An Apple Pie From Scratch, Part IVa: Planets and Moons: Formation and Orbits

Concept of Trappist-1 system. ESO/M. Kornmesser

If there is any one rule governing the structure of planetary systems, it is diversity. Both observations of exoplanetary system¹ and theoretical models² have shown a bewildering variety of possible systems, with planets of wildly varying size and compositions appearing just about anywhere within them. System architectures (that is, the particular arrangements of orbits and distributions of mass) thought impossible a few decades ago are old news today. So we’ve got quite a lot of freedom here.

Nevertheless, there are some patterns and restrictions to how planetary systems form and how the resulting architectures develop, and so there are some rules we should follow in building a system and designing the planets within it. This is a large and complex topic, so I’ve split this part into 3 sections: In section a, we’ll go over the current theories of how planetary systems form and talk about how to build a realistic system architecture based on those theories; in section b, we’ll talk more about what these planets will be like, what their physical characteristics are likely to be and what different types of surfaces they might have; finally in section c, we’ll focus in on the specific question of habitability, and what exact conditions a planet likely needs in order to have a good shot at developing complex life.

• Formation of a Planetary System
• Gas Giant Formation
• Terrestrial Planet Formation
• Post-Formation Instability
• Patterns of System Architecture
  
• Placing Orbits
• Co-orbital Planets
• Binary Planets 
• Trojans
• Tadpole Orbits
• Horseshoe Orbits
• Eccentric Resonance
• Inclined Resonance
  
• Rotation
• Tides
  
• Tidal Locking
  
• Moons
• In Situ Formation
• Capture
• Impact
• Pull-Down Capture
• Fission
• Limits and Stability
  
• Rings
  
• Engineering the Architecture
• Keeping Time
• In Summary
• Notes
  

Back to Part III

Formation of a Planetary System

Concept of OTS 44 NASA/JPL-Caltech

Stars typically form not individually but in clusters of thousands within nebulae, immense clouds of gas. Calling them “clouds” is a bit misleading, though; they are thousands of times denser than the typical interstellar medium, but that still leaves them orders of magnitude less dense than near-Earth space. For the most part they wouldn’t even be visible from up close, though bright stars may illuminate the surrounding gas³ and particularly large, dense regions may obscure the stars behind them. A spaceship traveling at a significant portion of light speed would notice the increased drag from the denser medium, so nebulae might provide barriers to the expansion of nearby interstellar species.

At any rate, at first the high relative velocity of molecules within the nebula holds it back from gravitational collapse (as any particles that fall towards a denser region gain enough speed to shoot back out again), but eventually it fragments and collapses into star-mass clumps. Exactly how is still debated—it may be due to magnetic interactions or turbulent flow patterns within the gas—but it seems to be helped along by the formation of a few early massive stars that then soon supernova and produce shockwaves in the gas that can catalyze the formation of other stars⁴.

Once the clumps have begun collapsing, the formation of stars and planets is pretty fast compared to the mind-boggling stretches of time that we’ve been dealing with so far. Within about 10,000 years⁵ the cloud has collapsed to form a protostar, a core of gas dense enough to resist further gravitational collapse. Though fusion hasn't begun yet, the compression causes the gas to heat up to interior temperatures of 2,000 Kelvin or more; hot enough to ionize the hydrogen and helium and emit some light, though in the early stages this is obscured by surrounding gas and it can only be observed from afar by radio, X-ray, or gamma ray emissions.

NASA/JPL-Caltech

Mass continues to fall into the star from the surrounding gas cloud, but some of the outlying material has too much angular momentum to fall directly into the core, and instead orbits it ever faster as it falls inward. Collisions between these gas particles tend to average out their angular momentum so that they converge on a common axis of rotation, transitioning from a spherical cloud to a disk shape where collisions are minimized—the protoplanetary disk, which is initially hundreds to thousands of AU in diameter.

Image of HD 163296. ESO, ALMA (ESO/NAOJ/NRAO); A. Isella; B.Saxton (NRAO/AUI/NSF)

After around 100,000 years, the core is dense enough to begin fusing deuterium, somewhat expanding the protostar. Jets of plasma from the protostar’s poles may impact the surrounding gas, creating bright Herbing-Haro objects. The exact mechanism for these jets is poorly understood, but as with pulsars and active black holes they are believed to arise⁶ from interactions between infalling gas and the protostar’s magnetic field. Though the star may lose as much as 10% of its mass

⁷ this way, the jets also carry away much of its angular momentum, reducing the centrifugal acceleration of matter in the protostar and thus helping it to collapse further.

 

Image of HH24 with large Herbig-Haro objects (the long jets). NASA/ESA

By about 1 million years after collapse begins, enough of the gas envelope has cleared for the protostar to be observed in the infrared and sometimes the visible spectrum. At this point it is referred to as a T Tauri star (technically the term only refers to the predecessors of F to M-type stars; pre-main-sequence A and B-type stars are referred to as Herbig Ae and Herbig Be stars respectively, and O-type stars have already entered the main sequence by the time they can be observed). The star has accreted most of its final mass and the remaining disk material typically amounts to 1 to 10 times the mass of Jupiter for a sunlike star, though it can be⁸ as much as 10% of the star's mass. More massive stars generally have more massive disks, but there’s plenty of individual variation and some stars can lack disks entirely. The disk may still extend several hundred AU, or it may have been cut down to 50 AU or less by encounters with other stars that pulled away outer parts of the disk.

 

Light and heat from the star creates a temperature gradient in the disk, from over 2000 K near the inner edge to under 20 K at the outer edge. The point where the disk reaches 170 K is called the iceline (A.K.A. snowline or frostline); beyond the iceline, water can condense to form grains of ice, but within the iceline it remains as a gas and so is more likely to be scattered by solar wind rather than contribute to the formation of solid bodies. This migrates inwards over time as the disk cools and condenses; in our solar system it may have started at around 8 AU, then moved in to around 2.7 AU at its closest⁹

Source

The dynamics of ice forming and sublimating at the iceline forms a gap in the disk¹⁰, and inward-migrating material piles up outside the gap, forming a dense ring of solid bodies. Similar gaps and rings form where the disk reaches 1400 K (~0.7 AU in the solar system), forming a similar iceline for silicates, and where it reaches 30 K (~20 AU), the iceline for carbon monoxide (methane, ammonia, nitrogen, and other volatiles—materials with low melting temperatures—each have icelines too, but these don't seem to impact planet formation as much, though they do influence the composition of the bodies that do form).

 

Izidoro et al. 2021

By 10 million years of age, hydrogen fusion finally starts in the star’s core, putting it on the main sequence. For stars like the sun or larger, the solar wind blows away any remaining gas in the disk or surrounding cloud not bound to a solid object within another 5-20 million years on average, but smaller stars may retain their disks¹¹ for 50 million years or more. Solid planetesimals may continue to combine into larger rocky or icy planets after the disk clears, but gas giants must have collected their entire gaseous outer layers by this point. And indeed, it seems that many stars⁵ lose their protoplanetary disks within just a few million years as the material is consumed by planet formation.

Gas Giant Formation

Concept of HD 100546 b, a gas giant forming around a Herbig Ae/Be star. ESO/L. Calçada

 

How exactly planets form is an area of ongoing research. There are two plausible models for the formation of large planets in the protoplanetary disk: The core accretion model, whereby solid dust grains join together to form planetesimals, and only after large cores are formed do gasses accrete onto them to form gas giants; And the disk instability model, whereby the gas in the disk fragments and collapses, just as in the formation of stars (indeed the smaller star in a binary often forms by much the same process¹²).

The disk instability method requires no metals¹³, but does require a massive, dense protoplanetary disk, and generally forms only gas giants larger than Jupiter and on very distant orbits from the star. Thus this may have been how the first planets in the universe formed, but now is likely only responsible for a small minority of observed planets (though there may be a selection bias there in which planets we observe).

In the core accretion model, formation of a gas giant requires first the formation of a large rocky core of about 10 Earth masses, which then has enough gravity to collect hydrogen and helium gas from the surrounding disk; increasing its gravity further in a runaway process that eventually forms a mostly gaseous planet 10s to 100s of times the mass of the solid core. For such a large solid body to form so quickly requires a substantial portion of metals in the disk, and so these gas giants can only form in systems with some minimum threshold of metallicity, and are generally more common around more massive and metal-rich stars.

 

Core accretion is generally agreed to be the dominant method for the formation of giant planets around sunlike stars today, though researchers are still working on understanding the whole process. In particular, meter-sized boulders formed by electrostatic forces may have trouble¹² joining together into kilometer-sized planetesimals that can continue growing through gravitational forces, as collisions in that size range can just as easily result in bodies fragmenting or bounce apart as joining together. But recent modeling¹⁴ seems to show that this gap can be bridged by pebble accretion, whereby a few initially larger bodies will grow by accretion of smaller, centimeter-sized bodies, rather than collision with each other.

This tends to be easier outside the iceline, where the ice both provides more solid material for planetesimal formation and makes for “stickier” debris that more easily crosses the meter-kilometer size gap (some research¹⁵ suggests that pebble accretion may only work outside the iceline). The lower temperature also helps gasses accrete to the core once it forms, such that the initial mass necessary for a gas giant can drop to as little as¹⁶ 3.5 Earth masses at 100 AU from a sunlike star. But the best place for formation¹⁷ of gas giants is in the aforementioned ring¹⁰ of dense material just outside the iceline; around 3 to 10 AU in the solar system.

 

Farther out, large planets may reach the threshold to form a giant atmosphere later and so only have a short time to accrete gasses before the disk clears; thus we get ice giants like Uranus and Neptune that have significant hydrogen/helium atmospheres, but are still mostly composed of other volatiles like methane, ammonia, and water. Large planets that form very close to their stars¹⁸ or impact other larger planets¹⁹ may be too hot to grow further (the hot gas escapes to space as quickly as more gas accretes) and so also end up with atmospheres only ~10-20% the mass of their cores.

Internal structure and composition of gas giants (Jupiter, Saturn) and ice giants (Uranus, Neptune). NASA

But just because a giant planet forms outside the iceline, doesn't mean it will stay there. One of the big surprises in the early days of exoplanet research was the discovery of hot jupiters like 51 Pegasi b, a gas giant with at least half Jupiter’s mass orbiting just 0.05 AU from a star slightly larger than the sun.

Concept of HD 189733b. ESA/C. Carreau

Gas giants may occasionally form¹⁷ somewhat inside the iceline, and smaller gaseous bodies¹⁸ might form in close orbits, but no gas giant that massive could have formed that close in, and so they must have first formed further out in the protoplanetary disk, beyond or at least closer to the iceline, and then migrated into their current position. This planet migration can actually be quite rapid when the disk is still present, due to a feedback between the planet gravitationally attracting disk material towards it and the disk material pulling back on the planet in turn. Faster-moving material inside the planet’s orbit will pull it forwards along its orbit, and slower-moving material outside will pull it back. For a typical disk we expect the latter effect to win out, sapping orbital energy from a planet and causing it to drift inwards towards the star.

Indeed, many planets may migrate in too fast; chemical evidence²⁰ indicates that roughly 1/4 of sunlike stars contain the remnants of planets that spiraled in to collide with the star early on. Some models of planet migration struggle to explain why this doesn't happen to all planets that form in this stage²¹, or suggest that they should all pile up²² at the inner edge of the disk, but it's an area of ongoing research. 

Terrestrial Planet Formation

Intriguing though gas giants may be, we expect that the majority of planets will be smaller bodies with mostly solid interiors. Very low-mass stars, with long-lived disks and short orbital periods within the disk, may form these planets before the disk clears²³ through the same core accretion process as gas giants. Though they rarely grow large enough for runaway growth to gas giants, they may still form substantial hydrogen atmospheres, often ending up as "mini-neptunes" ambiguously in between the terrestrial planets and gas giants of the solar system.

 

But for sunlike stars, terrestrial planets—with solid surfaces and (relatively) thin atmospheres—can take their time forming. After the gas of the protoplanetary disk is cleared, the system will still be filled with hundreds to thousands of planetesimals 0.01 to 0.1 times the mass of the Earth. While the disk was present they were mostly kept on circular orbits and so out of each other’s way, but with it gone they can pull each other into eccentric orbits and collide, thus eventually consolidating into a small number of terrestrial planets over the next 100 million years²⁴, a period researchers call the oligarchy stage.

Concept of NGC 2547-ID8 system. NASA/JPL-Caltech

If any gas giants are present, they will perturb the orbits of objects near resonant orbits (that is, objects with orbital periods that are a simple fraction of the gas giants’ orbital periods) which may encourage the formation of asteroid belts rather than planets, but otherwise planets generally seem to pack into the inner system²⁵ about as closely as they can without destabilizing each other. At this point, interactions with the little remaining debris settle the planets into circular, low-inclination orbits, though a few last large impacts—such as the one believed to have produced Earth’s moon—can still occur.

 

With the obscuring dust and gas of the protoplanetary disk gone, the icelines all move further out; our solar system's iceline has moved from 2.7 AU to its current position²⁶ at around 5 AU. But any icy planetesimals that formed before then can be shielded from sublimation by a surface layer of dust, and so the original positions of the icelines still largely determine the composition of the planets that form: Planets in the inner system will be composed mostly refractory rocks and metal, while those further out will have compositions increasingly dominated by ice. But the same processes that cause planet migration can also²⁷ pull icy material into the inner region of the disk, and the migration of giant planets itself can also scatter planetesimals far from where they form, so volatile-rich bodies can still form in the inner system.

Post-Formation Instability

Even once the oligarchy stage ends and the planets have all formed, they may still not be in their final positions. The planets may continue to interact with each other or with remaining debris to slowly shift their orbits, eventually reaching an unstable state where they shift around more rapidly, possibly passing close to each other or even colliding, until they finally achieve a more stable state again.

Most models for the development of the solar system predict that there would have been at least one such instability event within the first billion years after it formed, though so far they can't quite agree on what that event looked like. Some variations (which may not be mutually exclusive) include:

• The Nice Model²⁸ (named for the city in France, though I’m sure it’s friendly at parties) which proposes that early system was more compact than it is now, with Neptune (or perhaps Uranus in the outermost position) orbiting only 17 AU out, and as much as 50 Earth masses of material remaining in planetesimals in the Kuiper belt outside it. Planetesimals at the inner edge of this belt would occasionally encounter Neptune (or Uranus) and be flung inwards, pushing the planet outwards, and then continue down the chain of giant planets until they reached Jupiter, which was massive enough to turn these planetesimals around and fling them deep into the Oort cloud or out of the system entirely, pushing itself inwards.

After about 700 million years (though some researchers suggest²⁹ it was just a few tens of millions of years), Jupiter and Saturn entered a 1:2 orbital resonance, increasing their orbital eccentricities and wrecking hell with the rest of the system: Saturn was shoved outwards and the ice giants pushed out in turn (Neptune passing Uranus if it was previously inside it), plowing into the Kuiper belt and scattering it. Interactions with these bodies would have then stabilized the planet’s orbits again.

Orbits of the gas giants (colored orbits) and smaller bodies (white dots) in different stages of the Nice Model instability event. AstroMark, Wikimedia

• The Grand Tack Model³⁰ proposes that Jupiter initially formed at 3.5 AU, then migrated in while the disk was still present to as little as 1.5 AU until the influence of newly-formed Saturn helped stabilize it. Once the disk had cleared, they both migrated back out together towards their current positions. Jupiter's gravitational influence dragged along a good deal of debris along behind it, ultimately forming the asteroid belt and reducing the mass available to form Mars (which often forms with too much mass in simulations of the solar system without this movement).

