oklo

In a comment on yesterday’s core-accretion post, a reader anticipated that all is not hunky-dory with the core-accretion scenario for the formation of the gas giant planets in our solar system, and asked if is there any support for Alan Boss’ disk instability model. In the Boss model (described here by Alan, see also the buff 137-strong citation list) gas giant planets condense directly out of the protostellar disk as the result of gravitational instability in the disk.

The handy thing about an extrasolar planet web log is that you can express your opinions on the formation of extrasolar planets. In my opinion, there are a number of very serious difficulties with the hypothesis that gravitational instability is the dominant mechanism for giant planet formation. Here are three:

(1) In order to have gravitational instability work in the manner shown in the fragmentation simulations, you need to start with an axisymmetric disk that has a sufficiently low value for the Toomre Q parameter. That is, in order for the initial conditions in the successful Boss simulations to be valid, a growing protostellar disk needs to remain completely stable with respect to low-level non-axisymmetric disturbances until BOOM, it reaches a threshold Q value where it is prone to spiral instabilities that exponentiate on a near-orbital timescale.

In reality, I think that a growing (or alternately, a cooling) protostellar disk will be prone to low-level spiral disturbances that steadily transport mass inward and angular momentum outward, allowing the disk to avoid ever reaching the state where instabitilies can grow on an orbital timescale. (For a bulked-up version of this argument, see the papers (one and two) that I wrote with Vladimir Korchagin and Fred Adams on this issue).

(2) The core accretion model provides a very natural explanation for both the planet-metallicity correlation, as well as the paucity of Jovian-mass planets found in orbit around low-mass M type stars. The gravitational instability model predicts that the incidence of Jovian-mass planets should be independant of both the stellar metallicity and the parent star mass.

(3) There’s simply no way that the gravitational instability model can produce the 72 Earth Masses of heavy elements in HD 149026 b. (See this paper for a thorough discussion).

To be fair, there are also some thorny problems associated with core-accretion. In the next few posts of the giant planet formation series [1, 2, 3, 4 and 5] that we’ve been running, I’ll describe these in more detail.

Photo credit: ESO (VLT/NACO)

Another important point to stress is that Alan’s simulations certainly aren’t in error in the sense of being computationally wrong. It’s just that I don’t agree with the generic validity of the initial conditions. Indeed, I do think that gravitational instability sometimes plays a role in giant planet formation. The best example is probably the 5 Jupiter-mass companion to the brown dwarf 2M1207 discovered by Chauvin et al. last year. (The ESO press release on this system is here.) I see no way in which the core-accretion process could have made any headway at 55 AU in this particular system.

Finally, GJ 876, which is by far the best RV-characterized extrasolar planetary system, provides a tough challenge to both the gravitational instability and the core-accretion theories. The inner 7.5 Earth Mass planet in the GJ 876 system is almost certainly an accreted protoplanetary core (regardless of whether it formed in-situ, or migrated from a larger radius). It would be nearly impossible to form lil’ D via gravitational instability. The outer two planets, on the other hand, contain more than three Jupiters worth of mass, and stand in embarrassing conflict with the notion that core-accretion process is difficult to carry through to Jovian-mass completion in red-dwarf protostellar disks.

I would very much like the 411 on what went down in GJ 876’s protostellar disk.

[A continuation of posts 1, 2, 3, 4 and 5 on the formation of Jovian planets.]

Phoebe (photographed by Cassini). The cores of the giant planets were built from millions of these objects.

In the primitive solar system, ice formed at the expense of water vapor wherever the temperature was lower than 150 degrees above absolute zero. The 150 K isotherm in the disk was located roughly at Jupiter’s current distance from the Sun, and is known colloquially as the “snowline”. Just beyond the snowline, the planetesimals achieved their greatest ability to rapidly build themselves into larger bodies. It was cold enough for ice to be stable, yet close enough for the overall density of the disk to be high, and the planetesimals were prone to frequent collisions.

Low-speed planetesimals collisions were sticky events that can be simulated to wonderful effect with fast desktop computers. Imagine two Michelin Men, each loosely glued together, heading toward each other in a headlong embrace. A spare tire or two is lost in the collision, but one remains with a jumbled, combined mess. The first planetesimal collisions that seeded Jupiter’s core looked something like that.

