The Origin of the Solar System

0. In the Beginning... A Summary of Stellar Formation
Stars start out as interstellar clouds of gas and dust. If you were inside such a cloud, you probably couldn't tell that there was anything at all there, because the gases which make up the clouds are incredibly thin. Each cubic inch of these clouds contains only a few dozen to a few hundred atoms, while each cubic inch of our atmosphere contains almost a billion trillion atoms. If you had to expand a single cubic inch of our atmosphere until it was as thin as the gases in an interstellar cloud, it would be almost 200 miles on a side.
Although the clouds are incredibly rarefied, they are also incredibly big, stretching for trillions of miles in all directions. Because of their huge size, even though there is practically nothing at any given place within them, the huge extent of practically nothing adds up to substantial masses, hundreds of thousands of times greater than the mass of the Earth, like that of the Sun.
Because the material of the cloud is spread out over such a huge volume of space, the gravity caused by its mass is incredibly small, and under normal circumstances, it cannot force the cloud to contract to a smaller size. But under some circumstances, the clouds ARE forced to contract to smaller sizes. During such contractions, gravity gradually increases, and if its force becomes large enough, the thin gases within the cloud will not be able to exert enough outward pressure to prevent the gravitational pull from contracting the cloud to still smaller sizes, and so the cloud will continue to contract.
Although gravity is trying to make the cloud smaller, the pressure of the gases within the cloud is trying to stop the contraction. At first the pressure is negligible, since the gases are so incredibly thin, but as the cloud gets smaller, the gases become denser and hotter. The inward pull of gravity tries to make the gases move inward with greater and greater speeds, but random collisions between the atoms of the gas tend to convert this inward motion into a random sub-microscopic movement, which we perceive as heat. As the cloud contracts, greater and greater amounts of inward movement are converted into faster and faster microscopic motions, or greater and greater amounts of heat. The greater temperatures which result, combined with the greater density of the gas, create a continually increasing pressure, which fights against gravity.
As the heat generated by the contraction of the gases increases, pressure gradually rises, until it equals the gravitational forces, stopping the inward motion of the cloud. If this occurs while the cloud is still very large, and not very warm, no further contraction will occur. But if this balance does not occur until the cloud has contracted a long way, and has therefore generated a large amount of heat, some of the heat will be radiated away in the form of infrared light. The heat lost in this way reduces the ability of the gas to hold up against the pull of gravity, and causes a slow, semi-equilibrium contraction of the cloud. At each stage of the contraction, pressure and gravity are in balance, and if no more heat were radiated away, the contraction would stop, but the continual radiation of infrared light at the outside of the gradually warming cloud prevents this, and allows gravity to have a slow but steady victory over pressure.
Although the loss of heat at the outside of the cloud forces it to contract, there is a limit to how far the contraction can go. As the cloud continues to contract, temperatures within the cloud continue to rise. By the time that the cloud is as small as a star like the Sun, the central temperatures have risen to many millions of degrees, and a conversion of hydrogen to helium begins, in a process known as thermonuclear fusion. At first, this conversion is slow, and produces only a small amount of energy, but as the star continues to contract, the rate of nuclear fusion increases, producing more and more energy. The heat generated by this fusion helps replace the heat being lost at the outside of the star, slowing the rate of contraction. The closer the star gets to a stable size, the closer the core approaches an equilibrium temperature at which the nuclear reactions produce exactly as much heat as is being lost on the outside. When the star reaches that temperature, there is no longer any net loss of heat, and so the star's contraction finally ends.

