I thought that what you desire is impossible. After investigation, I decided that there is a slight chance that it might possibly be possible. My long answer ends with a recommendation of someone who might be willing to investigat further.tiaf
Part One: The Scale.
If you decide that your story can have a low number in the sliding scale of science fiction hardness there is no problem with your planetary system. But if you want your story to have a high number on the scale you have more difficulty.
Part Two: Separation Necessary for Orbital Stability.
For a long time I believed that the realtive separatinsof stars and planets in a bnary star system necessary for long term orbital stabiity would make it possible for one or both of hte stars to have habitable planets in S-type orbits, or possible for a binary system to have habitable planets in P-type orbits around both of hte stars.
But not a combination of a habitable zone around one of the stars and a habitable zone areound both of the stars.
Off hand, I would have to say that the separation between the two stars Genesis and Exodus should be several times the semi-major axis of the orbit of the planet in an S-type orbit around Genesis. So if a planet orbits Genesis with a semi-major axis of 10 million units, the semi-major axis of the orbis of Genesis and Exodus around each other should be at least maybe 50 million units. And the more eccentric the orbits of the stars are, the greater the minimum possbile semi-major axis will have to be for the planet orbiting Genesis to have a long term stable orbit and become habitable.
And similarly, the orbits of all the planets - including the habitable ones -orbiting both stars in P-type orbits will have to have semi-major axis at least several times the semi-major axis o the orbit of the two stars around each other.
Maybe five times and thus at least 250 million units, or 25 times the separation between the planet orbiting Genesis and Genesis.
So even if the orbits are all very close to circular, the planet orbiting Genesis will receive changing amounts of Illumination from the two stars.
It will always be 10 million units from Genesis. As it orbits around Genesis its average distance from Exodus will be about 50 million units but the distance will vary be between 40 million and 60 million units from Exodus. Sometimes the distance will be only 0.8 the average distance and it will receive 1.258 times the average illumination from Exodus, and sometimes the distances will 1.2 times the average and the planet will receive 0.694 the average illumination.
Thus the changing distance of the planet from Exodus could be the main factor causing the seasons on the planet.
To prevent the seasonal temperature changes on the planet orbiting Genesis from being too extreme, the planet could receive more radiation from Genesis than from Exodus. That is fairly easy to have. If Genesis and Exodus have the exact same luminosity the planet will receive an average of 25 times as much radiarion from Genesis as from Exodus if Exodus is an averge of 5 times as far from the planet as Genesis is.
Assuming that the habitable planets in P-type orbits around both stars have to be at least 250 million units from the center of gravity of Genesis and Exodus, they will be about 25 times as from from the two stars as the planet in the S-type orbit is from Genesis, and about 5 times as far from the two stars as the planet in the S-type orbit is from Eoxodus.
Thus they will recieved only 1/25, or 0.04 times, as much illumination from Exodus as the planet in an S-type orbit gets from Exodus, and only 1/625, or 0.0016 times, as much illumination from Genesis as the planet in an S-type orbit receives from Genesis.
So if the planet in an S-type orbit around Genesis receives enough illumination from Genesis and Exodus to be warm enough to be habitable (but not too much), the planets in P-types orbits around both Genesis and Exodus should receive much less radiation from the stars as the planet in an S-typed orbit around Genesis receives from the stars.
They should not receive enough radiation from the two stars to be warm enough to be habitable.
Part Three: PoOssible Lesser Oribital sEparation.l
Possibly the minimum orbital separation could be less than the factor of five mentioned above.
A number of exoplanets have in detected in binary systems, some in S-Type orbits and some in P-Type orbits.
According to this list, https://en.wikipedia.org/wiki/List_of_exoplanet_extremes
the smallest ratio between a P-type orbit and the orbits of the binary stars is that of Kepler 16 b. Kepler 16 A & B orbit each other with a separation of 0.22 astronomical Units (AU). Kepler 16 b orbits the two stars with a separation of 0.7048 AU, about 3.203 time the separation of the two stars.
Kepler-16b is also unusual in that it falls inside the radius that was thought to be the inner limit for planet formation in a binary star system.4 According to Sara Seager, a planetary expert at the Massachusetts Institute of Technology, it was thought that for a planet to have a stable orbit around such a system, it would need to be at least seven times as far from the stars as the stars are from each other.4 Kepler-16b's orbit is only about half that distance.4
It is believed that Kepler 16 b formed elsewhere and migrated to its pesent orbit. But it could have been in its present orbit for hundreds of millions or bilions of years.
Thus I guess that a science fiction writer could have a planet in a p-type orbit with a semi-major axis only 3.25, or 3.2, the semi-major axis of the two stars's orbit without being too improbable or too unrealistic.
