Modeling the Askaryan signal
(in dense dielectric media)
Jaime Alvarez-Muñiz
Univ. Santiago de Compostela, Spain
In collaboration with:
J. Bray, C.W. James, W. R. Carvalho Jr. , A. Romero-Wolf , R.J. Protheroe, M. Tueros, R.A. Vazquez, E. Zas
Outline
• Motivation& quick reminder of radio technique
• Methods to model the signal
– Monte-Carlo (microscopic) simulations
– Finite Difference Time Domain methods
– Very simple models
– Semi-analytical models
• Comparisons: MC-MC, MC-models, MC-FDTD,…
• Conclusions
Motivation: UHEn detection
Expected events
(Ahlers model)
Auger IceCube
0.6
0.4
How big a detector is needed ?
• Small fluxes of cosmogenic neutrinos:
≈ 0.5 per km2 per day over 2π sr above EeV
• Small neutrino interaction probability:
≈ 0.1 - 0.2 % per km of water at EeV
• Small budget … typically these days…
• Fix rate at ≈ 20 – 40 events/ km2 / year
(say)
≥ 100 km3 of instrumented volume of water
How to achieve this?.
Detection of coherent Cherenkov radiation in dense media.
Reminder:
Basics of radio-emission
in dense media
The radio technique in dense media
GLUE
LORD
NuMoon
MOON
REGOLITH
LUNASKA
dense → r ≈ 1 g cm-3
LOFAR
ANITA2
RICE - ARA - ARIANNA
Many experimental initiatives & hopefully
more to come (see this meeting)
The source: n-induced showers
“Mixed” showers
<Eelectromagnetic > ≈ 80% En @ EeV
<Ehadronic > ≈ 20% En @ EeV
Hadronic showers
<E> ≈ 20% En @ EeV
EM or Hadronic
Dimensions & speed of the source
Longitudinal spread
(Radiation length X0 ≈ 39 m in ice)
1 TeV electron shower
ice
Lateral spread
(Moliere radius rM ≈ 10 cm in ice)
Longitudinal spread in ice increases as:
log E
E < 1 PeV
E0.3-0.5
E > 1 PeV can reach ≈ 100 m
(LPM)
Lateral spread varies slowly with E
Ultra-relativistic electrons above K ≈100 keV
v > c/n (n=1.78)
Zas-Halzen-Stanev (ZHS) MC, PRD 45, 362 (1992)
Net negative charge: Askaryan effect
G. Askar´yan, Soviet Phys. JETP 14, 441 (1962)
• “Entrainment” of electrons from the medium as shower penetrates
Excess negative charge develops (electrons) →
N(e )  N(e )
Δq 
 25%
N(e )  N(e )
• Main interactions contributing:
Moeller
Compton
atomic
“Low” energy
processes ~ MeV
atomic
e+ annhilation
G.A. Askaryan
Bhabha
atomic
Askaryan effect confirmed
in SLAC experiments
Askaryan effect present in any medium with bound electrons (for instance in air).
Modeling the signal
(1) Monte Carlo simulations
• Basic idea:
– Obtain signal from 1 particle track from 1st principles (Maxwell).
– Simulate shower & add contributions from individual particle tracks.
• Advantages:
– Full complexity of shower development accounted for.
• Shower-to-shower fluctuations
– Accurate calculation of radiation from different primaries: e, p, n
• Disadvantages:
– Time consuming – “Thinning” techniques @ ≈ EeV and above
• Main codes & refs.:
– ZHS (Zas-Halzen-Stanev et al.), GEANT3.21 & 4, ZHAireS (ZHS+Aires)
Zas, Halzen, Stanev PRD 45, 362 (1992)
J. A-M, W. R. Carvalho, M. Tueros, E. Zas, Astropart. Phys. 35, 287-299 (2012)
Razzaque et al. PRD 65 , 103002 (2002)
Hussain & McKay PRD 70, 103003 (2004)
Radiation single particle track: Frequency domain
Maxwell´s equations → Fourier-components of Electric field E(ω,x) emitted by
charged particle traveling along finite straight track at constant speed v:
phase factor
global phase
diffraction



exp ( kR )
Eω e
exp[ i ( ω - k v ) t1 ] v  δt
R
frequency
v┴
t1
1.
2.
3.
4.
