KEPE S AWS AND PANETAY OBITS 1. Selected popeties of pola coodinates and ellipses Pola coodinates: I take a some what extended view of pola coodinates in that I allow fo a z diection (cylindical coodinates even though the motion is entiely in the xy-plane. Fom the dawing we obseve that: d d d = d ˆ + θ [1.1] Note also that: d = kˆ and ˆ θˆ = kˆ et. cyc. We can wite: = xiˆ + yˆj fom which we have immediately ˆ = cos( θ ˆ i + sin( θ ˆj. [1.3] dˆ = sin( θ ˆ i + cos( θ ˆj. but kˆ ˆ = sin( θ ˆ i + cos( θ ˆj so: dˆ = kˆ ˆ = ˆ θ [1.4] Ellipses: The ellipse shown below has the fom: = ecos( θ [1.5] whee and e ae constants. The quantity e has the popety that the focus, f, is displaced fom the oigin by the amount ea. Fom this we see that: f a f 1 1 a = + e 1 e a = [1.6] 1 e 3/8/007 1 F:\WEB PAGE STUFF\KEPES AWS AND PANETAY OBITS.doc
. Newton s laws of motion and gavity In this section I will set foth the fundamental laws and definitions that will be used to deive the esults of the following sections. I begin with Newton s law of gavitation: mm D F = G ˆ = ˆ whee: [.1] D = G m M [.] D is a constant with no physical meaning. It has dimensions of Kg m 3 /s Newton s nd aw states: dp F = [.3] dt The definition of angula momentum gives: d = p = m = m kˆ [.4] dt dt Whee we have used equation [1.] 3. Keple s nd aw In this section I show that the angula momentum is conseved fo a cental foce field and then show that this tanslates into Keple s nd aw. Since the foce in a cental-foce situation is diected diectly fom one body to the othe, the quantity τ = F is identically zeo and hence the angula momentum is constant. Keple s nd aw In pola coodinates the incemental aea swept out by an incemental change in diection is: 1 da = since is the height of the incemental tiangle and is the base. Hence: da 1 = [3.1] dt dt Theefoe by [.4] the quantity on the ight of [3.1] is constant which is Keple s nd aw. 3/8/007 F:\WEB PAGE STUFF\KEPES AWS AND PANETAY OBITS.doc
4. Keple s 1 st aw In this section I deive the elationship between and θ and show that the tajectoy of an object in an invese-squae-law cental-foce field is a conic section. Conside the coss poduct: F dp D = ˆ m kˆ dt dt whee on the left side we have substituted fom [.3] fo F and on the ight side [.] and [.4] fo F and espectively. Simplifying the two sides of this equation independently gives: d( p = md ˆ kˆ dt dt whee on the left we have used the fact that is a constant. We can simplify the ight hand d side futhe by noting that kˆ ˆ dˆ dˆ ˆ = [1.4] and = to give dt dt d( p dˆ = md [4.1] dt dt Now we can integate both sides of [4.1] to get: p = md( ˆ + e [4.] whee e is the constant of integation. It will tun out to be a vecto paallel to the semimajo axis of the ellipse with magnitude equal to the eccenticity, e. If we dot into both sides of [4.] we get: p = md ( ˆ + e p = md( + e cosθ = md( [4.3] 1 = [4.4] md This is the desied esult. This is the equation fo a conic section, If e < 1 we have an ellipse, if e = 1 a paabola and if e > 1 a hypebola. We can define the quantity = [4.5] md and wite [4.4] as = [4.6] Hee we see that has the dimensions of length and the physical meaning of the adius of the obit if e = 0. 3/8/007 3 F:\WEB PAGE STUFF\KEPES AWS AND PANETAY OBITS.doc
5. Potential, Kinetic and Total Enegy In this section I show the elationship between enegy and angula momentum fo planetay motion. Potential Enegy: Fo an invese squae law we can wite the potential enegy: D md U = = ( [5.1] Kinetic Enegy: Conside the quantity v. Fom the definition of angula momentum these two vectos ae pependicula to each othe so fom [4.]: v = D + e [5.] Total Enegy: Thus the total enegy is: ( D ( ecos e 1 md K = ( + e [5.3] v = θ + ( 1 e 1 md E = K + U = [5.4] 3/8/007 4 F:\WEB PAGE STUFF\KEPES AWS AND PANETAY OBITS.doc
6. Keple s 3 d aw In this section I elate the semi-majo axis of the planetay ellipse with the Peiod of the planet. Fom [.4] we have: = dt m [6.1] Substituting fo fom [4.4] md = ( 3 dt [6.] md We can define the constant : ω 0 = 3 [6.3] which has units of time -1 and the physical intepetation of the angula velocity of the cicula obit if e = 0. Then we can wite: = ω ( 0 dt [6.4] dt = ω ( 0 [6.5] 1 π T = ω 0 ( + θ 0 1 ecos [6.6] This integal can be found in Meye zu Capellan 3.4.3 page 16 and 3.1.3 page 161. Fo ou paametes it becomes: 1 π T = 3 / ω e [6.7] T = 0 1 ( (π 6 3 4 ( 1 e m D [6.8] Now fom [1.6] and [4.5]: 1 a = md 1 e [6.9] 6 3 1 a = 3 3 m D ( 1 e 3 [6.10] eliminating between [6.8] and [6.10] gives: 3 DT a ( π m [6.11] 3 GM a = T (π [6.1] 3/8/007 5 F:\WEB PAGE STUFF\KEPES AWS AND PANETAY OBITS.doc
7. Enegy and Angula momentum in tems of the Semi-majo axis In this section I elate the total enegy and the angula momentum of a planet to the semimajo axis of the elliptical obit. Combining [6.9] and [5.4] to eliminate e, we find: eaanging tems in [6.9] gives: D 1 GmM E = = [7.1] a a ( 1 e = Gm Ma( 1 e = mda [7.] = Gm M = Gm M 3/8/007 6 F:\WEB PAGE STUFF\KEPES AWS AND PANETAY OBITS.doc