1. The Formation of the Solar Nebula
A cloud which is trying to contract to become a star has several problems to overcome before it can do so. One of these is the outward pressure, discussed above. In this case, the solution to the problem is to heat the cloud to a high enough temperature so that some of the heat is being continually radiated away, reducing the amount of heat left over to create pressure.
Another problem is angular momentum. Even while the cloud is huge, it must have some tiny amount of rotational motion. The random motions which occur in different parts of the cloud will probably nearly cancel each other out, so that any overall motions are small, but it is not likely that they will exactly cancel, and so some tiny rotational motion is to be expected.
In many cases, the rotational motion of the cloud may be so large that it is impossible for the gravity of the cloud to overcome it, and the cloud remains as a large cool blob of gases, but the existence of so many stars in our Galaxy shows that there must be various ways in which stars can overcome this problem. One way is for the cloud to break up into two or more blobs, revolving around each other--a binary or multiple star system. Such systems are in fact quite common. Between one-third and one-half of all the stellar systems in our Galaxy are thought to consist of such multiple stars.
Since we do not know of any companions to the Sun, it seems that our Solar System solved the problem in a different way. Presumably the amount of rotation in the cloud was fairly small, and as a result, those parts of the cloud which happened to have a smaller amount of rotation could fall inwards more-or-less uniformly, forming a large, roughly spherical ball near the center, which became the Sun, while those parts of the cloud which were rotating the fastest formed a flattened circular disk rotating around the central ball, the Solar Nebula. As the cloud contracted, parts of it which were moving parallel to the axis of rotation (getting closer to the plane of rotation) would not have had their inward motion affected by the rotation, but parts which were moving in the plane of rotation (getting closer to the axis of rotation) would have gradually increased their rotational speeds, just as ice skaters spin faster by pulling their arms towards their bodies. The same thing can be seen in the motion of the planets around the Sun; Kepler's Law of Areas is mathematically equivalent to the Law of Conservation of Angular Momentum which determines how rotating objects speed up as they get closer to their axis of rotation.

Below, an artist's impression of the formation of a circumstellar disc around a star. Rotation of material falling toward the star causes it to form a circular disc surrounding the star (in the case of our Sun, this is referred to as the Solar Nebula). As material falls toward the star, it is heated and compressed. For smaller stars, most of the material may become part of the star; but for hot, bright massive stars (such as represented by this image) much of the infalling material is ejected into space, at very high velocity. Because of the density of the material in the circumstellar disc, the ejected material is forced to follow a path perpendicular to the plane of the disc, forming two "polar" jets.

Diagram showing probable temperatures in the early solar system

Temperatures in the early solar system (shown in red) must have been much hotter inside the orbit of Jupiter, than now (shown in black). Midway between Mars and Jupiter (vertical line), current temperatures are 200 degrees below zero Fahrenheit, but during the formation of the planetesimals, temperatures in that region must have been around 600 degrees above zero Fahrenheit.

In the Sun and other stars, and in the interstellar medium from which they formed, 90% of the atoms are hydrogen atoms, and almost all of the remaining 10% are helium atoms. Under the conditions in the Solar Nebula, these materials could only exist as gases. The remaining 1/10th of 1% of the atoms are mostly carbon, nitrogen, and oxygen. Their compounds with each other, and with hydrogen, can only be solid at low temperatures, and therefore could not exist as solids in the inner solar system. Only metals, such as silicon, iron and titanium, which make up only one ten-thousandth of 1% of the Solar Nebula, can form oxides and other compounds which can withstand the high temperatures in the inner part of the solar system. Within each part of the Solar Nebula, whatever grains could survive the temperatures in that region would be building up into planetesimals. In the outer solar system the temperatures would be low enough so that almost all of the original interstellar dust materials could remain solid, and so in that region dirty snowballs would be forming, but in the inner solar system, only rocky objects could form. Because the metal oxides which the rocks are made of are so rare, the rocky planetesimals would have been relatively small at any stage of the planets' formation, but in the outer solar system, where not only rocky materials, but also carbon compounds and ices could exist as solids, the larger amount of solid materials should have allowed the planetesimals to grow much faster. As a result, late in the formation of the planets the inner solar system must have contained fairly small rocky bodies, similar to the current asteroids, while the outer solar system must have contained fairly large dirty snowballs, perhaps even as large as the satellites of the Jovian planets.

