L1 planet

Posted on March 14th, 2014 by george.
Categories: Uncategorized.

Q: Is it possible for humans to exist on a planet at the L1 Lagrange point of a gas giant orbiting a star?

A: It’s definitely possible, but it’s not likely to be a stable orbit over the lifetime of a solar system. In fact, no Lagrange point is inherently stable except in special circumstances. Most of the time they require station keeping to counteract gravitational perturbations from other planets. Orbits around L4 and L5 last the longest, which is why you see so many Trojan asteroids leading and trailing Jupiter. L4 and L5 can be stable for billions of years if the body in the Lagrange point is 25 times less massive than the primary. Earth is 318 times less massive than Jupiter, so Earth could be a Jupiter trojan long term without interference. But the L1 point is not as stable as L4 and L5. And since there’s no such thing as station keeping for a planet, your planet would only stay in the L1 point for something on the order of millions, not billions, of years. This may sound like a long time, but it’s the blink of an eye compared to the necessary timescales for evolution, geology, and solar system lifespan. Intelligent life would not have time to evolve on a planet in L1 before it drifted out to wander the solar system. So you’ll need to explain the presence of humans there. A primitive species could have been dropped off by an intelligent race. Otherwise life would have had to evolve in a different orbit (likely a stable one to set the stage for life), and then once humans existed, the planet somehow migrated to this L1.

The problem is it’s difficult to destabilize a planet in a circular orbit, get it to a Lagrange point, then make it stay there. You would need to conjure a very unlikely scenario. Say, for instance, that another star passed nearby the Oort cloud of this hypothetical solar system and perturbed several planet-sized frozen bodies hanging out in the boonies. One massive interloper comes zipping in from the Oort cloud and whizzes by your planet, kicking it out to where it starts interacting with the gas giant. But that Oort cloud planet wasn’t the only one kicked loose, and another one comes careening through the inner solar system at the precise speed and trajectory such that when your planet reaches the gas giant L1, it whizzes by your planet and robs it of the exact amount of energy to stabilize its orbit in L1. This is a one-in-a-quadrillion shot (I made up those odds), but it is strictly possible. It also avoids the need for an impact. Impacts are often what would cause a planet to move around, spin up, form moons, and stabilize in a different orbit. The only other origins I can think of are: 1) if your planet starts out as a binary planet, the binary system is perturbed to L1, and another perturbation kicks off the twin, leaving your planet alone and quasi-stable at L1 and 2) a hyper-advanced alien race has the ability to move planets in their orbits without disturbing them, and does this to your planet just for fun.

Long story short if you want your humans to evolve on this planet they won’t be able to survive an impact-based restructuring of their orbit to L1. Most of the time an impact large enough to change orbital parameters will result in partial to total destruction of the original planet. Even if a collision could move the planet to L1 without destroying it, the impact would likely turn the entire surface of the planet into molten lava. So you’ll need to explain the migration with momentum transfers. This may sound fanciful, but keep in mind we know this happens for capturing moons. Our Moon is the result of a giant impact, as is Pluto’s moon Charon. Phobos and Deimos may be asteroids captured into Martian orbit, or may be the result of a giant impact. There is evidence that many small moons of Jupiter and Saturn are captured asteroids or Kuiper Belt Objects, and we are all but certain that Neptune’s moon Triton is a captured KBO. Triton is in a retrograde orbit, meaning it spins the opposite way around Neptune from the direction Neptune is spinning and from the direction Neptune is moving around the Sun. There is no way to explain a retrograde moon except capture, even though capture is very difficult to piece together gravitationally. Scientists are still studying how Triton was stabilized in its current orbit, whether it ejected a moon of its own, suffered an impact near Neptune, grazed Neptune’s atmosphere, or circularized due to tidal dissipation.

