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Calibration Access and Data Handbook

Procedure

These formulae are derived from the description in [4]. The distribution is computed according to the inner ($R_1$) and outer ($R_2$) radii of the XRT assemblies, so that the contribution appears like the projected telescope aperture for finite defocus distance $\Delta x$ (which is provided by CAL_getRFCdefocus).

$$D(\Delta\beta) = \frac{ 2 F L} {\pi R_2 \left [ 1 - \left (\frac{R_1}{R_2} \right )^2\right ]\Delta x} \times$$

$$\left [\sqrt{1-\left ( \frac{F L \Delta\beta}{R_2 \Delta x}\right )^2} \right . \times \left . \Theta\left ( \left \vert \frac{R_2\Delta x}{F L} \right \vert - \vert\Delta\beta\vert\right ) \right . -$$ (25)

$$\left . \frac{R_1}{R_2}\sqrt{1-\left ( \frac{F L\Delta\beta}{R_1 \Delta x}\right )^2} \right . \times \left . \Theta\left ( \left \vert \frac{R_1\Delta x}{F L} \right \vert - \vert\Delta\beta\vert\right ) \right ] \ ,$$

with $F=F'$, which is the result from Eq. (24), and $L$ as in Eq. (21), $\Theta(x)$ is the Heaviside-step function with its derivative $\delta(x)$, and $\Delta\beta_i=\beta_i-\beta_{\rm m}$. Eq. (25) needs to be integrated for each bin of $\Delta\beta_i$ , and thus the probability for bin $i$ is

$$P_i = \left . Q(R_1) \times \Theta\left ( \left \vert \frac{R_2\Delta x... ... x_i}{F L} \right \vert - \vert\Delta\beta_i\vert\right ) {\rm d}\Delta\beta_i}$$ (26)

$$- \left . G(R_2) \times \Theta\left ( \left \vert \frac{R_1\Delta x... ... x_i}{F L} \right \vert - \vert\Delta\beta_i\vert\right ) {\rm d}\Delta\beta_i}$$

with

$$Q(R) = \frac{-1}{\pi \Delta x (R_1^2-R_2^2)} \left [ \Delta\... ...^2 \arcsin\left(\frac{FL\Delta\beta}{R\Delta x}\right) \right]$$ (27)

and $\Delta x_i = \frac{1}{2}\left[\Delta x(\Delta\beta_1)+\Delta x(\Delta\beta_2)\right]$. $P_i$ is filled such that the first moment of $P_i$ is 0.