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An air shower is a cascade of secondary particles produced by an energetic particle interacting in the atmosphere. Both a charged particle and a gamma ray can start one.^[ref]

An atmospheric neutrino is a neutrino produced by particle decays in a cosmic ray air shower.

A cosmic ray is a high-energy charged particle propagating through space. Galactic cosmic-ray electrons emit synchrotron and inverse-Compton radiation, while cosmic-ray nuclei produce gamma rays through collisions with gas.

In modern terminology, a primary cosmic ray arriving at Earth's atmosphere is usually a proton or another atomic nucleus, with an Electron component as well. Atmospheric collisions generate secondary cosmic rays and air showers, including muons, Electrons and photons. A gamma ray can also initiate an air shower, although it is electrically neutral. Consequently, detecting atmospheric ionization does not establish that an incoming primary cosmic ray passed through the detector. Magnetic fields bend charged particles, while gamma rays provide a more direct directional connection to their sources.^[ref]

The Hess cosmic-ray balloon experiment asks how atmospheric shielding and separation from the ground affect ionization. Its power comes from changing the environment around an enclosed detector instead of identifying each incoming particle.

An original cosmic-ray altitude attenuation model illustrates how competing origins can produce a dip followed by a rise. It is a hypothetical absorption calculation, not a reconstruction or fit of Hess's measurements, and it excludes air shower production. Put $z$ for height above a flat ground surface and assume gas density $\rho (z)={\rho }_0{e}^{-z/H}$ with constant scale $H$. The overlying atmospheric mass per unit area isLet $G$ denote a ground-origin contribution attenuated by material below the chamber, and $E$ a downward extraterrestrial contribution attenuated by material above it. Assume constant attenuation mass columns ${\Lambda }_g,{\Lambda }_e>0$:These are rates at a fixed detector gas density, in arbitrary common units. The constant $B$ represents residual background. The equation $dI/dX=-I/\Lambda$ gives exponential function attenuation along each propagation direction. Thus $G$ decreases and $E$ increases as $z$ increases; their sum need not be monotone. The different arguments of the two exponential functions encode the entire geometric idea.

The figure chooses $H=8\mathrm{km}$, $G_0=0.8$, $E_{\mathrm{top}}=6$, $B=0.15$, ${\Lambda }_g=0.08X_0$ and ${\Lambda }_e=0.45X_0$, all for illustration. Differentiation givesThe minimum occurs where $E/{\Lambda }_e=G/{\Lambda }_g$, at approximately $1.13\mathrm{km}$ for these invented parameters. At $6\mathrm{km}$ the fixed-gas response is about $1.41$ times its ground value, but multiplying by $\rho (z)/{\rho }_0$ gives an uncorrected vented-chamber response of only $0.66$ times its ground value. A real increase in radiation response per gas mass can therefore be hidden by a decrease in target gas density. This is the reason to distinguish radiation from the amount of gas being irradiated.

The model treats propagation as attenuation without generating new particles. Real primary cosmic rays interact, redistribute energy and produce air showers; an atmospheric profile need not follow either simple exponential function. The model also holds backgrounds and detector response fixed. Its value is to expose the sign of competing effects and the density confound, while Hess's experimental argument depends on his actual comparisons and instrument limitations.

The 2013 selection rejected early boundary light. Tagged muons checked leakage; air shower companion muons also vetoed downward atmospheric neutrinos.^[ref]

A simple original conditional probability model explains why a veto changes the selected mixture. Let $A$ mean that an atmospheric neutrino interaction would pass all other cuts, $M$ that an accompanying muon reaches the veto, and $V$ that the veto fires. Set $a=P(M\mid A)$ and $ϵ=P(V\mid M,A)$, and neglect unrelated veto hits. The law of total probability givesFor chosen $a=0.8$ and $ϵ=0.95$, the passing fraction is $0.24$. Neither this value nor either input is an IceCube measurement. With equal incident candidate populations, this mechanism alone reduces the atmospheric contribution relative to a hypothetical population having $a=0$. It does not make every surviving event astronomical: $a=0$ or $ϵ=0$ gives no suppression, while $a=ϵ=1$ removes the entire chosen atmospheric population. If unrelated light vetoes a surviving event independently with probability $u$, its passing fraction becomes $(1-u)(1-aϵ)$; independence is an additional assumption.