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(PDF file)		 Planetary Formation The Planets After Formation

Aki Roberge
May 5, 1997
First Year Seminar Paper
Johns Hopkins University


Planetary Formation and Our Solar System



Introduction:


Planetary astronomy is a young science, and until recently, was essentially devoted to the study of planetary bodies in our own Solar System. The discovery of non-stellar objects orbiting other stars has suddenly changed that and has opened a whole new realm of planetary science. But still, our own Solar System is by far the easiest to study, and there is still a great deal that is unknown. In this paper, I would like to introduce some important basic characteristics of our Solar System and discuss the currently accepted theory of planetary formation, the Nebular Disk Model.


Age of the Solar System:


The age of the Solar System is roughtly 4.6 billion years. This age is determined primarily from radioactive dating of meteorites and the oldest rocks from the Earth's crust. Radioactive dating has also been used to determine the time-frame of the Solar System formation process. Relatively speaking, it happened very rapidly.

Our Solar System contains a relatively large amount of elements heavier than iron, which are created in novae or supernovae. This element-forming event (or events) happened only a few million years before the formation of solid planetary material. Most meteorites formed during a 20 million year period, and the Earth formed within 100 million years after that. Shown at right is an oxygen-rich supernova remnant located in the Large Magellanic Cloud.

In 1977, A.G. Cameron and J. Truran theorized that the element-forming event was a supernova and this became widely accepted. But in 1985, Cameron retracted this statement and argued instead that the element-forming event was an ordinary nova (or novae.) I believe the question remains open at this time. But at any rate, a good model of Solar System formation must allow rapid formation of planetary material.


Characterictics of the Solar System:


There are several important general characteristics of the Solar System that must be explained by any theory of its origin.

General Characteristics Of The Solar System To Be Explained

1. All the planet's orbits lie roughly in the same plane, referred to as the ecliptic plane.






2. The Sun's rotational equator co-incides with this plane.
3. The planets and the Sun all revolve in the same west-to-east direction, called prograde revolution.
4. Planetary orbits are nearly circular.
5. The planets contain much more of the net angular momentum of the Solar System. (The Sun rotates slower than expected by some early formation theories.)
6. The distances between the planets mostly obey the simple Bode's Rule.

To apply Bode's rule, one simply writes down a sequence of 4's, one for each planet and the asteroid belt, except Neptune. Then add the sequence 0, 3, 6, 12, 24, ... to the list of 4's. Dividing each of the results by 10 gives the mean distance of the planets from the Sun, in terms of astronomical units, (AU). One astronomical unit is the mean distance of the Earth from the Sun.




Reference: Hartmann, Moons and Planets, 3rd ed., 1993 p.17

This rule was developed in 1772 by Johann Bode and lacked any theoretical justification at the time. But it was confirmed in 1781 when William Herschel discovered Uranus at 19.2 AU. After this, a search was made for the "missing planet," which should lie between Mars and Jupiter. This led directly to the discovery of the asteroid belt in 1801. While Bode's Rule is not exact, it is too accurate to be entirely accidental. Theoretical justification of this rule is one of the goals of planetary formation models.
7. The planets differ in composition, roughly corelating with distance from the Sun.

The mean density of the planets decreases with increasing distance from the Sun. High density rocky planets (Mercury, Venus, Earth, Mars) lie close to the Sun, while the low density gas giants (Jupiter, Saturn, Uranus, Neptune) are further away.






As can be seen from the above plot of mean density vs. distance from the Sun, the planet with the highest mean density is Earth, while Saturn, which actually has a density less than that of liquid water, has the lowest. The densities decrease with distance, then increase again for the outermost planets.

Solar Formation:


To preface discussion of planetary formation, it is necessary to first discuss the formation of our Sun itself. Our Sun is an average star, currently in its long, stable, mid-life. The Sun is referred to as being on the main sequence in this phase of its life.

Stars are initially formed from the interstellar medium (ISM), which consists of gas and small dust grains of condensed material. The details of the process are something of a theoretical problem, but roughly, a local density increase in the ISM occurs, which allows the gravitational collapse of the interstellar cloud.

