The Record-breaking Building Project
In August 2013, the US city of Wilmington in Delaware saw the birth of a record-breaking monument: a 34.44m (113ft) tower made entirely from interlocking plastic LEGO bricks. The brightly coloured creation was assembled by pupils from 32 schools in the Red Clay district, each of whom pieced together separate sections of the tower. These larger segments were then combined by a construction crew and crane in a grand finale that was officially recognised by the Guinness Book of Records. In total, almost 500,000 LEGO bricks were used, and the end product beat the previous world-record holder by nearly 2m (6½ft).
It was a project that demonstrated something the Universe has known for billions of years: in order to build something truly big, you start small and work upwards. In the case of our own Solar System, this involves taking the microscopic dust grains around a young star and sticking them together to form a planet.
Despite the fact that we know this assembly process must work, planetary scientists were faced with two baffling problems. First, it was not remotely obvious how dust particles would stick together. The Itokawa asteroid kept its rubble pile of rocks in place by its own gravity. But the strength of gravity depends on the object’s mass. A rocky body smaller than roughly a kilometre in diameter would not have enough bulk to produce a decent sticking force. The result would be like pressing dry sand together on a beach; the grains slide apart as soon as you release your grip.
Second, it was unclear how any sticking mechanism could work fast enough to build our Solar System before the Sun destroyed the protoplanetary gas disc. The observations of protoplanetary discs around young stars gave them an upper time limit of 10 million years. Within this span, an assembly process would have to take dust grains a tenth of the size of sand, and create a young planet with enough mass to hold on to a gas atmosphere as the rest of the disc evaporated.
In short, this process was akin to being given a box of bricks and told to build a tower, only to find that the bricks were completely smooth and the box would be confiscated straight after lunch.
While even a record-breaking tower of building blocks can be comfortably measured in metres, the Universe’s construction projects happen on a rather larger scale. To avoid having to deal with numbers of a ridiculous size, let’s take a small detour to find a more practical unit of distance to explore the Solar System.
It is perfectly possible to discuss the positions of the planets in terms of metres or kilometres, but it is hard to make sense of numbers once they reach an obscenely large number of digits. For example, the Earth is 149,600,000km (92,960,000mi) from the Sun, while Jupiter sits out at 778,340,000km (483,640,000mi). Compared with your average run to the supermarket, both these distances come under the category of disturbingly big, and it is difficult to quickly tell how much deeper into our Solar System Jupiter is sitting relative to us.
To tackle this problem, astronomers measure distances compared to the Earth’s distance from the Sun. They call this unit an astronomical unit, or au, and by its definition, the Earth is on average 1au away from the Sun. Jupiter’s distance can then be written as 5.2au, telling you that it is just over five times as far from the Sun as the Earth’s position.
These distances are important because the distance from the Sun controls the type of dust that will be used to build the planets. Heated by its young star, the protoplanetary disc is much hotter near its centre than far from the Sun’s rays. This temperature gradient determines which elements are able to condense into solids. In the same way that water becomes ice at 0°C (32°F), other molecules change from gas to solid dust particles at lower or higher temperatures. Closer to the Sun than even Mercury’s orbit, the temperature exceeds 2,000°C (3,600°F) and evaporates all solids to form a region free of dust. As we step outwards, the temperature drops to 1,500°C (2,700°F), and the first dust particles form from metals such as iron, nickel and aluminium. At the Earth’s orbit of 1au, silicates join in the mix, and when the temperature drops below freezing, ices appear. The first ice to solidify is pure water, made from hydrogen and oxygen. As the temperature cools still further, other hydrogen-based ices form, including solid methane and ammonia. These ices comprise much more common elements than the metals of the inner disc, leading to a burst of extra material where they can solidify. The point where ices appear is commonly referred to as the ice line, frost line or snow line, and it separates the terrestrial planets such as Earth and Mars from the gas giants like Jupiter. What is more, it helps explain their key differences.
