Some applications

The 3D shower structure allows a study on lateral proles. Indeed, minimum and medium lateral dimen- sionsrmin and rmed may be dened for each shower. The positions of the near points (np) and central

(a) (b)

(c)

Figure 5.3: Comparison between the R_p, χ₀ and T₀ values as reconstructed by the standard and 3D approaches. Data collected from January 2006 until September 2006 with KG and 3D prole reconstruc- tions,χ^3D₀ ≥45^◦ and passing cut (5.3) withd^∗_CP−eye= 10km was used to produce the plots.

points of the volumes yield these variables directly:

rmin= P

i,kd_{P L}(~r_np,ik, ~s, ~r_s) Nvol

rmed= P

i,kd_{P L}(~r_ik, ~s, ~r_s) Nvol

where the sums are over all good volumes, N_vol is the number of those volumes and d_{P L}(~r, ~s, ~r_s) is the distance from point~rto the shower axis given by~sand~rs.

Another lateral parameter is the so-calledrmaxwhich is of an essentially dierent kind fromrmin and r_med and represents roughly the radial distance that includes most detected volumes. Its denition is as follows. A Monte Carlo method was designed to determine the volume of the irregular solid seen by each pixeliat each time slotk. Each solidikis surrounded by a cylinder of volumeVcyl,ikinside which N_tot,ik points are randomly generated according to the volume elementrdrdθdh=d

r² 2

dθdh. Of the totalN_tot,ik, onlyN_in,ik points~r_ik^(l) are in the detected volumeikand, therefore,

Vik '

P^Nin,ik

l=1 1 Ntot,ik

Vcyl,ik= N_in,ik Ntot,ik

Vcyl,ik (5.4)

σ(V_ik) = ∂Vik

∂N_in,ik

pN_in,ik+ ∂Vik

∂N_tot,ik

pN_tot,ik=V_ik 1 pNin,ik

+ 1

pNtot,ik

!

(5.5)

where Ntot,ik was xed to 1000 independently fromiandk. Then, a cylinder of radius r^(m)_cyl around the shower axis is dened and the following estimator computed:

η r^(m)_cyl

= P

i,kV_act,ik P

i,kVik

being V_act,ik the active volumeik, i.e. the volume of the solidik inside the cylinder of radiusr_cyl^(m):

V_act,ik = Z

V_ik

f_actdV '

N_in,ik

X

l=1

f_act

~

r^(l)_ik Vik

N_in,ik =

N_in,ik

X

l=1

f_act

~r_ik^(l)Vcyl,ik

N_tot,ik (5.6)

withf_act

~ r^(l)_ik

=







1 ifdP L

~ r^(l)_ik, ~s, ~rs

≤r_cyl^(m)

0 otherwise .

Physically, η r^(m)_cyl

is simply the fraction of the detected volume within r^(m)_cyl from shower axis.

Finally,r_maxis found by looping on r_cyl^(m)with step∆r_cyl= 10m:

rmax=r^(m)_cyl −∆rcyl

2 :

η r^(m)_cyl

−η

r_cyl^(m−1)

∆rcyl

≤10⁻³∧η r^(m)_cyl

≥0.1 (5.7)

With this denition, all the signicant volumesik of the event are inside the cylinder of radiusrmax

around the shower axis. It is obviously possible that there are volumes further away thanrmax, but they are assumed not to be physically signicant.

The comparison betweenrmin,rmedandrmaxis presented in gure 5.4(a). As expected,rmin∼0since the near points are supposed to be close to shower axis if this was correctly reconstructed, whilermed and r_maxhave more sparse distributions. These three parameters and their dependence on geometric variables constitute an useful tool in the identication of non-standard events. To begin with, the behaviour of rmin,rmed andrmaxwithχ0 is plotted in gure 5.4(b). There is a decrease of bothrmed andrmaxuntil χ₀=^π₂ and a stabilisation afterwards. Ther_min parameter, however, is close to 0 along the whole range in χ0. In other words, events with χ0< ^π₂ present large, spread volumes containing parts of the shower axis −the 3D structure of such an event is shown in gure 5.5. The reason behind this situation is the

(a) (b)

Figure 5.4: Distributions ofrmin,rmed andrmax in (a) and their dependence onχ0 in (b). The data set used is the same as in gure 5.3.

