= Cosmic-ray altitude attenuation model
{title2=$q_{\rm fixed}(z)=B+G(z)+E(z)$}
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> $\rho(z)=\rho_0e^{-z/H}$ with constant scale $H$. The overlying atmospheric mass per unit area is
$$
X(z)=\int_z^\infty\rho(h)\,dh=\rho_0H e^{-z/H},\qquad X_0=\rho_0H.
$$
Let $G$ 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 $\Lambda_g,\Lambda_e>0$:
$$
G(z)=G_0\exp\left[-\frac{X_0-X(z)}{\Lambda_g}\right],\qquad E(z)=E_{\rm top}\exp\left[-\frac{X(z)}{\Lambda_e}\right],\qquad q_{\rm fixed}(z)=B+G(z)+E(z).
$$
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.
\Image[/cosmic-ray-competing-altitude-models.png]
{title=Competing altitude trends and the gas-density confound}
{description=Original hypothetical curves for the <cosmic-ray altitude attenuation model>, with no historical data points. Left: ground-only and ground-plus-extraterrestrial explanations at fixed detector gas <density>. Right: the same mixed radiation field with fixed gas <density> or a vented chamber, each normalized to its own ground value.}
The figure chooses $H=8\,\mathrm{km}$, $G_0=0.8$, $E_{\rm top}=6$, $B=0.15$, $\Lambda_g=0.08X_0$ and $\Lambda_e=0.45X_0$, all for illustration. Differentiation gives
$$
\frac{dq_{\rm fixed}}{dz}=\frac{X(z)}H\left(\frac{E(z)}{\Lambda_e}-\frac{G(z)}{\Lambda_g}\right).
$$
The 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.
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