So your planet is possible, though it will only remain at L1 for a few million years at best. Based on our discovery of super-Earths you could make it much bigger than Earth and have all your humans be super strong. Or you could leave it the same size and avoid having to answer questions about all the hydrogen in its super-thick atmosphere. If you give your planet a 24-hour rotation period it will have the same day/night cycle as Earth, and if you give it the same axial tilt it should enjoy the same seasons. If the planet evolved in one part of a habitable zone and migrated to this L1 in another part of the zone (we’ll leave alone the stability of such an orbit so close to the L1 of a gas giant), your humans will have had to weather unfathomable climate change, either from a frozen planet to a steamy jungle/dry desert or vice versa, with snowball periods if the migration doesn’t start and end in a single orbit. It’s probably easier to go from warm to cold since you don’t have to pass the gas giant in the process and risk ejection from the solar system. If you make it a super-Earth it may be difficult to explain away the radiation from the gas giant. You might try claiming that the magnetic field is much stronger that Earth’s because the molten iron core is larger (if this planet happened to enjoy a core donation similar to the one Theia gave Earth). The problem is that the higher gravity of a super-Earth may preclude the interior differentiation that powers Earth’s magnetic-field-generating dynamo. Or you could just invoke magnesium oxide and say, “Trust me, it has a magnetic field strong enough to withstand gas giant radiation.” If this were the case, you could get amazingly intense aurorae on this planet, perhaps visible at much lower latitudes than we see them on Earth.

The phase of the gas giant would never change; it would always appear full at night, would always reach its zenith at midnight, and would never be visible at the same time the sun was visible. If the planet ends up following a Lissajous orbit around L1, you could see some interesting effects: the shadow of your planet might fall on the gas giant from time to time and trace different paths across its face. You may be aligned so perfectly that you’re in perpetual transit, your shadow a permanent beauty mark on the face of the gas giant.

It’s a very interesting idea. It just takes some explaining and your planet won’t stay there very long. But we’ve evolved from hunter gatherers to spacewalkers in only 50,000 years, so your humans could definitely go through their formative stages and reach Type II status before the planet was kicked out of L1.

1 comment.

Out on the range

Posted on March 10th, 2014 by george.
Categories: Uncategorized.

Got the following question from my friend Sean and thought I’d repost my answer here for those of you looking for a relatively simple explanation of Lagrange points.

Question:

Hypothetically, lets say there are two bodies of equal mass. We’ll assume they’re identical in every respect (size, density, etc). We’ll also assume that the distance between them is always constant. It seems reasonable that exactly half way in-between them their gravitational pull would be negated by the opposite mass. Therefore, if an object were to fly precisely in-between them at that point, it would not be affected by either’s gravitation pull. To the best of your knowledge is this presumption true? And if so, is there a term to describe such a zone of gravitational equilibrium? Or would an object be caught in between?

Answer:

Your intuition is correct. Such a spot in space is known as a libration point. Where it gets interesting is when you mix in orbital mechanics. God doesn’t reach down with two giant hands and hold two spheres motionless in space. The laws of gravitation tend to put everything into motion, always accelerating unless balanced by another force. On Earth, our bodies want desperately to accelerate toward the center, but the surface of the Earth provides a force that exactly balances that gravitational pull and holds us in one place. The only way two massive bodies can maintain a constant distance between each other is to orbit one another in a shared circular orbit (as opposed to an elliptical orbit, where the distance would be constantly changing).

This is the simplest case, called a two-body system. Each object wants to accelerate directly toward the other, but each also has a large amount of momentum directed ninety degrees away from the other body. The centripetal (outward) acceleration due to that momentum exactly balances the gravitational pull between the two objects. So as the sphere moves forward, instead of traveling in a straight line (which it would do if the other body were not there) it gets pulled a little toward its companion. Simultaneously, the companion traveling in the opposite direction gets pulled a little toward the original sphere. Add each increment of forward and lateral motion as you march forward in time step by step, and eventually the two bodies trace a path about a common central point called a barycenter. Here’s how two identical bodies can travel in non-circular orbits around a common barycenter:

This is what it looks like when the masses are different.

The barycenter of the Earth-Sun system is inside the Sun itself because the Sun is so much more massive. It starts to look like the maps of the solar system you’re accustomed to seeing.

Even though the planet is small, you can see that it still causes its central star to wobble. This is one of the methods we use to find planets around other stars. When a planet is large and close to its sun, we can see the wobble with our current telescopes.

In the special case of symmetric bodies in a circular orbit, the barycenter (center of mass) is a libration point. If you placed a third body there, it would not accelerate toward either of the original two spheres. This is not because it is unaffected by the gravitational pull of either primary body, but because those two pulls balance equally.

