Modeling the Askaryan signal(in dense dielectric media)

Jaime Alvarez-MuñizUniv. 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: UHE detection

Expected events (Ahlers model)

Auger IceCube0.6 0.4

• 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 big a detector is needed ?

How to achieve this?. Detection of coherent Cherenkov radiation in dense media.

Reminder: Basics of radio-emission

in dense media

MOON REGOLITH

RICE - ARA - ARIANNA

GLUE

ANITA2

LORD

LUNASKA

NuMoon

The radio technique in dense media

Many experimental initiatives & hopefully more to come (see this meeting)

LOFARdense → ≈ 1 g cm-3

The source: -induced showers

Hadronic showers <E> ≈ 20% E @ EeV

“Mixed” showers <Eelectromagnetic > ≈ 80% E @ EeV

<Ehadronic > ≈ 20% En @ EeVEM or Hadronic

Dimensions & speed of the sourceLongitudinal spread

(Radiation length X0 ≈ 39 m in ice)

Ultra-relativistic electrons above K ≈100 keV v > c/n (n=1.78) Zas-Halzen-Stanev (ZHS) MC, PRD 45, 362 (1992)

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

atomic

atomic

• “Entrainment” of electrons from the medium as shower penetratesExcess negative charge develops (electrons) →

• Main interactions contributing:

Net negative charge: Askaryan effectG. Askar´yan, Soviet Phys. JETP 14, 441 (1962)

Askaryan effect present in any medium with bound electrons (for instance in air).

G.A. Askaryan

Compton Moeller

Bhabha

e+ annhilation

Askaryan effect confirmed in SLAC experiments

atomic

25%)N(e)N(e

)N(e)N(eΔq

“Low” energy processes ~ MeV

Modeling the signal

• 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,

• Disadvantages:– Time consuming – “Thinning” techniques @ ≈ EeV and above

• Main codes & refs.:– ZHS (Zas-Halzen-Stanev et al.), GEANT3.21 & 4, ZHAireS (ZHS+Aires)

(1) Monte Carlo simulations

J. A-M, W. R. Carvalho, M. Tueros, E. Zas, Astropart. Phys. 35, 287-299 (2012)

Zas, Halzen, Stanev PRD 45, 362 (1992) Razzaque et al. PRD 65 , 103002 (2002)Hussain & McKay PRD 70, 103003 (2004)

Radiation single particle track: Frequency domain

1. E(ω,x) increases with frequency2. E(ω,x) ≈ length of particle track perpendicular to direction of observation3. E(ω,x) 100% polarized (perpendicularly to observer´s direction)4. E(ω,x) : diffraction pattern exhibiting central peak around cos θC = 1/nβ

(=0) & angular width Δθ ~ (ω δt)-1

Maxwell´s equations → Fourier-components of Electric field E(ω,x) emitted by charged particle traveling along finite straight track at constant speed v:

sin

δt v ] t) vk - ω ( i [ exp R

) kR ( exp e ω E 1

frequency

diffraction

tracklength

phase factor

) θ cos βn -1 (δt ω

global phase

v

v┴

θt1

t1 + t

k

track

v┴t

far-fieldobserver

Radiation from a shower: frequency domain

particlescharged

iE E

Contributions to E-field from all charged particles tracks

i

i1iiiii

sin ] t) cosθ βn - 1 ( iω exp[ δt ve ω

Phase factors (different for each particle)

If θ ≈ θC or obs >> 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 δtielectrons positrons charge excess

Charge of each particle) θ cos βn -1 ( δt ω iii

50 % of excess track due to electrons with Ke < 6 - 7 MeV

[E. Zas, F. Halzen, T. Stanev PRD 45, 362 (1992)]

keV-MeV electrons contribute most to radio emission

Excess negative tracklength: T(e-) – T(e+) vs Kthreshold

Modeling with MC: Low energies matter…

See also “end-points” algorithm: T. Huege et al., PRE 84, 056602 (2011)C. James talk at this meeting