Assembling from dust grains in the disc, each planet will consist of the solids that surrounded it as it formed. In the case of Mercury, this leads to a body predominantly made from iron. 1 After adjusting for its small size, which makes gravity squeeze it less than the Earth, Mercury’s heavy material gives it a density that is the highest in the Solar System. As more molecules join in the mix of available dust particles, the density of the planets further from the Sun drops slightly but they remain similarly rocky. However, as we hit the ice line, the disc is swamped by low-density ices. With this extra boon of material, bigger objects can form that will one day become the cores of giant planets.
However, while this fits into the picture of a planet assembled from its local dust grains, it doesn’t explain how the connection process actually works.
The glue stick
Suspended in the gas, dust grains are more easily led astray than a child in a sweet shop. This is actually a good thing for planet formation, since if the dust remained on militaristic circular orbits then collisions would happen rarely and large objects would never form. It is fortunate for us that the dust has a wild side that bumps grains off their circular orbit and into the path of other grains.
This type of deviant motion was first observed in 1827 by a botanist named Robert Brown, who was studying pollen grains suspended in water. Brown noticed that the grains appeared to be moving randomly, but he could not decipher what was causing this motion. It took until the turn of the following century for the problem to be untangled by Albert Einstein, who recognised that the water molecules were bumping about the pollen. Einstein might have won a Nobel Prize for this discovery since it confirmed the existence of atoms and molecules, but he had already scooped one five years earlier for an entirely different project. The award instead went to the French physicist Jean Baptiste Perrin, in 1926, who experimentally confirmed Einstein’s explanation. While Robert Brown’s observations were insufficient for him to get an award, the effect was named after him and became known as Brownian motion.
In the protoplanetary disc, the gas plays the part of the water molecules that buffet about the small dust grains. In addition to the Brownian motion, dust grains are moved into collision courses by the gas’s own slightly non-circular motion, which is encouraged by the magnetic field threading through the disc. Small pockets of slightly higher-density gas can also give a weak gravitational tug on the easily influenced tiny grains.
At the absolute beginning of the planet-assembly process, the sticking force between two colliding grains is less of a mystery. The dust grains that have condensed in the protoplanetary disc are less than a tenth of the size of sand grains, at only micrometres in size. Moving at speeds below 1m/s, these grains can be loosely held together by the electric charge of their atoms.
A dust grain consists of molecules such as ice or silicate, which are neutral with no overall positive or negative electric charge. Each of these molecules is made up of two or more atoms, which contain a central positively charged nucleus surrounded by equally negatively charged electrons. However, the electrons are not stationary. Instead, they scoot around the molecule and cause whichever side they briefly cluster at to gain a slight negative charge, while on the opposite side the molecule becomes positive. The charge from the negative end of the molecule can attract the positive end of a neighbouring molecule, holding the two together. This force from the slight asymmetry in electric charge is know as the van der Waals force, named after the Dutch scientist Johannes Diderik van der Waals. The force itself is actually fairly weak and can only work while the collision between dust grains is very gentle. When that is not the case, we begin (metaphorically and literally) to hit problems.
At micrometre sizes, the initial dust grains’ random motion is slow enough that the van der Waals forces are sufficient to stick colliding dust together. The problem is that as the dust particles grow, so does their collision speed. Once the micrometre-sized dust grains reach the princely size of a millimetre, the van der Waals forces fail to provide a sufficient amount of stick. Instead, grains hitting one another bounce.
When two dust grains bounce, neither increases in size. The grains therefore increase from micrometre to millimetre sizes and then get stuck, forming a sea of millimetre particles.
This is a disappointing dead end to the planet-formation process unless by some chance a few dust grains make it to the centimetre scale. Experiments conducted in the laboratory have shown that when the two particles colliding have sufficiently different sizes, the smaller grain will rebound but leave up to half its mass behind. This is akin to throwing a jelly at your brother. A large amount of the jelly may fall to the floor, but there will still be a satisfying amount sticking to his face. Centimetre-sized dust grains could therefore collide with the pool of millimetre dust, gaining mass with each interaction.
While this is promising, it does leave unresolved the question of how to first get centimetre-sized dust grains. In fact, there are two methods that can be used to bypass the bouncing barrier. The first is dumb luck. While the average collision speed between the dust grains increases with their size, there is still a range of values. This makes it possible for a handful of collisions to have velocities sufficiently low that the van der Waals forces can build a centimetre-dust grain. The second option is that the bouncing barrier may be much less of a problem if you are fluffy.
