1 Introduction

The search for the origin of the planets is surely almost as old as humanity itself. Our predecessors explained the origin of the Earth in a myriad of ways. Few saw the process that created our own Earth as nearly identical to that which created the other planets (until recently, the only known planets were Mercury, Venus, Mars, Jupiter and Saturn), of course, but explaining the Earth's origins was every bit as important three thousand years ago as it is to us today. Some of our ancestors believed the Earth was formed from a primordial Chaos (not so far from the truth, in fact!). Many thought the Earth simply always was, the First and Ultimate Mother, who, along with the sky god, begot the other gods and was the source of all things.

Ironically, science now is fairly certain that the cosmos formed in almost the exact opposite way, with planets forming almost last in the grand scheme of things (life, of course, must come after the host planet, but that is beyond the scope of this discussion). The model we now use dates back to Descartes at the earliest, although it is usually Kant who is given credit since he expanded on it after Descartes. The theory was first brought out of the philosophical realm into the scientific one by the great mathematician, physicist and astronomer Laplace. Their basic idea was that as the Sun formed, a ``protoplanetary'' disk was created with it, as a by product of the creation of the Sun. This disk fragmented and in the same manner as the Sun formed from a collapsing cloud of gas, so did the planets.

While elegant in many respects, this theory has quite a few problems. The first, and perhaps difficult, is that if the planets formed so much like our Sun, why are they so different from the Sun? Even the very composition of the planets differs radically from our Sun's composition. There must be some kind of selection effect at work, so the simple collapse of the cloud model fails. This, and other flaws, require redress. However, the model was a very good start, and its appeal (stemming from the fact that the planets were a simple by-product of star formation) made scientists examine it further; and indeed, astrophysicists are still probing the theory, trying to narrow down what conditions give rise to the observed characteristics of the planets.

2 The Planets

Solving the problem of the origin of our solar system is quite akin to solving a differential equation problem. Taking on the role of the differential equation are the known laws of physics. Assuming the role of the boundary conditions are the observed properties of the planets. The test of any theory will be whether it accounts for the planets, so we would be wise to take a quick look at the planets today. For later reference, see Appendix A for tables of relevant quantities.

For the purposes of this paper, we will only consider Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus and Neptune as planets. (We leave out Pluto for a large number of reasons, the bulk of which imply that it followed a different path in its formation then the other major planets.) The most obvious property of the planets is that their orbits all lie in almost the same plane and that the orbits are almost all very circular (see ``Orbital Data'' in Appendix A). These two observation alone eliminates many potential theories of planet formation. Additionally, ones quickly notices that six of the planets rotate in the same sense (that is, their angular momentum vectors all point in the same direction). Even more interesting is that these vectors are pretty much perpendicular to the ecliptic, so that the rotational angular momenta are in the same direction as the orbital angular momenta (and also the same direction as the Sun's rotational angular momentum). Six out of eight may not sound terribly impressive, but when we examine the two exceptions (Venus and Uranus) we see that Venus has an anomalously slow rotation period (240 days, as opposed to most other planets' one day or even a few hours) and that Uranus's rotational axis is very nearly in the ecliptic plane. Both of these anomalies lead us to suspect that some other mechanism may be responsible for their odd behaviors and that the rotation of the other planets can be thought of as a byproduct of planet formation.

The other properties of the planets that we notice are their differences. The most obvious is that the inner four planets are all rocky silicates and the outer planets appear to be gaseous, with Jupiter and Saturn being mostly hydrogen and helium and Uranus and Neptune consisting of hydrogen and helium along with methane. However, upon closer inspection, we have come to realize that most of the inner planets (generally referred to as terrestrial planets) have silicate crusts and all have, to one degree or another, nickel-iron cores. Even more interesting was the later discovery that the Jovian (i.e.-outer) planets appear to have rocky (silicates and nickel-iron) and icy cores as well. Thus, the differences in composition are not nearly as extreme as a first glance would imply. A further observation is that some planets have magnetic fields (Earth has the only truly notable field among the terrestrial planets, while all four Jovian planets have fairly strong fields by planetary standards). This may be either a formational effect or an evolutionary one, or both.

The last point of interest is moons. Among the terrestrial worlds, only Earth and Mars have moons. Mars's moons are almost certainly captured asteroids, given their size, irregular shapes, orbits and compositions. Earth's Moon is more of a puzzler. It is relatively large in comparison to Earth and has a similar composition to Earth's crust, but it lacks all but the smallest nickel-iron core. Our Moon has been a sticking point for formation theories for many years, and only recently did a reasonable explanation emerge (see section 4.4). On the other hand, the Jovian planets all have moon systems. Again, many moons are almost certainly captured asteroids and comets, as can be told by their compositions and odd orbits. But many moons are in circular, prograde orbits (that is, in the same direction as the planetary rotation) and appear to have formed near their planets. The presence and absence of moons, then, is another phenomenon we will want our planet formation theory to account for.

With this brief introduction to our solar system in mind, let us now examine the generally held theory of solar system formation.

