Depth of maximum of air-shower profiles at the Pierre Auger Observatory. I. Measurements at energies above $10^{17.8}\mathrm{eV}$

We report a study of the distributions of the depth of maximum, $X_{\max }$, of extensive air-shower profiles with energies above $10^{17.8}\mathrm{eV}$ as observed with the fluorescence telescopes of the Pierre Auger Observatory. The analysis method for selecting a data sample with minimal sampling bias is described in detail as well as the experimental cross-checks and systematic uncertainties. Furthermore, we discuss the detector acceptance and the resolution of the $X_{\max }$ measurement and provide parametrizations thereof as a function of energy. The energy dependence of the mean and standard deviation of the $X_{\max }$ distributions are compared to air-shower simulations for different nuclear primaries and interpreted in terms of the mean and variance of the logarithmic mass distribution at the top of the atmosphere.

Figure 1

Reconstruction of event 15346477.

Figure 2

Illustration of the influence of the FD field of view on the sampling of the $X_{\max }$ distribution. The slant-depth axes in ${\mathrm{g}/\mathrm{cm}}^2$ are shown on the left panel for three different examples of event geometries (A), (B) and (C) with different ground distances $R$, zenith angle $\theta$ and azimuthal angle $\varphi$. The FD field of view is indicated by the hatched area inside the dashed lines. Examples of correspondingly truncated $X_{\max }$ distributions are shown on the right panel together with their sum. For the purpose of this illustration, the same number of events for each geometry has been assumed.

Figure 3

$⟨X_{\max }⟩$ for showers binned in $X_{\mathrm{l}}$ and $X_{\mathrm{u}}$ in the energy interval $10^{18.1}$ to $10^{18.2}\mathrm{eV}$. The solid line shows a fit with the truncated mean of an exponential function folded with a Gaussian [76], and the dashed line indicates the field-of-view value at which this function deviates by more than $5{\mathrm{g}/\mathrm{cm}}^2$ from its asymptotic value.

Figure 4

Upper panel: $X_{\max }$ and energy of the events used in this paper. Lower panel: Number of events in bins of energy.

Figure 5

Upper panel: Measured $X_{\max }$ distribution [full selection, $19.0<\lg (E/\mathrm{eV})<19.1$]. Lower panel: Relative acceptance after quality cuts only (open markers) and after quality and fiducial cuts (filled markers). The parametrizations with Eq. (7) are indicated by lines.

Figure 6

$X_{\max }$ resolution as a function of energy. Bands denote the estimated systematic uncertainties.

Figure 7

Systematic uncertainties in the $X_{\max }$ scale as a function of energy.

Figure 8

Efficiency of the quality and fiducial selection for data and Monte Carlo simulation (MC). The ${\chi }^2$ of the sum of the (data-MC) residuals is quoted on the top right.

Figure 9

Distribution of $X_{\max }$ differences for events measured by more than one FD station. The quoted uncertainties for the standard deviation $\sigma$ are statistical for data and systematic for MC. The latter are dominated by the uncertainty in the contribution of the alignment and aerosols to the resolution (cf. Sec. 6).

Figure 10

Reconstructed $⟨X_{\max }⟩$ and $\sigma (X_{\max })$ (symbols) obtained from simulated data for different compositions using the Sibyll2.1 interaction model. The moments of the generated events before detector simulation are shown as solid lines.

Figure 11

Cross-checks. (a) Difference of moments obtained from each FD site separately to the results using data from all sites. The global ${\chi }^2$/ndf with respect to zero is given. (b) Subdivision of the data set in showers with near-vertical and inclined arrival directions. Parameters of a linear fit in $\lg (E)$ are shown with supporting points $c_{17.8}$ and $c_{19.6}$ at the centers of the first and last bin of $10^{17.85}$ and $10^{19.62}\mathrm{eV}$. (c) Difference of results with and without fiducial field-of-view selection. Parameters are the same as in panel (b). (d) Comparison of different methods to estimate $⟨X_{\max }⟩$ and $\sigma (X_{\max })$. The difference from the default method is shown. The average from the two deconvolution methods (SVD and Bayesian) is shown without error bars (see text). For the weighting method, the ${\chi }^2$/ndf with respect to zero is given.

Figure 12

$X_{\max }$ distributions for different energy intervals.

Figure 13

Energy evolution of the first two central moments of the $X_{\max }$ distribution compared to air-shower simulations for proton and iron primaries [80, 81, 95, 96, 97, 98].

Figure 14

Average of the logarithmic mass and its variance estimated from data using different interaction models. The nonphysical region of negative variance is indicated as the gray dashed region.

Figure 15

Fit to the tails of the $X_{\max }$ distribution [$18.1<\lg (E/\mathrm{eV})<18.2$]. The region of constant acceptance ${ϵ}_{\text{rel}}=1$ is indicated by arrows.