Satellites
et planètes
I. Mouvement circulaire uniforme
1. Définition
On dit qu’un
solide a une mouvement circulaire uniforme si sa trajectoire est un cercle et
si la valeur de sa vitesse est constante.
Remarque : Si le mouvement d’un mobile est
circulaire, il est possible de repérer sa position par son abscisse angulaire θ ou par son abscisse curviligne s (voir schéma
ci-contre).
2. Vitesse
Soit
s
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaam4Caaaa@329E@
l’abscisse curviligne du mobile à l’instant
t
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamiDaaaa@329F@
et soit
s'=s+Δs
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaam4CaiaacEcacqGH9aqpcaWGZbGaey4kaSIaeuiLdqKaam4Caaaa@3887@
l’abscisse curviligne du mobile à l’instant
t'=t+Δt
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamiDaiaacEcacqGH9aqpcaWG0bGaey4kaSIaeuiLdqKaamiDaaaa@388A@
.
|
La
vitesse du mobile s’écrit
v=
lim
Δt→0
Δs
Δt
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamODaiabg2da9maaxababaGaciiBaiaacMgacaGGTbaaleaacqqHuoarcaWG0bGaeyOKH4QaaGimaaqabaGcdaWcaaqaaiabfs5aejaadohaaeaacqqHuoarcaWG0baaaaaa@408D@
<=>
|
v=
ds
dt
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamODaiabg2da9maalaaabaGaamizaiaadohaaeaacaWGKbGaamiDaaaaaaa@377A@
|
Remarques :
· Le vecteur vitesse
v
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaaaaa@32B3@
est tangent à la trajectoire.
· Le vecteur vitesse
v
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaaaaa@32B3@
n’est pas constant
car sa direction n’est pas constante.
Repère
de Fresnet :
· Soit un vecteur unitaire
τ
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGafqiXdqNbaSaaaaa@337D@
orienté dans le sens positif de la tangente à
la trajectoire.
· Soit un vecteur unitaire
n
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmOBayaalaaaaa@32AB@
normal à la trajectoire et orienté vers le
centre O de celle-ci.
(
τ
→
,
n
→
)
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaaiikaiqbes8a0zaalaGaaiilaiqad6gagaWcaiaacMcaaaa@368B@
est appelé « repère de Fresnet ».
Dans le
repère de Fresnet, on peut écrire :
Remarque :
|
s=r.θ
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWGZbGaeyypa0JaamOCaiaac6cacqaH4oqCaaa@32C7@
=>
|
v=
d(r.θ)
dt
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWG2bGaeyypa0ZaaSaaaeaacaWGKbGaaiikaiaadkhacaGGUaGaeqiUdeNaaiykaaqaaiaadsgacaWG0baaaaaa@36FE@
|
|
=>
|
v=
r.d(θ)
dt
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWG2bGaeyypa0ZaaSaaaeaacaWGYbGaaiOlaiaadsgacaGGOaGaeqiUdeNaaiykaaqaaiaadsgacaWG0baaaaaa@36FE@
|
|
=>
|
|
avec {
v : vitesse du mobile (en m
.s
-1
)
r: rayon de la trajectoire (en m)
ω: vitesse angulaire du mobile (en rad
.s
-1
)
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@8FEA@
|
|
v=r.ω
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWG2bGaeyypa0JaamOCaiaac6cacqaHjpWDaaa@32E1@
|
|
|
3. Vecteur
accélération
On admettra que dans un mouvement
circulaire le vecteur accélération est normal
à la trajectoire et de valeur
a=
v
2
r
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWGHbGaeyypa0ZaaSaaaeaacaWG2bWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOCaaaaaaa@324B@
.