3 Formation: The Planetesimal Theory

Most planetary scientists hold to what is known as the Planetesimal Theory of planet formation. In brief, it holds that the planets formed out of a protostellar disk, first by forming small grains which then grow by pairwise collisions ultimately forming planets.

3.1 The Disk

To begin our tale, we need to roll back time more than 4.5 billion years to the formation of our Sun from a Galactic cloud of gas and dust. Our Sun, it is generally believed, formed when this cloud collapsed under its own gravity. A simple collapse, with no rotation or magnetic fields, would not only have been uninteresting, but also unlikely. We will neglect magnetic fields, although it is probable that they play a role in this process. Since the original cloud of gas almost certainly had some slight rotation to it, rotation simply cannot be ignored. As the cloud collapsed, in order to conserve angular momentum, the protostellar cloud had to spin ever faster.

A simple calculation of the angular momentum of the original cloud can be done by assuming the cloud to be a ball of uniform density and spinning at a single rate. For example, if we assume the cloud has mass equal to that of the sun (1.989 ×10³³ gm), radius 0.1 parsec¹ (or approximated 2.0 ×10⁴ AU) and a reasonable rotation period of 100 million years, using that L = I_ball &omega, where I_ball = ²/₅ M R² and &omega = [(2 &pi)/( period)] is the angular velocity, we find an angular momentum of 1.51 ×10⁵¹ [(cm² gm)/( sec)]. Most people find this figure meaningless, so let us see what it implies for a collapsing cloud. If we assume conservation of angular momentum and check the rotation rate when the cloud has collapsed to 30 AU (the size of the solar system), still using L = I_ball &omega, we get that the rotation period is now 287 years, a considerable drop. Thus, it is not at all surprising that rotation is important in the star forming and planet forming processes.

Eventually, the center of the cloud picked up enough mass to start accreting matter onto itself. However, much of the solar nebula (the name given to the cloud that formed the Sun and planets) had far too much angular momentum to fall straight into the protosun. Thus, the nebula was no longer spherically symmetric, but rather had flattened into an accretion disk, a commonly seen phenomenon in many parts of astrophysics. Some computer models have implied that the disk may have lacked the very central portion, making it a very wide ring instead; however, I will use the term ``disk'' and let the term encompass both the solid disk and the wide ring, as it makes little difference to the process anyhow. Due to frictional forces within the disk, the disk heated up, and in doing so lost energy via blackbody radiation. This energy came at the expense of the orbital energy, so that the particles in the disk made their way toward the protosun and eventually fell onto it. However, this infall could take millions of years, so there is plenty of time for interesting things to happen in the disk. Indeed, it is in this disk that the planets form. See fig.  1 for a breakdown of the composition of the disk.

Figure 1: The Composition of the Protoplanetary Disk (from Wood 1999)

3.1.1 Disk Temperature

To understand planet formation, an understanding of the accretion disk is vital. Perhaps the most important physical parameter of interest is the temperature of the gas at a given point in the disk, since this governs what kind of matter can condense out of the disk as we shall see. Near the protosun, the disk would have been hot and dense. As the distance from the protosun increased, the temperature would have fallen off, however. To see how this temperature fall-off went, we must use energy considerations.

We start with the energy of a piece of matter in the disk. Under the influence of gravity, the potential energy goes as U = [(-G Mm)/( r)], where m is the mass of the particle in the disk, M is the mass of the protosun since the protosun possessed the vast majority of the mass of the system, r is the distance from the protosun's center of gravity and G is Newton's constant. With the help of the Virial Theorem (which states that < E > = ¹/₂ < U > for bound systems; see Carroll and Ostlie 1996), we know that

E = -G M m 2 r . (1)

Consider now a thin ring in the disk of radius r, mass m and width dr. Matter enters the ring from the outer edge at rate [(M)\dot]. In a steady state, matter must leave the inner edge at the same rate, so that in time t, the mass that passes through the annulus is m = [(M)\dot] t. Thus,

3.1.2 Planetary Temperatures

Thus, we see that the temperature of the disk rose much more rapidly near the center then does the temperature of planets at comparable distances. See fig.  2 for a look at how the functions differ.

Figure 2: A comparison of the temperature in the disk (solid) and the planets (dashed)

3.1.3 The Largest Planets Start as the Smallest Grains

In the inner part of the disk, where the temperature was high, only materials with high melting points, like silicates and metals, could condense out of the hot solar nebula. At larger distances, probably of order 5 AU or more, ices (water or methane, in particular) could also condense. Typically sized particles at this stage would have been of order no more than 0.1 microns in diameter, mere dust by any standards. Also formed were chondrites, small beads of silicates with diameters on the order of 1 mm or less. The origins of the chondrites is not entirely clear, but laboratory experiments indicate that they must have cooled rapidly, which implies that they were not heated by ambient temperatures in the protoplanetary disk (the cooling would be much too slow), but rather by some violent event. What events caused this is a mystery, however.

