The original question asks for two double planets orbiting in each other's L4 and L5 positions. There is no known example of such an orbit, whether the two bodies are single or binary themselves.
The only known examples of Trojan orbits have a very massive primary, a much less massive secondary object that orbits the primary, and one or more tertiary objects that orbit in the L4 and/or L5 positions ahead or or behind the secondary object, and are much less massive than the secondary objects.
As a rule of thumb, the system is likely to be long-lived if m1 > 100m2 > 10,000 m3 (in which m1, m2, and m3 are the masses of the star, planet, and trojan).
The possible mass range of habitable planets and the stars that they could orbit were estimated by Stephen Dole in Habitable Planets for Man (1964, 2007)
Dole estimated that The range in mass of stars which could have habitable planets is thus 0.72 to 1.4 solar masses, corresponding to main-sequence stars of spectral types F2 to K1, and that a habitable planet should have a mass between 0.4 and 2.25 of earth's mass, which should be about 0.000001201 to 0.000006757 Solar masses.
If a star m1, has to be at least 100 times the mass of the secondary object, m2 in the equation, with a star having mass between 0.72 to 1.4 solar masses, the secondary object m2 thus has to have a mass of 0.0072 to 0.014 solar masses -or lower. Thus the tertiary objects, m3 in the equation, would have to have masses of 0.00000072 to 0.0000014 solar masses - or lower.
Jupiter, the most massive planet in the solar system, has a mass 317.8 times as massive as Earth and about 0.001 times as massive as the Sun. The line between the most massive planets and the least massive brown dwarfs is believed to be about 13 times the mass of Jupiter, or about 0.013 solar masses, which is close to the 0.014 solar masses limit for the secondary objects. So the most massive planet would have about 4,131.4 times the mass of the Earth, and a brown dwarf small enough for the system could have a mass as high as 4,449.2 earth masses.
If the tertiary objects, the m3 in the equation, have to have less than 0.001 times the mass of the secondary objects, they would have to have masses less than 0.41314 or 0.44492 Earth masses, which is a little more than Dole's estimated minimum mass for a habitable planet of 0.4 Earth masses.
Thus if Dole's estimated minimum mass for a habitable planet is not too low, and if the rule of thumb is correct, it is just barely possible for a habitable planet to orbit in the L4 or L5 position of a massive super Jupiter planet or brown dwarf orbiting around an F2 spectral class star.
But if a double planet has to have less than 0.41314 or 0.44492 Earth mass, each member of the double planet would have to have less than 0.20657 or 0.22246 times the mass of Earth. Which is far below the minimum mass estimated by Dole.
Furthermore, the lowest known ratio between the masses of a secondary object and a tertiary object in a Trojan relationship, that between Dione, a large moon of Saturn, and Helene, a moon in a Trojan relationship with Saturn and Dione, is about 36,000 times. If the largest possible mass for the secondary object in an imaginary solar system is 4,131.4 to 4,449.2 Earth masses, the largest possible mass that was no more than one thirty six thousandth or 0.0000277 of that would be 0.1147611 to 0.1235888 Earth masses. Which is less than Dole's minimum mass for a habitable planet.
So I would not feel secure in setting up a system with a star as the Trojan primary, and a giant planet or a brown dwarf as the Trojan secondary, and a planet large enough to be habitable as the Trojan tertiary, unless:
1) There was an example of an known Trojan system with the mass ratio between the secondary and the tertiary objects was much less than 36,000 times, or:
2) Sophisticated computer simulations proved that such a system would be stable for billions of years, or:
3) It was proved that one of Dole's estimates was wrong and that stars more massive that 1.4 solar masses could have habitable planets, or:
4) It was proved that one of Dole's estimates was wrong, and that planets less massive than 0.4 Earth masses could be habitable,or:
5) Two or more of the above.
So I might be a little hesitant to create a star system with two double habitable planets - one in the L4 position and one in the L5 position - orbiting along with a giant planet and with the primary a F2 class star.
I found a blog with a post that seems to be confident that a co orbital pair of plants co orbiting in each other's L4 and L5 positions would not need a much larger planet to keep them in place.
In the next blog:
A system is designed with six orbits, each orbit having two planets in Trojan positions, and each planet a binary planet, making a total of 24 habitable planets in the star's circumstellar habitable zone.
