Did you make it yourself?

We’ve known for a long time the broad details of how the Solar System formed. The Nebular Hypothesis was first proposed back in the 18^th century and has been refined over time with the aid of detailed observations of areas where other star systems are in the process of being born into its current general state. The theory goes that you start with what is called a molecular cloud – a big clump of gas that is mostly hydrogen. At first the matter in this cloud will be very diffuse and spread out, but since it isn’t exactly evenly distributed the cloud will have areas where there is a slightly higher concentration of gas molecules than the average. This is enough to get the gravitational ball rolling: those small concentrations of gas molecules are heavy enough to attract and absorb other gas molecules, which makes them heavier, which lets them attract yet more molecules etc. etc. It’s a runaway process that eventually draws most of the surrounding gas cloud in towards a common centre of mass. While this is happening the Brownian motion of the gas molecules in the cloud will average out so that the cloud starts to rotate in the direction of its net angular momentum¹, causing the outer regions of the cloud to flatten into a disc shape. The centre of the cloud collapses to the point where the pressure and temperature are enough to kickstart a nuclear fusion process, creating a star. This leaves a disc-shaped remnant of gases that is rotating around the star in the direction of the original cloud’s rotation, and this is where we eventually get the planets from.

This protoplanetary disc then goes on to form the planets in much the same way as the molecular cloud formed the star. This time the starting point is a dust grain within the disk. This dust grain collides with another dust grain in the disk, and the two stick together. Repeat this process a couple of million times and you now have a clump of matter several hundred metres on a side hurtling around the proto-star. Smash many of these clumps together and eventually a planetesimal will be created; this is a chunk of stuff about 1 km in diameter. The planetesimal stage is the point where the process becomes runaway since planetesimals are large enough to attract one another through their own self-gravity – they collide, merge and the resulting boost in mass means they attract even more planetesimals.

Eventually you end up with a load of planets that have cleared all the matter immediately surrounding their orbits (hence the inclusion of this condition as part of the IAU’s definition of a planet in 2006). Any gas remaining in the protoplanetary disc either falls onto the Sun or else is blown away by the solar wind. So far, so good. But this is where the Nebular Hypothesis runs into a few problems; namely, that the Solar System structure predicted by this model of accretion from a protoplanetary disc does not match what we see in the Solar System today. There should be a lot more planetesimals, the gas giants should be located closer to the Sun than they are today (this goes back to the “hot Jupiters” we often find orbiting other stars), and there should be a fairly dense cloud of leftover material outside the orbit of Neptune. Since none of these things are true, something obviously happened in the four billion years separating the formation of the planets and the present day to account for the difference in what we should see and what we do see.

Current best candidate for that something is the Nice Model. It explains the following:

• Where all the planetesimals went.
• Why the gas giants are where they are.
• Why the Kuiper Belt, Scattered Disc and Oort Cloud are structured the way they are.
• The Late Heavy Bombardment, an approximately 300 million year-long period just after the formation of the Solar System during which all the inner planets were subjected to a much, much higher rate of asteroid and comet impacts than they are today (as inferred from the very large number of impact craters on the surface of the Moon which date from around this time).

As the Nice Model would have it, none of these things makes sense on its own but when you put them all together you can in fact come up with a model of the evolution of the Solar System that works.

The starting point is the one suggested by the Nebular Hypothesis: all the planets clustered in a narrow 15 AU radius from the Sun, with a very dense disc of icy detritus containing many planetesimals that didn’t quite make it as planets surrounding them from 15 up to about 30 AU. One of the planetesimals gets nudged inwards and approaches the outermost gas giant. The gas giant gives it a gravitational velocity kick by exchanging angular momentum with it; conservation laws dictate that this leads to an equivalent loss of angular momentum – and thus orbital speed – in the gas giant, shifting it to an orbit further away from the Sun.

The planetesimal is tossed from gas giant to gas giant like a gigantic mystery parcel, getting a gravity boost that shifts the orbit of each gas giant outwards. Then it encounters Jupiter, fattest of all the planets. Jupiter has no time for tiny planetesimals and gives it such a smack that the planetesimal is ejected from the Solar System altogether; this moves the orbit of Jupiter inwards rather than outwards. One single planetesimal doing this will have next to no effect on the orbits of the gas giants – they’re sodding huge, after all – but if the process is repeated several thousand times as new planetesimals are leached from the outer debris disk, it will all add up and start to produce a significant effect. Over millions of years the orbits of the outer three gas giants will slowly migrate outwards, while Jupiter migrates inwards. Eventually they get to the point where Jupiter reaches a 1:2 orbital resonance with Saturn, at which point all hell breaks loose.