Imagine throwing a rubber ball at a wall. If your aim is any good, it will bounce straight off the wall and smack you on the nose. Now imagine that the wall is replaced by a giant ball of dust and fluff, of the sort that lurks under sofas. The ball you throw will penetrate such a dust ball rather than rebounding. If the dust ball is big enough, your ball will remain stuck inside its fluffy interior and become part of its structure.
Protoplanetary dust grains may not consist of dust mixed with cat hair and fluff, but in the low gravity of space they can be fluffy. This is especially true for those made of lighter elements such as ice. Collisions between such fluffy particles are difficult to perform in the laboratory, since Earth’s gravity will compress the grains. To get around this, the collisions can be performed virtually using computer simulations. This modelled reality showed that collisions between micrometre icy grains could stick, rather than bounce, for speeds of up to 60km/s (38mi/s). If the grains were still fluffy but made from silicates (as is more likely around the point where Earth forms), then sticking would still be effective up to 6km/s (3.8mi/s).
This seems like the solution to all our planet-formation problems. The micrometre slow-moving dust grains stick together via the weak van der Waals electric forces to form millimetre grains. The fluffiest of these aggregates can then stick together to form centimetre sizes, whereupon both fluffy and compact grains can gain mass during collisions with smaller grains. If this goes on for a few million years, Itokawa-sized objects can be formed that hold themselves together by gravity.
This would be the perfect solution if it were not for the gas disc.
Travelling in their orbit around the young Sun, the gas and solid particles feel different forces. For the smallest dust grains below a centimetre in size, this difference does not matter. Tiny grains are suspended in the gas, which carries them along like a baby in a sling, synchronising their velocities. As the dust grains grow in size to larger solids, they become more like hand-held toddlers. They are still orbiting the star, but their motion is no longer tightly linked to the surrounding gas. This is a problem because grains are solid while the gas is a fluid, and a fluid feels pressure.
In the absence of the gas disc, solid objects feel the tug from the Sun’s gravity and the reverse supporting force from their own rotation. Their resulting motion is said to be Keplerian, a term named after Johannes Kepler, who described this orbit in his laws of planetary motion. The gas, meanwhile, feels these two forces and an additional pressure force. The pressure originates because the accretion of material on to the Sun makes the disc denser near the centre. While the solids are not affected, this gradient produces an extra outward force on the gas that makes it flow about 0.5 per cent slower than the Keplerian speed. The result is that the solids feel a headwind in the same way a cyclist does, with the slower gas pushing against their motion. And like a cyclist in a strong wind, the solids start to lose speed.
As the solids’ speed drops, their rotation can no longer support them against the gravitational pull of the Sun and they begin to spiral inwards. This happens fastest for dust grains that have grown near to a metre in size, and can cause these boulders to hit the star within a few hundred years when starting from the Earth’s position. The only way to completely avoid this is to get bigger.
Anyone who has to clutch their stomach while taking a flight in a propeller plane knows that small aircraft are much more susceptible to air turbulence than a large commercial Boeing 747. This is because the drag from the surrounding air currents has a much bigger effect when your mass is small compared to the size of your surface. Likewise, once the dust agglomerates into kilometre-sized objects, it is no longer greatly bothered by the drag from the gas headwind. Unfortunately, the 100 years it would take for a metre-sized boulder to crash and burn into the Sun is a great deal shorter than the time needed for it to grow by collisions into a drag-oblivious, kilometre-sized rock. This problem is referred to as the metre-size barrier, and to stop it removing all our planet-building work, we need a way to pause this infall.
When competing in a bicycle race, a team of cyclists groups together to form a peloton, in order to mitigate against the exhausting force of the wind drag. An athlete cycling alone must battle against the wind while biking along the route, but when behind other cyclists, he is shielded and expends much less energy. Cyclists in a peloton take turns battling at the front of the formation, often allowing their champion cyclist to conserve strength for the final stretch near the rear of the formation.
