Possible Cross-Section for a Coronal Loop of Given Shape?

We aim to answer the question about the cross-section of a planar coronal loop with a prescribed shape. We restrict the analysis to coronal loops embedded in a planar potential magnetic field. Then we carry out the analysis in the leading-order approximation with respect to the small parameter ϵ equal to the ratio of the characteristic size of the loop cross-section to the loop length. We show that, in this approximation, the loop cross-section can be prescribed arbitrarily at one of its footpoints. Then the loop cross-section at any other point is obtained by stretching or compressing the prescribed loop cross-section in the direction that is perpendicular to the loop axis and in the plane of the loop. The variation of the coefficient of stretching or compression along the loop can be chosen arbitrarily. In particular, it follows from this result that we can consider a planar loop of arbitrary shape and assume that its cross-section is circular everywhere and has a constant radius.