Chapte 4 Intoduction to Systes of Diffeential Equations Poject 4.1 Keple's aws and Planetay Obits The Section 4.1 poject in the text stats with Newton's invese-squae law of gavitation and outlines a deivation of the pola-coodinate foula ( θ) = 1+ εcos( θ α) (1) descibing an elliptical planetay obit with eccenticity ε and sei-latus ectu. The angle α is the planet's pola coodinate angle at peihelion when it is closest to the sun. If the nueical values of, ε, and α ae known, then the ellipse can be plotted in ectangula coodinates by witing xt () = t ()cos, t yt () = t ()sin, t 0 t 2π (2) (with paaete t in place of θ). Hee we want to use foulas (1) and (2) to plot soe typical planetay obits, stating with data found in a coon souce like a wold alanac whee a planet's axiu and iniu distances and (espectively) fo the sun typically ae listed, but its sei-latus ectu is unlikely to be entioned. But if we take α = 0 in (1) then it should be clea that = and =. (3) 1 ε 1+ ε Upon equating values of, these two equations ae easily solved fist fo and then fo ε = + (4) = ( 1 ε) = ( 1+ ε ). (5) Poject 4.1 87
The initial coluns in the table below list the axiu and iniu distances fo the sun (in astonoical units, whee 1 AU = 93 illion iles is the ean distance of the Eath fo the sun) of the nine planets and Halley's coet. The last two coluns list values of ε and calculated using Eqs. (4) and (5). Planet ε ecuy 0.467 0.308 0.2056 0.371 Venus 0.728 0.718 0.0067 0.723 Eath 1.017 0.983 0.0172 1.000 as 1.667 1.382 0.0935 1.511 Jupite 5.452 4.953 0.0480 5.190 Satun 10.081 9.015 0.0558 9.518 Uanus 19.997 17.949 0.0540 18.918 Neptune 30.341 29.682 0.0110 30.008 Pluto 48.940 29.639 0.2456 36.919 Halley 35.304 0.587 0.9673 1.155 In the paagaphs below we illustate the use of aple, atheatica, and ATAB to plot typical planetay obits. You can ty these and othes. Using aple To plot the obit of the planet ecuy, we fist ente its axiu and iniu distances fo the sun. 1 := 0.467: 2 := 0.308: Then we calculate its eccenticity and sei-latus ectu using Eqs (4) and (5). e := (1-2)/(1 + 2): := 1*(1-e): := /(1+e*cos(t)): x := *cos(t): y := *sin(t): plot([x,y,t=0..2*pi], view=[-0.5..0.5, -0.5..0.5], 88 Chapte 4
thickness = 2, colo = ed, scaling = constained, title = "ecuy Obit"); The option scaling=constained insues equal scales on the x- and y-axes. We see that the elliptical obit of ecuy actually looks quite cicula. The nonunifoity of the otion of ecuy consists in the facts that this "cicle" is off-cente fo the sun at the oigin, and that its speed vaies with its position on the obit, with the planet oving fastest at peihelion and slowest at aphelion. Using atheatica To plot the obit of the planet ecuy, we fist ente its axiu and iniu distances fo the sun. 1 := 0.467: 2 := 0.308: Then we calculate its eccenticity and sei-latus ectu using Eq. (4) and (5). e := (1-2)/(1 + 2): := 1*(1-e): := /(1+e*cos(t)): Poject 4.1 89
x := *cos(t): y := *sin(t): PaaeticPlot[{x, y}, {t, 0, 2*Pi}, PlotRange -> {{-45, 5}, {-5, 5}}, AspectRatio -> 0.2, Axesabel -> {"x", "y"}, PlotStyle -> {Thickness[0.0065],RGBColo[1,0, 0]}, Plotabel -> "Obit of Halley's Coet"]; Obit of Halley 's Coet -40-30 -20-10 y 4 2-2 -4 x The option AspectRatio->1 seves to ensue equal scales on the x- and y- axes. The plotted obit cetainly looks athe eccentic, though pehaps not so uch as the actual eccenticity of ε 0.97 ight lead on to expect. Using ATAB To plot the Eath's obit, we fist ente its axiu and iniu distances fo the sun. 1 = 1.017; 2 = 0.983; Then we calculate its eccenticity and sei-latus ectu using Eqs. (4) and (5). e = (1-2)/(1 + 2); = 1*(1-e); t = 0 : pi/100 : 2*pi; =./(1+e*cos(t)); x =.*cos(t); y =.*sin(t); 90 Chapte 4
h = plot(x,y,'b'); set(h,'linewidth',2); w = 0.025; % to set viewing window axis(w*[-1 1-1 1]), axis squae hold on plot(w*[-1 1],[0 0],'k') plot([0 0],w*[-1 1;],'k') title('eath Obit') 1 Eath Obit 0.8 0.6 0.4 0.2 0-0.2-0.4-0.6-0.8-1 -1-0.5 0 0.5 1 The fact that the Eath's obit is elliptical is so engained in oden inds that the seeingly cicula appeaance of the obit (on any easonable scale) ay coe as a supise. We have plotted the obit in the squae viewing window 1. 025 xy, 1. 025 so that a caeful exaination of the figue will eveal that this "cicula" obit is visibly off-cente fo the sun at the oigin. This fact, togethe with the non-unifoity of the Eath's speed in its obit it oves fastest at peihelion and slowest at aphelion constitutes the actual ellipticity of the otion. Poject 4.1 91