Étude de l'énergie et du point d'émission radio des rayons cosmiques détectés dans...

Ahmed REBAI

Mémoire présenté en vue de l’obtention du grade de Docteur de l’Université de Nantes sous le label de L’Université Nantes Angers Le Mans

École doctorale : MOLÉCULES, MATIÈRES ET MATÉRIAUX EN PAYS DE LOIRE (3MPL)

Discipline : Physique nucléaire Spécialité : Astroparticules Unité de recherche : SUBATECH

Soutenue le 28 Juin 2013 Thèse N° :

Étude de l'énergie et du point d'émission radio des rayons cosmiques détectés dans l'expérience

CODALEMA

JURY

Rapporteurs : Mme. Christelle ROY, Directrice de recherche, CNRS, IPHC, Strasbourg M. Alain LECACHEUX, Directeur de recherche, CNRS, Observatoire de Paris, Meudon

Examinateurs : M. Georges LE VEY, Maître assistant, Ecoles des mines de Nantes IRCCyN, Nantes Mme. Cristina CARLOGANU, Chargée de recherche, CNRS, LPC Clermont Ferrand M. Olivier RAVEL, Maître de conférences, Université de Nantes, Subatech

M. Pascal LAUTRIDOU Directeur de recherche, CNRS, Subatech

M. Pascal LAUTRIDOU, Directeur de recherche, CNRS, Subatech Directeur de Thèse :

Encadrant scientifique : M. Olivier RAVEL, Maître de conférences, Université de Nantes, Subatech

1020 km2 1015 eV

km2 1000 km2

Ep 0 20 %

1020 eV 13 eV 5, 5.1019 eV

> EeV ( E, B) F = q( E + v ∧ B) 1013 V

100EeV

(∼ 10 km.s−1) 30 parsec 1011 ans!

107m.s−1 E = α.E n

E = E0(1 + α)n

n =ln( E

E0)

ln(1 + α)

n P n

N = N0Pn

N0

ln(N

N0) = n.ln(P ) = ln(

E

E0)

ln(P )

ln(1 + α)= ln(

E0

E)s

s = − ln(P )ln(1+α) N ≥ n

≥ E

dN(E)

dE= constante×

E0

E

(1+s)

s 1, 1 −2, 1 −2, 7 (1−P )

z|e| 100.z.T eV

90 100EeV ”X” 1020 eV

∼ 1016GeV 4 − 5 1024 eV mX σνN 246GeV σνN > 10−27 cm2 50 g.cm−2 10−26 cm2 ∼ 1000 g.cm−2 36000 g.cm−2 1021 eV

11 109 eV 1020 eV 32 m2.sr.s 109 eV 1 km2.sr.siecle 1020 eV

F (E) =d4N

dE.dS.dΩ.dt= E−n

N E S Ω t n

”X” 1016GeV γ γ 3.1019 eV γ

2, 7 n ∼ 2, 7 ∼ 3 3 − 5.1015 eV ∼ 3, 3 5.1017 eV 2, 7 − 3, 0 3.1018 eV Ep < 1GeV

11 ans

1977 1995

1Tesla R = p.c

Z.|e| p Z.|e|

E2,5p

γ 1018 eV

GeV 105GeV

∼ 2, 7 m2

m2 sr s m2 sr

s

m2 sr jour

m2 sr an

km2 sr an

km2 sr Siecle

g.cm−2

5 Z > 17

3.1015 eV 2, 7 3 m2 an

[0, 40]

3.1017 eV ∼ 3 3, 3

14TeV

2, 7 3.1018 eV ∼ 3µG

1964 3 1966 (T = 2, 73K) 1020 eV π0 π+ Δ+

p+ γ → Δ+ → n+ π+

p+ γ → Δ+ → p+ π0

E3

1020 eV ∼ 100Mpc ∼ 100Mpc Mpc

90 1991 3.1020 eV 11 1020 90 100Mpc

2006 2008 EGZK 5.σ 40 EeV 20 J

1021 eV 100 Mpc

Xmax Xmax < Xmax > RMS(Xmax) Xmax Xmax Xmax

< Xmax > RMS(Xmax)

< Xmax > RMS(Xmax) Xmax

< Xmax > < Xmax > RMS(Xmax)

1020 eV

∼ 100Mpc 57EeV 75Mpc 12eme

Ep ≥ 55EeV 3, 1 60

61% 33% 3σ

2σ

Ep ≥ 55EeV 60 3, 1 318 75Mpc

Xmax 1018 eV −1018,5 eV

√s =

57TeV

Xmax Xmax

20% dNdXmax

∝ e−XmaxΛ20 Λ20

σp−Air σp−Air = <mair>Λ20

σp−Air

σp−Air = (505± 22stat (+19−14)sys)mb

1000 g.cm−2 80 g.cm−2 (∼ 700m 10 km) 1938 Xmax (g.cm−2)

g.cm−2

e+/e− (γ −→ e+ + e−) (e± −→ e± + γ) λ X

N(X) = 2Xλ

2 Ep 2

E(X) =Ep

2Xλ

(Ec = 80MeV )

Xmax

dNdXmax

∝ e−XmaxΛ20 σp−Air σp−Air = <mair>

Λ20

σp−Air

Xmax = X0.ln(Ep

Ec)

X0 = 37 g.cm−2 Nmax Xmax

Nmax =Ep

Ec

N Ep A t X0 = 37, 15 g.cm−2 t = X−X1

X0

N(Ep, A, t) =Ep

Enet−tmax−2t.ln(s)

En = 1, 45GeV tmax = 1, 7+ 0, 76.(ln(EpEc

)− ln(A)) Xmax Ec = 80MeV s = 2t

t+tmax

N t N(t)

Xmax = X0.tmax Nmax Ep N(t)

Xmax

Ec

(π+, π−, π0)

(K+, K−, K0) λN ∼ 70−80 g.cm−2 π0 → 2γ

Xmax Xmax Xmax = Xsimu

0 ln(Ep

Esimu ) Xsimu

0 Esimu Xmax Ne(t)

ρ(r) = Nec(s)

r20(r0r

)s−2(1 +r

r0)s−4,5

avec c(s) = 0, 366s2(2, 07 − s)1,25 et s =3t

t+ 2tmax

Ne ρ r0 r0 = 80m 1016 eV

1016 eV

N = 1, 43.105 s = 0, 7 θ = 14, 2 N = 1, 28.105 s = 0, 8 θ = 15, 5

x ω v n(ω)

d2E ∝ ω

1− 1

β2.n(ω)2

dx dω β =

v

c

ω.dω ∼ 1014Hz ∼ 107Hz

100MHz e+/e−

20MeV δ δ 10%

N 108 .N 0, 1

Iradio = (.N)2Ipar

Ipar

Ivisible = N.Ipar0

Ipar0

Ivisible

Iradio=

N.Ipar0

(.N)2Ipar=

1

2.N

Ipar0

Ipar 107

= 0, 1

> 1016 eV 44MHz 2, 75MHz

6µs

B v B i v ∧ B α i ∝ sin(α) = ||v ∧ B||

0 ∝

Ep

0 ∝ Ep

0 Ep

i

di i i ∝ e−did0

30 − 300m θ < 35 d0 d0(f =55MHz) = 100 ± 10m d0(f = 32MHz) = 140m i θ i ∝ cos(θ) 0 32−55MHz

i(f) = 20.

Ep

1017 eV

.sin(α).cos(θ).e

− did0(f, θ) enµV.m−1.MHz−1

1017 eV θ < 35 < 300m

O(100)ns

GHz

10−240MHz

2650m 1017 eV ντ

20 km2

20% 30% (Ne−−Ne+ )

(Ne−+Ne+ )

E±(r, t) = ± 1

Δt

q

c

r ∧ (r ∧ β)

R(1− β.n)

+ β − β

E(r, t) =1

4π0

n.q(tret)

R2(1 − β.n)

ret

+1

c.∂

∂t

n.q(tret)

R(1− β.n)

ret

− 1

c2.∂

∂t

v.q(tret)

R(1− β.n)

ret

t = tret + |r−r0(tret)|c β R

Jµ(t, r, h) Jµ

Aµ(t, r) =µ0

4π

d3r

Jµ(t, r, h)|D| |t=t

h

> 30 MeV < 30 MeV γ τ = d

γc

Emax = γq

4π0d2

d

GeV Ep = 1017 eV 108 30 MeV (γ = 60) 10−4 V.m−1 10 ns 100 m

1 µs

Ep = 1017 eV

1 MHz 100 MHz 100 MHz

4 µV/m/MHz Ep = 1017 eV

Ep = 1017 eV

20 %

5 200 100 m

(d) = 0.exp(−d/d0)

100 m

φp = tan−1(ENS

EEO) =

1

2tan−1(

U

Q)

