13 - Tetrahedron Geometric Factors

The geometrical shape of the tetrahedron formed by the spacecraft is an essential criterion in the choice of scientific investigation which can be performed with data from a multi-spacecraft mission, such as Cluster. The shape of this polyhedron evolves continuously along the orbital trajectory of the spacecraft, and this shape has a major impact on the accuracy of the determination of scientific parameters related to the spatial gradient, such as the current density, which is discussed in Chapter 16. The scientific importance of the shape, combined with its variability, inevitably led to many proposals for “quality factors” to attempt to describe the geometric shape of the tetrahedron, or for “performance indicators”, to indicate the likely error of a particular scientific parameter. While these early geometric factors were all one-dimensional, 2-D parameters to characterise the geometrical shape of the tetrahedron have also been proposed, i.e., the “elongation” E and “planarity” P defined in terms of the eigenvalues of the volumetric tensor described in Chapter 12. In this chapter, we use the E and P parameters to define five characteristic types of tetrahedra and we check the validity and the meaning of the 1-D geometric factors by a numerical simulation using an “homogeneous reservoir of tetrahedra” in the E-P configuration space. As a practical application, we present an example of the Cluster orbit, and the associated computation of the 1-D and 2-D geometric factors. We represent these quantities in the E-P diagram, which allows a better understanding of their meaning. Finally, we demonstrate the limits of the 1-D geometric factors and point out the advantages of a 2-D geometric factor.