• One new model³¹ proposes that the solar system formed with several additional planets at closer orbits than Venus, and after half a billion years at most these experienced a rapid series of collisions that left Mercury as the only survivor, with much of its outer layers stripped off to leave the dense, iron-rich planet we see today.  

Aside from rearranging the orbits of planets, these instability events may also have scattered many of the smaller asteroids and comets left over after the oligarchy stage. Some may have been captured as highly-inclined moons or Trojan asteroids, but many may also have directly impacted the planets themselves. The Nice model in particular has been cited as a possible cause³² for the Late Heavy Bombardment, a period of massive impacts for all the inner bodies; cratering their surfaces, eroding their atmospheres, and perhaps killing off any life that may have been present (though there is some debate²⁹ regarding whether there was a distinct bombardment event rather than just a gradual decline of impacts after planet formation).

Concept of the Eta Corvi system, which is currently experiencing a similar instability event. NASA/JPL-Caltech

 

Observations of debris disks and unusual planetary orbits in other star systems indicate that such post-formation instability may be common, sometimes leading to planets colliding or being ejected from the system entirely. They need not happen only in young systems, either. Some models³¹ suggest that inner planets may commonly form in tightly-packed, unstable configurations, with half of these systems experiencing planet collisions within 100 million years and over 90% in 5 billion years.

 

Indeed, even the solar system may not be fully stable; models of its future evolution³³ predict a roughly 1% chance that interactions with Jupiter may alter Mercury's orbit enough that it crashes into the sun or collides with another of the inner planets within the next few billion years. The rest of the planets appear stable for now, but as the sun evolves into a white dwarf and loses mass they will migrate outwards, which may leave them in an unstable configuration³⁴ that will result in them all being ejected into empty space within 50-100 billion years.

Patterns of System Architecture

So now that we understand how planetary systems form (as best we can given current research), what does all this mean in terms of the sort of system architectures we should expect to see in mature systems? Overall, it appears that the arrangement we see in our solar system—several small terrestrial planets within the iceline and several gas giants beyond it—is not the norm, but neither is it especially unusual.

 

Gas giants in particular aren't actually all that common, forming mostly around more massive and metal-rich stars which are more likely to form sufficiently massive solid cores before the protoplanetary disk clears. Precise numbers are hard to pin down, but modelling suggests³⁵ that they almost never form around M-type red dwarfs, form around about 10% of sunlike stars, and as much as 20% or more of A-type stars. For even larger stars, protoplanetary disks don’t last long enough for gas giant formation, and even around A-type stars the lower time for migration leads to a paucity of hot jupiters³⁶.

 

When gas giants do occur, they do seem to form more commonly near or outside the iceline; planets at least 1/10 the mass of Jupiter are significantly more common³⁷between 1 and 10 AU from the star, compared to closer or more distant orbits, and hot jupiters are only expected to occur in³⁸ about 0.5% of all systems.

Effect of metallicity and star mass on the formation of planets of different sizes. Mulders 2018.

 

By comparison, about half of sunlike stars should have systems of small planets³⁹, and these are inversely more common around smaller stars, becoming almost ubiquitous⁴⁰ around M-types. These planets are also more likely⁴¹ to form in groups of planets of similar size, while giants more often form alone or of varying size. "Small" is a relative term here, mind; many of these planets are superearths with 2 to 20 times the mass of Earth, and limitations to current exoplanet detection methods have made it hard to say much for sure about the occurrence of planets smaller than Earth (though modelling⁴² indicates they're probably also more common around smaller stars). Planets around Neptune’s size appear to be around equally common for small and large stars.

Occurrence rate (portion of all stars) of planets (measured in Earth radii) around M-type versus larger stars. Mulders 2018

There's likely some relationship between these reversed trends: The earlier a gap forms at the water iceline, the less material moves into the inner solar system and the more is available to from a gas giant, that then opens another gap in the protoplanetary disk that further blocks inward migration⁴³ of disk material and planetesimals. Lower-mass stars with smaller and slower-evolving disks form these gaps later or less strongly and are less likely to form gas giants, and so more material moves into the inner system to form volatile-rich superearths.

How this pans out for more sunlike stars is less clear: while we might expect that formation of a gas giant should prevent the formation of inner superearths, so far observations seem to indicate that systems with "cold" gas giants near the iceline are actually more likely⁴⁴ to have at least one superearth. It may be that while gas giant formation does reduce the mass available in the inner system, any disk massive enough to form a gas giant is more likely to already have enough mass in the inner disk to form superearths even without additional material from the outer system. Thus a cold gas giant doesn't prevent inner superearths forming entirely, but may still reduce the size and number of them, or at very least cause them to form with lower proportions of volatiles⁴⁵ (in this context, Earth counts as "volatile-poor", having only partial ocean cover a few kilometers deep rather than global oceans hundreds of km deep; the "drier" option here is probably better for life for reasons we'll discuss another time).

Likely influence of disk mass on planet formation. Schlecker et al. 2021

There is clearer evidence that a "warm" gas giant well within the iceline likely prevents⁴⁴ the formation of any other inner planets, save perhaps on very close orbits of the star. But this is not the case⁴⁶ for a "hot" Jupiter very close to the star, as even though it migrates through the inner system, it typically does so quickly and leaves enough material and time for small planets to form afterwards.

The rings and gaps at icelines may also influence the formation of asteroid belts: the gap at the water iceline may provide a space between the inner and outer planets where asteroids can settle into stable orbits after being scattered by encounters with the planets, and the ring formed at the CO iceline may be too low-density and slowly-evolving (due to long orbital periods) to form planets, instead leaving a belt of icy material like our Kuiper Belt. But either belt could be scattered by later planet migration, and new belts may form due to collisions between large bodies.

After planet formation, instability events may not be ubiquitous, but are common enough to impact the populations of planets we see in mature systems; within systems that contain gas giants, there's a stark contrast between systems with giants on circular orbits, often with multiple other planets, and systems with a single giant in a highly eccentric orbit and few or no companions. The latter systems likely formed with more circular orbits initially, but interactions between the giant and other planets or stars increased its eccentricity⁴⁷ until it crossed the orbits of the other planets, either colliding with them or ejecting them out of the solar system until only the giant remained. This becomes increasingly likely if a system forms with multiple large giants⁴⁸ of similar mass.

 

More moderate instability events may also occur within the inner system. Inner planets seem to commonly form in very tightly packed resonant chains (we'll discuss mean-motion resonance shortly). For low-mass stars this is likely because the inner planets form within the protoplanetary disk and then migrate inwards; outer planets migrate faster until they reach a resonance with an inner planet, at which point they continue migrating together until either the disk clears entirely or a gap opens in the disk around the star and the outermost planet in the chain passes the inner edge of the disk, ceasing it's migration. In some cases⁴⁹ the inward migration of an outer gas giant may influence the inner planets to cluster even closer together.

These tightly-packed inner systems may occasionally survive for billions of years, but in most cases³¹ (especially if there's a gas giant present) they may eventually become unstable and encounter and collide with each other until several of the planets are destroyed, ejected out of the inner system, or fall into the star, leaving a smaller number of planets in a looser configuration (as mentioned, this may even have happened in the solar system).

So far, little is known about the formation of planets far beyond the iceline, like the ice giants in our system. Some analysis of debris disks⁵⁰ suggests that planets of Neptune to Jupiter mass should be common between 10 and 100 AU. Much as they block the inward migration of disk material, gas giants near the iceline may also block the inward migration⁵¹ of large planetary cores, such that those planets end up as ice giants in the outer system rather than superearths in the inner system.

Taking it all together, a rough guess of the population of mature planetary systems might go something like so:

• <1% of systems (mostly massive and metal-rich stars) with hot giants, often with additional small planets orbiting further out and sometimes even closer in (WASP-47, 55 Cancri A).
• ~1-5% of systems (mostly massive and metal-rich stars) with warm giants, occasionally with small planets on very close orbits (Gliese 876, Kepler-65).
• ~5-10% of systems (mostly massive and metal-rich stars) with highly eccentric gas giants and few or no other planets (TOI-677, 54 Piscium).
• ~5-10% of systems (moreso massive and metal-rich stars) with low-eccentricity, cold gas giants—rarely more than one large one—and small inner planets, possibly also with outer superearths or ice giants (Solar system, Kepler-90).
  
• ~5-10% of systems (moreso lighter stars) with large numbers of smaller planets, often superearths of similar size, tightly packed into the inner system, often in resonant chains, occasionally with an outer gas giant (TRAPPIST-1, K2-138)
  
• ~20-60% of systems (moreso lighter and metal-poor stars) without gas giants, with small and superearth planets of similar size in a looser arrangement (Kepler-186, 82 G. Eridani).
• ~10-40% of systems (moreso metal-poor or very massive stars) without large planets, though there should usually be at least some small bodies in orbit.
  

As our study of exoplanets continues, these estimates may shift around considerably, or we may find whole new system configurations as our ability to detect lower-mass or farther-orbiting planets improves (you may also note how vague I'm being about what counts as a "large" or "small" planet, given current uncertainties). And regardless, we can expect more than a few oddball systems that don't quite fit into any of these categories, due to unusual formation histories.

Placing Orbits

Now that we have a sense of the overall "shape" of most star systems, let's start digging into specifics. First, the boundaries: how close or far can a planet orbit? I've already mentioned that there is a silicate iceline (I've also heard it called the rockline) within which rock cannot solidify, which might have been at around 0.5-0.7 AU when the planets in the solar system began forming. But metals might still solidify within this boundary, and it also migrates significantly inwards as the inner disk cools. There is a harder boundary to formation at the inner edge of the protoplanetary disk, at about 0.1 AU for a sunlike star; any material that passes this limit⁵ during the star’s T Tauri stage will be pulled in by the magnetic field and either fall into the star or be thrown out in polar jets.

But we already know that planets can migrate in past this limit later in the formation process, and not only hot jupiters; recent research⁵² has discovered a new class of extremely close-orbiting terrestrial planets with periods measured in hours and surfaces so hot that their rocky surfaces are actually sublimating and being lost to space. The rate of mass loss becomes significant⁵³ when surface temperatures surpass roughly 2,000 K, which for a planet with a surface like Mercury would be at 0.02 AU from a sunlike star; though even here a large planet may take a long time to sublimate away completely.

The ultimate inner boundary is set by the Roche limit, the point at which the tidal forces from the star become stronger than the planet’s gravity and it is torn apart. For a rigid body, the calculation is straightforward:

d = Roche limit (any unit so long as r is the same)

r = radius of the planet

M = mass of the star (any unit so long as m is the same)

m = mass of the planet

But planets are not rigid, and in fact are often better modeled as fluids; a proper calculation for the Roche limit of fluid bodies is rather complex, but for most cases it can be reasonably approximated as 1.94 times the rigid body Roche limit.

The actual Roche limit for any body will be between these two values (by way of example, Saturn's innermost moon Pan orbits at 1.42 times its rigid Roche limit and 0.7 times its fluid Roche limit); given that the fluid limit approximation will always be larger, it can be taken as the safe limit.

 

Note that the rigid Roche limit formula can be equivalently written as:

 

d = Roche limit (any unit so long as R is the same)

R = radius of the star (not the planet)

Ρ = density of the star (any unit so long as ρ is the same)

ρ = density of the planet

 

I.e., it is the same for bodies of different masses but equal densities; thus we can define a Roche limit for specific materials, like rock or ice (though a given material will be more compressed and thus denser for a larger planet; more on that in the next post).

This isn’t a particularly stringent limit; the fluid Roche limit for a body of Earth's density orbiting the sun is at 0.007 AU, less than 1/50 the orbit of Mercury. But some possible exoplanets⁵⁴ are observed near their Roche limit, and are expected to be distorted by tidal forces into an elongated, American-football-like shape.

There are no strict outer limits for planetary orbits (except for in wide binary systems) but the further you get from the star, the less material there is available and collisions become less frequent. Neptune is the furthest-out known large planetary body at 30 AU, but decent evidence exists⁵⁵ for a 9^th planet at 400-800 AU. Were that the case, it would almost certainly be the last planet—and in a more crowded part of the galaxy, it could easily have been ejected by an encounter with another star long ago.

That settled, where might planets orbit within these limits? Across all the many architectures, there are a few common patterns. For one, most systems with multiple planets generally seem to have⁵⁶ very low average eccentricity (~0.04) and inclination between planetary orbits (~1.4°), though as mentioned there are a minority of systems with lone gas giants on highly eccentric orbits (~0.3), likely the remnants of past instability events. Planets in extremely close orbits often tend⁵⁷ to have orbits inclined by around 10-20° relative to the rest of the system—in some cases⁵⁸ higher than 90°, effectively on a perpendicular orbit—possibly due to either tidal interactions with the star or interactions with other planets that caused them to lose eccentricity and gain inclination as they migrated inwards.

 

There's also a pretty consistent trend for planets orbiting further from their star to be more spaced out in their orbits. 19^th century astronomers attempted to quantify this trend within our solar system with the Titius-Bode Law, but “law” may be too lofty a title for a rule that only works if you’re willing to count the asteroid belt as a planet and just forget about Neptune. A softer and more reliable rule, based on observations of exoplanetary systems, is that as you move outward each successive planet tends to have an orbital period between 1.5 and 3 times that of the previous planet. As per Kepler’s 3^rd Law, this corresponds to 1.3 to 2.1 times the semi-major axis:

a₁, a₂ = semi-major axes

P₁, P₂ = orbital periods

Where the bodies’ masses are negligible compared to the common parent body.

(See my previous post on orbits if any of the terms here are unfamiliar.)

 

But forget averages, what if we want to really pack these planets in there? The smallest period ratio observed⁵⁹ so far is 1.17 for two planets in Kepler-36, which appear to be in a stable configuration. But the theoretical limits of stability are best described not by period ratios, but by the radius of the Hill sphere, the region around a planet (or any other orbiting body) where smaller bodies would tend to orbit the planet rather than following their own orbits around the star (or other parent body). This Hill radius approximates to:

R_H = Hill radius

a = semi-major axis

m = mass of the orbiting body

M = mass of the parent body

Two bodies of similar mass will have a stronger mutual attraction and so tend to orbit each other at a greater distance than either of their individual Hill radii, so we can define a mutual Hill radius:

R_Hmut = mutual Hill radius

a₁, a₂ = semi-major axes

m₁, m₂ = masses of the orbiting bodies

M = mass of the parent body

Two planets can only follow independent orbits so long as they stay at least 1 mutual Hill radius apart; closer approaches won't immediately enter them into stable orbit of each other due to their initial relative velocity (in a sense they briefly enter a mutual hyperbolic "orbit", but not a closed elliptical orbit), but they will attract each other enough to shift into dramatically different orbits of the star. But even if they don't get so close together at first, their mutual attraction can gradually shift their orbits until they do have a close encounter.

 

Because of the chaotic nature of gravitational interactions over many orbits, it's difficult to place hard limits on which orbits are stable, but as a general trend a pair of bodies with a minimum separation below 8.6 mutual Hill radii are significantly less likely⁶⁰ to remain stable for billions of orbits, and a minimum separation below 6 mutual Hill radii virtually guarantees⁶¹ instability within millions of orbits (once systems do go unstable, planets are also more likely and quicker⁶² to collide the lower the initial separation between them). These limits can be about halved⁶³ for two bodies orbiting retrograde to each other, though a retrograde-orbiting body would likely have to be captured from outside the system (not unusual for moons, likely very rare for planets).

