Josh, 7, from London, writes:

Dear Hentzau,

            I went to the beach yesterday. The weather was nice. I had ice cream. When I was splashing around in the sea with my rubber ring wedged firmly around my waist to stop me sinking, I thought about the planets. Planets have rings, but they do not have to be prevented from sinking. Why do planet rings always go around the middle?

(And they do always go around the middle: see Uranus, which is tilted almost ninety degrees on its side but which has rings which still orbit around the equator.)

Well, little Josh, it’s because of three factors which shall be tackled in numerical order.

1)      Planetary formation from a protoplanetary disc.

2)      The Roche limit.

3)      Planets are fat.

When a planet forms, it does so out of a big cloud of stuff. Much of this stuff is fairly hefty planetesimals (stuff one kilometre plus on a side) but there is lots of smaller stuff as well. The planetesimals along with the smaller stuff collapse towards the centre of the cloud towards the proto-planet core. As we’ve seen in the post on the Nice model, when you have a big cloud of stuff that’s shrinking it will start to rotate. Rotation produces centrifugal force, and centrifugal force will flatten the cloud out into a disc. Eventually you end up with the proto-planet at the middle surrounded by a big disc of stuff. What happens to the leftover stuff is determined by how close it is to the proto-planet.

• Stuff close to planet: falls onto planet.
• Stuff far away from planet: forms moons and stuff.
• Stuff in the middle: forms rings.

Now, clearly there are some niggling factors which mean it isn’t quite as simple as that, chief of which is that only the gas giants have rings (yeah some people think Rhea might also have rings but honestly screw those people). Earth, Mars and the other terrestrials don’t. This is because the size of the “stuff in the middle” category is dictated by something called the Roche limit.

The Roche limit is, broadly speaking, the orbital radius inside which the gravitational tidal force of the planet is greater than the gravitational force keeping whatever object has been unwise enough to venture inside it in one piece. In other words, if you go inside the Roche limit and gravity is the major force holding you together¹ – so I don’t recommend this at all if you happen to be a planet or a moon — you get pulled apart. I like the Roche limit because it’s one of the few physical terms that I actually understand the derivation of, but I won’t go into that here. It’s expressed as an approximation by:

where R is the radius of the primary (i.e. the planet), ρ_M is the density of the primary, ρ_m is the density of the thing orbiting it, and d is the Roche limit.

So the Roche limit isn’t strictly a hard physical constant; it’s going to vary from body to body depending on the ratio of the densities of the primary and the orbiting body. However, because R is also in there, if you have a really, really big planet – like a gas giant, say – the Roche limit is going to be a factor no matter how dense your orbiting body is.

Using FABULOUS EXCEL TECHNOLOGY, I have created a graph which shows how the Roche limit varies with the density of the orbiting body for Saturn and Earth.

 
This makes it very easy to see that the Roche limit is big for less dense things and small for more dense things, and also that as you increase the density more and more it becomes practically static.

Now, to get the Roche limit for your proto-planet you have to put a number on the average density of protoplanetary material. I’m going to use the mass of a comet, which is about 500 kg m^-3. This gives Roche limits of:

• Saturn: 160,000 km.
• Earth: 35,000 km.

35,000 km is sod all, in orbital terms. It’s just slightly less than the altitude at which we orbit geostationary satellites; any protoplanetary material orbiting inside 35,000 km is overwhelmingly likely to fall onto the proto-Earth anyway, so that explains why the Earth has no rings. The Roche limit of 160,000 km for Saturn on the other hand matches very nicely the observed outer edge of Saturn’s ring system.

This is where the bulk of planetary ring material comes from, then; they’re formed from the leftovers of planet formation that are far enough away to avoid falling onto the planet outright, but too close to the planet to form themselves into moons. Since this disc was rotating around the “equator” of the proto-planet anyway thanks to the cloud collapse, that explains why the rings are found there. However, this doesn’t answer an important question: what about ring material that was captured after planet formation? Say a comet ventures inside the Roche limit on a highly inclined trajectory. Why isn’t it pulled apart into a ring orbiting the planet at that inclination?

Well, this is where we come to the third thing on our list: planets are fat. Literally. Because they’re rotating objects centrifugal force distorts their shape away from the perfect sphere of hydrostatic equilibrium, making them slightly fatter around the equator than they are around the poles. For example, Earth has a radius 6,378 km around the equator but only 6,356 km around the poles; those 22 km may not seem like much but they make a big difference as far as orbiting bodies are concerned, because it means things that aren’t orbiting around the equator will nevertheless be pulled in that direction thanks to the extra mass that’s present there. Over thousands of years they’ll slowly migrate to the equator, at which point they’ll probably collide with some of the other junk that’s orbiting there which robs them of their orbital momentum and stops them where they are. Planets may be a little bit flabby around the waist, but if you like to look at ring systems a little extra weight is no bad thing.

In conclusion:

• The majority of planetary ring material forms around the equator of the planet in a disc.
• The inner edge of the rings is created by the planet, which drags material that gets too close towards it.
• The outer edge of the rings is created by orbiting moons, which sweep up material outside of the ring area.
• The rings themselves are defined by the Roche limit, which dictates the nominal area where ring material cannot form itself into moons because the planet is gravitationally dominant.
• All material added to the rings after planet formation migrates towards the equator thanks to the planet not being a perfect sphere and having more mass – and therefore gravity – distributed around the equator.

BULLET POINTS.

1. You might think of the Earth as a solid object but it isn’t really. It’s held together by gravity; no material could possibly provide the same amount of bonding force to keep all of its mass in one place. This goes for most large bodies out there; they’re so big that the only thing that can keep them in one piece is gravity, so if they come across something with a larger gravitational force then it’s bad news for them.

The post The Rochefort Limit. appeared first on The Scientific Gamer.

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.

The post Nice Model. appeared first on The Scientific Gamer.