## 3 Results and Discussion
In other words, in addition to the AtOx fluence amount that depends on predefined conditions (such as orbit altitude and solar activity), the expected erosion relies on the effective fluence impinging on the surface considered: this continues to shrink at higher attitudes (see right images in Figure 4); in other words, the radiation intensity must be considered to decrease at higher-tilting angles. Thus, referring to Figure 5, the effective flux in Equation (3) may be expressed in a straightforward way as:  
\[f(t)=f_{0}[\tilde{\alpha}*\hat{\mathbf{n}}]_{t}=f_{0}\cos[\alpha(t)] \tag{4}\]  
where \(f_{0}\) represents the given (maximum) value of the AtOx flux (at fixed external conditions, as mentioned above), whereas the inner product between normal to the exposed surface and the velocity unit vector (spacecraft movement) requires knowing precisely the attitude variation along the time span considered (obviously, Equation (4) holds for \(-\pi/2<\alpha<\pi/2\), because the AtOx flux would be ineffective elsewhere--i.e., when the surface is ideally'shaded' by the vehicle motion itself). The combination of Equations (4) and (5) and the modeling extension shifts readily to express, for instance, the amount of mass eroded during a mission consisting of several steps '\(i\)' at predefined altitudes and known solar activities (\(f_{0,i}\)), as well as scheduled orbit inclinations (\(\alpha_{i}\)), written as:  
\[\Delta M=\sum_{i}\Delta M_{i}=\rho\cdot A\cdot E_{Y}\sum_{i}f_{0,i}\cdot\cos \alpha_{i}\cdot\Delta t_{i} \tag{5}\]  
Returning to the AtOx erosion measurements, with these assumptions a first attempt to retrieve the rate vs attitude values reported in Table 1, the value is given by the trivial expression:  
\[E_{Y}(\alpha)=E_{Y}(0^{\circ})\cdot\cos\alpha \tag{6}\]  
Figure 5: Mutual orientation schematics between the spacecraft outer surface and the AtOx flux. where the material 'zero' erosion rate is calculated from the pre-assessed value of the AtOx fluence specific to the adopted setup: because the latter is obtained by the frontal irradiation of reference specimens, Equation (6) is actually complementary to Equation (4), following the conventional physical concept of 'field flow' across a surface (\(f_{0}\) representing the scalar intensity, that is, the flux at the frontal incidence and \(\alpha\) being the tilt angle between normal to surface and local field direction). From the plots reported in Figure 6, it is clear that a simple (half) sinusoidal law (red-dotted curve) only partially fits the experimental results (black dots sequence), which is significant for both under- and over-estimating discrepancies. In particular, the sharp decrease in erosion at higher attitudes suggests that more effects of the AtOx damping arise by surface tilting than the simple one just considered. That is, the reduction in the effective flux occurred not only because the AtOx radiation intensity--in terms of 'number of impacting particles per unit of time/surface'--actually continued to decrease when moving off the normal incidence, but also because the erosion capability itself was fairly restrained, given that (for the same energy) the AtOx impact momentum was transferred less effectively to the exposed surface due to a non-negligible tangential component. We will not here go into the complex details of the chemical reactions involved in the erosion mechanism because they are intuitively comprehensible, given that the penetration depth of a colliding oxygen atom is statistically higher for a direct bump, while tending to be lower for a slitter interaction. This same trend holds for the probability of matter extraction from the surface, passing from inelastic impact dynamics to scattering behavior.