 

But we can push past even these limits with mean-motion resonances (MMRs): cases where two (or more) bodies orbit with exact ratios between their orbital periods, e.g. a 1:2 resonance means one body has exactly twice the orbital period of the other, 2:3 means one body completes 3 orbits in the same time the other completes 2, and so on (though there are more complex cases where orbital periods oscillate but are in resonance on average, or resonance exists in a rotating reference frame, etc.; the dynamics get very complicated quickly if you look into the details).

 

1:2:4 Laplace resonance between Jupiter's moons WolfmanSF, Wikimedia

 

This means that they pass through a regular cycle in their relative positions and closest approaches, exerting the same forces on each other over and over again. This generally results in a gradual exchange of momentum⁶⁴: one of the bodies is pulled into a wider or more eccentric orbit while the other reduces its semimajor axis or eccentricity (they can exchange inclination as well if their orbits are initially inclined to each other). In many cases, this eventually results in their orbits crossing with each other or with that of another body, leading to a close encounter that flings them into new orbits, which may lead to further encounters, and so on, potentially cascading into destabilizing the whole system if these are large bodies.

But as their orbits shift, the cycle of their influences on each other also changes. If everything lines up just right, they can eventually begin exchanging momentum back the other way, bringing them back towards their initial arrangement, at which point momentum exchange shifts again. This back-and-forth exchange of momentum can be very regular, ensuring that these bodies never destabilize each other even in very close orbits (for similar-mass bodies the oscillations can also be very minor, a tiny regular change in semimajor axis or eccentricity). They can even have orbits that cross each other (as is the case for Neptune and Pluto in a 2:3 resonance), as the resonance ensures that they're never near each other while one crosses the orbit of the other. I've also mentioned how, early in planet formation, planets in resonant orbits can migrate together; as one planet orbits towards or away from the other, the other planet will tend to migrate back towards a stable resonance, effectively following its partner's motion.

Hence, MMRs are something of a double-edged sword; sometimes destabilizing whole systems, sometimes allowing for planets or moons to remain stable in very closely packed orbits. Predicting whether or not a particular resonant pair of bodies will be stable is near impossible without detailed modelling, but there are a few major influencing factors:

• Resonance Order: The difference between the integers in the ratio; a 1:2 or 2:3 resonance is first-order, a 1:3 or 3:5 resonance is second-order, a 2:5 resonance is third-order, and so on. In general, lower-order resonances are more likely to become stable. But even MMRs cannot prevent instability if bodies pass within their mutual Hill radius, hence why very low-ratio MMRs like, say, 21:22 are rare, as they imply a very small difference in semimajor axis.
  
• Mass Ratio: The ratio between the masses of the two bodies. If the ratio is very large, then a small momentum change for the larger body will cause a very large momentum change for the smaller body, which can result in huge changes in eccentricity for the smaller body, greatly increasing the chance of an encounter with another body. Similar-mass bodies are more likely to be stable, though the picture is a bit more complicated when considering the stability of the whole system: Two Jupiter-mass planets may enter a stable resonance, but their regular oscillations may destabilize other planets in the system (which can ultimately feedback into destabilizing their resonance); hence why systems with multiple large gas giants are significantly less likely to remain stable. By contrast, a gas giant destabilizing an asteroid's orbit is unlikely to have many broader consequences, and if there are many asteroids in the system then there's a good chance that at least a few will happen into stable resonances with the larger bodies even if most are scattered away.
  
• Tidal Effects: As we'll discuss later, the tidal influence of a star or planet can act to reduce the eccentricity of an orbiting planet or moon. Thus, if two bodies in resonance are experiencing strong tidal forces from their mutual parent body, the tidal forces may counteract any eccentricity increase caused by the resonance, making it stable where it wouldn't be otherwise.
• Orbit Alignment: Much as they can orbit closer together out of resonance, bodies orbiting retrograde to each other are significantly more likely⁶⁵ to capture into stable resonances, even from initially high inclinations or eccentricities. but, again, retrograde-orbiting planets are likely very rare, though retrograde moons aren't unusual.
  

Within our solar system, these factors have a clear influence on the structure of the asteroid belt: there are major groups of asteroids in 2:3 and 3:4 resonances with Jupiter, but gaps have opened at the less-stable 1:3, 1:4, 2:5, and 3:7 resonances; any asteroids that may once have orbited there had their eccentricity increased until they impacted another body, either destroying them or shifting them into a different orbit (these are "gaps" in the distribution of orbital periods, mind; because many asteroids have elliptical orbits, there are no visible gaps in the positions of asteroids in the belt at any one time).

Wikimedia, based on plot by Alan Chamberlain, JPL/Caltech

None of the planets in our solar system are in resonant orbits (Jupiter and Saturn are close to a 2:5 resonance but unlikely to properly enter it any time soon), but many are in resonance with minor bodies; Aside from Pluto, Neptune has a number of smaller partners in 1:1, 1:2, 2:3, 2:5, 3:5, 3:10, 4:7, and 7:12 resonances, Jupiter has its aforementioned resonances with many asteroids, and even Earth has a few small asteroids in 1:1 or 5:8 resonances. Resonances are also fairly common for moons:

• Io, Europa, and Ganymede have a 1:2:4 resonance (1 orbit of Ganymede corresponds to 2 of Europa and 4 of Io), an arrangement called a Laplace resonance.
  
• Hyperion and Titan have a 3:4 resonance.
• Pluto's moons Styx, Nix, and Hydra appear to be in an 18:22:33 resonance.
  
• Some of Saturn's inner moons have a pair of overlapping resonances: by semimajor axis they're ordered Mimas, Enceladus, Tethys, and Dione, and though none are resonant with their immediate neighbor, there's a 1:2 resonance between Mimas and Tethys and between Enceladus and Dione.
• There are also several 1:1 resonances among Saturn's moons that we'll discuss shortly.
  

In exoplanetary systems¹ resonances are common among giant planets, but curiously rare for terrestrial worlds. However, there are a suspicious number

⁶⁶ of terrestrial pairs with period ratios slightly larger than 1.5 or 2.0, indicating that they probably formed in 2:3 or 1:2 resonances but the inner planet migrated slightly inwards due to interactions with the last remnants of the protoplanetary disk. When small planets do fall into resonances they can allow for extremely tight systems, such as the TRAPPIST-1 system which manages to pack 7 planets within 0.062 AU of the star with a 2:3:4:6:9:15:24 resonance chain. As mentioned, these resonance chains may be common in young systems, but liable to eventually destabilize in the majority of them.

Co-orbital Planets

The motion of 4 common types of co-orbital bodies from the perspective of the partner. Morais and Namouni 2017

But let's say we want to pack in even more planets. We can only get their orbits so close together, but can we put more than one planet in each orbit? As it turns out, we can, in several different ways.

Binary Planets

This one is fairly straightforward: rather than following separate orbits and never approaching each other, two planets could instead closely orbit a common barycenter (center of mass directly between them; see my discussion of multiple star systems for more information on how these orbits work) that itself orbits the star. Arguably this is true for any planet-moon system, and indeed there's no strict distinction between those and binary planets; much of what I'll say for moons later applies to binary planets as well. A common proposal is to define the two bodies as a planet-moon system if their barycenter is inside the more massive body and binary planets if their barycenter is outside either body—by this standard, Earth and Luna are a planet and moon while Pluto and Charon are binary dwarf planets. I have my issues with this definition, though; for one, it's awkwardly a bit dependent on the distance between the bodies rather than just their relative size, so if some outside influence were to cause Pluto and Charon to migrate closer together, their barycenter would eventually move into Pluto and Charon would be "demoted" to a moon; and for another, the status of Pluto's minor moons is made a bit ambiguous here, as they all orbit the Pluto-Charon barycenter rather than any point inside Pluto and it would be odd to count all of them—some under 20 km across—as dwarf planets of equal standing.

 

Given their similarities, we'll discuss possible mechanisms for the formation of both planet-moon systems and binary planets together in a bit. But the short version is that, while small planetesimals under ~1,000 km across may form as binaries, larger planets would likely have to form independently and then capture into a binary orbit. This can happen a few different ways, most of which are likely to leave them orbiting close together. Tidal interactions between the binary and their star will then cause them to gradually migrate towards each other⁶⁷, eventually causing them to collide—though this may take many billions of years.

And even if the planets do "collide", the may not do so cataclysmically, at least at first. If the planets have very similar mass and density, the Roche limit can become smaller than the combined radii of the planets (sort of; the dynamics here are a bit more complicated than simple Roche limit approximations can represent); the planets could stably orbit so close together as to be in contact, forming a contact binary (or rocheworld, if you prefer). To avoid intense friction that would pull them into a faster collision, the planets would need to be mutually tidally locked (not rotating relative to each other) and have no eccentricity in their orbit around their barycenter (which at this point would have a very short orbital period; something like 3-4 hours for a pair of Earth-mass planets). The intense gravity would distort them into teardrop shapes pointing towards each other. As they drew closer together, first their atmospheres would intermingle, and then, once very close together, a solid bridge of rock could form between them.

 

Cover of the novel The Flight of the Dragonfly, later renamed Rocheworld, which is actually a fairly decent representation of the likely shape of such a body (though they need not be such distinct colors).

Some asteroids and comets appear to be contact binaries between formerly distinct bodies, and some stars are also believed to be contact binaries, but whether a contact binary planet can form has yet to be seen. I have yet to find any formal treatment of the possibility, so I can only speculate: One possible concern is that the same tidal influence that brought them into contact would likely continue pulling them into a merger (though how gradual and non-cataclysmic that process could be is unclear). Even were that not the case, minor perturbations from outside or imperfections in their orbit or the flow of fluids could impose frictional forces that sap away orbital energy, with the same result. Without proper modelling, I simply can't be sure of how stable these bodies are likely to be.

 

486958 Arrokoth, a contact binary comet in the Kuiper Belt. NASA/Johns Hopkins University Applied Physics Laboratory/Southwest Research Institute/Roman Tkachenko.

As a last thought, if two planets can orbit each other in a binary, could a third planet be added, forming a trinary planet? Or a fourth? Stars manage to cluster in groups of up to 8, so it seems possible in principle, but there are a couple major challenges: first, as we'll discuss, the capture mechanisms that could form a binary planet are most likely to place them in a very close orbit, which is fine for a binary but problematic for a third planet that would need to orbit from further out for the whole system to remain stable. Perhaps the greater tidal influence of a binary rather than a single planet could allow for a more distant capture, but therein lies the second problem; these tightly packed multiple-planet systems would experience much stronger tidal forces between bodies than is typical for multiple-star systems, and are also subject to strong external influences from the star and any other planets in the system. Once again, there's no published literature investigating the plausibility of such a system, but at the very least I'd expect them to be rare and less likely to form in close orbits of the star (a binary planet with significantly smaller moons may be easier to achieve, though).

Trojans

These planets would still orbit the central star (or other body) independently, staying well outside their mutual Hill radius, but do so within the same orbital path and with the same orbital period, existing effectively in a 1:1 mean-motion resonance. As discussed in Part III, in any star-planet (or planet-moon) system there are 5 Lagrange points where their combined forces balance out to allow a third body to orbit in 1:1 resonance with the planet, but only the L4 and L5 points, 1/6 of an orbit ahead and behind the planet, are stable on astronomical timescales.

 

Anynobody, Wikimedia

Venus, Earth, Mars, Jupiter, Uranus, and Neptune are all known to have small asteroids at their L4 and/or L5 points, called trojans, and clouds of dust have gathered at the Moon's L4 and L5 points. Two of Saturn's moon, Tethys and Dione, each have a pair of smaller trojans at their L4 and L5 points.

 

But there need not be such a great mass ratio between planet and trojan; in fact, two planets of similar mass could act as mutual trojans, occupying each other's lagrange points. The main limit on mass is that the combined mass of the orbiting bodies (planet and trojan) cannot be more than⁶⁸ 4% the mass of the central body (star); For the sun, this is roughly 40 times the mass of Jupiter. Some models⁶⁹ indicate that trojan pairs of planets should form fairly often, though large, similar-mass trojans may usually drift into different orbits or collide due to migration⁷⁰ or tidal interactions⁷¹, though how ubiquitous this is for planet pairs of different masses, orbital periods, or eccentricities remains unclear. Prospects seem better for a gas giant with a smaller but still substantial trojan; planets as large as superearths⁷² may form directly in the giant's Lagrange point and more reliably stay with it through migration.

Of course, a planet has two stable Lagrange points, so why not place trojans at both? Indeed Jupiter likely has⁷³ thousands if not millions of trojan asteroids (mostly in tadpole orbits as we'll discuss next, so they're not all just crowded into the Lagrange points). A trio of equal-mass planets along the same orbit appears to be possible in principle⁷⁴, orbiting 47.4° apart rather than 60°.

Necessary spacing (as angles along orbit) of similar-mass planets placed within the same orbit; all are symmetric across the middle planet or pair of planets. Salo and Yoder 1988

In fact, anywhere up to 8 equal-mass planets lined up in this way could be stable in theory, and at least 7 planets evenly spaced out in a ring across the whole orbit are also stable, so long as⁷⁵ they remain at least 12 mutual Hill radii apart and⁷⁶ the mass of each planet is less than:

m = Individual planet mass (any unit so long as M is the same)

M = Star mass

n = Number of co-orbiting planets in ring (at least 7)

 

The planet masses need not be exactly the same, nor their positions perfectly lined up from the outset, but even so it's increasingly improbable for larger numbers of similar-mass planets to line up together in this way in any natural system. But an advanced civilization building their own planetary systems could take these principles to some fairly impressive extremes⁷⁷.

Tadpole Orbits

Strictly speaking, the L4 and L5 points are not fully stable; a perturbation by an outside body can shift the trojan off the Lagrange point, and it won't return directly back to it. But the influence of the planet and star will cause it to oscillate around the Lagrange point, still keeping it in a stable 1:1 resonance on average. Much as with other mean-motion resonances, this operates by a sequence of momentum exchanges; if the Trojan is, say, shifted into a slightly higher orbit from the L4 point, it will, per Kepler's laws, have a longer orbital period and move slower than the planet in its orbit, thus gradually drifting back towards it. As it gets closer, the planet's gravity will pull it down into a lower orbit, meaning that its orbital period decreases and it speeds up, drawing away from the planet. As it gets further from the planet it eventually moves back up into a higher orbit, and so on in a closed loop. From the planet's perspective, this loop looks something like a curved tadpole, hence the name.

 

Relative motion of a body in a tadpole (dark paths) or horseshoe (light path) orbit, though the variation in distance from the star is greatly exaggerated here. Murray 1997

 

These oscillations can be fairly small, such that we can say the trojan is effectively static at the Lagrange point relative to the planet (arguably the case for Tethys's trojans, for example), but they can also be fairly large, bringing the trojan as close as 23.5° from the planet in their orbit and as far as nearly the opposite side of the orbit.

 

Many trojans also have some orbital eccentricity and some inclination relative to the planet as well, though generally less than 0.3 and 20°, respectively; the maximum allowable eccentricity for a trojan is about 0.7 for very small bodies and then generally decreases⁷⁸ as the ratio of the combined planet/trojan mass to star mass increases, though with some oddities close to the aforementioned 4% limit; at a ratio of 3%, a trojan must have negligible eccentricity, and at a ratio of 4.6%, a trojan is actually possible⁷⁹ but only in a tadpole orbit and with around 0.314 eccentricity.

