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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.

Tomáš
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1 Answers1

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Fist of all, Welcome to the site !.

In my humble opinion, do not waste your time. All the work is purely numerical.

  1. Generate a detailed table of $K$ as a function of $\theta$
  2. Build an interpolating function
  3. Numerical integration

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 $\theta_j=\theta_{j-1}-\Delta$.

Claude Leibovici
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