U Q

Ep Ep 0 0 Ep 0 10 % Ep 30 % Ep, 0 0

|v ∧ B|EO = |sin(θ).cos(φ).cos(θb) + cos(θ).sin(θb)| B = 47µT θb

315 (0, 0) 10 m

14% a = |Er|

|Eg|sin(α)

α

(Ep, 0)

Xmax

Xmax 150 g.cm−2 30 g.cm−2

200 g.cm−2

[40, 60] MHz 10 %

3, 7 σ

η = −ln(

(τ2 + ρ2

2))

ρ τ

200 g.cm−2

50 g.cm−2

1GHz 2000 2002

2006

2010 2006 2008

1, 5 × 1, 5 km2

600m×500m 100m

∼ 100ns mV.m−1

[0, 200MHz]

Zant(f) = Rant(f) + j.Xant(f)

Rant = R + Rrad Rrad R Rant

Rant(f) = 20(π2L.f

c)2

L c Xant Lant Cant

Xant = Lant.2π.f − 1

Cant.2π.f

Cant =π.0.L

ln(La )

Lant =1

4.π2.Cant.f20

Xant

Zant Xant(f) = 0 f0

Q =Lω0

R=

1

2.π.Rant

Lant

Cant

L = λ

2 (f0 = c2.L )

L [37, 70]MHz f0 = 125MHz 1, 2m Vin

(θ, φ) P (θ, φ) P0

G(θ, φ) =P (θ, φ)

4πP0

P (θ, φ) ≥ 2.P0

70MHz

70MHz 30 (θ = 90) 100MHz

f0

f05

f0 Vout E

Vout

E= leff .

Vout

Va=

c

f.

Rrad(f).G(θ, φ, f)

120.π2.

A

1 + j.2π.Za(f).Cin.f

leff G(θ, φ, f) A

0, 8µm

leff = VaE

Va E

V.m−1

MHz GHz σbruit =

√< V 2 >

6 0LST 5 LST

24mV

0, 25W 30 dB

−3 dB 80 kHz 230MHz 10 pF 19µF

23h56mn4s

5 913 17 13 (θp, φp)

Ep

1015,7 eV 1016 eV 85m 5 7mn

γ 80 cm×80 cm×4 cm 1600V 1200V 104 0, 3 3000

θp 0 90 φp 0 90 180 270 p

200muons parm2.s

5 650ns θp = 90 1GHz 12 bits 2, 52µs

V (t) = V0 + Vmax((t− t0).e

n.τ)n.e(

−(t−t0)τ )

4 V0 Vmax n.τ t0

(θp, φp) χ2

χ2 =N

i=1

(c(t0i − T0) − (u.xi + v.yi)

σi)2

(u = sinθp.cosφp, v = sinθp.sinφp) (θp, φp) θp = sin−1(

√u2 + v2) φp = tan−1(uv )

θp

(σθp , σφp) (θp, φp)

θp ∼ 20

dN

dθ=

a.sin(θp).cos(θp)

1 + e(θp−bc )

θp < b = 27, 1

c = 7, 44 b a

φp 50 150

(σθp , σφp)

(xpc , y

pc , z

pc = 0)

4

ρ(r) = Ne.c(s)

r20.(r0r

)2−s.(1 +r

r0)s−4,5

r r0 80m s s = 1, 2 c(s) c(s) =0, 366.s2.(2, 07 − s)1,25 χ2

χ2 =

N

i=1

(ni − ρ(ri))2

σ2ni

N ni ri ri =

(xi − xprc )2 + (yi − yprc )2 (xi, yi)

(xprc , yprc ) (xpc , y

pc )

60 1, 5 km2 31

2

1

1, 2m 20MHz 100MHz [30− 40]MHz

220V 24V

ns

[45−55]MHz

1GS/s 14 bits

24 21 3 2520 1Gs/s 2, 52µs 5

2, 56µm

(θ, φ)

27 2006 2 2010

[0− 200MHz]

(Attenuation(dB/m) = 0, 0817.

f(GHz) +

0, 025.f(GHz)) [0, 200MHz]

4, 5ns metre

(10ns a ∼ 100ns) 100µV.m−1 Ep ∼ 1016 eV 1000µV.m−1 Ep ∼ 1018 eV [0− 100MHz] [1, 20MHz] [90, 110MHz] [0−120MHz]

[1 − 120MHz]

[24−82MHz] 1 0 300ns

[23 − 82MHz]

2, 52µs 300ns

7 [24, 82MHz]

µb =< V 2 >b

σb =< (V 2 − µb) >2

seuil ≥ k ∗ µb k k 25

Ep

Vp(k) V n

Vp(k) = a1V (k − 1) + a2V (k − 2) + ...+ anV (k − n) =

n

i=1

aiV (k − i)

V (k) Vp(k)

e(k) = V (k) − Vp(k) = V (k) − (a1V (k − 1) + a2V (k2) + ...+ anV (k − n))

a1, ..., an Np

χ2 =

Np

k=1

e2(k) =

Np

k=1

(V (k) −n

i=1

aiV (k − i))2

e(k)

≥ 3 u.xi + v.yi + w.zi + constante = 0

n =

u = sin(θ).cos(φ)v = sin(θ).sin(φ)

w = cos(θ)

[23 − 80MHz]

σt = 10ns

[45 − 55MHz]

σt = 5ns

σt = 10ns

σt = 1ns

[30 − 80MHz]σt = 1ns

MHz

χ2

χ2 =N

i=1

(c.ti − c.t0 + xi.u+ yi.v + zi.w

c.σi)2

(xi, yi zi) i σi i 10ns t0 zi = 0m χ2

χ2 =N

i=1

(c.ti − c.t0 + xi.u+ yi.v

c.σi)2

ti, xi, yi u, v, t0

χ2

12 .

∂χ2

∂u = 012 .

∂χ2

∂v = 012 .

∂χ2

∂c.t0= 0

Sxt − c.t0.Sx + u.Sxx + v.Sxy = 0Syt − c.t0.Sy + u.Sxy + v.Syy = 0−St + c.t0.S1 − u.Sx + v.Sy = 0

⇒

Sxx Sxy −Sx

Sxy Syy −Sy

−Sx Sy S1

uvc.t0

=

−Sxt

−Syt

St

Sβγ =

iβ.ασ2i Sxt =

ixi.c.tiσ2i

Sxx =

ix2i

σ2i Sxy =

ixi.yiσ2i Syt =

iyi.c.tiσ2i

S1 =

i1σ2i

J.A = B J χ2

J =

Sxx Sxy −Sx

Sxy Syy −Sy

−Sx Sy S1

A B A =

uvc.t0

B =

−Sxt

−Syt

St

J A = J−1.B

u2 + v2 + w2 = 1

θ = sin−1(

(u2 + v2))φ = tan−1( vu )

10ns

1015 eV 5.1016 eV 1018 eV 1 6.1017 eV v∧ B

α () α < 20 |dt| < 100ns σ(α) = 1, 6

i di

i = 20.(Ep

1017 eV)

0

.sin(α).cos(θ).exp(− did0(ν, θ)

)µV

m.MHz

d0 α θ 0 Ep

50% Ep = 2.1017 eV 1 1018 eV

B v ∧ B 2030 1708 322 0, 188 0, 17 15σ

0 E0 E0 Ep

log10(E0) log10(Ep) log10(E0) = a ∗ log10(Ep) + b

(θ, φ) 2030 (θ = 27, φ = 0) 5 1

PeV Xmax

Xmax Xmax

Xmax 40 g.cm−2

< Xmax >

Xmax Xmax Xmax g.cm−2

Ecal = Nmax

<

dE

dX> f(X)dX =

Ecrit

λ0Nmax

f(X)dX (1)

f(X) = (X −X0

Xmax −X0)Xmax−X0

λ .exp(−X −Xmax

λ) (2)

(2) < dE

dX >

g.cm−2 Ecritλ0

Ecrit = 80MeV λ0 = 37, 1 g.cm−2 2, 18MeV.e−1.g.cm−2 10% Eem <∼ 100MeV

100ns g/cm2 Ep = 3, 0 ± 0, 2.1019 eV

8% Ep 22% 14% 9, 5% 4% 10% 6%− 8%

40MeV Ep 11% 11% 10% 12% 22%

Ne Nµ Ep

Ecrit

θ

S(d) V EM/m2 m2 (d) 600m 800m 1000m S(d) = f(Ep, θ)

S(1000) S(1000) ∼ P (cos(θ)).E0,95

p P 2 cos2(θ) − cos2(< θ >)

S(1000) θ

S(d) S Ep N0

dN

dcos2(θ)= Constante

N0 CIC(θ) θ N(> CIC(θ)) = N0 S38

S38 =S(1000)

CIC(θ),

θ = 38 θ 60 S38 Ep

θ

CIC(θ) x = cos2(θ) − cos2(< θ >)