 

At any rate, this means that there is both an annual cycle in the Trojan's relative position to the planet due to their different orbital paths and a more gradual cycle in the Trojan's average position that loops around the Lagrange point.

Motion of the asteroid 2010 TK7 relative to Earth over the course of a single orbit (left, NASA), over the course of many orbits (middle, Phoenix7777, Wikimedia), and the motion of both 2010 TK7 and Earth relative to the Sun (right, same author). Note how the actual orbital path of the trojan changes little, with the "tadpole" motion due to very subtle shifts in semimajor axis (and thereby orbital period) causing large shifts in its relative position along its orbit.

Of course, all of these examples are large planets with small trojan asteroids. Similar-mass planets will have a more symmetric relationship, both oscillating in semimajor axis and more likely to share any eccentricity between them.

Horseshoe Orbits

Though the tadpole motions of trojans are driven by a slight mismatch in semimajor axis and orbital period, this is necessarily only a small difference and the trojan's orbit shifts very little over the cycle of motion. But a similar type of orbit cycle with an average 1:1 resonance can also occur between two bodies with a somewhat more substantial mismatch in semimajor axis. 

Much as with tadpole orbits, the body on the inner orbit—smaller semimajor axis—will have a shorter orbital period and move faster in its orbit, gradually moving ahead relative to the other body. Eventually it loops around the whole orbit and then approaches the slower-orbiting body from behind. As they near each other, there is a relatively rapid exchange of momentum between bodies; the faster-orbiting body is pulled out into a wider orbit and the slower-orbiting body is pulled in to a tighter orbit, essentially swapping position. The formerly faster-orbiting body now orbits slower than its partner and so falls behind it, until its partner loops around the star and approaches it from behind, exchanging momentum again and completing the cycle.

The name comes from the apparent horseshoe-shaped path that each body follows from the perspective of the other. Two of the small moons of Saturn, Epimetheus and Janus, are in a horseshoe orbit such that they swap order roughly every 4 years, and there are a few asteroids in horseshoe orbits with Earth with cycles over hundreds of years.

 

Modelling of such orbits suggests⁸⁰ that they should be possible for a pair of planets with a combined mass up to around 0.08% the mass of their star; for a sunlike star, around 80% the mass of Jupier, or a bit over twice the mass of Saturn. In truth, the dynamics of horseshoe orbits are a bit more complex than I've presented here (In addition to their swapping of orbits, their semimajor axes vary slightly throughout the rest of the cycle, getting closest when they're 180° apart in their orbit—near each other's L3 points—and farthest apart when 60° apart—near each other's L4 or L5 points—though in most cases these are negligible differences) but there are a few reliable approximations for stable orbits. First off, the critical ratio between semimajor axes (when the bodies are 60° apart) at which the bodies switch from a tadpole to horseshoe orbit can be approximated⁸¹ like so:

 

r_min =  Minimum ratio between semimajor axes of planets for horseshoe orbit (and maximum for tadpole orbits)

m = Combined mass of planets (any unit so long as M is the same)

M = Mass of star

For bodies far below the maximum mass for a stable horseshoe, the maximum difference between semimajor axes for stability can be generally approximated as 1.2 times their mutual hill radius, but the limit is stricter for more massive bodies and modelling of of these orbits⁸² indicates a it can be approximated as:

 

r_max =  Minimum ratio between semimajor axes of planets

m = Combined mass of planets (any unit so long as M is the same)

M = Mass of star

The difference between semimajor axes will of course determine the difference in periods, per Kepler's laws. The time between closest approaches, when their orbits switch, can be roughly estimated as their synodic period, (the time between conjunctions for any two bodies orbiting the same parent when they reach the same mean anomaly):

 

T_Syn = Synodic period (any unit so long as T₁ and T₂ are the same)

T₁ = Period of shorter orbit

T₂ = Period of longer orbit

But really this time will be slightly less, depending on the distance between bodies at closest approach. I haven't found any way to estimate exactly what this distance will be in any given case, but as general guidelines: the greater the difference between semimajor axes, the smaller the distance, with a minimum of 5 mutual Hill radii and a maximum of 23.5° along their orbits.

So, for example, a pair of Earthlike planets orbiting a sunlike star could have horseshoe orbits with a ratio of semimajor axes between 1.004 and 1.028; were one placed in Earth's orbit with an orbital period of 365 days, the other could have an orbital period of 350 days, with the orbits switching roughly every 23 years, with the bodies approaching within roughly 0.063 AU of each other; 25 times the distance from Earth to the Moon.

But we can get a more dramatic effect if we increase the mass of one of the planets to roughly 0.8 times the mass of Jupiter, the limit for stability; the maximum ratio of semimajor axes increases to 1.053, and the large difference in masses means that the large planet remains at roughly the same orbit while the smaller switches from 0.95 to 1.053 times its semimajor axis. So with the larger body in a 365-day orbit, the smaller would switch between 338-day and 395-day orbits roughly every 12 years; this represents about a 20% change in insolation, similar to a mild seasonal shift, which we can imagine would have some interesting impacts on climate. And of course the variation in year length could potentially lead to a rather confusing calendar.

 

In case you're wondering, a pair of mutual trojan planets can have a third body⁸³ in a tadpole or horseshoe orbit, in the latter case essentially bouncing around from encounters with one and then the other without ever getting between them.

Eccentric Resonance

In this case, two bodies have equal periods and orbit in phase (same mean anomaly) such that they remain near each other, but one or both bodies have high eccentricity, which offsets their motion enough that they aren't near each other when they cross each other's orbits. A few asteroids exist in such resonances with Earth—and Venus, Ceres, Vesta, Jupiter, Saturn, and Neptune—and are sometimes called quasi-satellites because, from our perspective, they appear to circle around Earth over the course of a year. But even though Earth influences their orbits, they remain outside Earth's Hill sphere and follow kidney-bean shaped rather than elliptical paths around it, and so cannot be said to truly orbit Earth. Pluto appears to have⁸⁴ a single "accidental" quasi-satellite that remains in rough resonance and phase with it not because of Pluto's influence but because both are in resonance with Neptune.

 

Motion of the asteroid 2016 HO3 relative to Earth (left) and motion of both relative to the Sun (right). Phoenix7777, Wikimedia

Much as with other resonances, there is a gradual exchange of momentum between the bodies, effectively passing eccentricity back and forth between them. For a planet in resonance with an asteroid, this won't do much to the planet's orbit, but for two planets of similar mass this could create a cycle where one planet shifts from a circular to highly eccentric orbit while the other does the reverse.

Orbital evolution of 2 Jupiter-mass planets in eccentric resonance. Laughlin and Chambers 2002

 

Modelling indicates that these orbits should be possible for a pair of planets with a combined mass up to 3.5% the star’s mass. This planets/star mass ratio also determines⁸⁵ the time for eccentricity to pass from one planet to the other and back again: For a pair of planets orbiting the sun (at moderate maximum eccentricity) it can be as little as 100 orbits for Jupiter-mass planets and as much as 100,000 orbits for Earth-mass planets. The maximum eccentricity also impacts the period, though only significantly when very high; for maximum eccentricities of 0.1 or 0.5 the periods are about the same, while for 0.9 the periods are roughly 10 times greater.

Unlike tadpole or horseshoe orbits, eccentric resonances could also be stable for bodies orbiting retrograde to each other, and at least one asteroid⁸⁶ is known to be in a retrograde resonance of Jupiter. From Jupiter's perspective the co-orbital body appears to follow a sort of double-loop called a trisectrix, but the actual path of the body and other dynamics are much the same as for prograde eccentric resonances. As with other resonances, capture into eccentric resonance is likely actually easier⁶⁵ for retrograde bodies, but they're still likely to be rarer just because retrograde-orbiting bodies are rare.

 

Orbit of the asteroid 2015 BZ₅₀₉ relative to the Sun and Jupiter, as seen from "above" Jupiter's orbit (top) and within it (bottom). Wiegert et al. 2017

Inclined Resonance

This is a fairly new configuration observed⁸⁷ in Neptune's moons Naiad and Thalassa, which orbit with a 69:73 resonance. Were they orbiting in the same plane, this would bring them within 1850 km on each other, but in fact their orbits are inclined by a bit over 4° and aligned such that they always stay over 3500 km apart during their closest approaches, reducing the strength of their mutual attraction that might reduce their orbital stability.

 

Path of Naiad's motion over many orbits relative to Thalassa. NASA/JPL-Caltech

Of course, these bodies aren't quite co-orbital—and such a resonance wouldn't be possible for co-orbiting bodies—but it does allow them to have semimajor axes that vary by under 4% from each other, which is within the same range as horseshoe orbits, so I thought it might be worth considering.

 

One final note on co-orbital motion before we move on: many of the examples of co-orbital bodies in the solar system I've given are not actually in stable configurations; many of Earth's partners in trojan, horseshoe, or eccentric resonances orbits will leave their resonances⁸⁸ within a few thousand years, some within a few hundred. But often they move from one type of co-orbital resonance to another: objects can transition between tadpole orbits and horseshoe orbits, between horseshoe orbits and eccentric resonances, and the reverse in both cases. How well this might work between more similar-mass bodies, I'm not sure, but it seems possible in principle that an Earthlike planet could oscillate between different 1:1 resonances with a larger gas giant.

Rotation

We cannot yet observe the obliquity or rotation rate of exoplanets (save for some rare cases⁸⁹ of giant planets spinning fast enough to generate notable blueshift) but so far as we’re aware there are few hard constraints on either, other than the influence of tides. Based on our solar system, initial rotation times of 5-30 hours seem to be reasonable; elsewhere, rotation periods up to thousands of hours may be possible⁹⁰, but recent models of planet formation¹⁵⁴ tend to predict a bias in impacts during accretion that results in fast, prograde rotation (spinning the same way as they orbit). More massive planets will have accreted more mass and so tend to spin faster, as we see with our gas giants. But still, a few large impacts or close encounters in the final stages of formation can shift a planet's rotation¹⁵⁵ to basically any orientation or speed, potentially causing retrograde rotation—as has happened to Venus—or high obliquity—as has happened to Uranus.

 

But a planet can only rotate so fast before the rotational velocity at the surface exceeds the velocity required to orbit above the surface, at which point centrifugal acceleration will tend to tear the planet apart. For Earth with its current radius, this would occur at a rotational period of 1.43 hours.

 

But as a planet rotates faster, the centrifugal acceleration causes its equator to bulge outwards, and at very fast rotation it will start to stretch into 2 lobes of material reaching out from the equator. I'll leave the specific math for this for the next post (once I update it; you can read ahead here⁹¹) but in short, the actual minimum period for Earth's rotation is probably closer to 2 hours.

Concept of Haumea, a dwarf planet elongated by its rapid rotation (once per 3.9 hours). Stephanie Hoover, Wikimedia

 

For a given rotation rate, the day length can be defined two ways: The sidereal day, which is the actual period it takes for the planet to rotate a full circle; and the synodic day, the period it takes for the sun to return to the same point in the sky (e.g. noon to noon). Earth’s 24-hour “day” is a synodic day; the sidereal day is 4 minutes shorter. This is because the Earth moves slightly in its orbit over the course of a day, and so any single point on the surface needs to rotate slightly more than a full circle to face the sun again.

 

A prograde-rotating planet will have one more sidereal day a year than synodic days to offset the star’s apparent rotation around the planet, and a retrograde-rotating planet will have one less (assuming a prograde-orbiting planet):

d_syn = synodic day length (any unit, so long as all three use the same)

d_sid = sidereal day length

P = year length

This is an average over the year; planets with eccentricity will experience slightly longer synodic days near periapsis and shorter ones near apoapsis. However, eccentricity would have to be pretty extreme—or the days very long compared to the orbital period—for this to be significant. Figuring out the specifics requires an annoying iterative algorithm (in short, at any given point of the year you have to find a solution for [sidereal day] / [year] * (360° + [change in true anomaly]) = [change in mean anomaly], then multiply [change in mean anomaly] * [year] / 360° = [synodic day]) but for example, if Earth had 0.9 eccentricity, synodic days would be 24 hours, 35 minutes at periapsis and 23 hours, 58 minutes at apoapsis.

 

While we're at it, a moon will also have a sidereal month—corresponding to the actual orbital period—and a synodic month—corresponding to how often it appears to move past the star, and thus the cycle of phases—with the same mathematical relation between them. The time between overhead transits of the moon is a synodic period (it's a general term for conjunctions between different astronomical cycles), which I already mentioned for horseshoe orbits but I'll repeat here:

 

T_Syn = Synodic period (any unit so long as T₁ and T₂ are the same)

T₁ = Month (sidereal or synodic, so long as T₁ is the same)

T₂ = Planet's day

This means that (for the surface of a prograde-rotating planet) all retrograde and high-orbiting prograde moons with months longer than the planet's day will appear to move retrograde, east to west, while low-orbiting prograde moons will appear to move prograde, west to east. Mars's moons both orbit the same way but, from the surface, appear to move in opposite directions due to their different orbital periods.

Tides

A planet orbiting close to another large body (star, planet, moon) feels tidal forces from that body due to the difference in gravitational force across its diameter. The side closest to the other body is pulled more strongly towards it and so bulges out, and the far side is pulled more weakly and so is, in a sense, left behind as the planet accelerates towards the other body (we could also say that it feels a stronger centrifugal force from the planet's orbit around the other body), bulging out away from it.

Source

 

We can roughly approximate⁹² the height of the resulting tidal bulges like so:

h = tidal height (any unit so long as a and r are the same)

a = distance between centers of mass of the 2 bodies

r = radius of planet

m = mass of other body (any unit so long as M is the same)

M = mass of planet

If the planet doesn't rotate relative to the other body, these bulges will be static on the surface and equally deform the crust, oceans, and atmosphere together, so from the surface these bulges wouldn't look any different from any other part of the planet in terms of relative ocean height. But if the planet does rotate, the bulges circle around the planet, creating a tidal cycle with a period half as long as the synodic period between the planet's rotation and other body's motion. The solid crust only deforms slightly as the bulges pass over (unless the tidal forces are very strong) but the fluid oceans and atmosphere rise to the full height of the bulge; hence, the rise and fall of the oceans we see due to the tidal influence of the Moon.

Each nearby body imposes its own tidal cycle on the planet (and the planet imposes tides on those bodies in turn): Earth experiences a 12.42-hour tidal cycle from the Moon and a 12-hour cycle from the Sun. The Moon's is about twice as high and so more apparent, but the total tidal range—total shift in water level between high and low tide—is significantly greater when the tidal bulges are aligned (Sun and Moon either on the same side or opposite sides of the Earth). A planet with multiple moons would have overlapping tidal cycles from all of them, and closely-packed planets or moons should be able to induce tides on each other, though the dynamics there will be more complex due to the variations in distance and I'm less confident in the accuracy of the above estimation for those cases.

Lookang, Wikimedia

Actual tidal ranges will vary across the surface due to a variety of factors: The average tidal range on Earth's coastlines is around 60 cm, twice the above estimation, and it varies locally from almost nothing to 16 meters. 

 

But even were there no landmasses in a way, the tidal motion of water wouldn't quite match an ideal tidal bulge: the planet's rotation pulls the tidal bulge slightly out of alignment with the body inducing it. The tidal forces continue to drag it back towards alignment, in effect fighting against the planet's rotation. The other body is, in turn, pulled towards the tidal bulge.