The Virial Theorem states that for a bound, stable system, the gravitational potential energy equals the negative of twice the kinetic energy. This result may be derived from quantum mechanics or classically, from the equation of hydrostatic equilibrium. Thus, for collapse to occur, the gravitational porential energy of the cloud must exceed twice its thermal kinetic energy. This allows one to calculate a critical density which must be exceeded for collapse to occur. The details of how this density fluctuation arises are not well understood, but one theory is that it can be caused by a supernova shock wave.

The collapsing cloud is now referred to as cocoon nebula, with a protostar forming at the center. Once the collapse is well under-way, the now-isolated cloud roughly obeys the Virial Theorem. This indicates that half of the potential energy lost during a decrease in radius goes into heating the cloud, and half must be radiated away. The growing density and therefore opacity of the cloud severely impedes energy loss at optical wavelengths, so the important cooling mechanism is infrared radiation by molecular hydrogen and dust grains. A cooling mechanism must exist or the pressure due to increasing kinetic energy of the particles would resist further collapse.

Additionally, the cloud will have some net angular momentum due to galactic rotation. As the cloud shrinks, its moment of inertia decreases and its angular velocity must increase in order for angular momentum to be conserved. So, the cloud will begin to rotate about some axis.

Material above and below the plane of rotation may collapse toward the plane while conserving angular momentum. But for motion about the central axis, the centrifugal force may balance the gravitational force, giving a stable orbit. Shrinkage to a smaller radius is precluded unless the centrifugal force is reduced, which may only happen if angular momentum is ejected from the system. This can happen, as the inner parts of the cloud pass angular momentum to the outer parts through viscosity in the gas, but it is much slower than the collapse of material to the plane of rotation. Thus, the initial spherical nebula collapses into a flat, rotating disk within a few million years.



Reference: Hartmann, Moons and Planets, 3rd ed., 1993 p.99

Accretion of material onto the protosun has now slowed and disk is almost in equilibrium, with gravity, gas pressure, and centrifugal force all in balance. Collisional interactions and tidal forces will tend to circularize the orbits of the particles in the disk. This discussion has so far explained characteristics 1 through 4, relating to the orbital motions of the Solar System bodies.

There is one further effect which acts to resist collapse of the interstellar cloud. Some small proportion of the cloud consists of ionized gasses and there are magnetic field lines associated with the moving charges. As the cloud shrinks, it carries the field lines with it, increasing the magnetic field density. This translates into a build-up of magnetic pressure that resists collapse just as thermal gas pressure does. Any models of the birth of the Solar System must therefore take this into account and the analysis would require magnetohydrodynamics. A brief discussion however of one consequence of the magnetic field interactions in the early Solar System will help to explain characteristic 5, the fact that most of the angular momentum of the Solar System is contained in the planets, not the Sun.

Initially, as a result of the contraction process, the Sun should have been spinning much faster than it does today. It is now believed that the magnetic field lines of the protosun would have tended to grip the nebula, causing a drag on the Sun. As the Sun slowed down, its angular momentum was transferred to the nebula. This would also explain the fact that T-Tauri stars, which are newly formed stars, rotate much faster than main sequence stars. T-Tauri stars have strong magnetic fields and nebulae, criteria for magnetic braking.

Now, to recap, we are at a stage where the Solar System consists of a newly formed protosun surrounded by a flat, rotating nebular disk of dust and gas. Shown at right is the Orion Nebula, a stellar birthplace, containing dozens of newly formed stars. Many of these stars have dust disks, indicating that they may be in the process of forming planetary systems. One is shown below at right.

From here on in, the Sun is of lesser interest and planetary formation theory is primatily concerned with processes in the nebular disk. But it is necessary to wrap up the Sun before moving on the the details of planetary formation.

