Simple Nature - Light and Matter

momentum tells us that L = mrv sin φ is a constant. Since theplanet’s mass is a constant, this is the same as the conditionrv sin φ = constant .Conservation of energy gives12 mv2 − G Mm = constant .rWe solve the first equation for v and plug into the second equationto eliminate v. Straightforward algebra then leads to the equationclaimed above, with the constant p being positive because ofour assumption that the planet’s energy is insufficient to escape fromthe sun, i.e., its total energy is negative.Proof of part (3)We define the quantities α, d, and s as shown in figure ad. Thelaw of cosines givesad / Quantities referred to inthe proof of part (3).d 2 = r 2 + s 2 − 2rs cos α .Using α = 180 ◦ −2φ and the trigonometric identities cos(180 ◦ −x) =− cos x and cos 2x = 1 − 2 sin 2 x, we can rewrite this asd 2 = r 2 + s 2 − 2rs ( 2 sin 2 φ − 1 ) .Straightforward algebra transforms this into√(r + s)sin φ =2 − d 24rsSince r + s is constant, the top of the fraction is constant, and thedenominator can be rewritten as 4rs = 4r(constant − r), which isequivalent to the desired form..264 Chapter 4 Conservation of Angular Momentum