He doesn't think small.
And if he is correct about two objects of equal mass being able to be each other's Trojan's, and the system being stable for billions of years, that would be an entirely plausible star system.
But I would feel a lot more confident if he wrote than he ran computer simulations which indicated that such a system would be stable for billions of years.
You might want to consider making your double or single planets co orbital planets like the co orbital moons in our solar system.
Such orbits would be different in some ways and similar in others to the Trojan orbits you imagined.
Four more or less Earth size planets orbit in the circumstellar habitable zone of the star TRAPPIST-1; there are 8 known planets in the system.
The orbits of the TRAPPIST-1 planetary system are very flat and compact. All seven of TRAPPIST-1's planets orbit much closer than Mercury orbits the Sun. Except for b, they orbit farther than the Galilean satellites do around Jupiter, but closer than most of the other moons of Jupiter. The distance between the orbits of b and c is only 1.6 times the distance between the Earth and the Moon. The planets should appear prominently in each other's skies, in some cases appearing several times larger than the Moon appears from Earth. A year on the closest planet passes in only 1.5 Earth days, while the seventh planet's year passes in only 18.8 days.
Can the planets in the habitable Zone of TRAPPIST-1 actually be habitable? Orbiting so close to their dim star, the planets would probably be tidally locked, so that one side always faced the star and one side was in eternal darkness. And it is uncertain whether such a planet could become or remain habitable.
Of course, if a double planet orbited close to a dim red star, the two planets in the double planet might become tidally locked to each other, and not to the star. Thus they would have days equal to their orbital periods around their center of gravity and not to their orbital periods around the star.
In this post:
The author is inspired in part by this paper:
That claims that seven to forty two astronomical bodies of he same mass could share a stable orbit if equally spaced along the orbit.
So theoretically seven to forty two Earth sized planets could have a stable orbit around a star equally spaced.
If habitable planets can have between 0.78 to 1.25 the radius of Earth, and if the radius of Earth is about 3.6669 times that to the Moon, a habitable planet would have between 2.860 and 4.583717 times the radius of the Moon, and thus would appear as wide as the Moon when it was 2.86 and 4.58 times as distant as the moon, and thus about 1,099,384 to 1,760,552 kilometers from the next planet in the ring.
With seven to forty two single planets in the ring, the total circumference of the ring of planets would be 7,695,688 to 73,943,184 kilometers, and thus the planets would orbit their star at a distance of about 1,224,807.8 to 11,768,433 kilometers, which would be a very close orbit for a not very spectacular view.
So it would be better to make each of the seven to forty two habitable planets that shared the orbit a double habitable planet. Thus the double planets could be spaced tens or hundreds of millions of kilometers apart along the orbit, and each planet in a double planet would still have its twin close by and looking very large in its sky.
There are problems with having Trojan planets.
The relative masses work out alright as long as the primary body has a mass like a star, the secondary body has a mass like a planet, and the bodies in Trojan obits have masses like asteroids.
As a rule of thumb, the system is likely to be long-lived if m1 > 100m2 > 10,000m3 (in which m1, m2, and m3 are the masses of the star, planet, and trojan).
So how massive would a habitable planet, the m3 in the equation, have to be in order to be habitable?
Back in 1964, Stephen Dole made estimates in Habitable Planets for Man (1964, 2009).
On pages 53 to 67 he discussed planetary properties necessary for habitability.
On page 53 Dole said that since a surface gravity of about 1.5 g seemed like the maximum that humans would tolerate, and that corresponded to a planet with a mass of 2.35 earth masses, a radius of 1.25 Earth radii, and an escape velocity of 15.3 kilometers per second.
On page 54 Dole calculated the minimum size of a planet that could retain a breathable atmosphere for billions of years as 0.195 Earth's mass, with of 0.63 of Earth's radius and a surface gravity of 0.49 g. But Dole believed such a planet would be unable to produce an atmosphere dense enough to be breathable.
Dole calculated two figures for the minimum mass necessary to produce a breathable atmosphere, 0.253 Earth mass, which he believed too low, and 0.57 Earth Mass, which he believed too high:
With 0.25 being too low, and 0.57 being too high, the appropriate value of mass for the smallest habitable planet must lie between those figures, somewhere in the vicinity of 0.4 Earth mass.
