A. How did they manage to melt?
      The Earth has a dense metal core and a less dense rocky mantle. This is referred to by saying that it is differentiated, or of different composition in different places, rather than homogeneous, or about the same composition everywhere. We believe that this differentiation implies that the Earth more-or-less completely melted at some time early in its history, so that heavy materials could sink to the bottom, and light materials float to the top.
      There are three major heat sources that could have allowed the Earth to melt:
     (1) the heat of the Sun
     (2) the heat of gravitational accretion and differentiation
     (3) radioactivity

1. Solar heating and the composition of primordial solids
     As previously discussed, the inner Solar System was very hot during the formation of the planets. At the present time, temperatures at the Earth's orbit are relatively comfortable; but during the formation of the planets 4.5 billion years ago, temperatures would have been between 1500 and 2500 Fahrenheit degrees. At the orbit of Mercury, temperatures would have been even higher; and even in the asteroid belt, temperatures were several hundred degrees above zero.
     In any given part of the Solar Nebula, the kind of minerals that condensed out of the gaseous material depended critically upon the temperatures in that region. In the inner region, where the Terrestrial planets are now found, high temperatures made it impossible for any materials to survive except high-melting-temperature compounds such as metal oxides: the silicate minerals that make up the Terrestrial planets. Far from the Sun, low temperatures allowed not only rocky materials, but also relatively volatile carbon compounds and ices to exist as solids. Aside from these large differences in the composition of the solid component of the Solar Nebula, there would have been differences between the kinds of materials that could be solid within each region.
     For instance, very far from the Sun, methane can exist as an ice; but closer in, it vaporizes to become a gas (as is now happening on Pluto). Water ice can exist as a solid near and beyond the orbit of Jupiter, but partway in towards the orbit of Mars, it too would begin to melt. As a result, small objects, such as comets, which contain ices cannot get much closer to the Sun than the orbit of Jupiter before most of their ices begin to vaporize; and even far from the Sun more volatile gases, such as methane, may begin to boil away from them.
     Similarly, in the inner Solar System, differences in temperature between the orbits of the different planets would significantly affect the kinds of materials contained in them. At the orbit of Mars, where temperatures would only be around 1000 Fahrenheit degrees, many rocky materials which would vaporize at temperatures of 1500 degrees or more could condense and remain solid. In the outer asteroid belt, where temperatures were near zero degrees, carbon compounds could survive which would have vaporized at the higher temperatures near the orbit of Mars. Because of this, Mars can contain large amounts of low-melting-temperature minerals which would be rare or non-existent in the materials which were forming inside the orbit of the Earth, and the asteroids in the outer part of the asteroid belt can be rich in carbon compounds, while those in the inner part are deficient in such materials.
     Conversely, if the high-melting-temperature materials which formed at the orbit of Mercury were moved to the orbit of Mars, chemical reactions between them and the (still relatively hot) gases in that region would gradually convert some of them to lower-melting-temperature minerals, increasing the percentage of volatile minerals, and decreasing the percentage of refractory minerals.
     With this in mind, we can see that in any region of the Solar Nebula, if the temperatures were raised just a few hundred degrees, a large part of the solid materials in that region would begin to melt. As a result, other factors only need to raise the temperature by a few hundred degrees (at most) to produce substantial melting and differentiation.

2. Heating by Collisions and Gravitational Acceleration
     When the planets first started forming, they consisted of small dust grains, pebbles and rocks. There were so many of these particles orbiting the Sun that collisions between them were extremely frequent, and they quickly built up into fairly large objects (like asteroids and comets), generally referred to as planetesimals (small things, -esimals, which, through collisions, end up as the planets). In the early stages of this buildup, because collisions were very frequent, all of the solid grains in a given region would have been orbiting the Sun with almost identical velocities, and collisions between them would have been relatively gentle.