θ v
track
far-field
k observer
v┴ dt
sin j
j
tracklength
j  ω δt ( 1 - n β cos θ )
t1 + dt
E(ω,x) increases with frequency
E(ω,x) ≈ length of particle track perpendicular to direction of observation
E(ω,x) 100% polarized (perpendicularly to observer´s direction)
E(ω,x) : diffraction pattern exhibiting central peak around cos θC = 1/nβ
(j=0) & angular width Δθ ~ (ω δt)-1
Radiation from a shower: frequency domain
Contributions to E-field from all charged particles tracks
Phase factors (different for each particle)
E
 E ω  e
i
i
vi  δt i
exp[iω ( 1 - n βi cosθ ) t1i ]
sin ji
charged
particles
Charge of each particle
ji  ω δt i ( 1- n βi cos θ )
If θ ≈ θC or lobs >> shower dimensions (small enough ω) at θ
→ Phase factors ≈ equally small → COHERENCE (@ MHz-GHz)
E ≈ ω Σ (-e) vi δti + ω Σ (+e) vi δti ≈ ω Σ (-e) vi δti
electrons
positrons
charge excess
ji
Modeling with MC: Low energies matter…
keV-MeV electrons
contribute most to
radio emission
50 % of excess track
due to electrons with
Ke < 6 - 7 MeV
Excess negative tracklength:
T(e-) – T(e+) vs Kthreshold
[E. Zas, F. Halzen, T. Stanev PRD 45, 362 (1992)]
Radiation single particle track: Time domain
Maxwell´s equations → Radiation comes from instantaneous acceleration at start &
deceleration at end of particle track
Vector potential
[J. A-M, A. Romero-Wolf, E. Zas, PRD 81, 123009 (2010)]
See also “end-points” algorithm:
T. Huege et al., PRE 84, 056602 (2011)
C. James talk at this meeting
v┴
θ
t1
Acceleration
track
v
t2
E-field
Time
Acceleration
Deceleration
Comparison algorithm with Jackson
A-M, Romero-Wolf, Zas, PRD 81, 123009 (2010)]
A. Romero-Wolf & K.Belov
(Proposal to test geosynchrotron in SLAC)
The Zas-Halzen-Stanev (ZHS) code
ZHS-”multi-media”
– Based on ZHS (1993).
– Electromagnetic showers only (E up to 100 EeV – “thinned”)
• All EM processes included (bremss, pair prod., Moeller, Compton, Bhabha,
e+ annhilation, dE/dX,…)
– Multi-media: (Almost) any dense, dielectric & homogeneous
medium can be used:
• Ice, sand, salt, Moon regolith, alumina,…
– Tracking of particles in small linear steps + ZHS algorithm
– E-field can be calculated in:
~ 1 m from
shower axis
• Time-domain & Frequency-domain.
• Far-field (Fraunhofer) and “near”-field (Fresnel – see later).
[J. A-M, C.W. James, R.J. Protheroe, E. Zas , Astropart. Phys. 32, 100 (2009)]
[J. A-M, A. Romero-Wolf, E. Zas, PRD 81, 123009 (2010)]
Also GEANT 3.21 & 4
Razzaque et al. PRD 65 , 103002 (2002)
Hussain & McKay PRD 70, 103003 (2004)
The ZHAireS code
ZHAireS = ZHS + Aires
– Shower simulation with Aires
– Electrom., hadronic & n showers (E up to 100 EeV – thinned sims)
• All relevant processes included.
• Different low & high-E hadronic interaction models available.
– Ice only (so far). Can be extended to other dense media.
• Works in air (see Washington R. Carvalho talk – this meeting)
– Tracking of particles in small linear steps + ZHS algorithm
– E-field can be calculated in:
• Time-domain & Frequency-domain.
• Far-field (Fraunhofer) and “near”-field (Fresnel).
J. A-M, W. R. Carvalho, M. Tueros, E. Zas, Astropart. Phys. 35, 287-299 (2012)
J.A-M, W.R. Carvalho, E.Zas, Astropart. Phys. 35, 325-341 (2012)
Also GEANT4
Hussain & McKay PRD 70, 103003 (2004)
(2) Finite Difference Time-Domain (FDTD)
• Basic idea:
techniques
– Discretize space-time into lattice
– Calculate electric field approximating Maxwell´s differential equations
by difference equations
• Advantages:
– Very flexible: effects of dielectric boundaries, index of refraction
gradients, far & near field,…
– 1 single run produces field “everywhere” in space-time
– Can be linked to shower MC simulation predicting excess charge
• Disadvantages:
– Computationally intensive: grid size < lateral dimension/10
• So far only radiation from unrealistic “shower” was obtained (?)