...This corresponds to a planet having a radius of 0.78 Earth radius and a surface gravity of 0.68 g.
So Dole believed a habitable planet would have to have a mass between 0.4 and 2.25 of earth's mass, a radius of 0.78 to 1.25 Earth radii, and a surface gravity of 0.68 to 1.5 g.
The largest asteroid is Ceres, which is large enough to count as a dwarf planet. It has a mass of 9.3835±0.0001 times ten to the 20th power kilograms, which is 0.00016 Earths, and 0.0128 Moons. Ceres is about 1,000,000,000 times the size and mass of a typical asteroid, so it is millions of times the mass of large asteroids and thousands of times the mass of really large asteroids, and so on.
The masses of planets in our solar system range from 3.301 times ten to the 23rd power kilograms for Mercury to 1.899 times ten to the 27th power kilograms for Jupiter. The largest that a planet could get before having a little fusion in its core and thus becoming a brown dwarf is estimated to be roughly 13 times the mass of Jupiter.
One of the most massive stars known is Eta Carinae, which, with 100–150 times as much mass as the Sun, will have a lifespan of only several million years. Studies of the most massive open clusters suggests 150 M☉ as an upper limit for stars in the current era of the universe. This represents an empirical value for the theoretical limit on the mass of forming stars due to increasing radiation pressure on the accreting gas cloud. Several stars in the R136 cluster in the Large Magellanic Cloud have been measured with larger masses, but it has been determined that they could have been created through the collision and merger of massive stars in close binary systems, sidestepping the 150 M☉ limit on massive star formation.
With a mass only 80 times that of Jupiter (MJ), 2MASS J0523-1403 is the smallest known star undergoing nuclear fusion in its core. For stars with metallicity similar to the Sun, the theoretical minimum mass the star can have and still undergo fusion at the core, is estimated to be about 75 MJ. When the metallicity is very low, however, the minimum star size seems to be about 8.3% of the solar mass, or about 87 MJ. Smaller bodies called brown dwarfs, occupy a poorly defined grey area between stars and gas giants.
Thus it appears that the most massive stars are about 1,800 times the mass of the least massive stars.
But if a writer wants some of the planets in his fictional star system to have native intelligent beings, and/or other advanced multi celled lifeforms, and/or to be habitable for human visitors or colonists, or otherwise be interesting for science fiction stories, then there are strict limits on the possible masses of the star (or stars) in the system.
Earth is about 4,600,000,000 years old. It took billions of years for Earth lifeforms to become multi celled and to colonize dry land. It took billions of years for Earth to develop an atmosphere with enough oxygen to be breathable for humans a few hundred million years ago.
If someone assumes that apes, and proboscideans, and cetaceans are intelligent enough to count as intelligent beings, that would still take the first intelligent beings on Earth back only about ten to thirty million years. There is a theory that an intelligent Cephalopod killed ichthyosaurs and arranged their spinal discs in patterns on the sea floor. Naturally that theory has little acceptance and 228 million years ago is only about 5 percent of Earth's age ago, anyway.
So most planets that are interesting for science fiction stories will have to orbit stars which remain on the main sequence and shine rather steadily for billions of years.
And what are the masses of stars which can shine steadily for billions of years?
Back in 1964, Stephen Dole made estimates in Habitable Planets for Man (1964, 2009). On pages 67-72 he discussed the properties need in a habitable planet's primary (the star). Dole estimated that the minimum possible age for a habitable planet would be three billion (3,000,000,000) years, which was generous, since it was about a billion years less than it took Earth to become habitable.
The only stars that conform with the requirement of stability for at least 3 billion years are main-sequence stars with a mass less than about 1.4 solar masses -spectral types F2 and smaller - ...
The less massive and luminous a star is, the closer to it its circumstellar habitable zone (which Dole calls an ecosphere) will be. But the closer a habitable planet has to be to its star, the stronger the tidal forces of the star on the planet will be, since the tidal forces will increase with closer distances more rabidly than the planetary temperature will. If the tidal forces are strong enough, the planet's rotation will be slowed until it is tidally locked to the star with one side always facing the star and having eternal light and heat and the other side always facing away from the star and having eternal darkness and cold. Whether a tidally locked planet could be habitable is a subject of speculation, and Dole assumes that tidally locked planets can not be habitable in his estimates.