Vector potential

E-field

Time

[J. A-M, A. Romero-Wolf, E. Zas, PRD 81, 123009 (2010)]

Radiation single particle track: Time domainMaxwell´s equations → Radiation comes from instantaneous acceleration at start & deceleration at end of particle track

Acceleration

DecelerationAcceleration

A. Romero-Wolf & K.Belov(Proposal to test geosynchrotron in SLAC)

A-M, Romero-Wolf, Zas, PRD 81, 123009 (2010)]

Comparison algorithm with Jackson

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:

• Time-domain & Frequency-domain.• Far-field (Fraunhofer) and “near”-field (Fresnel – see later).

The Zas-Halzen-Stanev (ZHS) code

[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 & 4Razzaque et al. PRD 65 , 103002 (2002)

Hussain & McKay PRD 70, 103003 (2004)

~ 1 m from shower axis

ZHAireS = ZHS + Aires– Shower simulation with Aires– Electrom., hadronic & 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).

The ZHAireS code

Also GEANT4Hussain & McKay PRD 70, 103003 (2004)

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)

• Basic idea:– 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)

(2) Finite Difference Time-Domain (FDTD) techniques

• 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 .

(3) (Very) simple models

Far-field observer at Cherenkov angle θC

t1→2 = L/v = t1→3 = L cosθC / (c/n)

Wavefronts in phase

Time-domain: 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

1D “line” model of shower development

Time

Assumptions:a.1D line of current (excess charge Q) spreading over length L. b.Charge travels at v > c/n

radiation

wavefronts in phase

θc≈ 56o ice

L

1

2

3

z

J(z,t) = v Q (z - vt)

“Huygens approach”

Far-field observer

1D “line” model of shower development

Far-field observer at θ ≠ θC

Wavefronts NOT in phase (due to longitudinal shower spread L)

Time-domain Observer sees radiation in a finite interval of timedepending on angle (sensitivity to long profile):

Δt (θ) ≈ L (1 – n cosθ) / c ≈ few 10 ns

Frequency domain Spectrum increases linearly with frequency up to: Frequency cut-off

ωcut(θ) ≈ Δt-1 ≈ few 100 MHz

radiation

wavefrontsout of phase

L

1

2 3

θ < θc

Time

z

J(z,t) = v Q (z - vt)

Far-field observer

2D “box” model of shower development

Assumptions:1.2D current: longitudinally over L & laterally over R. 2.Uniform excess charge & travels at v > c/n

Far-field observer at Cherenkov θC

Wavefronts NOT in phase (due to lateral spread R of shower).

Time-domain: Observer sees radiation in a finite interval of time:

Δt ≈ R sinθc / (c/n) < ns

Frequency domain: Spectrum increases linearly with frequency:

Frequency cut-off ω ≈ Δt-1 ≈ GHz

L

R

wavefronts in phase

out of phase

θc radiation

R sinθc

Time

z

Far-field observer

2D “box” model of shower development

Far-field observer at θ ≠ θC

Wavefronts NOT in phase (due to longitudinal L & lateral spread R of shower)

Time-domain: Observer sees radiation in finite interval of time:

Δ t = max [R sinθ/ (c/n), L (1 – n cosθ)/c]

Frequency domain: Spectrum increases linearly with frequency:Frequency cut-off

ω ≈ Δ t-1 ≈ few 100 MHz - GHz

L

R

wavefronts out of phase

radiation

R sinθ

θ < θc

Time

z

Far-field observer

• 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).

Conclusions from “box” model

• 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 .