A protoplanetary version of the cycling peloton is behind an idea called the streaming instability. The concept is that the solid boulders’ doomed walk towards the Sun would be paused if the gas drag could be switched off. Somewhat like in the cycling peloton analogy, this can be achieved by having enough solids in one place.
The spiralling path of the boulders as they move inwards through the disc is inevitably not going to be a homogenous affair. Knocked about by the gas, points along the route end up with higher concentrations of rocks. These clusters act like a peloton, reducing the headwind from the gas within that region. As new boulders are dragged inwards from further out in the disc, they hit the peloton and also slow as the gas drag is reduced. This adds to the peloton members and further reduces the headwind. The result is a runaway process, as the larger peloton is able to more easily collect incoming boulders.
Computer models of the streaming instability suggest that this protoplanetary peloton could gather together solids amounting to a total size of a few tens to a few hundred kilometres, comparable to the size of the dwarf planet Ceres. At this point, complicated gluing finally becomes unnecessary. Gathered together in the protoplanetary peloton is enough material to enable gravity to act and pull the rocks together in kilometre-scale objects. These solids now have a sufficiently respectable size that we can call them planetesimals.
In constructing the record-breaking tower, school students in Delaware went from bricks around 1cm (⅖in) in length to a tower 1,000 times larger. It was undeniably impressive, but this change in scale was outdone by the Solar System. In building a planetesimal out of dust, the protoplanetary disc assembled creations 1,000,000,000 times larger than its starting pieces. What is more, it is not yet done. We have now let gravity loose.
Gravity: the power tool
Getting gravity involved in the planet-building process is the equivalent of switching a pot of weak, school-safe glue for power tools. Planetesimals are now jostled in their orbits by the gravitational tug of the neighbouring rocks, which causes their paths to cross and the planetesimals to collide. While smaller bodies may still shatter or rebound from these impacts, their speed is not great enough to escape the gravitational pull of the largest planetesimals and they are pulled back in. The largest objects begin to eat everything in their path.
How fast a planetesimal grows hinges on the number of rocks it hits and adds to its mass. In the same way that a snowplough is more effective with a large shovel than a small one, a planetesimal will gather more material if it is bigger. As smaller planetesimals merge into larger bodies and increase in size, this rocky sweep-up becomes increasingly efficient, until the density of the small objects begins to drop. It sounds delightfully successful, but as it stands, it is not fast enough.
At the Earth’s position of 1au, it would take 20 million years for a planetesimal to snowplough into enough rocks to become our planet. If we take into account the sweep-up becoming less productive as the number of surrounding rocks thins, this time increases still further to 100 million years. As we move further from the Sun, the planetesimals become more spread out and their density drops. By Jupiter’s distance, 100 million years becomes the minimum time in which the giant planet’s solid central core can be formed. This is longer than the life of the gas disc, which must persist until the core has formed in order to source Jupiter’s massive atmosphere. Once we reach Neptune, the planet core would need longer than the life of the Solar System to gather its mass. This means that we need to give this growth process a speed kick.
Fortunately, gravity’s sticking power does not begin at an object’s surface. While gravity’s attractive force weakens away from the planetesimal, it can still drag nearby objects into a collision course. This results in an enhanced effective size for the planetesimal, equal to its geometric size plus a booster factor from gravity’s reach. This boost is proportional to the planetesimal’s mass, and therefore increases along with the geometric area as the planetesimal gets larger. The process becomes so efficient that the speed at which the planetesimal sweeps up new material gets faster with its size, leading to an ever-increasing growth rate. In this runaway phase, the biggest planetesimals rapidly accrete their surrounding neighbours. It is the planet-building version of ‘the rich get richer’.
A planetesimal in runaway growth mode might continue to expand until it has eaten the entire disc, if it was not for the central star. A nearby small planetesimal passing close to a larger body will feel two forces: the gravitational pull of its neighbouring massive planetesimal and also that of the star that it is orbiting. The point where these two forces balance is known as the Hill radius of the massive planetesimal. Inside the sphere of this radius, the planetesimal’s gravity is a stronger force than the pull from the star.
