Properties of the boundary integral equation for solar non-constant-α force-free magnetic fields

A boundary integral equation representation for the non-constant-a force-free field in space outside the Sun was investigated by Yan & Sakurai, who introduced a local parameter λ instead of the force-free factor a which should satisfy a proposed condition. Then the field at any point in space can be represented by the boundary integral equation, which is determined by the field and its normal derivative over the boundary. In the present paper, it is justified that, for a closed-form non-constant-a force-free field problem with finite energy content in free space around the Sun, as in Low & Lou, such real λ solutions do exist. Using numerical integration, it is found that the λ values that satisfy the condition at some point are not unique. However, this non-uniqueness of the λ solutions does not influence the computation of the field at that location, as demonstrated by numerical results. This is remarkable because the new parameter - λ has a local property at each position. Therefore, the calculation of λ at one point has no influence on that of λ at another point. The distributions of λ x , λ y and λ z have the same symmetrical or antisymmetrical features as B x , B y and B z , respectively. These properties of the boundary integral equation for a non-constant-a force-free field will promote the application of the integral technique for practical solar magnetic field problems.