• Main refs.:
– Talks by C.-C. Chen et al. this meeting
C.-Y. Hu, C.-C. Chen, P. Chen, Astropart. Phys. 35, 421-434 (2012)
(3) (Very) simple models
• Basic idea:
– Simple models of charge development (line current, “box” current,
constant charge,…)
• Advantages:
– Gain insight into features of radio-emission in time & freq-domains
– Help understanding complex Monte Carlo simulations
• Disadvantages:
– Too simplistic… but a necessary “academic” exercise…
• Main refs.:
– See this talk .
1D “line” model of shower development
1
Assumptions:
a. 1D line of current (excess charge Q) spreading
over length L.
b. Charge travels at v > c/n
“Huygens approach”
θc≈ 56o
ice
Far-field observer at Cherenkov angle θC
t1→2 = L/v = t1→3 = L cosθC / (c/n)
Time
 Observer sees whole shower “at once”
(sensitivity to longitudinal profile lost…)
Frequency domain:
 Constructive interference at ALL wavelengths
 Spectrum increases linearly with frequency:
NO frequency cut-off
radiation
L
Wavefronts in phase
Time-domain:
J(z,t) = v Q d(z - vt)
3
z
2
wavefronts
in phase
1D “line” model of shower development
1
Far-field observer at θ ≠ θC
Wavefronts NOT in phase
(due to longitudinal shower spread L)
J(z,t) = v Q d(z - vt)
θ < θc
Time
radiation
Time-domain
 Observer sees radiation in a finite interval of time
depending on angle (sensitivity to long profile):
Δt (θ) ≈ L (1 – n cosθ) / c ≈ few 10 ns
L
wavefronts
out of phase
z
Frequency domain
 Spectrum increases linearly with frequency up to:
Frequency cut-off
ωcut(θ) ≈ Δt-1 ≈ few 100 MHz
2
3
2D “box” model of shower development
R
Assumptions:
1. 2D current: longitudinally over L & laterally over R.
2. Uniform excess charge & travels at v > c/n
Time
Far-field observer at Cherenkov θC
Wavefronts NOT in phase
(due to lateral spread R of shower).
z
θc
radiation
L
Time-domain:
 Observer sees radiation in a finite interval of time:
wavefronts
in phase
Δt ≈ R sinθc / (c/n) < ns
Frequency domain:
 Spectrum increases linearly with frequency:
Frequency cut-off
ω ≈ Δt-1 ≈ GHz
R sinθc
out of
phase
2D “box” model of shower development
R
Far-field observer at θ ≠ θC
θ < θc
Wavefronts NOT in phase
(due to longitudinal L & lateral spread R of shower)
Time
L
Time-domain:
 Observer sees radiation in finite interval of time:
Δ t = max [R sinθ/ (c/n), L (1 – n cosθ)/c] z
Frequency domain:
 Spectrum increases linearly with frequency:
Frequency cut-off
ω ≈ Δ t-1 ≈ few 100 MHz - GHz
R sinθ
wavefronts
out of phase
radiation
Conclusions from “box” model
• Far-field observer at Cherenkov angle (θc ):
– Spread in time of pulses and frequency cut-off determined
by lateral spread of shower (R).
• Far-field observer at θ ≠ θc :
– Spread in time of pulses and frequency cut-off (mainly)
determined by longitudinal spread of shower (L).
(4) Semi-analytical models
• Basic idea:
– Obtain charge distribution from complex MC simulations.
– Use them as input for analytical calculation of radio pulses.
• Advantages:
–
–
–
–
–
Accurate & computationally efficient.
Full complexity of charge distributions (LPM effect,…)
Shower-to-shower fluctuations.
Different primaries (e, p, n)
Gain insight into features of radio-emission in time & freq-domains
• Disadvantages:
– None ! (well maybe that they need input from MC)
• Main refs.:
– This talk .
J.A-M, R.A. Vazquez, E.Zas, PRD 61, 023001 (1999)
J. A-M, A. Romero-Wolf, E. Zas, PRD 81, 123009 (2010)
J. A-M, A. Romero-Wolf, E.Zas, PRD 84, 103003 (2011)
1D “line” model with variable Q(z)
Assumptions:
a. 1D line of current (excess charge Q) spreading over length L.
b. Charge varies with depth & travels at v > c/n (obtained from MC)
Frequency domain:
slit
E-field can be obtained Fouriertransforming the longitudinal
profile Q(z) of excess charge
E( )    dz Q(z) e i pz
J(z,t) = v Q(z) d(z - vt)
Angular
distribution
of E-field
θc
L
p  (1  n cosq ) / c
Shower
Dq
Far-field
observers
Diffraction by a slit
Δθ ≈ (ω L)-1
[J. A-M & E. Zas, PRD 62, 063001 (2000)]
Time domain:
Far-field
observers
Current
J(z´,t´) = v Q(z´) d(z´ - vt´)
Longitudinal
development
from MC sims.