signal pile up that occurs for showers approaching the FD eye, i.e. χ₀< ^π₂. In fact, in the extreme case where a shower evolves towards the eye parallel to the viewing direction of a pixel i, both the standard and the 3D methods are problematic. In the standard procedure, asχ0=χi, (4.6) yieldsti=T0: all the photons produced along the shower axis arrive at pixeliat the same instantt_i. Notice that the standard geometry reconstruction does not fail because Rp,χ0 andT0 are known (Rp = 0, χ0=χi andT0=ti), but the shower prole determination becomes impossible. The 3D approach, on the other hand, diverges in the extreme case since α_i= 0and (5.1) readsr_ik→ ∞ −that is why smallerχ₀ correspond to larger volumes. Another important remark on events withχ0<^π₂ is that the ƒerenkov radiation emitted during cascade development may arrive directly at the FD telescopes and, consequently, this contribution must be taken into account together with uorescence light in the reconstruction of the shower prole and in energy estimation.

Figure 5.5: 3D visualisation of event SD 2553671 (FD 2/1081/2704). In this case, R_p ' 9.7 km and χ₀'33.1^◦. Note the dierence between the reconstructed volumes here and those of the event presented in gure 5.1(b).

Figure 5.6 shows the relation betweenr_maxandR_p with the quality cutχ₀≥^π₂ in order to eliminate the events described in the previous paragraph. The plot seems to indicate that rmax ∝ Rp which is natural since a pixel overviews greater volumes at greater distances from the eye. The deviation from this behaviour is a possible criterion to classify cosmic ray events and identify strange ones.

Figure 5.6: Dependence ofr_maxonR_p for the same data set as in gure 5.3 but passing the quality cut χ0≥^π₂.

Also the inertia coecients Ixx, Iyy and Izz, dened in the last section, play an important role in the characterisation and classication of events. From the analysis of these parameters in data, the rst conclusion is that Ixx/Iyy ∼1as shown in gure 5.7. Therefore, the shower 3D structure is cylindrically symmetric within a very good approximation. Moreover, the fact that Ixx ∼Iyy means the use of the observation time as a third dimension in FD analysis introduces no signicant bias in the 3D geometry reconstruction.

Figure 5.7: The I_xx/I_yy and I_zz/r²_med distributions for the same data set as in gure 5.3 but requiring r_med6= 0 and at least oneI_ii6= 0.

A study of the three dimensional shape of shower development may as well be performed using inertia relations. If the shower structure were a massive, uniform cylinder of radius r and height h, then the moments of inertia along the main inertia axes would be I_xx^cyl(r, h) =I_yy^cyl(r, h) = ^r₄² +^h₁₂² and I_zz^cyl(r) = ^r₂². In the case of a cone of radius rand height h, I_xx^cone(r, h) =I_yy^cone(r, h) = ₂₀³r²+^h₁₀² and I_zz^cone(r) = ₁₀³r². The typical lateral dimension of a shower is given byrmed and so one may verify if the 3D structure is similar to that of a cylinder of radius rmed, a cone of radius rmed or a cone of radius 2r_medin which cases the following relations are expected: _r^I2^zz

med

=^Izzcyl_r^(r2^med) med

=¹₂, _r^I2^zz med

=^Izzcone_r2^(rmed) med

= ₁₀³ and _r^I2^zz

med

= ^Izzcone_r2^(2rmed)

med

= ⁶₅. Figure 5.7 shows the distribution of _r^I2^zz

med in events from August 2006. The data is consistent with cylinders of radiusrmedand not with any of the cones considered above. However,

(a) (b)

Figure 5.8: The distributions ofp

I_xx+I_yy−I_zz (a) and√

I_zz (b) for the same data set as in gure 5.7.

the considerable dispersion around the mean value indicates a mix of dierent geometries − probably not only cylinders or cones − present in the data set. Note that the inertia coecients, namely Izz, are calculated with weightNγ,ik that is not uniformly distributed as the mass is in a massive, uniform cylinder or cone. Thus, the coecients are a convolution of the N_γ,ik distribution and the geometric structure and should not be misunderstood as simple geometric parameters such as rmin, rmed orrmax. Anyway, the trail of the _r^I2^zz

med histogram, this is, events far away from cylinder-like behaviour, corresponds to two dierent populations of showers: those badly reconstructed and those potentially interesting that present remarkable moments of inertia.

Finally, the longitudinal and lateral dimensions of a shower may be accessed usingp

Ixx+Iyy−Izz

and √

I_zz respectively, since p

I_xx+I_yy−I_zz ∝ hand √

I_zz ∝r in both a cylinder and a cone. The distributions of such parameters are presented in gure 5.8 and they allow the identication of particularly 'fat' and/or extense events.

Many more geometric tests and studies will be done in the near future.