Protoplanets form in the nebular disk and at some point, the protosun is massive enough and hot enough for hydrogen fusion to begin. The Sun turns on, vaporizing the dust in the inner Solar System. The disk continues to cool and planetary formation continues. Finally, the Sun goes into the T-Tauri phase with strong solar wind, which blows away the remaining gas, clearing the nebula. This phase occurs a few million years after the onset of fusion and is the event that ends planetary formation.


Planetary Formation:


To return to the nebular disk, it is in hydrostatic equilibrium in its vertical structure, with the force of gravity toward the plane of the disk balanced by the thermal gas pressure. This leads to the fact that the density doesn't vary much along the plane of the disk, but decreases with distance from the ecliptic plane. We also find that the scale height H, which is the distance above the ecliptic over which the density decreases by e, increases with distance from the Sun. Thus the disk thickness also increases with distance from the Sun.

We now need to consider the chemical evolution of the disk. Remember, contraction of the nebula warmed it up to about 2000 K. But once the disk formed, contraction slowed and the disk began to cool. As the disk cools, the various elements in it condense.

At around 1500 K, the refractory elements condense. These are elements like calcium, titanium, and aluminium. Then the more abundant iron, nickel, and silicates condense, which mostly make up what we call rocky material. Eventually, at low temperatures, the volatiles condensed (water, ammonia, and methane ices), which are the most abundant components of the nebular disk.

The temperature of the nebular disk decreases with distance from the Sun. Therefore, the rough compositional gradient of the planets can be explained by the fact that it never got cold enough in the inner Solar System for the volatile elements to condense before being blown away during the T-Tauri phase.

So, the inner planets, Mercury, Venus, Earth, and Mars, are essentially hunks of rock. It is thought that a significant portion of the water and other volatiles on the Earth's surface were delivered from the outer Solar System by comets and asteroids, early in the history of the Earth.

Now the question is how to build up huge solid bodies out of tiny grains of condensed material.

Steps in Accretion of Planetary Bodies

Step 1: 1 micron sized particles of condensed material collide slowly and stick electrostatically. This process can produce fluffy particles up to 1 cm in size.
Step 2: Inelastic collisions of the 1 cm sized particles allow bodies to grow up to approximately 1 km. At this point, two further processes come into play, one relating to the terrestrial planets and one to the gas giants.
Step 3a: (Terrestrial Planets) Further growth occurs by collisional accretion. In each zone of the disk, a single large planetesimal begins to dominate, sweeping up the remaining smaller bodies as they collide with it. As the planetesimal gets larger, the efficiency of the sweep-up increases. This is due to gravitational focussing, where the effective collisional cross-section of the body is greater than the actual radius, because of gravitational attraction.

This allows the single largest planetesimal to grow up to the thousand km scale. The impacts of the last few large planetesimals could account for the slight variations in the orbital characteristics of the final planets.
Step 3b: (Jovian Planets) Growth mainly occurs through gravitational accretion. Collisional accretion of km sized bodies allowed some to grow, as discussed for the terrestrial planets. These could grow to be 10 to 20 times the mass of the Earth however. This is because the more abundant volatile condensates are available in the outer regions of the nebula, due to the lower temperature of the solar nebula there.

Once the body got that large, it had such strong gravity that it began to draw in large amounts of uncondensed gaseous material directly from the solar nebula. This is a much more efficient process, leading to much more massive planets. Also, these planets will have lower mean densities. They will contain the same net amount of rocky material as the terrestrial planets, but will have lots and lots more low density gas.

In this region, large terrestrial planet-sized satellites may form as secondary condensations around the massive primary.


Computer models by Isaacman and Sagan have shown that a variety of planetary systems can be constructed using these principles, given varying initial conditions. Many of these strongly resemble our Solar System. In these models, the competition to sweep up planetesimals leads to a Bode's rule-like spacing between the final planets, helping to explain item 6 in our list of characteristics.

Returning to our plot of mean density with distance, we can now explain almost all of the qualitative features. The inner terrestrial planets have similar mean densities; none of them accreted much low density material, which was still vaporized in the hot inner Solar System.

As we move to Mars and the Asteriods, more volatile condensates like water were available, lowering the mean density. But these bodies were not able to grow large enough to gravitationally accrete any gasses. Jupiter and Saturn have low densities due to their high proportion of gas.