     As the planetesimals grew, however, there would be fewer and fewer of them, and collisions would become less frequent. This would allow small differences in velocity, fed by the pull of the Sun's gravity, to gradually increase. As an example, in the current asteroid belt, we could have collisional speeds of several thousand miles per hour. Collisions at such velocities could create a substantial amount of heat, vaporizing small fractions of the colliding bodies.
     For larger objects, like the Earth, collisional velocities can be even higher, because as an object approaches us, our gravity helps pull it into us. Even if the object had hardly any speed at all (relative to us) before it approached us, it would hit our upper atmosphere with a speed equal to our escape velocity, or 25000 miles per hour. An object running into Jupiter, which has five and a half times our escape velocity (for example, Comet Shoemaker-Levy 9, which ran into Jupiter a few years ago), would hit its atmosphere at nearly 150000 miles per hour.
     These huge impact velocities generate such immense amounts of heat that when an asteroid or comet runs into the Earth, it creates a hole that is ten to twenty times larger than the impacting object, vaporizing almost all of the object and a considerable part of the surrounding countryside.
     During the last stages of the formation of the planets, when they were close to their present sizes, the millions of collisions still to come would have heated up the outer parts of the planets tremendously, allowing at least substantial parts of their outer regions to melt, and then differentiate. Since, as already discussed, the heat of the Sun, by controlling the types of materials which existed in a given region, would already have made the average temperature of the planets only a few hundred degrees below the average melting temperature of their constituent materials, all of the larger planets, by obtaining substantial extra heating from the heat of collision and gravitational acceleration of any impacts, would have certainly become hot enough to substantially melt (as we already know to be true for the Earth).

3. Radioactivity and the Melting of the Smaller Rocky Bodies
     The two factors already discussed are adequate to explain the melting of large bodies, such as the planets. But for small objects, like our Moon and the asteroids, the force of gravity is small, and gravity cannot help accelerate incoming bodies to sufficient speed to guarantee substantial melting of these small bodies. And yet, we know that at one time, there must have been an object (or objects) in the asteroid belt which was (or were) differentiated, because the stony and iron meteorites which originate there must have come from such a differentiated body.
     Many of you have probably heard of the obsolete theory that there was once a large planet, similar in size to the Earth, in the asteroid belt, which was somehow destroyed, resulting in the formation of the asteroids as we know them. This theory dates back to the discovery of the asteroids. Prior to their discovery, it was recognized that the fairly uniform orbital spacing of the planets was different in the area between Mars and Jupiter. In every other region of the Solar System, each planet is between 1/2 and 2/3 as far from the Sun as the next planet out; but Mars is less than 1/3 Jupiter's distance from the Sun, leaving plenty of room for another planet at about 2.8 AUs distance. This is almost exactly the orbital size of Ceres, the first asteroid discovered, so when Ceres was found it was assumed that it was the "missing" planet that belonged in that region. Within just a few years, however, several other smaller asteroids were discovered, and of course we now know that there are several thousands or tens of thousands, depending upon how small you are willing to go to keep counting them.
     When it was realized that there were many small bodies in this region, instead of a single large body, it was suggested that perhaps once there was a large body there, but it was somehow destroyed, and colliding with the other planets lost most of the pieces. How an object of planetary size could be destroyed was never satisfactorily explained, but it fit the pre-conception that there should have been a "normal" planet in that region.
     As previously mentioned, small objects like the asteroids don't have enough gravity to help accelerate incoming bodies, and because of this, shouldn't have gotten hot enough to melt and differentiate. And yet, the meteorites that come from the asteroid belt show that there must have once been a differentiated body in this region. How can we explain this? The answer is, through radioactivity. As radioactive materials decay, they release heat. If they were buried deep inside a solid rocky body, this heat would be trapped, and gradually build up. If there were enough such heat, it could eventually melt the object, allowing it to differentiate.