Misalignment of tidal bulges (due here to counterclockwise rotation relative to other body), and resulting (clockwise) torque. GrNephrite, Wikimedia

 

A high-orbiting prograde moon will "spin down" a planet in this way, lengthening the planet's day while pulling itself into a higher orbit, gradually drifting away from the planet; A close-orbiting prograde moon with a month shorter than the planet’s day will “spin up” the planet instead, shortening its day and pulling itself into a closer orbit; and a retrograde moon will spin down the planet and pull itself into a closer orbit. Since the Moon formed, Earth has been spun down from 4-hour days to 24-hour days and the moon has spiraled out from 6-hour months to 27-day months, although the rate of both changes down has slowed. Resonances between the orbital period of the moon and rotation period of the planet or between the orbital periods of multiple moons may pause this process, but usually not permanently. Given enough time, this could eventually lead to the moon spiraling down inside the Roche limit or escaping out of the planet's orbit.

 

Tidal migration of a moon that is spinning down (left) or spinning up (right) its planet. Cmglee, Wikimedia

 

The moon's rotation is, of course, also affected by tides from the planet, spinning down as well (or spinning up if it initially rotates slower than it orbits), and much the same can happen for a planet in close orbit of a star. A similar process will also tend to reduce a moon or planet's orbital eccentricity, obliquity, and inclination relative to the parent body's equator (any of which cause oscillations in the relative orientation of the two bodies even without rotation).

 

All this movement of material and tugging on tidal bulges can also cause a good deal of internal friction within a planet, heating its interior. We can see the potential consequences of this on Jupiter's moon Io; though it has completely spun down to synchronous rotation, the other moons induce some slight eccentricity in its orbit and this causes enough internal friction to give it over 20 times the rate of geothermal heating at the surface as Earth and cause widespread volcanism.

Tidal Locking

Should spin-down continue for long enough, it will eventually result in tidal-locking, A.K.A. synchronous rotation: a resonance between the planet's rotation period and orbital period (if spun down by its parent body; if spun down by its moon, the orbital period of that moon). The most familiar and stable case of this is the 1:1 spin-orbit resonance, where rotation and orbital periods are the same and one side of the planet constantly faces the other body. The moon is tidal-locked to the Earth in this way, hence why we only ever see one side of it, and as we've discussed, many exoplanets are expected to be tidal-locked to their stars, such that one hemisphere receives constant daylight and the other constant night. Pluto and Charon are mutually tidal-locked, both with the same rotation and orbital periods, which prevents them from migrating either towards or away from each other.

But this isn't the only possible outcome of tidal-locking (though whenever I refer to tidal locking without further explanation you can assume I mean a 1:1 resonance). If the planet still has some orbital eccentricity when it tidal-locks, it could settle into a 1:2 state, where it rotates only once every 2 orbits; a 3:2 state, where it rotates 3 times in 2 orbits (which Mercury is currently locked in); a 2:1 state, where it rotates twice an orbit; or 5:2, 3:1, 7:2, 4:1, and so on. Every integer (n:1, where n is an integer) and half-integer (n:2, where n is an odd integer) is at least marginally plausible⁹³ over some range of eccentricity, with higher eccentricities more likely to result in higher-order resonances (save for a 0:1 resonance—no rotation at all—which doesn't appear to be stable, and I know a -1:1 resonance—one retrograde rotation per orbit—is possible⁹⁴ but I'm not sure about other retrograde resonances).

 

Probability for a planet (or moon) to lock into different resonances (marked here as spin/orbit ratios) depending on its eccentricity and whether it initially spins faster (left) or slower (right) than the resonances. All integer and half-integer resonances have their own probability curve, following the trends you can see here. Dobrovolskis 2007.

 

The above formula relating a sidereal day to a synodic day still applies so, for example, a planet tidal-locked to its star in a 2:1 resonance experiences one day an orbit, one in 3:2 resonance experiences half a day per orbit—each day takes two orbits—and one in 1:2 resonance experiences half a retrograde day per orbit—the star moves west to east in the sky. You can imagine the sort of interesting climate patterns that could result from this, but we'll leave that discussion for another time.

But as mentioned, tidal forces will tend to reduce eccentricity as well as relative rotation, making lower-order resonances more likely. Even if a planet does adopt a higher-order resonance initially, given strong enough tidal forces⁹⁵ the tidal friction cause by its eccentricity and rotation may heat it enough to melt a significant portion of its interior, altering its tidal dynamics enough to break it from resonance and continue spin down to 1:1 resonance. As such, this may be the only stable spin-orbit resonance for the closest-orbiting planets and moons (including habitable-zone planets of late M-type stars).

 

Setting aside these special cases for the moment, we can estimate⁹⁶ the time it takes a planet in a circular orbit with tides predominantly imposed by one other body to lock into a 1:1 resonance with that body:

t = time to tidal-lock (billion years)

Q = dissipation factor (~100 for typical solid planet of Earthlike size)

m = mass of planet (Earth masses)

a = semimajor axis (AU)

P = initial rotation period (hours)

M = mass of other body (sun masses)

r =  radius of planet (Earth radii)

 

Semimajor axis is the dominating factor here, hence why almost all moons in the solar system are tidal-locked to their planets, and we expect most planets of small red dwarf stars to be tidal-locked as well. If we're willing to assume arbitrarily slow initial rotation then there's no strict outer limit on where a planet can be tidal-locked to its star but even with an initial rotation period of 1,000 hours, an Earthlike planet more than about 1.1 AU from a sunlike star would still not be tidal-locked after 4.5 billion years. On the other hand, rotation can only get so fast: even with an initial period as low as 2 hours, a similar planet would tidal-lock in the same time within 0.4 AU of a sunlike star or in the habitable zone of any star below about 0.3 solar masses (the dissipation factor here also varies—for modern Earth it's actually around 12, but 100 better replicates the evolution of Earth's rotation and the Moon's orbit—but that only makes a small difference compared to the semimajor axis).

 

But there may be a few ways to avoid tidal-locking even in such states. A large moon⁹⁷ can have a stronger tidal influence on the planet than its star, as with Earth, though close-orbiting planets are less likely to have large moons for reasons we'll discuss shortly. A resonance with another planet or moon can also interfere with tidal-locking, though rather more dramatically: Saturn's moon Hyperion, due to its 3:4 resonance with Titan (as well as orbital eccentricity and oblong shape), rotates chaotically, with no fixed period or axis of rotation. Pluto-Charon's minor moons are also expected⁹⁸ to rotate chaotically due to tidal effects of the rotating binary.

Close-orbiting planets in orbital resonance may have similar effects on each other⁹⁹, causing them to rotate slightly slower or faster than 1:1 resonance, such that they effectively experience a synodic day several years long (Earth years; perhaps hundreds of their own orbits) or adopt a "quasi-stable" state where they remain in 1:1 resonance (though with significant oscillation of their alignment over periods of several years) for thousands of years and then abruptly flip around to face the opposite side to the star within tens of years—and they can also switch between these two behaviors.

 

Planets that have a softer interior than Earth (icy bodies) or large fluid layers (waterworlds, ice-covered bodies with subsurface oceans) and some eccentricity may also be able¹⁰⁰ to settle into pseudosynchronous rotation slightly raster than 1:1 resonance. A planet tidal-locked to its star with a substantial atmosphere may experience a thermal tide (expansion of the atmosphere due to solar heating) out of sync with its gravitational tide, also allowing for rotation slightly faster or slower than 1:1 resonance.

Approximate limits for where atmosphereic effects could allow asynchronous rotation for different star masses and atmospheric pressures; note that a 10-bar atmosphere is more likely to do so than either a thinner Earthlike (1 bar) or thicker Venus-like (93 bar) atmosphere. Leconte et al. 2015

Even if a planet does become locked into a 1:1 spin-orbit resonance, it does not necessarily remain perfectly aligned with its star (or moon). If the planet has any obliquity or eccentricity, this will cause it to librate over the course of its orbit, turning slightly back and forth.

Libration of the Moon with respect to Earth. Tomruen, Wikimedia

Tidal forces will generally work to reduce both obliquity and eccentricity, but they can be maintained with the help of an additional body. An additional outer planet or star (or moon) can help a planet achieve⁹⁴ a Cassini state where different aspects of its orbital and rotational motion oscillate together to give it a constant obliquity of up to 100° (or close to 180°, retrograde rotation), though strong enough tidal forces can overcome this influence¹⁰¹. Orbital resonance with another body can also increase eccentricity, though if the tidal forces are particularly strong this can lead to extreme heating of the planet's interior. So, much as with non-1:1 resonances, we are unlikely to see significant librations for the closest-orbiting planets and moons.

A couple final notes of interest: First, because the Hill radius and the strength of tidal forces are determined by the same parameters, they are related such that any body necessarily feels stronger tidal forces from its orbital parent than from its parent's parent; so a moon may be tidal-locked to the planet it orbits, but not to the star the planet orbits. Perhaps brief resonances may set in between the moon's rotation and planet's orbit, but the planet's tidal influence should pull the moon out of these in short order.

Second, planets locked into 1:1 resonance have, ironically enough, effectively no tides from the body they are locked to, in the sense of motion of the ocean surface. Water will be pulled up into a tidal bulge, but that bulge won't move across the surface, and before long the crust will also deform to fill that bulge, such that the depth of the ocean above the top of the crust doesn't vary across the surface (ignoring local topography, of course). Libration could still cause some tides, which would vary in height in complex patterns over the surface.

Special cases aside, this means that tidal height is, in a way, self-limiting. High tides requires strong tidal forces, which implies rapid tidal-locking. Based on the above equations, the ideal case for high tides but slow tidal-locking is a large planet with tides induced by a massive and distant partner; so, a star, but if we want this planet to be habitable, that restricts our options of star mass and distance. The highest tides achievable for an Earth-sized, habitable-zone planet that takes at least 4.5 billion years to tidal-lock (given initial 2-hour days) seems to be about 3 meters for a planet orbiting 0.35 AU from a star a touch under 0.7 solar masses. A planet 10 times Earth's mass (with the same composition) under the same restrictions could get 4-meter tides orbiting a slightly smaller star (0.31 AU from an 0.67-solar-mass star), and if we require only 1 billion years to tidal-locking, we could get 6-meter tides for the Earth-mass planet (0.26 AU from an 0.61-solar-mass star) and almost 8-meter tides for the 10-Earth-mass planet (0.23 AU from an 0.57-solar-mass star).

(Sidenote: in all these cases, rotation is fairly rapid for most of the time to tidal-locking, and then the final spin-down from 24-hours days to locking occurs within a few hundred million years. A planet that retains rapid rotation for long enough for complex life to develop and then gradually spins down as it evolves could be an interesting scenario).

Higher tides are probably possible, at least locally, with competing tidal forces from multiple bodies, libration, or some other special case, but the limits of those scenarios (and their implications for geothermal heating) are harder to determine.

Moons

To a great extent, moons are just planets in a different context and planets with orbiting moons are planetary systems in miniature. Most of what I say about "planets" here and in future posts applies equally well to moons, including the previous sections on limiting factors for orbits, possible co-orbital configurations, rotation, and tidal-locking (some researchers have proposed using the term "planemos" for both planets and large moons, which is a fine idea but I just don't like how the word sounds).

But, much as the process of planetary formation tends to form a few characteristic system architectures, the ways in which planets acquire moons creates some patterns in how they tend to appear, so let's discuss those mechanisms first. As mentioned, much of this will also apply to the formation of binary planets, which in many ways are essentially just a special case of planet-moon systems.

In Situ Formation

In this case, the moons form with the planet, orbiting it from the outset. Curiously enough, this tends to be how the smallest and largest bodies acquire their moons, but not those in between.

First off, as clumps of dust and debris collapse within the protoplanetary disk around a young star, many should form not one object but a binary pair of planetesimals, and we can still see many such binaries in the Kuiper belt today. According to modelling¹⁰², roughly 1/10 of these binaries should also form with additional outer moons orbiting their barycenter. But these binary planetesimals are each typically only 10s to 100s of km across. A few might occasionally reach over 1000 km, but formation of an Earthlike body over 10,000 km across requires mergers between many such planetesimals, during which any binary partners will likely either also impact the forming planet or be ejected out of orbit, leaving the planet initially moonless.

 

Formation of binary planetesimal and subsequent contact binary. NASA / JHUAPL / SwRI / James Tuttle Keane

But if a planet becomes large enough—as in, becomes a gas giant—it will begin to gather gas and dust around it and form its own circumplanetary disk, like a scaled-down version of the protoplanetary disk. The disk even has its own temperature gradient and iceline due to varying rates of collisions and heat radiating from the still-forming planet.

Planets of similar scale to Jupiter will clear out a gap between their surface and the innermost edge of the disk, much as stars do. Planetesimals (moonesimals?) will then form in the disk and tend to migrate inwards¹⁰³ until the innermost moon reaches the inner edge of the disk, where it will stop, and other moons will capture into a chain of 2:1 resonances outside it. But if more than 3 or 4 large moons lines up in this resonance chain, the orbit of the innermost moon may be destabilized enough to send it crashing into the planet. The next innermost moon will then migrate to the inner edge and the resonant chain of moons will follow, and this cycle can repeat several times. Thus, the dry inner moons are lost and a small number of water-rich moons in resonant orbits remain—plus often one non-resonant large outer moon formed at the very end of the process, like our Callisto.

Moon formation and migration in a typical model for a Jupiter-sized planet. "Rp" is radii of the planet, "Tk" is ~0.03 years; the orbital period at 20 Rp. Ogihara and Ida 2012

The resulting moons will have similar masses, depending on the mass of the planet and distance from the star: In a Jupiter-like orbit (5 AU from a sunlike star, or orbits with equivalent light levels for other stars) they should usually be around 1/10,000 that of the planet; 0.03 Earth masses for a Jupiter-mass planet. The mass ratio should roughly scale¹⁰⁴ with distance from the star (planets orbiting twice as far should have moons roughly twice as massive), except that moons like this are unlikely to form at all inside the water iceline—though a planet could form with moons outside the iceline and then migrate inwards. Larger moons up to Earth's mass are possible but likely rare except perhaps on very wide orbits¹⁰⁵. The moons will usually have very high water mass contents—up to half or more of the total mass—though the inner moons will tend to be somewhat drier and could lose most of their water due to intense tidal heating, as has happened to Io.

A planet closer to Saturn in mass likely forms slower and doesn’t clear a large gap in the protoplanetary disk, so the inner edge of the circumplanetary disk reaches the planet’s surface¹⁰⁶ and allows the inner moons to migrate into collision with the planet, leaving just one or two large icy moons in wide orbits, like Titan; though Saturn still has many smaller moons that didn't grow large enough to migrate down to impact.

Now, given how often exoplanets have surprised us before, we probably shouldn't be too confident in claiming that the architectures of moon systems elsewhere will always follow the same patterns we see in our own system, so let’s simply say that we expect the patterns seen for Jupiter and Saturn to be common, but not necessarily ubiquitous. A plausible alternative¹⁰⁷ is that moons could form just outside the Roche limit from a circumplanetary disk that is spreading outwards, and then migrate out to their current positions. More recent modelling¹⁰⁵ also suggests that very massive (~10 Jupiter masses) gas giants on very distant (~50 AU) orbits are less prone to forming moons in Laplace resonances, and that especially massive circumplanetary disks may occasionally¹⁰⁸ form larger moons.

