# Lagrange L4 L5 points and perifocal plane

I have 2 satellites at the L4 and L5 points and these are watching an object. Each satellite provides the angle to the object from its own position from a line parallel to the $\text{x-axis}$ of Earths perifocal plane (assume that the orbit of earth is circular and system Sun-Earth).

My questions are:

1. Using earths perifocal frame, the positions of the satellites are: $r=[r\cos f,r\sin f,0]$ ?

2. If i have the L4 and L5 positions (angles for example $(55,45)$) and after some period the angles are $(58,40)$ how can I find the coordinates of the object?

• Hi George. Welcome to Physics.SE. Here, we use an unique TeX markup called MathJax. The markup is very much helpful in understanding equations, etc. Please have a look here for an introductory, or atleast have a look at our FAQ for an overview. For now, I'll help revising your post. Commented Feb 2, 2013 at 13:05

Question 1: If we assume that the center of mass coincides with the origin of coordinates and is centered on the Sun, we know that coordinates of Sun-Earth-$L_4$ and Sun-Earth-$L_5$ are the vertices of equilateral triangles(Proved by Lagrange 1772). So the angle between the line joining the Sun and Earth and the line joining Earth and $L_4$ or $L_5$ are $\pm 60\deg$. Measuring the position of $L_4$ from the Sun gives $(r\cos(60),r\sin(60),0)$ and $L_5$ gives $(r\cos(60),-r\sin(60),0)$. Notice there is no difference in this case between measuring from Earth or the Sun. The numerical value of the $L_4$ and $L_5$ coordinates are the same in either case resulting from the fact that they are vertices of an equilateral triangle. The x-coordinate splits the distance between the Earth and Sun in half. You can see this by rotating a Sun centered frame of reference at the orbital frequency of the earth or alternatively by rotating an Earth centered coordinate system at the same frequency. Everything of interest in this problem is stationary in these rotating frames.
Question 2: Care must be taken now on the location of the origin of coordinates. Assuming we measure coordinates from the Earth, Locating the object is accomplished by drawing 2 lines and finding their intersection. These lines emanate from $L_4$ and $L_5$ and pass through the object. Let the Earth be located at point $E$, and the object at point $O$. Initially the problem states that $\angle OL_4E = 55\deg$ and $\angle OL_5E = 45\deg$. Now use the properties of equilateral triangles, some Euclidean geometry and the radius of the circular orbit to calculate the coordinates of the object. Hope this helps.