Drawing portraits of the population

As a newborn planet, Jupiter shone brightly in the sky and eclipsed today’s sun from the perspective of the gas giant’s largest moons. This early glow and upcoming visits by several spacecraft could help solve a 40-year-old mystery about the composition of these moons.

For decades, scientists have been trying to understand the strange differences in density of Jupiter’s four Galilean moons, which, in order from closest to the planet to farthest, are Io, Europa, Ganymede, and Callisto. Although these natural satellites must have formed from the same source material and therefore have a similar composition, density measurements show that Callisto and Ganymede are much icier than Europa, and Io has no ice at all. disclosed At a conference last month, Carver Birson, a planetary scientist at Arizona State University, can shed some light on the subject.

Giant planets are formed by the merger and compression of huge volumes of gas and dust. This process releases a lot of excess energy and gives the newborn giants a literal youthful glow that can last for millions of years. This is more than a theory: astronomers regularly use this glow to image young giant exoplanets that would otherwise be lost in the glow of nearby stars. But the less glaring question of how such a glow could shape accompanying moons has been largely unexplored. In the case of Jupiter, computer simulations by Birson and colleagues suggest that the planet’s early glow illuminated its newborn moons and evaporated most of their water over a period of several million years.

“It gave people the opportunity to think about a completely new process,” says Francis Nimmo, who studies icy moons at the University of California, Santa Cruz and was not involved in the study.

Four satellites, one source

The differential compositions of the four Galilean moons have puzzled researchers for decades, ever since the first high-quality satellite density measurements were made. Locked inside Jupiter’s radiation belt and heated from within by the planet’s powerful tidal forces that knead the Moon’s interior like dough, Io is a completely ice-free world of hyperactive volcanoes. A little more distant Europe is also in the grip of the radiation and tides of Jupiter. But more modest levels of internal heating have given the Moon a subsurface ocean and icy crust, rather than lava-spewing calderas. Ganymede and Callisto are relatively inert, rich in ice, and much farther from Jupiter than Io and Europa.

Although the differences in Jupiter’s gravitational hold clearly explain some of the differences between the moons, planetary scientists were still trying to understand how these objects could have a common origin, yet be so different from each other. Just as planets form from spinning protoplanetary disks of gas and dust around nascent protostars, large moons can form from smaller mini-disks around gathering gas giant worlds. Current thinking calls for Jupiter to gain most of its mass very quickly, during the first 10 million years of the solar system’s life, before the light and stellar winds from the ever-brightening sun swept all the gas from the protoplanetary disk.

This relatively compressed timeline means that Jupiter would have had to greedily, quickly gulp gas to reach its current size, which would have caused it to heat up and glow, reaching a temperature estimated at 1,160 degrees Fahrenheit (627 degrees Celsius). For the Galilean moons, which supposedly formed around the same time as Jupiter itself, the planet would have shone like a star in the sky and overwhelmed the light coming from the more distant sun. By carefully modeling the effects of Jupiter’s increased luminosity on the Galilean moons, Birson and his colleagues found that this beam of light could clearly solve the riddle of today’s diverse satellite composition.

Fragrant fresh baked moons

Torn apart by Jupiter’s gravity, Io today is a hellish landscape of volcanic eruptions and is the most active body in the solar system. But the team found that Jupiter’s youthful glow may have originally given Io Earth’s temperatures and perhaps even the ocean. “I think it’s likely that either during Io’s formation or just after Io’s completion, there is some water on the surface,” Birson says.

That would change quickly, Birson said, as Io received about 30 times more energy from Jupiter than it receives from the Sun today. If Io had as much water as its cousin Ganymede currently holds, all that moisture would be quickly removed, and any remaining ocean would evaporate in the first million years of the moon’s existence.

Europa, further away than Io, would have had slightly cooler surface conditions – although they might still have been hot enough for this moon to lose a significant amount of its water. Farther away, on Ganymede, Jupiter would appear barely brighter than today’s Sun, a level of isolation without significant impact on lunar ice. The distant Callisto, sent to the outskirts of the Jupiter system, would not have been impressed by the radiant youth of Jupiter. (All of this assumes the moons were in their current positions. They probably formed closer before migrating to their current locations, however, this means the results of the study are probably only a lower limit on how much each moon was baked by Jupiter. .)

“The advantage of this hypothesis is that there are several tests that you can apply,” says Nimmo.