2 Ep

14TeV 1017 eV

20 % Ep (d) Ep

S38 Ep

% 3EeV 6EeV 13% 6EeV 10EeV % 10EeV S38 Ep

Ep Ep

60 − 70

Ep 0 > 1018 eV

2 17 1035 g.cm−2 Ep ρ(r)

ρ(r) = Ne.c(s)

r20.(r0r

)2−s.(1 +r

r0)s−4,5

r0 r s s = 1, 2

c(s) = 0, 366.s2.(2, 07− s)1,25

Ne e−/e+

χ2 =

Nscintillateurs

i=1

(ni −NKG(ri))2

σ2ni

σni ni i ri

ri =

(xi − xpc)2 + (yi − ypc )2

Ne Ne θ Ne(0

)

Ne(θ) = Ne(0).exp(−Xprofondeur−Nancay

Λatt.(cos(θ)−1 − cos(0)−1))

Λatt Ne(θ) θ

< Ne(0

) > S(E)

S(E) =< Ne(0) >= Ne(θ).exp(−

Xprofondeur−Nancay

Λatt.(cos(θ)−1 − cos(0)−1))

Λatt Ep < Ne(0) >

Λatt [1015 eV − 1017.5 eV ] Λatt = 188 g.cm−2 (190 g/cm2)

Ep < Ne(0) >

< Ne(0) >

100 6 1016 − 1018 eV

Ep(eV ) = 2, 138 · 1010· < Ne(0) >0.9

Ep θ

Ep(eV ) = 2, 138.1010.(Ne(θ).exp(−Xprofondeur−Nancay

Λatt.(cos(θ)−1 − 1)))0,9

30% 1017 eV σEp

Ep< 30%

Ep 0 Ne Ep Ep 0 0 ∝ E0,96

p 60 5400m (0, Ep) 0 ∝ Ep

0, 3 E2,7 E2,7 · φ(E)

GeV 1.7 ·m−2 · s−1 · sr−1

E(eV )

4 20 ∝ Ep ⇒ 0 ∝

Ep

2000

Ep

i = 0.exp(− di

d0)

i = 20.(Ep

1017 eV)

0

.sin(α).cos(θ).exp(− did0(ν, θ)

)µV

m.MHz

0 10MHz Ep 20m 100m 180m 300m 500m 0 ∝ E0,96

p

0

0

i [µV.m−1] (xi, yi, zi = 0) 0 [µV.m−1] d0 [m] di [m] i

di = AB =AC2 −BC2,

AC

AC2 = (xi − xc)2 + (yi − yc)

2 + (zi − zc)2 = (xi − xc)

2 + (yi − yc)2

∀i zi = zc = 0m

BC = | AC.n| =

xi − xcyi − yczi − zc

.

cos(φ).sin(θ)sin(φ).sin(θ)

cos(θ)

= |(xi − xc).cos(φ).sin(θ) + (yi − yc).sin(φ).sin(θ) + (zi − zc).cos(θ)|= |(xi − xc).cos(φ).sin(θ) + (yi − yc).sin(φ).sin(θ)|

di =

(xi − xc)2 + (yi − yc)2 − ((xi − xc).cos(φ).sin(θ) + (yi − yc).sin(φ).sin(θ))2.

(Ep, 0) 4 ∝ Ep

φ = 0 φ = 90 θ = 0 θ = 90

dpi

19 Ep = 8.1016 eV 16 Ep = 1, 6.1018 eV

(φp, θp) (xpc , y

pc , z

pc = 0)

dpi

dpi =

(xi − xpc)2 + (yi − ypc )2 − ((xi − xpc).cos(φp).sin(θp) + (yi − ypc ).sin(φp).sin(θp))2.

0 d0 χ2

χ2 =

Nant−tag

i=1

i − 0.exp(− di

d0)

σi

2

σi i Nant−tag

χ2r

4 (0, d0) (xc, yc)

i = 0.exp

−

(xi − xc)2 + (yi − yc)2 − ((xi − xc).cos(φ).sin(θ) + (yi − yc).sin(φ).sin(θ))2

d0

χ2r =

Nant−tag

i=1

i − 0.exp(−di(xc, yc)

d0)

σi

2

(θ, φ) 4 0 d0

(xpc , ypc )

Ep = 8.1016 eV Ep = 1, 6.1018 eV ( i

σi )2

σi :

[23 − 83]MHz σi (f < 100MHz)

AM kHz 23MHz

FM 87, 5MHz 108MHz

[23 − 83]MHz σi

σi =< (V 2

i − < V 2i >)2 >

[23−83MHz] Ep Ep = 8.1016 eV Ep = 1, 6.1018 eV

d0 0 destim0 = 140, 7m σd0 = 5, 2m 0 estim0 = 381, 9µV.m−1 σ0 = 8, 5µV.m−1

σ0

0 σ0

0

σ0 : 0

MatlabTM (0, d0) (σ0 , σd0)

1000 MC

i µ = i σ = σi ±3.σi

1000 σ0 0 0 d0 0

22% 0

σd0 d0 d0 20% d0 600m d0 156m d0 2%

d0

0 ∝ Ex

p x 0 ∝ E1.03

p

0 = a.Ep + b χ2

0 Ep 30% Ep 22% 0 E1,03

p

χ2 =N

i=1

(0i − α.Epi − β)2

Ep 0 χ2 0i − α.Epi − β

V (0i − α.Epi − β) = V (0i − α.Epi)= V (0i) + α2.V (Epi) − 2.α.Cov(0i, Epi)

Epi 0i Cov(0i, Epi) = 0 (Epi, 0i) V (0i) = (σ0i)2 V (Epi) = (σEpi)2

χ2 =N

i=1

[0i − (α.Epi + β)]2

[(σ0i)2 + α2.(σEpi)2]

χ2 (Epi, 0i) 0 Ep

0 = a.Ep + b E0 Ep

E0 =0α

− β

α= a.0 + b

Ep−E0

Ep E0 Ep

Ep − E0

Ep

Ep−E0

E0

Ep E0 31% [−1, 1]

4.σ 5.σ

σ(Ep−E0

Ep) =

1

N−1

([

Epi−E0i

Epi] − µ)2

avec µ = 1N

Epi−E0i

Epi

[−1, 1]

|(v ∧ B)Est−Ouest| 0, 67 0, 40 0, 64 0, 77 0, 88

v∧ B v B

σ((EP − E0)/EP ) |(v ∧B)EO|

|(v ∧B)EO|

< |(v ∧B)EO| >= 0.4

< |(v ∧B)EO| >= 0.64 < |(v ∧B)EO| >= 0.77

< |(v ∧B)EO| >= 0.88

Ep−E0

Ep

c 0 10

√2σ2

√N−1

N ≈ 70

0 Ep |(v ∧ B)EO| + c

c c c.|sin(θ).sin(φ)| 0

0 ∼ Ep.|(v ∧ B)EO| + Ep.c.|sin(θ).sin(φ)|

E0−EpEp

1

(|(v∧ B)EO+c.|sin(θ).sin(φ)||) c |(v ∧ B)EO|

c = 10

|(v ∧ B)EO + 0, 95|

E0 E0 Ep−E0

Ep σEp

Ep

E0

(Ep, E0)

E Ep

E ±σEp σEp

Ep= 30%

E0 ±3.σE0 σE0

E0 Ep−E0

Ep

c E0 σEp

Ep= 30%

σ(Ep−E0

Ep) = 0, 34% 20% E0

2000

E0−EpEp

σ(E0)E0

5

σ(Ep)Ep

(0, 25, 0, 275, 0, 3 0, 325 0, 35) 30%

0 = (10, 9± 1, 1)[ µVm.MHz ](1

+(0, 16 ± 0, 02) − cos(α)).

cos(θ).exp( −RSA(202±64)m )(

Ep1017 eV )(0,94±0,03)

Ep = 1015,1.( 0|(v∧ B).eEW |.cos(θ) )

0,96

0|(v∧ B).eEW | = 3.10−17.Ep

0µV/m/MHz ∝ (3, 0± 0, 9)(

Ep1017 eV )0,95±0,04

Ep = (1, 10± 0, 35).1015.( 110mµV/m )1,02±0,05 eV 3

0 (|(v ∧ B).eEW |) = (sin(θ).cos(φ).cos(27) + cos(θ).sin(27))

(|(v ∧ B)|.eEW ).cos(θ) cos(θ)

