We have developed a code that semi-analytically calculates the radio
footprint (intensity and polarization) of an extensive air shower subject to
atmospheric electric fields. This can be used to reconstruct the height
dependence of atmospheric electric field from the measured radio footprint. The
various parameterizations of the spatial extent of the induced currents are
based on the results of Monte-Carlo shower simulations. The calculated radio
footprints agree well with microscopic CoREAS simulations.Comment: Contribution to the proceedings of the ARENA conference, Groningen,
  The Netherlands, June 7-10, 201

de Vries, Krijn D.

Scholten, Olaf

Trinh, Gia

van Sloten, Lucas

EPJ Web of Conferences

English

arXiv

We have developed a code that semi-analytically calculates the radio footprint (intensity and polarization) of an extensive air shower subject to atmospheric electric fields. This can be used to reconstruct the height dependence of atmospheric electric field from the measured radio footprint. The various parameterizations of the spatial extent of the induced currents are based on the results of Monte-Carlo shower simulations. The calculated radio footprints agree well with microscopic CoREAS simulations

Olaf Scholten

Gia Trinh

Krijn D. de Vries

Lucas van Sloten

EDP Sciences OAI-PMH repository (1.2.0)

Analytic Calculation of Radio Emission from Extensive Air Showers subjected to Atmospheric Electric Fields

