User:Margrain

the method I used to figure out the eccentric anomaly is something I came up with in one day. first to define the orbit take a variable 'a' this will mark the locus points of the ellipse being: (a,0) and (-a,0). then another variable 'b' this will mark the height of the orbit. the ellipse was calculated by taking sqrt(x^2+y^2+a^2+2xa)+sqrt(x^2+y^2+a^2-2xa)<2|a|+2|b|. then I took the circle that encompases the orbit being: sqrt(x^2+y^2)=|a|+|b|. I then took the highest point from the orbit which was calculated by finding the midpoint between the locus points which would have a distance of a, then say that 2sqrt(a^2+s^2)=2|a|+2|b| which can be turned into sqrt(a^2+s^2)=|a|+|b|, now it's time to factorize it. a^2+s^2=a^2+b^2+2|ab| s^2=b^2+2|ab| s=sqrt(b^2+2|ab|) now that I have 's' which is the highest point of the orbit then a have to scale it with the circle by finding the highest point of the circle which is |a|+|b|, this means that the scaling factor is (|a|+|b|)/(sqrt(b^2+|2ab|)). I use this to change the formula for the ellipse to being: sqrt(x^2+s^2*y^2)=|a|+|b|. now what I have to do is say that I have the variable 'p' p is the angle of rotation focused on (a,0) so that means that I have to take (cos(p)t+a,sin(p)t), this will have to be on the ellipse so this turns into sqrt((cos(p)t+2a)^2+(sin(p)t)^2)+t=2|a|+2|b|. now I have to start factorizing it. sqrt((cos(p)t+2a)^2+(sin(p)t)^2)=2|a|+2|b|-t (cos(p)t+2a)^2+(sin(p)t)^2=4a^2+4b^2+t^2+|8ab|-4t|a|-4t|b| t^2+4a^2+4cos(p)ta=4a^2+4b^2+t^2+|8ab|-4t|a|-4t|b| 4cos(p)ta+4t|a|+4t|b|=4b^2+|8ab| cos(p)ta+t|a|+t|b|=b^2+|2ab| t=(b^2+|2ab|)/(cos(p)a+|a|+|b|) now that we have t we can say that: (cos(p)t+a,sin(p)t) will align with a point on the orbit that has a rotation of p focused at a. to find E we have to scale the planet onto the circle so by multiplying the y cordinate of the planet by s we get: (cos(p)t+a,sin(p)ts), now all we need to do is find the arc of the angle. since the midpoint of the locus's are at (0,0) we can just take the arctan of (sin(p)ts,cos(p)t+a) this will give us the angle E solving for the eccentric anomaly using an algebraic formula.