The production spectrum of high-energy muons as a function of depth in the atmosphere is relevant for understanding properties of event rates in deep detectors. For a given atmospheric profile, cascades of heavy nuclei develop at higher altitude than proton showers, giving rise to larger separation of muons at depth. For a given type of primary cosmic ray, seasonal variations in muon rates reflect the fact that higher temperatures correspond to lower densities and to a relative increase in the ratio of decay to re-interaction of the parent mesons. In this paper, we present a generalization of the Elbert formula that tracks meson decay to muons along the trajectory of the primary cosmic-ray nucleus. The convolution of the production spectrum with a changing atmospheric profile provides the dependence of event rates and sizes of muon bundles on temperature and primary mass. We consider applications to IceCube and also to multiple muon events in the compact underground detectors of MINOS and the NOvA Near Detector.

作为大气深度函数的高能μ子的产生谱与理解深度探测器中事件率的特性有关。对于给定的大气剖面，重核级联会在比质子雨更高的高度发展，从而在深度产生更大的 μ 子分离。对于给定类型的初级宇宙射线，μ 子速率的季节性变化反映了这样一个事实，即较高的温度对应于较低的密度以及衰变与母介子重新相互作用的比率的相对增加。在本文中，我们提出了 Elbert 公式的推广，该公式跟踪介子衰变到沿初级宇宙射线核轨迹的介子。产生光谱与不断变化的大气剖面的卷积提供了事件率和 μ 子束大小对温度和初级质量的依赖性。我们考虑应用到 IceCube 以及 MINOS 的紧凑型地下探测器和 NOvA Near Detector 中的多个 μ 子事件。