12 ORBITAL MOTION 12.6 Planetary orbits
represents the position of the Sun. The lines SP and SPJ are both approximately
of length r. Moreover, using simple trigonometry, the line PP J is of length r δθ,
where δθ is the small angle through which the line joining the Sun and the planet
rotates in the time interval δt. The area of the triangle PSP J is approximately
1
δA = × r δθ × r : (12.34)
2
i.e., half its base times its height. Of course, this area represents the area swept
out by the line joining the Sun and the planet in the time interval δt. Hence, the
rate at which this area is swept is given by
limδA 1
= r^2
limδθ r^2 θ ̇
= =hδt (^0) δt 2 δt 0 δt 2 2.^ (12.35)^
Clearly, the fact that h is a
→
constant of th
→
e motion implies that the line joining the
planet and the Sun sweeps out area at a constant rate: i.e., the line sweeps equal
areas in equal time intervals. But, this is just Kepler’s second law. We conclude
that Kepler’s second law of planetary motion is a direct manifestation of angular
momentum conservation.
Let
r =
1
, (12.36)
u
where u(t) ≡ u(θ) is a new radial variable. Differentiating with respect to t, we
obtain
̇r = −
u ̇
θ
̇ (^) du
= −
= −h
du
. (12.37)
u^2 u^2 dθ dθ
The last step follows from the fact that θ ̇^
with respect to t, we obtain
= h u^2. Differentiating a second timëd
du!(^) ̇ d^2 u 2 2 d^2 u
(^)
(^) (12.38)
r = −h
dt dθ =^ −h^ θ^ dθ 2 =^ −h^ u^
dθ^2
.
Equations (12.27) and (12.38) can be combined to give
d^2 u
dθ^2
- u =
G MⓈ
h^2
. (12.39)