The top-hat electrostatic analyzer is a well known type of ion deflector and has been introduced by Carlson et al. (1982). It can be schematically represented by two concentric hemispheres, set to opposite voltages, with the outer one having a circular aperture centered around the symmetry axis (see Figure 121). This entrance allows charged particles to penetrate the analyzer for being detected at the base of the electrostatic plates by the anodes, which are connected to an electronic chain. To amplify the signal, between the base of the plates and the anodes are located the Micro-Channel Plates (not shown in this picture). The MCP is made of a huge amount of tiny tubes, one close to the next one, able to amplify by a factor up to the electric charge of the incoming particle. The electron avalanche that follows hits the underlying anode connected to the electronic chain. The anode is divided in a certain number of angular sectors depending on the desired angular resolution.
The electric field generated between the two plates when an electric potential difference is applied to them, is simply obtained applying the Gauss theorem and integrating between the internal () and external () radii of the analyzer
In order to have the particle to complete the whole trajectory between the two plates and hit the detector located at the bottom of the analyzer, its centripetal force must be equal to the electric force acting on the charge. From this simple consideration we easily obtain the following relation between the kinetic energy of the particle and the electric field :
Replacing with its expression from Equation (128) and differentiating, we get the energy resolution of the analyzer
Such an energy filter would be able to discriminate particles within a narrow energy interval and coming from a small element of the solid angle. Given a certain energy resolution, the 3D particle velocity distribution function would be built sampling the whole solid angle , within the energy interval to be studied.
In this section, we will show how to reconstruct the average density of the distribution function starting from the particles detected by the analyzer. Let us consider the flux through a unitary surface of particles coming from a given direction. If is the particle distribution function in phase space, is the number of particles per unit volume with velocity between and and and , the consequent incident flux through the unit surface is122).
The transmitted flux will be less than the incident flux because not all the incident particles will be transmitted and will be multiplied by the effective surface , i.e.,
Since for a top-hat Equation 130 is valid, then
We have that the counts recorded within the unit phase space volume would be given byGeometrical Factor and is a characteristic of the instrument. Then, from the previous expression it follows that the phase space density function can be directly reconstructed from the counts
Once we are able to measure the density particle distribution function , we can compute the most used moments of the distribution in order to obtain the particle number density, velocity, pressure, temperature, and heat-flux (Paschmann et al., 1998).
If we simply indicate with the density particle distribution function, we define as moment of order of the distribution the quantity , i.e.,
- the number density
- the number flux density vector
- the momentum flux density tensor
- the energy flux density vector
Once we have computed the zero-order moment, we can obtain the velocity vector from Equation (137). Moreover, we can compute and in terms of velocity differences with respect to the bulk velocity, and Equations (138) and (139) become
The new Equations (140) and (141) represent the pressure tensor and the heat flux vector, respectively. Moreover, using the relation we extract the temperature tensor from Equations (140) and (136). Finally, the scalar pressure and temperature can be obtained from the trace of the relative tensors
There are two classes of instruments to measure the ambient magnetic field: scalar and vector magnetometers. While nuclear precession and optical pumping magnetometers are the most common scalar magnetometers used on board s/c (see Pfaff et al., 1998b, for related material), the flux-gate magnetometer is, with no doubt, the mostly used one to perform vector measurements of the ambient magnetic field. In this section, we will briefly describe only this last instrument just for those who are not familiar at all with this kind of measurements in space.
The working principle of this magnetometer is based on the phenomenon of magnetic hysteresis. The primary element (see Figure 123) is made of two bars of high magnetic permeability material. A magnetizing coil is spooled around the two bars in an opposite sense so that the magnetic field created along the two bars will have opposite polarities but the same intensity. A secondary coil wound around both bars will detect an induced electric potential only in the presence of an external magnetic field.
The field amplitude produced by the magnetizing field is such that the material periodically saturates during its hysteresis cycle as shown in Figure 124.
In absence of an external magnetic field, the magnetic field and produced in the two bars will be exactly the same but out of phase by since the two coils are spooled in an opposite sense. As a consequence, the resulting total magnetic field would be 0 as shown in Figure 124. In these conditions no electric potential would be induced on the secondary coil because the magnetic flux through the secondary is zero.
On the contrary, in case of an ambient field , its component parallel to the axis of the bar is such to break the symmetry of the resulting (see Figure 125). represents an offset that would add up to the magnetizing field , so that the resulting field would not saturate in a symmetric way with respect to the zero line. Obviously, the other sensitive element would experience a specular effect and the resulting field would not be zero, as shown in Figure 125.
In these conditions the resulting field , fluctuating at frequency , would induce an electric potential , where is the magnetic flux of through the secondary coil.
At this point, the detector would measure this voltage which would result proportional to the component of the ambient field along the axis of the two bars. To have a complete measurement of the vector magnetic field it will be sufficient to mount three elements on board the spacecraft, like the one shown in Figure 123, mutually orthogonal, in order to measure all the three Cartesian components.