5.3.2 Coned Cosine Distribution

Next, the distribution introduced in (2.17) is considered, for which ${g}({\theta})=\cos({\theta}/{a})$ for all ${\theta}\in\left[0,{\theta}_{\text{cone}}\right]$ . Hence, all angles are within a cone with apex angle $2{\theta}_{\text{cone}}$ . This is the reason for the name of the distribution used in this work. ${a}$ and ${\theta}_{\text{cone}}$ are related by $\frac{\pi}{2}{a}={\theta}_{\text{cone}}$ . To restrict the emission of particles to one hemisphere, ${\theta}_{\text{cone}}$ must satisfy ${\theta}_{\text{cone}}\leq\frac{\pi}{2}$ , which is equivalent to ${a}\leq1$ . The probability density of the polar angle is given by

$${f}_{\theta}({\theta}) = \cos({\theta}/{a})\sin{\theta}.$$ (5.40)

The corresponding cumulative distribution function is

$${F}_{\theta}({\theta}) = \begin{cases}\frac{ \sin{\theta}\sin({\t... ...{\pi}{2}{a})-{a} } & {a}<1, \\ \left(\sin{\theta}\right)^2 & {a}=1, \end{cases}$$ (5.41)

Since it is not possible to calculate an explicit expression for the inverse function ${F}_{\theta}^{-1}({\theta})$ , which is useful for the inverse method, the rejection technique is chosen instead [25]. For the rejection method an instrumental distribution ${f}'_{\theta}({\theta})$ is necessary, which is an upper bound approximation of ${f}_{\theta}({\theta})$ , and which leads to an invertable cumulative distribution function.

Using the inequalities

$$\cos{x} \leq 1-\left({\textstyle\frac{2}{\pi}}{x}\right)^2 \quad \forall {x}\in{\textstyle\left[-\frac{\pi}{2},\frac{\pi}{2}\right]}$$ (5.42)

and

$$\sin{x} \leq {x} \quad \forall{x}\geq0$$ (5.43)

(see Inequality 1 and Inequality 2 in Appendix C) such an instrumental probability density function for (5.40) is given by

$${f}'_{\theta}({\theta}) := \left(1-\frac{{\theta}^2}{{\theta}_{\t... ...theta}({\theta}) \quad \forall{\theta}\in\left[0,{\theta}_{\text{cone}}\right].$$ (5.44)

The corresponding cumulative distribution function is

$${F}'_{\theta}({\theta}) = 1-\left(1-\left(\frac{{\theta}}{{\theta}_{\text{cone}}}\right)^2\right)^2.$$ (5.45)

According to the rejection method a random polar angle following ${f}_{\theta}$ can be generated by feeding the inverse of the instrumental cumulative distribution function

$${\theta} = {F}'^{-1}_{\theta}({u}) = {\theta}_{\text{cone}}\sqrt{1-\sqrt{1-{u}}}$$ (5.46)

with uniformly distributed random variates ${u}\in\left[0,1\right]$ and accepting ${\theta}$ , if

$${u}'{f}'_{\theta}({\theta})\leq{f}_{\theta}({\theta}),$$ (5.47)

where ${u}'$ denotes another uniformly distributed variate on $\left[0,1\right]$ . Algorithm 5.3 summarizes the entire procedure for determining random polar angles, which are distributed according to the coned cosine distribution.

The efficiency ${\gamma}$ of rejection sampling for this case, thus the fraction of successful attempts satisfying (5.47), is given by

$${\gamma} =\frac{ \int_{0}^{{\theta}_{\text{cone}}}{f}_{\theta}({\... ...\left(1-{a}^2\right)} & \text{if}\ 0<{a}<1,\\ 1 & \text{if}\ {a}=1. \end{cases}$$ (5.48)

The presented algorithm shows a very high efficiency, since ${\gamma}\geq\frac{8}{\pi^2}$ holds for all ${a}\in\left]0,1\right]$ (see Inequality 3 in Appendix C), which corresponds to a success rate of approximately $\SI{81}{\percent}$ in the worst case.