University of Groningen

   University of GroningenAnalytic Calculation of Radio Emission from Extensive Air Showers subjected to AtmosphericElectric FieldsScholten, Olaf; Trinh, Gia; de Vries, Krijn D.; van Sloten, LucasPublished in:7th International Conference on Acoustic and Radio EeV Neutrino Detection ActivitiesDOI:10.1051/epjconf/201713503004IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2017Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Scholten, O., Trinh, G., de Vries, K. D., & van Sloten, L. (2017). Analytic Calculation of Radio Emissionfrom Extensive Air Showers subjected to Atmospheric Electric Fields. In 7th International Conference onAcoustic and Radio EeV Neutrino Detection Activities: ARENA 2016 (Vol. 135). [3004] EPJ Web ofConferences. https://doi.org/10.1051/epjconf/201713503004CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Analytic Calculation of Radio Emission from Extensive Air Show-ers subjected to Atmospheric Electric FieldsOlaf Scholten1,2,?, Gia Trinh1, Krijn D. de Vries2, and Lucas van Sloten11University of Groningen, KVI Center for Advanced Radiation Technology, 9747 AA Groningen, The Nether-lands2Vrije Universiteit Brussel, Dienst ELEM, IIHE, Pleinlaan 2, 1050 Brussels, BelgiumAbstract. We have developed a code that semi-analytically calculates the radio footprint(intensity and polarization) of an extensive air shower subject to atmospheric electricfields. This can be used to reconstruct the height dependence of atmospheric electric fieldfrom the measured radio footprint. The various parameterizations of the spatial extent ofthe induced currents are based on the results of Monte-Carlo shower simulations. Thecalculated radio footprints agree well with microscopic CoREAS simulations.1 Introduction and summaryAtmospheric electric fields during thunderstorm conditions can drastically modify the radio footprintof extensive air showers (EAS). This fact has been used [1, 2] to extract from the measured radiofootprint the altitude dependence of the electric fields. In contrast to conventional weather-balloonmeasurements this offers a non-intrusive way to determine these fields that are essential for under-standing lightning initiation and development [3].The extraction of the atmospheric fields relies on a model calculation of radio emission from anEAS. Accurate codes exists that perform reliable microscopic calculations [4]. One such code thatallows for setting arbitrary atmospheric electric fields is CoREAS [5] that uses results of a CORSIKAshower evolution. This uses a Monte-Carlo simulation and thus the results of two calculations withvery similar but unequal electric fields may differ considerably due to random shower-to-shower fluc-tuations. As a result it is close to impossible to use CoREAS in a systematic optimization procedure.To resolve this problem we have developed a deterministic semi-analytic calculation that predicts theradio-footprint given a certain altitude dependence of the induced currents in the EAS. An additionaladvantage is that the analytic code is much less CPU intensive than the microscopic calculation (a fewseconds compared to many hours). As a result it now becomes practical to solve the inverse problemfrom radio emission, i.e. to determine from a measured radio footprint (intensity and polarization dis-tributions on the ground) of a shower the configuration of currents in the atmosphere that producedthis.The code is based on the macroscopic formulation of radio emission as developed in refs [6–10] and incorporated in the EVA-code [11]. An important difference between the EVA-code and thepresent code (called EVA-light) is that here we use a direct parametrization of the current distribution?e-mail: Scholten@KVI.nl     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 3004 © The Authors,  published  by EDP Sciences.  This  is  an  open  access  article  distributed  under  the  terms  of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). in the EAS. Certain parameters, such as the distribution of the current density as function of distanceto the shower axis, are kept fixed, while other parameters related to the altitude dependence of thecurrents are optimized to fit the data. In addition the new code is optimized for a fast calculation ofthe complete radio footprint.Work is still ongoing to refine the parametrizations of the charge-excess and the induced-currentclouds where the realistic distributions as determined in Monte-Carlo shower evolution calculationsare guiding. Preliminary results show good agreement with those of microscopic CoREAS calcula-tions.2 Modelling Radio emission from EASThe motion of the charged particles in the EAS are modeled as a charge cloud with a parameterizeddensity profile moving towards Earth with the light velocity. Radio emission is subsequently calcu-lated using the retarded vector potential where we follow the a macroscopic approach to construct theretarded potentials. Furthermore we have put some effort in optimizing the code for the calculation ofa complete radio footprint.In this paper we work with the four-current jµ(ts, xs, ys, h) where zs = −ts (taking c = 1) is thedistance from the shower front to the ground, called height and always measured along the showeraxis, xs, and ys are the transverse directions, and h is the distance from the shower front to a pointin the charge cloud. We use the notation where µ = 0 denotes the time (charge) components andµ = x, y, z the space (current) components of a four vector. The shower front is moving with the lightvelocity. A particular point in the charge cloud is thus at an height of ζ = zs + h.Following the usual notation, the vector potential for an observer at a point (xo, yo, zo = 0) on theshower plane, defined as the plane perpendicular to the shower axis going through the point of impactof the shower on the ground, is taken asAµ(to, ~xo) =∫d3 ~x′jµ(tr, ~x′)L∣∣∣∣∣dtrdt∣∣∣∣∣ , (1)where L is the optical path-length, L = c(to − tr). Only for a homogeneous medium with a constantindex of refraction n it is given by L = nR = n| ~xo − ~x′| = c(to − tr). We introduce the retarded distanceDdef= L∣∣∣∣∣ dtdtr∣∣∣∣∣ = nR (1 − n~β · n̂)∣∣∣∣ret = (t − tr) − n2β(h − βtr) = n √(−βt + h)2 + (1 − β2n2)d2 , (2)for a moving point charge with velocity cβ. The vector potential can now be written compactly asAµ(to, ~xo) =∫d3 ~x′jµ(tr, ~x′)D. (3)The index of refraction depends on the height in the atmosphere through it density dependence asgiven by the Gladstone-Dale relation, nGD = 1 + nρ ρ(z) where ρ is the air density. For the evaluationof the retarded distance given in Eq. (2), the average value of the index of refraction over the photontrajectory will be used, assuming the straight-line approximation for the photon path [11],n = (1/z)∫ z0(1 + nρρ(ζ)) dζ , (4)which does not depend on the distance of the observer to the shower axis for vertical showers.     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 30042To evaluate the integration over z′ in Eq. (3), we follow the same approach as used in Ref. [11] andreplace the integral in the z-direction by an integral over λ = hc − h whereD = 0 at the critical heighthc. This substitution allows for an easier calculation of derivatives of the vector potential that arenecessary to calculate the electric field since the derivatives vanish on the limits of the λ integration,as shown in Ref. [11].2.1 Electric field, charge excessThe charge excess in the shower front is given by Q(z) propagating with the light velocity in the−ẑ-direction thus contributing to the zero and the z component of the vector potential. Substitutingthe expression for the vector potential, integrating over the spatial extent of the charge cloud andconverting to an integration over λ, the radial component of the electric field can be written asECXr = −∂A0∂ro= −∫dxs dys∂w(rs)/rs∂rsICXλ ; ICXλ =∫ hc0dλ f (hc − λ)QnD∣∣∣∣∣λ=hc−h, (5)where r2s = (x2s + y2s) is an integration point in the shower front and the position of the observer isdenoted with the subscript o. The integral is separated into an integration over λ, a line-sub-shower,along a line parallel to the axis of the full shower, followed by one over the radial extent. In thenumerical calculation the results line-sub-showers, ICXλ (t), are stored for a grid of rs and d values,where the latter is the distance of the observer to line-sub-shower.In order to obey charge conservation a residue charge line-density Q′(z) = (dQ/dz) is left behindat rest in the atmosphere which can be ignored as shown in Ref. [7].2.2 Electric field, Transverse currentAlong a similar line of reasoning as for the charge-excess radiation, the transverse current contributionis written asETCx = −∂Ax∂to=∫dxs dysw(rs)rsITCλ , (6)ITCλ =∫ hc0dλ f ′(hc − λ)JxnD∣∣∣∣∣λ=hc−h−∫ hc0dλ f (hc − λ)J′xnD∣∣∣∣∣λ=hc−h. (7)where J′x = dIx/dtr and f ′(h) = d f /dh. As before the results for line-sub-showers, ITCλ (t) arecalculated once on a grid of rs and d values to be used subsequently in the calculation of the completefootprint of the electric field.Due to the continuity equation we have dJx/dx = dQ/dt, thus for r2s = x2s + y2s we writeQDx (tr, xs, ys) =[∫ tr−∞Jx(t)dt]×d w(rs)/rsdxsand a similar term for Jy. This corresponds to a dipole chargedistribution moving with the shower front. In reality these particles thermalize and will trail well be-hind the front and thus not contribute to coherent emission. To account for this we include a decreaseproportional to e−X/aDX0 ,~JD(tr) =∫ tr−∞~J(t) e−[X(tr)−X(t)]/(aDX0) dt , (8)where X depends on t through its dependence on the height in the atmosphere. In Eq. (8) X0 is theelectron mean free path and aD a parameter that is adjusted to get best agreement with microscopicshower calculations. Following a similar notation as before the contribution to the electric field isEDx = −∫dxs dysd w(rs)/rsrs drsIxD ; IDx =∫dλ f (hc − λ)JDxnD∣∣∣∣∣∣λ=hc−h, (9)     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 30043where a similar expression is obtained for EDy .2.3 ParameterizationsThe density distribution of the four current is parameterized asjµ(t, x, y, h) =w(r)rf (h, r) Jµ(t); w(r) = Nw r/M0(r/M0 + 1)−2.5 ; f (h, r) = N f uαe−2u , (10)where r =√x2 + y2, u = h/α λ, M0 the Moliere radius, and h the distance behind the shower front.The normalization constants Nw and N f (r) in the functions w and f are determined by∫w(r) dr = 1and∫ ∞0 f (h, r) dh = 1 ∀ r. In this way Jµ(t) is the charge and current at time t integrated over thewhole shower front. Note that w(r) corresponds to r times the NKG function for s = 2. The twoparameters in f , the pancake thickness λ(r) and α(X), depend on distance to the shower axis, r, andpenetration depth, X, asλ(r) = max[λ0, λ1 (r/r1)] ; α(X) = 1 + 0.5 ×√X/Xmax . (11)The radial dependence of the pancake thickness, λ(r), is such that for r = 0 we have λ(0) = λ0increasing linearly with distance on an exponential grid where at a distance of r1 =100 m from thecore we have λ(100) = λ1. With increasing α the maximum in the particle density shifts to largerdistances behind the shower front.The optimization of the parameters is still ongoing. We note that choosing M0 = 30 m,λ0 = 0.15 m, λ1 = 3 m, reproduces the main features of the charge cloud as deduced from mi-croscopic calculation. This also gives rise to a radio footprint that agrees well with that obtainedfrom CoREAS simulations. We are still investigating the extent to which the distributions need to bechosen differently for charge excess and transverse current in view of the findings in Ref. [2]. Wenote that changing the pancake thickness with distance to the shower axis is essential to be able toreproduce the (not frequency filtered) pulse shape as obtained from the CoREAS calculations. Thepenetration-depth dependence of α does not have a strong influence on the calculations.References[1] P. Schellart et al., Phys. Rev. Lett 114, 165001 (2015), arXiv:1504.05742.[2] G. Trinh, O. Scholten, et al., Phys. Rev. D 93, 023003 (2016), arXiv:1511.03045.[3] J.R. Dwyer and M.A. Uman, Phys. Rep. 534, 147 (2014).[4] T. Huege, Phys. Rep. 620, 1 (2016), arXiv:1601.07426.[5] T. Huege, M. Ludwig, and C. James, AIP Conf. Proc. 1535, 128 (2012).[6] O. Scholten, K. Werner, and F. Rusydi, Astropart. Phys. 29, 94 (2008), arXiv:0709.2872.[7] Klaus Werner and Olaf Scholten, Astropart. Phys. 29, 393-411 (2008), arXiv:0712.2517.[8] K.D. de Vries, et al., Astropart. Phys. 34, 267 (2010), arXiv:1008.3308.[9] K. D. de Vries, et al., Phys. Rev. Lett 107, 061101 (2011).[10] Olaf Scholten, Krijn D. de Vries, Klaus Werner, EPJ Web of Conf. 53, 08005 (2013),arXiv:1207.1874.[11] K. Werner, K.D. De Vries, O. Scholten, Astropart. Phys. 37, 5 (2012), arXiv:1201.4471.     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 30044