The greatly-increased gravitational effect of the resonance of the two largest planets in the Solar System throws everything else into chaos. The video above illustrates quite nicely the slow migration and subsequent catastrophic interactions of the planets. Saturn gets shunted out into a wider orbit, which in turn moves Uranus and Neptune outwards³. This sends Neptune careening into that huge, densely populated disc of icy planetesimals that had previously been outside its orbit, scattering them in every direction like ninepins. Many planetesimals are scattered inwards towards the terrestrial planets, turning that region of space into a shooting gallery for the next couple of hundred million years: this is how the Nice Model explains the Late Heavy Bombardment. Others are scattered outwards at all inclinations and eccentricities, explaining the Scattered Disc and the Oort Cloud. Finally, the stuff at the innermost edge of the debris disc – the stuff closest to Neptune – is booted out of the Solar System entirely unless it is fortunate enough to fall into one of the narrow orbital bands defined by stabilising resonances with Neptune, as with Pluto and the rest of the classical Kuiper belt population.

That’s the Nice Model. I think it’s a very convincing piece of work, but it’s worth bearing in mind that the only criteria for the correctness of the Nice Model we have is that it produces something close to what we see today. It’s entirely possible that it happened entirely differently — we’ll never know for sure exactly how without a time machine — and even the Nice Model has several significant question marks hanging over it, such as the inability to explain the two distinct population types found in the Kuiper Belt and the fact that the Late Heavy Bombardment might not have even happened; the evidence for it consists of an extremely limited sampling of a few lunar impact sites and it’s possible that their common origin is just a rather large coincidence.

 As far as meshing with the currently prevailing scientific theories on the history of the Solar System goes, though, the Nice Model is the best we’ve got. Next week: Oort? Kuiper? What hell they? I attempt to explain.

1. Don’t worry if you don’t understand the specifics of this; you’re in good company. As far as I can make out exerting a unidirectional gravitational force on an object that is already undergoing Brownian motion that has a component lateral to that force will create a torque force on the molecule, which results in the gas molecules in the cloud acquiring a rotational motion about the axis of the cloud’s centre of mass. That makes sense to me, and past there I guess since random Brownian motion of all the gas molecules in the cloud won’t average out exactly to zero (it would be pretty goddamn amazing if it did) the sum of the angular momentum of the cloud will also not be zero. Hence the cloud as a whole rotates one way or the other, and this rotation will get faster as the cloud gets smaller due to conservation of angular momentum (same principle as an ice skater pulling their arms in closer to their torso to spin faster) which causes the cloud to flatten into a disc thanks to centrifugal force². This could be complete rubbish, but hey – I’m just a poor Solar System scientist. Can’t expect me to know everything.

2. One of the things they try to drill into you at Physics School is that centrifugal force isn’t a real force. It’s an approximation that only “exists” to make calculations within rotating reference frames (i.e. the surface of the Earth) easier. So whenever somebody wrote down the word “centrifugal” on their work it’d come back with red pen all over it and eventually the practice was stamped out, but I secretly keep the flame alive in my heart.

3. It’s suspected that Neptune may actually have formed as the seventh planet from the Sun, with Uranus as the outermost planet, and that when Saturn boosted Neptune and Uranus outwards they switched places. The model seems to work just as well either way.

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How do atmospheres work? As it turns out, the same way as everything else; that is, the gas molecules making up a planet’s atmosphere are bound to it by the planet’s gravity. However, there are some key differences. Inside the atmosphere all the different gas molecules are constantly moving around, colliding with each other, exchanging energy and then moving off in different directions. This means there are a couple of special rules that apply thanks to the difference in behaviour between a gas molecule and the unwieldy mass of squishy cells and organs that makes up the average human being. The general behaviour of a gas is described by the equation

where P is the pressure of its gas, V is its volume, N is the number of molecules of gas enclosed in that volume, k is something called the Boltzmann constant and T is the temperature of the gas in Kelvin.

