Trojan planets are the best! I am a big fan. I love the idea of two planets sharing the same orbit around a star. To me, it’s where physics meets magic. (Remember, we’re talking planets not condoms! See here for a little refresher, and here for another article about Trojans).
As a planet orbits a star there are five special places — called Lagrange points L1 through L5 — in which another body orbits along with the planet. The points we care about are the ones that are stable, called L4 and L5. These are the Trojan points:
Two planets separated by 60 degrees along the same orbit are in each other’s mutual L4/L5 point. It’s a nice stable setup that I used to my advantage in building the Ultimate Solar System.
Now, what if the blue dot in the image was a star, not a planet?
First, we need a little mathematical interlude because there is a limit to when the L4 and L5 points are stable. It’s simple: the big yellow dot in the image must be at least 25 times more massive than the blue dot. That’s it.
The tiniest stars are about 8% as massive as the Sun. Below that mass live the brown dwarfs. The most massive stars are more than 100 times the Sun’s mass. So we can imagine a system of two stars in which a high-mass star is 25 ore more times more massive than another, puny star. From there we can place a planet in a Trojan orbit with the puny star:
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.
The puny star is really faint. If we take the planet-hosting star TRAPPIST-1 as a proxy, it is about 2000 times fainter than the Sun. That makes it 24000 times fainter than the high-mass star.
The planet is always the same distance from each star because of the 60 degree angle between the puny star and planet. Let’s make the planet’s (and puny star’s) orbit around the high-mass about 3.4 times wider than the Earth’s around then Sun (that is, 3.4 Astronomical Units). That will make the planet receive the same amount of energy from the high-mass star as the Earth does from the Sun. The planet and the puny star both orbit in the high-mass star’s habitable zone.
Meanwhile, the puny star would simply be a very bright star in the planet’s sky. The puny star is only 11% the size of the Sun. It would appear 30 times smaller in the sky than the Sun does from Earth! That makes it 17 times brighter than the full Moon but only 3% the size. Like a blazing flashlight in the sky!
You know that I can’t stop there! Let’s go nuts and build the Ultimate Trojan 2-star system.
First, we can have 2 Trojan star-star planets, one each at L4 and L5. That doubles our planetary real estate.
Second, each “planet” can be more than one planet. Let’s take a page from our earlier exploits. At L4 we’ll put a binary Earth. That’s two Earths orbiting each other. At L5 we’ll put a gas giant planet orbited by five large moons. Boom! That’s 7 possibly life-bearing worlds in our system!
Third, the puny star can also have its own planets. Given the wide (3.4 Astronomical Unit) orbit of the puny star around the high-mass star, planets can stably orbit the puny star out to a separation of about 0.4 Astronomical Units, the size of Mercury’s orbit around the Sun. Let’s put the innermost planet at one one-hundredth of an Astronomical Unit. This is the same orbital separation as TRAPPIST-1b, an Earth-sized planet orbiting a similar puny star. If we assume that planets are spaced evenly by period ratios (see here), then there is enough dynamical space for between 9 and 14 planets on stable orbits, depending how closely we pack them (9 planets for a chain of 3:2 resonances, 14 planets for a chain of 3:2 resonances). Let’s be conservative and choose 9 planets.
Here is what our Ultimate Trojan 2-star system looks like:
This system contains 2 stars and 16 worlds! All of the planets orbit within the high-mass star’s habitable zone, but not all are likely to bear life. The binary Earth receives the right amount of energy from the high-mass star for liquid water. Same for the gas giant and its moons. Those worlds get basically zero energy from the puny star.
But the puny star’s planets are so close-in that they do get energy. The closest planets are roasted by the puny star. The planets in the puny star’s habitable zone are also too hot because they get a double-dose of energy, from both the puny and high-mass star. Only planets beyond the puny star’s habitable zone are cool enough to have liquid water and maybe life. Those planets orbit the puny star but get most of their energy from the high-mass star. The puny star’s inner 4 or 5 planets are probably ruled out for life, but the outer ones remain potentially habitable.
But there’s another twist! The puny star’s planets, especially the hottest ones, are certainly tidally-locked. They always show the same face to the puny star.
Imagine a person on the dark side of a planet orbiting the puny star. That whole half of the planet remains dark, receiving no light from the puny star. But as the planet orbits the puny star, the person sees the high-mass star rise and set in the sky. But the length of the day for that person is set by how long the planet takes to circle the puny star.
Here is what it looks like as the planet goes around (see here for a more complex example):
Now imagine a person on the side of the planet always facing the puny star. For that person, the puny star is always overhead in the sky. The high-mass star also rises and sets as the planet orbits the puny star. But if it’s too close to the puny star, this side of the planet is really hot, given the extra energy received from the high-mass star.
Could the planets close to the puny star be two-faced, with one side that’s too hot and the other that’s just right?
If their atmospheres are thin, yes.
A thick atmosphere transfers energy and smoothes out a planet’s surface temperature. For example, Venus’ atmosphere is 90 times thicker than Earth’s and its surface is a uniform (ridiculously hot) temperature. With a thin atmosphere a planet can have large but stable temperature differences. For example, our Moon, with no atmosphere to speak of, has a roasting day side and a freezing night side.
For our two-faced planet to be habitable, it needs an atmosphere that is thin but not too thin. This is entirely plausible. It would be a Dune-style planet. An Eyeball-like planet (see here and here for more on Eyeball planets)
So there you have it: the Ultimate 2-star Trojan planetary system! Boom!
I think these planets would make a good setting for a science fiction story so I’ll throw this post in with the Real-life Sci-Fi Worlds