## 3 Instrument Performance  
### Velocity Resolution  
The size of the aperture and of each pixel are two parameters that play key roles in the velocity resolution of the MPA instrument. In addition, the detector sensitivity has an effect on the instrument performance. We discuss a trade-off between different detector technologies (micro-channel plates or multiple stacked channel electron multipliers) in Section 5.2. We calculate the velocity resolution of MPA in the case of a particle flying in a plane parallel to the magnets (\(\mathbf{v}_{0}=v_{0\mathbf{x}}\mathbf{e}_{x}\)). We determine the maximum velocity of a particle hitting pixel \(i\) at the distance given by Equation (23) from the center of the aperture with a velocity \(v_{i}=q\,B_{0}\,R_{i}\,/\,m\) expressed in Equation (24). The fastest particle that this pixel measures comes from the top of the aperture and hits the bottom of the pixel. Thus,  
\[2\,R_{max}=2\,R_{i}+\frac{a}{2}+\frac{l}{2}, \tag{31}\]  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Parameter** & **Low** & **Medium** & **High** \\ \hline Energy range & 20 keV–240 keV & 80 keV–1200 keV & 800 keV–4800 keV \\ Magnetic field strength & 0.055 T & 0.16 T & 0.48 T \\ Magnetic chamber gap & 7 mm & 7 mm & 12 mm \\ Field of view & \(20^{\circ}\times 10^{\circ}\) & \(20^{\circ}\times 10^{\circ}\) & \(16^{\circ}\times 19^{\circ}\) \\ Apperture geometry & 2 mm \(\times\) 5 mm & 2 mm \(\times\) 5 mm & 10 mm \(\times\) 5 mm \\ Mass & 8.5 kg & 8.5 kg & 8.5 kg \\ \hline \hline \end{tabular}
\end{table}
Table 1: Key parameters for MagEIS Low, Medium and High [16; 23]. The mass of each instrument is estimated given the total mass of the suite containing four analyzers (one Low, one High and two Medium) [23].  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Aperture shape & Cone of half-angle 2.5\({}^{\circ}\) \\ Aperture entrance & Disk of radius 1 mm \\ Magnetic field & Uniform at 0.1 T \\ Instrument length & 30 cm \\ Instrument height & 15 cm \\ Instrument gap width & 14 mm \\ Pixel size & 1 mm \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of the MPA characteristics.  
which leads to a maximum speed associated with this pixel of  
\[v_{max}=v_{i}+\frac{a+l}{4}\frac{qB_{0}}{m}. \tag{32}\]  
By analogy, the minimum speed associated with pixel \(i\) is  
\[v_{min}=v_{i}-\frac{a+l}{4}\frac{qB_{0}}{m}. \tag{33}\]  
Consequently, the velocity resolution of the instrument is  
\[\Delta\,v=v_{max}-v_{min}=\frac{a+l}{2}\frac{qB_{0}}{m}. \tag{34}\]  
With MPA's characteristics presented in Table 2, we obtain a resolution of 14.4 km/s, which yields for a 500 km/s solar wind a relative resolution of \(\Delta\,v/v_{0}=2.8\%\). This phenomenon is illustrated in Figure 3 where the hit positions of five beams of protons, separated by velocity steps of 7.2 km/s, are depicted. Pixels of 1 mm width are represented between the black lines. A pixel positioned to measure particles with speed \(v_{0}\) can in fact detect particles with speed in \([v_{0}-\Delta v/2\,;\,v_{0}+\Delta v/2]\). Thus, the uncertainty on the speed of one detected particle is \(\Delta v\).  
Figure 3: Five beams of particles are simulated using the SIMION software. Each beam has a set speed and a cylindrical (uniform) distribution of positions at the aperture of the instrument. The radius of the beam is 1 mm (2-mm disk aperture). Pixels are the areas separated by black horizontal lines.