     The problem with this explanation is that different radioactive materials decay at different rates, and the ones that decay quickly, releasing large amounts of heat in a fairly short period of time, are rare. Only the types that decay slowly, releasing relatively small amounts of heat at any given time, are relatively common.      As an example of how this works, Uranium has two fairly common isotopes (atoms of the same element with different atomic weights, due to having different numbers of neutrons in their nuclei). Most of the Uranium on Earth is Uranium 238 (so-called because it has 92 protons, which makes it Uranium, and 146 neutrons, which adds up to 238 nucleons); only a small percentage is Uranium 235 (which also has 92 protons, but only 143 neutrons, for a total of 235 nucleons). Uranium 238 is not very radioactive (its half-life is over 4 billion years, meaning that it takes over 4 billion years for half of it to decay into lead), while Uranium 235 is much more radioactive (its half-life is only 700 million years). If each of these two kinds of Uranium were equally abundant, the much faster decay rate of Uranium 235 would cause it produce over 6 times as much radioactive heating (ignoring any differences in the amount of heat produced per decay). However, since Uranium 238 is about 100 times more abundant than Uranium 235, it actually produces most of the radioactive heating of the Earth.
     4.5 billion years ago, however, this result would have been completely different. Since Uranium 238 decays very slowly, the current amount is almost half as much as existed in the Earth when it was forming, and the radioactive heating due to this material at that time would have only been about twice as great as now (and not very significant compared to the other forms of heating already discussed). The much faster decay of Uranium 235, however, means that it has decreased in quantity by many, many times, and is now only a few percent of its original amount; or, that it was once over 40 times more abundant than it now is. With a much greater abundance, and a much faster decay rate, instead of producing far less radioactive heating than Uranium 238, it would have actually produced much more radioactive heating when the Earth was young.
     This result is pretty much typical of calculations made for other types of radioactive materials. Things which decay slowly are still pretty abundant (because they do decay slowly), and things which decay quickly are now pretty rare (because they decay so much faster), but when the Earth was young, the faster-decaying materials would have been much more important to the Earth's heat budget, because they would have been both abundant and intensely radioactive. Extending this idea, the most significant radioactive heat source during the formation of the planets would have been radioactive materials which no longer exist at all, because they decay so quickly that all of their atoms would have long since turned into non-radioactive decay products.
     The problem with this result is, that in the case of the two Uranium isotopes, we know how much of each there is now, and we know their decay rates, and so we can calculate how of them there was in the distant past. But with the presumably extinct short-lived materials just discussed, we are supposing that there aren't any atoms of them left at all; so how can we possibly calculate how many there once were? Strictly speaking, we can't, and so any guess as to what their abundances were, and how much radioactive heating they would have produced, would be just that: a guess. And although astronomers aren't at all averse to making wild speculations which might someday be capable of being proven right or wrong, they are (normally) loath to solve problems by making suggestions which have no hope of being verified or disproved. So, if the solution to melting the asteroids is the one-time existence of short-lived radioactive materials whose supposed existence can't be proven, then that solution would have to be rejected. And that is exactly what was done, earlier in this century, when radioactivity was suggested as being of some importance for this problem.
     As it turns out, however, we now know that during the formation of the Solar System, there were indeed huge amounts of short-lived radioactive materials. As you will see (in detail) later in the semester, the formation of our Solar System was almost certainly triggered by the explosion of one or more massive stars which formed in the same region as the Sun but at a very slightly earlier time. In the destruction of these massive stars, a catastrophic implosion of the central core creates an even more catastrophic explosion of the core and the regions surrounding it. Temperatures soar into the trillions of degrees, causing the materials in the core to undergo incredibly fast, unstable nuclear reactions. All of the materials in our Solar System, other than hydrogen and helium, which date back to the beginning of the Universe, are created in such massive stars. The relatively larger or smaller abundances of different materials are to a great extent reflective of how long they were being created: the elements lighter than iron were created during the "normal" lifetime of the stars, which lasted a few million years, and are relatively abundant, while the elements heavier than iron are rare, because they were created only in the few seconds of the firestorm which destroyed the stars.
