First law Kepler's laws of planetary motion
figure 2: kepler s first law placing sun @ focus of elliptical orbit
figure 4: heliocentric coordinate system (r, θ) ellipse. shown are: semi-major axis a, semi-minor axis b , semi-latus rectum p; center of ellipse , 2 foci marked large dots. θ = 0°, r = rmin , θ = 180°, r = rmax.
mathematically, ellipse can represented formula:
r
=
p
1
+
ε
cos
θ
,
{\displaystyle r={\frac {p}{1+\varepsilon \,\cos \theta }},}
where
p
{\displaystyle p}
semi-latus rectum, ε eccentricity of ellipse, r distance sun planet, , θ angle planet s current position closest approach, seen sun. (r, θ) polar coordinates.
for ellipse 0 < ε < 1 ; in limiting case ε = 0, orbit circle sun @ centre (i.e. there 0 eccentricity).
at θ = 0°, perihelion, distance minimum
r
min
=
p
1
+
ε
{\displaystyle r_{\min }={\frac {p}{1+\varepsilon }}}
at θ = 90° , @ θ = 270° distance equal
p
{\displaystyle p}
.
at θ = 180°, aphelion, distance maximum (by definition, aphelion – invariably – perihelion plus 180°)
r
max
=
p
1
−
ε
{\displaystyle r_{\max }={\frac {p}{1-\varepsilon }}}
the semi-major axis arithmetic mean between rmin , rmax:
r
max
−
a
=
a
−
r
min
{\displaystyle \,r_{\max }-a=a-r_{\min }}
a
=
p
1
−
ε
2
{\displaystyle a={\frac {p}{1-\varepsilon ^{2}}}}
the semi-minor axis b geometric mean between rmin , rmax:
r
max
b
=
b
r
min
{\displaystyle {\frac {r_{\max }}{b}}={\frac {b}{r_{\min }}}}
b
=
p
1
−
ε
2
{\displaystyle b={\frac {p}{\sqrt {1-\varepsilon ^{2}}}}}
the semi-latus rectum p harmonic mean between rmin , rmax:
1
r
min
−
1
p
=
1
p
−
1
r
max
{\displaystyle {\frac {1}{r_{\min }}}-{\frac {1}{p}}={\frac {1}{p}}-{\frac {1}{r_{\max }}}}
p
a
=
r
max
r
min
=
b
2
{\displaystyle pa=r_{\max }r_{\min }=b^{2}\,}
the eccentricity ε coefficient of variation between rmin , rmax:
ε
=
r
max
−
r
min
r
max
+
r
min
.
{\displaystyle \varepsilon ={\frac {r_{\max }-r_{\min }}{r_{\max }+r_{\min }}}.}
the area of ellipse is
a
=
π
a
b
.
{\displaystyle a=\pi ab\,.}
the special case of circle ε = 0, resulting in r = p = rmin = rmax = = b , = πr.
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