Section 8.7

Planetary Motion – Kepler' s Problem

AP E3200x
Columbia University

Preliminaries

Plot[1/Sqrt[(ϵ^2 - 1) + 2 x - x^2] /. ϵ -> 0.1, {x, 1.0-0.1,1.0+0.1}]

Solve[(ϵ^2 - 1) + 2 x - x^2 == 0, x]

Indefinite Integral

Integrate[1/Sqrt[(ϵ^2 - 1) + 2 x - x^2], x, Assumptions-> 0 < ϵ < 1]

Plot[-ArcTan[(((-1+x) (Sqrt[-1+2 x-(x^2)+(ϵ^2)]))/(1-2 x+(x^2)-(ϵ^2)))] /. ϵ -> 0.1, {x, 0.7,1.3}]

Definite Integral

Integrate[1/Sqrt[(ϵ^2 - 1) + 2 x - x^2], {x, 1 + ϵ,xx}, Assumptions-> 0 < ϵ < 1 && xx > 1 + ϵ]

Integrate[1/Sqrt[(ϵ^2 - 1) + 2 x - x^2], {x, 1 + ϵ,xx}, Assumptions-> 0 < ϵ < 1 ]

Plot[π/2 +ArcTan[((-1+xx)/(Sqrt[-(((-1+xx))^2)+(ϵ^2)]))] /. ϵ -> 0.1, {xx, 1.0+0.1-0.00001,1.0-0.1+0.00001}]

ParametricPlot[{r Cos[π/2 +ArcTan[((-1+1/r)/(Sqrt[-(((-1+1/r))^2)+(0.1^2)]))]], r Sin[π/2 +ArcTan[((-1+1/r)/(Sqrt[-(((-1+1/r))^2)+(0.1^2)]))]]}, {r, 1/( 1.0+0.1-0.00001),1/(1.0-0.1+0.00001)}, PlotStyle -> Blue];
ParametricPlot[{r Cos[π/2 +ArcTan[((-1+1/r)/(Sqrt[-(((-1+1/r))^2)+(0.2^2)]))]], r Sin[π/2 +ArcTan[((-1+1/r)/(Sqrt[-(((-1+1/r))^2)+(0.2^2)]))]]}, {r, 1/( 1.0+0.2-0.00001),1/(1.0-0.2+0.00001)}, PlotStyle -> Red];
ParametricPlot[{r Cos[π/2 +ArcTan[((-1+1/r)/(Sqrt[-(((-1+1/r))^2)+(0.3^2)]))]], r Sin[π/2 +ArcTan[((-1+1/r)/(Sqrt[-(((-1+1/r))^2)+(0.3^2)]))]]}, {r, 1/( 1.0+0.3-0.00001),1/(1.0-0.3+0.00001)}, PlotStyle -> Green];
ParametricPlot[{r Cos[π/2 +ArcTan[((-1+1/r)/(Sqrt[-(((-1+1/r))^2)+(0.5^2)]))]], r Sin[π/2 +ArcTan[((-1+1/r)/(Sqrt[-(((-1+1/r))^2)+(0.5^2)]))]]}, {r, 1/( 1.0+0.5-0.00001),1/(1.0-0.5+0.00001)}, PlotStyle -> Magenta];
Show[%,%%,%%%, %%%%, AspectRatio -> Automatic, PlotLabel -> "Orbits with α = 1"]

Solving for Radius

eqn = θ == π/2 +ArcTan[((-1+xx)/(Sqrt[-(((-1+xx))^2)+(ϵ^2)]))]

Solve[eqn, xx]
xx /. Last[%]
% // Simplify
% // PowerExpand
% // TrigFactor