The list of exoplanet extremes also has a record for the closest orbit of two stars with a planet orbiting one of hte stars in an S-type orbit. That planet orbits one star at a distance of 0.7 AU while the two stars orbit at 12 to 17 AU, 17.14 to 24.28 times the distance.
But the theoretical limit is a ratio of about 5 times, as suggested above.
In non-circumbinary planets, if a planet's distance to its primary exceeds about one fifth of the closest approach of the other star, orbital stability is not guaranteed.4
So if the planet in an S-type orbit orbits Genesis at 10 million units,the distance between Genesis and Exodus should be at least 50 milllion units, and the distance between the two stars and the nearest planets in P-type orbits should be at least about 160 million to 162.5 million units.
Thus the planet in the nearest P-type orbitshould receive about 1/256, or 0.0039062, times the radiation from Genesis as the planet in the S-Type orbit does, and only 1/10.24, or 0.0976 times, the radiaiton from Exodus as the planet in the S-type orbit does.
That would help with the problem a bit.
Part four. Masses and Luminosities of Stars with Habitable Planets.
And if a writer wants to find out whether such a system could have planets in those orbits be habitable, they can try different masses, luminosities, and distances of the stars and planets to see if they can get all four planets habitable.
Even though the masses of stars only vary by about 3,200 times (from 0.07 to 226 tims the maso fhe Sun)
They vary in lumnosity by many millions of times.
Unfortunately, the masses, and thus the luminosities, of stars with planets with oxygen rich atmospheres habitable for have much more limited range.
To be generous the range of stellar luminosities might range from F0V class stars with 7.24 the luminosity of the Sun to M5V stars with 0.003 the luminosity of the sun, a range of 2,400 times. Many people would prefer a range from about G0V to about K5V, 1.35 to 0.17 times the luminosity of the Sun, a range of only 7.94 times.
Of course, if that limits your attempts to design such a star system too much, you could decide that at least one of the stars in the system could not naturally have habitable planets and that long ago an advanced society moved planets around or even built them in your system and terraformed them to give them breathable oxygen atmospheres.
That is the plot element to use if your star system needs to have a set up which can possibly exist, but which could not possibly happen naturally and thus has to be artificial.
Part Five. Problem of the Width of the Circumbinary Habitable Zone.
To add to the problems, the innermost possible P-type planetary orbit might be about 3.2 to 3.25 the orbital separation of the two stars. And the two planets orbiting beyond it would have to orbit farther out and thus receive even less illumination from the two stars.
It would seem simple to find the width of the circumbinary habitable zone for the star pair Genesis and Exodus. Find the width of the Sun's circumstellar habitable zone and adjust to account for the relative luminosity of the star pair Genesis and Exodus compared to the Sun.
So what is the width of the Sun's circumstellar habitable zone between its inner and outer edges?
Here is a link to a list of about a dozen estimates of the inner or outer edges, or both, of the Sun's circumstellar habitable zone made in the last 60 years:
Note that the narrowest habitable zone from combining estimates would have its outer ege only 1.014 times as far the Sun as its inner edge, and the broadest habitable zone from combining estimates would have its outer edge about 26.315 times as from the Sun as its inner edge.
And some of the estimates that extend the inner and outer edges of the habitable zone do so for planets with special types of atmospheres, whihc wouldproably be unbreathable for humans.
The question asks for:
The atmospheres are all similar to Earth's, meaning humans can freely walk upon them with little to no need for supporting gear (the S-type can be excused from that if necessary.)
So the planet in the S-type orbit around the star Genesis might have to be too hot for liquid water using life, except for having a special type of atmopshere which humans couldn't breath. Humans might have to use breathing gear and possibly refrigerated suits to walk on the surface of the planet in the S-type orbit.
In fact it might be hard to fit the three planets in circumbinary or P-type orbits into a habitable zone, depending on the actual width of habitable zones which nobody has yet proven conclusively.
Suppose that the first planet in P-type orbit orbits at 3.2 times the separation of the two stars, and the next planet orbits about 1.7 times as far as the first P-typte orbit, and the third planet orbits about 1.7 times as far as the second P-type orbit. The second planet would orbit at about 5.44 times the separation of the two stars and the third planet would orbit at about 9.248 times the separation of the two stars.
Thus the third planet in a P-type orbit would orbit about 2.89 times as far from the center of the system as the first planet in a P-type orbit.
And in some systems of exoplanets planetary orbits seem to increase their semi-major axis by very roughly 1.73 in each successive orbit.
Part Six. Closer Planetary Orbits.
However, some star systems do have exoplanets orbiting with smaller orbital ratios.