On en déduit :
|
a
→
=
v
2
r
n
→
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaaceWGHbGbaSaacqGH9aqpdaWcaaqaaiaadAhadaahaaWcbeqaaiaaikdaaaaakeaacaWGYbaaaiqad6gagaWcaaaa@3361@
|
Remarque n°1: De façon générale, dans le repère
de Fresnet, le vecteur accélération a pour expression
a
→
=
v
2
r
n
→
+
dv
dt
τ
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaaceWGHbGbaSaacqGH9aqpdaWcaaqaaiaadAhadaahaaWcbeqaaiaaikdaaaaakeaacaWGYbaaaiqad6gagaWcaiabgUcaRmaalaaabaGaamizaiaadAhaaeaacaWGKbGaamiDaaaacuaHepaDgaWcaaaa@39F1@
.
Dans le cas d’un mouvement uniforme, la valeur de la vitesse est constante et
dv
dt
=0
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWcaaqaaiaadsgacaWG2baabaGaamizaiaadshaaaGaeyypa0JaaGimaaaa@3300@
.
Le vecteur accélération s’écrit alors
a
→
=
v
2
r
n
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaaceWGHbGbaSaacqGH9aqpdaWcaaqaaiaadAhadaahaaWcbeqaaiaaikdaaaaakeaacaWGYbaaaiqad6gagaWcaaaa@3362@
.
Remarque n°2: Le mouvement est périodique de
période de révolution :
|
T=
2πr
v
=
2π
ω
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWGubGaeyypa0ZaaSaaaeaacaaIYaGaeqiWdaNaamOCaaqaaiaadAhaaaGaeyypa0ZaaSaaaeaacaaIYaGaeqiWdahabaGaeqyYdChaaaaa@391F@
|
II. Mouvement des planètes autour du Soleil
1. Loi
de gravitation universelle
Deux
corps A et B, de masses respectives
m
A
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWGTbWaaSbaaSqaaiaadgeaaeqaaaaa@2F4E@
et
m
B
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWGTbWaaSbaaSqaaiaadkeaaeqaaaaa@2F4F@
,
à répartition sphérique de masse (en abrégé RSDM), sont soumis à des forces
d’attraction :
|
F
A/B
→
=−
F
B/A
→
=−
G.
m
A
.
m
B
r
2
.
u
AB
→
avec {
F
A/B
=
F
B/A
: forces d'attractions existant entre les corps A et B (en N)
r: distance séparant les centres des corps A et B (en m)
G: constante de gravitation universelle (G=6,67×
10
-11
N.
m
2
.k
g
−2
)
u
AB
→
: vecteur unitaire orienté de A vers B
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@18CF@
|
2. Etude
du mouvement d’une planète
Système
étudié : la
planète de masse
m
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWGTbaaaa@2E5C@
.
Force
extérieure appliquée au système: la force d’attraction gravitationnelle
F
→
=
−G.m.
M
S
r
2
.
u
SP
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaaceWGgbGbaSaacqGH9aqpdaWcaaqaaiabgkHiTiaadEeacaGGUaGaamyBaiaac6cacaWGnbWaaSbaaSqaaiaadofaaeqaaaGcbaGaamOCamaaCaaaleqabaGaaGOmaaaaaaGccaGGUaWaa8HaaeaacaWG1bWaaSbaaSqaaiaadofacaWGqbaabeaaaOGaay51Gaaaaa@3C79@
exercée par le Soleil (de masse
M
S
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWGnbWaaSbaaSqaaiaadofaaeqaaaaa@2F40@
) sur la planète.
Référentiel : héliocentrique supposé galiléen par
approximation.
|
D’après
la deuxième loi de Newton,
F
→
=m.
a
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWhcaqaaiaadAeaaiaawEniaiabg2da9iaad2gacaGGUaWaa8HaaeaacaWGHbaacaGLxdcaaaa@352D@
=>
|
−G.m.
M
S
r
2
.
u
SP
→
=m.
a
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWcaaqaaiabgkHiTiaadEeacaGGUaGaamyBaiaac6cacaWGnbWaaSbaaSqaaiaadofaaeqaaaGcbaGaamOCamaaCaaaleqabaGaaGOmaaaaaaGccaGGUaWaa8HaaeaacaWG1bWaaSbaaSqaaiaadofacaWGqbaabeaaaOGaay51GaGaeyypa0JaamyBaiaac6caceWGHbGbaSaaaaa@3E38@
|
|
=>
|
a
→
=
−G.
