III. Corrigé : Applications diverses : pression, extinction

m= 3 ·1041
n= 0.1 Mpc−3= 10−19 pc−3= 3.40 ·10−69 m−3
ucin =Z∞
0n(p)E(p)dp E (p) = p2
2m
n(p) = n δ(p−mv)
ucin =Z∞
0E(p)n δ(p−mv)dp =n(mv)2
2m=n1 2mv2
P=2
3ucin =1 3n mv2= 3.4·10−18 Nm−2
2.725
urad =4σ
cT4=aT 4

a= 7.566 ·10−16 −3−4
Prad =1
3urad =1 3aT 4= 1.39 ·10−14 Nm−2
a(z)a airmass 1
a(z) = sec(z) + Ohsec(z2)i
z
z > 60◦
Iν=I0
νe−τν
ν λ
Iλ(z) = I0
λe−τλa(z)
τλI0
λ
mλ(z)−m0
λ=−2.5 log 
Iλ(z)
I0
λ
= +2.5 log(e)τλa(z)
m0
λKλ.
=
2.5 log(e)τλ= 1.086 ·τλa(z)
mλ(z) = m0
λ+Kλ·a(z)∼m0(λ) + Kλ·sec z

Kλ
U B V
mλ(z)a(z)
a(z)=0 m0
λ
M
D
mλ(a(z)) D
M
K[U]= 0.59

K[B]= 0.25
K[V]= 0.12
d
x1=dcos φsin θ , y1=dsin φsin θ , z1=dcos θ
d
N
x=PN
i=1 dcos φisin θi
y=PN
i=1 dsin φisin θi
z=PN
i=1 dcos θi
r2=x2+y2+z2=d2
N
X
i=1
cos φisin θi
2
+
N
X
i=1
sin φisin θi
2
+
N
X
i=1
cos θi
2
N
X
i=1
cos φisin θi
2
= (cos φ1sin θ1+cos φ2sin θ2+...+cos φNsin θN)2
=
N
X
i=1
cos2φisin2θi+ 2 ·X
i6=j
cos φicos φjsin θisin θj
θi
φiN

N
X
i=1
cos φisin θi
2
'
N
X
i=1
cos2φisin2θi
N
X
i=1
sin φisin θi
2
'
N
X
i=1
sin2φisin2θi
N
X
i=1
cos θi
2
'
N
X
i=1
cos2θi
r2=d2
N
X
i=1
cos2φisin2θi+
N
X
i=1
sin2φisin2θi+
N
X
i=1
cos2θi
=d2
N
X
i=1
sin2θihcos2φi+ sin2φii+ cos2θi
=Nd2
N
r=d√N
l=1
κρ =1 10·1400 ∼7·10−5
R∼7·108
N= R
l!2
=
7·108
7·10−5
2
= 1026
N
s=Nd = 1026 ·7·10−5m = 7 ·1021 m∼7·105a.l.
1 a.l.= 9.46 ·1015 m∼1013
τ=s/c ∼7·105ans ∼106ans
τ∼107