Particle distribution functions

There are several options for the description of the particles on the ground that are measured by e.g. Cherenkov tanks and scintillator panels. They can be initialize in a python script using, e.g. ldf = rock_bottom.ldf.NKG() and can then be given to a signal model to be used in a reconstruction.

The available models can be found in rock_bottom/private/rock_bottom/functions and the functional forms are listed below.

Lateral distribution functions

Log-log parabola

\(\lg{S} = \lg{S_\text{ref}} - \beta x - \kappa x^{2}\)

where \(x = \lg{r / r_\text{ref}}\)

NKG-like

\(\lg{S} = \lg{S_\text{ref}} + (\alpha - \beta) x + (\alpha - \gamma) \lg \left( \frac{r + r_{0}}{r_\text{ref} + r_0} \right)\)

where \(x = \lg{r / r_\text{ref}}\)

Lagutin (see: Roman Raikin et al 2019 J. Phys.: Conf. Ser. 1181 012032)

\(S = S_\text{ref} \left( \frac{r}{r_\text{ref}} \right)^{-\alpha} \left(1 + \frac{r}{r_\text{ref}} \right)^{\alpha - \beta} \left( 1 + \left(\frac{r}{10\, r_\text{ref}}\right)^{\gamma} \right)^{-\delta}\)

Muon LDF

\(\lg{\rho} = \lg{\rho_{\mu}} - \beta \lg \left( \frac{r}{r_\text{ref}} \right) - \gamma \lg \left( \frac{r + r_\text{Moliere}}{r_\text{ref} + r_\text{Moliere}} \right)\)

Shower-front shape

The shape of the shower front, specifically the time of arrival of the particles as a function of distance to the shower axis. They can be constructed using e.g. rock_bottom.showerfront.TimeGauss() These models are all given as a delay with respect to a shower plane front.

Plane \(t(\vec{r}) = t_\text{plane}(\vec{r}) = T_{0} + \frac{(\vec{r} - \vec{r}_\text{axis}) \cdot \hat{n}}{c_\text{light}}\)

where the shower axis intersects the ground plane at \(\vec{r}_\text{axis}\) and originates from \(\hat{n}\).

Parabola

\(t(\vec{r}) = t_\text{plane}(\vec{r}) + p_0 r^2\)

Parabola with Gaussian nose

\(t(\vec{r}) = t_\text{plane}(\vec{r}) - p_0 \left( \frac{-r^{2}}{p_1} - 1 \right) + p_2 r^2\)