Capture

Even if a planet forms without moons, the young system is likely to have many bodies on unstable orbits and encounters between them may be common, providing opportunities for a planet to capture another body into its orbit, including anything from a small asteroid to a binary partner of similar mass to itself.

 

Now, if a planet or smaller body simply passes through the hill sphere of a larger planet, it will not be captured. Some event has to remove momentum from the object during that flyby—or over the course of several sequential flybys—such that it moves from a fast flyby trajectory to a slower closed orbit. Fortunately, there are a few different ways this could happen:

• For one, while the protoplanetary disk is still present it may provide some drag that helps slow planets during encounters. Young planets may commonly capture into binary pairs this way, though usually in very wide orbits such that they drift away from each other and separate again after just a few thousand years. But on rare occasions¹⁰⁹ a fortuitous encounter with a third body may push the planets closer together into a more stable orbit.
• Drag may also allow a planet to capture much smaller bodies that pass through their circumplanetary disk or the outermost layers of their atmosphere.
• Pairs of gas giants (and perhaps other bodies with large fluid layers like waterworlds) may capture into mutual orbit⁶⁷ through strong tidal forces during encounters, likely capturing into very close orbits initially.
• If the planet already has moons, they may gravitationally interact with the body and absorb some of its momentum, possibly but not necessarily leading to their own escape. The body may also directly impact one of the moons, possibly breaking apart initially but eventually reforming into a larger moon.
• If a binary pair of bodies encounters a larger planet, the influence of the planet’s gravity could pull the pair apart¹¹⁰, causing one to be ejected with most of the pair’s original momentum and the other to remain behind in orbit of the planet.
• Were a body to break apart on passing inside a planet's Roche limit, some of the fragments may end up moving slow enough to be captured. This is likely to be rare as bodies can usually survive briefly passing inside the Roche limit, but notably Shoemaker-Levy 9 did break up when passing Jupiter (not that any of those fragments were captured).

Most of these scenarios result in a moon (or binary partner) on a highly eccentric orbit, but tidal interactions with the planet will act to circularize the orbit over time, such that the moon has a good chance¹¹¹ of settling into a stable, circular orbit within a few million years, usually with low inclination and a roughly even chance of ending up in a prograde or retrograde orbit. And indeed, our system appears to include many captured moons, judging by their compositions and irregular orbits. Triton, the moon of Neptune, is the largest such moon, at 0.0036 Earth masses, but there’s no particular reason a moon of similar mass to Earth or larger couldn’t be captured. The recently described¹¹² exomoon Kepler-1625b I appears to be roughly Neptune-mass (17 Earth masses) and if confirmed may have been captured¹¹³ into its current orbit.

Impact

Of course, the most obvious way for two planets to slow down relative to each other is to simply smash into each other, and indeed such impacts are likely fairly common during the formation of solid planets.

NASA/JPL-Caltech

Most of these impacts are likely to result in complete or partial breakup of the planets. Most of the resulting debris will gather back together into a single larger planet but some may be thrown out into a ring that can then coalesce into one or multiple moons just outside the Roche limit. Recent work¹¹⁴ also suggests that the debris may form a red-blood-cell-shaped cloud called a synestia within which a moon forms before it collapses back into a round planet within a few thousand years.

Cross-sectional views through a post-impact synestia during moon formation. Locke et al. 2018

 

An impact between the early Earth and a roughly Mars-sized planet called Theia is widely believed to be responsible for our moon’s formation, and an impact between Mars and a dwarf planet may have formed¹¹⁵ its bitesize moons

Phobos and Deimos. Models suggest that¹¹⁶ something like 1 in 12 terrestrial planets may form substantial moons in this way, and that the resulting moons are likely no more than¹¹⁷ 6% the mass of the planet (about 5 times the mass of our moon for an Earth-mass planet), though subsequent collisions may form additional moons that then merge together¹¹⁸. Even gaseous planets may form moons this way, but the higher-energy impacts of larger planets will cause more of the ejected debris to vaporize and disperse, such that formation of a large moon in this way is likely only possible¹¹⁹ for solid planets with less than 1.6 times the radius of Earth (around 6 times the mass for a rocky planet).

 

However, a glancing impact may actually allow both bodies to remain largely intact, allowing for capture of a much larger moon. Pluto and Charon (with a mass ratio of 12%) may have had such a collision¹²⁰, perhaps throwing off enough debris for form their smaller moons in the process, and models suggest¹²¹ that similar-mass binary planets could form this way too, even between¹²² gaseous superearths.

 

Simulated Pluto-Charon impact over the course of ~23 hours. Barr 2017

Pull-Down Capture

This is somewhat of a variant capture model that proposes that, as a forming gas giant reaches the stage of runaway growth, its Hill radius rapidly expands, allowing it to capture other planets orbiting nearby as moons. The idea has fallen out of favor¹²³ as an explanation for any of the solar system's moons due to the extreme rate at which a planet would have to grow to capture a passing body, but it may be viable¹²⁴ if the two bodies are already co-orbital, and in particular may allow for capture of large moons into more distant orbits than is typical for other capture mechanisms like tidal interactions or impacts.

Fission

One last alternative for smaller bodies: A non-spherical asteroid may reflect sunlight unevenly, causing its rotation rate to increase until it’s torn apart by centrifugal forces, forming one or more moons. Such a scenario is implausible for a spherical planet with much more mass relative to its surface area (and so less acceleration from sunlight), though it might be marginally plausible for an encounter with another planetary body to greatly increase the planet's rotation rate to the point of tearing itself apart.

 

Another curious proposal¹²⁵ is that a planet may form with rapid rotation and then the centrifugal acceleration could help concentrate enough fissile material at the core-mantle boundary to trigger a nuclear explosion that throws off part of the surface. The idea has not caught on as an explanation for the formation of Earth's moon (largely due to issues with the "concentrating fissiles" step) but whether or not it might be possible for an exoplanet hasn't been explored.

Limits and Stability

That all settled, let's go over some of the specific limits on moon orbits. Much of the constraints are the same as for planets: the Roche limit is calculated the same way and much the same rules apply for separation of orbits from each other by mutual Hill radii, with exceptions for resonances and co-orbital bodies, and there's a similar tendency towards low mutual inclination. But overall irregularly orbiting moons are likely more common than irregularly orbiting planets, because capture of a moon from elsewhere in the solar system is more frequent than capture of a planet from elsewhere in the galaxy.

Though any object within a planet’s Hill radius will tend to orbit it for the moment, pertubations from the star or other planets will cause the ejection of objects in the outer region into orbits of the star. The maximum stable semimajor axis¹²⁶ for a moon can be approximated like so for a prograde-orbiting moon:

 

a_max = Maximum stable semimajor axis (Hill radii)

e_p = Planet's eccentricity

e_m = Moon's eccentricity

And for a retrograde-orbiting moon:

So if both moon and planet have no eccentricity, a prograde orbit is stable up to around 0.49 Hill radii, while a retrograde orbit is stable out to 0.93 Hill radii (though stability becomes increasingly unlikely over 0.4 Hill radii for prograde orbits¹²⁷ and 0.67 Hill radii for retrograde orbits¹²⁸).

 

Likelihood of stability (white: no stable orbits, black: all orbits stable, not accounting for influence of other moons, planets, or stars) depending on semimajor axis relative to parent's Hill radius and eccentricity for prograde-orbiting moons (left, Rosario-Franco et al. 2020), retrograde-orbiting moons (middle, Quarles et al. 2021), and prograde-orbiting moonmoons (right). Dashed lines are the limits given by the above formulas, red lines are similar formula that can be found in the linked sources.

 

Thus in our solar system the moons of gas giants can be largely divided into prograde, close-orbiting, low-inclination regular moons, likely formed in situ or by impact, and mostly retrograde, far-orbiting, high-inclination irregular moons, likely formed by capture.

The orbits of some of Jupiter's irregular moons (the red line is the planet's orbit, and the perspective shifts to show their inclination). Kieff, Wikimedia

Hill radii are, of course, smaller for planets in closer orbits to their stars. The above stability limits are still larger than the Roche limit for an Earth-moon pair to as little as 0.025 AU from a sunlike star, but a smaller range of stable orbits likely still reduces the chances of a moon forming or capturing.

 

And even if a moon does form, no moon is totally stable in its orbit: all are gradually migrating due to tidal interactions with their planet and will eventually spiral out of the Hill radius and enter their own orbit of the star (some researchers¹²⁹ have proposed we call these escaped moons-cum-planets ploonets) or spiral in below their Roche limit and break apart or, if dense enough to avoid breakup, merge with the planet. Even if planet and moon become mutually tidal-locked, tides from the star will sap energy from the system¹³⁰ and cause them to spiral in towards each other. More distantly-orbiting planets with greater Hill radii and weaker solar tides will hold onto their moons for longer, and at a given distance the longest-lived moons are those that are small relative to their parent (such that the impose small tides), orbit initially fast-spinning planets (which have more energy that the star must remove to cause a collision), or are large and orbit slow-spinning planets (causing faster mutal tidal-locking). Denser planets also retain their moons¹³¹ for longer.

(Some formulas for estimating the lifetime of a moon can be found here¹³², but they're rather daunting; I may attempt to incorporate them into my spreadsheet in the future.)

Maximum time before collision or escape for a moon of Earth, depending on the mass ratio and the planet's initial rotation. Sasaki and Barnes 2014

Close encounters between planets can also strip away moons¹³³, ejecting them onto their own orbits of the star, though at least some moons may be retained even in dramatic scenarios such as¹³⁴ a planet being totally ejected from the star system. If a planet migrates in towards its star, its Hill radius will shrink and it may lose some outer moons. There is also some risk¹³⁵ that a close-orbiting moon may be pushed into collision with the planet due to resonances that arise during inward migration, though this can be avoided with rapid migration, tidal interactions, or the influence of other moons. If a planet migrates in to a very close orbit, the intense heating and rapid atmospheric escape may cause it to lose enough mass¹³⁶ that its moons spiral out of orbit and escape.

 

All of this is problematic for planets orbiting smaller stars. Large planets in the habitable zone of an M0 star are roughly half as likely¹¹¹ to pull a captured moon into a stable orbit compared to those in an analogous orbit of a sunlike star. Tighter systems are also more likely¹³⁷ to have close encounters between planets or between planets and the star. And of course, smaller Hill radii and a stronger tidal forces from the star will cause faster losses of moons due to tidal migration, which may limit the lifetime of moons¹³⁰ to less than the Earth's current age for planets in the habitable zone of stars less than about half the mass of the sun.

Limit at which any moon would be lost due to tidal migration within 5 billion years for planets of given mass and composition in Earthlike orbits of different stars. Given some of this model's assumptions regarding habitable orbits and stable moon orbits, these are likely conservative estimates (i.e., a more optimistic analysis may allow moons to be retained for smaller stars). Sasaki and Barnes 2014

This doesn't necessarily imply that close-orbiting planets in old systems can never have moons; the planet may have recently migrated into its orbit, the moon may have formed or been captured long after the planet formed, or, as we'll discuss later, a moon may break up inside the Roche limit and some of its mass may later reassemble into a smaller moon. Tidal interactions between multiple moons could also have more complex dynamics that might keep at least one of them in orbit for longer.

 

On a brighter note, the presence of one large moon doesn't seem to preclude the presence of other moons, though with some caveats. The Pluto-Charon binary has an additional 4 outer moons and modelling suggests¹³⁸ Kepler-1625b would have no trouble hosting an Earth-mass moon in a closer orbit than its existing Neptune-sized moon. A larger number of moons does require¹³⁹ a lower total mass relative to the planet for stability (though we can probably exclude small asteroid-like bodies from this count of moons), so more than two moons over 1% the mass of the planet are unlikely to be stable.

Rough probability of stability for different numbers of moons (generally with the largest no more than 10 times the mass of the smallest) in orbit of a planet of Jupiter's mass (left), random mass (middle), and random mass with the moons placed initially in orbital resonance (right), depending on their combined mass relative to the planet (on a log scale, e.g. "-3" would be 1/1,000). Teachey 2021

Other planets on neighboring orbits are also unlikely to cause major problems: Even a very close neighboring planet is unlikely¹⁴⁰ to destabilize moons orbiting prograde within 0.4 Hill radii of the planet or retrograde within 0.6 Hill radii, though a large moon can destabilize a very close resonance (e.g. 11:12) between planets. A binary star partner has a similar moderate effect¹²⁸ on the stability of moons around S-type planets.

 

Incidently, because a planet's Hill radius relies on many of the same values as its orbital period,  the two are fundamentally linked such that, assuming the planet's mass is negligible compared to the star's (such that it doesn't have to be accounted for in the planet's orbital period), the orbital period for a moon orbiting at the planet's Hill radius should always be 0.577 times the planet's orbital period. Given the stability limits we've established the relationship between semimajor axes and periods, the maximum orbital period for a stable moon is therefore 0.198 times the planet's orbital period for a prograde-orbiting moon and 0.519 times for a retrograde-orbiting moon.

 

Now, if planets can orbit stars and moons can orbit planets, can moons have their own satellites? In some cases, possibly so¹⁴¹. There’s no agreed term for such a body though moonmoon seems to be the most popular right now, because I guess none of us are feeling more creative. In principle Callisto, Iapetus, and our own moon could support small moonmoons—some have proposed¹⁴² that Iapetus may once have had rings that later fell to the surface, forming its equatorial ridge. But these moonmoons may not be stable over billions of years due to the tidal influences of their planets and the sun. A larger moon like the aforementioned Kepler-1625b I could do a better job of holding onto moonmoons. As a rule of thumb, a moonmoon should be no more than 1/10⁵ times the mass of the moon and orbit within¹²⁷ 1/3 of the moon's Hill radius to remain in a stable orbit.

But why stop there? Could a moonmooon have its own satellite, a moonmoonmoon? In principle, yes, but we may be pushing our luck. The planet Kepler 1625b is around 3 Jupiter masses, so if we assume a planet 4 times as massive (close to the maximum size before a planet becomes a brown dwarf) could support a moon 4 times as massive, that gives us a moon of 68 Earth masses. If we stick to the 1/10⁵ rule thereafter, that implies there could be a moonmoon of 0.00068 Earth masses, similar to the dwarf planet Haumea, and a moonmoonmoon of 0.0000000068 Earth masses (4.1*10¹⁶ kg) similar to the asteroid Ida—which has a satellite, Dactyl, implying the possibility of a moonmoonmoonmoon, but by this point I expect the complex resonances and tidal interactions that would appear in this crowded hierarchy could be problematic.

Rings

Saturn’s moon Daphnis raising waves in its rings. NASA/JPL-Caltech/Space Science Institute

In addition to large moons, all our gas giants also have rings of debris, though for Jupiter and Neptune these are just thin, diffuse rings of dust. Saturn’s famous main rings are a near-continuous disk composed primarily of gravel (1 cm) to boulder (10 m) size chunks of ice. Despite covering an orbital space over 70,000 kilometers wide, the rings may be as little as 10 meters thick, though they contain a number of variations in structure at different scales:

• At the smallest scales, the slight gravitational attraction between ring particles gathers them together into elongated clumps a few meters across, which soon fall apart due to Saturn's tidal influence or impacts between clumps.
  
  

  

  Concept of likely small-scale appearance of Saturn's rings. NASA/JPL/University of Colorado

  
  
• Orbital resonances with moons orbiting outside the rings create spirals of increased density, though so tightly wound that they can easily be mistaken for distinct rings.
  