Thousands of years passed, punctuated by these (initially) slow-motion catastrophes. The planetesimals gradually become fewer in number and individually larger. Those that experienced a few extra collisions in the beginning were able to take advantage of their burgeoning self-gravity to collide more often, and were thus able to grow faster (sound familiar?) Inevitably, a few big winners, called oligarchs, began to emerge. These oligarchs, with radii thousands of miles across, were massive enough to simply haul in their neighboring small-fry kilometer-sized brethren. The more an oligarch gets, the more it wants, and the farther its reach. A runaway occurs. Somewhere in the current vicinity of Jupiter, 4.54 billion years ago, an oligarch reached an Earth mass.

What would this oligarch have been like? Certainly, it would be something that we would have little difficulty calling a planet in distress. All riled up. A five hundred mile wide core of molten iron, surrounded by perhaps a thousand miles of pressurized plastic rock, not unlike the mantle of the Earth. Above that, thousands upon thousands of miles of hot, pressurized, water ocean. Floating atop the ocean, a tarry layer of hydrocarbons, perhaps with a smell like hot asphalt, and with an indistinct surface merging into a choking thick noxious atmosphere.

The atmosphere bulks up fast. Liberated hydrogen and helium gas bubbles up from the layers of denser materials in the interior. The oligarch passes several Earth masses in size, and grows massive enough to grab gas directly from the disk. Meanwhile, new planetesimals are arriving all the time. Kilometer-sized projectiles streak through the exosphere, exploding as they slam into the atmosphere. The unsettled skies are continuously ablaze with meteors. The temperature rises, becoming so warm that the atmosphere glows a dull coal-red in the darkness of the nebula.

When the growing oligarch, now a full-fledged protoplanet, reaches seven or ten times the mass of the Earth, it is pulling in gas as fast as it can. The atmosphere has swelled and bloated to a thickness of literally hundreds of thousands of kilometers. The gas glows fire-engine orange, and pours infrared light out into space. This radiation is accompanied by slow settling of the lower layers, providing room at the top for more gas to flow in.

Finally, the growth experiences its first taste of a slowdown. The consumption of rocky icy planetesimals has been so rapid that the total reservoir of these objects in the annular region of the nebula occupied by the planet is depleted. The planet has managed to effectively clear out the solid material from a vast ring-like region of the nebula. The region around the protoplanet still contains vast quantities of gas, but this gas is prevented from accreting onto the bloated planet. The planet can only add new gas as fast as the older gas is able to settle, and the gas can only settle by radiating and cooling.

For the next million years, the planet grows slowly. Gas flows in from the disk as fast as the cooling of the planet will allow, and as the mass of the planet increases, the annular ring of the disk from which planetesimals can be drawn also slowly increases in size…

[A continuation of posts 1, 2, 3, and 4 on the formation of Jovian planets.]

The idea that the planets in our solar system arose from a flattened, rotating cloud of gas and dust dates back to Kant and Laplace in the 1700s. Their so-called nebular hypothesis drew part of its original support from the spurious suggestion (by William Herschel and others) that the spiral “nebulae” such as M31 in Andromedae might be solar systems caught in the early phases of formation.

By the late 1800s, however, it had become clear that spiral galaxies are most certainly not protoplanetary disks. This realization removed a primary pillar of observational support for the nebular hypothesis, and forced theories of planet formation to rest largely on assumptions and theoretical arguments. For the majority of the twentieth century, astronomers trying to figure out how the planets came to be were forced to work backward from the more or less static clues that are provided by the condition of the solar system today. Not a happy situation. Science works best with direct observations. To really understand how planets form, we really need to see the formation process in action.

As it turned out, protostellar disks around newborn stars were observed before the discovery of the first extrasolar planets. In the early 1980’s, the IRAS infrared satellite discovered that Beta Pictoris, a young, apparently ordinary, sun-like star 53 light years from Earth, was glowing unexpectedly brightly in infrared light. When Beta Pictoris was examined with careful follow-up observations, it was found to be orbited by a large flattened disk of dusty particles. After this first discovery, many more protoplanetary disks were discovered. Beautiful examples occur, for example, in the Orion Nebula, where they are imaged by the Hubble Space Telescope in stark rigid detail against a glowing backdrop of nebulosity. From careful study of these disks, we know that they generally contain anywhere from 1 to 100 times the mass of Jupiter, and are composed primarily of hydrogen and helium gas, along with swarms of dust and icy particles.

