# What are LaGrange Points, and Why do they Matter?

We just celebrated 20 years of continuous human presence in space. Plans to replace the ISS are well underway (the station was designed to last 15 years and is starting to show problems).

That presence, however, is in low earth orbit. This makes sense. The further from Earth we go, the more expensive it is in terms of fuel and time spent in transit by astronauts.

In the long term, as we come up with technologies to reduce those costs, low earth orbit is not the best place to put a space habitat. Especially a space city.

So, where is? The answer boils down to math one by a brilliant man long *before* we even dreamed of going to space.

# Who is this Lagrange Guy, Anyway?

Joseph-Louis Lagrange was a French-Italian mathematician born in Piedmont-Sardinia (which is now part of Italy) in 1736.

Yes.

*1736*. This guy was working in the 18th century when we didn’t even have manned flight. We didn’t even have hot air balloons until 1783.

He was a theoretical mathematician and he was undoubtedly a genius. He wrote papers on algebra, calculus, and number theory.

But he also had an interest in astronomy. He turned his mind to the three body problem, cometary orbits and in one paper he did work that combined multiple astronomical observations.

This man was looking up at the stars.

And back to that three body problem. He didn’t solve it. He accidentally found something *else*. And that something is key to permanently inhabiting space.

What is the Constant-Pattern Solution?

The constant-pattern solution is a special application of mathematics to any three masses with circular (or reasonably circular orbits). He applied this to both his own work and that of a man named Leonard Euler, an he discovered that there is a place where the gravitational forces of two bodies form a balance.

We call those balance points Lagrange Points.

# Where are the Lagrange points?

There are lots of Lagrange points in the solar system, but we care primarily about five of them, which are numbered one through five because humans are boring. (As a note. These should be subscripts, but I have tried everything to get Medium’s interface to acknowledge that).

For our purposes, the L1, L2, and L3 points aren’t much use. They aren’t stable, and the Earth-Sun L3 lies on the far side of the sun, a good distance away.

What we are mostly looking at are the L4 and L5 points, particularly the Earth-Moon L4 and L5.

Draw an equilateral triangle with its base being the line between the two objects. Actually, draw two, one pointing up, one pointing down. The one that’s ahead of the moon is L4, the one behind is L5.

Objects that find their way to these points tend to *stay there*. This does mean they’re full of a lot of debris, but if we’re building space habitats of any size, that’s a trivial problem. Some of it could even be mined for the construction.

A space habitat at L4 or L5 would stay there with no to minimal use of fuel once you get everything there. You can just park the thing. While it would cost more to get there than to Earth orbit, it would cost less to *maintain* your space habitat.

And if you’re building a permanent space habitat where you intend people to live and raise children, something that will be constantly rebuilt to maintain its life, that’s not trivial.

If it ever happens, of course, there must be a plaque there not just to Joseph-Louis Lagrange but to another, less well known mathematician, Leonard Euler, who was the guy who actually realized the L1, L2, and L3 points exists. Lagrange built on that work.

If it ever happens.