= 1. For a hyperbolic orbit with an attractive inverse square force, the polar equation with origin at the center of attraction is

Transcription

1 15. Kepleian Obits Michael Fowle Peliminay: Pola Equations fo Conic Section Cuves As we shall find, Newton s equations fo paticle motion in an invese-squae cental foce give obits that ae conic section cuves. Popeties of these cuves ae fully discussed in the accompanying Math fo Obits lectue, hee fo convenience we give the elevant pola equations fo the vaious possibilities. Fo an ellipse, with eccenticity e and semilatus ectum (pependicula distance fom focus to cuve) : = 1+ ecos θ. Recall the eccenticity e is defined by the distance fom the cente of the ellipse to the focus being ae, = a 1 e = b / a. whee a is the semi-majo axis, and ( ) Fo a paabola, ( θ ) = 1+ cos. Fo a hypebolic obit with an attactive invese squae foce, the pola equation with oigin at the cente of attaction is = 1 ecosθ wheeθ asymptote < θ < π θ asymptote. (Of couse, the physical path of the planet (say) is only one banch of the hypebola.) The (, θ ) oigin is at the cente of attaction (the Sun), geometically this is one focus of the hypebola, and fo this attactive case it s the focus inside the cuve. Fo a hypebolic obit with a epulsive invese squae foce (such as Ruthefod scatteing), the oigin is the focus outside the cuve, and to the ight (in the usual epesentation): with angula ange θ < θ < θ. asymptote asymptote = ecosθ 1,

2 Summay We ll begin by stating Keple s laws, then apply Newton s Second Law to motion in a cental foce field. Witing the equations vectoially leads easily to the consevation laws fo angula momentum and enegy. Next, we use Benoulli s change of vaiable u = 1/ to pove that the invese-squae law gives conic section obits. A futhe vectoial investigation of the equations, following Hamilton, leads natually to an unsuspected thid conseved quantity, afte enegy and angula momentum, the Runge Lenz vecto. Finally, we discuss the athe supising behavio of the momentum vecto as a function of time. Keple s Statement of his Thee Laws 1. The planets all move in elliptical obits with the Sun at one focus.. As a planet moves in its obit, the line fom the cente of the Sun to the cente of the planet sweeps out equal aeas in equal times, so if the aea SAB (with cuved side AB) equals the aea SCD, the planet takes the same time to move fom A to B as it does fom C to D. D A B S C Fo my Flashlet illustating this law, click hee. 3. The time it takes a planet to make one complete obit aound the sunt (one planet yea) is elated to the length of the semimajo axis of the ellipse a : T a 3.

3 3 In othe wods, if a table is made of the length of yea T fo each planet in the Sola System, and the 3 length of the semimajo axis of the ellipse a, and T / a is computed fo each planet, the numbes ae all the same. These laws of Keple s ae pecise (apat fom tiny elativistic coections, undetectable until centuies late) but they ae only desciptive Keple did not undestand why the planets should behave in this way. Newton s geat achievement was to pove that all this complicated behavio followed fom one simple law of attaction. Dynamics of Motion in a Cental Potential: Deiving Keple s Laws Conseved Quantities The equation of motion is: m = f. ( ) ˆ Hee we use the hat ^ to denote a unit vecto, so f ( ) gives the magnitude (and sign) of the foce. Fo f = GMm /. Keple s poblem, ( ) (Stictly speaking, we should be using the educed mass fo planetay motion, fo ou Sola System, that is a small coection. It can be put in at the end if needed.) Let s see how using vecto methods we can easily find constants of motion: fist, angula momentum just act on the equation of motion with : m = f ( ) ˆ Since ˆ = 0, we have m = 0, which immediately integates to m = L, a constant, the angula momentum, and note that L = 0 = L, so the motion will always stay in a plane, with L pependicula to the plane. This establishes that motion in a puely cental foce obeys a consevation law: that of angula momentum. (As we've discussed ealie in the couse, conseved quantities in dynamical systems ae always elated to some undelying symmety of the Hamiltonian. The consevation of angula momentum comes fom the spheical symmety of the system: the attaction depends only on distance, not angle. In quantum mechanics, the angula momentum opeato is a otation opeato: the thee components of the angula momentum vecto ae conseved, ae constants of the motion, because the Hamiltonian is invaiant unde otation. That is, the angula momentum opeatos commute with the Hamiltonian. The classical analogy is that they have zeo Poisson backets with the Hamiltonian.)

