# Erosion Rate Measurements for DART Spacecraft Ion Propulsion System +
## 3 Results
### Absolute Density Model Based on Near-Field LIF and Far-Field QCM Data  
As already discussed, integrated Doppler profiles are proportional to molybdenum density at each probed location. Each density obtained was treated as a quantity with arbitrary units. The relationship between relative and absolute density was expressed as  
\[\rho_{abs}=k\rho_{LIF}, \tag{8}\]  
where \(\rho_{abs}\) is the absolute sputter density, and \(\rho_{LIF}\) is the relative density obtained via LIF. The constant \(k\) is the conversion factor from relative to absolute units. LIF data collected at 0\({}^{\circ}\), 4\({}^{\circ}\), 8\({}^{\circ}\), 12\({}^{\circ}\), and 17\({}^{\circ}\) from the thrust axis were fit to Gaussian curves. The five fitted curves were integrated and linearly interpolated, after subtracting baselines, to produce \(\rho_{LIF}\) as a multivariable function of angle and LIF frequency. The scans of sputtered molybdenum LIF produce Doppler profiles corresponding to the velocity distribution component that is colinear with the laser beam. Molybdenum velocity components orthogonal to the laser beam do not produce a Doppler shift, therefore the two-dimensional velocity distribution is projected into one-dimension. Nonetheless, the 2D distribution can be reconstructed approximately by assuming it is rotationally symmetric about an axis through the peak of the observed LIF profile and azimuthally symmetric with respect to the grid. Off-center grid locations have non-zero directed velocity toward the periphery of the grid, enhancing their flux contributions at high angles with respect to the thrust axis. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Location** & **Background Density** & **Source Density** & **Total Density (cm\({}^{-3}\))** \\  & **(cm\({}^{-3}\))** & **(cm\({}^{-3}\))** & **(cm\({}^{-3}\))** \\ \hline Near Accel Grid, & & & \\ Downstream Side & & & \\ \hline Discharge Chamber & \(<\)9 \(\times\) 10\({}^{10}\) \# & 2 \(\times\) 10\({}^{12}\) & 2 \(\times\) 10\({}^{12}\) \\ \hline Intergrid Region & \(\leq\)9 \(\times\) 10\({}^{10}\) \# & 5 \(\times\) 10\({}^{11}\) & 6 \(\times\) 10\({}^{11}\) \\ \hline \hline \end{tabular} \# Density may be elevated due to high flux of backstreaming facility neutrals toward grids.  
\end{table} 
Table 2: Rough estimates for average DTL28-H-H background and source densities. \begin{table}
\begin{tabular}{c c} \hline \hline
**Downstream CEX Average Collection Length, \(l_{d}\) (cm)** & **Background Density Enhancement Factor \({}^{*}\)** \\  & 1.7 & \(\approx\)3 \\ \hline \hline \end{tabular} \({}^{*}\) Relative to wall ionization gauge measurement.  
\end{table} 
Table 3: Results of accel current collection model for DTL28. The collection length is also a measure of separation distance between ion beam neutralization front and accel downstream face. The two-dimensional distribution was assumed to be Gaussian with the form  
\[\rho_{rel}\big{(}V_{x},V_{y}\big{)}=\frac{1}{2\pi\sigma^{2}}e^{-\frac{(|V_{x}-y_{ x}|^{2}+(V_{y}-y_{y})^{2})}{2\sigma^{2}}} \tag{9}\]  
where \(\sigma\) is standard deviation, a measure of profile width, \(V_{x}\) and \(V_{y}\) are two orthogonal velocity components in a plane with its normal defined by a ray pointing in the direction of maximum sputter density, and ray origin is at the grid location tangent to the laser beam. This plane is not necessarily tangent to the grid because the peak position of the LIF profile shifts to higher frequencies across the grid face, from apex to edge, indicating increasing radial velocity with distance from the apex. The shifted peak was assumed to only be shifted in one dimension along the axis of the laser beam. The variable \(V_{y}\) is the velocity component normal to the axis of the laser beam; \(V_{x}\) is in the same plane as the plane formed by the laser beam axis and grid normal. Because the plane of \(V_{x}\) and \(V_{y}\) may not be tangent to the grid, the component \(V_{x}\) may be angled with respect to the laser beam.