(4) Semi-analytical models

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)

Diffraction by a slit

Δθ ≈ (ω L)-1

L

Angular distribution of E-field

θc

1D “line” model with variable Q(z)

[J. A-M & E. Zas, PRD 62, 063001 (2000)]

Frequency domain:E-field can be obtained Fourier-transforming the longitudinal profile Q(z) of excess charge

pz ie Q(z) dz)E(

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)

J(z,t) = v Q(z) (z - vt)

Shower

slit

cnp /)cos1(

Far-field observers

Time domain:

[J. A-M, A. Romero-Wolf, E. Zas, PRD 81, 123009 (2010)]

J(z´,t´) = v Q(z´) (z´ - vt´)

Current

A(tobs , θ) ≈ v Q(ζ) / RVector potential

E(tobs , θ) = dA(tobs , θ)/dtobs

Electric field

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

E-field Bipolar pulses

Vector potential = Re-scaling of longitud. profile

Longitudinal developmentfrom MC sims.

Far-field observers

Coulomb gauge

Relativistic effectsSource 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

• Modelling signal away from Cherenkov simple & straightforward– Time-domain: vector potential = rescaling & time-

transforming longitudinal profile. – Freq.-domain: Electric field = Fourier transform of

longitudinal profile.

(Longitudinal profile easy/fast to obtain with MC simulations)

Conclusion from 1D line model

3D model with variable Q(z)

Current:

J(r´, φ´,z´,t´) = v(r´,φ´,z´) f(r´,z´) Q(z´) (z´ - vt´)

Lateral spread

Longitudinal spread

1D model fails close to Cherenkov angle

where lateral spread is of utmost importance for radio

emission

[J. A-M, A. Romero-Wolf, E. Zas, PRD 84, 103003 (2011)]

Dealing with the lateral spread

Convolution of longitudinal & lateral contributions

LateralLongitudinal

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´,´,z´) ≈ v(r´,´)

[J. A-M, A. Romero-Wolf, E. Zas, PRD 84, 103003 (2011)]

• A(Cher 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.

“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 ?)

Vector potential at θC :

obtained in MC sims.

Form factor

Shower tracklength:obtained in MC sims.

Integrals decouple at Cher

Observer´s time [ns]

R x

Vec

tor

pote

ntia

l [V

s]

[J. A-M, A. Romero-Wolf, E. Zas, PRD 84, 103003 (2011)]

contribution to A due to lateral spread at z weighted by Q(z)/4R tobs = z (1 - ncosθ)/c

z

tobs

Time reverses < Cher

Depth z [m]

Cher - 20o

Electron 1 EeV

tobs = z (1 - ncosθ)/c Large compression in time when is close to Cher

Depth z [m]

z

tobs

Cher + 0.1o

Electron 1 EeV

• Modelling signal at any 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, showers– Simply use lateral contribution corresponding to hadronic

showers or a mixture in case of e Charged Current

Conclusion from 3D model

Comparison of methods

MC vs MC

Frequency spectrum – EM showersElectron 1 PeV ice

θC

θC - 10o

θC - 20o

sho

wer

observer

θ

E-field

Frequency spectrum – EM showers

sho

wer

observer

θ

E-field

Electron 1 TeV ice

Semi-analytical models vs MC

Electron 1 EeV

Time-domain: away from Cher

Vector potential traces shape of longitudinal profile

Time reversal

Time-domain: close to Cher

Electron 1 EeVVector potential traces shape of longitudinal profile

Time reversal

Time-domain: even closer to Cher

Electron 1 EeV

Sensitivity to longitudinal profile lost

Time-domain: proton showersA(Cher t) proton- showers

(obtained in MC sims. with ZHAireS)

A(Cher t) proton vs electron-showers

ZHAireS MC

Proton – 100 TeV

Time-domain: -induced showers

A(Cher,t) (e) =

0.83 A(Cher,t) (e @ 1 EeV) +

0.17 A(Cher,t) (p @ 1 EeV)

e + N → e + jet

E(e) = 1 EeVE(electron) = 0.83 EeV

E(hadronic jet) = 0.17 EeVZH

Air

eS

MC

e – 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(-z2/2z

2) symmetric (instead of Greisen-like)

• lateral(r) ≈ exp(-r2/2r2) with r=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 > L2sin2/

– 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

More asymmetric waveform as R decreases

Observer sees different slices of shower at different distances, angles,…

1/sqrt(R) in near-field

1/R in far-field

Electron 10 TeV - ZHS MC

Electron 10 TeV - ZHS MC

~ Cher

Near-field in MC & FDTDSh

ower

(≈ 2

0 m

long

)