3. The Formation of the Jovian Planets and the End of the Solar Nebula
In the early part of their formation, the planetesimals would be too small to have any significant gravity. Even though 99.9% (or more) of the materials in any part of the Solar Nebula would be gases, the planetesimals could not hold on to any part of these gases. As they moved around among the gases, those might pile up temporarily around the planetesimals, but their gravities would be too weak to permanently hold onto the gases. This would be especially true in the inner solar system, where the smaller sizes of the rocky planetesimals would cause them to have smaller gravities, but even in the outer solar system, the dirty snowballs would be mostly too small to have any chance of gravitationally holding on to the gases through which they were moving.
As they became larger, however, the gravities of the planetesimals would increase, and in some cases, they might become large enough to hold on to the gases through which they were moving. As soon as they did so, they would begin to accumulate mass at a much faster rate, since there were far more gases than solid materials in any region. While they were too small to hold on to such gases, they could only accumulate the less than 1/10th of 1% of the material that they ran into that happened to be solid, but once they could hold on to gases, they could accumulate a much greater fraction of the material that they ran into, allowing them to grow as much as a thousand times faster than before. Within a short period time they would mushroom to much greater sizes and masses.
This is exactly what must have happened to the Jovian planets. In the last stages of their formation they must have become very large dirty snowballs, with gravities large enough to hold on to even hydrogen and helium. As they moved around the Sun, sweeping up everything in their paths, they grew at an enormous rate. This would be especially true for Jupiter and Saturn. Closer to the Sun, they would be running around in a denser part of the Solar Nebula than Uranus and Neptune, and at a faster velocity. So Jupiter and Saturn would accumulate much more gaseous material, ending up with a composition almost equal to that of the huge amounts of gases swept up in the last stages of their formation, while Uranus and Neptune would accumulate smaller amounts of gases, ending up with a composition intermediate between that of the gases and their original cores.
At the end of the Sun's formation, strong "winds" developed, probably driven by a very rapid rotation of the young star, carried out into the Solar Nebula by the Sun's magnetic field. Within a hundred thousand years or so, virtually all of the gases in the Nebula would be driven out of the solar system, pushed back into interstellar space. This cannot have happened before the cores of the Jovian planets became large enough to capture light gases, or they would still be just large dirty snowballs. Some lines of evidence seem to indicate that most of the gases of the Jovian planets accumulated in a very short period of time, possibly less than ten thousand years. Therefore, the Jovian planets' cores must have become large enough to accumulate gases just before the end of the dispersal of the Solar Nebula.
This also appears to be verified by the characteristics of the Earth and Venus. At the time that the cores of the Jovian planets were approaching the critical sizes required to accumulate gases, the cores of the Terrestrial planets must also have been approaching their final sizes, but although these planets are too small to hold on to the hydrogen and helium which made up most of the mass of even the inner Solar Nebula, their finished sizes are big enough to hold on to heavier gases, such as carbon dioxide and water vapor. These heavier gases are not nearly as abundant as the lighter gases, but they are still many times more abundant than the rocky materials which make up the Terrestrial planets. As a result, if the Earth and Venus had reached their current size while the Solar Nebula still existed, they would have become several times more massive.
As a result, it appears that the planetary system could have ended up quite differently than it did. The buildup of the planetesimals into the final sizes of the planets must have finished at almost the same time that the Solar Nebula was destroyed by the strong solar winds of the newly-formed Sun. If the planets had formed a little later, there would be no Jovian planets. If they had formed a little sooner, there would be one or two more Jovian planets (the Earth and Venus, or a combined single object).

