A Method of Shower Reconstruction from the Fluorescence Detector M.Giller, G.Wieczorek and the Lodz Auger group GZK-40 Moscow Workshop, May 2006

All large showers are similar when described by the age parameter s s = 3X / (X + 2Xmax) Studying EAS by CORSIKA simulations X – depth in g/cm^2 in the atmosphere Xmax- depth of shower maximum

Shower characteristics at given level s Energy & angle distribution of electrons Lateral distribution function r* is Moliere radius at the level

Energy spectra of electrons x E Each curve -average of 10 proton showers with E 0 =10 19 eV s= shower maximum for different shower ages s

Energy spectra – shower-to-shower fluctuations three showers shower ages s fluctuations are negligible

–Energy spectra - comparison of proton and iron initated showers no differences: electron energy spectrum at a given age s does not depend on primary mass.

Electron energy spectrum at a given shower age s does not depend on primary particle: neither on its mass, nor on its energy! s = 3X/(X+2Xmax) Shower age : Calculate fraction of electrons emitting Cherenkov light as function of s and height in the atmosphere (Ethr(h))

Effective fraction of electrons F(s, h) emitting Cherenkov light as a function of shower age s and height h above Auger level. P 10^20 eV P 10^19 Fe 10^20 Fe 10^19 s

Not to be distinguished from each other Angular distribution of electrons f(  ) age

Are there any fluctuations of f(  ) from shower to shower ? 4 curves for 4 showers Very small !

f(  for various ages s S=

Average  (in °) (in °) eV eV RMS

Angular distribution of electrons at a given age s does not depend on the primary particle, neither on its energy, nor on its mass !

Angular distributions of shower electrons g(  ;E). electron energy

Angular distribution of electrons of a given energy E does not depend on anything else than this energy E. Angular distribution of ALL electrons at a given age s depends on energy because their energy spectrum does.

Angular distributions of shower electron for several values of their energy. Analytical expression fitted

Lateral distributions for different ages

Lateral distribution function when expressed as function of depends on s only ! ( neither on E 0 nor on A)

scattered Cherenkov light fluorescence light isotropic direct Cherenkov light collimated along shower

N(s) = { for s < 1 for s > 1 Application to shower reconstruction, cont. or Gaiser-Hillas gamma function of X

Application to shower reconstruction Number of fluorescence photons n i,fl Number of photons  n i emitted toward the camera ith pixel:

Number of fluorescence photons  N ph (x) emitted by all electrons on shower path  X (in g  cm -2 ) number of photons emitted by one electron where Finally

Average energy deposit per particle = is a function of s only !

Application to shower reconstruction, cont.. Number of Cherenkov photons - n i,Ch Number of direct Cherenkov photons - n i,Ch- d Number of scattered Cherenkov photons n i,Ch- sc

Application to shower reconstruction The procedure to find shower cascade curve N(X(s)) ● Guess initial values of N max and X max for minimizing procedure; ● Having X max, the dependence X(s) can be determined; ● From t correlation “ 1 and X max find initial values for “ 1 ● From correlation  1 and “ 2 find “ 2 ● From the initial parameters, calculate the number of photons emitted ● Compare with those measured; ● Minimizing procedure the parameters of the curve can be found.

Summary Age parameter gives a universal description of electrons in big showers; Having age and N(X(s)) easy to predict both fluxes of light: - fluorescence and - Cherenkov (scattered and direct). No need to use iteration methods applicable only when Cherenkov contribution is small. Work on reconstruction of real events - in progress.....