A Trojan relationship is when an astronomical object A is orbited by astronomical object B, and a third object, C, orbits A at the same distance as B and 60 degrees ahead of or behind B, in the L4 or L5 position.

In our solar system, hundreds of asteroids (C) have Trojan orbits relative to Jupiter (B) and the Sun (A), and there are 17 known Neptunian Trojans, 4 known Martian Trojans, 2 known Uranian Trojans, 1 Known Earth Trojan, and 1 temporary Venusian Trojan.

Thetys, a moon of Saturn, has two Trojan moons, Telesto and Calypso, and Dione, another moon of Saturn, has two Trojan moons, Helene and Polydeuces.

It is obvious that in these cases object A is many times more massive than object B, which in turn is many times more massive than object C.

For example, Mars, the smallest planet with Trojan asteroids, has a mass 0.3227 X 10 to the minus 6th power, or 0.0000003277 the mass of the Sun, while Jupiter, the most massive planet with Trojan asteroids, has a mass of 0.0009547919 the mass of the Sun.

Tethys and Dione, moons of Saturn, are thousands and tens of thousands of times as massive as their Trojan moons.

The largest Trojan asteroid, 624 Hektor has a diameter of 225 kilometers. Earth should be over 100,000 times as massive as 624 Hektor and Jupiter is 317.7 times as massive as Earth.

So in our solar system the object A (the Sun or Saturn) ranges from thousands to millions of times as massive as object B (a planet or a large Saturnian moon) and object B ranges from thousands to millions (and possibly billions) of times as massive as object C (an asteroid or a tiny, asteroid-size moon of Saturn).

Obviously, a Trojan orbit can be stable for millions and even billions of years if there are such vast differences in mass between objects A, B, & C.

But in science fiction there are many examples were object C is much larger than an asteroid, and in fact is a planet, often one habitable for Humans.

The common types are systems where A and B are both stars and C is a planet, and systems where A is a star and B and C are both planets.

And for a long time, I believed that it was impossible for object C in a Trojan orbit to be as massive as a planet, regardless of whether object B was a star or a planet.

It is said that as a rule of thumb, a Trojan orbit can be stable if the mass of object A is greater than 100 times the mass of object B and greater than 10,000 times the mass of object C.

[https://en.wikipedia.org/wiki/Trojan\_(astronomy)[1]](https://en.wikipedia.org/wiki/Trojan_(astronomy)%5B1%5D)

In our solar system the least massive planet, Mercury, is 0.00017 the mass of the most massive planet. Thus Jupiter is 5,882.35 times as massive as Mercury.

Of the five official dwarf planets in our solar system, Ceres has the smallest known mass, 0.0015 the mass of Earth, meaning Earth has 66,666.66 times the mass of Ceres. Since Jupiter has 317.7 times the mass of Earth, it has 2,117,999.9 times the mass of Ceres.

An object has planetary mass if it is massive enough to pull itself into a regular, more or less spherical shape, and is less massive than a star.

An object more massive than about 13 times the mass of Jupiter will have great enough core pressure and temperature to fuse hydrogen, and thus be a star. But the least massive stars can only fuse the rare hydrogen isotope deuterium and thus are very, very dim. They are called brown dwarfs and can be classified as planets, stars, or neither.

The minimum mass of a brown dwarf might be between 11 and 25 times the mass of Jupiter, which is enough to make the largest possible planet tens of millions of times as massive as the least massive possible planet.

The maximum mass for a brown dwarf, and the minimum mass for a full-fledged star is believed to be about 75 to 80 times the mass of Jupiter, and thus about 0.0716 to 0.076 times the mass of the Sun. The least massive known normal star, VB10, or Van Biesbroeck's Star, has a mass of about 0.075 that of the Sun, or the Sun is 13.333 times as massive as VB10.

It has been calculated that a star above about 150 times the mass of the Sun, would have fusion reactions so strong they would blow the star apart. But there a few stars which might have masses higher than 150 times the mass of the Sun, listed here:

[https://en.wikipedia.org/wiki/List\_of\_most\_massive\_stars[2]](https://en.wikipedia.org/wiki/List_of_most_massive_stars%5B2%5D)

RMC 136a1 is both the most luminous known star and the most massive, allegedly having 315 times the mass of the Sun - or between 265 to 375 times the mass of the Sun.

Thus the most massive stars should be 1,999.99 times as massive as the least massive stars, or possibly even as much as 4,999.99 times as massive.

So the possible mass range of stars, and the possible mass range of planets is such that as a rule of thumb, there could be a stable Trojan system with a star as object A, a planet as object c, and either a star or a planet as object B.

So far, so good.

We already know there can be a Trojan system were object B is a habitable planet and object C is a tiny asteroid. But there is nothing glamorous or interesting about tiny asteroids in Trojan orbits relative to a habitable planet, except for the possible advantages of mining those asteroids.

