GraphingCalculator 4;
Window 52 5 849 1426;
PaneDivider 548;
FontSizes 18;
BackgroundType 0;
StackPanes 1;
Slider 0 1;
SliderSteps 400;
SliderControlValue 0;
2D.Scale 0.1 0.1 5 5;
2D.BottomLeft -2.19375 -1.125;
2D.Axes 0;
2D.GraphPaper 0;
Text "Keplerian orbits.  Illustrates both the physical two-body problem and its relationship to the equivalent one-body problem in terms of the reduced mass.

Version 0.7 4-28-10
To do:
(o) Make time dependence physical rather than uniform angular velocity [theta-dot=l/mr^2=> theta(t)=?]
[Turns out this is analytically unsolved, except in implicit form, even for Kepler – can separate variables and do the ungodly integral to find t(theta), but can't invert it.]
(i) Area law – more carefully.


m1,m2 = masses; M = total mass; m = reduced mass; k = Gm1m2;  l= angular momentum; E = eccentricity; A = l^2/mk; T0 = line of apsides; 
";
Color 8;
MathPaneSlider 10;
Expr b=slider([0,1,40]);
Text "Fix product km = C so that relative scale of orbit does not change as relative mass b is adjusted.  [The major and minor axes are proportional to A=l^2/km times powers of sqrt(1-E^2).  With m=m1m2/(m1+m2) and m1=bm2, a little algebra shows that m2=([1+b/b^2] km))^(1/3).]  Other schemes are of course possible, but this is the least distracting when trying to show effect of adjusting relative masses.";
Color 7;
Expr C=1,m_2=[(1+b)/b^2*C]^(1/3),m_1=b*m_2,M=m_1+m_2;
Expr G=1,k=G*m_1*m_2,m=C/k,A=l^2/C;
Color 5;
MathPaneSlider 19;
Expr l=slider([0,10]);
Color 2;
MathPaneSlider 66;
Expr E=slider([0,2]);
Color 3;
Expr T=slider([0,1]);
Color 5;
Expr T_0=2*pi*T;
Text "Equation of ellipse; +/- in denominator determines which direction section opens/which focus is pericentral (+ is right focus; - is left).
""If"" conditional is a kludge [due to Chris Young & Ron Avitzur] to prevent ""crossing"" of hyperbola when E>1 that occurs at zeros of denominator.";
Expr K=0;
Color 2;
Expr function(R,a)=[if(A/(1-(E*cos([a-T_0]))),abs(1-(E*cos([a-T_0])))>0.1)];
Text "Orbit of equivalent reduced mass:";
Color 7;
Expr r=function(R,theta);
Color 4;
Expr function(X,s)=vector(function(R,s)*cos([s]),function(R,s)*sin([s]));
Expr function(X,2*pi*n);
Color 17;
Expr vector(0,0),function(X,2*pi*n);
Text "Left focus/center of mass:";
Color 2;
Expr vector(0,0);
Text "Positions of physical masses; center of mass is at origin:";
Color 6;
Expr function(X_1,s)=function(X,s)*m_2/M;
Color 6;
Expr function(X_2,s)=[-function(X,s)]*m_1/M;
Text "Inverts positions if necessary:";
Color 8;
MathPaneSlider 2;
Expr p=slider([-1,1,2]);
Color 17;
Expr function(X_1,2*pi*n)*p;
Color 17;
Expr function(X_2,2*pi*n)*p;
Text "Relative separation vector:";
Color 17;
Expr function(X_2,2*pi*n)*p,function(X_1,2*pi*n)*p;
Text "Orbits of physical masses:";
Color 17;
Expr function(X_1,2*pi*t)*p;
Color 17;
Expr function(X_2,2*pi*t)*p;
Text "Kepler's Area Law – equal areas swept out in equal times.  With a small time interval d, l=mR^2 theta-dot means dtheta = (l/mR^2)dt:";
Color 4;
Expr S=slider([0,2*pi]);
Color 3;
Expr d=slider([0,1]);
Color 3;
Expr r<function(R,theta),S<theta<S+d/(m*function(R,S)^2);
Text "


<size=16>Author: David A. Craig <<http://web.lemoyne.edu/~craigda>";