In the TRAPPIST-1 system the ratios of orbits are smaller than about 1.7
Planet b orbits at 0.01154 AU.
Planet c orbits at 0.01550 AU, 1.34 times as far as b.
Planet d orbits at 0.02227 AU, 1.436 times as far as c, and 1.929 times as far as b.
Planet e orbits at 0.02925 AU, 1.313 times as far as d, and 1.887 times as far as c.
Planet f orbits at 0.03849 AU, 1.316 times as far as e, and 1.728 times as far as d.
Planet g orbits at 0.04683 AU, 1.217 times as far as f, and 1.601 times as far as e.
Planet h orbits at 0.06189 AU, 1.32 times as far as g, and 1.608 times as far as f.
So planet TRAPPIST-1 G orbits only about 1.601 times as far from the center as planet TRAPPIST-1 e. If three planets in a system have that relative spacing and are in the habitable zone that would require that the habitable zone had a width of only 1.601 or a little more.
The list of exoplanet extremes gives Kepler-36 b and Kepler-36 c as the exoplanets with the smalles ratio between their orbits. Kepler-36 c has a semi-major axis of 0.1153 AU, which is 1.1127 times the semi-major axis of the orbit of Kelpler-36 b, 0.1283 AU.
So with a ratio of 1.1127 in successive plaenatry orbits, if the innermost circumbinary planet in a P-type orbit is at 3.2 times the seperation between the two stars, the second planet in a P-type orbit can be at 3.560 times the seperation of the two stars, and the third planet in a P-type orbit could be at 3.962 times the separation of the two stars.
The third planet would orbit at 1.238 times as far as the first planet from the center of gravity of the two stars, requiring only a modestly thick circumbinary habitable zone.
And if the planets in P-type orbits are close to the outer edge of the habitable zone, maybe the planet in an S-type orbit around the star Genesis could be within the habitable zone but close to its inner edge.
Part Seven. A frame challenge:
I would put four habitable planets in a binary star system in S-type orbits around the individual stars, each of the stars with two habitable planets in S-type orbits, or else one star with one habitable planet in an S-type orbit and the other star with three habitable planets in in S-type orbits. If the two stars in a binary systema re separated widely enough, each star can have its own habitable zone for planets in S-type orbits.
And If the stwo stars were close enough together, there could be a circumbinary habitable zone around both the stars.
But never before today, October 4, 2022, while working on this answer, did I even think that the relative distances involved for stable orbits around a single star and around the pair of stars would permit planets to have stable orbits in S-type orbits in the habitable zone of one star in a binary while other planets have stable P-type orbits in the circumbinary habitable zone around both of the stars.
My calculations today don't convince me that such a situation is possible. They do convince me that it might possibly be possible for such a situation to exist. But someone else would have to do far more detailed calculations before I would be convinced that such a situation is possible.
Part Eight. Trojan Orbits.
So if you really want to have one habitable planet in an S-type orbit around the star Genesis, and three habitable planets in P-type orbits aroun d both the stars, maybe you should reduced the number of P-type orbits to one orbit for all three planets.
So there could be three planets in one set of Trojan orbits around the pair of stars. One planet would orbit in the L4 point 60 degrees ahead of the middle planet, which would be separated by 60 degrees from each of the other two planets, and the third planet would orbit in the L5 point 60 degrees behnd the center planet.
The problem with that is that human habitable planets have a rather narrow mass range, while the object in a Trojan relationship have very widely differing masses.
The mass range between the Sun, and a planet, and even the most massive asteroid in a trojan orbit, is very great.
More complicated caclulations may vary the possible mass range in specific conditions, but there is a general rule of thumb which writers who don't like calculations might as well follow when designing Trojan obits.
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).
The mass range for habitable planets is way t0o small for the center planet to be at least 10,000 the mass of each of the planets in the trojan points.
Part nIne. A Giant Planet:
So make the center planet a giant planet. Jupiter has 317.8 times the mass of Earth, and the dividing zone between the most massive possible planets and the least massive possible brown dwarfs is about 13 times the mass of Jupiter, or about 4,131.4 timess the mass of Earth. So that would mean that even if the center planet was the most massive possible planet, the planets in the L4 and L5 points would have to be no more than 0.41314 the mass of Earth.
According to the calculations by Stephen H. Dole, in Habitable Planets for Man, 1964, page 54, the minimum mass of a planet whichcould retain an oxygen rich atmosphere for geological eras of time would be about 0.195 Earth Mass.
Dole didn't think that such a small planet would be able to produce an oxygenrich atmosphere. In the following pages he tried to calculate the minimummass to for a planet to be able to create an oxygen rich atmosphere. On pages 56 to 57 he decided that about 0.4 Earth mass would be the minimum to produce an oxygen rich atmosphere breathable for humans.