4. Why Did the Terrestrial Planets End Up Different From the Jovian Planets?
In some discussions of the origin of the planets it is stated (or at least implied) that the Terrestrial planets formed in a region which was swept clear of gases by the young Sun, while the Jovian planets formed in a region which had lots of gases. The reasons for this statement are that if the Earth and Venus had reached their present sizes while they were still surrounded by gases, they would have ended up with large amounts of those gases, and if the Jovian planets had reached a core size large enough to accumulate those gases only after they were gone, they would still be just large dirty snowballs.
The trouble with this idea is that it is easy to take it too far, and to imagine that throughout most of the time that the planets were forming, there were significant differences between the different parts of the solar system. So you may easily get the incorrect impression that during the time that the Terrestrial planets were forming, the inner solar system had very little, if any gases, and that most of the gases were in the outer solar system.
Now there must indeed have been many differences between different parts of Solar Nebula. The gases (and other materials) must have been relatively dense close to the Sun, and much more rarefied far from it, with the outermost traces of reasonably dense gases not much more than 30 AUs from the Sun. The gases must have been moving around the Sun at relatively high speeds near the orbit of Mercury, and ten times lower speeds near the orbit of Pluto. And the temperatures must have been very high near the Sun, and very cold far from it, so that close to the Sun, only rocky materials could exist as solids, while far from the Sun carbon compounds and even ices could exist as solids, but although there were differences in the conditions at different distances from the Sun, we don't have to assume that there were significant compositional differences. There are no physical or chemical reasons why there should be more of any given type of material in one part of the cloud than in another part. Just because water ice couldn't exist near our orbit doesn't mean the materials which make up water didn't exist here. They would simply have had to exist as gases.

The Composition of the Solar Nebula

One common error in thinking about differences between different parts of the Solar System is that heavy things must have fallen towards the Sun, and lighter things floated away from it. This error is based on the fact that the inner planets are dense, rocky objects, while the ones further away are made of lighter ices and gases. This idea is reinforced by the fact that when the Earth was young, it must have melted and differentiated (heavy things sank to the middle, to form the dense metallic core, and lighter things floated to the top, to form the lighter mantle and crust). But in the case of the Earth, everything is just "sitting" here, and the effects of buoyancy and density can act on its materials with no complications. In the case of the Solar Nebula, everything is actually in orbit around the Sun, and for orbital motions, it doesn't make any difference what things are made of, because gravity acts the same on all orbiting objects, whatever their mass, size, density or composition. As a result, there is absolutely no reason for light objects to end up in one place, and dense objects to end up in another. (To realize that this must be the case, remember that the Sun, which is in the center of the Solar System, is made almost entirely of hydrogen and helium, which would not be the case if heavy things tended to fall towards the Sun more than light things.) As a result, the proportions of different materials in different places should be just the same as in the Sun. Only the percentages which are solid -- which depends upon temperature -- would vary from place to place, and the fact that the denser solid materials are closer to the Sun is not due to the fact that they are dense, but that they can remain solid at high temperatures. You can see how this works in this diagram:

Diagram of Composition of Solar Nebula
Amount of material rapidly increasing upwards,
Distance from forming Sun increasing to right

The relative proportions of various materials are the same at all distances from the Sun, as indicated by the parallel nature of the curves showing the amounts of different substances -- primarily hydrogen, with lesser amounts of helium, still lesser amounts of medium-weight atoms (carbon, oxygen, and nitrogen) and their compounds with each other and hydrogen, and still smaller amounts of heavier atoms, such as silicon and iron, and their oxides. However, the solid proportion of these materials, shown by the red curves, is extremely variable. Far from the Sun, where it is cold, there would be tens of Earth masses of icy materials. Closer in, the materials that make up these ices would still be common, but they are all vaporized by the higher temperatures. Rocky materials would exist almost everywhere, because they can withstand higher temperatures, but in the outer Solar System, the relative rarity of the materials which make up those rocky materials would make them relatively unimportant, in comparison to the much more common icy materials. So, far from the Sun, we would have large amounts of snow, and small amounts of rocks; and close to the Sun, small amounts of rock; and everywhere, large amounts of gas -- but the gas is of minor importance until, late in the game, the planetesimals get large enough to gravitationally attract it (as discussed below).