M
S
r
2
.
u
SP
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaaceWGHbGbaSaacqGH9aqpdaWcaaqaaiabgkHiTiaadEeacaGGUaGaamytamaaBaaaleaacaWGtbaabeaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaOGaaiOlamaaFiaabaGaamyDamaaBaaaleaacaWGtbGaamiuaaqabaaakiaawEniaaaa@3AF0@
|
Le
vecteur accélération de la planète est donc radial (dirigé vers le centre du Soleil).
Lorsque
le vecteur accélération est radial, le mouvement circulaire uniforme est l’une
des solutions possibles. En réalité, le mouvement des planètes est elliptique
avec une faible excentricité.
Dans la
suite du cours, nous considérerons que le mouvement des planètes est
circulaire. Dans ce cas, le vecteur accélération est centripète (dirigé vers le
centre de la trajectoire).
L’accélération
tangentielle est alors nulle (
a
τ
→
=
0
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWhcaqaaiaadggadaWgaaWcbaGaeqiXdqhabeaaaOGaay51GaGaeyypa0JabGimayaalaaaaa@33D1@
) et d’après le paragraphe I. 3., on peut écrire
dv
dt
=0⇒v=cte
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWcaaqaaiaadsgacaWG2baabaGaamizaiaadshaaaGaeyypa0JaaGimaiabgkDiElaadAhacqGH9aqpcaWGJbGaamiDaiaadwgaaaa@3A28@
.
Le mouvement est uniforme et le vecteur accélération est normal (
a
→
=
v
0
2
r
n
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmyyayaalaGaeyypa0ZaaSaaaeaacaWG2bWaa0baaSqaaiaaicdaaeaacaaIYaaaaaGcbaGaamOCaaaadaWhcaqaaiaad6gaaiaawEniaaaa@39FA@
).
Vitesse
de la planète :
La valeur de la vitesse étant constante, nous la noterons
v
0
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIWaaabeaaaaa@3387@
dans la suite.
|
Le
vecteur accélération est normal :
a
→
=
v
0
2
r
n
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmyyayaalaGaeyypa0ZaaSaaaeaacaWG2bWaa0baaSqaaiaaicdaaeaacaaIYaaaaaGcbaGaamOCaaaadaWhcaqaaiaad6gaaiaawEniaaaa@39FA@
=>
|
−G.
M
S
r
2
.
u
SP
→
=
v
0
2
r
.
n
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWcaaqaaiabgkHiTiaadEeacaGGUaGaamytamaaBaaaleaacaWGtbaabeaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaOGaaiOlamaaFiaabaGaamyDamaaBaaaleaacaWGtbGaamiuaaqabaaakiaawEniaiabg2da9maalaaabaGaamODamaaDaaaleaacaaIWaaabaGaaGOmaaaaaOqaaiaadkhaaaGaaiOlaiqad6gagaWcaaaa@3F5E@
|
|
=>
|
−G.
M
S
r
2
.
u
SP
→
=
−
v
0
2
r
.
u
SP
→
(car
n
→
=−
u
SP
→
)
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWcaaqaaiabgkHiTiaadEeacaGGUaGaamytamaaBaaaleaacaWGtbaabeaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaOGaaiOlamaaFiaabaGaamyDamaaBaaaleaacaWGtbGaamiuaaqabaaakiaawEniaiabg2da9maalaaabaGaeyOeI0IaamODamaaDaaaleaacaaIWaaabaGaaGOmaaaaaOqaaiaadkhaaaGaaiOlamaaFiaabaGaamyDamaaBaaaleaacaWGtbGaamiuaaqabaaakiaawEniaiaabccacaqGOaGaae4yaiaabggacaqGYbGaaeiiaiqad6gagaWcaiabg2da9iabgkHiTmaaFiaabaGaamyDamaaBaaaleaacaWGtbGaamiuaaqabaaakiaawEniaiaabMcaaaa@50BC@
|
|
=>
|
G.