3.2 Accumulation of Matter

3.2.1 First Stage

In any case, it is with these grains and chondrites that we begin the first of three stages of planet formation. During the first stage, these motes of dust struck one another and stuck. The sticking could have occurred in several ways, ranging from Van der Waals forces, to surface cohesion to electrostatic forces. For example, looking at electrostatic forces, consider what would happen if two particles were to collide, one with a slight (of order a few tens of electron-charges) positive charge, the other with a slight negative charge. We can solve the velocity that would be required to escape their mutual attraction. As with gravitational escape velocity, we can assume the critical velocity requires that E=0 at infinity (where U=0 at infinity, as usual). Thus, in contact the two particles feel a potential of - [(q₁ q₂)/( r)] , where r is the distance between the centers of charge and q₁ and q₂ is the charge on each particle. To keep E=0, then, we have

E = U + T 0 = q1 q2 r + 2 ( 1 2 m v2) (8)

(the factor of 2 is for the existence of two particles, with roughly the same velocity). Assuming q₁ = - q₂, we can solve for v:

v =     æ √ q2 m r   . (9)

With a density roughly equal to the density of Earth's silicate crust, 3    gm/cm³ and assuming the dust motes are two spheres of radius r/2, an approximate calculation yields escape velocities of order 10 cm/sec to 100 cm/sec.

Unfortunately, this escape velocity is much less than the probable relative velocities of the particles in the disk, so it seems unlikely that two colliding particles would stick together purely based on electrostatic attraction. This is, in fact, a major problem in the current model of planet formation. While there are several theories as to how these particles stuck and accumulated, there is no consensus as yet. However, based on meteorites we have collected, we are reasonably sure that such accumulations must have taken place (see below in section 3.3).

An important factor in this stage is the gas in the protoplanetary disk. At this point, we assume that the particles of dust are bound in roughly circular Kelperian orbits about the protosun (rather like the planets now are). This constrains the velocities of the particles once their distance from the Sun is given. The gas ought to be likewise bound, with one caveat. The gas also feels gas pressure in the anti-protosun direction, thus reducing the velocity of the gas needed to maintain a circular orbit. The gravitational force felt by both gas and dust particles goes as - [(G M_protosun m)/( r²)] while the gas feels a force in the anti-protosun direction equal to -[(m)/( &rho_gas)] [(d P)/( dr)] (this is the same gas-pressure force felt by particles in the interior of stars; see Carroll and Ostlie 1996, p 316 for a derivation of this). In order to maintain a circular orbit, this must equal -[(m v²)/( r)]. Equating and solving for v yields:

v =     æ √ G Mprotosun r2 + 1 &rhogas d P dr   (10)

We can get dP/dr using the Ideal Gas Law, P = n k T, and the disk temperature derivation in 3.1.1: T = ([(G M [(M)\dot])/( 8 &pi &sigma)])^1/4 r^-3/4 Using typical disk parameters ( &rho_gas = 10^-9 gm/cm³, M_protosun = 1.989 ×10³³ gm, ([(G M [(M)\dot])/( 8 &pi &sigma)])^1/4 = 5.34 ×10¹² (roughly typical for most models) and r= 1 AU), we find that the gas moves with velocity of about 2.97 ×10⁶ [(cm)/( sec)], which is less than the pressure-less velocity by about 7 percent. A more exact look at the effect of pressure yields a difference of order 0.5 percent (Lisseur 1993). So the solid grains move through a slower gas medium, thus feeling a drag force. This drag causes the particles to spiral in toward the protosun. Ultimately, once the solids reach a diameter on the order of meters to kilometers, the drag forces becomes negligible and the orbits stabilize (Lisseur 1993). Because the drag can be significant for medium sized particle (of size about 1 meter), such particles must make the transition from small grains (which are coupled strongly to the gas, hence feel less drag) to kilometer-size objects (which feel little drag due to their size) fairly quickly.

3.2.2 Second Stage: Planetesimals

The lesson in this is that the more massive the planetesimal, the larger the escape velocity, hence the more matter the planetesimal was able to capture out of the disk. Thus, the largest bodies were able to grow most rapidly.

3.2.3 Final Stage: Protoplanets and T Tauri Stars

Eventually, this whole process must have drawn to a close when the protosun started nuclear fusion in its core, thus reaching the so called T Tauri stage. At this point, observational evidence tells us that the young star developed a powerful wind. This stellar wind blew off its nebula and disk, leaving only the largest objects (major and minor planets), which were less affected by the stellar wind. The planetesimals may continue to merge for some time after this, until they are are all out of the gravitational reach of planets or have been flung out of the solar system (Jupiter-sized planets have been shown to be very effective at removing smaller bodies). The time scales believed to be involved in this process are of order 10⁵ to 10⁷ years. Estimates vary depending on the planet (larger ones form faster, as we shall soon see) and the model (most of the numerical results come from computer models, which tend to be sensitive to choices of parameters, so estimates are expected to vary). See fig.  3 for a rough time scale for a fairly standard model.

