THE SOLAR SYSTEM We now want to apply what we have learned to the olar ytem. Hitorially thi wa the great teting ground for mehani and provided ome of it greatet triumph, uh a the diovery of the outer planet. We will ue a implified model of the ytem baed on the fat that the ma of the un i muh greater than that of any of the planet or in fat of all of them ombined. We will alo neglet the effet of the planet on eah other, although it i preiely thi whih led to the diovery of the outer planet. Hene we will uppoe that the only objet in the univere are the un and the planet other objet we are onidering. The firt tep i to hooe a atifatory oordinate ytem. It mut be inertial if we want to avoid the ue of the fititiou fore we have diued earlier. Ideally we would hooe a oordinate ytem either entered on the earth (Pre Coperniu) or on the un (Pot Coperniu). At firt glane neither would be atifatory beaue of Newton third law. Eah exert a fore on the other and thu neither i unaelerated. However thi problem an be overome, a we will now ee. We begin with an inertial ytem and loate the planet and the un with repet to it. Then Rp R r Ap A a F F m A m A a p p p p p F F Fp ma A m p Then Fp Fp mp a m
mp Fp 1 mpa m and mm p Fp a a m m p where μ (the redued ma) = m m p /(m +m p). Thu we an hooe the enter of the un a the origin of our oordinate ytem if we merely replae the ma of the planet by it redued ma. In the ae of the olar ytem all mae are mall ompared to that of the un, and hene the orretion i uually negligible. However if we were to onider the moon going around the earth thi would not be the ae. The phyial reaon for thi effet i eay to undertand. An oberver on either objet ee both her aeleration and the aeleration of the other objet toward her, a diued in la. We now have a very imple fore diagram for planet. There i only one fore the gravitational attration of the un. Sine it i direted through the axi it produe no torque: RFp 0 Then beaue of Newton eond law written in term of angular quantitie: We mut have the angular momentum ontant. But ine angular momentum i a vetor it diretion mut be ontant. Sine: dr dt J rp rmv and both r and v are in the plane of the orbit we ee that J i perpendiular to the plane of the orbit. Then ine that diretion i fixed we get our firt onluion the planet mut move in a plane. We will find the orbit exatly in a moment, but firt we onider a peial ae in whih the orbit i irular. It turn out that mot of the planet move in nearly irular orbit o thi i a good tarting point. In thi ae the problem i very imple. The aeleration toward the enter of the irle mut be provided by the gravitational attration:
mp vp FG m p R R FM Rv p But vp R P where P i the period of the planet the time to go one around the un. Then or 3 4 R P 3 R p 4 (the ame for all objet in the olar ytem). Thi reult i Kepler third law, and provide a mean of meauring the ma of the un. We now turn to the general problem where the orbit are not irular. Beaue the orbit may repeat it will be muh more onvenient to ue polar rather than Carteian oordinate (note that we are in a plane o pherial oordinate are not needed). We now need to alulate the aeleration in polar oordinate. We do thi a before. To get ˆ dr / dt onider the keth r rrˆ dr dr drˆ v rˆ r dt dt dt
Then drˆ d ˆ dt dt Hene dr d v rˆ r ˆ dt dt Of oure thi i obviou. dr/dt i jut the omponent of veloity along the radiu d r r dt i jut the tangential veloity along the irumferene. Next dv d r dr drˆ dr d d d dˆ a rˆ ˆ r ˆ r dt dt dt dt dt dt dt dt dt Hene we need d ˆ /dt. We find thi from the keth Thu dˆ d ˆr dt dt
Then d r d ˆ d dr d a rˆ r r dt dt dt dt dt Again mot of thee term are obviou. d r/dt i jut the aeleration along r. i jut the familiar entripetal aeleration. d v v r r r dt r r d d r r r dt dt i jut the tangential aeleration along the urve. The lat term dr d dt dt i new beaue we have not previouly onidered aeleration in both the radial and tangential diretion at the ame time. Sine we know the fore we an now write Newton eond law in our hoen oordinate ytem. m F r p rˆ d r dr dt dt r (1) d dr d r 0 dt dt dt () Thi look a bit intimidating until we ue our knowledge of the phyi of the ituation. We know that angular momentum i ontant. We have already ued the fat that the diretion i ontant to ee that the motion i in a plane. We now ue the fat that the magnitude i ontant to olve ().