M
S
r
2
=
v
0
2
r
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWcaaqaaiaadEeacaGGUaGaamytamaaBaaaleaacaWGtbaabeaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaOGaeyypa0ZaaSaaaeaacaWG2bWaa0baaSqaaiaaicdaaeaacaaIYaaaaaGcbaGaamOCaaaaaaa@3777@
|
|
=>
|
v
0
=
G.
M
S
r
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIWaaabeaakiabg2da9maakaaabaWaaSaaaeaacaWGhbGaaiOlaiaad2eadaWgaaWcbaGaam4uaaqabaaakeaacaWGYbaaaaWcbeaaaaa@3917@
|
|
|
|
On
remarquera que la vitesse ne dépend pas
de la masse de la planète.
|
Période
de révolution :
T=
2πr
v
0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamivaiabg2da9maalaaabaGaaGOmaiabec8aWjaadkhaaeaacaWG2bWaaSbaaSqaaiaaicdaaeqaaaaaaaa@38E5@
=>
|
T=
2πr
G.
M
S
r
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamivaiabg2da9maalaaabaGaaGOmaiabec8aWjaadkhaaeaadaGcaaqaamaalaaabaGaam4raiaac6cacaWGnbWaaSbaaSqaaiaadofaaeqaaaGcbaGaamOCaaaaaSqabaaaaaaa@3B85@
|
|
=>
|
T
2
=
4
π
2
r
3
G.
M
S
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamivamaaCaaaleqabaGaaGOmaaaakiabg2da9maalaaabaGaaGinaiabec8aWnaaCaaaleqabaGaaGOmaaaakiaadkhadaahaaWcbeqaaiaaiodaaaaakeaacaWGhbGaaiOlaiaad2eadaWgaaWcbaGaam4uaaqabaaaaaaa@3D35@
|
|
=>
|
T=2π
r
3
G.
M
S
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamivaiabg2da9iaaikdacqaHapaCdaGcaaqaamaalaaabaGaamOCamaaCaaaleqabaGaaG4maaaaaOqaaiaadEeacaGGUaGaamytamaaBaaaleaacaWGtbaabeaaaaaabeaaaaa@3B5D@
|
La période ne dépend pas de la masse
de la planète.
On
remarquera que
T
2
=
4
π
2
r
3
G.
M
S
⇔
T
2
r
3
=
4
π
2
G.
M
S
=cte
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamivamaaCaaaleqabaGaaGOmaaaakiabg2da9maalaaabaGaaGinaiabec8aWnaaCaaaleqabaGaaGOmaaaakiaadkhadaahaaWcbeqaaiaaiodaaaaakeaacaWGhbGaaiOlaiaad2eadaWgaaWcbaGaam4uaaqabaaaaOGaeyi1HS9aaSaaaeaacaWGubWaaWbaaSqabeaacaaIYaaaaaGcbaGaamOCamaaCaaaleqabaGaaG4maaaaaaGccqGH9aqpdaWcaaqaaiaaisdacqaHapaCdaahaaWcbeqaaiaaikdaaaaakeaacaWGhbGaaiOlaiaad2eadaWgaaWcbaGaam4uaaqabaaaaOGaeyypa0Jaam4yaiaadshacaWGLbaaaa@4F15@
.
On retrouve la 3ème loi de Képler (voir paragraphe V.)
III. Mouvement d’un satellite autour de la Terre
La
raisonnement est identique :
Système
étudié : le
satellite de masse
m
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWGTbaaaa@2E5C@
.
Force
extérieure appliquée au système: la force d’attraction gravitationnelle
F
→
=
−G.m.