If we assume that giant planets do not condense directly out of these disks as the result of gravitational instability, then we need a coherent picture for forming the planets that we know actually do exist. The current best-guess scenario for forming Jovian-mass planets is called the core accretion theory.

In the core accretion picture, planets start small, through the buildup of dust.

If you have a hardwood floor, you can develop a hands-on sense of how dust agglomeration works by refraining from vacuuming under the bed. If you do this, you will notice that the dust does not accumulate in a uniformly thick layer with time. Rather, the presence of slight air currents swirls the dust around, and causes it to build up into dust bunnies. Look at a dust bunny under a magnifying glass, or put it on a flat-bed scanner and import it into photoshop (that is what I did to generate the image at the top of this post). It’s mostly air. The dust – hair, dandruff, unidentifiable strands of ticky-tacky, has a structure that takes up a large volume in comparison to its mass. This property makes it effective at scooping up more material. Once dust agglomerations begin to grow, their subsequent growth becomes easier. A similar agglomerative process may be at work in building up the dust agglomerations that are present in protostellar disks.

Even so, the initial growth of dusty, icy objects in a proto-planetary disk seems fraught with difficulty. The problem is that as the dust-ice agglomerates become larger and larger, they experience a headwind from the gas in the disk. This headwind causes them to spiral inward, eventually vaporizing as they get close to the central star. Some mechanism must exist to concentrate the dusty debris and allow it to build in size more quickly than it can be destroyed through spiraling inward. There seem to be two reasonable candidates for sequestering dust. The protoplanetary nebula might contain vortices, that is, storm systems in the disk itself, in which regions of the disk participate in a hurricane-like flow pattern. Numerical simulations show that disk vortices, if they live long enough, can trap and concentrate solid particles in their centers. Another possibility is gravitational instability (of a more restricted type than dramatic variety described in post #3). If the gas in the disk is flowing very smoothly, then the solid particles in the flow will have a tendency to settle to a thin layer at the disk mid-plane. If this mid-plane layer grows dense enough and massive enough, then a gravitational runaway can occur. The solid particles, the dust, the ice, the gravel can rapidly form larger and larger objects. Once these objects attain a certain size, several tens of kilometers, say, they are safe from the drag force exerted by the nebular gas. A best-guess scenario has tens of trillions of kilometer-size planetesimals emerging in the disk a hundred thousand years or so after the disk forms.

Trillions of planetesimals sounds like a lot. Nevertheless, the disk at that stage would not have seemed particularly crowded. The density of gas would have been thousands of times less than the density of air, and the distance between kilometer-sized bodies would be measured in thousands of miles. If you could transport yourself to a random point in the middle reaches of the disk, there would seem to be only relentlessly empty blackness. No view of the stars, no view of the young forming sun. It would seem as if nothing had changed from the earlier molecular cloud phase.

A thermometer, however would indicate that a difference does exist. Whereas the molecular cloud was incredibly cold, 5 or 10 degrees above absolute zero, the temperatures in the protostellar disk are much warmer, ranging from hundreds, even thousands of degrees very near the star, down to several tens of degrees above absolute zero in the farthest reaches of the disk.

This post continues the oklo.org posts: (1) the black cloud, and (2) disks.

There are two competing, completely distinct theories that describe how a giant planet like Jupiter can be generated from a protostellar disk of gas and dust. The first theory, formation via gravitational instability, lends itself to large-scale hydrodynamical simulations and extraordinary animations that can be downloaded over the Internet. It’s an easy theory to grasp. The second theory, formation via core accretion, presents a more complicated chain of events, but nevertheless contains the story that seems (in my opinionated opinion) to be most nearly correct. Let’s look at what these two theories say, and let’s examine the evidence in favor of and against each.