University of Groningen Research Database

  
 University of Groningen
Analytic Calculation of Radio Emission from Extensive Air Showers subjected to Atmospheric
Electric Fields
Scholten, Olaf; Trinh, Gia; de Vries, Krijn D.; van Sloten, Lucas
Published in:
7th International Conference on Acoustic and Radio EeV Neutrino Detection Activities
DOI:
10.1051/epjconf/201713503004
IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from
it. Please check the document version below.
Document Version
Publisher's PDF, also known as Version of record
Publication date:
2017
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Scholten, O., Trinh, G., de Vries, K. D., & van Sloten, L. (2017). Analytic Calculation of Radio Emission
from Extensive Air Showers subjected to Atmospheric Electric Fields. In 7th International Conference on
Acoustic and Radio EeV Neutrino Detection Activities: ARENA 2016 (Vol. 135). [3004] EPJ Web of
Conferences.  EPJ Web of Conferences, DOI: 10.1051/epjconf/201713503004
Copyright
Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the
author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).
Take-down policy
If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately
and investigate your claim.
Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the
number of authors shown on this cover page is limited to 10 maximum.
Download date: 11-04-2018
Analytic Calculation of Radio Emission from Extensive Air Show-
ers subjected to Atmospheric Electric Fields
Olaf Scholten1,2,?, Gia Trinh1, Krijn D. de Vries2, and Lucas van Sloten1
1University of Groningen, KVI Center for Advanced Radiation Technology, 9747 AA Groningen, The Nether-
lands
2Vrije Universiteit Brussel, Dienst ELEM, IIHE, Pleinlaan 2, 1050 Brussels, Belgium
Abstract. We have developed a code that semi-analytically calculates the radio footprint
(intensity and polarization) of an extensive air shower subject to atmospheric electric
fields. This can be used to reconstruct the height dependence of atmospheric electric field
from the measured radio footprint. The various parameterizations of the spatial extent of
the induced currents are based on the results of Monte-Carlo shower simulations. The
calculated radio footprints agree well with microscopic CoREAS simulations.
1 Introduction and summary
Atmospheric electric fields during thunderstorm conditions can drastically modify the radio footprint
of extensive air showers (EAS). This fact has been used [1, 2] to extract from the measured radio
footprint the altitude dependence of the electric fields. In contrast to conventional weather-balloon
measurements this offers a non-intrusive way to determine these fields that are essential for under-
standing lightning initiation and development [3].
The extraction of the atmospheric fields relies on a model calculation of radio emission from an
EAS. Accurate codes exists that perform reliable microscopic calculations [4]. One such code that
allows for setting arbitrary atmospheric electric fields is CoREAS [5] that uses results of a CORSIKA
shower evolution. This uses a Monte-Carlo simulation and thus the results of two calculations with
very similar but unequal electric fields may differ considerably due to random shower-to-shower fluc-
tuations. As a result it is close to impossible to use CoREAS in a systematic optimization procedure.
To resolve this problem we have developed a deterministic semi-analytic calculation that predicts the
radio-footprint given a certain altitude dependence of the induced currents in the EAS. An additional
advantage is that the analytic code is much less CPU intensive than the microscopic calculation (a few
seconds compared to many hours). As a result it now becomes practical to solve the inverse problem
from radio emission, i.e. to determine from a measured radio footprint (intensity and polarization dis-
tributions on the ground) of a shower the configuration of currents in the atmosphere that produced
this.
The code is based on the macroscopic formulation of radio emission as developed in refs [6–
10] and incorporated in the EVA-code [11]. An important difference between the EVA-code and the
present code (called EVA-light) is that here we use a direct parametrization of the current distribution
?e-mail: Scholten@KVI.nl
     
DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of Conferences
ARENA 2016
0 03004 3004
 © The Authors,  published  by EDP Sciences.  This  is  an  open  access  article  distributed  under  the  terms  of the Creative
 Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). 
in the EAS. Certain parameters, such as the distribution of the current density as function of distance
to the shower axis, are kept fixed, while other parameters related to the altitude dependence of the
currents are optimized to fit the data. In addition the new code is optimized for a fast calculation of
the complete radio footprint.
Work is still ongoing to refine the parametrizations of the charge-excess and the induced-current
clouds where the realistic distributions as determined in Monte-Carlo shower evolution calculations
are guiding. Preliminary results show good agreement with those of microscopic CoREAS calcula-
tions.
2 Modelling Radio emission from EAS
The motion of the charged particles in the EAS are modeled as a charge cloud with a parameterized
density profile moving towards Earth with the light velocity. Radio emission is subsequently calcu-
lated using the retarded vector potential where we follow the a macroscopic approach to construct the
retarded potentials. Furthermore we have put some effort in optimizing the code for the calculation of
a complete radio footprint.
In this paper we work with the four-current jµ(ts, xs, ys, h) where zs = −ts (taking c = 1) is the
distance from the shower front to the ground, called height and always measured along the shower
axis, xs, and ys are the transverse directions, and h is the distance from the shower front to a point
in the charge cloud. We use the notation where µ = 0 denotes the time (charge) components and
µ = x, y, z the space (current) components of a four vector. The shower front is moving with the light
velocity. A particular point in the charge cloud is thus at an height of ζ = zs + h.
Following the usual notation, the vector potential for an observer at a point (xo, yo, zo = 0) on the
shower plane, defined as the plane perpendicular to the shower axis going through the point of impact
of the shower on the ground, is taken as
Aµ(to, ~xo) =
∫
d3 ~x′
jµ(tr, ~x′)
L
∣∣∣∣∣dtrdt
∣∣∣∣∣ , (1)
where L is the optical path-length, L = c(to − tr). Only for a homogeneous medium with a constant
index of refraction n it is given by L = nR = n| ~xo − ~x′| = c(to − tr). We introduce the retarded distance
D def= L
∣∣∣∣∣ dtdtr
∣∣∣∣∣ = nR (1 − n~β · nˆ)∣∣∣∣ret = (t − tr) − n2β(h − βtr) = n√(−βt + h)2 + (1 − β2n2)d2 , (2)
for a moving point charge with velocity cβ. The vector potential can now be written compactly as
Aµ(to, ~xo) =
∫
d3 ~x′
jµ(tr, ~x′)
D . (3)
The index of refraction depends on the height in the atmosphere through it density dependence as
given by the Gladstone-Dale relation, nGD = 1 + nρ ρ(z) where ρ is the air density. For the evaluation
of the retarded distance given in Eq. (2), the average value of the index of refraction over the photon
trajectory will be used, assuming the straight-line approximation for the photon path [11],
n = (1/z)
∫ z
0
(1 + nρρ(ζ)) dζ , (4)
which does not depend on the distance of the observer to the shower axis for vertical showers.
     
DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of Conferences
ARENA 2016
0 03004 3004
2
To evaluate the integration over z′ in Eq. (3), we follow the same approach as used in Ref. [11] and
replace the integral in the z-direction by an integral over λ = hc − h whereD = 0 at the critical height
hc. This substitution allows for an easier calculation of derivatives of the vector potential that are
necessary to calculate the electric field since the derivatives vanish on the limits of the λ integration,
as shown in Ref. [11].
2.1 Electric field, charge excess
The charge excess in the shower front is given by Q(z) propagating with the light velocity in the
−zˆ-direction thus contributing to the zero and the z component of the vector potential. Substituting
the expression for the vector potential, integrating over the spatial extent of the charge cloud and
converting to an integration over λ, the radial component of the electric field can be written as
ECXr = −
∂A0
∂ro
= −
∫
dxs dys
∂w(rs)/rs
∂rs
ICXλ ; I
CX
λ =
∫ hc
0
dλ f (hc − λ) QnD
∣∣∣∣∣
λ=hc−h
, (5)
where r2s = (x
2
s + y
2
s) is an integration point in the shower front and the position of the observer is
denoted with the subscript o. The integral is separated into an integration over λ, a line-sub-shower,
along a line parallel to the axis of the full shower, followed by one over the radial extent. In the
numerical calculation the results line-sub-showers, ICXλ (t), are stored for a grid of rs and d values,
where the latter is the distance of the observer to line-sub-shower.
In order to obey charge conservation a residue charge line-density Q′(z) = (dQ/dz) is left behind
at rest in the atmosphere which can be ignored as shown in Ref. [7].
2.2 Electric field, Transverse current
Along a similar line of reasoning as for the charge-excess radiation, the transverse current contribution
is written as
ETCx = −
∂Ax
∂to
=
∫
dxs dys
w(rs)
rs
ITCλ , (6)
ITCλ =
∫ hc
0
dλ f ′(hc − λ) J
x
nD
∣∣∣∣∣
λ=hc−h
−
∫ hc
0
dλ f (hc − λ) J
′x
nD
∣∣∣∣∣
λ=hc−h
. (7)
where J′x = dIx/dtr and f ′(h) = d f /dh. As before the results for line-sub-showers, ITCλ (t) are
calculated once on a grid of rs and d values to be used subsequently in the calculation of the complete
footprint of the electric field.
Due to the continuity equation we have dJx/dx = dQ/dt, thus for r2s = x
2
s + y
2
s we write
QDx (tr, xs, ys) =
[∫ tr
−∞ Jx(t)dt
]
× d w(rs)/rsdxs and a similar term for Jy. This corresponds to a dipole charge
distribution moving with the shower front. In reality these particles thermalize and will trail well be-
hind the front and thus not contribute to coherent emission. To account for this we include a decrease
proportional to e−X/aDX0 ,
~JD(tr) =
∫ tr
−∞
~J(t) e−[X(tr)−X(t)]/(a
DX0) dt , (8)
where X depends on t through its dependence on the height in the atmosphere. In Eq. (8) X0 is the
electron mean free path and aD a parameter that is adjusted to get best agreement with microscopic
shower calculations. Following a similar notation as before the contribution to the electric field is
EDx = −
∫
dxs dys
d w(rs)/rs
rs drs
IxD ; I
D
x =
∫
dλ f (hc − λ) J
D
x
nD
∣∣∣∣∣∣
λ=hc−h
, (9)
     
DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of Conferences
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0 03004 3004
3
where a similar expression is obtained for EDy .
2.3 Parameterizations
The density distribution of the four current is parameterized as
jµ(t, x, y, h) =
w(r)
r
f (h, r) Jµ(t); w(r) = Nw r/M0(r/M0 + 1)−2.5 ; f (h, r) = N f uαe−2u , (10)
where r =
√
x2 + y2, u = h/α λ, M0 the Moliere radius, and h the distance behind the shower front.
The normalization constants Nw and N f (r) in the functions w and f are determined by
∫
w(r) dr = 1
and
∫ ∞
0 f (h, r) dh = 1 ∀ r. In this way Jµ(t) is the charge and current at time t integrated over the
whole shower front. Note that w(r) corresponds to r times the NKG function for s = 2. The two
parameters in f , the pancake thickness λ(r) and α(X), depend on distance to the shower axis, r, and
penetration depth, X, as
λ(r) = max[λ0, λ1 (r/r1)] ; α(X) = 1 + 0.5 ×
√
X/Xmax . (11)
The radial dependence of the pancake thickness, λ(r), is such that for r = 0 we have λ(0) = λ0
increasing linearly with distance on an exponential grid where at a distance of r1 =100 m from the
core we have λ(100) = λ1. With increasing α the maximum in the particle density shifts to larger
distances behind the shower front.
The optimization of the parameters is still ongoing. We note that choosing M0 = 30 m,
λ0 = 0.15 m, λ1 = 3 m, reproduces the main features of the charge cloud as deduced from mi-
croscopic calculation. This also gives rise to a radio footprint that agrees well with that obtained
from CoREAS simulations. We are still investigating the extent to which the distributions need to be
chosen differently for charge excess and transverse current in view of the findings in Ref. [2]. We
note that changing the pancake thickness with distance to the shower axis is essential to be able to
reproduce the (not frequency filtered) pulse shape as obtained from the CoREAS calculations. The
penetration-depth dependence of α does not have a strong influence on the calculations.
References
[1] P. Schellart et al., Phys. Rev. Lett 114, 165001 (2015), arXiv:1504.05742.
[2] G. Trinh, O. Scholten, et al., Phys. Rev. D 93, 023003 (2016), arXiv:1511.03045.
[3] J.R. Dwyer and M.A. Uman, Phys. Rep. 534, 147 (2014).
[4] T. Huege, Phys. Rep. 620, 1 (2016), arXiv:1601.07426.
[5] T. Huege, M. Ludwig, and C. James, AIP Conf. Proc. 1535, 128 (2012).
[6] O. Scholten, K. Werner, and F. Rusydi, Astropart. Phys. 29, 94 (2008), arXiv:0709.2872.
[7] Klaus Werner and Olaf Scholten, Astropart. Phys. 29, 393-411 (2008), arXiv:0712.2517.
[8] K.D. de Vries, et al., Astropart. Phys. 34, 267 (2010), arXiv:1008.3308.
[9] K. D. de Vries, et al., Phys. Rev. Lett 107, 061101 (2011).
[10] Olaf Scholten, Krijn D. de Vries, Klaus Werner, EPJ Web of Conf. 53, 08005 (2013),
arXiv:1207.1874.
[11] K. Werner, K.D. De Vries, O. Scholten, Astropart. Phys. 37, 5 (2012), arXiv:1201.4471.
     
DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of Conferences
ARENA 2016
0 03004 3004
4


ARTS repository - University of Groningen

Crossref

   University of GroningenAnalytic Calculation of Radio Emission from Extensive Air Showers subjected to AtmosphericElectric FieldsScholten, Olaf; Trinh, Gia; de Vries, Krijn D.; van Sloten, LucasPublished in:7th International Conference on Acoustic and Radio EeV Neutrino Detection ActivitiesDOI:10.1051/epjconf/201713503004IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2017Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Scholten, O., Trinh, G., de Vries, K. D., & van Sloten, L. (2017). Analytic Calculation of Radio Emissionfrom Extensive Air Showers subjected to Atmospheric Electric Fields. In 7th International Conference onAcoustic and Radio EeV Neutrino Detection Activities: ARENA 2016 (Vol. 135). [3004] EPJ Web ofConferences.  EPJ Web of Conferences https://doi.org/10.1051/epjconf/201713503004CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Download date: 12-11-2019Analytic Calculation of Radio Emission from Extensive Air Show-ers subjected to Atmospheric Electric FieldsOlaf Scholten1,2,?, Gia Trinh1, Krijn D. de Vries2, and Lucas van Sloten11University of Groningen, KVI Center for Advanced Radiation Technology, 9747 AA Groningen, The Nether-lands2Vrije Universiteit Brussel, Dienst ELEM, IIHE, Pleinlaan 2, 1050 Brussels, BelgiumAbstract. We have developed a code that semi-analytically calculates the radio footprint(intensity and polarization) of an extensive air shower subject to atmospheric electricfields. This can be used to reconstruct the height dependence of atmospheric electric fieldfrom the measured radio footprint. The various parameterizations of the spatial extent ofthe induced currents are based on the results of Monte-Carlo shower simulations. Thecalculated radio footprints agree well with microscopic CoREAS simulations.1 Introduction and summaryAtmospheric electric fields during thunderstorm conditions can drastically modify the radio footprintof extensive air showers (EAS). This fact has been used [1, 2] to extract from the measured radiofootprint the altitude dependence of the electric fields. In contrast to conventional weather-balloonmeasurements this offers a non-intrusive way to determine these fields that are essential for under-standing lightning initiation and development [3].The extraction of the atmospheric fields relies on a model calculation of radio emission from anEAS. Accurate codes exists that perform reliable microscopic calculations [4]. One such code thatallows for setting arbitrary atmospheric electric fields is CoREAS [5] that uses results of a CORSIKAshower evolution. This uses a Monte-Carlo simulation and thus the results of two calculations withvery similar but unequal electric fields may differ considerably due to random shower-to-shower fluc-tuations. As a result it is close to impossible to use CoREAS in a systematic optimization procedure.To resolve this problem we have developed a deterministic semi-analytic calculation that predicts theradio-footprint given a certain altitude dependence of the induced currents in the EAS. An additionaladvantage is that the analytic code is much less CPU intensive than the microscopic calculation (a fewseconds compared to many hours). As a result it now becomes practical to solve the inverse problemfrom radio emission, i.e. to determine from a measured radio footprint (intensity and polarization dis-tributions on the ground) of a shower the configuration of currents in the atmosphere that producedthis.The code is based on the macroscopic formulation of radio emission as developed in refs [6–10] and incorporated in the EVA-code [11]. An important difference between the EVA-code and thepresent code (called EVA-light) is that here we use a direct parametrization of the current distribution?e-mail: Scholten@KVI.nl     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 3004 © The Authors,  published  by EDP Sciences.  This  is  an  open  access  article  distributed  under  the  terms  of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). in the EAS. Certain parameters, such as the distribution of the current density as function of distanceto the shower axis, are kept fixed, while other parameters related to the altitude dependence of thecurrents are optimized to fit the data. In addition the new code is optimized for a fast calculation ofthe complete radio footprint.Work is still ongoing to refine the parametrizations of the charge-excess and the induced-currentclouds where the realistic distributions as determined in Monte-Carlo shower evolution calculationsare guiding. Preliminary results show good agreement with those of microscopic CoREAS calcula-tions.2 Modelling Radio emission from EASThe motion of the charged particles in the EAS are modeled as a charge cloud with a parameterizeddensity profile moving towards Earth with the light velocity. Radio emission is subsequently calcu-lated using the retarded vector potential where we follow the a macroscopic approach to construct theretarded potentials. Furthermore we have put some effort in optimizing the code for the calculation ofa complete radio footprint.In this paper we work with the four-current jµ(ts, xs, ys, h) where zs = −ts (taking c = 1) is thedistance from the shower front to the ground, called height and always measured along the showeraxis, xs, and ys are the transverse directions, and h is the distance from the shower front to a pointin the charge cloud. We use the notation where µ = 0 denotes the time (charge) components andµ = x, y, z the space (current) components of a four vector. The shower front is moving with the lightvelocity. A particular point in the charge cloud is thus at an height of ζ = zs + h.Following the usual notation, the vector potential for an observer at a point (xo, yo, zo = 0) on theshower plane, defined as the plane perpendicular to the shower axis going through the point of impactof the shower on the ground, is taken asAµ(to, ~xo) =∫d3 ~x′jµ(tr, ~x′)L∣∣∣∣∣dtrdt∣∣∣∣∣ , (1)where L is the optical path-length, L = c(to − tr). Only for a homogeneous medium with a constantindex of refraction n it is given by L = nR = n| ~xo − ~x′| = c(to − tr). We introduce the retarded distanceD def= L∣∣∣∣∣ dtdtr∣∣∣∣∣ = nR (1 − n~β · nˆ)∣∣∣∣ret = (t − tr) − n2β(h − βtr) = n√(−βt + h)2 + (1 − β2n2)d2 , (2)for a moving point charge with velocity cβ. The vector potential can now be written compactly asAµ(to, ~xo) =∫d3 ~x′jµ(tr, ~x′)D . (3)The index of refraction depends on the height in the atmosphere through it density dependence asgiven by the Gladstone-Dale relation, nGD = 1 + nρ ρ(z) where ρ is the air density. For the evaluationof the retarded distance given in Eq. (2), the average value of the index of refraction over the photontrajectory will be used, assuming the straight-line approximation for the photon path [11],n = (1/z)∫ z0(1 + nρρ(ζ)) dζ , (4)which does not depend on the distance of the observer to the shower axis for vertical showers.     