Could a Trappist-like planetary system exist around a sun like ours?

Question

So, a couple of years ago, the TRAPPIST sun was discovered to have 7 different earth-like planets orbiting it – three of them in the habitable zone. Nothing like this has ever been seen before, or found again since. And the main reason it's possible at all is because of the orbital resonance of the planets – they orbit in a pattern which causes the system to be stable long-term. And this whole system of planets take place closer to the TRAPPIST star than Mercury orbits our sun.

This is interesting for a lot of reasons, but one that appealed to me was that so many planets could fit – and be stable – in such a small space. I was under the impression that you couldn't fit that many planets so close together.
(Another interesting fact: The distance between TRAPPIST-1B and TRAPPIST-1H is less than the distance between the two edges of our sun's Goldilocks zone.)

So my question is this: could such a system be stable around a larger sun – a sun like ours? And could such a system work with the planets further out – in the habitable zone? Would it be stable or am I missing something?

Answer 1

There are a couple of systems around solar-mass stars that follow the pattern of tightly-packed planets orbiting close to the star. These include:

• Kepler-11 (0.961 solar masses), a system of six low-density sub-Neptune planets orbiting within 0.5 au of the star
• HD 10180 (1.06 solar masses), a system of six Neptune-size planets of which four are within 0.5 au of the star.
• Kepler-90 (KOI-351) (1.2 solar masses), a system with eight planets within 1 au of the star ranging in size from Earth-sized to Jupiter-sized.

In the case of HD 10180, planet g is located in the liquid water zone, though its mass indicates that it is an ice or gas giant planet and thus not habitable. Kepler-90's outermost planet is located at 1 au, but the star is more luminous than our Sun so it might be too hot to support liquid water and is in any case a Jupiter-sized planet. So systems like TRAPPIST-1 with packed inner planetary systems clearly exist around stars like the Sun, although the difficulty of finding small planets at sufficient distance from the star means that so far we haven't found such systems with terrestrial planets in the liquid water zone.

The question of how many planets you could pack into the habitable zone has been investigated by Kane et al. (2020), "Dynamical Packing in the Habitable Zone: The Case of Beta CVn", where they investigate the maximum number of planets that can dynamically fit in the habitable zone by spectral type, and also consider in detail the specific case of the G-dwarf Beta Canum Venaticorum. From the paper:

Our analysis presented here demonstrates that in fact the dynamical limitations to the packing of HZ terrestrial planets is ∼5 planets for most spectral types, and ∼6 planets for stellar masses $\gtrsim 0.7 M_\odot$. Packing 7 planets in the HZ is possible within certain specific stellar mass and architecture regimes, but becomes vulnerable to MMR perturbations that compromise the dynamical stability of such configurations.

(MMR = mean motion resonance)

Whether real systems actually reach such a dense packing in the habitable zone is unknown, and they note some caveats to their analysis:

There are some caveats to the analysis presented here that are worth repeating. First, Section 2.1 describes the use of 3 Gyr isochrones, which are adopted in order to include stars at the higher mass end of the stellar mass range. However, the star will evolve with time, as will the HZ (Gallet et al. 2017), resulting in an increase in both the width and orbital distance of the HZ. Subsequently, planets located at the inner edge of the HZ at 3 Gyr will transition into the Venus Zone (Kane et al. 2014) where runaway greenhouse scenarios may become dominant. Second, our simulations account for Earth-mass planets and circular orbits. Higher masses and non-zero eccentricities will reduce orbital stability within the HZ, but lower mass planets may allow for additional planets, with the exception of strong resonance regions.

(HZ = habitable zone)

Answer 2

That is a fascinating question due to the fact that planetary orbits have both absolute and relative spacing.

The absolute spacing between planetary orbits is the number of kilometers that the semi major axis of the orbit of the outer planet exceeds the semi major axis of the orbit of the inner planet by.

The relative spacing is the ratio between the semi-major axis of the two orbits, one orbital radius divided by the other.

So is the minimum possible space between planetary orbits determined by absolute spacing or by relative spacing, or maybe determined by a complex formula which might include both relative and absolute spacing as well as other factors?

Astronomers have been trying to calculate factors which affect the stability of the orbits of planets, moons, comets, asteroids, etc. for many decades, more than a century, and no doubts there have been calculations about the minimum possible stable distances between orbits.

Here is a link to an online copy of Habitable Planets for Man, Stephen H. Dole, 1964. 2007, which attempts to calculate the percentage of stars with planets habitable for humans:

https://www.rand.org/content/dam/rand/pubs/commercial_books/2007/RAND_CB179-1.pdf[1]

On pages 49 and 52 Dole discusses the forbidden regions around the orbit of each planet in our solar system where another planet could not have a stable orbit, according to calculations on the restricted three body problem.

Dole notes that the solar system consists of about 50 percent forbidden regions and 50 percent empty regions.

So going by the data in Dole's discussion, it seems possible that in a system where the planets orbited with their forbidden regions almost touching, there might possibly be zero, one, two, three, or four planets within the circumstellar habitable zone. Depending in large part on how wide the circumstellar habitable zone of the star was.

The way to calculate the size of a star's circumstellar habitable zone is to take the size of the Sun's circumstellar habitable zone and multiply it by the ratio of the star's luminosity to the Sun's luminosity.

Unfortunately, the size of the Sun's circumstellar habitable zone is not known. This chart gives several estimates of the inner or outer edges or both:

https://en.wikipedia.org/wiki/Circumstellar_habitable_zone#Solar_System_estimates[2]

Note the vast difference between the narrowest and the widest estimates.

Over 4,000 exoplanets have been discovered, some in systems with more than one exoplanet. There is a wide range in the spacing of planets in those systems.

My answer to this question:

https://worldbuilding.stackexchange.com/questions/151028/what-is-the-theoretical-maximum-number-of-habitable-planets-in-one-solar-system%5B1%5D[3]

Discusses the question of whether the spacing of planetary orbits is determned by the absolute distance between orbits or the relative distance ratio between orbits.

I point out that if the spacing of planetary orbits in the habitable zone of a star is determined by the absolute spaceing between orbits, there could be many planetary orbits if they were spaced as close together as the closest known examples, while if the spacing of planetary orbits in the habitable zone of a star determined by the relative spacing, the ratio of planetary orbits, there would be room for far fewer orbits.

I believe there could be room for no more than 6 planetary orbits in Kasting's optimistic habitable zone for the Sun if they were at the closest known relative spacing, but room for 518 or 519 planetary orbits if they were at the absolute spacing between Kepler-70 b and c.

Since Kepler-70 b and C might not exist, we could use the spacing between the orbits of TRAPPIST-1 f and g, 1,250,000 kilometers, as the minimum absolute spacing. Kasting's optimistic habitable zone for the Sun is 0.83 AU, or 124,16,232.7 kilometers wide, and thus would have space for 99.33 planetary orbits spaced 1,250,000 kilometers apart.

And there is a blog called PlanetPlanet about planetary formation. It has some sections about science fiction worlds.

It has a section called Ultimate Solar System with posts designing solar systems with successively more habitable planets.

https://planetplanet.net/the-ultimate-solar-system/[2]

And the more habitable planets there are in one of those solar systems, it less likely it would be for such a solar system to form naturally, and the more likely it would be that such a solar system would have been constructed or engineered by a highly advanced civilization.

So we can be pretty certain that systems like The Ultimate Retrograde Solar system, The Ultimate Engineered Solar System, The Black Hole Ultimate Solar System, and The Million Earth Solar System Would have been deliberately constructed by advanced civilizations.