PLANETPLANET

This post starts off with some world-building, jumps into eclipses and moons’ orbits, and finishes with a brand new Kalgash system that Isaac Asimov would be proud of (dropped into darkness every 2000 years!).

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More than one planet can share the same orbit around a star. This is not big news: the concept of Trojan planets has been around for decades. In a Trojan setup, two planets orbit the same star separated by 60 degrees along their orbit. Jupiter has two populations of asteroids that share its orbit, centered on the Lagrange points 60 degrees in front of and behind its orbit.

In Cohorts of co-orbital planets I took this concept two step farther. First I showed that Trojan setups can have more than two planets. A full ring of 6 Earths or 6 Neptunes can share the same orbit, all separated by 60 degrees along an orbit. The stability of these “mega-Trojan” systems depends on the planet masses: only four Saturns can share an orbit, only three Jupiters, and only two super-Jupiters.

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Second, I showed that planets don’t need to be 60 degrees apart to be stable. Planets can be a lot closer together along their orbit, especially if they are ~Earth-mass. I called these “cohorts” of planets. A cohort can be made up of two to about twenty planets all sharing the same orbit.

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Now let’s level-up the cohort concept with black holes!

I’ve built planetary systems around black holes before, like the Black Hole Ultimate Solar System and the million-Earth Solar System. Sounds weird but the concept is simple.

A star orbiting a massive black hole behaves essentially the same as a planet orbiting a star.

In celestial mechanics what matters is the ratio of the masses of the central object and the smaller orbiting body. For Earth around the Sun, that ratio is about 1 to 300,000. If we switch out the Earth for the Sun, everything will stay the same (in a relative sense) if we replace the Sun with a black hole that is 300,000 times more massive than the Sun. This is actually pretty wimpy for a supermassive black hole (like the ones at the centers of galaxies).

Let’s make the central black hole a little more massive and boost it up to a million times the mass of the Sun. The slightly-higher mass of the black hole will actually stabilize orbits with more than one Sun (because their Hill radii will be smaller).

We can use the star-planet cohort systems we’ve already created as a blueprint for black hole-star systems. Here is a nice simple one that we’ll build on:

Now the fun part begins. What kind of interesting systems can we create by throwing planets into the mix?

Let’s start by plopping an Earth around the central star in the 3-star cohort above (side note: have you ever “plopped” a planet before? Well now you have!).

What is the scale of the system? Assuming a Sun-like star then the star-planet separation should be the Earth-Sun distance of 1 astronomical unit. To ensure that the planet’s orbit around the star remains stable, the star’s Hill sphere must be at least 2-4 times as big as this (so, 2-4 astronomical units). This is a direct analogy to moons orbiting planets around the Sun (see here and here for more). In order for the three stars’ orbits around the black hole to remain stable, the distance between the stars along their shared orbit must be at least 20-30 Hill radii (see here).

Put together, this means that the stars are about 50-100 astronomical units apart along their orbits. That’s a few times bigger than Neptune’s orbit around the Sun, but we’re dealing with stars and black holes, not just little chunks of ice and rock! It’s worth remembering that we can spread the system out and it will stay nice and stable. But we can’t shrink it down without it possibly going unstable.

From the point of view of a person sitting on the planet, being in a cohort like this would not be shockingly different from simply Earth orbiting the Sun. The main difference would come from the other stars in the cohort, each of which would look brighter than most stars in the sky, and even brighter than the full Moon. But these stars would not provide a big source of energy compared with the Sun — they would just be for show. And they are far enough to be mainly just points of light — about the size of Saturn in our sky.

Let’s spice things up and switch out the other two yellow stars for more colorful ones. We’ll put a red giant star on one side and a blue supergiant on the other. These stars are more massive than the Sun so their Hill spheres are bigger, meaning that we need to scale up the size of the system such that the stars are at least a few hundred astronomical units apart. [Technical note: the main cohort systems whose stability I tested previously were all the same mass. Here I am switching up the masses of different members of the cohort. I haven’t tested this comprehensively (yet), but a few test N-body simulations seem to indicate that it’s stable.]