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 30042To evaluate the integration over z′ in Eq. (3), we follow the same approach as used in Ref. [11] andreplace the integral in the z-direction by an integral over λ = hc − h whereD = 0 at the critical heighthc. This substitution allows for an easier calculation of derivatives of the vector potential that arenecessary to calculate the electric field since the derivatives vanish on the limits of the λ integration,as shown in Ref. [11].2.1 Electric field, charge excessThe charge excess in the shower front is given by Q(z) propagating with the light velocity in the−zˆ-direction thus contributing to the zero and the z component of the vector potential. Substitutingthe expression for the vector potential, integrating over the spatial extent of the charge cloud andconverting to an integration over λ, the radial component of the electric field can be written asECXr = −∂A0∂ro= −∫dxs dys∂w(rs)/rs∂rsICXλ ; ICXλ =∫ hc0dλ f (hc − λ) QnD∣∣∣∣∣λ=hc−h, (5)where r2s = (x2s + y2s) is an integration point in the shower front and the position of the observer isdenoted with the subscript o. The integral is separated into an integration over λ, a line-sub-shower,along a line parallel to the axis of the full shower, followed by one over the radial extent. In thenumerical calculation the results line-sub-showers, ICXλ (t), are stored for a grid of rs and d values,where the latter is the distance of the observer to line-sub-shower.In order to obey charge conservation a residue charge line-density Q′(z) = (dQ/dz) is left behindat rest in the atmosphere which can be ignored as shown in Ref. [7].2.2 Electric field, Transverse currentAlong a similar line of reasoning as for the charge-excess radiation, the transverse current contributionis written asETCx = −∂Ax∂to=∫dxs dysw(rs)rsITCλ , (6)ITCλ =∫ hc0dλ f ′(hc − λ) JxnD∣∣∣∣∣λ=hc−h−∫ hc0dλ f (hc − λ) J′xnD∣∣∣∣∣λ=hc−h. (7)where J′x = dIx/dtr and f ′(h) = d f /dh. As before the results for line-sub-showers, ITCλ (t) arecalculated once on a grid of rs and d values to be used subsequently in the calculation of the completefootprint of the electric field.Due to the continuity equation we have dJx/dx = dQ/dt, thus for r2s = x2s + y2s we writeQDx (tr, xs, ys) =[∫ tr−∞ Jx(t)dt]× d w(rs)/rsdxs and a similar term for Jy. This corresponds to a dipole chargedistribution moving with the shower front. In reality these particles thermalize and will trail well be-hind the front and thus not contribute to coherent emission. To account for this we include a decreaseproportional to e−X/aDX0 ,~JD(tr) =∫ tr−∞~J(t) e−[X(tr)−X(t)]/(aDX0) dt , (8)where X depends on t through its dependence on the height in the atmosphere. In Eq. (8) X0 is theelectron mean free path and aD a parameter that is adjusted to get best agreement with microscopicshower calculations. Following a similar notation as before the contribution to the electric field isEDx = −∫dxs dysd w(rs)/rsrs drsIxD ; IDx =∫dλ f (hc − λ) JDxnD∣∣∣∣∣∣λ=hc−h, (9)     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 30043where a similar expression is obtained for EDy .2.3 ParameterizationsThe density distribution of the four current is parameterized asjµ(t, x, y, h) =w(r)rf (h, r) Jµ(t); w(r) = Nw r/M0(r/M0 + 1)−2.5 ; f (h, r) = N f uαe−2u , (10)where r =√x2 + y2, u = h/α λ, M0 the Moliere radius, and h the distance behind the shower front.The normalization constants Nw and N f (r) in the functions w and f are determined by∫w(r) dr = 1and∫ ∞0 f (h, r) dh = 1 ∀ r. In this way Jµ(t) is the charge and current at time t integrated over thewhole shower front. Note that w(r) corresponds to r times the NKG function for s = 2. The twoparameters in f , the pancake thickness λ(r) and α(X), depend on distance to the shower axis, r, andpenetration depth, X, asλ(r) = max[λ0, λ1 (r/r1)] ; α(X) = 1 + 0.5 ×√X/Xmax . (11)The radial dependence of the pancake thickness, λ(r), is such that for r = 0 we have λ(0) = λ0increasing linearly with distance on an exponential grid where at a distance of r1 =100 m from thecore we have λ(100) = λ1. With increasing α the maximum in the particle density shifts to largerdistances behind the shower front.The optimization of the parameters is still ongoing. We note that choosing M0 = 30 m,λ0 = 0.15 m, λ1 = 3 m, reproduces the main features of the charge cloud as deduced from mi-croscopic calculation. This also gives rise to a radio footprint that agrees well with that obtainedfrom CoREAS simulations. We are still investigating the extent to which the distributions need to bechosen differently for charge excess and transverse current in view of the findings in Ref. [2]. Wenote that changing the pancake thickness with distance to the shower axis is essential to be able toreproduce the (not frequency filtered) pulse shape as obtained from the CoREAS calculations. Thepenetration-depth dependence of α does not have a strong influence on the calculations.References[1] P. Schellart et al., Phys. Rev. Lett 114, 165001 (2015), arXiv:1504.05742.[2] G. Trinh, O. Scholten, et al., Phys. Rev. D 93, 023003 (2016), arXiv:1511.03045.[3] J.R. Dwyer and M.A. Uman, Phys. Rep. 534, 147 (2014).[4] T. Huege, Phys. Rep. 620, 1 (2016), arXiv:1601.07426.[5] T. Huege, M. Ludwig, and C. James, AIP Conf. Proc. 1535, 128 (2012).[6] O. Scholten, K. Werner, and F. Rusydi, Astropart. Phys. 29, 94 (2008), arXiv:0709.2872.[7] Klaus Werner and Olaf Scholten, Astropart. Phys. 29, 393-411 (2008), arXiv:0712.2517.[8] K.D. de Vries, et al., Astropart. Phys. 34, 267 (2010), arXiv:1008.3308.[9] K. D. de Vries, et al., Phys. Rev. Lett 107, 061101 (2011).[10] Olaf Scholten, Krijn D. de Vries, Klaus Werner, EPJ Web of Conf. 53, 08005 (2013),arXiv:1207.1874.[11] K. Werner, K.D. De Vries, O. Scholten, Astropart. Phys. 37, 5 (2012), arXiv:1201.4471.     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 30044

