3.4. KEPLER S LAWS 145 3.4 Keple s laws You ae familia with the idea that one can solve some mechanics poblems using only consevation of enegy and (linea) momentum. Thus, some of what we see as objects move aound in the wold is a diect consequence of these consevation laws athe than being the esult of some detailed mechanism. It is nice to give an example of how consevation of angula momentum has similaly poweful (and pehaps moe famous) consquences. We ecall that oughly 500 yeas ago, Keple made one of the geat beakthoughs (not just in physics, but in human thought), poviding evidence that planet motions as descibe by Tycho Bahe ae much moe simply descibed in a wold model with the sun (athe than the eath) at the cente. I don t think we can ovestate the impotance of ealizing that we ae not at the cente of the univese. Quantitatively, Keple noticed seveal things (Keple s laws): The obits of planets aound the sun ae elliptical, the peiods of the obits ae elated to thei adii, and as the obit poceeds it sweeps out equal aea in equal times. Of these, the equal aea law is the one which is elated to consevation of angula momentum. If the obits wee cicula it would be tivial that they sweep out aea at a constant ate. The equal aea law is, in a sense, all that is left of the pefection that people had sought with cicula obits. If we ae at a distance fom the cente of ou coodinate system (the sun), and we move by an angle θ, then fo small angles the aea that is swept out is A = 1 ( ) 1 dθ 2 2 θ = 2 t. (3.59) 2 The equal aea law is the statement that the tem in paentheses, da = 1 dθ 2 2, (3.60) is a constant, independent of time. We know that angula momentum is conseved, so let s see if this has something to do with the equal aea law. The vecto position of the planet can always be witten as = ˆ, whee ˆ is a unit vecto pointing outwad towad the cuent location. The velocity consists of components in the ˆ diection and in the ˆθ diection, aound the cuve, d = d dθ + ˆ ˆθ. (3.61)
146 CHAPTER 3. WE ARE NOT THE CENTER OF THE UNIVERSE Hence p m d = m d dθ + m ˆ ˆθ (3.62) ( L p = (ˆ ) m d ) ( ˆ +(ˆ ) m dθ ) ˆθ. (3.63) To finish the calculation we pull all the scalas out of the coss poducts, ( L = m d ) ( (ˆ ˆ )+ m 2 dθ ) (ˆ ˆθ), (3.64) and then we note that ˆ ˆ = 0, (3.65) ˆ ˆθ = ẑ. (3.66) Thus we find that the angula momentum is given by ( L = m 2 dθ ) ẑ. (3.67) Compaing Eq. (3.60) with Eq. (3.67), we see that da = 1 2m ( L ẑ ), (3.68) so that consevation of angula momentum ( L = constant) implies that da/ is a constant the equal aea law. To go futhe in deiving Keple s laws we need to know about the actual foces between the sun and the planets. You pobably know that one of Newton s geat tiumphs was to ealize that if gavity obeys the invese squae law, then the ate at which the moon is falling towad the eath as it obits is consistent with the ate at which objects we can hold in ou hands fall towad the gound. In moden language we say that the potential enegy fo two masses M and m sepaated by a distance is given by V () = GMm, (3.69) whee G is (appopiately enough) known as Newton s constant. We ae inteested in the case whee m is the mass of a planet and M is the mass of the sun. We choose a coodinate system in which the sun is fixed at the oigin. To undestand what happens it is useful to wite down the total enegy of the system. We have the potential enegy explicitly, so we need the kinetic
3.4. KEPLER S LAWS 147 enegy. We know that the velocity has two components, one in the adial diection and one in the angula diection, d = d dθ + ˆ ˆθ, (3.70) so that 1 2 mv2 1 2 m d 2 = 1 2 m [ (d ) 2 ( + dθ ) ] 2. (3.71) So the total enegy of the system, kinetic plus potential, is given by [ (d E = 1 ) 2 ( 2 m + dθ ) ] 2 GMm. (3.72) But we know that angula momentum is conseved, so we can say something about the tem that has dθ/ in it: L z = m 2 dθ (3.73) dθ = L z m 2. (3.74) Substituting into ou expession fo the total enegy this becomes [ (d E = 1 ) 2 ( 2 m + L ) ] 2 z m 2 GMm (3.75) = 1 ( ) d 2 2 m + L2 z 2m 2 GMm (3.76) = 1 ( ) d 2 2 m + V eff(), (3.77) whee in the last step we have intoduced an effective potential V eff () = L2 z 2m 2 GMm. (3.78) Notice that by doing this ou expession fo the total enegy comes to look like the enegy fo motion in one dimension (), with a potential enegy that has one pat fom gavity and one pat fom the indiect effect of the angula momentum. Notice that the contibution fom angula momentum is positive, and vaies as 1/ 2. This means that the coesponding foce F = V/
148 CHAPTER 3. WE ARE NOT THE CENTER OF THE UNIVERSE 100 80 60 gavitational potential!!1/ centifugal potential! +1/ 2 total effective potential hamonic appoximation 40 potential V() 20 0!20!40!60!80!100 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 adius (abitay units) Figue 3.1: Effective potential enegy fo planetay motion, fom Eq (3.78).
3.4. KEPLER S LAWS 149 1/ is positive it pushes outwad along the adius. This foce is what we expeience when we sit in a ca going aound a cuve: the centifugal foce. Imagine that we tie a weight on the end of a sting and swing it in a cicle ove ou heads. The sting will stay taught, and this must be because thee is a foce pulling outwad; again this is the centifugal foce, and is geneated by this special tem in the effective potential. Notice that we have eliminated any mention of the angle θ, and in the pocess have changed the potential enegy fo motion along the adial diection. This is a much moe geneal idea. We often eliminate coodinates in the hope of simplifying things, and ty to take account of thei effects though an effective potential fo the coodinates that we do keep in ou desciption. This is vey impotant in big molecules, fo example, whee we don t want to keep tack of evey atom but hope that we can just think about a few things such as the distance between key esidues o the angle between two big ams of the molecule. It s not at all obvious that this should wok, even as an appoximation, although in the pesent case it s actually exact. Recall that the total enegy is the sum of kinetic and potential, and this total is conseved o constant ove time. Thee is a minimum effective potential enegy fo adial motion, as can be seen in Fig 3.1, If the total enegy is equal to this minimum, then thee can be no kinetic enegy associated with the coodinate, hence d/ = 0. Thus fo minimum enegy obits, the adius is constant the planet moves in a cicula obit. If we look at obits that have enegies just a bit lage than the minimum, we can appoximate V eff () as being like a hamonic oscillato. Then the adius should oscillate in time, but time is being maked by going aound the obit, so eally the adius will be a sine o cosine function of the angle, and this is the desciption of an ellipse if it is not too eccentic. In fact if you wok hade you can show that the obits ae exactly ellipses fo any value of the enegy up to some maximum. This is anothe of Keple s laws. Once you have the ellipse you can elate its size (the analog of adius fo a cicle) to the peiod of the obit, and this is the last of Keple s laws. Notice that if the enegy is positive then it is possible fo the planet to escape towad at finite velocity, and then the obit is not bound. But if the total enegy is negative, thee is no escape, and the adius moves between two limiting values, namely the points whee the total enegy intesects the effective potential. We eally should say moe about all this, but it is teated in many standad texts.
150 CHAPTER 3. WE ARE NOT THE CENTER OF THE UNIVERSE