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

1m

55MHz

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

v ∧ B θ φ

27 |(v ∧ B)Est−Ouest|

|(v ∧ B)Est−Ouest| = | − sin(θ).cos(φ).cos(27) − cos(θ).sin(27)|

v v ⊥ B B

0

0|(v∧ B)EO| Ep

(|(v∧ B)EO| < 0.1) Ep

0|(v∧ B)EO| Ep

|(v ∧ B)EO| < 0, 1

c 0

0 → 0

|(v ∧ B)EO| + c

E0 = |(v ∧ B)EO|.E → E0 = |(v ∧ B)EO|.E + c.E

c |(v ∧ B)EO| 0 1

c = 0 c c.E

Ep−E0

Ep 1

(|(v∧ B)EO|+c)

c |(v ∧ B)EO|

c

c Ep−E0

Ep c

|(v ∧ B)EO| Ep−E0

Ep c = 0.95

B c c

|(v ∧ B)EO| B c = 0

|(v ∧ B)EO| c c Ep−E0

Ep

c

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

Ep−E0

Ep c |(v ∧ B)EO|

|(v ∧ B)EO| + c

v ∧ B

|(v ∧ B)EO|+ c |(v ∧ B)EO|.(1+c

|(v∧ B)EO| ) 0 |(v∧ B)EO|

0 = Ep.|(v ∧ B)EO|(1 +c

|(v ∧ B)EO|+

d

|(v ∧ B)EO|2+ ...)

c d c |(v∧ B)EO|

293

(EeV/mV ) (EeV )

1.03 ± 0.05 0.0083 ± 0.0004 |(v ∧B)EW |

c = 0

0.59 ± 0.06 0.0230 ± 0.0005

|(v ∧B)EW | + 0, 95

1.59 ± 0.09 0.0152 ± 0.003

(c = 0, 95)

(a, b) 0

E0 =a

(|(v ∧ B)EO + c|).0 + b

a 37% b 36%

(a, b)

|(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)

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

3

α p(X|α) X α X L(α) =

Ni=1 p(Xi|α) p(Xi|α)

Xi α Xi Y (X) σi L(α) =

Ni=1 exp(− 1

2 .(Yi−Y (Xi)

σi)2).dY

95% 68% 99.7%, χ2

α p(α|X) X p(α|X) = p(X|α).p(α)

p(X) p(α) α X

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

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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