Angular distribution of gamma rays produced in proton-proton collisions

Accurate modeling of how high-energy proton-proton collisions produce gamma rays through the decays of pions and other secondaries is needed to correctly interpret astrophysical observations with the Fermi-LAT telescope. In the existing literature on cosmic-ray collisions with gas, the focus is on the gamma-ray yield spectrum, <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>d</a:mi><a:msub><a:mi>N</a:mi><a:mi>γ</a:mi></a:msub><a:mo>/</a:mo><a:mi>d</a:mi><a:mi>E</a:mi></a:math>. However, in some situations, the joint energy and angular distribution can be observed, so one needs instead <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>d</c:mi><c:mn>2</c:mn></c:msup><c:msub><c:mi>N</c:mi><c:mi>γ</c:mi></c:msub><c:mo>/</c:mo><c:mi>d</c:mi><c:mi>E</c:mi><c:mi>d</c:mi><c:mi mathvariant="normal">Ω</c:mi></c:math>. We provide calculations of this distribution over the energy range from the pion production threshold to 100 GeV, basing our results on simulations. We provide the results in tabular form and provide a ython tool on itub to aid in utilization. We also provide an approximate analytic formula that illuminates the underlying physics. We discuss simplified examples where this angular dependence can be observed to illustrate the necessity of taking the joint distribution into account.