3.3 Evidence in Support of the Planetesimal Theory

The best evidence today is recent observations of protoplanetary disks about other stars such as Vega and Arcturus. The disks have been sighted by looking for infrared excess in the emissions of the stars² (Holland et al. 1998). Within the past year, tests have been run with nulling interferometers (Angel and Woolf 1997; Hinz, Angel et al. 1998)) and actual images of these disks have been obtained. The most intriguing support may come from Beta Pictoris, a star whose disk we see edge-on. Many astronomers claim that the warp seen in the disk is evidence that a protoplanet is actually in the disk (see fig.  4; both images are of Beta Pictoris, the centerline of the disk has been added to the lower image for reference).

An indirect piece of evidence is unfolding on computers. With the advent of computers with sufficient power to handle the motions of thousands of bodies, planetary scientists have been modeling various stages of planet formation. Different stages are generally handled in slightly different ways (for example, early in the second stage, the planetesimals are treated using statistical mechanics by analogy to the gas in a box problem), so computer models do not start from the early disk and work all the way to planets. However, later stages can be modified and tested using results from previous stages. While such modeling is tricky and not truly conclusive, the fact that models generally result in the formation of planets similar to the ones we see is very encouraging.

3.4 Complications

Another complication is the sticking that occurred early on in the process. As I mentioned above, no one is entirely sure how the sticking occurs, merely that it does occur. While working out the details of this process to everyone's satisfaction would be a major step toward understanding planet formation better, such a resolution has yet to occur.

There is also the question of time scales. While we are fairly certain that the process took place over no more than hundred million years (this from simple observational restraints like the ages of chondrites as compared to the ages of the planets), many computer models take far longer than this to play out. In fact, some take longer than the present age of our solar system (4.5 billion years), which is somewhat problematic to say the least.

4 Planetary Properties: Where Do They Come From?

It is my intention to devote most of the remainder of this paper to considerations of the various properties of the planets any how the Planetesimal Theory explains them. While the theory can successfully explain most to all of the observed properties of the planets, I will single out a few of the most interesting.

4.1 Orbital Properties

The planets' orbits are all roughly circular since when two planetesimals collided and merged, their momenta were added (in order to conserve momentum). The result was that the radial velocities all largely averaged out, since there was no net radial motion among the planetesimals. Orbits with no radial motion are, of course, circular, so the planets ended up with orbits very near circular.

4.2 Spin

For example, when we imagine a smaller planetesimal striking a protoplanet, if the planetesimal strikes one edge of the protoplanet (say the sun-ward side, as seen from behind the protoplanet) it will tend to impart a slight retrograde rotation, while striking the other side will yield a slight prograde rotation. If a large imbalance in the numbers of protoplanets striking one side vs. the other occurs, then the resulting planet will have net spin. I find this explanation unsatisfactory, however (as do many planetary scientists) because it would seem that the planetesimals would strike both sides of the protoplanet with equal likelihood.

In examining angular momentum, I have come to the following theory: if we consider the angular momentum of a ring that will eventually form a planet via collisions and gravitational capture, and then we look at the orbital angular momentum of the resulting planet, we find that the ring has an excess of angular momentum which must go into spin of the planet.

For a crude look at this process, let us assume that the planet forms out of an annulus of radius R (where we measure the radius to the center of the annulus), which will also become the orbital radius of the planet, width 2r, density &rho, height and h. As a rough first approximation, let us assume that the angular velocity is constant across the annulus (that is, the annulus rotates as a rigid body) and let us further assume that the final orbital angular velocity of the planet is this same quantity. Calling this common angular velocity &Omega, the angular momentum of a disk is

1 2 M R2 &Omega = &piR2 h &rho 2 R2 &Omega, (17)

so the density of moment of inertia of an annulus, L_annulus, is

Lannulus = &pi(R4outer - R4inner) h &rho 2 &omega > = &pi(R2outer - R2inner)(R2outer + R2inner) h &rho 2 &Omega = R2outer + R2inner 2 M &Omega (18)

Using R_outer = R+r and R_inner = R-r, we get:

R2outer + R2inner = R2 + 2 R r +r2 + R2 - 2 R r + r2 = 2 R2 + 2 r2 (19)

Thus,

Lannulus = R2 M &Omega+ r2 M &Omega, (20)

under the approximation that the whole annulus rotates as a rigid body.

Now, for the planet, the total angular momentum is L_orbit +L_spin, where L_orbit is the angular momentum due to the orbital motion and L_spin is the angular momentum due to the spin. The orbital angular momentum is simply given by:

Lorbit = R2 M &Omega, (21)

and the spin angular momentum is:

Lspin = 2 5 M R2planet w,

where &omega is the spin angular velocity of the planet.

Knowing that angular momentum is conserved,

Lannulus = Lplanet = Lorbit + Lspin R2 M &Omega+ r2 M &Omega = R2 M &Omega+ Lspin (22)

Cancelling terms on both sides, we see that in order to conserve angular momentum, the planetary spin angular momentum must be L_spin = r²M &Omega. Using this to solve for the period of rotation, period = [(2 &pi)/( &omega)]. With &omega = [(5 r² M &omega)/( 2 M R²_planet)], we get:

period = 4 &piR2planet 5 r2 &omega (23)

Using Jupiter parameters (R_planet = 7.02 ×10⁹        cm , r=0.1 AU, T_orbit = 12        years), we get a spin rate of 1 hour, which is one order of magnitude off.

