## 2 Instrument Design
### From Counts to a VDF  
We assume a constant conical aperture with a half angle of \(2.5^{\circ}\) for the remainder of the article. Based on the experience with the MagEIS instrument, this aperture parameter guarantees a roughly constant viewing angle of the distribution function at a reasonable error on count numbers. This aperture parameter is also the starting point of our discussion of the focusing property in Section 2.2. The aperture at the edge of the magnetic chamber is a disk of area \(A\) and diameter \(a\) (see Figure 1). The number of particles entering the conical aperture of area \(A\) and semi-angle \(\Theta_{0}/2=2.5^{\circ}\) during an acquisition time \(\tau\) is  
\[N=\tau\,A\,\int_{0}^{\infty}\int_{0}^{2\pi}\int_{0}^{\Theta_{0}/2}v\cos(\theta )\,f(v)\,v^{2}\sin(\theta)\,d\theta\,d\phi\,dv, \tag{17}\]  
\[N=\tau\,A\,\frac{2\pi}{4}\left(1-\cos\left(\Theta_{0}\right)\,\right)\,\int_{ 0}^{\infty}v^{3}f(v)\,dv, \tag{18}\]  
where \(f(v)\) is the (assumed to be) spherically symmetric VDF, which simplifies to:  
\[N=G\,\int_{0}^{\infty}v^{3}f(v)\,dv, \tag{19}\]  
where  
\[G=\tau\,A\,\frac{\pi}{2}\,\left(1-\cos\left(\Theta_{0}\right)\right) \tag{20}\]  
is the \"geometric factor\" of the instrument. The factor \(G\) accounts for the key parameters of the instrument: aperture area, acquisition time and FOV. Since \(\Theta_{0}\) is small, we use  
\[1-\cos\left(\Theta_{0}\right)\sim\frac{\Theta_{0}^{2}}{2}, \tag{21}\]  
leading to  
\[G=\tau\,A\,\frac{\pi}{4}\,\Theta_{0}^{2}. \tag{22}\]  
We note that the geometric factor derived here is not identical to the definition of the geometric factor of ESAs (see Equation (3) in [17] or Equation (3.11) in [18]) since \(dE/E\) is not constant in the case of MPA. The position of the center of a detecting pixel \(i\) in the position-sensitive sensor plane is defined as  
\[z_{i}=\left(i-\frac{1}{2}\right)l+\frac{a}{2}\qquad i\in\left[\!\left[1;n\right] \!\right], \tag{23}\]  
where we place the origin of our coordinate system at the center of the aperture. According to Equation (4) and using the pixel setup shown in Figure 1b, the position of the center of pixel \(i\) is linked to the velocity of a particle detected by this pixel as  
\[v_{i}=\left[\left(i-\frac{1}{2}\right)l+\frac{a}{2}\right]\frac{qB_{0}}{2m}, \tag{24}\]  
where \(v_{i}\) is the central velocity associated with pixel \(i\) and \(l\) is the width of the pixel. The difference of the speeds associated with two neighboring pixels is given by  
\[\delta v=v_{i+1}-v_{i}=\frac{qB_{0}l}{2m}, \tag{25}\]  
so that the number of counts (assuming a detection efficiency of 100%) by pixel \(i\) during an acquisition time \(\tau\) is  
\[C_{i}=G\,\int_{v_{i}-\frac{qB_{0}l}{2m}}^{v_{i}+\frac{qB_{0}l}{2m}}v^{3}f(v)\,dv. \tag{26}\]If we design the instrument so that the velocity step between two neighboring pixels is small compared to the measured velocities (\(\delta v/v_{0}\sim 0\)), we can re-write the number of counts in pixel \(i\) for a given input distribution as  
\[N=\tau\,A\,\frac{2\pi}{4}\left(1-\cos\left(\Theta_{0}\right)\,\right)\,\int_{ 0}^{\infty}v^{3}f(v)\,dv, \tag{18}\]0  
The estimated distribution function obtained from the number of counts per pixel by the instrument is then simply  
\[N=\tau\,A\,\frac{2\pi}{4}\left(1-\cos\left(\Theta_{0}\right)\,\right)\,\int_{ 0}^{\infty}v^{3}f(v)\,dv, \tag{18}\]1  
This result is in agreement with the method used by [PERSON] et al. (2019) where the output VDF is obtained by inverting Equation (27). Equations (27) and (28) do not account for finite detection efficiencies and other non-ideal behaviors of the instrument. In general, the sensitivity of any type of particle detector depends on the speed, mass, charge and hit angle of the detected particles ([PERSON], 2016; [PERSON], 2016). We introduce a finite, dimensionless efficiency parameter \(\epsilon_{i_{\text{MCP}}}<1\), where the subscript \(i\) indicates pixel \(i\) associated with speed \(v_{i}\).