Distance of L3 Langarian point

Description

In celestial mechanics, the Lagrangian points (also Lagrange points, L-points, or libration points) are positions in an orbital configuration of two large bodies, wherein a small object, affected only by the gravitational forces from the two larger objects, will maintain its position relative to them. The Lagrange points mark positions where the combined gravitational pull of the two large masses provides precisely the centripetal force required to orbit at the same angular velocity (essentially, the speed of the orbit) and thus remain in the same relative position. There are five such points, labeled L1 to L5, all in the orbital plane of the two large bodies. The first three are on the line through the two large bodies; the last two, L4 and L5, each form an equilateral triangle with the two large bodies. The two latter points are stable, which implies that objects can orbit around them in a rotating coordinate system tied to the two large bodies.

Several planets have satellites near their L4 and L5 points (trojans) with respect to the Sun, with Jupiter in particular having more than a million of these. Artificial satellites have been placed at L1 and L2 with respect to the Sun and Earth, and Earth and the Moon, for various purposes, and the Lagrangian points have been proposed for a variety of future uses in space exploration.

Considering the mass of the smaller object (M2) being much smaller than the mass of the larger object (M1), then L2 is at approximately the radius of the Hill sphere, we get an approximation of r for L3, by solving the equation of gravitation providing the centripetal force(with parameters defined as for the L1 and L2 cases except that r now indicates how much closer L3 is to the more massive object than the smaller object.).

Related formulas

Variables

rdistance of L1 from smaller object (m)
Rdistance between 2 main objects (m)
M2mass of small obgect (kg)
M1mass of large object (kg)