# Erosion Rate Measurements for DART Spacecraft Ion Propulsion System +
## 3 Results
### Absolute Density Model Based on Near-Field LIF and Far-Field QCM Data
measurements were assumed to occur at this radius, even though in the experimental setup the measurements were taken at points approximately 1 cm distance from the grid face. The range of polar angles that describes the grid active area was assumed to be \(+/-15^{\circ}\). The grid was discretized into elements with size \(1^{\circ}\) polar and \(1^{\circ}\) azimuthal, relative to the stated coordinate system. 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 mask was assumed to be conical in shape, where the edge of the cone approximates the furthest edge of the grid mask, and the cone apex is at the origin. This assumption was made because any particle with trajectory above this grid mask edge is unaffected, whereas any particle below is obstructed by the grid. This is identically accomplished with the conical geometry. The cone was discretized into elements with size 1 cm radially by \(10^{\circ}\) azimuthally. The grid mask element coordinates were then triangulated to produce a triangle mesh. The far field was approximated as an arc along a spherical surface with radius equal to 1 m and origin at the center of the grid exit plane. The arc corresponds to a path along the spherical surface at azimuthal angle \(0^{\circ}\). The points in the far field were parameterized as a function of their angle from the thruster exit plane. Again, the origin of each point was set at the center of curvature of the grid. The plane of the QCM was assumed to be normal to the line drawn from the center of the grid exit plane to the given point in the far field. The flux of mJy sputter in the far field was predicted from the LIF grid sputter densities by summing the differential flux from each differential grid area for any given point in the far field. The differential flux for any given differential grid area was determined by first finding the sputter flux at the grid and then calculating the mass flow from the grid element to the far field. The sputter flux at the grid was found from the product of the local sputter density and speed,  
\[dj_{g}=\rho_{rel}V. \tag{13}\]  
The local sputter density, \(\rho_{rel}\), was found from Equation (12), where the density is a function of \(V_{x}\) and the peak shift angle, \(\psi\). Both are functions of the given differential grid area, \(dA_{g}\), and the polar angle of the given point in the far field, \(\theta_{f}\); \(dj_{g}\) is the differential flux at the grid. The sputter speed, \(V\), is assumed constant. The velocity distribution of sputtered particles was assumed to be independent of angle--both polar and azimuthal with respect to the grid aperture normal. The constant speed value was found by averaging the speeds of sputter with incidence angle \(30^{\circ}\) and field angle \(-45^{\circ}\), weighted by the relative sputter densities [15]. The constant speed used was 0.497729 cm\({}^{-1}\) or 5.83 \(\times\)\(10^{5}\) cm/s, assuming an unshifted wavenumber of 25,614.357 cm\({}^{-1}\). The local sputter density, \(\rho_{rel}\), is a function of the given grid differential area position and the given point in the far field. The position vectors of each are used to find the sputter density from Equation (12) and the interpolated LIF data. The vector that describes the path of a sputtered alloy particle is \(\mathbf{w}\), and it can be found for any given point in the far field and any given differential area on the grid by subtracting the position vector of the grid differential area, \(\mathbf{u}\), from the position vector of the far field point, \(\mathbf{v}\). Put concisely:  
\[\mathbf{w}=\mathbf{v}-\mathbf{u}. \tag{14}\]  
With the assumption of constant speed sputter particles, the sputter velocity is given by  
\[\mathbf{V}=\mathbf{V}\mathbf{\hat{w}}, \tag{15}\]  
where \(\mathbf{\hat{w}}\) is the unit sputter vector, normalized from Equation (14).