## D Reference Systems

Interplanetary magnetic field and plasma data are provided, usually, in two main reference systems: RTN and SE.The RTN system (see top part of Figure 118) has the axis along the radial direction, positive from the Sun to the s/c, the component perpendicular to the plane formed by the rotation axis of the Sun and the radial direction, i.e., , and the component resulting from the vector product .

The Solar Ecliptic reference system SE, is shown (see bottom part of Figure 118) in the configuration used for Helios magnetic field data, i.e., s/c centered, with the -axis positive towards the Sun, and the -axis lying in the ecliptic plane and oriented opposite to the orbital motion. The third component is defined as . However, solar wind velocity is given in the Sun-centered SE system, which is obtained from the previous one after a rotation of around the -axis.

Sometimes, studies are more meaningful if they are performed in particular reference systems which result to be rotated with respect to the usual systems, in which the data are provided in the data centers, for example RTN or SE. Here we will recall just two reference systems commonly used in data analysis.

### D.1 Minimum variance reference system

The minimum variance reference system, i.e., a reference system with one of its axes aligned with a direction along whit the field has the smallest fluctuations (Sonnerup and Cahill, 1967). This method provides information on the spatial distribution of the fluctuations of a given vector.

Given a generic field , the variance of its components is

Similarly, the variance of B along the direction would be given by

Let us assume, for sake of simplicity, that all the three components of fluctuate around zero, then

Then, the variance can be written as

which can be written (omitting the sign of average ) as

This expression can be interpreted as a scalar product between a vector and another vector whose components are the terms in parentheses. Moreover, these last ones can be expressed as a product between a matrix built with the terms , , , , , and a vector . Thus,

where

and

At this point, is a symmetric matrix and is the matrix of the quadratic form which, in turn, is defined positive since it represents a variance. It is possible to determine a new reference system such that the quadratic form does not contain mix terms, i.e.,

Thus, the problem reduces to compute the eigenvalues and eigenvectors of the matrix . The eigenvectors represent the axes of the new reference system, the eigenvalues indicate the variance along these axes as shown in Figure 119.

At this point, since we know the components of unit vectors of the new reference system referred to the old reference system, we can easily rotate any vector, defined in the old reference system, into the new one.

### D.2 The mean field reference system

The mean field reference system (see Figure 120) reduces the problem of cross-talking between the components, due to the fact that the interplanetary magnetic field is not oriented like the axes of the reference system in which we perform the measurement. As a consequence, any component will experience a contribution from the other ones.

Let us suppose to have magnetic field data sampled in the RTN reference system. If the large-scale mean magnetic field is oriented in the direction, we will look for a new reference system within the RTN reference system with the -axis oriented along the mean field and the other two axes lying on a plane perpendicular to this direction.

Thus, we firstly determine the direction of the unit vector parallel to the mean field, normalizing its components

so that is the orientation of the first axis, parallel to the ambient field. As second direction it is convenient to choose the radial direction in RTN, which is roughly the direction of the solar wind flow, . At this point, we compute a new direction perpendicular to the plane

Consequently, the third direction will be

At this point, we can rotate our data into the new reference system. Data indicated as in the old reference system, will become in the new reference system. The transformation is obtained applying the rotation matrix

to the vector , i.e., .