30%

0

Ep−E0

Ep

20%

Xmax

km2 (10%) Xmax

Xmax

km2

3

(1865 − 1963)

km rlim = 2D2/λ D D = 300 m rlim rlim

rlim = 2D2/λ f λ

i Δtexpi Δttheoi

tmax

i Δtexpi =tmaxi − tref tref

Δttheoi Δtexpi

c Δttheoi

Δttheoi = tref − ux+ vy

c,

u = sin θ cosφ v = sin θ sinφ u v

i (Δtexpi − Δttheoi )2

Δtexpi Δttheoi Rs 3 5 10 km Δttheoi Δtexpi

10ns

ti = ts +

(xi − xs)2 + (yi − ys)2 + (zi − zs)2

c

(xi, yi, zi, ti) i (xs, ys, zs, ts) c

34 [45 − 55MHz] σt = 5ns 4

χ2 =

N

i=1

(x0 − xi)

2 + (y0 − yi)2 + (z0 − zi)

2 − c2(t0 − ti)22

χ2

(θplan, φplan) (θsphe, φsphe) 5 60

χ2 =N

i=1

((τi − τ0) − (ti − t0))

2

(σi)2

+

(1 − γ)2

(σγ)2

v = γc γ = 1 (1−γ)2

(σγ)2

τ0 χ2 13× 13 km2

1642

χ2 =

N

i=1

ti − t0 −

|| Xi − X0||c

2

Rs =

x2s + y2s + z2s

θ = arcsin(

( xsRs

)2 + ( ysRs

)2) φ = arctan(ys/xs)

Xmax

5 ri = (xi, yi, zi) (4)

ts S rs = (xs, ys, zs) ti i ∈ 1, ..., N

ti = ts +

(xi − xs)2 + (yi − ys)2 + (zi − zs)2

c+G(0, σt)

G(0, σt) 1σt

√2πe−

12 (

tσt

)2 t = 0 s

σt

3 σt = 0ns : σt = 3ns : 1ns 1m

σt = 10ns : [23 − 84]MHz

5 200m 1m

(θp, φp)

1 km 10 km φ = 0 θ = 90 1m

χ2 =N

i=1

c.(ti − ts) − (xi.u+ yi.v)

c.σt

2

u = sin(θp).cos(φp), w = sin(θp).sin(φp) ts

(θp, φp) ti rs = (xs, ys, zs) ts

f(rs, t∗s) =

1

2

N

i=1

||rs − ri||22 − (ct∗s − ct∗i )

22

fi = ||rs − ri||22 − (ct∗s − ct∗i )2

θP

f(rs, t∗s) rs ts f

TM

2 3

(∇f)k

(∇2f)k

f(x+ t.d)

f(x+ t.d, d) d

xk+1 = xk+tkdk tk dk

O(2n)

Rs 0, 1 km 20 km Rs

(> 104)

1 10 km Rvraie = 1 km Rvraie = 1 km

1 10 km

σt(ns) Rvraie(m) Rmoyen(m) Rmode(m) σR(m)

0

1000 (10071) 1002 (1081) 1081 (5763) 102

1198 1133 1477

3000

10000

3

1000

3000

10000

10

1000

3000

10000

2 Rmode Rmoyen

rs ri ts ti

t∗s t∗i t∗ = c.tσti

Xs Xi : ∇f ∇2f f

M =

1 0 0 00 1 0 00 0 1 00 0 0 −1

2

Q Li < .|. >

XT

ri = (xi, yi, zi)T

rs = (xs, ys, zs)T ts

ti

f (X) =1

2

N

i=1

−→rs −−→ri 22 − (t∗s − cti)

22

=1

2

N

i=1

f2i

σt

i = σ = constante ∀ i ∇f ∇2f

f

f f f

f f

∇f(X) ∇f(Xs) = 0

f

f (Xs) =1

2

N

i=1

f2i (Xs)

fi (Xs) = (Xs −Xi)T · M · (Xs −Xi) = −→rs −−→ri 2 − (t∗s − t∗i )

2 M f

1

2∇f (Xs) =

fi (Xs)

M ·Xs −M ·

fi (Xs)Xi

∇2f (Xs) =4M ·NXsX

Ts +

XiX

Ti −Xs

Xi

T−

Xi

XT

s

·M + 2

fi (Xs)

·M

f C∞

R4,R

R [X1, . . . , X4]

Xs = (−→rs , t∗s)T R4 −→ε =

−→h , t∗

T

R4

Ki = −→rs −−→ri 22 − (t∗s − t∗i )2 Xs

Li (−→ε ) =

−→rs −−→ri | −→h− (t∗s − t∗i ) · t∗

Q−→h , t∗

=−→h

2

2− t∗2

f (Xs + −→ε ) =1

2

i

−→rs +−→h −−→ri

2

2− (t∗0 + t∗ − t∗i )

2

2

=1

2

i

−→rs +−→h −−→ri | −→rs +

−→h −−→ri

− (t∗s + t∗ − t∗i )

22

=1

2

i

−→rs −−→ri 22 +

−→h2

2+ 2

−→rs −−→ri | −→h− (t∗s − t∗i )

2 − t∗2 − 2t∗ (t∗s − t∗i )

2

f

f−→rs +

−→h , t∗s + t∗

≈ 1

2

i

K2i + 2

i

Ki · Li

−→h , t∗

+ 2

i

L2i

−→h , t∗

+

i

Ki

·Q

−→h , t∗

12

i

K2i

∇f (Xs)T · −→ε =

2 ·

i

Ki

−→rs −−→rit∗i − t∗s

T· −→ε f

(−→rs , t∗s) Xs

1

2Q

Xs,Xi

=

i Ki + 2

ixs − xi

2 2ixs − xi

ys − yi

2ixs − xi

zs − zi

2ixs − xi

t∗i − t∗s

∗ i Ki + 2

iys − yi

2 2iys − yi

zs − zi

2iys − yi

t∗i − t∗s

∗ ∗ i Ki + 2

izs − zi

2 2izs − zi

t∗i − t∗s

∗ ∗ ∗ −i Ki + 2

i

t∗i − t∗s

2

f (−→rs , t∗s) ∗

f

f R4

∇2f (X) d R4 f

f

f ⇔

⇔ ⇔

Soit f ⇔ ∀d, ∀X, dT · ∇2f (X) · d 0

X d dT · ∇2f (X) · d < 0 f

Q dT = (0 0 0 1)

dT · ∇2f(X) · d = (0 0 0 1) ·Q(Xs, Xi) ·

0001

= −

i

Ki + 2

i

(t∗i − t∗s)2

4

Xs

ys = zs = t∗s = 0 4

i

(xs − xi)2>

i

−y2i − z2i + 3t∗2i

x2s −2

i xiN

xs +1

N

i

x2i + y2i + z2i − 3t∗i

2> 0

xs xs xs f

A = aij ∈ Mn(K) AJJ AJJ q(A(X)) =t X.A.X xj JJ x1 x1, x2x1, x2, x3

Cond(Q) = ||Q||.||Q−1|| Q

100 m

f ∇f(Xs) = 0

12∇f(Xs) = (

fi(Xs))M.Xs −M.(

fi(Xs)Xi)

∇f(Xs) = 0

Xs =

N

i=1

fi(Xs)j fj(Xs)

Xi (∗)

Mi mi

OG =N

i=1

miNj=1mj

OMi

(∗) fj

∇f(Xs) = 0

(xs, t∗s) = arg minx,t∗ (

1

2

N

i=1

(xs − xi)

2 − (t∗s − t∗i )22

) 1 i N

Contraintes de propagation : |xs − xi| = |t∗s − t∗i |Contraintes de causalite : t∗s < mini(t

∗i )

N YM ⊂ XN M < N

(xs, t

∗s) (xs, t

∗s)

3 x1 = −200m, x2 =0m, x3 = 200m xs = 60m ts = 0 s 1 2 ts = 60−xs 3 ts = −60 + xs

Xs = (xs, t∗s) f ∇f(Xs) = 0

L =

LL

Xs −L ∇f(Xs −L) = 0

3 t∗s = 60 − xs

f

∇f (xs, t∗) = 2

i

(xs − xi)(xs − xi)

2 − (t∗s − t∗i )2

i

(t∗i − t∗s)(xs − xi)

2 − (t∗s − t∗i )2

Xs

i (xs − xi)

(xs − xi)

2 − (t∗s − t∗i )2

= 0 (1)

i (t∗i − t∗s)

(xs − xi)

2 − (t∗s − t∗i )2

= 0 (2)

Xs − L

i (xs − xi − L)

(xs − xi − L)

2 − (t∗s − t∗i − L)2

= 0 (3)

i (t∗i − t∗s + L)

(xs − xi − L)

2 − (t∗s − t∗i − L)2

= 0 (4)

(3) ⇒

i (xs − xi)(xs − xi)

2 − (t∗s − t∗i )2 − 2L [(xs − xi)− (t∗s − t∗i )]

− L

i (xs − xi)

2 − (t∗s − t∗i )2

. . . + 2L2 i [(xs − xi)− (t∗s − t∗i )] = 0

⇒ −L

i (xs − xi)2 + L2

i [(xs − xi)− (t∗s − t∗i )]− L

i (xs − xi)2 − (t∗s − t∗i )