Giant stars are hundreds to thousands of times more luminous than the Sun (see the Earth with five Suns). At separations of a few hundred astronomical units, these stars provide a significant source of energy on the Earth orbiting the central yellow star. Plus, since these stars are super big, they would really look like Suns.

The illumination pattern on the Earth orbiting the yellow star is the sum of three different patterns. The day/night cycle of the yellow Sun is pretty standard, lighting up half of the planet at a time. At any time of day you might get significant energy from the blue supergiant or the red giant, but never from both at the same time (except when each is just at the horizon). See the illumination map in the image above.

From the surface of the planet there would always be at least one star in the sky. Each star would spend half of the day in the sky. In the configuration from the image, the yellow star would rise when the red giant was overhead. The red star would set when the yellow star was overhead, and the blue star would rise at the same time. The yellow star would set leaving only the blue star in the sky.

Perhaps the most magical time of day would be midnight (relative to the yellow star in this setup). The red star would rise as the blue star set, starting the cycle again.

The illumination cycle would change as the planet orbited the yellow Sun. The red and blue stars remain on opposite sides of the planet, so one of the two would always be present. But the relative position of the yellow Sun would change.

In Isaac Asimov‘s classic book and short story Nightfall, night only falls on the planet Kalgash every 2049 years. Kalgash is in near-permanent daytime, illuminated by six different stars in its system.

In two previous posts I first debunked Asimov’s envisioned setup for Kalgash (sorry Asimov — I’m still a huge fan). Then I built a system that did produce a planet in near-permanent daytime, using rings of stars orbiting a supermassive black hole.

The planet in the three-star cohort system is actually a much more natural analog to Kalgash. And this setup is a lot simpler than the one I came up with previously, and it only uses half of the six stars from Asimov’s original story. A weakness of my setup is that it was not easy to envision a blackout that happened every 2000 years.

Let’s use our three-star cohort system to make a true Kalgash analog on which darkness does fall…. once every long while. We’ll use eclipses, so we need a moon. And we’ll need to get three-dimensional and eccentric.

Our Moon orbits the Earth once a month. We don’t have an eclipse every month because the plane of the Moon’s orbit around the Earth is tilted compared with the plane of the Earth’s orbit around the Sun, by about 5 degrees. Total eclipses only happen when the Moon crosses the Earth’s orbital plane at the right phase to cast a shadow on Earth. Most of the time Earth misses the Moon’s shadow. Even when it does pass through the shadow, it most often is a near-miss, creating a partial (not total) eclipse).

Below is an illustration of why there is not a total lunar eclipse every month, due to the Moon’s inclination. This is analogous to our situation (dealing with solar eclipses), although the image shows the Moon passing through Earth’s shadow, a solar eclipse happens when Earth passes through the Moon’s shadow.

Now let’s imagine that the Sun our three-star cohort system has a moon. By analogy with moonmoons (moons of other moons), a moon of a planet orbiting a star orbiting a supermassive black hole can often be stable (even in rhyme). Let’s assume that we’re okay in that regard.

Let’s keep it simple and imagine the moon is the same size as Earth’s. Let’s also assume that all three stars share the same orbital plane around the black hole, and that the Moon also shares that plane. In other words, the system is 2-dimensional. Let’s also assume that we have chosen the stars’ sizes and distances carefully such that they are each the size size in the sky (about half a degree on the sky, as is the case for the Sun and Moon in Earth’s sky).

Since the Moon, stars and planet always stay in the same plane, the Moon would pass in between the planet and each star over the course of a month. But not all of these eclipses would result in darkness on the planet, because a lot of the time there are more than one star in the planet’s sky.

If one star is eclipsed but another is still lighting up the planet, it doesn’t create darkness.

The image on the left shows the basic setup and the one on the right shows the illumination pattern on the planet. At each snapshot in time (at 3, 6, 9 and 12 on the “clock face” that tracks different positions along the planet’s orbit), each star illuminates slightly more than half of the planet. This is because twilight keeps the sky bright until the Sun is about 15 degrees below the horizon (and we’ll assume an Earth-like atmosphere on this planet).

