Simple Nature - Light and Matter

ad/Quantities referred to in
the proof of part (3).

momentum tells us thatL =mrvsinφis a constant. Since the

planet’s mass is a constant, this is the same as the condition

rvsinφ= constant.

Conservation of energy gives

1

2

mv^2 −G

Mm

r

= constant.

We solve the first equation forvand plug into the second equa-

tion to eliminatev. Straightforward algebra then leads to the equa-

tion claimed above, with the constantpbeing positive because of

our assumption that the planet’s energy is insufficient to escape from

the sun, i.e., its total energy is negative.

Proof of part (3)

We define the quantitiesα,d, andsas shown in figure ad. The

law of cosines gives

d^2 =r^2 +s^2 − 2 rscosα.

Usingα= 180◦− 2 φand the trigonometric identities cos(180◦−x) =

−cosxand cos 2x= 1−2 sin^2 x, we can rewrite this as

d^2 =r^2 +s^2 − 2 rs

(

2 sin^2 φ− 1

)

.

Straightforward algebra transforms this into

sinφ=

√

(r+s)^2 −d^2

4 rs

.

Sincer+sis constant, the top of the fraction is constant, and the

denominator can be rewritten as 4rs= 4r(constant−r), which is

equivalent to the desired form.

270 Chapter 4 Conservation of Angular Momentum