A 'full ecosphere can exist around primaries of stellar mass greater than about 0.88 solar mass, but the ecosphere is narrowed by the tidal braking effect for primaries of lesser mass until it disappears when the stellar mass reaches about 0.72. The range in mass of stars which could have habitable planets is thus 0.72 to 1.4 solar masses, corresponding to main-sequence stars of spectral types F2 to K1.
Thus according to Dole's calculations, the most massive star with a habitable planet would be about 1.94444 times as massive as the least massive star with a habitable planet.
As we know, the planet Jupiter orbits the Sun and has many asteroids in its L4 and L5 Trojan positions. The mass of Jupiter is 1.8982 times ten to the 27th power kilograms, or 317.8 times the mass of Earth and 1/1047 the mass of the Sun or 0.0009551 solar masses. 7,040 asteroids have been discovered orbiting in Jupiter's Trojan positions by 2018.
The largest Jupiter trojan is 624 Hektor, which has an mean diameter of 203 ± 3.6 km.11
624 Hektor is believed to have a mass of about 8 to 10 times ten to the 18th power kilograms. Thus the mass of Jupiter is roughly a hundred million times the mass of Hektor, its most massive Trojan.
The total mass of the Jupiter trojans is estimated at 0.0001 of the mass of Earth or one-fifth of the mass of the asteroid belt.14
The planet Neptune has a mass of 1.02413 times ten to the 26 kilomgrams, which is 17.147 times that of Earth and thus about 1/194200 or 0000514 the mass of the Sun. There are Neptunian Trojan asteroids. The known Neptunian Trojans are a bit smaller than 624 Hektor, but since Neptune is so much less massive than Jupiter the mass difference should be a bit less.
Earth has a mass of 5.97237 times ten to the 24th power kilograms, which is 1/330,000 or 0.000003 the mass of the Sun.
2010TK7 is the only know Earth Trojan. With a diameter of about 300 meters, it has roughly one billionth the mass of 624 Hektor, so it has even less mass relative to Earth than 624 Hektor has to Jupiter.
Mars has a mass of about 6.4171 times ten to the 23rd power kilograms, or 0.107 times the mass of Earth, or about 0.000000324 times the mass of the Sun.
The largest Martian Trojan, 5261 Eureka, has an estimated diameter of about 1 to 4 kilometers.
Thus 624 Hektor should have about 117,000 to 11,400,000 times the volume and mass of 5261 Eureka. That is many times the difference in mass between Jupiter and Mars - Jupiter is only 2,970.0934 times as massive as Mars.
Uranus has a mass of 8.6840 times ten to the 25th power kilograms, which is 14.536 times the mass of Earth, and 0.0000436 times the mass of the Sun.
One of the two Uranian Trojan asteroids, 2014YX49, is calculated to be between 40 and 120 kilometers in diameter. If 624Hektor has a diameter between 147 and 231 kilometers, it should have 1.225 to 5.775 times the diameter and thus it should have about 1.8382 to 192.59985 times the volume of 2014YX49. Since Jupiter has about 21.862 times the mass of Uranus, it seems impossible to calculate which Trojan asteroid is more massive relative to its planet.
Two moons of Saturn, Tethys and Dione, have smaller moons in Trojan positions.
Tethys has a mean radius of about 531 kilometers, while its slightly larger Trojan, Telesto, has a mean radius of about 12 kilometers. So Tethy has about 44.25 times the radius of Telesto, and about 86,644 times the volume of Telestro, and thus possibly about 86,000 times the mass of Telesto.
Dione is a little larger than Tethys, with a mean radius of 561 kilometers,and its larger Trojan, Helene, has a mean radius of about 17 kilometers. So Dione has about 33 times the radius, and 35,937 times the volume of Helene, and thus possibly 36,000 times the mass of Helene.
So the ratio of mass between Dione and Helene might be only about 36,000 times, which would be over a thousand times smaller than the ratio of mass between Jupiter and 624 Hektor, and the lowest known ratio of mass between a secondary object and a tertiary object in a Trojan relationship.