2

. . . + L

i (xs − xi) (t∗s − t∗i ) = 0

i

(xs − xi)2 − (t∗s − t∗i )

2

L

i

(xs − xi) − (t∗s − t∗i ) =

i

(xs − xi) ((xs − xi) − (t∗s − t∗i ))

xs − xi < 0 ∀i (xs − xi) − (t∗s − t∗i ) = 0 xs − xi < 0 ∀i (3)

(4) (4) ⇒

i (t∗i − t∗s)

(xs − xi)

2 − (t∗s − t∗i )2 − 2L [(xs − xi)− (t∗s − t∗i )]

+ L

i (xs − xi)

2 − (t∗s − t∗i )2

. . .− 2L2 i [(xs − xi)− (t∗s − t∗i )] = 0

⇒ −2L

i (t∗i − t∗s) (xs − xi)− 2L

i (t

∗i − t∗s)

2 + L

i (xs − xi)2 − (t∗s − t∗i )

2

. . .− 2L2 i (xs − xi)− (t∗s − t∗i ) = 0

L

i

(xs − xi) − (t∗s − t∗i ) =

i

(t∗i − t∗s) ((xs − xi) − (t∗s − t∗i ))

3 t∗s = t∗i −

(xs − xi)2 + y2s ∀ i

3

(xs, t∗s) = arg minx,t∗ (

1

2

N

i=1

((xs − xi)2 + (ys − yi)

2 − (t∗s − t∗i )2)2)

Contraintes de propagation : (xs − xi)2 + (ys − yi)

2 = (t∗s − t∗i )2

Contraintes de causalite : t∗s < mini(t∗i )

R (R, t)

Xmax 1017 eV −1018 eV

(θ, φ)

(θ, φ) 0, 1 (c2σ2

t )2 m4

Rs 0, 1−20 km

10 km Rmode Rmoyen

σt(ns) Rtrue(m) Rmean(m) Rmode(m) σR(m)0

10

300 m 50 m

c = 3.108m.s−1

1c = 3, 33ns.m−1

v < 3000m.s−1

1

v = 0, 33ms.m−1

4π steradian

Xmax

0 0 = Ep|(v ∧ B)| + Epc 20%

1016 eV

χ2

minimiser f(x)

soumis a ci(x) ≤ bi, i = 1, ...,m

x = (x1, ..., xn) f : Rn → R ci : Rn → R i = 1, ..., m b1, ..., bm xs

∀ z, c1(z) ≤ b1, ..., cm(z) ≤ bm, f(z) ≥ f(xs)

χ2 =

N

i=1

[0i − (α.Epi + β)]2

[(σ0i)2 + α2.(σEpi)2]

χ2 =N

i=1

(c.ti − c.t0 + xi.u+ yi.v

c.σi)2

x

f : Rn → R 0 ≤ α ≤ 1 x y f

f(α.x+ (1− α).y) ≤ α.f(x) + (1 − α).f(y)

f f

ζ E

∀x, y ∈ ζ : (1 − α).x+ α.y ∈ ζ, ∀α ∈ [0, 1]

E P E E P P P A, B [AB] E P

1902 3

A.x = b A cond(A) = λmax

λmin= A.A−1 . 1 : A1 =

maxjn

i=1 |aij | 2 : A2 =ρ(AtA) AFroebenius =

i,j |aij |2 aij ∞ : A∞ = maxin

j=1 |aij |

10−10 10−20

.1, .2, .∞, .Froebenuis 3000 1

λmax A λmin A ρ(A) = max|λi| A λi

.1 √.

f (Xs) =1

2

N

i=1

f2i (Xs)

fi (Xs) = (Xs −Xi)T · M · (Xs −Xi) = −→rs −−→ri 2 − (t∗s − t∗i )

2 M

∇f (Xs) =

fi (Xs) · ∇fi (Xs)

∇fi (Xs) = 2M · (Xs −Xi)

∇f

∇f (Xs) =

fi (Xs) · ∇fi (Xs)

=

fi (Xs) · 2M · (Xs −Xi)

f

1

2∇f (Xs) =

fi (Xs)

M ·Xs −M ·

fi (Xs)Xi

∇2f (Xs) =

∇fi (Xs) · ∇fi (Xs)T

+

fi (Xs) · ∇2fi (Xs)

∇2fi (Xs) = 2M (AB)

T= BTAT

∇2f (Xs) =

∇fi (Xs) · ∇fi (Xs)T

+

fi (Xs) · ∇2fi (Xs)

=

2M · (Xs −Xi) · (2M · (Xs −Xi))T

+

fi (Xs) · 2M

=4M ·

XsXTs −XsX

Ti −XiX

Ts +XiX

Ti

·M + 2

fi (X)

·M

=4M ·NXsX

Ts +

XiX

Ti −Xs

Xi

T−

Xi

XT

s

·M + 2

fi (Xs)

·M

Bibliographie[1] Physics and astrophysics of ultra high energy cosmic rays / M.Lemoine; G. Sigl (ed.) : Springer, 2001 (Lecture notes in physics ; Vol. 576).[2] P. K. F. Grieder. Extensive Air Showers High Energy Phenomena and Astrophysical Aspects, vol. 1. Springer (2010). doi :10.1016/B9780444507105/ 500038.[3] V. Hess. “Uber Beobachtungen der durchdringenden Strahlung bei sieben Freiballonfahrten. Z. Phys., 13 : p. 1084 (1912).[4] R. Ghandi et al. Astropart. Phys. 5, 81 (1996) ; Phys. Rev. D 58, 093009 (1998).[5] M. Kachelrieb et al. Phys. Rev. D 62, 103006 (2000), L. Anchordoqui et al., eprint hepph/0011097.[6] P. Bhattacharjee et al. , 2000, Phys. Rep. 327, 109.[7] E. Fermi : Phys. Rev. 75, 1169 (1949).[8] K. Kotera et al. The Astrophysics of Ultrahigh Energy Cosmic Rays. Annu. Rev. Astron. Astrophys 2010 49.[9] G. F. Krimsky : Sov. Phys. Dokl. 23, 327 (1977), A. R. Bell : Mon. Not. R. Astron. Soc. 182, 147 (1978), W. I. Axford et al. : Proc. 15th International Cosmic Ray Conference (1977) et R. D. Blandford et al. : Astrophys. J. 221, L29 (1978).[10] T. Stanev, High Energy Cosmic Rays, Second edition, Springer.[11] M.S. Longair, High Energy Astrophysics (Cambridge University Press, Cambridge) 1992.[12] W. R. Webber, The Astrophysical Journal, 506, pp. 329334, A new estimate of the local interstellar energy density and ionization rate of galactic cosmic rays, 1998.[13] The Swarthmore/Newark neutron monitor is constructed and operated by the Bartol Research Institute of the University of Delaware.[14] M. Casolino et al. (PAMELA Collaboration). Launch of the space experiment PAMELA. Advances in Space Research arXiv 0708.1808.[15] Aguilar, M. and others, „AMS on ISS, Construction of a particle physics detector on the International Space Station“ , Nucl. Instrum. Meth. 2005.[16] E. Parizot. Rayons cosmiques et rayonnement du cosmos. Mémoire d’Habilitation à diriger des recherches, Université de Paris 7.[17] The KASCADE collaboration. KASCADE measurement of energy spectra for elemental groups of cosmic rays : results and open problems. Astroparticle Physics, 24 :1, 2005.[18] K. Greisen. End of the cosmic ray spectrum ? Physical Review Letters, 16 :748, 1966.[19] V.A Kuzmin and G.T Zatsepin. Soviet physics JETP Letters, 10 :146, 1966.[20] (CREAM experiment) H. Ahn, P. Allison, M. Bagliesi, J. Beatty, G. Bigongiari, P. Boyle, J. Childers, N. Conklin, S. Coutu, M. DuVernois, et al., “The Cosmic Ray Energetics And Mass (CREAM) Instrument”, Nuclear Inst. and Methods in Physics Research A 579(2007) no. 3, 1034–1053.