Now here’s where it gets fun. The gravitational forces and the centripetal acceleration don’t just balance between the two bodies. When you invoke Kepler’s equations of motion and Newton’s law of gravitation and solve for the points in space where the gravitational pull due to both bodies matches the centripetal force needed to orbit in lockstep with them, you arrive at FIVE libration points. Euler discovered the first three that exist in a straight line traced between the two main bodies, but Lagrange discovered four and five, so they named the libration points of a three-body system after him. The one in the center of two identical bodies is the easiest to conceive of, since equal forces pull in opposite directions. This is called the L1 point. As soon as you consider a system with bodies of different masses, however, the barycenter and the L1 point part ways; the former moves toward the heavier body, and the latter marches closer to the lighter one. But there are also points on the outside of the two bodies where all the forces balance.

Let’s switch to a Sun/Earth model so it’s easier to understand. The further you orbit from a central body (the Sun in this case), the slower your orbital speed (but the higher your orbital potential energy). If you aligned a Sun-orbiting satellite with Earth at a much greater distance from the Sun and let it go, the Earth would start outpacing it on the way around the Sun. For example, Mars does not co-rotate with Earth. It follows Kepler’s laws and orbits with a period of about 1.9 Earth years. This is not to say the Earth does not exert gravitational forces on Mars. It most definitely does. The Sun, all the planets, every asteroid and all objects in the solar system (and, to an unmeasurable degree, the universe) feel the Earth’s gravitational pull. The reason the Sun dominates this neighborhood is because it is the most massive. Remember, too, that the Sun is in orbit around the center of the Milky Way galaxy. The reason we don’t orbit the galaxy directly is because we are so close to the Sun relative to the galactic center. Long story short the Earth-generated forces felt on Mars are not as large as the pull of the Sun, and Earth is so far from Mars (even at closest approach) that the overall effect on Mars is tiny over the age of the solar system.

Now if you move the Sun-orbiting satellite close enough to the Earth, its gravitational pull eventually gets strong enough that it drags the satellite along behind it, even though the satellite is still orbiting the Sun in a higher orbit, and should be orbiting more slowly than the Earth. This is L2. For reference, it is outside of the Moon’s orbital distance. Move the satellite too close to Earth, though, and it will orbit Earth instead of the Sun.

L3 is a little harder to grasp because it’s all the way on the other side of the Sun from the Earth. But it’s still in a higher orbit than Earth, so it should be going slower. Once again, however, Earth pulls the satellite along and causes it to orbit at Earth speed in its higher orbit where it should have a longer period and be outpaced by Earth. This is in spite of the fact that Earth is two astronomical units (186 million miles) away and the force due to Earth’s gravity traces a line through the Sun itself! If the Earth were the only planet orbiting the Sun, L3 would be a stable libration point, meaning if the satellite drifted the forces would conspire to bring it back (as long as you ignore solar radiation pressure, the fact that the Earth is not a perfect sphere and that the Sun is squashed, among other perturbing effects). Maintaining your position near a Lagrange point (or in any desired orbit) is called “station keeping”. But remember how the Earth is on the other side of the Sun from L3? And remember that Venus exists? In real life the Earth-Sun L3 point is unstable, because the gravitational pull of Venus is much larger than Earth’s pull on a fictional satellite at L3, no matter where Venus is in its faster-moving orbit. Venus will quickly destabilize a satellite at L3.

L4 and L5 are wacky and hard to conceive, which is why Euler missed them and Lagrange got all the points named after him when he discovered four and five. These two are at the point where the distance between the satellite and the Sun and the satellite and Earth are equal to the Earth-Sun distance, forming an equilateral triangle. The forces once again balance and the satellite is held in a higher orbit while still rotating around the Sun at Earth speed. The best way I know to think about L4 and L5 is that the presence of the Earth and its small gravitational pull causes the satellite to orbit the Earth-Sun barycenter, not the center of the Sun. It’s not as easy to understand as the case where the satellite is “pulled along” as in L2 and L3, but if you stare at this diagram long enough, you’ll see that the presence of the Earth is effectively shortening the radial distance between the satellite and the center. As we know from Kepler, shorter radial distance equals faster orbital speed, so the satellite orbits as fast as Earth even though it’s further from the Sun. This diagram shows the Earth and the Moon, but the physics is the same.

Here are all the Lagrange points together. Haw haw haw haw.

Bonus: you can orbit a Lagrange point in what’s called a “halo orbit”. There’s nothing there for you to revolve around, of course, but your momentum and the gravitational forces of the other two bodies balance perfectly to conspire to make it seem as if there is. NASA’s multi-billion dollar James Webb Space Telescope plans to use a halo orbit around the Earth-Sun L2.

Bonus bonus: JWST looks like an imperial star destroyer.

0 comments.