# How do Lagrange points L2 and L3 form?

Five Lagrange points form between any two bodies in the space (say Sun and Earth). According to my understanding, Lagrange points L1, L4 and L5 can form because gravitation pull can cancel out here as these are between Sun and Earth (where gravitational pull is in two different directions).

However, I could not understand why L2 and L3 Lagrange points form in the first place.

I couldn't understand why would the gravitational pull cancel out at the L2 and L3 as they are not in between Earth and Sun and the gravitational pull (of Earth and Sun) at these points are at the same direction. As gravitation pull on L2 and L3 is in same direction shouldn't the gravitation pull at these points be high instead?

• Scott Manley just posted an outstanding video explaining Lagrange points and the physics of why they exist. Outstanding visuals. This is link-only, so just a comment: youtube.com/watch?v=7PHvDj4TDfM Oct 18 '21 at 13:16
• Luckily Scott Manley has done a video on this youtube.com/watch?v=7PHvDj4TDfM&t=633s Oct 18 '21 at 20:07
• @Stewbob +1 - this is one of those things where a good picture is worth even more than a thousand words.
– J...
Oct 18 '21 at 20:13
• @J... I'm not a fan of images, particularly at the SE network. One of the most brilliant people I've worked with is blind. Images without alt-text are meaningless to him. If an image truly is worth more than a thousand words, it had <expletive deleted> better have more than thousand words of alt-text so as to make the image accessible to the visually impaired. The default alt-text for an image on the SE network is "Enter image description". I have taught myself to try stay away from images. Oct 23 '21 at 14:26
• @DavidHammen One person's blindness does not invalidate the utility of a visual representation for the overwhelming majority of the remaining sighted people. Nobody suggested we stop writing things, or that we forsake using the alttext in a considerate way, but our eyes are an incredibly powerful tool and telling us it's wrong to use that tool just because a few people in the world are unfortunate enough to have been deprived of it and have learned to conceptualize the world in an entirely different way is ridiculous. If we could provide a tactile 3D model we would, but unfortunately we can't.
– J...
Oct 23 '21 at 14:42

According to my understanding Lagrange points L1, L4 and L5 can form because gravitation pull can cancel out here as these are between Sun and Earth (where gravitational pull is in two different directions).

This is a misconception. The net gravitational attraction is directed toward the Sun at the Sun-Earth L1 point. The point at which gravitational acceleration is canceled is closer to the Earth and is uninteresting.

It is the net gravitational attraction plus the centrifugal acceleration about the Sun-Earth barycenter that cancels, and then only when this is done the perspective of a frame that rotates at one revolution per sidereal year. (There is no centrifugal acceleration in an inertial frame.)

The existence of the linear Lagrange points is perhaps easier to understand from the perspective of an inertial frame whose origin is the center of mass of the two massive bodies (e.g., the Sun and the Earth). The only forces one needs to consider from such a perspective are gravitational acceleration of the third body (which has negligible mass) toward the two massive bodies.

The L1 point is where the net gravitational acceleration is just enough to make the third body orbit the center of mass at the same rate as the rate as the two massive bodies orbit one another. Keep in mind that the two massive bodies orbit one another at $$\omega^2 = \frac{G(M_1+M_2)}{r^3}$$. The net gravitational acceleration points toward to the less massive body at distances close to the less massive body, so obviously not good. There's a point between the two bodies where the net gravitational acceleration is zero. That once again is not good. With increasing distance from the less massive body (decreasing distance from the more massive body), acceleration toward the more massive body dominate. There's a point, the L1 point, at which the net acceleration is exactly enough to make an object of negligible mass orbit the center of mass at the same rate as the two massive bodies orbit one another.

The L2 point is perhaps even easier to understand from the perspective of an inertial frame. Beyond the less massive body, the acceleration vectors toward the two massive bodies both point toward the center of mass. At distances very close to the less massive body the net acceleration is too great. At extreme distances, the net acceleration is near zero. There's an intermediate point, the L2 point, where the net acceleration is exactly right.

A similar line of thinking applies to the L3 point: The acceleration vectors toward the two massive bodies both point toward the center of mass. There is a non-intuitive aspect regarding the L3 point, which is that the distance between the center of mass and the L3 point is greater than the distance between the center of mass and the less massive body, but is less than the distance between the two massive bodies.

Aside: To understand the instability of the linear Lagrange points it is much easier to look at things from the perspective of a rotating frame.

• > "It is the net gravitational attraction plus the centrifugal acceleration about the Sun-Earth barycenter that cancels" ah got it. I didn't consider the centrifugal force into consideration. Oct 18 '21 at 0:40
• @barath: I hadn't either, so I thank you for asking the question. I also think the super-short version of the answer is "Because they are rotating". Oct 18 '21 at 14:42
• It's worth noting the brief that these points don't offer 'free' positioning, they are merely 'more stable' and require less energy to stay stable as they orbit the center of mass. You don't want to fly off into space, but you don't want to fall into the bodies, either, so these points provide 'flatter' space, more consistently than surrounding areas. Oct 18 '21 at 16:02