137

[21] J. Blumer, R. Engel, and J. R. Horandel. Cosmic rays from the knee to the highest energies. Progress in Particle and Nuclear Physics, 63 :293–338, 2009.[22] W. Apel et al., Phys.Rev.Lett. 107, 171104 (2011).[23] A. A. Penzias and R.W.Wilson. A Measurement of Excess Antenna Temperature at 4080 Mc/s. ApJ, 142 :419–421, July 1965. 13.[24] J. L. Puget et al. Astrophys. J. 205, 638 (1976) ; L. N. Epele et al. , Phys. Rev. Lett. 81, 3295 (1998); J. High Energy Phys. 9810, 009 (1998); F. W. Stecker, Phys. Rev. Lett. 81, 3296 (1998); F. W. Stecker et al., Astrophys; J. 512, 521 (1999).[25] G. Sigl et al. Astropart. Phys. 2, 401 (1994) ; J. W. Elbert et al. , Astrophys. J. 441, 151 (1995).[26] Physics Briefing Book, Input for the Strategy Group to draft the update of the European Strategy for Particle Physics, Draft from 8 December 2012.[27] J. W. Cronin. The highestenergy cosmic rays. Nucl. Phys. Proc. Suppl. 138 (2005), 465–491.[28] A. LetessierSelvon et al. , Rev. Mod. Phys. 83, 907942 (2011). arXiv :1103.0031.[29] Takeda et al., Physical Review Letters, 81, Extension of the Cosmic Ray Energy Spectrum beyond the Predicted GreiseZatsepinKuzmin Cutoff, 1998.[30] C. C. H. Jui et al., Journal of Physics : Conf. Series 47, Result from the HIRES Experiment, 2006.[31] R. Abbasi et al. (HiRes Collaboration), 2008a, Phys. Rev. Lett. 100, 101101, arXiv :astroph/0703099.[32] J. Abraham et al. (Pierre Auger Collaboration), 2008b, Phys. Rev. Lett. 101, 061101, arXiv :0806.4302.[33] Abramowski et al. 2012, arXiv :1204.1964, Discovery of VHE γray emission and multiwavelength observations of the BL Lac object.[34] R. Maze et P. Auger. Comptes Rendus de l’Académie des sciences, 208, 1938.[35] W. Heitler, “The Quantum theory of radiation”, (1954) et J. Matthews, “A Heitler model of extensive air showers”, Astroparticle Physics 22 (2005) no. 56, 387–397.[36] O. Catalano et al. “The longitudinal EAS profile at E > 1019 eV : A comparison between GIL analytical formula and the predictions of detailed Monte Carlo simulations.”, In Proc. of 27th ICRC, Hamburg (Germany), vol. 2001, p. 498. 2001.[37] P. Abreu et al. (Pierre Auger Collaboration), Astropart. Phys. 34, (2010) 314.[38] J. Abraham et al (Pierre Auger Collaboration) Science 318 no.5852 (2007) 938. arxiv :0711.2256[39] K. H. Kampert, Proceedings of the 32nd International Cosmic Ray Conference, Aug. 1118, 2011, Beijing, arXiv :1207.4823.[40] T. AbuZayyad et al. Astrophys. J. 757, (2012) 26.[41] K.H. Kampert, State of the Collaboration and Summary of Science, Finance Board Presentation, Nova Gorcia, Septembre 2012.[42] J. Abraham et al. (Auger collaboration), Phys. Rev. Lett. 104, 091101 (2010).[43] R. U. Abbasi et al. (HiRes collaboration), Phys. Rev. Lett. (in press), available athttp ://arxiv.org/abs/0910.4184.[44] P. Abreu et al., PRL 109, (2012) 062002.

[45] B. Rossi et K. Greisen, “Cosmicray theory”, Reviews of Modern Physics 13 (1941) no. 4, 240–309.[46] The Pierre Auger Collaboration, “Correlation of the HighestEnergy Cosmic Rays with Nearby Extragalactic Objects”, Science 318 (2007) no. 5852, 938–943, astroph/0711.2256.[47] R. Ulrich et al. , Phys. Rev., 2011, D83 : 054026.[48] K., H. Kampert et al. (Pierre Auger Collaboration). Highlights from the Pierre Auger Observatory. 32nd international cosmic ray conference, Beijing 2011.[49] G. Aad, et al. (ATLAS Collaboration), Nature Comm. 2012, 2 : 463.[50] http ://www.physicstoday.org/1.2833184.[51] W. Galbraith et J. Jelley. Light pulses from the night sky associated with Cosmic Rays. 1953.[52] H.R Allan. Progress in elementary particle and cosmic ray physics. ed. by J.G. Wilson and S.A. Wouthuysen, page 169, 1971.[53] G. Askaryan. Soviet Phys. JETP, 14, (1962) 411.[54] J. Linsley, “Thickness of the particle swarm in cosmicray air showers,” Journal of Physics G : Nuclear P hysics, vol. 12, no. 1, p. 51, 1986.[55] J. Jelley, et al. , “Radio pulses from extensive cosmicray air showers”, NATURE 205 (1965) 327.[56] P. R. Barker et al. Physic Review Letter 18, 51, 1967.[57] S. N. Vernov et al. Soviet Physic JETP Letters 5, 126, 1967.[58] H. Falcke et al. (Lopes Collaboration) Detection and imaging of atmospheric radio flashes from cosmic ray air showers. Nature, 435 :313, 2005.[59] A. Haungs et al. (KASCADEGrande Collaboration) Latest Results and Perspectives of the KASCADEGrande EAS Facility. Nuclear Instruments and Methods in Physics Research Section A : Accelerators, Spectrometers, Detectors and Associated Equipment, 662(Supplement 1) 150–156, 2012. ARENA 2010.[60] H. Rottgering et al. (LOFAR Collaboration) a new radio telescope for low frequency radio observations science and project status. In Texas in Tuscany. XXI Symposium on Relativistic Astrophysics, pages 69–76, 2003.[61] A. Nelles et al. (LOFAR Collaboration). Detecting Radio Emission from Air Showers with LOFAR. Accepted for AIP Conference Proceedings, ARENA 2012, Erlangen, Germany.[62] S. Fliescher for the Pierre Auger Observatory. Radio detection of cosmic ray induced air showers at the Pierre Auger Observatory. Nuclear Instruments and Methods in Physics Research Section A : Accelerators, Spectrometers, Detectors and Associated Equipment,662(Supplement 1) :S124 – S129, 2012. ARENA 2010.[63] O. MartineauHuynh et al. (Trend Collaboration) First results of the TIANSHAN radio experiment for neutrino detection. Nuclear Instruments and Methods in Physics Research Section A : Accelerators, Spectrometers, Detectors and Associated Equipment,662(Supplement 1) :S29 – S31, 2012. ARENA 2010.[64] K. Werner et O. Scholten, “Macroscopic treatment of radio emission from cosmic ray air showers based on shower simulations”, Astroparticle Physics 29 (2008) no. 6, 393–411.[65] M. Ludwig et T. Huege. REAS3 : A revised implementation of the geosynchrotron model for radio emission from EAS. Nuclear Instruments and Methods in Physics Research, A, Article in

Press 2010.[66] V. Marin. “Radio détection des rayons cosmiques d’ultra haute énergie. Analyse, simulation et interprétation.” PhD thesis, Université de Nantes, Ecole Doctorale 3MPL, 2013.[67] T.Huege. Theory and simulations of air shower radio emission. Proceedings of the ARENA2012 conference (Erlangen, Germany), to be published in AIP Conference Proceedings.[68] M. G. Aartsen et al. (IceCube Collaboration) First observation of PeVenergy neutrinos with IceCube. (19 avril 2013) arXiv :1304.5356.[69] T. Saugrin. Radiodétection et caractérisation de l’émission radio des gerbes cosmiques d’énergie supérieure à 1016 eV avec l’expérience CODALEMA. PhD thesis, Université de Nantes, Ecole Doctorale STIM, 2008.[70] T. Garçon. Vers la radiodétection autonome des rayons cosmiques de très haute énergie. PHD thesis, Université de Nantes, Ecole Doctorale 3MPL, 2011.[71] J. D. Kraus. Antennas. McGrawHill Book Company, Inc., 1950.[72] Didier Charrier, For the CODALEMA Collaboration. Antenna development for astroparticle and radioastronomy experiments. Nucl. Ins. Methods in Physics Research Section A Volume 662, Supplement 1, 11 January 2012, Pages S142–S145.[73] Site du logiciel EZNEC payant http ://www.eznec.com/ et site du logiciel gratuit 4nec2 http ://home.ict.nl/~arivoors/[74] D. Charrier. Rapport de Stage de DESS : Conception d’un préamplificateur d’antenne à large bande et bas bruit, 2003.[75] S. Fliescher. Antenna Devices and Measurement of Radio Emission from Cosmic Ray induced Air Showers at the Pierre Auger Observatory. PhD thesis, Université de Aachen 2011.[76] J. Lamblin and the CODALEMA collaboration. Radiodetection of astronomical phenomena in the cosmic ray dedicated CODALEMA experiment. In Proceeding of the 30th International Cosmic Ray Conference, 2008.[77] M. Aglietta, et al., Nuclear Instruments and Methods in Physics Research, Section A (1993) 1, 31321. G. Agnetta, et al., Nuclear Instruments and Methods In Physics Research Section A 570 (2007) 1, 2235.[78] C. Rivière. Des signaux radio aux rayons cosmiques. PhD thesis, Université de Grenoble, Ecole Doctorale de Physique de Grenoble, 2009.[79] T. K. Gaisser : Cosmic Ray and Particle Physics, Cambridge, 1990.[80] D. Ardouin et al. (CODALEMA Collaboration). Geomagnetic origin of the radio emission from cosmic ray induced air showers observed by CODALEMA. Astroparticle Physics, 31(3) :192 – 200, 2009.[81] S. Valcarès. De la mesure des champs électriques par l’expérience CODALEMA aux caractéristiques des rayons cosmiques. PhD thesis, Université de Nantes, Ecole Doctorale STIM, 2008.[82] D. Ardouin et al. (CODALEMA Collaboration). Radioelectric field features of extensive air showers observed with CODALEMA. Astropart. Phys.26 :341350, 2006.[83] A. Rebai for the CODALEMA collaboration, Some recent results of the Codalema Experiment, Proc. of the annual meeting of the French Society of Astronomy & Astrophysics SF2A, Paris, June 2011, G. Alecian, K. Belkacem, R. Samadi and D. VallsGabaud (eds).[84] O. Ravel for the CODALEMA collaboration, NIM A, Proceedings of Arena conference,