The most likely explanation of the problem with the lack of gases near the Terrestrial planets is that the planets probably formed at slightly different rat (see diagram below). In the inner solar system, where only rocky bodies could form, the materials out of which they were forming represented only a very tiny fraction of one percent of the total mass of the Nebula, whereas in the outer solar system, where icy materials could also exist as solids, the materials out of which the proto-Jovian planets were forming would have been tens or hundreds of times more abundant. Thus, at the stage where the Jovian planets were getting close to their final size, and just becoming capable of holding on to light gases, the Terrestrial planets were probably no more than half their final sizes, and not capable of holding on to even heavy gases. As stated on the previous page, even the Jovian planets must not have gotten large enough to hold on to hydrogen and helium until near the end of the dispersal of the Solar Nebula by the solar wind, so the Terrestrial planets, which wouldn't have gotten large enough to hold on to gases until a bit later, would simply have reached their final size too late to have any gases left around them.
In other words, although we probably have to assume that the Terrestrial planets finished their formation in a region nearly free of any gases, we don't have to assume that they started their formation in such a region. Out of the few million years that it probably took to build up planetary bodies from the originally microscopic dust grains, a few hundreds of thousands of years lag in the final rate of formation of the inner planets would be quite adequate to explain how they could have finished their formation after the dispersal of the Solar Nebula, while the outer planets, forming just a little earlier, could have finished their formation just before the end of the dispersal, and been able to hold on to whatever gases were still left in the outer solar system. (More to come on this topic at a later date.)

Rates of Growth in Different Parts of the Solar Nebula

Diagram Showing Planetary Sizes as a Function of Time
(Bya = Billion years ago)

One way to look at the growth of planets is to consider how rapidly things would grow in different parts of the Solar System. As shown in the diagram above, in the asteroid belt, where there was relatively little material, things grew slowly, and, because there wasn't enough material to allow an effective sweeping up of any materials left over after the initial formation, continued to grow very slowly, all the way to the present time. In the case of the Earth and the other Terrestrial planets, the much larger amounts of material present in the regions where they were forming allowed them to grow to larger sizes in relatively short periods of time, and to eventually become large enough to hold onto heavy gases. In the case of the Jovian planets, which formed in regions with large amounts of solid ices, the growth to large size was even faster, and they were able to reach a size large enough to hold onto gases at just about the same time that the Sun blew away (or finished blowing away) the remnants of the Solar Nebula. Since there were, even at that time, far more gases than solid materials, the ability to hold onto these gases allowed the Jovian planets to mushroom in size at a very rapid rate.
Current estimates are that the Jovian planets got big enough to hold onto gases only a few tens of thousands of years before the Sun finished blowing away the gas in the Solar Nebula; if they had gotten big enough to hold onto gases a few hundred thousand years earlier, Jupiter in particular might have ended up as a small star, instead of as a large planet. If they had gotten big enough to hold onto gases a few hundred thousand years later, there would have been very few gases left for them to pile onto their icy cores, and the largest of them might not have even as much gas as Neptune. A few million years later, when the Terrestrial planets reached sizes large enough to hold onto gases, those gases had long since been blown out of the Solar System, and so they unable to accumulate any gases (this of course leads to the question of how the Terrestrial planets ended up with the atmospheres they have, which is discussed in The Formation and Evolution of Planetary Atmospheres).