So far so good. However, we now see that the densities of the outermost bodies increase again. This is due to a somewhat subtle chemistry effect. In the much colder outer Solar System, carbon is able to bind with oxygen to form carbon monoxide gas, which mostly gets blown away. So, most of the carbon gets tied up in this gas and is not available to form low density methane condensates.

Ordinarily, this low density methane ice would be accreted onto the planet; since it's not, the planet has a higher proportion of heavy materials, like water and silicates. So you get planets with lower total masses and higher mean densities. This roughly explains the increase in mean density at the edge of the Solar System, although this effect alone may not be large enough to explain the high density of Pluto.

Post-Formation Processes:


I would like now to briefly touch on some post-formation processes, particularly differentiation. For bodies larger than about a few hundred km, internal gravitational pressure and heating from radioactive decay is sufficient to melt the interiors. The body becomes segregated, as heavier elements sink to the center and lighter ones rise to the surface.

So, one ends up with a body which has mostly iron and iron-affinity elements in the core, and mostly silicon and silicon-affinity elements in the outer layers. These outer layers are called the mantle and crust, or lithosphere.

Formation of our Moon:


I haven't discussed any of the various satellites in the Solar System, but just to finish up, I'd like to discuss one that was a thorny problem and is of particular interest.

Our own moon is quite unusual; it has a much lower density than the Earth and is unusually large for an inner Solar System satellite.

density of Moon = 3.3 g/cubic cm
density of Earth = 5.2 g/cubic cm


Its radius is 1738 km. The satellites of Mars, for comparison, have radii on the order of tens of km.

Additionally, analysis of Moon rocks shows them to have similar composition to the Earth's crust, but depleted in volatile elements, like water. This indicates that the Moon was once very hot after it formed, at which time the volatiles were vaporized and driven off. Finally, the Moon's orbit lies in the ecliptic plane, not in the Earth's plane of rotation, which is slightly tilted with respect to the ecliptic (giving rise to seasons on Earth.)

Three early theories of the Moon's formation were:
1. Binary accretion, which says that the Earth and Moon formed up independently at about the same radius from the Sun. This is not consistent with the low mean density of the Moon though. Why should two bodies that formed right next to each other be so compositionally different in an initially homogeneous solar nebula?
2. The second theory is the fission hypothesis, which says that the Moon spun off the rapidly rotating Earth. This is not consistent with the orbital plane of the Moon. If it spun off, it should be orbiting in the plane of the Earth's rotation, not the ecliptic plane.
3. The third theory is that the Moon was a captured planetesimal, which was formed elsewhere in the Solar System. But this is not consistent with the fact that the Moon's composition is like the Earth's crust, but volatile-depleted.

The currently accepted theory was put forward in 1975 and became widely accepted around 1984. It postulates that after the Earth had formed and differentiated, a remaining Mars-sized body crashed into the Earth, splashing off a large amount of molten lithospheric material. This material went into orbit around the Earth and coalesced to form the Moon.

This theory is consistent with the low density of the Moon, as the lithosphere is depleted in heavy elements, which are mainly in the core of the Earth. Impact heating would cause the escape of volatiles, and finally, a projectile is more likely to be in the ecliptic plane. The splashed material would therefore orbit in the ecliptic plane as well.

Addendum: Recently, questions have arisen as to the feasibility of this theory about the Moon's formation. Detailed modeling of the possible impact process now implies that the impacting body would have to have been much larger than was previously thought. This would make the collision much less likely, as there would have been few planetesimals large enough remaining at the time of the Moon's formation. Clearly, there's still work to be done.

This problem is illustrative of many that remain in planetary astronomy. In particular, the study of newly discovered extrasolar planets and young protoplanetary disks has begun to indicate there probably are serious problems with the theory of planetary formation briefly described above.

For more information about our Solar System today, see my second year seminar web paper, The Planets After Formation.

The information in this web paper is based on Hartmann, Moons and Planets, 3rd ed., 1993.