There can be a plausible story involving mining operations or scientific research of some kind on a lifeless and uninhabitable planet in a Trojan orbit, with object B being either a star or another lifeless and uninhabitable planet.

But what about a system in which object C is a habitable planet?

If we arbitrarily assume that a planet habitable for Humans should have a mass between 0.5 and 2.0 that of Earth (planets outside that range might be habitable for other forms of life, including some from Earth), then a giant planet might be as much as 11,913.75 to 50,832 times as massive as an Earth-like and habitable planet.

So a Trojan system with a star as A, a giant planet as B, and an Earth-like and habitable planet as C, seems to fit within the 1:0.01:0.0001 rule of thumb.

And yesterday, March 5, 2018, I found some online scientific papers discussing hypothetical Trojan planets.

[http://www.univie.ac.at/adg/Research/exotro/exotro.html[3]](http://www.univie.ac.at/adg/Research/exotro/exotro.html%5B3%5D)

[https://arxiv.org/abs/1307.7161[4]](https://arxiv.org/abs/1307.7161%5B4%5D)

[https://link.springer.com/article/10.1007/s10569-009-9210-9[5]](https://link.springer.com/article/10.1007/s10569-009-9210-9%5B5%5D)

And those articles consider Earth-mass and potentially habitable Trojan planets possible.

So it appears that a system with a star, a giant planet, and an Earth-mass planet can have a stable Trojan orbital configuration, despite what I believed for a long time.

What about a system with two stars and an Earth mass and a habitable planet in a stable Trojan orbital configuration?

That is a bit more complicated. The Earth is about 4,550,000,000 years old. The first microscopic life appeared about 4,100,000,000 to 3,500,000,000 years ago, but Earth was not yet habitable for Humans.

Lifeforms began to produce oxygen about 2,000,000,000 years ago and oxygen levels in the atmosphere rose to breathable levels by about 500,000,000 years ago when Earth was about 4,000,000,000 years old.

Complex multicellular life appeared about 580,000,000 years ago and the first land organisms appeared about 480,000,000 years ago.

If one assumes that the evolution of life could be much faster or slower on different planets, one might assume that the minimum possible age for a planet to have a breathable oxygen-rich atmosphere and multicellular lifeforms on land, and thus be habitable for Humans, might be 3,000,000 years.

A F5V class main sequence star would have a lifetime on the main sequence of 3,440,000,000 years before becoming a red giant star. Thus some F5V stars can have planets over 3,000,000,000 years old with advanced life and breathable air, planets suitable for being colonized by Humans or having native intelligent beings. Such a star would have a mass of 1.4 times the Sun or 18.666 times the mass of the least massive stars.

[http://www.world-builders.org/lessons/less/les1/StarTables.html[6]](http://www.world-builders.org/lessons/less/les1/StarTables.html%5B6%5D)

This blog post says that in a Trojan system with two stars and a planet the larger star has to be at least 25 times as massive as the least massive star, a ratio of 1:04.

[https://planetplanet.net/2016/11/07/the-ultimate-trojan-2-star-planetary-system/[7]](https://planetplanet.net/2016/11/07/the-ultimate-trojan-2-star-planetary-system/%5B7%5D)

This is a much smaller mass difference that the 1:01 rule of thumb ratio.

In this system, the smaller star has 0.08 the Mass of the Sun and the larger star thus must have at least 2 times the mass of the Sun.

> 
> For my Trojan star-star systems I’m choosing a puny star at the border between brown dwarfs and stars, at 8% the mass of the Sun. This keeps the mass of the high-mass star as low as possible. This, in turn, will allow for a long-lived high-mass star. The high-mass star is an A star twice as massive as the Sun and about 12 times as bright (see here). This star has a lifetime of about 2 billion years as a “normal” main sequence star.
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[https://planetplanet.net/2016/11/07/the-ultimate-trojan-2-star-planetary-system/[7]](https://planetplanet.net/2016/11/07/the-ultimate-trojan-2-star-planetary-system/%5B7%5D)

With a lifetime of about 2,000,000,000 years, the brighter star will not remain on the main sequence long enough for the planet(s) in the Trojan positions to develop advanced life and become habitable for Humans, unless life on those planets develops more than twice as fast as life on Earth for some reason. That is why spectral type A stars are not considered suitable for having habitable planets.

So, from what I have learned today, I would say that a Trojan system with a suitable type star as object A, a habitable planet as object C, and either a giant planet or a brown dwarf as object B may be mathematically possible.

But it still seems that a Trojan system with a suitable type star as object A, a habitable planet as object C, and a smaller star as object B is mathematically impossible, because the two stars could not have the proper mass ratio for a stable Trojan system.

So can anyone clarify whether the two types of Trojan systems with habitable planets are stable or not?

[http://www.world-builders.org/lessons/less/les1/StarTables.html[6]](http://www.world-builders.org/lessons/less/les1/StarTables.html%5B6%5D)