A more exact approach would note that by Kepler's third law, p² = a³, so that, since p µ 1/ &omega >, &omega µ 1/r^3/2. The angular momentum will be the integral over r from the inner boundary to the outer boundary of 2 &pir &rhor^-3/2 r² = 2 &pi &rhor^3/2. Doing the math exactly, with the capture radius set to 0.1 AU, we get a spin rate for 10 hours, which is quite accurate.

4.3 Composition

The answer is somewhat surprising at first: ice. Looking at fig.  5 (see section 3.1.1 for a derivation of disk temperature) we see that the temperature in the inner solar system was too high for gases to condense, so that only rocky material and metals, like iron and nickel, could form and eventually accrete. But at about 5 AU, we see that the temperature of the disk dropped below the freezing point of water. It is no coincidence that Jupiter formed just beyond this point. With water ice to accrete as well as silicates and metals, the outer planets formed much more quickly then the inner planets did.

Once the outer planets reached a critical mass, they were able to start accreting gas from the protoplanetary disk, further increasing their masses. Around Jupiter and Saturn, the disk was dense enough so that they were able to accrete massive amounts of gas before exhausting the supply within their reaches. Uranus and Neptune had far less gas and dust close enough to accrete so they are smaller and more deficient in hydrogen and helium. Also, because of having less matter to build with in their area, they formed more slowly, so that the Sun may have reached the T Tauri stage and put a premature end to their growth by blowing away the disk from which they had been gathering gas. On the other hand, Neptune formed beyond the point at which methane freezes, so methane also was able to participate in the building of Neptune.

4.4 Satellites

It is natural to guess that the Jovian planets might have had accretion disks of their own, just as the Sun did, only on smaller scales. Such disks could equally well have given rise to moon systems in the same manner as the planets arose about the Sun. The trends seen in the solar system, such as composition, are even mirrored in the moon systems. For example, the innermost Galilean moon, Io, is mostly silicates and sulfur, while the outer moons, such as Ganymede and Europa are rich in ice.

The real question then becomes: at what point does the process of bodies having accretion disks and forming satellites have to stop? How large must a protoplanet be to form such a disk? Could a large moon form one? These questions are difficult to answer right now, as we have little evidence to work with, but they are intriguing to ask.

Earth's Moon is a bit of a enigma. Its orbit is far too circular to have been captured. Additionally, it shares its composition with Earths crust (except for being more depleted in volitiles), indicating a common origin. However, with a mass 1.2 percent Earth's, it seems unlikely to have formed out of a planetary accretion disk even if Earth had one (also unlikely due to Earth's small mass). Additionally, Earth's Moon is inclined a mere 6 ^° to the ecliptic, making its orbit well out of Earth's equatorial plane, furthering the suspicion that our Moon did not form in the usual manner. The consensus in the scientific community is with the ``Big Whack'' theory, positing that the proto-Earth was struck by another protoplanet of mass roughly 3 times Mars's present mass. The impact would have thrown up much crustal material, which could have formed the Moon. Computer models support this hypothesis (Carroll and Ostlie, 1996).

4.5 Magnetic Field

One might well ask the question, ``Why aren't the planetary interiors charge neutral?'' As a rule, large objects are charge neutral, so there should be no currents in the planetary interiors at all, rotation or no. In fact, the currents are caused by charge separation (positive charges moving outward and negative inward or vice versa), so that overall neutrality is preserved, but local net charges can exist.

This can come about in a few ways. One interesting, yet simple, one is caused by heat gradients. Consider a metal rod being heated at one end. The electrons in the hot end are more excited then those in the cold end so tend to travel to the cold end more then the cold end electrons travel to the hot end. However, the atomic nuclei, positively charged, are largely stationary, so the result is a net flow of electrons to the cold end of the rod, hence charge separation (equilibrium is reached when the Coulombic potential balances the thermal potential).

Another interesting (and instructive) route toward charge separation harks back to the creation of the planets. The Sun's magnetic field would have been stronger then (much of the field has been lost when charged particles were blown from the solar system). As the molten planetary interiors rotated, the nuclei and electrons, being separated due to the heat, would have drifted in opposite directions under the influence of the magnetic field. Again, this would cause charge separation. For a crude calculation of the Earth's magnetic field using a simple charge differentiation model with rotating spheres and shells of charges, see Appendix C.

The Jovian planets all have strong magnetic fields. This is probably because they all rotate quickly and are mostly fluid. Jupiter, the fastest rotator, has the strongest magnetic field, so this simplified model is not without its merit. It is thought that Jupiter and Saturn consist of an outer envelope of insulating hydrogen and helium within which lies a layer of conducting metallic hydrogen and helium (the layer is believed to extend from 0.2 to 0.8 of the radius of Jupiter and 0.4 to 0.5 the radius of Saturn). It is from the metallic hydrogen and helium that the magnetic field must arise. The case is probably similar for Uranus and Neptune.

