Atmospheric muons and their variations with temperature

Seasonal variations of atmospheric muons are traditionally interpreted in terms of an effective temperature that relates the atmospheric temperature profile at a given time to the dependence of muon production on atmospheric depth. This paper aims to review and generalize the treatment of muon production and effective temperature that has been used to interpret seasonal variations of atmospheric muons by many experiments. The formalism is developed both in integral form -- for application to compact detectors at a fixed depth that record all muons with $E_\mu > E_\mu^\mathrm{min}$ -- and in differential form -- for application to extended detectors like IceCube, KM3NeT, and Baikal-GVD, where the rates are proportional to energy-dependent effective areas.