In the gravitational instability picture, the outer lagging remnants of the molecular cloud core fall in and land on the protostellar disk, causing it to grow in mass. As the disk mass increases, it begins to be influenced by its own gravity. That is, it starts to feel a tendency to fragment in response to its own weight. Simultaneously, the pressure of the gas in each nascent fragment pushes back and partially offsets the fragment’s inclination toward collapse. Pressure thus acts as a small-scale stabilizing influence against collapse. In addition, the differential rotation of the disk (material closer to the star orbits faster) tries to sheer a growing fragment apart. Differential rotation thus acts as a large-scale stabilizing influence against gravitational collapse.

The question boils down to the following: Does gravity win, allowing a Jupiter-mass planet to rapidly form as a condensation in the disk, or do shear and pressure win, keeping the disk free of giant-planet fragments?

The situation lies within the general framework of a linearized hydrodynamical stability analyses, and can be analyzed mathematically. The analysis leads to a so-called stability criterion, the famous Toomre Q:

Where c_s is the sound speed in the disk, kappa is the epicyclic frequency, G is Newton’s gravitational constant, and sigma is the disk surface density. If Q<1 at any radius in the disk, then the disk is unstable with respect to m=0 (ringlike) disturbances. If Q is slightly greater than 1, computer simulations show that the disk is prone to strong non-axisymmetric instabilities, and hence experiences exponential growth of disturbances and eventual fragmentation.

As with any seemingly abstruse physical phenomenon, The disk instability analysis can be illuminated with an analogy. In this case, the appropriate analogy involves a rock band, a house party, kegs of free beer, and uninvited punks and thugs.

Neophyte rock bands need to attract audiences for their shows. Hence, they need to provide inducements. Free beer does the trick. Free beer, or more precisely, flyers posted all over a college campus advertising a party serving free beer, act in analogy to the self-gravity of a disk. As I have discovered (through direct experience, back in my reckless, rock-band fronting youth), such a course of action can lead to instability. If you flyer a campus with news of free kegs, then dozens to hundreds of punks and thugs, whom no-one has ever seen before, and whom no-one wants to see again, will descend upon the hapless band’s house-party show. Amplifiers are destroyed. Holes are kicked in sheetrock. The cops show up, and the band does not play. This outcome can be profitably compared to a disk that undergoes a gravitational collapse into Jovian-mass fragments.

[Above: Our bass player (at a show of ours that was shut down by the cops after several songs). He later graduated with a Ph.D. in Physics, after defending his dissertation on 2D quantum black holes.]

In practice, however, the police do not always show up at house-party shows. Sometimes, the band gets to play. This happier outcome is abetted by two stabilizing effects. Just as in the case of the disk gravitational instability, one of these stabilizing effects operates on large scales, and the other operates on small scales. On the large scale, one can create an analog of “differential rotation” with a lack of specificity on the flyers regarding the precise time of the show. Punks drift in. They see that they don’t particularly like how the band sounds. They see the long lines to the kegs. They drift away. The band plays its entire set to a modest audience, and the cops don’t show up. Support on small scales, the analog of “pressure” is provided by a quite different effect: body odor. The thugs that show up invariably smell poorly, and the unpleasantness associated with a sweaty throng of them will drive some away. If the pressure is high enough, that is, if the thugs smell badly enough, then the show proceeds, and instability is again averted.

For readers familiar with the linearized analysis that leads to the Toomre Q criterion, here’s an illustration of how the analogy can be applied to the standard WKB dispersion relation:

In a future post in this series, we’ll explain why the weight of observational and theoretical evidence seems to be shifting against the gravitational instability hypothesis. The computer simulations, which become ever more impressive with each inexorable tick of Moore’s law, show that in order for fragments to form and then last as planets, the rate of cooling in the disk must be extremely efficient. Rapid cooling robs a nascent fragment of its ability to produce pressure, and hence permits gravitational collapse. Perhaps more importantly, the computer simulations also show that a disk will suffer from a whole panoply of instabilities before its mass grows large enough to trigger the full-blown collapse of Jupiter-like planets.

These instabilities take the form of spiral waves of the same type that occur in spiral disk galaxies such as M51, shown in the HST photo at the top of the post. In a protostellar disk, the spiral wave action pushes pulse after pulse of gas out of the regions of the disk that are in the most danger of fragmenting directly into planets. Some of this gas is forced to large distances from the central star, while the majority flows inward and eventually winds up on the star. In all likelihood, most protoplanetary disks manage to avoid direct fragmentation.