1 Feb 6, 001 An unusual appoach to Keple s fist law Ameican Jounal of Physics, 69(001), pp. 106-8. Thomas J. Osle Mathematics Depatment, Rowan Univesity, Glassboo NJ 0808, osle@owan.edu Keple s fist law of planetay motion states that the obits of planets ae elliptical, with the sun at one focus. We pesent an unusual veification of this law fo use in classes in mechanics. It has the advantages of esembling the simple veification of cicula obits, and stessing the impotance of Keple s equation. I. INTRODUCTION Keple s fist law of planetay motion states that a point mass moving in a cental k foce field of the fom f = will have an obit which is elliptical in shape with focus at the oigin if the motion is bounded. Thee ae many ways to show this to student s who have masteed only calculus. The pupose of this note is to call attention to a method that does not seem to be used often, but is appealing because it esembles closely the veification of cicula obits. The position vecto descibing cicula motion (adius a) with unifom angula velocity ω and time t is descibed by the pai of equations = acos i+ asin j (1) = ω t. () Motion on an elliptical obit with eccenticity e and focus at the oigin is descibed by the moe complex pai = ( acos ae) i+ bsin j () esin = ω t. (4)
Notice that when e = 0 (and thus b= a), equations () and (4) educe to (1) and (). The ellipse with position vecto = acos i+ bsin jwould have the oigin of coodinates at point C in Figue 1. Ou ellipse () has been shifted the distance ae to the left so that the oigin is at a focus. quation (4) is called Keple s equation 1. Keple s equation is a vesion of Keple s second law, that the adius vecto sweeps out equal Figue 1 The elliptical obit
aeas in equal times. (Astonomes call the eccentic anomaly and ω t the mean anomaly. The tem anomaly has been used fo angle by astonomes fo hundeds of yeas because of the iegulaities in planetay positions.) A simple geometic deivation of (4) is given in the appendix to make this pape self contained. While (1) and () can be combined to expess the position vecto diectly in tems of time, = acosωti+ asinω t j, this cannot be done in the elliptical case with () and (4). We cannot solve (4) fo in tems of t using convenient elementay functions. II. CIRCULAR ORBITS We seek obits of point masses that satisfy Newton s second law of motion in a cental foce field attacting invesely as the squae of the distance: k. (5) = (We use the notation common in mechanics: d =, and dt d dt =.) Diffeentiating (1) twice we get = ω ω i + ω j = ω ( acos t asin t ). (6) Compaing this last esult with (5) we see that k ω =. (7) Fom (7) we conclude that the cicle of adius a (1) is a tue obit (satisfies (5)) if the angula velocity ω satisfies ω = k/ a. If T is the peiod of the obit, then ω = π /T and we get fom (7), T 4π = a. This is Keple s thid law. k
4 III. LLIPTICAL ORBITS We will now show that the elliptical obit descibed by () and (4) satisfies Newton s second law (5) in much the same way that the cicula obit given by the pai (1) and () does. The calculations ae a bit longe, but thee ae no ticks. Fom () we see that the length of the position vecto is given by Using b = a a e we get which simplifies to = ( acos ae) + ( bsin ). = ( acos ae) + ( a a e )sin = a( 1 ecos ). (8) (4) we get Now we obtain nice expessions fo and. Diffeentiating Keple s equation and using (8) we have ( 1 ecos ) = ω, (9) a ω =. (10) Diffeentiating (9) and solving fo we get esin e. 1 cos = F H I K Using (8) and (10) we can ewite this as = a ω esin. (11)
5 Now we can find the acceleation along the elliptical obit. Diffeentiating () twice we get = asin i+ bcos j, and F = F I I F F I I HG H K KJ + HG H K KJ asin acos i bcos bsin j. Next we use (10) and (11) to eplace and. We get F HG I F KJ HG 4 a ω esin a ω cos a ω esin cos abω sin = i j. Factoing out a ω / we have a ω b ( ) = aesin cos besin cos sin + i+ + j. a Using (8) to emove fom {...} in the above expession we get afte simplifying Finally using () we have b a ω = a ae i + b j m cos sin. g a ω =. (1) Notice how (1) fo the acceleation on the elliptical path compaes with (6) the acceleation on the cicula obit. Substituting (1) into Newton s second law (5) we get I KJ a ω k =. Theefoe the elliptical motion descibed by the equations () and (4) satisfies Newton s second law if
6 k a ω =. (1) IV. FINAL RMARKS 1. One shotcoming of this method is that it is a veification, not a deivation. We must know elations () and (4) (which ae mathematical statements of Keple s fist and second laws), befoe we begin.. Histoically, Keple s laws wee known befoe Newton s laws of motion and gavity. Keple s fist two laws make thei initial appeaance in his Astonomia Nova of 1609. Initially, Geman astonomes, as well as Galileo, wee eluctant to abandon obits composed of cicula motion fo Keple s ellipse. Typical was the eaction of David Fabicius, a clegyman and amateu astonome who wote: With you ellipse you abolish the ciculaity and unifomity of the motions, which appeas to me the moe absud the moe pofoundly I think about it. If you could only peseve the pefect cicula obit, and justify you elliptic obit by anothe little epicycle, it would be much bette. The fist to ealize the impotance of Keple s discoveies wee the Bitish. In Newton s Pincipia 4, (1687), he poves that if the obit of the planet is an ellipse, with one focus at the cente of foce, then that foce must vay invesely as the squae of the distance.. Ou method emphasizes the impotance of Keple s equation (4). This elation (4) enables us to locate the position of the planet on the elliptical obit as a function of time. While (4) is always featued in advanced woks on celestial mechanics, it seems to be omitted in most mathematical teatments of Keple s laws in couses in elemenay and intemediate mechanics. Duing the past 00 yeas, hunded s of papes have been
7 published giving methods of solving (4). The book by Colwell 5 taces this emakable histoy. 4. We ecommend Koestle s biogaphy of Keple fo a lively account of his emakable achievements. APPNDIX: A DRIVATION OF KPLR S QUATION We now deive Keple s equation (4). Ou deivation is simila to Moulton s 1. Refe to Figue 1. Keple s second law states that the adius vecto sweeps out equal aeas in equal times as the planet P moves along the ellipse. Let t be the time equied fo the planet to move fom D to P, and let T be the time fo a complete tavesing of the ellipse. Then we have fom Keple s second law whee we ecall that Aea ODP t =, (14) π ab T π ab is the aea of the full ellipse. Since ou ellipse is the esult of squashing the lage cicle of adius a in the vetical diection by the facto b/a, we see that Now b Aea ODP = Aea ODA. (15) a Aea ODA = Aea CDA Aea COA a ( ae)( asin ) =. (16) Combining (15) and (16) we see that ab eabsin Aea ODP =. Substituting this last elation into (14) gives us Keple s equation
8 π esin = t = ωt. T 1 F. R. Moulton, An Intoduction to Celestial Mechanics, nd ed., Dove Pub., New Yok, N.Y., 1970, p. 160. W. H. Donahue, Johannes Keple, New Astonomy, Cambidge Univesity Pess, Cambidge, 199. A. Koestle, The Wateshed, Doubleday, Gaden City, New Yok, 1960, p. 164. 4 I. Newton, The Pincipia, (tanslated by Andew Motte), Pometheus Books,Amhest, New Yok, 1995, pp. 5-. 5 P. Colwell, Solving Keple s quation Ove Thee Centuies, William-Bell, Inc., Richmond, Viginia, 199.