An eclipse can only cast darkness on a given part of the planet’s surface when just a single star is illuminating that part of the surface (and that star is being eclipsed). Places on the surface where darkness is possible are the pie wedges (on the right) that only have single color (red, blue or yellow), not a combination of different colors.

An eclipse of the yellow star will never result in darkness on the planet. There are no isolated yellow wedges anywhere. This is because the blue and red stars are coming from polar opposite directions. If the moon blocks light from the yellow star, no matter — one of the other stars is already up in the sky.

But there are blue and red wedges! The planet is sometimes illuminated only by the blue or red giant star. When the planet is at 3 on the clock face, the side of the planet he opposite the yellow star is only lit up by the blue star. If a shadow of the blue star landed on that side of the planet’s surface, it would create darkness. The same type of setup exists when the planet is at the 9 on the clock face, where one side is only illuminated by the red star.

Averaged over the planet’s orbit and its surface, there is just one star in the sky a little more than a third of the time. [Technical note: calculating an exact number depends on accounting for all of the single-color wedges in the image (and remember that no darkness can fall while the yellow Sun is in the sky, which is 14 hours out of every 24 hour day due to twilight).]

The single star in the planet’s sky is always the red or blue one. In a two-dimensional setup each star is eclipsed by the Moon every month. The eclipse of the yellow star doesn’t create darkness. There is a one in three chance that either of the other eclipses will lead to darkness, because that is how often (on average) there is just one star in the sky. So, on average, there is 2/3 of an dark eclipse per month, or two periods of eclipse-driven darkness every three months.

The time between blackouts is longer in a 3D system.

Imagine the Moon in the 3-star cohort system has the exact same orbit around its planet as our Moon around the Earth. On Earth there is on average one eclipse every 18 months. In our three-star system there are three times as many chances for eclipses but only twice as many opportunities for darkness, so there should be a darkening eclipse every 27 months or so. That’s not too shabby but it’s still 1000 times more often than on Kalgash.

If the Moon’s orbit was more inclined then the odds of it crossing the plane of the planet’s orbit at the right time for eclipse would be less. We could safely increase the Moon’s orbital inclination by about a factor of ten. That would make eclipses ten times less likely and stretch the time between dark episodes to 270 months, a little over 20 years.

Something that can make total eclipses less common is the shape of the Moon’s orbit.

Our own Moon’s orbit is slightly non-circular, with an average orbital eccentricity of about 5.5%. That means that it looks about 10% bigger at its closest approach to Earth (“perigee”) than its farthest approach (“apogee”). This is part of what makes a “super moon” so bright — the Moon is actually bigger in the sky because it’s closer to Earth than usual.

On the flip side, the shrinking of the Moon at apogee can make eclipses disappear. When the Moon is small in the sky it can’t quite block the entire Sun — this causes an “annular eclipse” that makes the Sun look like a ring. It’s beautiful of course (especially in Canada at sunrise — image by Kevin Baird). But it would not block enough light to cause the total darkness we are looking for on Kalgash.

If the Moon’s orbit were much more stretched out, total eclipses would be rare. The Moon would grow and shrink in the sky along its orbit, and it would only create a total eclipse if it was close enough to the Earth to be large enough in the sky. Here is an illustration of an extreme case with the Moon on a very stretched-out orbit. In this setup, the Moon would only cause a total eclipse very rarely, when it happened to be near its closest approach to the Earth while it was passing exactly between the Earth and Sun.

So a Moon on a stretched-out orbit around the Earth would usually have annular eclipses, with only the occasional total eclipse. Just how rare would total eclipses be? It depends on the Moon’s exact orbit. For our Moon around the Earth, there are about the same number of annular and total eclipses.

For different orbits of the Moon the fraction of total eclipses could vary a lot. The space of possible orbits is shown in the image below on the left. The x axis is the orbital size and the y axis is the orbital eccentricity, and the color corresponds to the fraction of each orbit that is close enough to the Earth for a total eclipse.

