I am trying to reproduce a result from https://arxiv.org/pdf/0811.2230.pdf Particularly, I am trying to compute the total inelasticity and make the same plot as in fig.1. However, I am unable to explicitly express $K_\theta$ from eq.21, and therefore I cannot make the integral as stated in eq.22.
The transcendental equation for $K_\theta$ is
$(1-K_\theta)\sqrt{s} = F + \beta\sqrt{(F^2-s_p)}cos(\theta) $ $\qquad \qquad \qquad \qquad $ (1)
where
$\beta = \sqrt{1-\frac{s}{E^2}}$
$F = \frac{1}{2\sqrt{s}} (s + s_p - s_x) $
and
$ s=2\sqrt{s} \, \epsilon + s_p $
$ s_p = 2\delta_pE_p^2 + m_p^2 $
$ s_x = 2\delta_xE_x^2 + m_x^2 $
$E_p = (1-K_\theta)E$
$E_x = K_\theta E$
When I plugged everything to (1) I obtained
full expresion (in this picture K means $K_\theta$, this is what I want to express)
$K_\theta$ is a function of $\epsilon,E$ and $m_p, m_x, \delta_p, \delta_x$ are known constants.
I tried to solve this with Maple and Matlab by using the function solve(), but I obtained solution on several pages and further I was unable to make the integral $\int_{0}^{\pi} K_\theta d\theta $ Thank you very much for any help.
Fist of all, Welcome to the site !.
In my humble opinion, do not waste your time. All the work is purely numerical.
For the first point, starting with $\theta_0=\frac \pi 2$ makes the problem simple and you get $K_0$. Now $\theta_i=\theta_{i-1}+\Delta$; solve the equation starting with $K_{i-1}$ as a guess to get $K_i$ and continue. Repeat the process with new values $\theta_j=\theta_{j-1}-\Delta$.