# How does an object at L1 stay at L1?

According to NOAA

A million miles away, DSCOVR orbits a unique location called Lagrange point 1, or L1. This point is a gravity neutral point in space, allowing DSCOVR to essentially hover between the sun and Earth at all times.

With it being at L1, it's orbit around the sun is slightly shorter than Earth's. So my uninformed gut feeling tells me it should orbit the sun in a few days less than Earth does, but it clearly doesn't.

How is this object at L1 kept at L1?

Could it be that the gravity of the Earth acting on it in a really convenient way? Or does it use fuel? What fuel? Or is it a property of the L1 point that I have misunderstood that an object will naturally stay there once in position?

• Does this answer your question? Why are telescopes positioned in Lagrange points? – usernumber Nov 12 '20 at 15:22
• @usernumber one of the answers describes the L2 point, and notes in passing that fuel is required to keep it there. That question in general is about why humans would choose to put satellites at these points, my question is maybe badly worded, but I really aimed to ask how it stays there, I will edit the question to make this clearer – Darren H Nov 12 '20 at 16:08
• Tosic's answer summarizes the situation. I'd use different words to say the same thing. These are orbits around the Sun, a little closer or farther than Earths. Due to their closeness to Earth (1.5 million km vs 150 km to the Sun) the Earth's gravity slightly speeds up the outer L2 and slightly slows down the inner L1 object. The effects happens if the objects are above or below the ecliptic by up to 0.5 million km, and those objects wiggle up and down. From the perspective of the Earth they seem to make circles, what we call "halo orbits". – uhoh Nov 12 '20 at 22:07

I'll give a more grasp the concept answer if you don't mind. Consider the discovery of the planet Neptune.

Neptune was discovered because it was observed that Uranus wasn't moving as fast as orbital mechanics suggested it should be moving. By calculating how much Uranus had slowed down, an estimate was made for where an undiscovered planet should be and Neptune was identified very soon after.

Uranus is well inside Neptune's orbit, nearly twice as close to the sun and very far inside of Neptune's L1, but Neptune was still able affect Uranus' orbit enough to get noticed. Props to Newtonian mechanics. It works exceptionally well for a lot of measurements.

Loosely speaking, this type of orbital variation, L1 semi-stability or orbital perturbations like the slowing of Uranus' movement is a subset of the 3-body problem and that's actually more real. Observed orbits don't precisely follow Newton's or Kepler's formulas because of other massive bodies and perturbations. 3-body problem math gets crazy complicated, but the specific example of L1 can be explained using basic arguments without much math.

Earth's L1 point is about 1% closer to the sun or about .99 Astronomical Units or "AU". 1% closer gives an orbital velocity of a little over 0.5% faster and an orbital period of a bit over 1.5% faster or an orbital period of about 359.8 days, about 5.4 days shorter. That's what .99 AU circular Kepler orbit should do.

Consider an object at .99 AU, and a period of 359.8 days orbiting the Sun just inside the Earth's orbit - assuming both with circular orbits. Now imagine that object passes the Earth. It's relatively small so it doesn't change the Earth's orbit much, but passing that close to the Earth would give it a considerable gravitational tug, basically tossing it out of it's orbit. .99 AU is too close for a 2nd Kepler orbit and as a result, it's quickly tossed into a new orbit.

Two bodies can't orbit a massive central body with just 1% variation in their distance unless both bodies are very low mass (for example, Janus and Epimetheus are a rare exception to this).

Earth doesn't have low mass, so an object with a Kepler orbit at .99 AU would get tossed about after just one pass. In other words, if the object orbited as fast as your "gut" tells you it should, then it wouldn't be stable at all.

Now consider an object at .99 AU but orbiting the sun too slowly to maintain it's orbit. Slower velocity means the object falls in towards the sun, speeding up as it falls, entering a new, shorter period and more elliptical orbit. That's where the balance comes in. The slower velocity means the object wants to fall towards the Sun but being between the Earth and the Sun, Earth's gravity pulls it towards the Earth. The point where those two forces are equal is the L1 point, where it's no longer a Kepler orbit, but a 3 body orbit, slower than a Kepler orbit, but pulled in opposite cancelling ways by the Earth and the Sun. The "too slow" orbit is still unstable, but it's considerably more stable than a Kepler orbit in the same location would be.

With Earth's perturbed elliptical orbit, L1 isn't really a point so much as a smallish region, but on a computer, if we give the Earth a perfect circular orbit around the Sun with no other planets, L1 becomes a point of perfect balance. For actual orbits, L1 is always unstable.

Despite it's instability, Lagrange points are very useful low maintenance areas to station telescopes because much less energy is needed to keep the telescopes there. This only works if the central body (the Sun) is at least 24.96 times the mass of the orbiting, less massive body (the Earth)

Tosic's answer summarizes the situation. I'd use different words to say the same thing. These are orbits around the Sun, a little closer or farther than Earths. Due to their closeness to Earth (1.5 million km vs 150 km to the Sun) the Earth's gravity slightly speeds up the outer L2 and slightly slows down the inner L1 object.

The effects still happens if the objects are above or below the ecliptic by up to 0.5 million km, and those objects wiggle up and down. From the perspective of the Earth they seem to make circles, what we call halo orbits or Lissajous orbits. These are the most commonly used terms, but there are other kinds of three-body orbits that can happen as well.

Since these orbits around the Sun are synchronized to Earth's we can call them Earth-resonant heliocentric orbits.

While space telescopes and Earth observation spacecraft use these, they need to execute small but regular propulsive maneuvers to stay there because these orbits are not long-term stable. Eventually they may drift off into other orbits around the Sun that are no longer connected to Earth.

Lagrange points are points where the angular speed of a body with negligible mass would be equal to that of the secondary body (which is orbiting the primary), here the sec. body is Earth and the primary is the Sun. This means the period is not a few days shorter due to the Earth acting on the body and decreasing the centripetal force it gets from the Sun. A diagram and more explanations (as well as a derivation of position of $$L_1$$) can be found on this link.