Nantes, 2010.[85] P. A. R. Ade et al. (Planck Collaboration) (2013), 1303.5062.[86] D. Torres (CODALEMA Collaboration), Latest results of the CODALEMA experiment : cosmic rays radio detection in a self trigger mode. 2013 J. Phys. : Conf. Ser. 409 012074.[87] A. Bellétoile. Développement et analyse des données d’une expérience de radiodétection des Rayons Cosmiques d’Ultra Haute Energie. PhD thesis, Université de Nantes, Ecole doctorale STIM, 2007.[88] S. R. Vernov et al. Detection of radio emission from extensive air showers with a system of single halfwave dipoles. Can. J. Phys., 46, 1968.[89] P.R. Barker et al. , Phys. Rev. Lett., 18, 51 (1967).[90] H. R. Allan et al. Radio Pulses from Extensive Air Showers. Nature, 227 : pp. 11161118 (1970). doi :10.1038/2271116a0.[91] P. Lautridou et al. Some possible interpretations from data of the CODALEMA experiment, Proc. of the 5th international workshop on Acoustic and Radio EeV Neutrino Detection Activities (ARENA 2012), Erlangen 2012.[92] C. Glaser for the Pierre Auger Collaboration. Energy Estimation for Cosmic Rays Measured with the Auger Engineering Radio Array. 5th ARENA workshop 2012, Erlangen Germnay.[93] A. Horneffer for the LOPES Collaboration. Primary Particle Energy Calibration of the EAS Radio Pulse Height. Proceedings of the 30th International Cosmic Ray Conference ICRC 2007, Mérida México.[94] S. Knurenko, V. Kozlov. Z, Petrov, M. Pravdin. Experimental results of radio observations at the Yakutsk EAS in 2009 − 2011 (2012). Proceedings of the ECRS, Moskov Russia.[95] H. Schoorlemmer, PhD thesis, Tuning in on cosmic rays. Polarization of radio signals from air showers as a probe of emission mechanisms[96] B. G. Kristiansen, et al. Cosmic Rays of Super high Energies. Atomizdat, 1975. (Russia).[97] K. Kamata J. Nishimura. The Lateral and the Angular Structure Functions of Electron Showers. Prog. Theor. Phys. Supp., 6 : pp. 93 155 (1958). doi : 10.1143/PTPS.6.93.[98] K. Greisen. Progress in Cosmic Ray Physics. vol III , NorthHolland Publishing Company, Amsterdam (1965).[99] T. K. Gaisser : Cosmic Ray and Particle Physics, Cambridge, 1990.[100] R. Pesce for the Pierre Auger Collaboration. Energy calibration of data recorded with the surface detectors of the Pierre Auger Observatory : an update. Proceedings of 32nd ICRC 2011, Beijing China. arxiv :1107.4809.[101] P. K. F. Grieder. Extensive Air Showers High Energy Phenomena and Astrophysical Aspects, vol. 1. Springer (2010). doi :10.1016/B9780444507105/ 500038.[102] M.Risse and D. Heck. Energy Release in Air Showers, Astroparticle Phys. 20 (2004) 661.[103] T. Shibata for the Telescope Array Collaboration. Absolute energy calibration of the Telescope Array fluorescence detector with an Electron Linear Accelerator. Proceedings of 32nd ICRC 2011, Beijing China.[104] T. Abu Zayyad et al. Telescope Array Collaboration. TA EnergyScale : Methods and Photometry. Proceedings of 32nd ICRC 2011, Beijing China.[105] T. Huege. Radio Emission from Cosmic Ray Air Showers. PhD thesis, Universitat Bonn, 2004.

[106] K. Kamata J. Nishimura. The Lateral and the Angular Structure Functions of Electron Showers. Prog. Theor. Phys. Supp., 6 : pp. 93155 (1958). doi : 10.1143/PTPS.6.93.[107] K. Greisen. Progress in Cosmic Ray Physics. vol III , NorthHolland Publishing Company, Amsterdam (1965).[108] C. Rivière. Des signaux radio aux rayons cosmiques. PhD thesis, Université de Grenoble, Ecole Doctorale de Physique de Grenoble, 2009.[109] S. J. Sciutto. AIRES A system for air shower simulations. version 2.6.0[110] H. Falcke, P. W. Gorham , Astropart. Physics 19 (2003) 477494.[111] M. Du Vernois et al. Proc. of the 29th ICRC, Pune (India), vol. 8. 2005.[112] P. W. Gorham et al., arXiv :0705.2589v1 17 may 2007.[113] O. Scholten, K. Werner, and F. Rusydi, Astropart. Phys. 29, 94 (2008).[114] N. MeyerVernet et al., Astronomy & Astrophysics, 480,1525 (2008).[115] J. Chauvin et al., Astroparticle Physics 33 (2010) 341350.[116] V. Marin et al., Asptropart. Phys., ISSN 09276505, 10.1016/ j.astropartphys 2012.03.007.[117] W. D. Apel et al. (Lopes Collaboration) Astroparticle Physics 32 (2010) 294303.[118] A. Rebai for the CODALEMA collaboration, Some recent results of the CODALEMA Experiment, Proc. of the annual meeting of the French Society of Astronomy & Astrophysics SF2A, Paris, June 2011, G. Alecian, K. Belkacem, R. Samadi and D. VallsGabaud (eds).[119] D.Ardouin, et al., Astroparticle Physics 31 (2009) 192.[120] A. Rebai for the CODALEMA collaboration, Radio Detection od Ultra High Energy Cosmic Rays With The CODALEMA Experiment, African School of Physics ASP2010, Stellenbosch Afrique du sud.[121] J. Abraham et al. (Auger collaboration), Phys. Rev. Lett. 104, 091101 (2010).[122] R. U. Abbasi et al. (HiRes collaboration), Phys. Rev. Lett. (in press), available at http://arxiv.org/abs/0910.4184.[123] A. Bellétoile et al. (Codalema collaboration), Evidence for the chargeexcess contribution in air shower radio emission observed by the CODALEMA experiment, submitted to the astroparticle physics journal.[124] A. Rebai et al., arXiv:1210.1739v1 [astroph.HE] 5 Oct 2012.[125] D. Barnhill et al. for the Pierre Auger Collaboration. Measurement of the Lateral Distribution Function of UHECR Air Showers with the Pierre Auger Observatory. In Proceedings of the 29th International Cosmic Ray Conference (2005), 101–106.[126] V. Marin for the CODALEMA collaboration, Proc. of the 32nd ICRC, Beijing, 2011.[127] K.D. de Vries et al. Phys. Rev. Lett.107 :061101, 2011.[128] K. Werner et al. arXiv :1201.4471v1[astro ph.HE], 21 Jan 2012.[129] F.G. Schroder for the LOPES Collabotation, Cosmic Ray Measurements with LOPES : Status and Recent Results, arXiv :1301.2557v1 11 Jan 2013.[130] J. Abraham, et al. The fluorescence detector of the Pierre Auger Observatory. Nucl. Instrum. Methods Phys. Res., Sect. A, 620(23) : pp. 227 251 (2010).doi :10.1016/j.nima.2010.04.023.[131] M. Ludwig, Comparison of LOPES measurements with CoREAS and REAS 3.1 simulations, ARENA conference 2012 – Erlangen.[132] Uman, M. A. (2001), The Lightning Discharge, Dover Publications, Inc.

