# Erosion Rate Measurements for DART Spacecraft Ion Propulsion System +
## 3 Results
### Absolute Density Model Based on Near-Field LIF and Far-Field QCM Data
The density of only sputter in the direction of \(\mathbf{\hat{w}}\) was found by inputting the velocity \(\mathbf{V}\) into the relative density function derived from the interpolated LIF data. This was done by taking the sine of the angle between \(\mathbf{w}\) and the density distribution's axis of symmetry. The result is the Doppler-shifting velocity component, which, converted to wavenumbers, can be inputtedinto Equation (12), where the LIF relative density function is the multivariable function from the interpolated LIF data. With the sputter speed and relative density at the grid in the direction of the given point in the far field, the differential flux can be found from Equation (13). The net mass flow from the grid differential area, \(dA_{g}\), to the far field differential area, \(dA_{f}\), is described by  
\[\rho_{rel}\big{(}V_{x},V_{y}\big{)}=\frac{1}{2\pi\sigma^{2}}e^{-\frac{(|V_{x}-y_{ x}|^{2}+(V_{y}-y_{y})^{2})}{2\sigma^{2}}} \tag{9}\]0  
where \(d\Omega_{f}\) is the solid angle with which \(dA_{f}\) is seen from \(dA_{g}\); \(R\) is the distance from \(dA_{g}\) to \(dA_{f}\), and \(\theta_{f}\) is the far field angle between the far field normal and the incident sputter vector. The mass flux in the far field was desired, and so \(dA_{f}\) was divided out of Equation (16). Therefore, the far field flux is  
\[\rho_{rel}\big{(}V_{x},V_{y}\big{)}=\frac{1}{2\pi\sigma^{2}}e^{-\frac{(|V_{x}-y_{ x}|^{2}+(V_{y}-y_{y})^{2})}{2\sigma^{2}}} \tag{9}\]1  
Combining Equations (12), (13), and (17) gives a final function for the far field mass flux using the described geometry and measured data,  
\[\rho_{rel}\big{(}V_{x},V_{y}\big{)}=\frac{1}{2\pi\sigma^{2}}e^{-\frac{(|V_{x}-y_{ x}|^{2}+(V_{y}-y_{y})^{2})}{2\sigma^{2}}} \tag{9}\]2  
The total sputter flux at a given point in the far field from the grid is given by the summation  
\[\rho_{rel}\big{(}V_{x},V_{y}\big{)}=\frac{1}{2\pi\sigma^{2}}e^{-\frac{(|V_{x}-y_{ x}|^{2}+(V_{y}-y_{y})^{2})}{2\sigma^{2}}} \tag{9}\]3  
where \(N\) is the number of discrete grid elements. Not all grid elements contribute sputter flux to a point in the far field--there are two factors preventing a given element's contribution: occlusion by the grid and occlusion by the grid mask. To calculate occlusion by the grid, all elements were neglected whose sputter vector had an obtuse included angle between it and the given element normal vector. Because the grid geometry is spherical, any sputter vector with an obtuse angle intersects the grid itself, and so the given element cannot contribute flux to the far field. Grid mask occlusion was calculated by finding the intersection point between the sputter vector and every plane of the mask triangulation. If the intersection point was within the triangle that defined the plane, then the sputter vector was obstructed, and any contributions from that grid element were neglected. Using these methods, the flux of only sputter for a given point in the far field was calculated. Iterating this calculation over the entire far field produced the complete flux prediction. The calculation was performed for angles \(-13^{\circ}\) to \(90^{\circ}\) from the thruster exit plane. With the complete prediction, the QCM data and far field prediction were correlated by assuming they were related by a single constant, as described by Equation (8). The conversion constant was found by correlating the two datasets at angles from the grid exit plane of \(12^{\circ}\) and \(16^{\circ}\) and averaging the conversion factor for each. This conversion factor was then applied to both the far-field flux prediction and the grid density plots. Figure 15 shows the 1-m distance predicted absolute flux profile based on the near-field LIF measurements calibrated with a selected portion of the raw QCM data, also plotted. The shape of the predicted only flux profile deviates strongly from a simple cosine approximation due to the variation of production efficiency across the grid and preferred directionality.