Since even a planetesimal in runaway growth mode will be vastly smaller than the star, the Hill radius is close to the planetesimal compared with the star’s distance, although potentially many times the rock’s size. Anything within the Hill radius will be pulled on to a collision course with the runaway planetesimal, but objects outside will still feel a tug. In fact, a planetesimal cannot keep to a safely stable orbit unless it is more than about three-and-a-half times the Hill’s radius from its neighbour. Once jostled from its orbit, the planetesimal’s path can cross the Hill radius and be devoured. A growing planetesimal can therefore feed from a path around 7 Hill radii in width as it orbits the star.
As the planetesimal grows, its Hill radius also expands to give a larger feeding zone, within which smaller planetesimals can be attracted. When the planetesimal and its Hill radius are small, the objects it accretes are those on close orbits. However, as the planetesimal goes through runaway growth, its increased Hill radius allows it to pull in bodies from a much wider area of the disc. These objects are initially moving at significantly different speeds from the main planetesimal, and are hauled off their orbits by its gravitational pull. Due to the strength of this force, these smaller planetesimals hurtle towards the central attractor at much higher speeds. This allows them to resist being channelled into a direct collision and instead they loop around the main planetesimal in chaotic orbits. This is much less efficient than everything neatly colliding. As a result, the runaway growth slows and a new phase of oligarchic growth (from the Greek, ‘rule of the few’) ensues.
Figure 6 Feeding zone. A growing planetesimal can feed on smaller planetesimals within roughly 3.5 x Hill radius as it orbits. The Hill radius is the region where the planetesimal’s gravity dominates over that of the star.
Oligarchic growth still allows the largest planetesimals to grow, but slower than their slightly smaller neighbours in the runaway growth phase. This results in a catch-up process, where smaller objects grow faster than the most massive ones.
As the number of small objects decreases, the fresh food supply entering the planetesimal’s expanding Hill radius dries up and the growth eventually stops. At this stage the planetesimal reaches a maximum mass known as the isolation mass, whereby it has eaten all the other objects in its orbital path. With a path width of around 7 Hill radii, the isolation mass is about 10 per cent of an Earth mass for an object at our 1au position, based on the estimate of the available mass from our Solar System’s MMSN. Near Jupiter, this increases to 1 Earth mass, since the extra distance from the Sun weakens our star’s gravitational pull and allows for a larger Hill radius. An Earth-sized core is not big enough to reel in a large gas atmosphere, which has led to speculation that the MMSN underestimates the mass in our giant planet region. This is not unreasonable, since the huge gravitational pulls of the big planets are capable of accelerating planetesimals so fast that they shoot out of the Solar System entirely. This is much harder near the Earth, where the Sun’s gravitational grip can keep rocks making a dramatic exit. If a young solar system did have more mass around the giant planets, then a typical core mass could reach around 10 Earth masses; the amount needed to start pulling in a massive atmosphere.
Out near Pluto at 40au, the pull from the Sun is so weak that the Hill radius becomes huge, giving an isolation mass of approximately 5 Earth masses. This is much larger than Pluto, which weighs in at only 0.2 per cent of the Earth’s mass. Such an inconsistency suggests that the time needed for Pluto to clear its orbit is still longer than the age of the Solar System. While even today it is true that Pluto remains embedded in a sea of smaller objects (a fact that led to it being designated a dwarf planet in 2006), the comparison with its isolation mass is not entirely fair, since these distant objects probably did not form in their current positions.
Our planet-forming planetesimals are now dubbed planetary embryos, and somewhere between 30 and 50 of them would have sat between the orbits of Mercury and Mars. While initially forming on different orbits, the paths of the embryos do not remain separate. Collisions occur both with one another and with fresh planetesimals scattered in from the outer Solar System. In a violent, gladiator-style end game, the collection of embryos merges to leave just four terrestrial worlds.
To trigger such giant impacts, a massive gravitational bully is needed to scatter the path of embryos and planetesimals – and for that, the planetary embryos beyond the ice line need to become the gas giants.
Notes
1. Mercury turns out to be even more dominated by iron than expected from its hot location. The planet was probably involved in a collision that stripped away part of its non-iron crust, to leave it even more iron dominated than before. However, even this does not entirely explain its composition, which remains an open question.