M
T
r
2
.
u
TS
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaaceWGgbGbaSaacqGH9aqpdaWcaaqaaiabgkHiTiaadEeacaGGUaGaamyBaiaac6cacaWGnbWaaSbaaSqaaiaadsfaaeqaaaGcbaGaamOCamaaCaaaleqabaGaaGOmaaaaaaGccaGGUaWaa8HaaeaacaWG1bWaaSbaaSqaaiaadsfacaWGtbaabeaaaOGaay51Gaaaaa@3C7E@
exercée par la Terre (de masse
M
T
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaacaWGnbWaaSbaaSqaaiaadsfaaeqaaaaa@2F41@
) sur le satellite.
Référentiel : géocentrique supposé galiléen par
approximation.
Remarque
préliminaire :
F
→
=
−G.m.
M
T
r
2
.
u
TS
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaaceWGgbGbaSaacqGH9aqpdaWcaaqaaiabgkHiTiaadEeacaGGUaGaamyBaiaac6cacaWGnbWaaSbaaSqaaiaadsfaaeqaaaGcbaGaamOCamaaCaaaleqabaGaaGOmaaaaaaGccaGGUaWaa8HaaeaacaWG1bWaaSbaaSqaaiaadsfacaWGtbaabeaaaOGaay51Gaaaaa@3C7E@
<=>
F
→
=
G.m.
M
T
r
2
.
n
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaaceWGgbGbaSaacqGH9aqpdaWcaaqaaiaadEeacaGGUaGaamyBaiaac6cacaWGnbWaaSbaaSqaaiaadsfaaeqaaaGcbaGaamOCamaaCaaaleqabaGaaGOmaaaaaaGccaGGUaGabmOBayaalaaaaa@3801@
car
n
→
=−
u
ST
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmOBayaalaGaeyypa0JaeyOeI0Yaa8HaaeaacaWG1bWaaSbaaSqaaiaadofacaWGubaabeaaaOGaay51Gaaaaa@3933@
|
D’après
la deuxième loi de Newton,
F
→
=m.
a
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWhcaqaaiaadAeaaiaawEniaiabg2da9iaad2gacaGGUaWaa8HaaeaacaWGHbaacaGLxdcaaaa@352D@
=>
|
G.m.
M
T
r
2
.
n
→
=m.
a
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWcaaqaaiaadEeacaGGUaGaamyBaiaac6cacaWGnbWaaSbaaSqaaiaadsfaaeqaaaGcbaGaamOCamaaCaaaleqabaGaaGOmaaaaaaGccaGGUaGabmOBayaalaGaeyypa0JaamyBaiaac6caceWGHbGbaSaaaaa@39C0@
|
|
=>
|
a
→
=
G.
M
T
r
2
.
n
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaaceWGHbGbaSaacqGH9aqpdaWcaaqaaiaadEeacaGGUaGaamytamaaBaaaleaacaWGubaabeaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaOGaaiOlaiqad6gagaWcaaaa@3678@
|
Le
vecteur accélération du satellite est radial (et centripète). L’accélération
tangentielle est donc nulle :
a
τ
→
=
0
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWhcaqaaiaadggadaWgaaWcbaGaeqiXdqhabeaaaOGaay51GaGaeyypa0JabGimayaalaaaaa@33D1@
et
dv
dt
=0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWcaaqaaiaadsgacaWG2baabaGaamizaiaadshaaaGaeyypa0JaaGimaaaa@32FF@
.
La vitesse est constante et le mouvement est uniforme.
|
Vitesse
du satellite:
a
→
=
v
0
2
r
n
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmyyayaalaGaeyypa0ZaaSaaaeaacaWG2bWaa0baaSqaaiaaicdaaeaacaaIYaaaaaGcbaGaamOCaaaadaWhcaqaaiaad6gaaiaawEniaaaa@39FA@
=>
|
G.
M
T
r
2
.
n
→
=
v
0
2
r
.
n
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWcaaqaaiaadEeacaGGUaGaamytamaaBaaaleaacaWGubaabeaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaOGaaiOlaiqad6gagaWcaiabg2da9maalaaabaGaamODamaaDaaaleaacaaIWaaabaGaaGOmaaaaaOqaaiaadkhaaaGaaiOlaiqad6gagaWcaaaa@3AE6@
|
|
=>
|
G.
