# Erosion Rate Measurements for DART Spacecraft Ion Propulsion System +
## 3 Results
### Relative Doppler and Density Profiles across the Grid Face  
Lineshape profiles at each of the five LIF probe angles across the grid face are shown in Figure 9. Profile shapes are due to the Doppler effect and were fit, in most cases, with a Gaussian function  
\[f(x)=a_{1}\exp\left(-\left(\frac{x-a_{2}}{a_{4}}\right)^{2}\right)+a_{3}, \tag{1}\]  
where the \(a_{i}\) are fitting parameters. The largest angle, 17 degrees, was the only exception. Here the profile was clearly asymmetric, and the existence of a long tail on the high frequency side significantly reduced the quality of fit for the Gaussian function. For this case, a modified function with Thompson character [13; 14] was utilized:  
\[f(x)=a_{1}(a_{2}(x-12807.0))^{2}/\left((a_{2}(x-a_{5}))^{2}+a_{3}^{2})^{a_{4} }\right)+a_{6}. \tag{2}\]  
Figure 8: DTL28 parameter variations. (**a**) Dependence of accel current on the \((1-\eta)/\eta\) function; (**b**) QCM signal dependence on background pressure near peak of DTL28-H-H deposition rate profile; (**c**) accel current variation with background pressure; and (**d**) molybdenum peak LIF signal variation with accel current and \((1-\eta)/\eta\) at grid center. As shown by Figures 9 and 10, lineshape width was reduced by nearly half at high grid angles compared to its maximum. Laser frequency at the peak of the profile shifted progressively higher with angle, corresponding to a velocity component directed radially outward. The observed velocity profile was symmetric at grid center, whereas near the perimeter it was substantially asymmetric. Near the edge, the Doppler profile shows that most molybdenum atoms are moving radially away from the thrust axis, with few atoms traveling in the opposite direction. The intensity of laser-induced fluorescence from sputtered molybdenum plume atoms, after integrating over the Doppler profile, is proportional to the particle density in the region of interrogation. Doppler profiles are Gaussian in nature for particles with a thermalized velocity distribution, and the averaged velocity vector is zero. Sputtered atom velocity  
Figure 10: Measured position of peak Doppler profile for DTL28-H-H, from center to edge of grid. The a4 coefficient is a measure of width, obtained from fitting profiles with Equation (1). Figure 9: Molybdenum Doppler profiles at the five grid angles probed by LIF for DTL28-H-H. Progressive peak shifting occurs from grid center to edge, together with narrowing of the profile. Profiles are pure Gaussian at grid center and acquire significant Thompson character near the edge. is more complex, with directed velocity away from the point of origin so that the sum of particle velocity vectors does not vanish. The Thompson distribution function works better to model such profiles if the laser beam is aligned with the most probable direction of motion from the sputter site. In the present case, the region of interaction between laser and interrogated molybdenum atoms is approximately 1 cm distance from the grid surface, and the laser beam passes tangent to the nearest grid point. Although the Doppler profile is really a hybrid with both Gaussian and Thompson character, we have been able to conveniently fit the measured Doppler profiles across the grid face using pure Gaussian functions, except for the 17-degree data points where the quality of fit is degraded. The width obtained from the Gaussian fit at 17 degrees is included in Figure 10 for comparison. The Doppler shift of the peak profile density from grid center to edge was approximately 0.10 cm\({}^{-1}\) at the fundamental laser frequency. The corresponding velocity shift was easily calculated, and we obtained 2.34 km/s, which is equivalent to 2.7 eV kinetic energy--about half the most probable energy for sputtered molybdenum atoms under low energy Xe\({}^{+}\) bombardment [15].