Hene dr d ˆ d J mr v mrrˆ rˆ r mr zˆ dt dt dt d r ontant dt Thu d d dr d d d dr d r 0 r r r r dt dt dt dt dt dt dt dt Thu d r ont A dt i the olution of (). We now ubtitute thi reult in (1) to get: A r dr dt r r (3) To olve (3) it i ueful to make a hange of variable. Let: 1 r u Then dr dt 1 du u dt dr du 1 du 3 dt dt u u dt and du 1 d u 3 Au dt u u dt 3 u (4)
Sine we are intereted in the hape of the orbit we really want u(θ) rather than u(t). We therefore make another hange of variable. We have: We now ue the hain rule to get: Subtituting thi in (4) give: or u(t) = u(θ(t)) du du d du A du Au dt d dt d r d d u d du d u du Au Au Au Au Au d d dt d d du 1 d u 1du 3 Au Au Au A u 3 u u d u d ud du 3 Au Au u d d u u d A But thi i an equation we know how to olve. We do it a the um of the general olution of the homogeneou equation plu one olution of the whole equation. The olution of the homogeneou equation i obviouly: uh Bo where B and φ are arbitrary ontant. A olution of the whole equation i: Thu u p A u uk up Bo A
To fix the arbitrary ontant we hooe θ to be zero at the point of loet approah. At that point we alo know the angular momentum i: But Hene Chooe φ = 0. Then rmv J A rv m 1 1 1 r r u Bo rv Bo rv r Br o 1 rv 1 r Br 1 rv r rv 1 r r rv B 1 r r r rv 1 o 1 o rv rv 1 where 1 rv rv 1 rv
But thi i jut the equation of a oni etion with eentriity ε. There are now everal ae to onider depending on the value of ε. We begin by aking whether or not the planet will return, i.e. i it in a loed orbit or doe it ultimately leave the olar ytem. To anwer thi we need only look at the energy. Thi i given at the point of loet approah by: m 1 1 E mv m v r r If the planet i to reah infinite ditane it will have to have a veloity greater than or equal to zero at infinite ditane. Hene E mut be greater than or equal to zero. 1 1 r v v 0 v 1 r r If thi ondition i not met the planet will return and alo we an predit how far from the un it will get. To do o we note that both the angular momentum and the energy are onerved. Thu: or rv r v m m 1 1 v vm r r 1 r v p m m E m r r E 1 rm rm rv 0 m p r m E r v E m p mp 1/ Sine E/m p < 0 we an ue either ign. One give r, the other r m (the maximum ditane).
The ae to be onidered are then: Cae 1: < 1 Cae : ε = 1 Cae 3: ε = 0 Cae 4: ε > 1 CASE 3 Thi i the implet ituation. We then have: r v r r v rv or r 4 P Thi i jut the ae of irular orbit diued above. CASE Thi i a bit more ompliated, but not muh. In thi ae we have: rv rv 11 1 Thi i the ituation in whih the planet would jut reah infinite ditane with zero veloity. We an readily find the equation of the orbit a follow. Swithing to Carteian oordinate we have rv r 1 o
rv 1/ x y x 1 1/ x y 1/ r v rv rv rv x y x x y x xx But thi i jut the equation of a parabola. y r r x y x r r CASE 1 Thi i the ae for whih the objet will return, and hene the one that applie for planet in the olar ytem. Thi i more ompliated to analyze, o ome patiene i required. Now we have: rv rv r r 1/ r x y 1o x 1 1/ x y 1/ rv x y x r rv rv rv x y r x r r x x rv rv x 1 r xy r
We now omplete the quare on the x term x rv r y rv r x 1 1 1 rv rv rv y r r r x 1 1 1 1 rv r rv 1 r 1 1 1 Sine < 1, thi an equation of the form x y 1 a b whih i the equation of an ellipe entered at x = -α with emi-major axi a and emi-minor axi b. Note that But rv rv rv rv rv rv r r 1 r 1 r 1 1 rv rv rv rv 1 1
r o r rm E 1 v m p r i the enter of the ellipe. Then rv rv o 1 1 v rv v r r r 1 r r r Thu we have the equation of an ellipe with origin at the enter of the ellipe. CASE 4 Now all that hange i the ign of (1 ε ). Thi hange the equation to: x y 1 a b whih i the equation of a hyperbola. We now turn to the quetion of the motion a a funtion of time. To do thi we note that: d A r d dt dt r A 3 r A d t d A 1 o o o 3 1 tan A in 1 1 tan 3/ 1/ o 1 1o 1 1 For example we an find the period of the orbit of a planet a:
P 3 1 tan 3 A 1 A tan 3/ 1/ 3/ 1 1 1 0 But 1/ r r v A A a 1 1 a 1 3 3/ P A 1 a A a 3/ 1/ 4 3 P a Thi i Kepler third law. In other word the planet have the ame period a irular orbit of radiu equal to the emi-major axi would have. It i now a trivial problem to find the period of any planet if we know it ditane and veloity relative to the un at loet approah.