The red zone is no good because the Moon is always big enough for a total eclipse. In the pink, green and blue zones there is a mix of total and annular eclipses. The total eclipse fraction drops for orbits that are more stretched-out and farther from the Earth. If the Moon was on an orbit that is too far (the black zone) then there never would be any total eclipses. The sweet spot is the purple region, where the Moon’s orbit only spends about one percent of its time close enough to Earth for a total eclipse. Of course, we need to make sure the orbit is stable. Any orbits above the dashed line are unstable (for prograde orbits; details here), so the region of rare total eclipses by a stable Moon is the narrow purple wedge below the unstable curve.

One example from the purple wedge is shown in the image on the right. In this case, the orbital size is 40% larger than Earth’s Moon and the orbital eccentricity of the orbit is 28.6%, putting it just on the right side of the stability limit. If the Moon was on this orbit, a total eclipse would only happen once every 100 times.

[Technical note: we could easily do the same exercise with a Moon that is a little smaller and a little closer to Earth. In that case, orbits that were a little more eccentric would be stable and it would be easier to find a Moon orbit that spends just a tiny speck of its time in the total eclipse zone. Moons can also remain stable farther out if their orbits are retrograde — in which case the entirety of the left-hand figure above is perfectly stable (see here). ]

We’ve got all the ingredients we need to put a new Kalgash system together.

The key piece of the puzzle: a Moon on a tilted, stretched-out orbit. The strong tilt (of 50 degrees or so) ensures that eclipses are rare. The stretched-out orbit is such that almost all eclipses are annular rather than total.

Let’s go through the numbers one more time. There is on average about one total eclipse every 18 months on Earth. Our system has three stars but only two (the red and blue ones) that can cause total darkness when eclipsed. Plus, the odds of one star being in the sky at any given time are only 1 in 3. So, this gives a typical time between dark eclipses of 27 months. By increasing the inclination of the Moon’s orbit we can extend this time by a factor of 10, to 270 months or more than 20 years. Now, by stretching out the Moon’s orbit so that it only very rarely passes close enough for a total eclipse, we can extend this time even further. A Moon orbit in the purple region that only spends ~1% of its time in the total eclipse zone will make the time between total eclipses 100 times longer, about 2000 years! This is Kalgash, straight out of Nightfall! Boom!

The inhabitants of Kalgash would know about eclipses of course. They would see partial and annular eclipses of each of their three stars on a yearly basis. But blackouts from eclipses would be unheard-of, and when one eventually happened, things would get bonkers!

Even though there are only three stars in the system (plus a supermassive black hole of course), I think Asimov would be pleased with the system we’ve built.

There are a bunch of ways to play with the system that could expand or shrink the time between dark eclipses. For instance, if this was a 4, 5 or more star cohort then the whole thing gets even more complicated. But if the central star was a red dwarf (as in Nightfall) then all of the timescales speed up.

[Technical note — of course, there are always nit-picky details. For example, the orbit of the Moon would not be perfectly fixed in time and could wobble and change shape from gravitational perturbations by the not-perfectly-spherical planet that it orbits or from external forces. And there’s always tides to worry about, which change a Moon’s orbit on long timescales…]

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I’ve kind of gone off the deep end with this Kalgash system. There are a zillion other systems to build that include cohorts of stars. Just take your pick of stars, planets and moons (and moonmoons if you’ve brave), put them in orbit around a black hole, and go nuts! If you need inspiration, feel free to send me a personal note.

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Additional Resources

• Cohorts of co-orbital planets
• The Black Hole Ultimate Solar System
• The Million-Earth Solar System
• My first attempt to use a black hole to re-imagine Kalgash from Asimov’s Nightfall — a planet in near-permanent daytime
• Can moons have moons?

Mustache bonus: A couple years ago I was interviewed for Japanese public television (NHK) about the possibility of planets orbiting black holes. By chance, the interview took place during the month of November, during which I often grow a mustache (for Movember). Hence, this photo:

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Finally, a big thank you to agmartin, whose great ideas and comments on my original cohorts of co-orbital planets post in large part motivated this one.