r ( θ ) = p 1 + e cos θ {\displaystyle r(\theta )={p \over 1+e\cos \theta }}
La distance au foyer de l'ellipse est minimale quand cos θ = 1 {\displaystyle \cos \theta =1} :
r m i n = r P = p 1 + e {\displaystyle r_{min}=r_{P}={p \over 1+e}}
La distance au foyer de l'ellipse est maximale quand cos θ = − 1 {\displaystyle \cos \theta =-1} :
r m a x = r A = p 1 − e {\displaystyle r_{max}=r_{A}={p \over 1-e}}
r m i n + r m a x = 2 a = p 1 + e + p 1 − e = 2 p 1 − e 2 {\displaystyle r_{min}+r_{max}=2a={p \over 1+e}+{p \over 1-e}={2p \over 1-e^{2}}}
a = p 1 − e 2 {\displaystyle a={p \over 1-e^{2}}}
c = a − r m i n = p 1 − e 2 − p 1 + e = p 1 − e 2 − p ( 1 − e ) ( 1 + e ) ( 1 − e ) = p − p + p e 1 − e 2 {\displaystyle c=a-r_{m}in={p \over 1-e^{2}}-{p \over 1+e}={p \over 1-e^{2}}-{p(1-e) \over (1+e)(1-e)}={p-p+pe \over 1-e^{2}}}
c = p e 1 − e 2 = a e {\displaystyle c={pe \over 1-e^{2}}=a\,e}
e = c a {\displaystyle e={c \over a}}
r P = a − c = a − a e = a ( 1 − e ) {\displaystyle r_{P}=a-c=a-ae=a(1-e)}
r A = a + c = a + a e = a ( 1 + e ) {\displaystyle r_{A}=a+c=a+ae=a(1+e)}
Pour une ellipse, on a :
a 2 = b 2 + c 2 {\displaystyle a^{2}=b^{2}+c^{2}}
b 2 = a 2 − c 2 = p 2 ( 1 − e 2 ) 2 − p 2 e 2 ( 1 − e 2 ) 2 = p 2 ( 1 − e 2 ) ( 1 − e 2 ) 2 = p 2 ( 1 − e 2 ) {\displaystyle b^{2}=a^{2}-c^{2}={p^{2} \over (1-e^{2})^{2}}-{p^{2}e^{2} \over (1-e^{2})^{2}}={p^{2}(1-e^{2}) \over (1-e^{2})^{2}}={p^{2} \over (1-e^{2})}}
b = p ( 1 − e 2 ) = p a p = a p {\displaystyle b={p \over {\sqrt {(1-e^{2})}}}=p{\sqrt {a \over p}}={\sqrt {ap}}}
Vitesse aérolaire
d S d t = C 2 {\displaystyle {dS \over dt}={C \over 2}}
Pendant une période T : S = π a b = 1 2 C T {\displaystyle S=\pi \,a\,b={1 \over 2}C\,T}
T 2 = 4 π 2 a 2 b 2 C 2 = 4 π 2 a 2 a p C 2 = 4 π 2 a 3 m k {\displaystyle T^{2}={4\pi ^{2}a^{2}b^{2} \over C^{2}}={4\pi ^{2}a^{2}ap \over C^{2}}={4\pi ^{2}a^{3}m \over k}}
T = 2 π m a 3 k {\displaystyle T=2\pi {\sqrt {ma^{3} \over k}}}
T 2 a 3 = 4 π 2 m k = 4 π 2 m k {\displaystyle {T^{2} \over a^{3}}={4\pi ^{2}m \over k}={4\pi ^{2}m \over k}}
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