NARCIS 

   University of GroningenAnalytic Calculation of Radio Emission from Extensive Air Showers subjected to AtmosphericElectric FieldsScholten, Olaf; Trinh, Gia; de Vries, Krijn D.; van Sloten, LucasPublished in:7th International Conference on Acoustic and Radio EeV Neutrino Detection ActivitiesDOI:10.1051/epjconf/201713503004IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite fromit. Please check the document version below.Document VersionPublisher's PDF, also known as Version of recordPublication date:2017Link to publication in University of Groningen/UMCG research databaseCitation for published version (APA):Scholten, O., Trinh, G., de Vries, K. D., & van Sloten, L. (2017). Analytic Calculation of Radio Emissionfrom Extensive Air Showers subjected to Atmospheric Electric Fields. In 7th International Conference onAcoustic and Radio EeV Neutrino Detection Activities: ARENA 2016 (Vol. 135). Article 3004 EPJ Web ofConferences. https://doi.org/10.1051/epjconf/201713503004CopyrightOther than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of theauthor(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons).The publication may also be distributed here under the terms of Article 25fa of the Dutch Copyright Act, indicated by the “Taverne” license.More information can be found on the University of Groningen website: https://www.rug.nl/library/open-access/self-archiving-pure/taverne-amendment.Take-down policyIf you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediatelyand investigate your claim.Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons thenumber of authors shown on this cover page is limited to 10 maximum.Analytic Calculation of Radio Emission from Extensive Air Show-ers subjected to Atmospheric Electric FieldsOlaf Scholten1,2,?, Gia Trinh1, Krijn D. de Vries2, and Lucas van Sloten11University of Groningen, KVI Center for Advanced Radiation Technology, 9747 AA Groningen, The Nether-lands2Vrije Universiteit Brussel, Dienst ELEM, IIHE, Pleinlaan 2, 1050 Brussels, BelgiumAbstract. We have developed a code that semi-analytically calculates the radio footprint(intensity and polarization) of an extensive air shower subject to atmospheric electricfields. This can be used to reconstruct the height dependence of atmospheric electric fieldfrom the measured radio footprint. The various parameterizations of the spatial extent ofthe induced currents are based on the results of Monte-Carlo shower simulations. Thecalculated radio footprints agree well with microscopic CoREAS simulations.1 Introduction and summaryAtmospheric electric fields during thunderstorm conditions can drastically modify the radio footprintof extensive air showers (EAS). This fact has been used [1, 2] to extract from the measured radiofootprint the altitude dependence of the electric fields. In contrast to conventional weather-balloonmeasurements this offers a non-intrusive way to determine these fields that are essential for under-standing lightning initiation and development [3].The extraction of the atmospheric fields relies on a model calculation of radio emission from anEAS. Accurate codes exists that perform reliable microscopic calculations [4]. One such code thatallows for setting arbitrary atmospheric electric fields is CoREAS [5] that uses results of a CORSIKAshower evolution. This uses a Monte-Carlo simulation and thus the results of two calculations withvery similar but unequal electric fields may differ considerably due to random shower-to-shower fluc-tuations. As a result it is close to impossible to use CoREAS in a systematic optimization procedure.To resolve this problem we have developed a deterministic semi-analytic calculation that predicts theradio-footprint given a certain altitude dependence of the induced currents in the EAS. An additionaladvantage is that the analytic code is much less CPU intensive than the microscopic calculation (a fewseconds compared to many hours). As a result it now becomes practical to solve the inverse problemfrom radio emission, i.e. to determine from a measured radio footprint (intensity and polarization dis-tributions on the ground) of a shower the configuration of currents in the atmosphere that producedthis.The code is based on the macroscopic formulation of radio emission as developed in refs [6–10] and incorporated in the EVA-code [11]. An important difference between the EVA-code and thepresent code (called EVA-light) is that here we use a direct parametrization of the current distribution?e-mail: Scholten@KVI.nl     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 3004 © The Authors,  published  by EDP Sciences.  This  is  an  open  access  article  distributed  under  the  terms  of the Creative Commons Attribution License 4.0 (http://creativecommons.org/licenses/by/4.0/). in the EAS. Certain parameters, such as the distribution of the current density as function of distanceto the shower axis, are kept fixed, while other parameters related to the altitude dependence of thecurrents are optimized to fit the data. In addition the new code is optimized for a fast calculation ofthe complete radio footprint.Work is still ongoing to refine the parametrizations of the charge-excess and the induced-currentclouds where the realistic distributions as determined in Monte-Carlo shower evolution calculationsare guiding. Preliminary results show good agreement with those of microscopic CoREAS calcula-tions.2 Modelling Radio emission from EASThe motion of the charged particles in the EAS are modeled as a charge cloud with a parameterizeddensity profile moving towards Earth with the light velocity. Radio emission is subsequently calcu-lated using the retarded vector potential where we follow the a macroscopic approach to construct theretarded potentials. Furthermore we have put some effort in optimizing the code for the calculation ofa complete radio footprint.In this paper we work with the four-current jµ(ts, xs, ys, h) where zs = −ts (taking c = 1) is thedistance from the shower front to the ground, called height and always measured along the showeraxis, xs, and ys are the transverse directions, and h is the distance from the shower front to a pointin the charge cloud. We use the notation where µ = 0 denotes the time (charge) components andµ = x, y, z the space (current) components of a four vector. The shower front is moving with the lightvelocity. A particular point in the charge cloud is thus at an height of ζ = zs + h.Following the usual notation, the vector potential for an observer at a point (xo, yo, zo = 0) on theshower plane, defined as the plane perpendicular to the shower axis going through the point of impactof the shower on the ground, is taken asAµ(to, ~xo) =∫d3 ~x′jµ(tr, ~x′)L∣∣∣∣∣dtrdt∣∣∣∣∣ , (1)where L is the optical path-length, L = c(to − tr). Only for a homogeneous medium with a constantindex of refraction n it is given by L = nR = n| ~xo − ~x′| = c(to − tr). We introduce the retarded distanceDdef= L∣∣∣∣∣ dtdtr∣∣∣∣∣ = nR (1 − n~β · n̂)∣∣∣∣ret = (t − tr) − n2β(h − βtr) = n √(−βt + h)2 + (1 − β2n2)d2 , (2)for a moving point charge with velocity cβ. The vector potential can now be written compactly asAµ(to, ~xo) =∫d3 ~x′jµ(tr, ~x′)D. (3)The index of refraction depends on the height in the atmosphere through it density dependence asgiven by the Gladstone-Dale relation, nGD = 1 + nρ ρ(z) where ρ is the air density. For the evaluationof the retarded distance given in Eq. (2), the average value of the index of refraction over the photontrajectory will be used, assuming the straight-line approximation for the photon path [11],n = (1/z)∫ z0(1 + nρρ(ζ)) dζ , (4)which does not depend on the distance of the observer to the shower axis for vertical showers.     