# Ordering of the Lagrange points

Is there any basis for the ordering of the L-points? Specifically, is there any particular reason for choosing L1 as the first L-point?

According to Wikipedia, L1-L2-L3 were discovered first, by Euler, prior to Lagrange's work, and L1 is "the most intuitively understood" of them.

• Fun facts: see the list of things named after Leonhard Euler, and especially the passage: Euler's work touched upon so many fields that he is often the earliest written reference on a given matter. In an effort to avoid naming everything after Euler, some discoveries and theorems are attributed to the first person to have discovered them after Euler. – Jules Oct 4 '17 at 15:40

The points are numbered according to their Jacobi constants. The zero velocity curves open first at L1, the libration point with with the lowest energy (highest value of Jacobi.) They open next at L2, and then at L3. The ZVCs leave the plane at last at the L4 and L5 points simultaneously.

JC_L1 > JC_L2 > JC_L3 > JC_L4 = JC_L5

• Interesting! I didn't know this. In the special case where mass of P1 = P2, the L3 and L2 would open simultaneously. – HopDavid Dec 15 '17 at 14:42
• You're right! And in the Hill restricted 3 Body case, where the mass of P2 is infinitesimal compared to the mass of P1, the L1 and L2 points open simultaneously. – Diane Dec 15 '17 at 18:05
• I've always assumed that to be true. For example both Mars Phobos L1 and L2 seem to be equidistant from Phobos, each about 3.5 km from the ends of Phobos if my math is right. The acceleration gradient seems to be less lopsided as you close in on the orbiting body. But I haven't seen a formal proof. Of course it is true at the limiting example of p2=0. – HopDavid Dec 16 '17 at 17:17
• Hi @Diane do you have the source handy for this Figure 2.3? If you find it, can you add a link or cite the reference? It's probably pretty standard, reminds me of Figure 2.4 in this thesis as well. Thanks! – uhoh Mar 19 '19 at 1:36
• @uhoh, if you can forgive me for referencing my own thesis: engineering.purdue.edu/people/kathleen.howell.1/Publications/… – Diane Mar 20 '19 at 12:40