## 2 Instrument Design
### Dependence of the Field of View on Speed and Determination of the Magnetic Chamber Width  
If the magnetic chamber is too narrow, fast particles with \(v_{0y}\
eq 0\) hit the magnet surfaces. These particles would not be detected by the sensor or could potentially experience back-scattering as neutrals after charge-exchange, generating false counts in the sensor. This effect leads to an effective velocity-dependence of the field of view. For this characterization, we assume a particle inside the chamber with a trajectory according to Equation (8). Since the Lorentz force does not affect the trajectory of the particle along the parallel direction (\(y\)), the \(y\)-position of the particle after a time \(t\) is \(y(t)=v_{0y}\,t+y_{0}\), where \(y_{0}\) accounts for a potential offset of the initial particle position. We define the half-width of the magnetic chamber as \(h\). Without loss of generality, we assume that \(v_{0y}\geq 0\). The following arguments apply likewise to the case in which \(v_{0y}<0\). A particle reaches the sensor plane without hitting a magnet surface if and only if  
\[v_{0y}\,t+y_{0}<h\,\,\forall\,t\,\in\left[0\,;\,t_{\text{hit}}\right]\,\,\text {i.e., }\forall\,t\,\in\,\left[0\,;\,\dfrac{\pi-2\alpha}{\omega_{0}}\right]. \tag{12}\]  
Thus, to fulfill the detection condition, the \(y\)-component of the initial velocity must follow this inequality:  
\[v_{0y}<\dfrac{(h-y_{0})\,\,\omega_{0}}{\pi-2\alpha}, \tag{13}\]  
which we re-write for small \(\alpha\) as  
\[v_{0y}<\dfrac{(h-y_{0})\,\,\omega_{0}}{\pi}. \tag{14}\]Defining \(\beta\) as the incoming angle of the particle in the \(x\)-\(y\) plane, we find from Equation (14) a relation between the out-of-plane angle and the measured velocity \(v_{0x}\):  
\[\beta=\arctan\left(\frac{v_{0y}}{v_{0x}}\right)<\arctan\left(\frac{q\left(h-y_{ 0}\right)\,B_{0}}{\pi\,m\,v_{0x}}\right). \tag{15}\]  
As a starting point, we choose to physically constrain the FOV of MPA to \(5^{\circ}\times 5^{\circ}\), setting a \(5^{\circ}\) upper limit for the out-of-plane view angle of the instrument. This choice is based on the effective angular acceptance of the MagEIS Low and Medium instruments. Although the MagEIS Low and Medium chambers have a physical \(20^{\circ}\times 10^{\circ}\) FOV, their effective out-of-plane acceptance ranges from \(10^{\circ}\) for the lowest to \(3^{\circ}\) for the highest detectable energies [16]. Considering the higher density of thermal ions measured by MPA compared to the highly energetic electrons measured by MagEIS, it is not necessary to use acceptance angles as large as \(20^{\circ}\). Such a large acceptance angle would lead to a degradation of the focusing of ions in the sensor plane and production of more secondary emissions due to increased collisions of ions with the magnetic chamber walls. With our choice of a \(5^{\circ}\times 5^{\circ}\) FOV, the out-of-plane view angle is given by  
\[\gamma=\min\left[5^{\circ},2\times\arctan\left(\frac{q\left(h-y_{0}\right)\,B _{0}}{\pi\,m\,v_{0x}}\right)\right]. \tag{16}\]  
Figure 2 shows the dependence of \(\gamma\) on speed for different chamber widths when particles are assumed to enter the instrument at the center of the aperture: \(y_{0}=0\). As expected, a larger gap between the magnets allows for a wider field of view at a given speed. MPA's effective out-of-plane view angle remains constant for particle speeds up to 500 km/s with a chamber gap of 14 mm. Faster particles will be detected with a narrower FOV. We note that the MagEIS instrument used a gap between its two magnets of approximately the same size (see Table 1). To obtain a FOV of \(5^{\circ}\times 5^{\circ}\) on a wider range of velocities, two parameters can be adjusted: the chamber width or gap (as seen in Figure 2) and the magnetic field strength. Wider magnetic chambers will induce higher levels of stray fields, as discussed by [PERSON] et al. [16]. On the other hand, we can keep the chamber narrow by increasing the magnetic field to the detriment of heavier magnetic materials.