Quadratic magnetic gradients from seven- and nine-spacecraft constellations

Abstract

To uncover the dynamics of magnetized plasma, it is crucial to determine the geometric structure of the magnetic field, which depends on its linear and quadratic spatial gradients. Estimating the linear magnetic gradient requires at least 4 simultaneous magnetic measurements, while calculating the quadratic gradients generally requires at least 10. This study focuses on deriving both linear and quadratic spatial gradients of the magnetic field using data from the nine-spacecraft (9S/C) HelioSwarm or seven-spacecraft (7S/C) Plasma Observatory constellations. Time series magnetic measurements, combined with transformations between reference frames, were employed to determine the apparent velocity of the magnetic structure and the quadratic magnetic gradient components along the direction of motion. The linear gradient and remaining components of the quadratic gradient were derived using the least-squares method with iterative calculations applied to ensure precision. The validity of the approach was demonstrated using magnetic flux ropes and dipole magnetic field models. The findings indicate that constellations with at least seven spacecraft in nonplanar configurations can successfully yield linear and quadratic spatial gradients of the magnetic field.