5. The Age of the Solar System and Its Early History
The Sun and planets must have been formed about 4.5 billion years ago. This date is determined by studying the characteristics of rocks which contain small amounts of radioactive substances. If the mineral grains which contain such materials have not been altered significantly since their formation, the decay products will be trapped in those minerals, but the decay products do not have the same chemistry as the original radioactive materials, and so they stick out like a sore thumb when detailed chemical and physical studies of the minerals are made. By comparing the fraction of radioactive materials which have already decayed to the total amount of such materials, and measuring the rate at which such materials decay in the laboratory, it is possible to determine the "age" of the rock. Of course, this is only an estimate in many cases, and if the rock has been altered in some significant way since the minerals were first formed, it may not be an accurate indication of how long ago that was, but if we look at many samples from various places, the overall results are almost certainly correct.
The age of the Solar System is determined by the study of Earth rocks, Moon rocks, and meteorites. The oldest rocks which we have discovered on the Earth only date back to 3.8-3.9 billion years ago. The Earth itself must be somewhat older than that, as these rocks are all sedimentary and metamorphic rocks, meaning that they were formed from the compression and alteration of sediments derived from the weathering and erosion of still older rocks, but no samples of those older rocks are known to still exist. It is therefore difficult to estimate the true age of the Earth from direct study of Earth rocks, but calculations based on the relative distribution of the decay products of radioactive materials in rocks of various ages seem to imply an age somewhere in the range of 4 to 5 billion years.
The rocks which the Apollo astronauts brought back from the Moon give us a slightly more accurate estimate of the actual age of the solar system. Most of these rocks are basaltic lavas from the lunar maria, and date only to 3.3 to 3.8 billion years ago, but some of them are heavily fractured granitic rocks which appear to have been blasted off the lunar highlands, and date to over 4.3 billion years ago. This implies that the Moon must have formed a little earlier than that, but again, just how much earlier could be difficult to estimate.
The best estimates of the age of the solar system seems to come from certain primitive meteorites, which appear not to have been significantly altered since they were formed. They exhibit a range of ages, but most of their ages cluster closely about a value of 4.5 billion years ago. Since this is in reasonable agreement with the best estimates that we can make from the Earth and the Moon, we believe that this is the true age of the Solar Nebula, the Sun, and all the other bodies in the solar system.
The total history of the accumulation of the planetesimals into planets and other solid bodies probably did not encompass more than a few million years, and in comparison to the 4500 million years or so back to the beginning of the solar system, represents only an instant. So we should probably consider all of the bodies which we now see as being of essentially the same age.
Towards the end of their formation, the planets must have undergone a period of melting. Certainly the differentiation of the Earth, with its heavy metallic core, and lighter rocky mantle, requires some such period of melting. The exact time that this occurred can only be estimated, but probably was very close to the initial formation of the planets, as even the Moon seems to have completed such a molten state at a very early date. Like the Earth, the Moon has a differentiated crust, with a low-density granitic "slag" forming the bulk of the highland surface of the Moon. This implies that the Moon must have melted, differentiated, and then begun to re-solidify before the date, 4.3 to 4.4 billion years ago, which we determine as the "age" of the rocks which were recovered from the Moon. This means that the period of melting must have been within the first 100 million years after the formation of the planetary bodies.