4 4 To get back to Keple s statement of his Laws, notice that when the planet moves though an incemental distance d it sweeps out an aea 1 d, so the ate of sweeping out aea is 1 da / dt = = L / m. Keple s Second Law is just consevation of angula momentum! Second, consevation of enegy: this time, we act on the equation of motion with : m = f This immediately integates to ( ) ˆ. 1 m f ( ) d = E. Anothe consevation law coming fom a simple integal: consevation of enegy. What symmety does that coespond to? The answe is the invaiance of the Hamiltonian unde time: the cental foce is time invaiant, and we e assuming thee ae time-dependent potential tems, (such as fom anothe sta passing close by). Standad Calculus Deivation of Keple s Fist Law The fist mathematical poof that an elliptic obit about a focus meant an invese-squae attaction was given by Newton, using Euclidean geomety (even though he invented calculus!). The poof is notoiously difficult to follow. Benoulli found a faily staightfowad calculus poof in pola coodinates by changing the vaiable to u = 1/. The fist task is to expess F = ma in pola, meaning(, θ ), coodinates. The simplest way to find the expession fo acceleation is to paameteize the plana motion as a i complex numbe: position e θ iθ iθ, velocity e + i θe θ, since thei ensues the, notice this means ( ) θ tem is in the positiveθ diection, and diffeentiating again gives a = = θ, θ + θ. ( ) Fo a cental foce, the only acceleation is in the diection, so θ + θ = 0, which integates to give the constancy of angula momentum. Equating the adial components, m θ = L, d θ = dt GM

5 5 This isn t eady to integate yet, because θ vaies too. But since the angula momentum constant, we can eliminate θ fom the equation, giving: L = m θ is d GM L = + dt m GM L = + m 3 This doesn t look too pomising, but Benoulli came up with two cleve ticks. The fist was to change fom the vaiable to its invese, u = 1/. The othe was to use the constancy of angula momentum to change the vaiable t to θ. Putting these togethe: so Theefoe L m θ = = d Lu d = dt mdθ u dt. m dθ d d 1 1 du L du = = = dt dt u u dt m dθ and similaly Going fom to u in the equation of motion we get o d Lu d u dt m dθ =. d GM L = + dt m 3 Lu d u Lu = + m dθ m 3 GMu, d u GMm dθ + = L u.

6 6 This equation is easy to solve! The solution is 1 GMm u = = + Ccosθ L wheec is a constant of integation, detemined by the initial conditions. This poves that Keple s Fist Law follows fom the invese-squae natue of the foce, because (see beginning of lectue) the equation above is exactly the standad (, θ ) equation of an ellipse of semi majo axis a and eccenticity e, with the oigin at one focus: a ( 1 e ) = 1+ ecos θ. Compaing the two equations, we can find the geomety of the ellipse in tems of the angula momentum, the gavitational attaction, and the initial conditions. The angula momentum is ( ) = 1 = /. L GMm a e GMm b a A Vectoial Appoach: Hamilton s Equation and the Runge Lenz Vecto (Mainly following Milne, Vectoial Mechanics, p 35 on.) Laplace and Hamilton developed a athe diffeent appoach to this invese-squae obit poblem, best expessed vectoially, and made a supising discovey: even though consevation of angula momentum and of enegy wee enough to detemine the motion completely, fo the special case of an invesesquae cental foce, something else was conseved. So the system has anothe symmety! Hamilton s appoach (actually vectoized by Gibbs) was to apply the opeato L to the equation of motion m = f : ( ) ˆ L m f ( ) = ( m ) ˆ Now ˆ ˆ ˆ d d d, dt dt dt ( ) = ( ) + = = = so ( ) dˆ. L m = m f dt

7 7 This is known as Hamilton s equation. In fact, it's petty easy to undestand on looking it ove: d ˆ / dt has magnitudeθ and diection pependicula to, m = f, m θ = L, etc. ( ) ˆ It isn t vey useful, though except in one case, the invese-squae: ( ) Then it becomes tactable: ( ) immediately to f = k / (so k = GM. ) dˆ dˆ L m = m f = km, and supise this integates dt dt L m = kmˆ A whee A is a vecto constant of integation, that is to say we find A = p L kmˆ is constant thoughout the motion! This is unexpected: we found the usual conseved quantities, enegy and angula momentum, and indeed they wee sufficient fo us to find the obit. But fo the special case of the invese-squae law, something else is conseved. It s called the Runge Lenz vecto (sometimes Laplace Runge Lenz, and in fact Runge and Lenz don t eally deseve the fame they just ehashed Gibbs wok in a textbook). Fom ou ealie discussion, this conseved vecto must coespond to a symmety. Finding the obit gives some insight into what s special about the invese-squae law. Deiving the Obital Equation fom the Runge-Lenz Vecto The Runge Lenz vecto gives a vey quick deivation of the elliptic obit, without Benoulli s unobvious ticks in the standad deivation pesented above. Fist, taking the dot poduct of A = p L kmˆ with the angula momentum L, we find A L = kmˆ L = 0, meaning that the constant vecto A lies in the plane of the obit. Next take the dot poduct of A with, and since p L = L p = L, we find L = km + A, o = 1+ ecosθ whee = L / km, e = A / km and θ is the angle between the planet s obital position and the Runge Lenz vecto A. This is the standad (, θ ) equation fo an ellipse, with the semi-latus ectum (the pependicula distance fom a focus to the ellipse), e the eccenticity. Evidently A points along the majo axis.