Rene Heller and Roy barnes discussed the mass range for habitable worlds on pages 3 to 4, of this 2013 article:
And their sources gave a mass of about 0.12 Earth mass as the minimum to retain a dense atmosphere for geological eras of time. But other factors considered to be necessary for habitablity led them to conclude that in most situations a minimum mass of about 0.25 Earth mass would be necessary for world to be habitble.
So if the central planet is as massive as a planet can be, the objects in the L4 and L5points can be just barely more than the minimummass of a habitable world.
Part Ten: A Brown Dwarf:
So why not make the center object in the orbit a brown dwarf? The dividing zone between the most massive brown dwarfs and the least massive stars is at about 75 times the mass of Jupiter, or about23,835 times the mass of Earth. If the brown dwarf was at the upper mass limit, the most massive possible objects at the L4 and L5 points could be as much as 2.3835 times as massive as Earth, which would probably be too massive to be habitable for humans.
Part Eleven. Mooning Around:
But if the object in the center point of the orbit is a giant planet or a brown dwarf, it will many times too massive to be habitable for humans. So there will be only two habitable worlds in that shared orbit.
If the center object is a giant planet, why not make the desired third habitable world a very large natural satellite of the giant planet, a giant moon large enough to be habitable.
And in fact the article mentioned above is a discussion of the potential habitability of hypothetical exomoons orbiting around giant exoplanets in the habitable zones of their stars.
Part Twelve: Natural Satellites of Brown Dwarfs:
For the planets in the L4 and L5 points to be massive enough to be habitable, the object in the center of the orbit would more probably be a brown dwarf. Just as planets have moons and stars have planets, brown dwarfs should have natural satellites orbitng them. I don't know what those natural satellites should be called, maybe planoons or moonets.
Anyway, some brown dwarfs might have natural satellites the right size to be habitable.
So with a giant planet or a brown dwarf in the orbit, you could have one habitable world orbiting the giant planet or the brown dwarf, and two others, one in the L4 point and one in the L5 point.
Or maybe you could have all there of those habitable worlds orbiting in around the giant planet or brown dwarf.
Part Thirteen. Ring of Planets:
Sean Raymond's blog PlanetPlanet has an ultimate solar system section. Raymond tries to design imaginary solar systems with as many habitable planets as possible in his posts there.
In the Ultimate Engineered Solar System post:
He mentions a science paper calcualtion that seven to fortytwo planets could share an orbit their star if they were equally spaced and had equal masses. So a naturally Raymond goes on to design solar systems with hundreds of habitable planets ins everal concentric rings aorund their star.
With seven to fortytwo planets equally spaced around anorbit, the seperation between planets on the orbit would be between 8.571 and 51.428 degress, instead of the sixty degrees in trojan orbits.
As Raymond says, it would be extremely improbable for such a ring of equally spaced planets to naturally form. You could say the odds against that would be "astronomical". So a system with such a ring of planets would have to built by a very advanced civilziation.
But that ring would give you seven to fortytwo habitable planets in a circumbinary orbit instead of the three you requested. What if you have your heart set on four and only four habitable planets in your system?
Part Fourteen. Arc of Triumph?
In a later post: https://planetplanet.net/2020/11/19/cohorts/ Raymond says that his computer similations show that a ring of planets doesn't have to be complete to be orbitally stable. He claims that a segment or arc or cohort of a ring of planets can be stable.
He ran similations using arcs of 2 or 3 planets to find out what separations would be necessary for long term orbital stability. An arc of three coorbital planets would have to have at least 20 hill radii between each planet.
Thus you can have three habitable planets in an arc along a P-type orbit just within the outer edge of the circumbinary habitable zone, and possibly have the planet in a an S-type orbit around the star Genesis be within the habitable zone, though probably close to the inner edge.
Part Fifteen. Caution:
I said possibly there were would be a configuration where the planet in the S-type orbit could possibly be habitable. That is not saying it is certain there are configurations where planets in S-type orbits are habitable while there are planets in P-type orbits. It is saying it might be possible that such a situtiaon might be possible.
Part Sixteen. Recommendation:
Possibly oyu might want to ask Sean Raymond mentioned in the previous few parts if the confuguration you desire is possible and if so how to design a star system with that configuration.
I strongly suspect that Raymond always thought, like me, that what you want would be impossible, and so would have to investigate the problem. And it might take a lot of calculation to find a possible system with a habitable planet in a stable S-type orbit around one star and three habitable planets in stable orbits (or sharing one orbit) around both stars.