The heat required to produce this melting appears to have been caused by the decay of short-lived radioactive materials. These materials are created inside supernova explosions, and one or more such explosions must have occurred in the region near the interstellar cloud which became the solar system within the last few tens of millions of years prior to the formation of the solar system, in order for any significant amounts of such radioactive substances to have still existed at the time that the solar system formed. But this would not be surprising, as we believe that the Sun, like most stars, probably formed in a group or cluster of stars, and if any of those were much more massive than the Sun, they could easily have formed, lived out their lives, and died, all during the time that the cloud which became our solar system was hovering on the edge of contraction. In fact, some primitive meteorites have unusual abundances of very heavy atoms which are, as a result, thought to be at least partly the decay products of extremely heavy atoms which cannot normally exist in nature, except for short times after they are created in supernova explosions, and before they have had a chance to decay. As a result, we feel certain that at the time the planets were forming, they contained significant amounts of short-lived radioactive substances which would soon decay and disappear. If those substances were permanently trapped inside small bodies, such as the primitive meteorites, then the heat generated by the decay of these materials would easily leak to the surface and be radiated away into interplanetary space, but if they were trapped inside larger bodies, such as the asteroids or planets, it would take a long time for the heat to leak through the thicker layers of rocky materials, and so heat would build up inside the larger bodies.
According to current estimates of the amounts of such radioactive substances in the early solar system, any bodies more than 50 to 100 miles in diameter would soon accumulate so much heat that they would start to melt, allowing the heavier metals to sink to the bottom and the lighter rocky materials to rise to the top. As a result, all of the Terrestrial planets, the Moon, and even the half dozen or so largest asteroids must have become completely molten, differentiated objects. As the short-lived radioactive materials died out, the heat created by their decay would also die out, and the molten bodies would gradually solidify. The crustal materials, being exposed directly to the relatively low temperatures of interplanetary space, would solidify first, while the rocky mantles, insulated by hundreds or thousands of miles of overlying materials, would take considerably longer. So the crust of the Moon could easily have formed within the 200 million years or so allowed by our current knowledge of the ages of highland rocks, but the deep interior of the Moon might well have still been molten at that time (in fact, "fossil" magnetism inside Moon rocks implies that it did have a molten core, and a magnetic field created by that core, for at least a short period of time).
Looking at the highland surface of the Moon, we can see that at the time that it solidified, not all of the rocky material in the inner solar system was inside the Moon and planets. At least some small fraction of the planetesimals must have still been moving around in independent orbits, and as these objects ran into the now-solid surfaces of the cooling planets, they blasted out huge craters. The number of objects left in between the planets must have been only a small fraction of the mass of the planets themselves, or else heat generated by the violence of their collisions would have re-melted the surfaces of the planets, but there must still have been a huge number of them, since all the truly ancient planetary surfaces still visible to us, such as the surfaces of our Moon, Callisto and Mercury, are completely covered with craters tens or hundreds of miles in diameter.
Eventually, of course, this stage of bombardment of the early planetary surfaces must have come to an end. As the planets gradually swept up the objects not yet in them, the numbers of such objects which were still left would have gradually declined, and so there would be fewer and fewer objects left to cause still other collisions. By around 4 billion years ago, about half a billion years after the start of the solar system, there were so few objects left that the early period of intense bombardment had essentially ended, and surfaces which are younger than that are relatively unscathed by cratering.