8 8 The point is that the diection of the majo axis emains the same: the elliptical obit epeats indefinitely. If the foce law is changed slightly fom invese-squae, the obit pecesses: the whole elliptical obit otates aound the cental focus, the Runge Lenz vecto is no longe a conseved quantity. Stictly speaking, of couse, the obit isn t quite elliptical even fo once aound in this case. The most famous example, histoically, was an extended analysis of the pecession of Mecuy s obit, most of which pecession aises fom gavitational pulls fom othe planets, but when all this was taken into account, thee was left ove pecession that led to a lengthy seach fo a planet close to the Sun (it didn t exist), but the discepancy was finally, and pecisely, accounted fo by Einstein s theoy of geneal elativity. Vaiation of the Momentum Vecto in the Obit (Hodogaph) It s inteesting and instuctive to tack how the momentum vecto changes as time pogesses, this is easy fom the Runge Lenz equation. (Hamilton did this.) Fom p L = kmˆ + A, we have L p L = kml + L A ( ) ˆ That is, L A kml p = + ˆ. L L Staing at this expession, we see that p goes in a cicle of adius km / L about a point distance A/ Lfom the momentum plane oigin. Of couse, p is not moving in this cicle at a unifom ate (except fo a planet in a cicula obit), its angula pogession aound its cicle matches the angula pogession of the planet in its elliptical obit (because its location on the cicle is always pependicula to the ˆ diection fom the cicle cente). An obit plotted in momentum space is called a hodogaph. Obital Enegy as a Function of Obital Paametes Using Runge-Lenz We ll pove that the total enegy, and the time fo a complete obit, only depend on the length of the majo axis of the ellipse. So a cicula obit and a vey thin one going out to twice the cicula adius take the same time, and have the same total enegy pe unit mass. Take p L = A + kmˆ and squae both sides, giving ˆ p L = A + k m + kma L km = A + k m + km = A k m + kml /.

9 9 Dividing both sides by ml, p k A km m ml =. Putting in the values found above, A = kme, L = km, = a ( 1 e ) we find p k k =. m a So the total enegy, kinetic plus potential, depends only on the length of the majo axis of the ellipse. Now fo the time in obit: we ve shown aea is swept out at a ate L/m, so one obit takes time T π ab /( L / m) b = a 1 e, L = kma 1 e, so =, and ( ) This is Keple s famous Thid Law: T T = a m k = a GM 3/ 3/ π / π /. a, easily poved fo cicula obits, not so easy fo ellipses. 3 Impotant Hint! Always emembe that fo Keple poblems with a given massive Sun, both the time in obit and the total obital enegy/unit mass only depend on the length of the majo axis, they ae independent of the length of the mino axis. This can be vey useful in solving poblems. The Runge-Lenz Vecto in Quantum Mechanics This is fully discussed in advanced quantum mechanics texts, we just want to mention that, just as spheical symmety ensues that the total angula momentum and its components commute with the Hamiltonian, and as a consequence thee ae degeneate enegy levels connected by the aising opeato, an analogous opeato can be constucted fo the Runge-Lenz vecto, connecting states having the same enegy. Futhemoe, this aising opeato, although it commutes with the Hamiltonian, does not commute with the total angula momentum, meaning that states with diffeent total angula momentum can have the same enegy. This is the degeneacy in the hydogen atom enegy levels that led to the simple Boh atom coectly pedicting all the enegy levels (apat fom fine stuctue, etc.). It s also woth mentioning that these two vectos, angula momentum and Runge-Lenz, both sets of otation opeatos in thee dimensional spaces, combine to give a complete set of opeatos in a fou dimensional space, and the invese-squae poblem can be fomulated as the mechanics of a fee paticle on the suface of a sphee in fou-dimensional space.