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In CR3BP, there are 5 equilibrium points, which is also called Lagrange Points. They stationary (motionless points) in the rotating frame and they can be computed with equation of motion of CR3BP, making velocity and acceleration components zero. The collinear points, L1, L2 and L3, are unstable, ans triangular points, L4 and L5, are stable; Which it is explained in Energy Surface section, why either they are stable or unstable.
In up, the Equilimrium points are given. With Change of mass ratio (μ), the places of them are change as in figure. Red dots are unstable ones (L1, L2 and L3), and Green dots are stable ones (L4 and L5). Also, Black dots are place of M1 (left black dot) and M2 (right black dot). If in μ=0.5 (so M1 and M2 have equal mass), then the formation of all dot become perfect symmetric, But if μ<0.5, they are shifted as in figure, and L1 and L2 points close M2 so much.
The equilibrium points and their dynamics provide great advantageous to space mission designers. There are some peroiodic (Halo) and quasi-periodic (Lissajous) orbits around these points, which can be used such as advance parking and science orbits. Furthermore, a bunch of asymptotic trajectories, which are asymptotic to Halo orbits, forms invariant manifold tubes, which are also very functional in space missions.
Matlab Code for ploting Lagrange Points is given below. Copy-paste the code in your matlab work folder, then write in the commant line as lagrange_points(μ,’L1′,’L2′,’L3′,’L4′,’L5′). μ is mass ratio, let it be o.2, for example.
function lagrange_points(mue,lp1,lp2,lp3,lp4,lp5) y1=@(x)x-(1-mue)/(x+mue)^2+mue/(x-1+mue)^2; y2=@(x)x-(1-mue)/(x+mue)^2-mue/(x-1+mue)^2; y3=@(x)x+(1-mue)/(x+mue)^2+mue/(x-1+mue)^2; Lp1=fzero(y1,0); Lp2=fzero(y2,0); Lp3=fzero(y3,0); Lp4x=0.5-mue; Lp4y=0.5*sqrt(3); Lp5x=0.5-mue; Lp5y=-0.5*sqrt(3); hold on switch lp1 case 'L1' plot(Lp1,0,'r.');text(Lp1,0,['L_1'],'HorizontalAlignment','right'); otherwise end switch lp2 case 'L2' plot(Lp2,0,'r.');text(Lp2,0,['L_2'],'HorizontalAlignment','right'); otherwise end switch lp3 case 'L3' plot(Lp3,0,'r.');text(Lp3,0,['L_3'],'HorizontalAlignment','right'); otherwise end switch lp4 case 'L4' plot(Lp4x,Lp4y,'g.');text(Lp4x,Lp4y,['L_4'],'HorizontalAlignment','right'); otherwise end switch lp5 case 'L5' plot(Lp5x,Lp5y,'g.');text(Lp5x,Lp5y,['L_5'],'HorizontalAlignment','right'); otherwise end grid on