6. The Formation of the Asteroids
In most cases, the planets seem to have regularly spaced orbits, with each planet being about 1/2 or 2/3 as far from the Sun as the next planet out. Even Neptune and Pluto follow this rule, if you only consider their distances from the Sun at the times that Neptune is lapping Pluto (at which times Pluto is always near aphelion, 45 to 50 AUs from the sun). There is, however, a notable exception to this rule, in the region which we call the asteroid belt. If we wanted the rule to always work, then we would have to suppose that there should be a planet between Mars and Jupiter, with an orbit size of 2.5 to 3 AUs, and in fact, when the first asteroid (Ceres) was discovered, it was thought to be that missing planet, because it orbits in that region, but Ceres turned out to be only the largest out of hundreds and thousands of smaller bodies, so that instead of a single large planet, we have a host of "minor: planets.
An early explanation of this difference was that perhaps a single large planet had once existed where the asteroids now are, and had somehow been blown or blasted to bits, and in the early part of this century, when it was realized that the Earth is differentiated, the stony and iron meteorites seemed to lend a measure of proof to this idea. At that time, it was not known that the Earth had been melted by short-lived radioactive materials, and other theories of how a planet could be melted depended on factors which work better for large bodies, which have substantial gravities, than for small objects like the asteroids. Since it can be shown with some certainty that the asteroids must be the parent bodies of the stony and iron meteorites, it seemed that they must have melted and differentiated, but since it was not known how to melt small objects like the asteroids, it was tempting to put some faith in the idea that there had been a single large planet in that region, so that the asteroids themselves could have started out as an object large enough to become differentiated. Now, of course, we know that even middling-size asteroids could have easily been melted by the large amounts of short-lived radioactive materials present in the early history of the solar system, and detailed studies of the compositions of various meteorites seem to divide them into various groups which do not seem likely to have originated in less than a half-dozen independent parent bodies. So we now think that the asteroids have always existed as separate objects, and were never a single large object. This leads us to the question of why they never accumulated into a single object, while the planets did.
The most likely explanation of this problem has to do with the mass of the asteroids. Although the space within the asteroid belt is often portrayed as being full of rocks, it is actually very nearly empty. There are several thousand asteroids that are a 1/4 mile or more in diameter, but they are spread out over a region almost a billion miles in diameter, and tens of millions of miles thick. Because they occupy such a vast region, they are usually quite far apart. In fact, during the several months that it takes a spacecraft to pass through the asteroid belt, an observer on the craft would probably not see any asteroid close enough to look like a tiny disk, and would see even faint distant asteroids only every few days.
Because the asteroids are fairly far apart, collisions between them are not nearly as frequent as people normally presume. Admittedly, there must be some collisions, as some "families" of asteroids have orbital characteristics which imply that they resulted from the collision of two larger objects at some time within the last few millions or tens of millions of years, but such individual collisions probably occur only once in every few millions of years now, while in the early solar system, collisions had to be so frequent that whole planets could be built up in a very short time, so the current rate of asteroid collisions doesn't seem to be a very useful method of building up planets.
To explain the planets' formations, we must assume that there were far more objects running around, and into each other, than we now see in the asteroid belt. Looking at just the surface of the Moon, we see more craters than would be caused by throwing all known asteroids into it, and we know that even the huge number of collisions required to form the Moon's craters cannot have been more than a fraction of the mass of the Moon itself.
The solution to our problem lies in the small amount of mass found in the asteroid belt. The total mass of all the asteroids is only one thousandth of the mass of the Earth. If we assume that, towards the end of the formation of the solar system, collisions between planetesimals had built up objects similar in size to the current asteroids, then in the asteroid belt there would have been about the same number of objects as now, but near the orbit of the Earth, with a thousand times the mass, there would have been a thousand times as many objects. For each collision in the asteroid belt, there must have been close to a million collisions in the Earth's orbit. (Each object near our orbit would see a thousand times as many targets, so each one would have a thousand times the chance for a collision, and with a thousand times as many objects, each a thousand times as likely to have a collision, the total number of collisions would be a thousand thousand times larger.)
At some point, each region would have built up a small number of relatively large objects, which contained most of the mass of that region (indeed, Ceres already contains half the mass of all the asteroids, and the next dozen largest asteroids contain most of the rest of the mass). At that point, since so few objects were left, the number of "accidental" collisions would be tremendously reduced in comparison to earlier stages in the accumulation of material. If we continued to rely only on such collisions, then as in the case of the asteroids we would be at a dead end. In the case of the planets, however, gravity, and their relatively large sizes, can help finish the job.
If two asteroids pass fairly close to each other, but do not collide, their small masses produce so little gravitational effect that they continue on with nearly the same orbital motions that they had prior to their encounter. But if objects tens or hundreds times larger were to pass so close, their large sizes would make it far more likely that they would collide, and even if they still missed each other, their larger masses would produce much larger gravitational effects, and their orbits would be significantly altered. In the case of the asteroids, the small changes that they produce in each others' orbital motions mean that unless they are already in intersecting orbits, they will have to wait a long time, if not forever, for the very minor rate of orbital change to allow them to have intersecting orbits and collide with each other. Larger masses, by changing each others' orbits faster, could end up with intersecting orbits and collide with each other much sooner. In other words, if the random collisions that are most important while there are still huge numbers of planetesimals can continue long enough to build up good-sized masses, then the size and gravity of those masses, by allowing significant interactions in the present, and additional interactions later on, will continue the process of accumulation until only a single large mass is left in a given region.
Of course, this theory explains how, if there is little mass in the asteroid belt, the asteroids can fail to complete the process of planetary growth, but it does not explain why there should have been so little mass. To explain this, we can rely on the fact that the asteroids are not far enough out to allow icy materials to be solid, but too far out for rocky materials to be thickly clustered together, and to a possible gravitational interaction between the huge masses of Jupiter and Saturn, which may have swept materials that might have existed in the asteroid belt into other regions. Since the Jovian planets must have grown faster than the inner planets, they probably affected the final stages of formation of the asteroids and possibly even the inner planets, so at least part of the reason for the low masses in the asteroid belt may be due to such gravitational interactions. However, if that were the only reason, then there should be at least a somewhat greater mass in that region, so part of the answer must be that there just wasn't much solid material in the region in the first place.