Phenomenology of Muon Showers Underground

Measurements of shower arrival directions (incident in a range of zenith angles between 40\ifmmode^\circ\else\textdegree\fi{} and 70\ifmmode^\circ\else\textdegree\fi{} and slant depth of rock between 2 \ifmmode\times\else\texttimes\fi{} ${10}^{5}$ and 6.5 \ifmmode\times\else\texttimes\fi{} ${10}^{5}$ g ${\mathrm{cm}}^{\ensuremath{-}2}$), structure, and intensities are used as the basis to develop a detailed descriptive phenomenology of muon showers underground. Distributions in right ascension and declination corresponding to minimum values of the median primary energy ranging from 25 TeV for single detected muons to 9000 TeV for events where six muons were detected revealed no compelling evidence for anisotropy in the primary radiation. Measurements of counting rates of groups of two and three of the nearly parallel muons in individual showers as a function of muon separation (structure) were found to be consistent with a shower radial density distribution having the form $\ensuremath{\rho}(r)=P(\frac{r}{\ensuremath{\sigma}})\ifmmode\times\else\texttimes\fi{}\mathrm{exp}(\ensuremath{-}\frac{r}{\ensuremath{\sigma}})$, where $r$ is the distance of the shower axis in m, $P$ is a polynomial in ($\frac{r}{\ensuremath{\sigma}}$), $\ensuremath{\sigma}=K\ifmmode\times\else\texttimes\fi{}\frac{{(sec\ensuremath{\theta})}^{1.3}}{{{E}_{\ensuremath{\mu}}}^{0.8}}$, $\ensuremath{\theta}$ is the zenith angle in degrees, ${E}_{\ensuremath{\mu}}$ is the threshold muon energy in units of TeV, and $K=3.6$ m. The empirical shower radial density distribution above reduces to one predicted by Adcock, Wdowczyk, and Wolfendale on the basis of a detailed shower development calculation with a value of $K=4$ m at an energy of 1 TeV and a zenith angle of 45\ifmmode^\circ\else\textdegree\fi{}. Assuming that the distribution of shower sizes is proportional to a power law in shower size, the exponent of the shower-size distribution was measured and found to lie in the range - 3.5 to - 4.5. The parameters of an empirical density spectrum were optimized by iterative fits to the numbers of events where one to three muons were detected with the result that best-fit parameters were close to those previously obtained by Porter and Stenerson using a smaller sample of shower data. The predicted energy and angular dependence is shown to be in reasonable qualitative agreement with intensities measured in 20-${\mathrm{m}}^{2}$ detectors.