     This theory of the creation of the elements predicts that some elements, such as oxygen and carbon, should be relatively abundant; that other elements, such as silicon and iron, should be less abundant; and that still other elements, such as uranium and gadolinium, should be quite rare. The abundances of all the elements, other than hydrogen and helium, match very closely their abundances as predicted by this theory.
     This theory also predicts that certain types of short-lived radioactive isotopes were produced in large amounts. These isotopes, because of their short half-lives, no longer exist, but their decay products do, and if they have unusual abundances, we could hope to notice this in rocks which have not been significantly changed since the origin of the Solar System. Such rocks would include the "primitive" meteorites (meteorites whose minerals have physical and chemical properties which suggest that they were never inside an asteroid, but are instead pieces left over from the formation of the Solar System), such as the chondrites and carbonaceous chondrites. It is now possible, using micro-chemistry (the chemical analysis of microscopic bits of material), to analyze the minerals in meteorites in extraordinary detail. As it turns out, a very few of these have been found to have peculiar abundances of some heavy atoms which can be most easily explained by assuming that, at the time these primitive meteorites formed, the Solar System contained extremely heavy atoms which could only have existed if they had been recently created in a supernova explosion. Because of this, we suspect that rocky materials in the early Solar System were indeed quite radioactive.
     Using microchemistry, we have also discovered that there are "families" of meteorites with similar isotopic compositions (a nearly exact match in the abundances of the isotopes of various elements). Within a given family, the abundances of the isotopes of different elements are so similar, that the most likely explanation of the similarity is that all of the meteorites involved were once inside the same parent body. If all meteorites had the same isotopic composition, this would point to a single primordial parent, as once suggested. But instead, there are about a half-dozen different isotopic families, implying that there must have been at least a similar number of parent bodies. This makes the idea that the meteorites originated in planet-sized bodies unattractive. Although one such object is a barely conceivable possibility, no one is willing to propose that there were half a dozen or more such objects which were all somehow destroyed.
     Instead, it appears that in the early history of the Solar System, there was so much short-lived radioactive material around that any rocky body much larger than 100 miles in diameter would have melted. This includes the Earth and the Terrestrial planets, our Moon, and the larger asteroids. The heat of the Sun, by making it relatively easy to melt the materials, would be a significant factor for all of these objects. For the larger objects, their gravity, by helping accelerate incoming bodies, would also have helped. But for all of them, and particularly the smaller bodies, the radioactivity present at that time would have been critical to their melting.

4. The Heating of the Jovian Planets
     Since the Jovian planets are made mostly of hydrogen and helium (almost entirely liquefied by the tremendous weight of the huge amounts of these materials), it hardly seems necessary to explain how they could melt and differentiate. When they started off, they were just dirty snowballs, but when they got big enough to gravitationally attract the gases surrounding them, they mushroomed in size to become the mostly liquefied gas objects that they now are. This process of formation would automatically put the heavy stuff in the middle and the lighter stuff on the outside. And if dirty snowballs or rocks subsequently ran into them, the heavier materials would tend to sink to the bottom with the original cores of the planets. However, it doesn't hurt to consider the heating of these bodies, because as you probably already know, Jupiter and Saturn are extremely hot on the inside.
     When you compress gases, they become heated by the work that is required to compress them. When you accumulate 300 Earth masses of gases, as Jupiter did, the weight of all that gas heats it up a lot. Depending on how long it took to accumulate all that gas, some of the heat could have been radiated away before the process was done, and so the temperatures reached in the cores of the Jovian planets would depend on how long they took to reach their present size; we currently believe that most of the gas was accumulated in well under 100000 years, and that during this time, Jupiter may have reached internal temperatures in excess of 250000 Fahrenheit degrees. Although it has cooled off in the 4.5 billion years since it formed, a substantial part of its original heat still remains, and is probably the main contributor to its current temperature of over 50000 Fahrenheit degrees.

B. The History of the Melting of the Planets and Smaller Bodies
     Having solved the problem of whether the various bodies could melt, it would also be nice to know when they did so. (To be finished later)