M
T
r
2
=
v
0
2
r
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaadaWcaaqaaiaadEeacaGGUaGaamytamaaBaaaleaacaWGubaabeaaaOqaaiaadkhadaahaaWcbeqaaiaaikdaaaaaaOGaeyypa0ZaaSaaaeaacaWG2bWaa0baaSqaaiaaicdaaeaacaaIYaaaaaGcbaGaamOCaaaaaaa@3778@
|
|
=>
|
v
0
=
G.
M
T
r
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIWaaabeaakiabg2da9maakaaabaWaaSaaaeaacaWGhbGaaiOlaiaad2eadaWgaaWcbaGaamivaaqabaaakeaacaWGYbaaaaWcbeaaaaa@3918@
|
|
=>
|
v
0
=
G.
M
T
R
T
+h
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamODamaaBaaaleaacaaIWaaabeaakiabg2da9maakaaabaWaaSaaaeaacaWGhbGaaiOlaiaad2eadaWgaaWcbaGaamivaaqabaaakeaacaWGsbWaaSbaaSqaaiaadsfaaeqaaOGaey4kaSIaamiAaaaaaSqabaaaaa@3BD6@
|
|
|
|
La vitesse ne dépend pas de la masse
du satellite.
|
Période
de révolution :
T=
2π(
R
T
+h)
v
0
MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamivaiabg2da9maalaaabaGaaGOmaiabec8aWjaacIcacaWGsbWaaSbaaSqaaiaadsfaaeqaaOGaey4kaSIaamiAaiaacMcaaeaacaWG2bWaaSbaaSqaaiaaicdaaeqaaaaaaaa@3CFC@
=>
|
T=2π
(
R
T
+h)
3
G.
M
T
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamivaiabg2da9iaaikdacqaHapaCdaGcaaqaamaalaaabaGaaiikaiaadkfadaWgaaWcbaGaamivaaqabaGccqGHRaWkcaWGObGaaiykamaaCaaaleqabaGaaG4maaaaaOqaaiaadEeacaGGUaGaamytamaaBaaaleaacaWGubaabeaaaaaabeaaaaa@3F75@
|
La période ne dépend pas de la masse
du satellite.
Remarque : Un satellite géostationnaire a une
position fixe par rapport au référentiel terrestre. Il tourne dans le plan de
l’équateur, dans le même sens que la rotation de la Terre. Sa période de
révolution est égale au jour sidéral soit
T
0
=23h56min=86400s
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamivamaaBaaaleaacaaIWaaabeaakiabg2da9iaaikdacaaIZaGaamiAaiaaiwdacaaI2aGaciyBaiaacMgacaGGUbGaeyypa0JaaGioaiaaiAdacaaI0aGaaGimaiaaicdacaWGZbaaaa@40DE@
.
Le calcul de son altitude donne
h=35800km (h≈36000km)
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamiAaiabg2da9iaaiodacaaI1aGaaGioaiaaicdacaaIWaGaam4Aaiaad2gacaqGGaGaaiikaiaadIgacqGHijYUcaaIZaGaaGOnaiaaicdacaaIWaGaaGimaiaadUgacaWGTbGaaiykaaaa@4354@
.
IV. Lois de Képler
Première loi de Képler (loi des
orbites)
Dans le
référentiel héliocentrique, la trajectoire du centre d’une planète est un
ellipse dont le centre du Soleil est l’un des foyers.
Deuxième loi de Képler (loi des aires)
Le
vecteur
SP
→
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaa8HaaeaacaWGtbGaamiuaaGaay51Gaaaaa@3507@
qui relie le centre du Soleil à celui de la
planète balaie des aires égales pendant des durées égales.
Troisième loi de Képler (loi des périodes)
Le
rapport
T
2
a
3
MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaWaaSaaaeaacaWGubWaaWbaaSqabeaacaaIYaaaaaGcbaGaamyyamaaCaaaleqabaGaaG4maaaaaaaaaa@3552@
est constant (voir paragraphe II.2.)
