# Lagrange systems: What is the mass range for a gas giant if I want to place a earth-like planet on L4 and L5

I am wanting to make a 3 world lagrange system around a Sol analog star that includes 1 gas giant and 2 terrestrial worlds with large moons. I want the terrestrial worlds to have a similar gravity to Earth (about 80-120% of Earth's gravity) and also be able to support liquid water and carbon-based life, the moons can range from the mass of Ceres to that of Mars for now.

My question is how large does the gas giant have to be to support 2 of these systems at L4 and L5?

Right now I am being safe and going for super-jupiters, but since I also want a gas giant that can support a habitable moon I am afraid something with the mass and magnetic field of Jupiter might just be far to hostile to life for my liking, so I really want to find the minimum mass of a gas giant that can support this type of system.

To address some of the concerns listed:

First off, the shape of the orbit of the gas giant. It has to be close enough to the star to support liquid water and can't stray out of that zone around the star. For simplicity's sake, it is roughly circular, it would have a super low eccentricity with the periapsis and apoapsis being relatively close together

Secondly, do not worry about whether this system could form over millions of millions of years. Pretty simply, this system is created by the author and there doesn't need to be a natural explanation for how it could've formed. It was born that way, you could say. Similarly, the period of stability doesn't need to be long, though I would like it to naturally be stable across a timescale of many million years simply because I don't want a distant large object, like another star less than 1/2 a lightyear away from destroying it. If it matters, the simple context is that it is part of a four/five-star system. I say four/five because it's really a four-star system, but a small red dwarf star is on a collision course with a distant sub-system of the four-star system and it will make contact in roughly 16 000 years. For the structure of the system, if this matters. We have a one-star Sol analog sub-system far away from an s-type sub-system with 2 sub-systems inside it, one of the sub-systems is a p-type two-star and the other has one star.

Basically: Y-*. If that doesn't make sense, the "Y" is the large subsystem and the "*" is the small sub-system, the "-" is a symbol showing that these two subsystems are connected. The subsystem we are discussing is the "*" subsystem, the P-type system is represented by the "V" part of the "Y" and the small super-subsystem is the "l" part of the "Y". We are discussing the small subsystem which is the "*" in the diagram.

If the fifth star is included, we have (not to scale) this diagram: >Y-* which basically tells you that the oncoming star is on the opposite side of the system as the small sub-system which we are talking about. Whether this structure works is admittedly it's own question but I won't get into that.

When it comes to the stability of this system, I want it to be stable enough that gravitational forces from around 6600 AU away won't ruin the darn thing in short order. The system should hopefully last about 24 000 years in these conditions because that is how long the history runs. Simply put, there doesn't need to be a system stable enough for life to develop, because life is placed there artificially.

Currently, I don't have time to address the rest of the concerns, but I will see if there is anything else I need to address later.

• For millions of years? For a few billion? What are you after here? Dec 18, 2019 at 18:37
• According to From orbitsimulator.com/formulas/LagrangePointFinder.html, "Objects placed on L4 and L5, when given the proper velocity will remain there indefinately as long as the primary object is at least about 25 times as massive as the secondary object." Not sure if that holds true if the secondary is 25x the combined mass of the L4 and L5 objects, or what... Anyway, 25x the mass of one of the L4/L5 planets is probably a lower bound. Dec 18, 2019 at 18:42
• @Matthew That's perhaps going to hold for systems without perturbation by other planets in the system or the moon around the gas giant that the OP referred to. Also the shape of the gas giant's orbit was not specified, nice circular calculations of a 5 body problem even - well there's no neat solutions. Alas that we don't have access to a supercomputer. Dec 18, 2019 at 18:50
• The main problem with this isn't really the stability of the lagrange points, it's that everything we know about planetary formation indicates that you can't have multiple planets form in the same orbital path no matter what their relative sizes are. Under normal conditions you'd never get a planet forming in those lagrange points, all that mass would have been swept up by the gas giant during its formation. Dec 18, 2019 at 19:24
• @MorrisTheCat, I was actually just thinking the same thing. It seems implausible that such a system could arise naturally. However, the OP didn't specify that as a requirement... Dec 18, 2019 at 19:50

Both this site and Wikipedia assert that L4 and L5 are stable for a mass up to about 25 times the mass of the secondary. Since plain old Jupiter is about 318 Earth-masses. Since a) 318 is more than 10x larger than 25, and b) L4/L5 are nowhere near any moons orbiting the secondary, I would have to guess that, if these points are stable at all (and I think they would be if you want to "cheat" and have no other planets in the system), then it is plausible that two Earth-mass planets could sit at L4 and L5 of a planet with "merely" Jovian mass... or even smaller; no need for a "Super-Jupiter".

As We Are Monica notes, I don't have a supercomputer¹ to prove this, but your audience won't either. Ergo, my "official" answer is that this appears to pass a "reasonable suspension of disbelief" test.

(¹ TBH, I'm not sure how a supercomputer would help.)

• Supercomputers (that I was referring to) are all about modeling chaotic systems - ie. those not easily subject to analysis by classical equations of motion, eg. weather systems, planetary motion of more than 2 bodies. +1 Dec 18, 2019 at 19:13
• ...but a discrete 3-body simulation doesn't need a supercomputer. You may argue that my time steps have to be so small that my simulation speed starts approaching real-time, but I don't know how to scale such a simulation unless we're talking about needing an N-body simulation for N larger than a dozen or so. Yes, supercomputers are great for simulations involving thousands (or more) of elements, such as e.g. fluid simulations. Unless you're simulating asteroids, I don't know offhand how an orbital simulation would parallelize. That was my point. Dec 18, 2019 at 19:24
• Well, I guess that all my comments point to the first (unanswered by the OP) comment about the timescale of system stability. Tiny increments of instability add up over time. The question needs to be clearer. (BTW. as it turns out it's at least a seven-body system) Millions of years or billions would still be my issue. Dec 18, 2019 at 19:30
• Sure, but a high-end overclocked gaming rig can crunch a mere 7 bodies faster than the largest (x86-based, at least!) supercomputer. The naïve simulation simply doesn't scale. (But maybe you know of a different approach that does?) Dec 18, 2019 at 19:39
• If we were to simulate this, one presumes that the objective would be to determine as quickly as possible if there are any early indications of long-term instability, rather than trying to determine what will actually happen over billions of years. As we are dealing with a system which we expect to remain relatively static, i.e. with the various bodies maintaining more or less constant relative positions, as opposed to a general three-body problem with chaotic orbits, it shouldn't be hard to spot if it starts to fall apart. Dec 18, 2019 at 19:43

able to support liquid water and carbon-based life

Well I see a problem with you... an estimated 240,000 to 160,000 problems.

These problems are the Trojans, the asteroids that already collect in the L4 and L5 points of planets already. Trojans have fewer large asteroids than the asteroid belts, suggested to be because of collisions between them.

The planet is probably going to survive as a planet. However, it will probably take far longer before becoming habitable.

• I'm not sure this is true. Having a (smaller) planet at the L4/L5 point(s) is going to tend to make any junk in that area very quickly run into the planet, at which point you cease to have asteroids there. Unless you are proposing some reason why more new asteroids would be attracted to L4/L5 than would be attracted to a planet in a different orbit? Dec 18, 2019 at 19:49