed curves correspond to the upper and the lower possible values of the speed. The vertical dotted lines give the limits of the loop density estimation using TRACE 171 Å and 195 Å images. The distance between the loop footpoints is estimated as $83 Mm$ (from Nakariakov and Ofman, 2001 ).

A similar estimation for the field strength (about $15 G$ ) 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 $6.6 s$ , which are associated with a global kink oscillation. Using Equation (32 ) and assuming $ne/n0 = 0.1$ , we find the loop to have a magnetic field strength of $400 G$ . This value is consistent with a magnetic field extrapolation (see Asai et al., 2001 , which found a magnetic field strength of $300 G$ ). For an alternative interpretation of this observation in terms of the global fast sausage mode, see Section 4.