SOK-y offer

If Europa had lost most of its ice in its lifetime, rather than formed from less ice than its siblings, the remaining hydrogen and oxygen would have a different isotopic fingerprint than the ice on Ganymede and Callisto. Thus, an isotopic comparison of Europa with one or both of the most distant moons could finally reveal the truth about how these moons diverged from their common origin. “The more comparisons you can make [among the chemistries of the moons]the more you will understand how things will develop at this very early time,” says Birson.

This is quite an attractive proposition, given the recent launch of the European Space Agency’s Jupiter Icy Moons Explorer (JUICE) mission. Between 2031 and 2034, JUICE will make 35 flybys of Europa, Callisto and Ganymede before entering orbit around Ganymede. An extended tour can go a long way in determining whether all Galilean moons were born with the same amount of ice. JUICE has a mass spectrometer, which Nimmo says can make important measurements of hydrogen and water vapor that can come into space from moons, specifically Ganymede.

“The question is whether Ganymede provides enough material for the heights that JUICE can sample,” says Nimmo. He remains confident that this will be the case.

Even if JUICE research fails to crack the case, it won’t be the only moon-exploring spacecraft hovering in the Jupiter system. NASA’s Juno mission is already in orbit around the gas giant, and the space agency’s Europa Clipper mission is set to launch next year to travel to the mission’s moon of the same name. The Clipper data should provide a clear comparison of JUICE’s views of Europa’s ice that will be enough to extrapolate and distinguish from what a European spacecraft sees on Ganymede and possibly Callisto.

“Comparisons between moons will be extremely important,” Birson says. “It’s so exciting that JUICE and Europa Clipper will appear almost at the same time and maybe overlap a bit.”

In his spare time, David Smith designs tiles. In particular, a retired printer and an amateur mathematician put together as many tiles as they can (no gaps allowed) before the pattern either repeats or fails to continue.

Until recently, every form anyone has ever tested has met one of these two fates, despite the scrutiny of many brilliant minds over the past 50 years. Then, one day last November, Smith discovered the only known exception.

13-sided shape

With an app called PolyForm puzzle solver, Smith built a jagged 13-sided shape. Vaguely resembling a cylinder, it began to fill the screen with copies of it. They connected smoothly and, to his surprise, without repetition.

“The tessellations were something I hadn’t seen before,” he says.

Eager to continue his investigation, he cut out dozens of paper copies and started over. One by one the tiles fell into place. They immersed him deeper into the striking visual pattern—and as he grew, so did his excitement.

He showed this promising creation to Craig Kaplan, a computer scientist at the University of Waterloo. “Almost immediately, it felt like he had stumbled upon something new and profound,” Kaplan says.

Although the mathematical proof took longer, their instinct was sound: as they and two other colleagues announced in March, paper which has not yet been peer-reviewed, Smith stumbled upon the long-awaited form of “Einstein”.

Einstein’s elusive tile

The word “einstein” used here has nothing to do with a German physicist. Instead, he evokes the literal meaning of his last name: “one stone”.

It’s a less yawn-inducing nickname for what is technically known as an “aperiodic mono-tile” – a single tile that can fill an endless plane, over and over again for eternity, in a pattern that never repeats.

On a typical bathroom floor, you will find distinct squares, triangles, or hexagons arranged in some sort of clearly visible pattern. Now imagine replacing those neat rows with a decidedly random jumble of blocks, and voila, aperiodic decor.

In other words, there is no section to cut and paste to complete the rest of the jigsaw puzzle.

Other aperiodic tilings

Before the 13-sided “hat” was revealed to Smith, no one could have guessed whether Einstein even existed.

Mathematicians have been after him since 1966, when Robert Berger designed the first set of tiles that could be laid out aperiodically. It was a landmark innovation, but with a bulky 20,246 different shapes, it wasn’t yet a viable option for home remodelers.

As the search for more elegant combinations continued, in just a few years this number dropped from five to single digits. Soon the monotile was within reach.

In 1973 Oxford mathematician and physicist Roger Penrose set a bar on two tiles by arranging a pair of figures called kites and darts in an aperiodic fashion. But then progress stalled, and the last challenge stood for five whole decades.

Read more: How a mathematician solved a problem that puzzled computer scientists for 30 years

In search of the unknown

After so many years, it may seem surprising that an amateur beat the professionals to the finish line. In fact, Smith wasn’t even looking for Einstein. He attributes his success to “mostly perseverance” and perhaps some luck, “although I feel like I was the chosen one,” he says.

