Formulary Kepler's laws of planetary motion

1 formulary

1.1 first law
1.2 second law
1.3 third law

formulary

the mathematical model of kinematics of planet subject laws allows large range of further calculations.

first law

the orbit of every planet ellipse sun @ 1 of 2 foci.

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.

second law

a line joining planet , sun sweeps out equal areas during equal intervals of time.

the same (blue) area swept out in fixed time period. green arrow velocity. purple arrow directed towards sun acceleration. other 2 purple arrows acceleration components parallel , perpendicular velocity.

the orbital radius , angular velocity of planet in elliptical orbit vary. shown in animation: planet travels faster when closer sun, slower when farther sun. kepler s second law states blue sector has constant area.

in small time



d
t



{\displaystyle dt\,}

planet sweeps out small triangle having base line



r



{\displaystyle r\,}

, height



r

d
θ


{\displaystyle r\,d\theta }

, area



d
a
=



1
2



⋅
r
⋅
r
d
θ


{\displaystyle da={\tfrac {1}{2}}\cdot r\cdot rd\theta }

, constant areal velocity






d
a


d
t



=



1
2




r

2





d
θ


d
t



.


{\displaystyle {\frac {da}{dt}}={\tfrac {1}{2}}r^{2}{\frac {d\theta }{dt}}.}

the area enclosed elliptical orbit



π
a
b
.



{\displaystyle \pi ab.\,}

period



p



{\displaystyle p\,}

satisfies

p
⋅



1
2




r

2





d
θ


d
t



=
π
a
b


{\displaystyle p\cdot {\tfrac {1}{2}}r^{2}{\frac {d\theta }{dt}}=\pi ab}

and mean motion of planet around sun

n
=
2
π

/

p


{\displaystyle n=2\pi /p}

satisfies

r

2



d
θ
=
a
b
n

d
t
.


{\displaystyle r^{2}\,d\theta =abn\,dt.}



third law

the square of orbital period of planet directly proportional cube of semi-major axis of orbit.

this captures relationship between distance of planets sun, , orbital periods.

kepler enunciated in 1619 third law in laborious attempt determine viewed music of spheres according precise laws, , express in terms of musical notation. known harmonic law.

according law expression pa has same value planets in solar system. here p time taken planet complete orbit round sun, , mean value between maximum , minimum distances between planet , sun.

the corresponding formula in newtonian mechanics is

p

2



a

3




=



4

π

2




g
(
m
+
m
)



≈



4

π

2




g
m



=

c
o
n
s
t
a
n
t



{\displaystyle {\frac {p^{2}}{a^{3}}}={\frac {4\pi ^{2}}{g(m+m)}}\approx {\frac {4\pi ^{2}}{gm}}=\mathrm {constant} }

where m mass of sun, m mass of planet, , g gravitational constant.

as sun more massive planet, kepler s third law approximately correct in newtonian mechanics.