Second law Kepler's laws of planetary motion

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.}






^ cite error: named reference wolfram2nd invoked never defined (see page).