Coulomb gauge
Vector potential
A(tobs , θ) ≈ v Q(ζ) / R
Vector potential =
Re-scaling of
longitud. profile
ζ → Retardation + time-compression effects:
Source position (z) mapped to observer time (tobs)
via θ –dependent relation:
tobs = z(1 - ncosθ)/c + t0
tobs = t0 @ θc
Electric field
E(tobs , θ) = dA(tobs , θ)/dtobs
[J. A-M, A. Romero-Wolf, E. Zas, PRD 81, 123009 (2010)]
E-field
Bipolar pulses
Relativistic effects
Source position (z) mapped to observer time
(tobs) via θ –dependent relation:
tobs = z(1 - ncosθ)/c + t0
When observing shower at angles:
θ = θc → observer sees shower at t=t0
As observer moves from θc shower
appears to last longer in time.
θ > θc observer sees first the start of
shower and then the end (causality)
θ < θc observer sees first the end of the
shower and later the start (non-causality)
(Many more interesting effects if observer
is in the near-field… see later in this talk)
Far-field
observers
Conclusion from 1D line model
• Modelling signal away from Cherenkov simple &
straightforward
– Time-domain: vector potential = rescaling & timetransforming longitudinal profile.
– Freq.-domain: Electric field = Fourier transform of
longitudinal profile.
(Longitudinal profile easy/fast to obtain with MC simulations)
3D model with variable Q(z)
1D model fails close to
Cherenkov angle
where lateral spread is of
utmost importance for radio
emission
Lateral spread
Current:
J(r´, φ´,z´,t´) = v(r´,φ´,z´) f(r´,z´)
Q(z´) d(z´ - vt´)
Longitudinal
spread
[J. A-M, A. Romero-Wolf, E. Zas, PRD 84, 103003 (2011)]
Dealing with the lateral spread
Lateral spread is difficult to model & deal with when obtaining vector potential.
However if 2 assumptions are made:
(a) Shape of lateral density depends weakly on depth: f(r´,z´) ≈ f(r´)
(b) Radial velocities depend weakly on depth´: v(r´,f´,z´) ≈ v(r´,f´)
Convolution of longitudinal
& lateral contributions
Longitudinal
Lateral
[J. A-M, A. Romero-Wolf, E. Zas, PRD 84, 103003 (2011)]
“Trick” to obtain form factor:
At Cherenkov angle only lateral spread is relevant
(all shower depths z´are seen at the same time – remember box model ?)
Integrals decouple at qCher
Vector potential at θC :
obtained in MC sims.
R x Vector potential [V s]
Form factor
Shower tracklength:
obtained in MC sims.
• A(qCher t) quasi-universal function
• Scales with primary energy
• Dependence with primary: e, p
• Asymmetryc due to observer´s position &
radial components of velocity
• Existing parameterisation.
Observer´s time [ns]
[J. A-M, A. Romero-Wolf, E. Zas, PRD 84, 103003 (2011)]
Electron 1 EeV
Dz
Depth z [m]
contribution to A due to lateral spread at Dz
weighted by mQ(z)/4pR
qCher - 20o
Dtobs
Time reverses
Dtobs = Dz (1 - ncosθ)/c q < q
Cher
Electron 1 EeV
Dz
Depth z [m]
Large compression in time
when q is close to qCher
Dtobs = Dz (1 - ncosθ)/c
qCher + 0.1o
Dtobs
Conclusion from 3D model
• Modelling signal at any q is simple & straightforward
– Vector potential = convolution longitudinal & lateral
contributions, with appropriate rescaling & time-compression.
– Lateral contribution = form factor (easily obtained in MC sims.
from vector potential at Cherenkov angle)
– Longitudinal profile modeled with MC sims. (fast !)
