Reciprocal Vectors of an Arbitrary Constellation of Spacecraft for Estimating Gradients and Wave Vectors
Authors: Chanteur Gérard M. 
Published: February 2023

At least four spacecraft are essential to investigate physical processes in space plasmas. CLUSTER was the first four-spacecraft mission orbiting the Earth, during its life time the size of the tetrahedron has been adjusted between hundred to thousands of kilometers accordingly to various scientific objectives. The size of MMS, the second four-spacecraft mission, is smaller. In the barycentric formalism [1, chapter 14] reciprocal vectors of the tetrahedron play a prominent role for estimating field gradients and analysing the propagation of waves or discontinuities. The method of least squares [1, chapter 12] underlines also the importance of the reciprocal vectors and both methods are identical &#160;for four spacecraft [1, chapter 15], nevertheless, meanwhile the barycentric approach is restricted to four spacecraft, the least squares approach is applicable to any number N of spacecraft, and more, weighted least squares offer a possibility of optimization. Generalized reciprocal vectors for N spacecraft with arbitrary weights have been defined [2] and it was announced [3] that gradients of fields estimated from CLUSTER or MMS observations could be improved by an appropriate choice of weights ; this is wrong, we hereby demonstrate that for N=4 the estimated gradient of a vector field is independant of the weights. This unfortunate false announcement was due to a programming error: an erratum has been recently presented [4]. Discriminating between synchronous and asynchronous measurements by the constellation helps clarifying the tools and their uses. Synchronous data which gather observations of the same vector field by all spacecraft at the same time are used to estimate the spatial gradient of the field, meanwhile asynchronous data gathering observations of the same field made by the spacecraft at different times are used to estimate wave vectors or the propagation of discontinuities. The generalized synchronous position tensor R1, built from the vertices of the constellation, is introduced to analyze synchronous data, meanwhile the generalized asynchronous position tensor R2, built from the couples of vertices of the constellation, is introduced to analyze asynchronous data. It worth noticing that the synchronous analysis can be optimized only for N > 4 and by contrast the asynchronous analysis for any number N of spacecraft. The corresponding generalized synchronous and asynchronous reciprocal vectors q are defined by applying the inverses Q1 and Q2 of R1 and R2 to the position vectors, and their properties are demonstrated. We also give in tensor form the estimated gradient of a vector field satisfying the solenoidal condition, initially given by components in[1, chapter 12] and we discuss briefly the aliasing of waves by the constellation, initially adressed for a tetrahedron in [1, chapter 14].ReferencesAnalysis Methods for Multi-Spacecraft Data, ISSI Scientific Report SR-001, Eds. G. Paschmann and P.W. Daly, 1998.
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