The Boltzmann constant is necessary because temperature is not a property that can be applied to an individual gas molecule. There is no such thing as a gas molecule with a temperature of 300K; instead, the molecule will be moving at such-and-such a speed which gives it a certain kinetic energy, which will be different from the speed and energy of an adjacent molecule in the same volume. This speed is constantly changing as the molecule collides with other molecules and loses or gains energy, making it impossible to get discrete speed/energy measurements for a single gas molecule. The only way we can deal with a gas in any meaningful sense is by measuring the average kinetic energy of all the molecules inside it; this is expressed as its temperature. So in order to convert temperature — a measure designed to describe the general behaviour of a whole bunch of gas molecules — to the energy of a single molecule of the gas, we need to chuck in the Boltzmann constant k. The general idea is that the quantity kT will be on the order of (that is, somewhere close to, but probably not the same as) the energy of a given gas molecule in the volume V.

Understanding the difference between temperature and the kinetic energy of a single molecule is important for understanding how atmospheres work. All the gas molecules in a planet’s atmosphere are whizzing around at completely different speeds and kinetic energies even though the atmosphere itself has a certain temperature T. Most of the molecule velocities will cluster around the molecule speed described by that temperature T, but there will be many, many outliers which travel slower or faster. The range of molecule speeds and how they change with temperature is described by something called the Maxwell-Boltzmann distribution.

That graph may look a bit bewildering, but hopefully once I’ve explained what’s going on it won’t be all that complicated. The number on the vertical (or y) axis is the probability function, while the number on the horizontal x-axis is the speed of the molecule. What the graph is describing is what the probability is of a certain molecule in a gas travelling at a certain speed given a temperature a. If a single molecule has a 0.6 (or 60%) chance of travelling at a certain speed, then it follows that 60% of all the molecules in the gas will be travelling at that speed. The three different coloured plots on the graph show how this probability distribution changes at three different relative temperatures, a = 1, 2 and 5.

So from the graph we can see that for a low temperature of 1, all the gas molecules will be tightly clustered around a speed of 1-2, with none of them exceeding a speed of five³. Increasing the temperature to 2 changes the shape of the distribution; the average speed of a gas molecule is now 3-4 but the range of speeds at which the gas molecules as a whole travel is now much larger, as shown by the wider base and shallow peak of the distribution. Finally, for a high temperature of 5 the speeds of the gas molecules are much more evenly distributed, with a small peak at about 7 but with a long tail that stretches off all the way up to 15.

This long tail is the thing that interests us. Even if the temperature of an atmosphere is low, and the majority of the gas molecules in it are travelling at low speeds, some of the molecules will be travelling faster. A few of them will be travelling much faster – and if they’re travelling fast enough, they’ll reach what is called the planet’s escape velocity.

The escape velocity is a measure of how fast something has to be going in order to escape the planet’s gravity well permanently. You could strap yourself into a rocket and blast yourself into space, but if your rocket wasn’t powerful enough to propel you up to escape velocity before it ran out of fuel you’d eventually plummet back down to Earth if you didn’t manage to get into a stable orbit. The escape velocity changes from planet to planet as every planet has a different mass and therefore a different level of gravity, and furthermore since the gravitational force a planet exerts on an object diminishes the further away that object is from it, the escape velocity is greatest at the surface of a planet and diminishes as you progress upwards into space.

Now that we have all the background information down, we can finally start to look at why atmospheres are the way are. In reality atmospheres are not composed of a single homogenous gas but instead a whole variety of different elements. Even inside an atmosphere with a single uniform temperature the heavier gas molecules such as nitrogen and oxygen might have the same kinetic energy as a hydrogen molecule, but they’ll be travelling at lower speeds thanks to their greater mass. This causes the various gaseous elements inside an atmosphere to differentiate themselves according to weight; the heavier, slower stuff can’t get very far off the ground and is found close to the surface, while the lighter molecules can and slowly migrate to the higher portions of the atmosphere over time. If a molecule is light enough and the air around it is thin enough (so that it doesn’t get its direction of travel changed by a collision with another molecule) then the velocity of that molecule can easily exceed the planet’s escape velocity and escape into space – and this is true even if the temperature isn’t high enough for the average velocity of a gas molecule to exceed the escape velocity because the long tail of the Maxwell distribution ensures that some of the molecules will be moving fast enough, causing the planet to slowly (or not so slowly) bleed portions of its atmosphere away into space.