• Procedure works in the far-field & “near”-field
(near-field = distances > lateral shower dimensions i.e. > 1 m )
• Procedure works also for p, n showers
– Simply use lateral contribution corresponding to hadronic
showers or a mixture in case of ne Charged Current
Comparison of methods
MC vs MC
Frequency spectrum – EM showers
Electron 1 PeV ice
θC
shower
θC - 10o
E-field
θC - 20o
θ
observer
Frequency spectrum – EM showers
Electron 1 TeV ice
shower
E-field
θ
observer
Semi-analytical models vs MC
Time-domain: away from qCher
Electron 1 EeV
Vector potential
traces shape of
longitudinal profile
Time reversal
Time-domain: close to qCher
Electron 1 EeV
Vector potential
traces shape of
longitudinal profile
Time reversal
Time-domain: even closer to qCher
Electron 1 EeV
Sensitivity to
longitudinal
profile lost
Time-domain: proton showers
A(qCher t) proton- showers
(obtained in MC sims. with ZHAireS)
Proton – 100 TeV
ZHAireS MC
A(qCher t) proton vs electron-showers
ne – 1 EeV
ZHAireS MC
Time-domain: n-induced showers
ne + N → e + jet
E(ne) = 1 EeV
E(electron) = 0.83 EeV
E(hadronic jet) = 0.17 EeV
A(qCher,t) (ne)
=
0.83 A(qCher,t) (e @ 1 EeV)
+
0.17 A(qCher,t) (p @ 1 EeV)
FDTD vs MC
• Quantitative comparisons not possible…
– FDTD → radiation for unrealistic shower dimensions
(memory limitations - size of space-time lattice)
• Q(z) ≈ exp(-z2/2sz2) symmetric (instead of Greisen-like)
• rlateral(r) ≈ exp(-r2/2sr2) with sr=1 m (instead of ≈ 0.1 m)
– Different dimensions alter coherence of emission in FDTD
compared to MC (ZHS, ZHAireS, G4).
• MC & FDTD predict same effects in the “near” field:
– Fields decreasing as 1/sqrt(R) (cylindrical symmetry)
– Dependence on frequency of transition 1/sqrt(R) → 1/R
(given by Fraunhofer condition: R > L2sin2q/l)
– More assymmetric waveforms than in far-field
– Transition from more bipolar waveform in near-field to
more multi-peaked in far-field (LPM showers)
“Near-field”
in MC & FDTD
1/sqrt(R) in near-field
Electron 10 TeV - ZHS MC
1/R in far-field
Electron 10 TeV - ZHS MC
q ~ qCher
More asymmetric waveform as R decreases
Observer sees different slices of shower at
different distances, angles,…
ZHAireS Monte Carlo vs 3D model
Observers
Shower (≈ 20 m long)
Near-field in MC & FDTD
Shower max. seen at qCher
→ time compression
→ “single” bipolar pulse
Shower NOT seen at qCher →
NO time compression
→ multi-peaked bipolar pulse
Near-field effects in MC & FDTD
Compression effects
very important
Observer may see 2 distinct
slices of shower longitudinal
development at once
Polarization depends on time.
Mixing of 2 polarizations
Data vs MC
Experiments at SLAC: sand, salt & ice
Bunches of ~ GeV bremss. photons dumped in
sand & salt & ice: E0 ~ 6 x 1017 – 1019 eV .
Frequency spectrum
• Askaryan effect seen !!!
• Linearly polarized signal
• Power in radio waves goes as E02
• Bipolar pulses in time-domain
• Agreement with theoretical expectations
ANITA
Angular distribution of electric field
ICE
target
D. Saltzberg et al. PRL 86 (2001);
P.Miocinovic et al. PRD 74 (2006),
P. Gorham et al. PRL 99 (2007)
More attempts: K. Belov, A. Romero-Wolf @ this meeting
Summary: Modeling Askaryan signal
– MC simulations: Achieved maturity
• Agree between each other : ZHS & ZHAireS & GEANT3.21 & 4
– ZHAireS most complete: time & freq. – far & “near” – e & p & n
• Validated by data !
– more tests – air in proposal stage.
– FDTD:
• More flexible than MC.
• Need to be applied to more realistic cases: comparison to MC
– Semi-analytic models:
• Reproduce complex MC simulations (get input from them)
• One of the best compromises: accuracy/fastness
Some things to-do
• Quantitative comparison of FDTD & MC
– Validation of algorithms (FDTD does not implicitely use
any algorithm)
• Propagation effects not included.