  

  

  Density waves formed at the 1:2 resonance with Janus. NASA/JPL-Caltech/Space Science Institute

  
  
• Moonlets around 100 m across disturb the surrounding material, creating wakes that are distorted by the variation in rings' orbital motion into structures called propellers.
  
  

  

  Images of propellers in Saturn's rings (top) and simulated structure (bottom). Tiscareno 2020

  
  
• More substantial shepherd moons 1 to 100s of km across clear out complete gaps in the rings, and still create wakes on the neighboring ring material that trail ahead of them on the lower-orbiting edge of the gap and behind on the higher-orbiting edge.
  
  

  

  Tiscareno 2020

Circumplanetary disks may be common in young systems, but any material outside the Roche limit should eventually coalesce into moons, and any material within it should collapse onto the planet's surface in the process of planet formation. Short-lived rings outside the Roche limit may form later due to collisions, but again will soon form moons.

Concept of J1407b; the system is only 16 million years old, so the rings are likely leftover debris still coalescing into moons. Ron Miller

 

Thus, long-lasting rings can only form within the Roche limit and must form after the planet. If the Nice Model is correct, they may have formed¹⁴³ due to close flybys of planetesimals from the Kuiper belt that were torn apart by tidal forces; some of the fragments were captured into orbit and experienced further collisions that ground them into the fine material of the disk today. Plausible alternatives

¹⁴⁴ include an impact event between 2 large moons or the stripping of material from a large moon that passed inside its Roche limit, leaving a rocky core that then plunged into the planet.

 

How long rings like Saturn's could last is a subject of some debate: As the rings age, they should spread inwards and outwards and eventually disperse, and also darken due as rocky dust is caught in the ring. Shepherd moons can slow the spreading, but many models suggest that Saturn's rings may be only¹⁴⁵ a few hundred million years old, though it remains a matter of debate¹⁴⁶, with some possibility that the rings are more massive than previously thought or material in the more diffuse outer rings may be somehow recycling back into the main rings.

 

Diffuse, dusty rings like those of Jupiter or the outer F ring of Saturn are less mysterious: These are continuously produced by dust kicked up from their small inner moons by solar winds or impacts or ejected from icy moons by cryovolcanism¹⁴⁷. Were these processes to stop, the rings would disperse within thousands of years.

Rings need not be icy or form only around gas giants. Even the asteroid Chariklo has a set of rings. Moons could also conceivably have rings¹⁴⁸, including, as mentioned, Iapetus in the past. When Phobos passes inside its Roche limit in the next 20-40 million years, it will give Mars a large set of rings¹⁴⁹, though one unlikely to last more than 100 million years thereafter. But some models¹⁵⁰ suggest that Mars may be experiencing a regular cycle wherein a moon migrates inside its Roche limit and breaks apart into rings, the rings spread out, around 4/5 of the ring material eventually falls to the surface, the remaining 1/5 reassembles into a moon just outside the Roche limit, and this moon migrates back in; Phobos may be the remnant of 3 to 7 of these cycles.

 

All planetary rings we've observed have very low orbital inclination, orbiting directly or almost directly over the equator of their planet. Largely this is a result of how these rings form—usually from low-inclination moons—but even if a ring did initially form at high inclination, it wouldn't stay that way for long; As mentioned, any rotating planet will be oblate, slightly bulged at its equator, and so the ring particles (which necessarily orbit pretty close to the planet given that they must be below the Roche limit) will be slightly tugged towards the planet's equatorial plane as they pass above and below it. This only causes a very gradual reduction in their inclination directly, but it also causes rapid precession¹⁵³; rotation of their orbital axis around the center of the planet. Different ring particles on different orbits will precess at different rates and so misalign with each other, rapidly spreading out from a single ring to a more diffuse cloud. Particles within this cloud will begin colliding with each other and, much as in the formation of the protoplanetary disk, will eventually converge back down to a single disk. Because their precession is centered around the planet's equator—and because precession will continue to scatter apart the rings so long as they're inclined—the resulting disk will inevitably end up at low inclination. Passing bodies could perhaps temporarily tilt outer rings, and a high-inclination moon just outside the Roche limit could perhaps help maintain a high-inclination ring—but such a ring would likely be pretty diffuse, and tidal interactions with the planet will reduce the moon's inclination over time.

Engineering the Architecture

There are a lot more factors we have to take into consideration before deciding what kind of planets we want to populate our example system. But because I’m a prophet and can see into the future, I already know what planets we’re going to use and I want to run through the process of placing them into a stable system architecture before we forget the important factors involved.

To help us out in this and future endeavors, I’ve put together a spreadsheet to handle most of the math of building planets and systems. A lot of it relates to elements we’ll discuss in the later posts, so don’t worry if it’s a bit bewildering for now—the “System builder” and “Moon builder” tabs contain most of the calculations related to orbits.

Alternatively, if you’re in a bit of a hurry or lacking inspiration a quick shortcut is to load up Space Engine, a free (for versions 0.980 and before), procedurally generated universe simulator that attempts scientific accuracy as much as possible, and explore around until you find a system you like. Even if you intend to build a system yourself it’s a good way to find some ideas and get a sense of the scale of different system architectures.

One last note before we begin: By convention exoplanets are named after their star with an appended letter in order of their discovery and then in order of the distance from the star if discovered simultaneously, starting with “b” (the star itself is supposed to be “a”, e.g. Teacup Aa, but I don’t hear that used much). We haven’t quite decided what to do with circumbinary planets yet but the most popular proposal is to use the letters of all the stars it’s orbiting in parentheses, e.g. Teacup (AB)b. Moons get their planet’s name plus a roman numeral, by the same order as planets, starting with “I”. I’m not too sure what we’d do with a binary planet; we could call them both planets, call them both moons and use the planet label for the barycenter, or just decide the more massive one is the planet and the other one the moon. In the case of the Teacup system I’ll take everything as having been “discovered” simultaneously and name it all in order of orbital distance, except for minor moons of the outer planets which I just won’t bother with right now.

Alright, a good first step is to place the iceline—both during formation and in the current system. Both should be at radii of roughly equivalent sunlight to their positions in our solar system, but the different albedos of ice¹⁵¹ under different stellar spectra will alter their position. Assuming the position of the iceline is tied to the surface temperature of small bodies, this effect can be roughly accounted for:

B_n = boundary distance in new system (any unit so long as B_s is the same)

B_s = boundary distance in solar system (~2.7 AU for early iceline, ~5 AU for current iceline)

L = star luminosity (relative to sun)

T_eff = star effective temperature (K)

But the actual position³⁵ of the early iceline depends heavily on the density and optical properties of the protoplanetary disk, which can vary quite a bit. Speaking very broadly, a system with a higher total mass of planets relative to the star should have an iceline closer in. And in the current solar system, icy objects can persist well inside the iceline if they are covered by dust or have atmospheres (higher surface pressure raises the temperautre of vaporization).

For Teacup A this formula predicts an early iceline at 1.28 AU and a current iceline at 2.37 AU, which seems fine enough.

That established, what should be the distribution of mass within the system? Planet mass plays into a lot of important factors we’ll discuss later, but for now we can use the solar system as a reference, and ours is a very top heavy system. The sun occupies 99.86% of the solar system’s mass, and of the remaining mass 71% is in Jupiter. But the further down you go, the more mass is divided among many different bodies. Going by orders of magnitude:

• There is 1 body (besides the sun) above 100 Earth masses: Jupiter
• There are 3 bodies between 10 and 100 Earth masses: Saturn, Neptune, and Uranus.
• There no bodies between 1 and 10 Earth masses (if we don't count Earth), though the elusive 9^th planet might be.
• Earth included, there are 3 bodies between 0.1 and 1 Earth masses: Earth, Venus, and Mars.
• There are 6 bodies between 0.01 and 0.1 Earth masses: Mercury, Ganymede, Titan, Callisto, Io, and the Moon.
• There are 4 bodies between 0.001 and 0.01 Earth masses: Europa, Triton, Eris, and Pluto.
• There are 15 bodies between 0.0001 and 0.001 Earth masses: Makemake, Haumea, Titania, Oberon, Rhea, Iapetus, 2007 OR₁₀, Charon, Ariel, Umbriel, Quaoar, Dione, Ceres, Tethys, and Orcus, though at this point it’s likely there are more unknown bodies in the Kuiper belt.
• Past that few bodies have reliable mass estimates, but presuming objects between 200 and 500 km radius are likely to have between 0.00001 and 0.0001 Earth masses, there are about 80 such objects known.
• Past that even detection becomes unreliable, but there are somewhere in the range of millions to billions of boulder-sized of larger objects within the system out to the Kuiper belt, and perhaps trillions in the Oort cloud beyond.

So if we want a system resembling ours, we should have a handful of gas giants, about 2 times as many terrestrial planets and very large moons, 2 or 3 times again as many dwarf planets and intermediate moons, and then many smaller objects. The overall mass of the system and the relative abundance of gas giants and terrestrial planets may vary, but this overall trend of mass distribution will probably be consistent.

Now, I want the Teacup A system to resemble ours in broad strokes: a single dominant giant near the iceline, a couple smaller giants in the outer system, and several terrestrial planets in the inner system. This both reduces the chance we’re messing with something necessary for complex life and makes sure the development of astronomy and space travel roughly parallels ours. I still want it to be different, though, so I’ve made a few changes that could have interesting consequences down the line.

Using the spreadsheet, I’ve spaced out a system of planets from 0.13 to 18 AU at intervals of around 1.5-2 orbital period ratios, though up to 4 in a couple places to keep the gas giants at a comfortable distance in Hill radii. Like our system, I’ve placed a couple gas giants outside the iceline and 4 terrestrial planets inside it—though to shake things up a bit, I placed one gas giant just inside the iceline and a super-earth in the outer system. I also placed some of the planets just wide of a 4:6:9 resonance chain, like we see in other systems.

So we’ve got a reasonable system, and everything’s either spaced out beyond 8.6 mutual Hill radii or in resonance with its neighbor. But is it really stable in the long term?

Researchers use a variety of programs to model system evolution, but a lot of them require some familiarity with programming and using command lines—not a lot, but enough to be offputting for people with zero experience (For those interested, REBOUND seems to be the standard, though SPOCK is another interesting project that attempts to predict stability through machine learning). From what I’ve seen, the most approachable is Orbe. The procedure is very straightforward: Input initial values for the masses and orbital elements of a set of planets into a config file, let the simulation run as long as you like, and then load the output values into a spreadsheet and chart the values to see the results (the authors specifically advise against using Excel to produce scientific charts, but I’m not exactly trying to get published in Nature here).

Typical result for eccentricity in stable system (it would probably be more regular in a mature system but this simulation resulted in no major instability events)

The speed of the simulation depends on the computer you’re using, but is particularly affected by the periods of the planets—shorter periods increase computation time. I found that my laptop, left to run the program overnight, can manage about 10 million years over 8 hours for the full Teacup A system and 100 million years using just the outer planets—long enough to be fairly confident of stability.

All the orbital elements will tend to vary periodically over time due to various subtle resonances between the planets’ motions: The semimajor axis, eccentricity, and inclination all tend to oscillate in sinusoidal patterns, and the other elements tend to circle around from 0 to 360° or the reverse. What we want to watch out for are sudden shifts in the semimajor axis or eccentricity of a planet that indicates that a close encounter has occurred, meaning the system architecture is not stable (Orbe’s predictions after such an event become unreliable, but it often seems to result in one or more planets being ejected from the system).

For Teacup A I took the SMAs and eccentricities from the spreadsheet, added low inclinations, and then picked the other elements more-or-less at random. I’ll spoil right now that I intend Teacup Ae to be our habitable world with intelligent life, so I set it’s inclination and true anomaly at epoch (at the start of the simulation) as 0°, with everything else defined in reference to that. We’ll say this epoch occurs at a northern vernal equinox at some point when civilization has developed on the planet—say, 6 billion years after formation of the system.

My initial sketch for the system was a bit too ambitious—I had a crowded outer system with a pair of dwarf planets in resonant orbits with my outermost gas giant, but I could never quite get it to be stable in the long term so in the end I cut them out. I also had an issue with the innermost planet gaining eccentricity until it encountered other planets; Orbe doesn’t seem to model the tidal forces that would work to lower the eccentricity of close-orbiting planets and prevent this, but I spaced out the inner 2 planets anyway.

 

Example of an instability event in one of the early simulations; the orbits of the inner 2 planets crossed over and they had several close encounters.

 With all that worked out, here is the resulting system: 

Body	Ab	Ac	Ad	Ae	Af	Ag	Ah	Ai	Aj
Mass (earths)	0.1	0.5	1.6	0.8	120	0.0001	400	3	25
SMA (AU)	0.13	0.25	0.34	0.45	0.9	1.6	4	10	18
Period (Years)	0.056	0.149	0.237	0.361	1.02	2.42	9.55	37.8	91.3
Eccentricity	0.05	0.15	0.03	0.1	0.02	0.06	0.005	0.05	0.08

I’ll put the full list of elements in the notes, but these are the most important values for determining the conditions of these planets in later sections. Presumably there are more dwarf planets beyond Teacup Aj, and Teacup B has its own system of planets, but this will do for now.

 

Orbital paths of all the planets, with Teacup A at the bottom right (I haven't accounted for eccentricity or inclination here).

Though I’ve put giants on either side of the iceline, I’ve also left a big enough gap between them to accommodate an asteroid belt straddling the iceline, like ours, and even placed Teacup Ag there as a Ceres analogue. The orbital space is a bit of a maze of resonances from Af and Ah, so to figure out where the asteroids are most likely to settle I’ve gone back to Orbe and placed a series of test masses spaced at regular intervals between the gas giants (I specified 0 mass for all of them, which tells the program to calculate the influence of other bodies on them but not their influence on each other or other bodies, which saves a lot of computation time) and checked which ones survived after a few million years.

The most stable orbits are those with low eccentricity—where the asteroids will experience the fewest collisions. So it looks like the main belt population should be between 1.5 and 2.5 AU.

There will also be a larger belt of material beyond Teacup Aj, but again we won’t worry about that now.

To finish up, let’s populate the system with some moons.

Ab and Ac are both small planets in close orbits with small Hill radii, so let’s leave them alone.

Ad is a super Earth, so let’s give it a large moon of 0.03 Earth masses—2.44 times the mass of Earth’s moon—and make the planet and moon mutually tidally locked, with an SMA of 146,334 km and a period of 122.34 hours (weird numbers, I know, but the result is they both have a synodic day of 130 hours)

Orbit of Teacup Ad I around Teacup Ad, with the sizes of the bodies to scale (I'll explain how I calculate those next time). These images for each moon system will not be to scale with each other, and do account for inclination but not eccentricity.

Let’s give Ae, our Earth analogue, 2 moons: one similar to our own but about a third the mass (0.004 Earth masses) and a bit over half the SMA (250,000 km); and a small captured asteroid about the size of Phobos (1.7*10^-9 Earth masses) in a highly inclined, retrograde orbit (105,000 km, 130°). Together these should give the surface a fairly interesting lunar cycle. Presuming a roughly earthlike radius for Ae, tides on the surface from the larger moon will be somewhat larger than those we get from ours—though the tides from the star will be about twice as high as those.

Af is our first gas giant, but because it formed inside the iceline it had fairly little material to form large moons—though, so close to the asteroid belt, it’s probably captured quite a few small ones. For now we’ll give it 4 smallish inner moons in a pair of overlapping 2:1 resonances—like Saturn’s moons—and one large captured outer moon of 0.2 Earth masses. Af also has a small ring of debris but nothing too impressive.