Marjorie Seneschal, a professor emeritus at Smith College who has studied tile since the 1970s, notes that the area’s history is littered with the contributions of untrained artisans. Remarkably, around the time Penrose introduced his kites and darts, an amateur and mail sorter named Robert Ammann independently invented a strikingly similar solution.

“It’s a subject where you can literally get down to business,” says the Seneschal, who profiled Ammann in 2004 For Mathematical intellectual. “If you have a good eye and an inquisitive mind, you can find things that other people trying to work with theory can’t find.”

Mathematics of patterns

Smith’s ingenuity sets things in motion. But since we’re dealing with an infinite plane, no amount of end tile manipulation can guarantee that the pattern won’t end up starting over again.

The only way to be sure? Mathematical proof. So Smith and Kaplan brought in two more experts: Chaim Goodman-Strauss, a mathematician at the University of Arkansas, and Joseph Myers, a British software engineer.

In fact, aperiodicity is child’s play. Plain old rectangles can satisfy the requirement of non-repeatability, even if you can easily collect them periodically.

The real trick is to find the shape that only works aperiodically, with the right balance of complexity—enough to break the periodic pattern, but not so much that the whole pattern degenerates.

“It’s the magic that makes aperiodicity interesting,” says Kaplan. “They have to do a very careful dance between order and chaos.”

Read more: Is it really impossible to “square the circle”?

Proof of aperiodicity

To make sure the hat hit the mark, Myers first applied a tried and tested method pioneered by Berger himself.

It starts with a set of “metatiles”, simple polygons that roughly resemble small groups of hats. From there, you can combine meta tiles into super tiles, super tiles into super super tiles, and so on ad infinitum.

Their article demonstrates that this hierarchy is the only way to tile the plane with hats, which is tantamount to proving that form will never slip into periodicity. But then Myers fabricated a new type of proof, and to understand it, we need to break the hat down into its main parts.

However exotic the form may seem, it is well known to geometers as polykit; start with a hexagon, draw three lines connecting opposite sides in the middle and you have six kites.

Combine two or more of them and you get a polykit. Add eight together in a particularly fortunate order and you have a groundbreaking mathematical discovery. As the team writes in their article, “the shape is almost mundane in its simplicity.”

Read more: form of insanity

Setting aperiodicity

The 13 sides of the hat have two lengths, and Myers realized that by adjusting the length of any set, he could create new shapes with the same properties. This means that there is not one Einstein, but an infinite family of them, each of which turns into another in a wide spectrum.

At the two extreme points (where the long and short sides disappear) and at the middle point (where they become equal) there are periodic figures that can be used to establish the aperiodicity of the rest.

This intriguing addition to the mosaic proof repertoire gives mathematicians food for thought, but the Seneschal explains that it is rooted in a more traditional strategy.

“There is a connection with a long-standing theory,” she says. “It puts their work on a continuum of not just mosaics, but thinking about mosaics.”

The first of many Einsteins

By the mere fact of its existence, the hat solved a riddle. But it also creates new ones. Could there have been more Einsteins, for example, apart from this family? The Seneschal suspects that they must be, and that this triumph may revive the search.

One of the most intriguing questions is the reason for its aperiodicity. The “special sauce” is still unknown, as Kaplan put it: “I can’t point to part of the mold and say, ‘That’s why.’

Nicholas de Bruijn, Dutch mathematician, after all explained Penrose tiles as two-dimensional projections of a five-dimensional mosaic. But as far as the theoretical description of the hat is concerned, Kaplan says “we’re not even close to that yet.”

Whether this abstract curiosity applies in the real world remains to be seen.

Read more: 5 interesting facts about Albert Einstein

hat animation

It certainly has a promising future in interior design, especially considering how tightly it fits into the grid of hexagons. They begin to lay it out on the kitchen floor; anyone with a hexagonal mosaic and enough determination can trace the pattern for themselves.

Many aperiodic tiles are roughly superimposed on an ordered background, “but this one,” Kaplan says, “just sits there really well. It’s almost ridiculously well-behaved.”

Assuming the hat does make it to your local home improvement store, keep in mind that the occasional event is not for the faint of heart.

“If you just go blind,” says Kaplan, “you will probably get stuck.” And if, by some miracle, you find a contractor willing to cheer you up, “there will be big labor costs.”

Read more: How mathematicians cracked the cipher of the zodiac killer