Merge pull request #8 from nj-vs-vh/main

Pion emissivity from pp interaction

sections/hadronic.tex

@@ -263,7 +263,7 @@ \subsection{The $\pi^0$ gamma-ray spectrum}
 Upon substitution, the pion emissivity simplifies to
 %
  \[
-q(E_\pi) \simeq 4 \pi n_{\rm H} I_0 \sigma_0 \int_{E_\pi}^\infty \frac{dE_p}{E_\pi} \left(\frac{E_p}{E_0} \right)^{-\alpha} f(x) = 4 \pi n_{\rm H} I_0 \sigma_0 \left(\frac{E_p}{E_0} \right)^{-\alpha} \underbrace{K_\pi^{-\alpha} \int_0^1 dx x^{\alpha-2} f(x)}_{Y(\alpha)}
+q(E_\pi) \simeq 4 \pi n_{\rm H} I_0 \sigma_0 \int_{E_\pi}^\infty \frac{dE_p}{E_\pi} \left(\frac{E_p}{E_0} \right)^{-\alpha} f(x) = 4 \pi n_{\rm H} I_0 \sigma_0 \left(\frac{E_\pi}{E_0} \right)^{-\alpha} \underbrace{ \int_0^1 dx x^{\alpha-2} f(x)}_{Y(\alpha)}
 \]
 %
 where $Y(\alpha)$ is the spectrum-weighted yield of pions produced from a power-law proton spectrum of spectral index $\alpha$.
@@ -539,4 +539,4 @@ \subsection{Neutrinos from cosmic accelerators}
 
 An underlying assumption for this model is that the source spectrum is steeper than an \(E^{-2}\) energy distribution. This assumption is based on typical observations and theoretical considerations of cosmic ray sources. The steeper spectrum implies that there is a relatively higher abundance of lower-energy particles compared to the higher-energy ones, which influences the overall intensity and energy distribution of emitted particles, including neutrinos.
 
-%It's important to note that these considerations are generalized and might vary with specific astrophysical environments and source characteristics. Additional factors, such as the local environment's opacity and the presence of other interaction processes, can also impact the overall neutrino production and escape mechanisms.
+%It's important to note that these considerations are generalized and might vary with specific astrophysical environments and source characteristics. Additional factors, such as the local environment's opacity and the presence of other interaction processes, can also impact the overall neutrino production and escape mechanisms.