Cosmic-ray altitude attenuation model ($q_{\mathrm{fixed}}(z)=B+G(z)+E(z)$)

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.