[133] D. Torres (for the CODALEMA Collaboration), Latest results of the CODALEMA experiment: cosmic rays radio detection in a self trigger mode. 2013 J. Phys.: Conf. Ser. 409 012074.[134] L. Mohrmann. Measurement of Radio Emission from Cosmic Ray induced Air Showers at the Pierre Auger Observatory with a Spherical Wave Reconstruction. Master thesis, Aachen university Physikalischen Institut A, septembre 2012.[135] D. Ardouin et al. (Trend Collaboration): First detection of extensive air showers by the TREND selftriggering radio experiment, Astropart, (34) 2011,p. 717.[136] A. Rebai et al., arXiv:1208.3539v2 [astroph.IM] 18 Sep 2012.[137] P. Lautridou (for the CODALEMA Collaboration), Nuclear Instruments and Methods in Physics Research A, Volume 692, p. 6571.[138] S. Fliescher. Antenna Devices and Measurement of Radio Emission from Cosmic Ray induced Air Showers at the Pierre Auger Observatory. PhD thesis, Aachen university Physikalischen Institut A, decembre 2012.[139] F.G. Schroder et al. (Lopes Collaboration) PhD thesis (2011).[140] F. Jedrzejewski Introduction aux méthodes numériques Deuxième édition.[141] J. P. Demailly Analyse numérique et équations différentielles.[142] J. Vallverdú. The False Dilemma: Bayesian vs. Frequentist arXiv:0804.0486.[143] D. T. Vu. Outils statistiques pour le positionnement optimal de capteurs dans le contexte de la localisation de sources. PhD thesis, Université de Paris Sud Paris XI 10/2011.[144] Peter Kall and Stein W. Wallace, stochastic programming, John Wiley & Sons, Chichester, 1994.[145] Convex Optimization, Stephen Boyd, Lieven Vandenberghe, Cambridge university press.[146] J. F. Bonnans et al. , Numerical Optimization: Theoretical and Practical Aspects, SpringerVerlag, Universitext, (second ed.) 2003.[147] J. Nocedal et S. Wright Numerical optimization springer series.[148] F. James, M. Roos, Comput. Phys. Commun. 10 (1975) 343. F. James, The Interpretation of Errors in Minuit, Cern Geneva, June 16, 2004. F. James, MINUIT Function Minimization and Error Analysis, CERN Program Library D506, Geneva, 1998.[149] J. Hadamard, Sur les équations aux dérivées partielles et leur signification physique. Bull. Univ. Princeton, 13 (1902) 4952. (OEuvres 3, 10991105).[150] F. DelpratJannaud and P. Lailly, IllPosed and WellPosed Formulation of the Reflection Travel Time Tomography Problem, J. of Geophysical Research, vol. 98, No. B4, p. 6589, April 10, 1993. (page 15).[151] R. A. Horn et C. R. Johnson, Matrix Analysis, Cambridge University Press, (first ed. 1985) 1999.[152] Lagarias, J.C., J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence Properties of the NelderMead Simplex Method in Low Dimensions," SIAM Journal of Optimization, Vol. 9 Number 1, pp. 112147, 1998.[153] Levenberg, K., "A Method for the Solution of Certain Problems in LeastSquares," Quarterly Applied Math. 2, pp. 164–168, 1944 et Marquardt, D., "An Algorithm for LeastSquares Estimation of Nonlinear Parameters," SIAM Journal Applied Math., Vol. 11, pp. 431–441, 1963.[154] Donald, K., The Art of Computer Programming. 3: Sorting and Searching (3rd ed.).

AddisonWesley. pp. 396–408. (1997) ISBN 0201896850.[155] A. Björck Numerical methods for least squares problems. SIAM, Philadelphia. (1996) ISBN 0898713609.[156] S. K. Godunov équations de la physique mathématique. Moscou : Éditions Mir.[157] T. Appelquist et al. (1987). Modern Kaluza–Klein Theories. Menlo Park, Cal.: Addison–Wesley. ISBN 0201098296.[158] Eugene, D. Is this least squares estimates?, Biometrika (2000), 87, 2, pp. 437452.[159] R. Abbasi et al. (IceCube Collaboration). The Design and Performance of IceCube DeepCore. Astroparticle Physics 35 (2012) 615624, May 2012.[160] P. Amram et al. (Antares Collaboration) Background light in potential sites for the ANTARES undersea neutrino telescope Astroparticle Physics 13 (2000) 127136 [astroph/9910170].[161] Tim Huege Theory and simulations of air shower radio emission AIP Conf. Proc. 1535, pp. 121127.[162] T. Huege et C. W. James, Full Monte Carlo simulations of radio emission from extensive air showers with CoREAS. 33rd International Cosmic Ray Conference, Rio De Janeiro 2013.[163] J.A. Muniz, et al., Monte Carlo simulations of radio pulses in atmospheric showers using ZHAireS. Astroparticle Physics, 2011.[164] N. MeyerVernet, A. Lecacheux et al. Radio pulses from cosmic ray air shower Boosted Coulomb and Cerenkov fields. Astronomy and Astrophysics.[165] J. Lamblin for the Codalema Collaboration, Radiodetection of astronomical phenomena in the cosmic ray dedicated CODALEMA experiment, 30th International Cosmic Ray Conference, Merida 2007.[166] Alain lecacheux et al., (for the CODALEMA Collaboration) Radio signature of extensive air showers observed with the Nançay Decameter Array. Proceeding of the 31st ICRC, LODZ 2009.[167] V. Marin for the CODALEMA collaboration. Charge excess signature in the CODALEMA data. Interpretation with SELFAS2. International Cosmic Ray Conference, Beijing 2011.[168] T. Huege for the Auger Collaboration, Probing the radio emission for cosmic ray induced air showers by polarization measurements. 33rd International Cosmic Ray Conference, Rio De Janeiro 2013.[169] G. steve, Finger on the pulse of cosmic rays. Dependence of the radio pulse shape on the air shower geometry. PhD thesis (2013).[170] W. D. Apel et al. (KASCADEGrande Collaboration) Nuclear Instruments and Methods in Physics Research A 620, 202 (2010).[171] P. Doll et al. Nuclear Instruments and Methods in Physics Research A 488, 517 (2002).[172] W. D. Apel et al. (LOPES Collaboration) Experimental evidence for the sensitivity of the airshower radio signal to the longitudinal shower development, Phys. Rev. D, 85.071101, 2012.

Study of the energy and the radio emission point of cosmic rays

Résumé

Le but de l’expérience CODALEMA, installée à l’bservatoire de radioastronomie de Nançay (France), est d’étudier la radiodétection des rayons cosmiques d’ultrahaute énergie. Le dispositif initial, réparti sur une aire de 0, 25 km2, utilise en coïncidence un réseau des détecteurs de particules et un réseau d’antennes. Utilisant cet appareillage, une nouvelle analyse visant à estimer l’énergie de la gerbe à partir des données radio est présentée. Nous déduisons qu’une résolution en énergie meilleure que 20% peut être atteinte et que, non seulement la force de Lorentz, mais aussi une contribution proportionnelle à la charge totale produite dans la gerbe atmosphérique joue un rôle significatif dans l’amplitude du champ électrique mesuré par les antennes (effet de cohérence ou d’excès de charge). Depuis 2011, un nouveau réseau de détection radio, constitué de 60 stations autonome auto-déclenchées est en déploiement sur une superficie de 1, 5 km2 autour du dispositif initial. Ce développement conduit à des défis spécifiques en termes d’identification des gerbes atmosphériques et de reconstruction de la courbure des fronts d’onde radio. Concernant la localisation de la source apparente de l’émission radio, pour les algorithmes de minimisation courants, nous soulignons l’importance du processus de convergence induit par la minimisation d’une fonction non linéaire des moindres carrés qui peut induire une dégénérescence des solutions. Une méthode alternative d’estimation est proposée.

Abstract

The purpose of the CODALEMA experiment, installed at the Nançay Radio Observatory (France), is to study the radio-detection of ultra-high-energy cosmic rays. Distributed over an area of 0,25 km2, the original device uses in coincidence an array of particle detectors and an array of short antennas. A new analysis of the energy reconstruction from radio data obtained with this device is presented. We suggest that an energy resolution of less than 20% can be achieved and that, not only the Lorentz force, but also another contribution proportional to all charged particles generated during the shower development, could play a significant role in the amplitude of the electric field measured by the antennas (as an effect of coherence or of charge excess). Since 2011, a new array of radio-detectors, consisting of 60 stand-alone and self-triggered stations, has been in deployment over an area of 1.5 km2 around the first device. This new development leads to specific challenges which are discussed in terms of recognition of cosmic rays and reconstruction of the curvature of radio wave fronts. For commonly-used minimization algorithms, we emphasize the importance of the convergence process induced by the minimization of a non-linear least squares function that affects the results in terms of degeneration of the solutions. We derive a simple method to obtain a satisfactory estimate of the location of the apparent emission source, which mitigates the problems previously.

L’Univesité Nantes Angers Le Mans

Etude de l’énergie et du point d’émission radio des rayonscosmiques détectés dans l’expérience CODALEMA

Ahmed REBAI

Étude de l'énergie et du point d'émission radio des rayons cosmiques détectés dans...

Education