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 30042To evaluate the integration over z′ in Eq. (3), we follow the same approach as used in Ref. [11] andreplace the integral in the z-direction by an integral over λ = hc − h whereD = 0 at the critical heighthc. This substitution allows for an easier calculation of derivatives of the vector potential that arenecessary to calculate the electric field since the derivatives vanish on the limits of the λ integration,as shown in Ref. [11].2.1 Electric field, charge excessThe charge excess in the shower front is given by Q(z) propagating with the light velocity in the−ẑ-direction thus contributing to the zero and the z component of the vector potential. Substitutingthe expression for the vector potential, integrating over the spatial extent of the charge cloud andconverting to an integration over λ, the radial component of the electric field can be written asECXr = −∂A0∂ro= −∫dxs dys∂w(rs)/rs∂rsICXλ ; ICXλ =∫ hc0dλ f (hc − λ)QnD∣∣∣∣∣λ=hc−h, (5)where r2s = (x2s + y2s) is an integration point in the shower front and the position of the observer isdenoted with the subscript o. The integral is separated into an integration over λ, a line-sub-shower,along a line parallel to the axis of the full shower, followed by one over the radial extent. In thenumerical calculation the results line-sub-showers, ICXλ (t), are stored for a grid of rs and d values,where the latter is the distance of the observer to line-sub-shower.In order to obey charge conservation a residue charge line-density Q′(z) = (dQ/dz) is left behindat rest in the atmosphere which can be ignored as shown in Ref. [7].2.2 Electric field, Transverse currentAlong a similar line of reasoning as for the charge-excess radiation, the transverse current contributionis written asETCx = −∂Ax∂to=∫dxs dysw(rs)rsITCλ , (6)ITCλ =∫ hc0dλ f ′(hc − λ)JxnD∣∣∣∣∣λ=hc−h−∫ hc0dλ f (hc − λ)J′xnD∣∣∣∣∣λ=hc−h. (7)where J′x = dIx/dtr and f ′(h) = d f /dh. As before the results for line-sub-showers, ITCλ (t) arecalculated once on a grid of rs and d values to be used subsequently in the calculation of the completefootprint of the electric field.Due to the continuity equation we have dJx/dx = dQ/dt, thus for r2s = x2s + y2s we writeQDx (tr, xs, ys) =[∫ tr−∞Jx(t)dt]×d w(rs)/rsdxsand a similar term for Jy. This corresponds to a dipole chargedistribution moving with the shower front. In reality these particles thermalize and will trail well be-hind the front and thus not contribute to coherent emission. To account for this we include a decreaseproportional to e−X/aDX0 ,~JD(tr) =∫ tr−∞~J(t) e−[X(tr)−X(t)]/(aDX0) dt , (8)where X depends on t through its dependence on the height in the atmosphere. In Eq. (8) X0 is theelectron mean free path and aD a parameter that is adjusted to get best agreement with microscopicshower calculations. Following a similar notation as before the contribution to the electric field isEDx = −∫dxs dysd w(rs)/rsrs drsIxD ; IDx =∫dλ f (hc − λ)JDxnD∣∣∣∣∣∣λ=hc−h, (9)     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 30043where a similar expression is obtained for EDy .2.3 ParameterizationsThe density distribution of the four current is parameterized asjµ(t, x, y, h) =w(r)rf (h, r) Jµ(t); w(r) = Nw r/M0(r/M0 + 1)−2.5 ; f (h, r) = N f uαe−2u , (10)where r =√x2 + y2, u = h/α λ, M0 the Moliere radius, and h the distance behind the shower front.The normalization constants Nw and N f (r) in the functions w and f are determined by∫w(r) dr = 1and∫ ∞0 f (h, r) dh = 1 ∀ r. In this way Jµ(t) is the charge and current at time t integrated over thewhole shower front. Note that w(r) corresponds to r times the NKG function for s = 2. The twoparameters in f , the pancake thickness λ(r) and α(X), depend on distance to the shower axis, r, andpenetration depth, X, asλ(r) = max[λ0, λ1 (r/r1)] ; α(X) = 1 + 0.5 ×√X/Xmax . (11)The radial dependence of the pancake thickness, λ(r), is such that for r = 0 we have λ(0) = λ0increasing linearly with distance on an exponential grid where at a distance of r1 =100 m from thecore we have λ(100) = λ1. With increasing α the maximum in the particle density shifts to largerdistances behind the shower front.The optimization of the parameters is still ongoing. We note that choosing M0 = 30 m,λ0 = 0.15 m, λ1 = 3 m, reproduces the main features of the charge cloud as deduced from mi-croscopic calculation. This also gives rise to a radio footprint that agrees well with that obtainedfrom CoREAS simulations. We are still investigating the extent to which the distributions need to bechosen differently for charge excess and transverse current in view of the findings in Ref. [2]. Wenote that changing the pancake thickness with distance to the shower axis is essential to be able toreproduce the (not frequency filtered) pulse shape as obtained from the CoREAS calculations. Thepenetration-depth dependence of α does not have a strong influence on the calculations.References[1] P. Schellart et al., Phys. Rev. Lett 114, 165001 (2015), arXiv:1504.05742.[2] G. Trinh, O. Scholten, et al., Phys. Rev. D 93, 023003 (2016), arXiv:1511.03045.[3] J.R. Dwyer and M.A. Uman, Phys. Rep. 534, 147 (2014).[4] T. Huege, Phys. Rep. 620, 1 (2016), arXiv:1601.07426.[5] T. Huege, M. Ludwig, and C. James, AIP Conf. Proc. 1535, 128 (2012).[6] O. Scholten, K. Werner, and F. Rusydi, Astropart. Phys. 29, 94 (2008), arXiv:0709.2872.[7] Klaus Werner and Olaf Scholten, Astropart. Phys. 29, 393-411 (2008), arXiv:0712.2517.[8] K.D. de Vries, et al., Astropart. Phys. 34, 267 (2010), arXiv:1008.3308.[9] K. D. de Vries, et al., Phys. Rev. Lett 107, 061101 (2011).[10] Olaf Scholten, Krijn D. de Vries, Klaus Werner, EPJ Web of Conf. 53, 08005 (2013),arXiv:1207.1874.[11] K. Werner, K.D. De Vries, O. Scholten, Astropart. Phys. 37, 5 (2012), arXiv:1201.4471.     DOI: 10.1051/,  (2017) 7135epjconf/201135EPJ Web of ConferencesARENA 20160 03004 30044

Analytic Calculation of Radio Emission from Extensive Air Showers subjected to Atmospheric Electric Fields

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