Among the terrestrial planets, only Earth and Mercury possesses notable global magnetic fields (Mars and the Moon are known to have localized magnetic fields, indicating that at one time a field was probably present and has since disappeared). In the case of Mars, the absence of such a field is probably because the core of the planet has become frozen, hence can no longer circulate. Venus's small field may be due to its very slow rotation, although there has been argument that its core ought to circulate much more rapidly than the planet rotates (Merrill et al. 1996).

The fact that Earth and Mercury have liquid cores is actually tremendously interesting. If left to themselves, both planets should have cooled by now. In Earth's case, we know that decaying radioactive isotopes have provided the heat needed to keep the core molten. In the case of Mercury, being much smaller than Earth, something more is needed to maintain a liquid core. We would expect the rate of cooling to be proportional the surface area of the planet over the mass. Since mass in expected to rise as radius³ and surface area goes as radius², the rate of cooling should fall off as [1/( radius)]. Thus Mercury, with a radius 0.382 times that of Earth would be expected to cool 2.6 times faster then Earth. Something has to keep Mercury's core from solidifying, so it is conjectured that the core is not pure iron but has a substantial component of FeS. This is interesting, because this would imply that FeS was able to condense out of the protoplanetary nebula in the area of Mercury, putting constraints on the temperature and composition of the disk in the region near Mercury.

With a such small handful of bodies in our solar system and with rotation appearing to be so important in the generation of magnetic fields, one is tempted to plot the dipole moment of each body vs. the rotation rate (fig.  6). Surprisingly, this yields a fairly well-ordered plot. There has been great debate whether this is indicative of an underlying physical cause or if this is just a coincidence spawned by too few data points. It is likely that only measurements of other stars and planets will clear this up.

Unfortunately, there is little else to tell about magnetic fields. More data from the other planets is needed, as well more accurate models. Furthermore, since the regions of interest are the cores of the planets, it is doubtful if we will ever be able to directly test our hypothesis on the origins of the fields. However, it does seem apparent that composition of the planets⁴ seems to be of the utmost importance in determining the planet's field strength.

5 Prospects for Life-Bearing Planets

While the study of the origin of planets and the discovery of extrasolar planets is scientifically fascinating, and holds public interest as well, it pales in fascination with the prospect of extra-terrestrial life. At this time, very little information about the creation of life exists, since we know of only one place where life developed (two if Mars did, in fact, once harbor life). But we make some tentative statements about what is needed to make a life-friendly planet, based upon what we believe is needed to support life ⁵.

The first and probably most important requirement we need to impose is that the planet should be terrestrial. It seems unlikely that life as we know it could ever form in a Jovian planet. Slightly more subtlety, we believe that life would require liquid water. The reason for this is two-fold. First, life on Earth uses water as a medium in which the biological processes can freely occur. Less geocentric but more important, it is likely that if Earth did not possess liquid water, it would have turned into a runaway greenhouse planet like Venus, making it too hot to harbor life (Earth's oceans trapped CO₂, greatly reducing the greenhouse effect). This puts a constraint on where the planet can be relative to its Sun, since water is only a liquid between 273 K and 373 K. Using equation (6), we can solve for what values of d will give temperatures in this range. For a Sun-like star (luminosity 3.8 ×10³³[(ergs)/( sec)]), planets can be expected to have liquid water between 8.33 ×10¹² cm (0.56 AU) 1.55 ×10¹³ cm (1.03 AU) ⁶.

Another important concern is a debris-free environment. If much of the planetesimal and smaller debris that was so vital in the formation of the planets remained in the vicinity of a potentially life bearing planet, frequent collisions would make evolution and development difficult at best and perhaps impossible. How does this debris get cleared away from potentially live-bearing planets? Surprisingly, Jovian planets are the answer. In our solar system, Jupiter (and to a lesser degree, Saturn) has perturbed the orbits of such small debris over the years, eventually causing it to either enter the Sun or to be flung from our solar system. Thus, life-bearing planets ought to be found in solar systems with Jovian planets ⁷

Other more subtle and (perhaps) less vital concerns include magnetic fields and large moons. The magnetic field of Earth protects its fragile life from many types of high-energy particles that could easily damage the organisms. Earth's large Moon may have stabilized Earth's spin, keeping the planet from precessing about as much as it otherwise might, thus maintaining steadier seasons ⁸. Armed with these and similar thoughts, future hunts for extra-terrestrial life may be able to focus on a few likely candidates rather than a myriad of improbable ones.

6 Conclusion

The origin of the planets is not easy to research, but thanks to careful detective work, computer modeling and a robust theory, we feel that we may have a good idea of what happens as worlds form. As old as the theory is, however, it is still being modified and adjusted as new ideas emerge and computer modeling capabilities improve. Just as in the past two decades the theory has made enormous strides, we can expect to see equally brilliant advances in the next few decades as our solar system (and others!) unlock their mysteries to human probes, telescopes and minds.