– Absorption other than 1/R or 1/sqrt(R)
– Effect of variable refractive index
• Curved paths from shower to observer
• Time delays
• NOTE: Other MC using params. of signal include
propagation effects (UDel MC, Ohio MC,…)
– Tailored to specific expts. (ANITA, ARA,…)
End
Backup slides
Why dense media & why radio?
n interaction probability scales with density
Excess charge (Askaryan effect) due to keV-MeV electrons.
MeV electrons travel at v > c/n in dense media
Observation wavelength l >> shower dimensions
Coherent emission → Power ≈ (Excess charge)2 ≈ (Shower energy)2
(In dense media ex. ice, L ≈10 m, R ≈ 0.1 m → coherence up to MHz – GHz)
Broad bandwidth ( MHz → GHz )
Cheap detectors: antennas (dipoles, etc…)
Large vols. of dense, radio “transparent” media exist in Nature: ice, moon regolith, salt, …
Information on n energy, direction, flavour,… preserved
Net charge
• Electromagnetic showers at high-E dominated by:
 pair production: g g → e+ e bremsstrahlung: e+/- + N → e+/- + N + g
“electrically neutral interactions” → no net charge
• Charge separation due to geomagnetic field unimportant in
dense media:
10 MeV e- traveling L=1 m deviates R≈ 0.05 cm laterally
(irrelevance of this mechanism checked in simulations).
• Net charge in dense media produced by “Askaryan effect”
Excess negative charge
N(e )  N(e )
Δq 
 25%


N(e )  N(e )
Δq scales with shower E.
Δq increases slowly with depth.
ZHS Monte Carlo simulations e-induced showers in ice
Δq depends on medium.
Frequency spectrum – EM showers
shower
observer
E. Zas, F. Halzen, T. Stanev, PRD 45, 162 (1992),
J. A-M & E. Zas, PLB 411, 218 (1997)
E-field/E0 V MHz -1 TeV -1
Angular distribution
Δθ
Cherenkov
peak
Δθ ≈ (Ln)-1
Angle w.r.t. shower axis [deg]
ZHS ice
E. Zas, F. Halzen, T. Stanev, PRD 45, 162 (1992),
J. A-M & E. Zas, PLB 411, 218 (1997)
LPM effect in EM showers
Screening effect on electron & photon interaction reduces bremsstrahlung &
pair-production cross sections w.r.t. Bethe-Heitler predictions
Electromagnetic showers having E > ELPM (~ 2 PeV in ice – medium dependant):
• Long. dimension L increases faster than ~ log E, typically as Eβ, β ~ ⅓ – ½
Produces multiple lumps in long. development at highest energies.
• Lateral dimension R does not change much with shower energy.
1 PeV
10 PeV
Above EeV other processes
(photoproduction) dominate
100 PeV
1 EeV
10 EeV
Field normalized to primary energy
Frequency spectrum in “LPM showers”
θC
θC - 10o
θC - 20o
• Elongated profiles at EeV induce smaller cut-off frequencies at θ≠θc
• Cut-off frequency at Cherenkov angle unaffected.
• Large shower-to-shower fluctuations
Freq. spectrum – Hadronic showers
• Slow elongation with energy → small cut-off frequencies at θ≠θc
• Cut-off frequency at Cherenkov angle increases slowly with energy
Contribution to radio-emission from:
protons + charged pions + muons + charged kaons < 2% above PeV
Radio emission in several
dense dielectric media
Ice vs Moon regolith vs Salt
Medium
Density
[g/cm3]
Radiation
length
[cm]
RMoliere
[cm]
Excess
Tracklength/E0
[m/TeV]
Ice
0.9
39.3
11.4
1980
Moon
1.8
13.0
6.9
1190
Salt
2.0
10.8
5.9
1105
Salt vs Ice vs Regolith
Longitudinal development of
excess charge (length units)
Radiation lengths:
L0 ≈ 39.3 cm
L0 ≈ 10.8 cm
L0 ≈ 13.0 cm
Lateral spread of excess
charge @ shower max.
Moliere radii:
RM ≈ 11.5 cm
RM ≈ 5.9 cm
RM ≈ 7.0 cm
Salt vs Ice vs Moon regolith
Medium
E field
V/MHz/TeV
@ 1 MHz
Cut-off
frequency
@ θc
[GHz]
Cut-off
frequency
@ θc – 50
[MHz]
Ice
2.1 x 10-10
≈2
≈ 200
Moon
9.2 x 10-11
≈5
≈ 600
Salt
8.6 x 10-11
≈4
≈ 500
Frequency spectrum – EM showers
shower
E-field
θ
observer