Observers

Shower max. seen at Cher → time compression → “single” bipolar pulse

Shower NOT seen at Cher → NO time compression → multi-peaked bipolar pulse

ZHAireS Monte Carlo vs 3D model

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

• Askaryan effect seen !!!• Linearly polarized signal• Power in radio waves goes as E0

2

• Bipolar pulses in time-domain• Agreement with theoretical expectations

Bunches of ~ GeV bremss. photons dumped in sand & salt & ice: E0 ~ 6 x 1017 – 1019 eV .

D. Saltzberg et al. PRL 86 (2001); P.Miocinovic et al. PRD 74 (2006), P. Gorham et al. PRL 99 (2007)

Angular distribution of electric field

Frequency spectrum

ICE target

ANITA

More attempts: K. Belov, A. Romero-Wolf @ this meeting

– MC simulations: Achieved maturity• Agree between each other : ZHS & ZHAireS & GEANT3.21 & 4

– ZHAireS most complete: time & freq. – far & “near” – e & p &

• 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

Summary: Modeling Askaryan signal

• 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,…)

Some things to-do

End

Backup slides

Why dense media & why radio?

Observation wavelength >> shower dimensionsCoherent 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 )

Excess charge (Askaryan effect) due to keV-MeV electrons.MeV electrons travel at v > c/n in dense media

interaction probability scales with density

Large vols. of dense, radio “transparent” media exist in Nature: ice, moon regolith, salt, …

Cheap detectors: antennas (dipoles, etc…)

Information on energy, direction, flavour,… preserved

• Electromagnetic showers at high-E dominated by: pair production: → e+ e-

bremsstrahlung: e+/- + N → e+/- + N + “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”

Net charge

Excess negative charge25%

)N(e)N(e

)N(e)N(eΔq

Δq scales with shower E.

Δq increases slowly with depth.

Δq depends on medium.ZHS Monte Carlo simulations e-induced showers in ice

Frequency spectrum – EM showers

E. Zas, F. Halzen, T. Stanev, PRD 45, 162 (1992), J. A-M & E. Zas, PLB 411, 218 (1997)

sho

wer

observer

Angular distribution

ZHS ice

E-f

ield

/E0

V M

Hz

-1 T

eV -1

E. Zas, F. Halzen, T. Stanev, PRD 45, 162 (1992), J. A-M & E. Zas, PLB 411, 218 (1997)

Angle w.r.t. shower axis [deg]

Cherenkov peak

Δθ ≈ (L)-1

Δθ

1 PeV

10 PeV

100 PeV

1 EeV10 EeV

LPM effect in EM showersScreening 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.

Above EeV other processes (photoproduction) dominate

θC

θC - 10o

θC - 20o

Frequency spectrum in “LPM showers”

• Elongated profiles at EeV induce smaller cut-off frequencies at θ≠θc • Cut-off frequency at Cherenkov angle unaffected.• Large shower-to-shower fluctuations

Fiel

d no

rmal

ized

to p

rimar

y en

ergy

Freq. spectrum – Hadronic showers

Contribution to radio-emission from:protons + charged pions + muons + charged kaons < 2% above PeV

• Slow elongation with energy → small cut-off frequencies at θ≠θc

• Cut-off frequency at Cherenkov angle increases slowly with energy

Radio emission in several dense dielectric media

Ice vs Moon regolith vs Salt

MediumDensity [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 RegolithLongitudinal development of excess charge (length units)

Radiation lengths:L0 ≈ 39.3 cmL0 ≈ 10.8 cmL0 ≈ 13.0 cm

Lateral spread of excess charge @ shower max.

Moliere radii:RM ≈ 11.5 cmRM ≈ 5.9 cmRM ≈ 7.0 cm

Salt vs Ice vs Moon regolith

MediumE 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

sho

wer

observer

θ

E-field