Fig.  7 backs this theory up, since you'll notice our solar system has a lot more angular momentum than it should based on the best-fit line to other stars' angular momenta. What is most intriguing, however, is that our solar system lies along the extrapolation of the best-fit line for the high mass stars, before the ``elbow'' in the graph. This could well be taken to imply that the elbow occurs because the angular momentum that we would expect to see in the lower mass stars, extrapolating from their high mass cousins, is locked away in planets that we cannot see (yet!). If this is the case, then planets should be expected around most lower mass stars⁹, a happy prospect indeed.

Appendix C

A Simple Model of the Earth's Dynamo

Figure 8: Diagram for finding the field due to a loop of current (from Griffiths 1989)

Consider a loop of radius R carrying current I. The magnetic field at a point P z unit up the axis (see fig.  8) due this current will be (from the Biot-Savart Law):

Now, the field at P due to a rotating disk of surface charge &sigma can be thought of as the integral over loops of radius 0 to R. Bearing in mind that the current at each r depends on r as I = &omegar &sigmadr, we get:

B(z) = 2 &omega &sigma &pi c ó õ R 0   r3 (r2 + z2)3/2 dr = 2 &omega &sigma &pi c ( R2 + z2    _______ √R2 + 2 z2   - 2 z) (26)

Again, a sphere is just a stack of disks, so we can perform the same trick as above to get:

B(z) = &rho &omega > 15 c z3 8 &pi(-30 R z4 +(R+z)3 (R2 - 3 z R +6 z2)- (R-z)3 (R2 + 3 z R + 6 z2)) (27)

Since the Earth rotates once every 23 hours and 56 minutes (one sidereal day), &omega = 7.29 ×10^-5 [1/( sec)] . Assuming that the negative charge migrates outward, and that the positive charges is confined to the inner 1/10 R_Å while the negative charge is in a spherical shell¹⁰ of radius 1/10 R_Å to 2/10 R_Å, we get that the field of the Earth is about one Gauss (which is what we in fact measure) if the charge density in the positive and negative regions is about 1 esu/cm³ which doesn't seem altogether unreasonable. For more complete models, see Merrill et al. 1996.

Glossary

Accretion Disk

A disk of matter spiraling down onto a central body.

AU

Astronomical Unit; the mean distance between the Earth and the Sun (1.4960 ×10¹³ cm).

Chondrite

A meteorite made up of chondrules.

Chondrule

A small bead of silicates condensed out of the early proto-stellar disk.

Ecliptic

The plane in which the Earth orbits about the Sun.

Keplerian Orbit

An orbit following Kepler's three laws of planetary motion:

1. Planets travel in elliptical orbits with the sun at one focus.
2. The area that a line from the planet to the sun sweeps out in any time is the same at any point in the orbit.
3. If a is the semi-major axis of the orbit and p is the period, then a³ ∝ p².

All of Kepler's Laws can be derived from from Newton's Laws of Motion and Newton's Law of Universal Gravitation.

Planetesimal

Solid bodies in the solar system that are smaller than planets but at least 1 km in size.

T Tauri

The stage is stellar birth in which fusion begins in the stellar core. This results in a strong stellar wind sweeping material away from the new star.

References

J. R. P. Angel and N. J. Woolf, Än Imaging Nulling Interferometer to Study Extrasolar Planets", ApJ, 475:373-379 (1997)

Bradely Carroll and Dale Ostlie, An Introduction to Modern Astrophysics, Addison- Wesley Pulbishing Company, Reading, MA, 1996

Guillot, Burrows, Hubbard, Lunine and Saumon, "Giant Planets at Small Orbital Distance", ApJ, 459:L35-L38 (1996)

Philip Hinz, J. Roger Angel, William F. Hoffman, Donald McCarthy, Patrick McGuire, Matt Cheselka, Joseph Hora and Neville Woolfe, ``Imaging Circum stellar Environments with an Nulling Interferometer'', Nature, 395:251-252, (1998)

Holland, Greaves, Zuckerman, Webb, McCarthy, Coulson, Walther, Dents, Gear and Robson, "Submillimeter Images of Dusty Febris Around Nearby Stars", Nature, 392:788-791 (1998)

David J. Griffiths, Introduction to Electrodynamics, Prentice-Hill Inc., Upper Saddle River, NJ, 1989

D. N. C. Lin, P. Bodenheimer , and D. C. Richardson, Örbital Migration of  the Planetary Companion of 51 Pegasi to its Present Location", Nature, 380:606- 607 (1996)

Jack Lissauer, "Planet Formation", Ann. Rev. Astron. Astrophys. (1993), 31:129-174

R. A. Rasio, C. A. Tout, S. H. Lubow, and M. Livio, "Tidal Decay of Close Planetary Orbits", ApJ, 470:1187-1191 (1996)

Ronald Merrill, Michael McElhinney, and Phillip McFadden, The Magnetic Field of the Earth, Academic Press, San Diego, 1996

Stuart Ross Taylor, Solar System Evolution: A New Perspective, Cambridge University Press, New York, 1992

John A. Wood, "Forging the Planets", Sky and Telescope, January 1999, pp. 36-48

Bibliography

Introductory

Ray Jayawardhana, "Spying on Planetary Nurseries", Astronomy, November 1998, pp. 63- 67

A nice article on where we now stand regarding studying the disks around other stars that are believed to give rise to planets.