How likely a given molecule of gas is to remain part of the atmosphere can be calculated mathematically. My notes here don’t go into the specifics of how the Maxwell distribution and the interactions between molecules dictate this, but if a molecular constituent’s thermal velocity is near one-third the escape velocity, then about half of that molecule type will have escaped from the atmosphere within weeks. If the thermal velocity is one-fifth of the escape velocity, then the planet will lose half of that molecule type after a billion years. And if the thermal velocity is one-tenth of the escape velocity, then the planet will retain that molecular constituent indefinitely.

Therefore we run two parallel equations.

where G is the gravitational constant and k is the Boltzmann constant.

Everything in these equations that isn’t a number or a constant is a property that affects the likelihood of a planet losing a certain type of gas – the mass of the planet, the radius of the planet (since gravity diminishes the further away you go from the centre of mass), the temperature of the atmosphere and the mass of the gas molecule. If, at the end of the day, v_thermal is more than 0.1 v_escape, the planet will eventually lose that gas type. How much more v_thermal is is what dictates how quickly this happens.

From this the compositions of the atmospheres of the planets become much clearer. Earth has very little hydrogen or helium in its atmosphere because it’s not massive enough to hold on to them. Jupiter has no such problems (not to mention forming in an area where light materials were much more abundant) and so it’s still got most of its primordial hydrogen – this is why Jupiter and the other outer planets are referred to as gas giants. Mars is a relatively light planet and so much of its atmosphere has seeped away over time, but it still retains a fair amount of the heavier stuff such as carbon dioxide. Poor old Mercury gets hit with a double whammy: not only is it situated very close to the Sun — raising its surface temperature and thus its escape rate — but it’s also constantly being blasted by the solar wind which also serves to strip away atmosphere, as a result of which it doesn’t really have one any more.

So how is this all relevant to the terraforming argument? Well, if we run the equations above for an oxygen molecule in the Martian atmosphere we get a mean thermal velocity of 630 metres per second. Mars has an escape velocity of 5 kilometres per second. While Mars is losing oxygen molecules to thermal escape over time it’s doing it very slowly. This means that the fact that Mars has a crappy atmosphere in terms of potential human habitation has very little to do with how small it is, and is in fact heavily related to two other factors:

• Attrition by the solar wind; something we don’t have to worry about so much here on Earth thanks to the magnetosphere.
• Mars getting absolutely creamed by asteroid/comet impacts (along with every other terrestrial planet) during the Late Heavy Bombardment. Like, so much so that there’s still chunks of Mars dating from around about then floating around the Solar System which occasionally fall to Earth. This is because the impacts were so violent they threw up ejecta and debris from the surface so high and so fast that they reached escape velocity and were never seen again. From the point of view of Mars, anyway. If the Late Heavy Bombardment did that sort of thing to solid rock, imagine what it did to the atmosphere.

The first point is something that any prospective terraformers would still have to worry about, and given the lack of any magnetosphere on Mars the solar wind is likely to contribute far more to atmosphere loss than thermal escape. Barring some sort of cataclysmic event we don’t have to worry about the second ever happening again, at least over human timescales. As a result, while any Martian atmosphere we generate would dissipate relatively quickly, that “relatively” is relatively to the lifetime of the planets and the Solar System. In timescales relevant to humans even the most pessimistic estimates have a usable Martian atmosphere sticking around for 200,000 – 300,000 years, and it’s more likely that it’d last for a million plus. Are we really going to say terraforming the Martian surface isn’t worth it because the atmosphere will “only” last for 300,000 years?

1. It was a talk for kids aged 12-16 so I couldn’t say what I really thought of them.

2. The moral of this story is either that you should never use even mild language like “crazy lunatics” in case there are some crazy lunatics in your audience, or else that no matter what you do you should be prepared to go through life inadvertently offending an awful lot of people.

3. This graph uses dimensionless measures because the shape of it will be the same no matter what system of measurement you use to measure the temperature and speed of the gas.

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