Satellites et planètes

 

Le télescope spatial Hubble

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 MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqabeaadaabauaaaOqaaiabeI7aXbaa@3983@  ou par son abscisse curviligne s (voir schéma ci-dessous).

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 Δt0 Δs Δt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipeea0xXdHaVhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaWaaeaaeaaakeaacaWG2bGaeyypa0ZaaCbeaeaaciGGSbGaaiyAaiaac2gaaSqaaiabfs5aejaadshacqGHsgIRcaaIWaaabeaakmaalaaabaGaeuiLdqKaam4Caaqaaiabfs5aejaadshaaaaaaa@40AF@  <=>

v= ds dt MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGaamODaiabg2da9maalaaabaGaamizaiaadohaaeaacaWGKbGaamiDaaaaaaa@377A@

Remarques :

·Â  Le vecteur vitesse v MathType@MTEF@5@5@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqee0evGueE0jxyaibaieYdNi=BH8wipC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=xfr=xb9adbaqaaeaacaGaaiaabaqaamaabaabaaGcbaGabmODayaalaaaaa@32B3@  est tangent à la trajectoire.

·Â  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 Frenet :

·Â  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.

·Â  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 ) est appelé  "repère de Frenet" MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgaruWqVvNCPvMCG4uz3bqefqvATv2CG4uz3bIuV1wyUbqee0evGueE0jxyaibaieYdf9irVeeu0dXdh9vqqj=hEeeu0xXdbba9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXdbPYxe9vr0=vr0=vqpWqaaeaabaGaaiaacaqaaeaadaabauaaaOqaaiaacIcacuaHepaDgaWcaiaacYcaceWGUbGbaSaacaGGPaGaaeiiaiaabwgacaqGZbGaaeiDaiaabccacaqGHbGaaeiCaiaabchacaqGLbGaaeiBaiaabMoacaqGGaGaaeiiaiaabkcacaqGYbGaaeyzaiaabchacaqGOdGaaeOCaiaabwgacaqGGaGaaeizaiaabwgacaqGGaGaaeOraiaabkhacaqGLbGaaeOBaiaabwgacaqG0bGaaeOiaaaa@580B@ .

Dans le repère de Frenet, on peut écrire :

Remarque: s=r.θ v= d(r.θ) dt v= r.d(θ) dt v=r.ω avec 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@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=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@C3A3@

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@+=feaafiart1ev1aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgarmWu51MyVXgarqqtubsr4rNCHbGeaGqipCI8VfYBH8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaaiaacaGaaeaabaqaaeaadaaakeaaceWGHbGbaSaacqGH9aqpdaWcaaqaaiaadAhadaahaaWcbeqaaiaaikdaaaaakeaacaWGYbaaaiqad6gagaWcaaaa@3362@

Remarque n°1: De façon générale, dans le repère de Frenet, 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@

Le système solaire (vue d'artiste)

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 =0v=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 (h36000km) 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.)