John A. Wood, "Forging the Planets", Sky and Telescope, January 1999, pp. 36-48

An excellent article on planet formation. A great place for the non-scientist (or even the scientist!) to start. The article is rather long, but also very thorough.

Intermediate

A. G. W. Cameron, Örigin of the Solar System", Ann. Rev. Astron. Astro phys. (1988), 26:441-472

A very good look at planet formation in our solar system. Not as good as Lissauer, but close.

Bradely Carroll and Dale Ostlie, An Introduction to Modern Astrophysics, Addison-Wesley Publishing Company, Reading, MA, 1996

A fine astrophysics text; I used it for two derivations, and it was quite good for that purpose.

Kisha Delain, ``Shining Beacons in Space: A Look at Active Galactic Nuclei'', Integrative Exercise, 1996

This really has nothing to do with my topic, except that I looked at her paper for style hints. But three years ago I promised Kisha that I would reference her comps here, so here it is!

Jack J. Lissauer, "Planet Formation", Ann. Rev. Astron. Astrophys. (1993), 31:129-174

I used this work heavily. The work is thorough and easy to read. Highly recommended.

James Pollack, Örigin and History of the Outer Planets: Theoretical Models and Observational Constraints", Ann. Rev. Astron. Astrophys. (1984), 22:389-424

A more in-depth look at only the Jovian planets and the special concerns that go into them in terms of planet formation. This work also has planetary data, which is very useful.

Stuart Ross Taylor, Solar System Evolution: A New Perspective, Cambridge University Press, New York, 1992

This work is full of material, but has little math, so it was useful for the descriptions and data, but not tremendously helpful overall.

George Wetherill, "Formation of the Terrestrial Planets", Ann. Rev. Astron. Astrophys. (1980), 18:77-113

Another look at planet formation, this time focusing on the terrestrial planets. I found it rather old and bit out of date.

Advanced

David Black and Mildred Matthews, (editors), Protostars and Planets II, Uni versity of Arizona Press, Tucson, AZ, 1985

See Gehrels 1978

S. F. Dermott, The Origin of the Solar System, John Wiley and Sons, New York, 1978

A lot like the Protostars and protoplanets series; this volume also suffers from age.

Tom Gehrels, Protstars and Planets, University of Arizona Press, Tucson, AZ, 1978

Overall, the Protostars and Protoplanets series was moderately useful. Many of the older articles seemed out of date, and over all they were often a little too focused to be really helpful.

Eugene Levy and Jonathan Lunine, Protostars and Planets III, University of Arizona Press, Tucson, AZ, 1993

See Gehrels 1978; this is the most current and the most helpful of the three volumes.

D. N. C. Lin, P. Bodenheimer , and D. C. Richardson, Örbital Migration of the Planetary Companion of 51 Pegasi to its Present Location", Nature, 380:606- 607 (1996)

A short news blurb on theories of how the ``hot Jupiters'' moved in so close to their parent stars. The explanations were, however, fairly terse, so they didn't alway clear up things terribly well.

Ronald Merrill, Michael McElhinney, and Phillip McFadden, The Magnetic Field of the Earth, Academic Press, San Diego, 1996

A very handy tool. It contains both material on dynamo theory and a chapter on the magnetic fields of the Sun and other planets. Very nice, although it could have been more thorough and better explained.

Footnotes:

¹ Observations of clouds that are beginning to collapse and form stars are typically of this size, so this is a perfectly reasonable value.

² The disks absorb radiation and re-emit it as black body radiation as we saw above; the temperatures of the disks would make the emission in the infrared.

³ A density wave is a density enhancement that propagates through the disk; it moves through the matter in the disk while the matter itself enters and leaves the wave as the wave passes. Such density waves on a grander scale are believed to also be responsible for spiral structure in galaxies.

⁴ Both in the sense of Jovian vs terrestrial and in the sense of the exact chemical composition

⁵ Of course, to have potentially life-bearing planets, one needs planets in the first place. Planets have already been found around other stars; for a brief look at the prospects of finding planets commonly around stars, see Appendix B

⁶ This figure is a bit naive; slow rotation and atmospheric effects can blur the line somewhat.

⁷ This is not a harsh constraint. Not only are Jovian planets easily and quickly formed, some research indicates that Jovian planets may be required in order to form terrestrial worlds.

⁸ This, of all the issues raised, would likely be the hardest condition to meet, given the Moon's unusual and likely improbable formation. Fortunately, it is probably the least important concern.

⁹ High-mass stars are unlikely to have planets, as a rule, since they form much more quickly then their low-mass cousins. Thus, planets don't have sufficient time to form before the T Tauri stage sets in.

¹⁰ To accomplish a shell instead of a ball, I merely find the field due to a ball of radius equal to the outer radius of the shell and then subtract off the field due to a ball of charge rotating in the opposite direction and with radius equal to the inner radius of the shell. This effectively creates a rotating shell of charge.