# Erosion Rate Measurements for DART Spacecraft Ion Propulsion System +
## 3 Results
### Absolute Density Model Based on Near-Field LIF and Far-Field QCM Data
This angle, \(\psi\), is the same angle by which the peak axis is angled from the grid normal. The quantities \(\mu_{x}\) and \(\mu_{y}\) correspond to peak position with zero Doppler shift. Figures 13 and 14 illustrate the geometry involved. The scans of sputter LIF describe Equation (9) integrated across the entire y-axis. Additionally, due to the shifted peak angles, the velocity component along the axis of the laser beam is equal to \(V_{x}/\cos\psi\). Therefore, the relative density described by magnitude of the LIF signal is  
\[\rho_{LIF}(V_{x},\psi)=\frac{1}{2\pi\sigma^{2}}e^{-\frac{(V_{x}/\cos\psi-\mu_{ x})^{2}}{2\sigma^{2}}}\int_{-\infty}^{\infty}e^{-\frac{(V_{y}-y_{y})^{2}}{2 \sigma^{2}}}dV_{y}=\frac{1}{\sqrt{2\pi}\sigma}e^{-\frac{(V_{y}/\cos\varphi-\mu _{x})^{2}}{2\sigma^{2}}}. \tag{10}\]  
Substituting the \(x\)-axis term in Equation (9) with Equation (10) gives the two-dimensional density distribution as a function of the measured relative densities and the \(y\)-axis term:  
\[\rho_{rel}\big{(}V_{x},V_{y},\ \psi\big{)}=\frac{1}{2\pi\sigma^{2}}\Big{(} \sqrt{2\pi}\sigma\rho_{LIF}(V_{x},\psi)\Big{)}e^{-\frac{(V_{y}-y_{y})^{2}}{2 \sigma^{2}}} \tag{11}\]  
Because the distribution is assumed to be axisymmetric, the function can be evaluated at the y-value that corresponds with the peak position of the two-dimensional distribution, i.e., the origin of the distribution, where \(V_{y}=\mu_{y}\). The relative density can be evaluated as a function of only \(V_{x}\) and the shifted peak angle:  
\[\rho_{rel}(V_{x},\psi)=\frac{\rho_{LIF}(V_{x},\psi)}{\sqrt{2\pi}\sigma}. \tag{12}\]  
The standard deviation was found from fitted curves for the interpolated LIF data. Equation (12) shows that the flattened two-dimensional Gaussian simply scales the true distribution by a constant. Because the data are relative with only arbitrary units, the constant multiple in Equation (8) can be neglected when correlating the far field flux prediction and the QCM data, but the value of the conversion factor is important when determining absolute densities across the grid face, and so the additional terms in Equation (12) are included for all calculations. To find the conversion factor to translate the relative mass units to absolute units, the LIF relative density data were correlated with mass flux QCM data. The QCM measurements were taken at a 1 m distance from the grid. Therefore, in order to correlate the relative LIF data with the absolute QCM data, the LIF data were used to predict the mass flux of sputter at the distances corresponding to the QCM measurements. This prediction provides insight into the mass flux across the entire 1 m far field, and it produces the conversion factor to compute absolute densities for sputter across the grid face. The prediction was calculated in MATLAB by approximating the geometry of the thruster grid, grid mask, and 1 m far field. The discrete elements of the geometries were then used to ray-trace-sputter trajectories from the grid face to the far field. Performing these calculations over the grid face area and summing their contributions at each far-field location produced a prediction for relative sputter mass fluxes at 1-m distance in the horizontal plane. This result could then be correlated with QCM data to produce the desired relative-to-absolute conversion factor. The grid and grid mask were discretized into matrices of points in vector space, where the coordinate system origin was set at the center of grid curvature. The grid geometry was assumed to be a spherical cap with defined radius of curvature. The LIF relative density  
Figure 14: Diagram showing an arbitrary area element near the NEXT grid face and coordinate system for analyzing its contribution at an arbitrary location along an arc 1 m from thruster. Figure 13: Diagram of one-half of the NEXT grid, and shift of average velocity vector for only sputter efflux with respect to local grid normal.