Ag is a small dwarf planet. It could still have moons, but I won’t bother.

Ah is our largest planet and comfortably outside the iceline, so we can give it a Jupiter-like set of large moons. Let’s start with 3 main moons in a 1:2:4 Laplace resonance. Like Jupiter’s moons their orbital phases will probably be locked into a specific pattern; as a shortcut, we can initially place the inner 2 moons at mean anomalies 180° apart—so on exact opposite points in their orbits—and give the outermost moon the same mean anomaly as either of the 2 middle ones—if they all have 0 eccentricity, they and the planet will all sit along one line. Rather than 1 Callisto-like outer moon, let’s give it a pair of smaller co-orbital moons acting as mutual trojans. And finally, we’ll give the planet a nice big set of rings with a couple of small shepherd moons.

I'm not showing the rings in these images, but here they'll be around the 2 innermost orbits

Ai is unlike any body in our system—a large solid body in the outer system. We can probably expect it to have a few moons, but not as many as our gas giants. Thus I’ve given it 3 notable moons and a small system of rings.

And finally Aj, our Neptune analogue. Neptune only has 1 large moon but Uranus has a few, so I’ll give Aj a smattering of moons, including one retrograde irregular moon, and modest rings.

 Given that Orbe doesn’t model tidal forces and is slow to calculate the evolution of short-period orbits, it’s not well equipped to predict the stability of our moon systems, so we’ll just have to trust our own intuitions for now. Here are all the established properties of the moons I’ve described:

Body	Ad I	Ae I	Ae II
Mass (Earth Masses)	0.03	1.7*10-9	0.004
SMA (km)	146,334	105,000	250,000
Period (days)	5.10	3.02	16.1
Eccentricity	0	0.1	0.01
Inclination (°)	0	130	5

Body	Af I	Af II	Af III	Af IV	Af V
Mass (Earth Masses)	0.00005	0.000009	0.0004	0.00015	0.2
SMA (km)	179,539	241,905	285,000	384,000	1,210,000
Period (days)	0.800	1.25	1.60	2.50	14.0
Eccentricity	0.003	0.01	0.001	0.0005	0.09
Inclination (°)	0.02	1.2	0.03	1.5	25

Body	Ah I	Ah II	Ah III	Ah IV	Ah V	Ah VI	Ah VII
Mass (Earth Masses)	10-7	10-8	0.025	0.02	0.03	0.005	0.0008
SMA (km)	150,000	180,000	494,000	784,176	1,244,802	4,000,000	4,000,000
Period (days)	0.335	0.440	2.00	4.00	8.00	46.08	46.08
Eccentricity	0	0	0.003	0.002	0.005	0.001	0.02
Inclination (°)	0	0	0.02	0.05	0.5	12	12

Body	Ai I	Ai II	Ai III	Aj I	Aj II	Aj III	Aj IV
Mass (Earth Masses)	0.0001	0.001	0.0003	0.00002	0.0006	0.01	0.000005
SMA (km)	80000	588200	933639	110000	144141	1500000	8000000
Period (days)	1.505	30	60	0.84	1.261	42.32	521
Eccentricity	0	0.002	0.05	0.001	0	0.008	0.2
Inclination (°)	0	2	2	0	0	1	160

Keeping Time

Constructing a calendar is a subject we’ll probably come back to somewhere down the line, but for the moment I want to quickly go over the astronomical cycles that inhabitants of our Earth-analogue world might have to work off of.

Given its semimajor axis of 0.45 AU and a star of 0.7 solar masses, Teacup Ae will have a year just over 1/3 our year in length—131.78 Earth days. For convenience I’ll give it a synodic day of 34 hours, which makes the year slightly over 93 days long (just 43.2 minutes longer); the inhabitants will only need a leap day every 47 of their years. To avoid confusion, I’ll use the terms Tyear and Tday to refer to these time periods from now on.

Teacup Ae II has an orbital period of 16.1 days, or 11.3 Tdays and a synodic month of 12.9 Tdays (18.3 days). That’s slightly over 7 months a year, so we might imagine the inhabitants of Ae could eventually come to divide their years into 13-Tday Tmonths.

Teacup Ae I has a period of just 105.2 hours. The synodic period is 100.1 hours, which is 3.02 Tdays. This is close to a 3:1 resonance, so any given spot on the surface below 40° latitude will see Teacup Ae I pass directly overhead about every 3 Tdays, at a slightly later time of day each time. Such a small object will not be as prominent in the sky as Ae II, but its odd motion should make it stand out, so it could become the basis for a 3-Tday…Tweek? That may not be necessary with so short a Tmonth, but we’ll decide that later.

And that should about wrap it up as far as orbital architecture is concerned. We’ll want to keep all this in mind in the next 2 posts, but in the next one we’re going to focus more on what these planets will actually be like; what types of surfaces they could have, and how might they appear either to someone standing on them or viewing from afar.

In Summary

• It takes roughly 10 million years for a collapsing gas cloud to become a main sequence star.
• Some outer material will form a protoplanetary disk around the star, which will form a series of gaps and rings at the icelines for silicates, water, and CO.
• Gas giants may form in just a few million years from runaway accretion of gas onto rocky cores.
  
• Gas giant formation is easier outside the iceline, but planets can migrate after formation.
• Terrestrial planets usually form after the disk has cleared through collisions between planetesimals
  
• Gas giants are rare but more common around larger and metal-rich stars, terrestrial planets are common especially around small stars.
• Formation of a giant near the iceline may cause smaller, drier planets to form in the inner system.
  
• Planets can continue to move and systems become unstable long after formation.
• Planets tend to have low eccentricity and mutual inclination, and are more spaced out further from the star.
• Planets will disintegrate if they pass within the Roche limit of their star, though this limit varies for different densities.
  
• Planet orbits usually become unstable if they approach within ~8.6 mutual Hill radii of each other, but can be packed closer with mean-motion resonances
• Planets can occupy the same orbit in a number of ways:
• They may form binary planets orbiting their common barycenter.
• They may be mutual trojans occupying each other's Lagrange points.
• They may occupy a horseshoe orbit and regularly swap orbits.
• They may occupy an eccentric resonance and gradually exchange eccentricity.
  
• Initial planet rotations are essentially random, but tidal forces can cause tidal-locking between moons, planets, and stars.
• Tidal-locking can cause many different spin-orbit resonances, but 1:1 spin-orbit resonance is the most likely.
  
• Moons (and binary planets) can form several different ways:
• In situ from a collapsing debris cloud or circumplanetary disk.
• Capture of a passing body.
• Direct or glancing impact with another body.
• Pull-down capture due to rapid growth of a gas giant's Hill sphere.
• Fission due to rapid rotation, though whether this could happen for large planets remains controversial.
  
• Moon systems are governed by most of the same laws and principles as planetary systems.
• Moon orbits become unstable past 0.49 Hill radii from their planet if orbiting prograde and 0.93 Hill radii if retrograde.
• Tidal forces cause all moons to eventually escape or merge with their planet, making long-lived moons unlikely in close orbits or around small stars.
• Long-lived rings can form within a planet or moon's Roche limit, though how long-lived is unclear.
  

Notes

Apparently the theory for the formation of the solar system out of a primordial cloud of gas and dust was first proposed by Emmanuel Swedinborg of all people, and then expanded on by Immanuel Kant. My grandfather was a Swedinborgian at a few points, between being a Buddhist, Episcopalian, and some variety of Pagan.

I found this nice graphic timeline of Earth’s early history.

“The orbital architecture of the Solar System presents a number of oddities. But like a polka lover’s musical preferences, these oddities only become apparent when viewed within a larger context.” (Raymond et al. 2018¹⁵²). I have no comment here.

I do intend to keep expanding and refining the worldbuilding spreadsheet as we explore more topics. If you spot an error or obvious improvement or you desperately want an addition, let me know.

As promised (Ae's argument of periapsis has been updated a few times in later posts to adjust its climate):

Body	Ab	Ac	Ad	Ae	Af	Ag	Ah	Ai	Aj
Mass (earths)	0.1	0.5	1.6	0.8	120	0.0001	400	3	25
SMA (AU)	0.13	0.25	0.34	0.45	0.9	1.6	4	10	18
Eccentricity	0.05	0.15	0.03	0.1	0.02	0.06	0.005	0.05	0.08
Inclination	2	5	2	0	1	8	1	4	1
Long. Of Ascending Node	20	260	150	0	60	310	80	110	250
Argument of Periapsis	50	80	140	240	90	60	340	280	10
Mean Anomaly at Epoch	20	320	10	120	270	80	190	100	350

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Part IVb

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29 Bottke, W. F., & Norman, M. D. (2017). The late heavy bombardment. Annual Review of Earth and Planetary Sciences, 45, 619-647.
30 Walsh, K. J., Morbidelli, A., Raymond, S. N., O'Brien, D. P., & Mandell, A. M. (2011). A low mass for Mars from Jupiter’s early gas-driven migration. Nature, 475(7355), 206-209.
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32 Gomes, R., Levison, H. F., Tsiganis, K., & Morbidelli, A. (2005). Origin of the cataclysmic Late Heavy Bombardment period of the terrestrial planets. Nature, 435(7041), 466-469.
33 Batygin, K., & Laughlin, G. (2008). On the dynamical stability of the solar system. The Astrophysical Journal, 683(2), 1207.
34 Zink, J. K., Batygin, K., & Adams, F. C. (2020). The great inequality and the dynamical disintegration of the outer solar system. The Astronomical Journal, 160(5), 232.
35 Kennedy, G. M., & Kenyon, S. J. (2008). Planet formation around stars of various masses: the snow line and the frequency of giant planets. The Astrophysical Journal, 673(1), 502.
36 Alibert, Y., Mordasini, C., & Benz, W. (2011). Extrasolar planet population synthesis-III. Formation of planets around stars of different masses. Astronomy & Astrophysics, 526, A63.
37 Fulton, B. J., Rosenthal, L. J., Hirsch, L. A., Isaacson, H., Howard, A. W., Dedrick, C. M., ... & Wright, J. T. (2021). California Legacy Survey. II. Occurrence of Giant Planets beyond the Ice Line. The Astrophysical Journal Supplement Series, 255(1), 14.
38 Mordasini, C., Alibert, Y., Benz, W., & Naef, D. (2009). Extrasolar planet population synthesis-II. Statistical comparison with observations. Astronomy & Astrophysics, 501(3), 1161-1184.
39 Winn, J. N., & Fabrycky, D. C. (2015). The occurrence and architecture of exoplanetary systems. Annual Review of Astronomy and Astrophysics, 53, 409-447.
40 Mulders, G. D., Pascucci, I., & Apai, D. (2015). An increase in the mass of planetary systems around lower-mass stars. The Astrophysical Journal, 814(2), 130.
41 Otegi, J. F., Helled, R., & Bouchy, F. (2022). The similarity of multi-planet systems. Astronomy & Astrophysics, 658, A107.
42 Burn, R., Schlecker, M., Mordasini, C., Emsenhuber, A., Alibert, Y., Henning, T., ... & Benz, W. (2021). The New Generation Planetary Population Synthesis (NGPPS)-IV. Planetary systems around low-mass stars. Astronomy & Astrophysics, 656, A72.
43 Mulders, G. D., Drążkowska, J., Van Der Marel, N., Ciesla, F. J., & Pascucci, I. (2021). Why do M dwarfs have more transiting planets?. The Astrophysical Journal Letters, 920(1), L1.
44 Rosenthal, L. J., Knutson, H. A., Chachan, Y., Dai, F., Howard, A. W., Fulton, B. J., ... & Wright, J. T. (2021). The California Legacy Survey III. On The Shoulders of (Some) Giants: The Relationship between Inner Small Planets and Outer Massive Planets. arXiv preprint arXiv:2112.03399.
45 Schlecker, M., Mordasini, C., Emsenhuber, A., Klahr, H., Henning, T., Burn, R., ... & Benz, W. (2021). The New Generation Planetary Population Synthesis (NGPPS)-III. Warm super-Earths and cold Jupiters: a weak occurrence correlation, but with a strong architecture-composition link. Astronomy & Astrophysics, 656, A71.
46 Mandell, A. M., Raymond, S. N., & Sigurdsson, S. (2007). Formation of Earth-like planets during and after giant planet migration. The Astrophysical Journal, 660(1), 823.
47 Mustill, A. J., Davies, M. B., & Johansen, A. (2017). The effects of external planets on inner systems: multiplicities, inclinations and pathways to eccentric warm Jupiters. Monthly Notices of the Royal Astronomical Society, 468(3), 3000-3023.
48 Carrera, D., Davies, M. B., & Johansen, A. (2016). Survival of habitable planets in unstable planetary systems. Monthly Notices of the Royal Astronomical Society, 463(3), 3226-3238.
49 Hands, T. O., & Alexander, R. D. (2016). There might be giants: unseen Jupiter-mass planets as sculptors of tightly packed planetary systems. Monthly Notices of the Royal Astronomical Society, 456(4), 4121-4127.
50 Pearce, T. D., Launhardt, R., Ostermann, R., Kennedy, G. M., Gennaro, M., Booth, M., ... & Stone, J. M. (2022). Planet populations inferred from debris discs: insights from 178 debris systems in the ISPY, LEECH and LIStEN planet-hunting surveys. arXiv preprint arXiv:2201.08369.
51 Izidoro, A., Raymond, S. N., Morbidelli, A., Hersant, F., & Pierens, A. (2015). Gas giant planets as dynamical barriers to inward-migrating super-Earths. The Astrophysical Journal Letters, 800(2), L22.
52 van Lieshout, R., & Rappaport, S. (2017). Disintegrating rocky exoplanets. arXiv preprint arXiv:1708.00633.
53 Perez-Becker, D., & Chiang, E. (2013). Catastrophic evaporation of rocky planets. Monthly Notices of the Royal Astronomical Society, 433(3), 2294-2309.
54 Price, E. M., & Rogers, L. A. (2020). Tidally distorted, iron-enhanced exoplanets closely orbiting their stars. The Astrophysical Journal, 894(1), 8.
55 Batygin, K., & Brown, M. E. (2016). Evidence for a distant giant planet in the solar system. The Astronomical Journal, 151(2), 22.
56 Xie, J. W., Dong, S., Zhu, Z., Huber, D., Zheng, Z., De Cat, P., ... & Zhang, Y. (2016). Exoplanet orbital eccentricities derived from Lamost–Kepler analysis. Proceedings of the National Academy of Sciences, 113(41), 11431-11435.
57 Becker, J., Batygin, K., Fabrycky, D., Adams, F. C., Li, G., Vanderburg, A., & Rodriguez, J. E. (2020). The Origin of Systems of Tightly Packed Inner Planets with Misaligned, Ultra-short-period Companions. The Astronomical Journal, 160(6), 254.
58 Bourrier, V., Lovis, C., Cretignier, M., Allart, R., Dumusque, X., Delisle, J. B., ... & Osorio, M. Z. (2021). The Rossiter-McLaughlin effect revolutions: an ultra-short period planet and a warm mini-Neptune on perpendicular orbits. arXiv preprint arXiv:2110.14214.
59 Carter, J. A., Agol, E., Chaplin, W. J., Basu, S., Bedding, T. R., Buchhave, L. A., ... & Winn, J. N. (2012). Kepler-36: A pair of planets with neighboring orbits and dissimilar densities. Science, 337(6094), 556-559.
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68 "Stability of Lagrange Points", Richard Fitzpatrick
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