Ray projection on ACIS

Lets assume the trajectory of the emergent ray is described by the following equation:  

$${\bf r = l_0 + {\rm t}l}$$ (57)

where ${\bf l_0}$ is an arbitrary point on the ray trajectory with LSI coordinates (X₀, Y₀, Z₀), ${\bf l}$ is the ray direction (specified by the three direction cosines $\alpha_X$, $\alpha_Y$,$\alpha_Z$), and t is a parameter which describes the position of the point on the line. Intersecting the line with the CCD translates into the geometrical problem of determining t such that the point lies on the CCD plane.

 

Figure 6.10: Projecting rays on ACIS. Case A: plane of the CCD orthogonal to the HRMA axis.  

We will distinguish two cases: A: The CCD plane is orthogonal to the HRMA axis. Let P be the intersection of the ray with the CCD plane, and (X_CCD, Y_CCD, Z_CCD) its coordinates in the LSI system on the CCD. The geometry is summarized in Figure 6.10. By imposing that P lies on the line, i.e., its coordinates satisfy eq. (6.3), we derive t:  

$$t=\frac{X_{CCD} - X_0}{\alpha_X}$$ (58)

and thus the position of P on the CCD:  

$$Y_{CCD} = Y_0 + \frac{X_{CCD}-X_0}{\alpha_X}\alpha_Y$$ (59)

 

$$Z_{CCD} = Z_0 + \frac{X_{CCD}-X_0}{\alpha_X}\alpha_Z$$ (60)

One needs to know a priori X_CCD, which is just the distance of the CCD from the LSI coordinate origin (Fig. 6.10).

B: The CCD plane is tilted with respect to the HRMA axis. In this case we will use the general expression of a point on the CCD plane in LSI system, given by eq. (6.2), and impose that the point belongs to the ray trajectory, eq. (6.3). Figure 6.11 visualizes the situation.

By equating eqs. (6.2) and (6.3), we derive:

 

$$t{\bf l} = ({\bf p_0 - l_0}) + Y_{CCD}{\bf e_Y} + Z_{CCD}{\bf e_Z}.$$ (61)

We now take the dot product of both sides of the equation with the normal ${\bf e_X}$ of the CCD (i.e., we project the vector $t{\bf l}$along the tilted X_CCD axis) and derive t:

$$t= \frac{{\bf (p_0 - l_0) \cdot e_X}}{{\bf l \cdot e_X}}.$$ (62)

Indeed, as Figure 6.10 shows, in the case of no tilt, ${\bf p_0 \cdot e_X} = X_{CCD}$, ${\bf l_0 \cdot e_X} = X_0$, and ${\bf l \cdot e_X} = \alpha_X$, so that eq. (6.7) becomes eq. (6.4). To derive the coordinates of the point in the LSI system we project P on the CCD axes, i.e.,

$$Y_{CCD} = {\bf (r-p_0) \cdot e_Y}$$ (63)

$$Z_{CCD} = {\bf (r-p_0) \cdot e_Z}$$ (64)

 

Figure 6.11: Projecting rays on ACIS. Case B: plane of the CCD tilted with respect to the HRMA axis.  

The ASC program SAOSAC will provide the necessary information about the ray trajectory, specifically the three coordinates of ${\bf l_0}$and the direction cosines defining ${\bf l}$. Since these coordinates are in the XRCF system, we will need to do the appropriate transformation into LSI coordinates before using the above formulas.

We are now working on incorporating the above alghorithms in an IDL program. The output (a FITS file) will be used directly for the CCD simulator and, eventually, for comparison with the calibration data. We can use the complete simulation system (SAO-sac + ray projection + CCD simulator) to estimate the orientation of the ACIS focal plane in XRCF coordinates, by fixing the chip spacing and tilts and other measured geometric quantities of the instrument/FAM combination, then comparing the photon positions inferred from the simulation with actual XRCF data. We can also use this system to obtain better estimates of the spacing and tilt of individual chips in the ACIS focal plane, iterating between XRCF data and the models to improve our picture of ACIS geometry. Ultimately, we will use this exercise to guide us in designing the most effective on-orbit tests to fix the geometry of ACIS in the spacecraft coordinate system, to be performed in the early calibration phase of on-orbit operations.