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namic flow that results in very rapid mass growth of the planet. As the planet mass undergoes a runaway growth, tidal torques exerted by the planet on the protoplanetary disk open a "gap" around the orbit of the planet. Subsequent mass in-flow to the planet continues at a reduced rate.

Growth by nucleated instability clearly involves two distinct timescales. First, the core must grow to critical mass. Second, the nebular gas must be accreted by the core. Core growth, which occurs by binary accretion as for the terrestrial planets, is the slower process. It is the principal cause of concern with the nucleated instability model and so has been the subject of much attention. The key issue is that the core must grow on a timescale that is short compared with the timescale for the dissipation of the gas nebula. Observations of young stars with dusts disks generally fail to reveal attendant gas, leading to the inference that the gas is quickly removed, probably on timescales of a few Myr for sun-like stars and almost certainly on timescales < 10 Myr [10]. This sets an upper limit to the core growth times and is a primary challenge to the core accretion model. One way in which core growth might have been accelerated is through an increase in the disk column density just beyond the snow-line, owing to the extra mass in solids added by the freeze-out of nebular water vapor [20]. Million year growth times at the orbit of Jupiter are not hard to obtain from current models, but more work is needed to induce Uranus and, especially, Neptune to grow on cosmically reasonable timescales.

A different giant planet growth scenario has been proposed in which the "slow step" of core accretion is side-stepped. In this model, the protoplane-tary disk is supposed to have been intrinsically unstable to collapse under its own gravity. Disk instabilities clearly favor higher than MMSN disks (models typically assume disk masses ~10 times the MMSN in order to produce spontaneous collapse), but even MMSN models have been reported to be susceptible to collapse under some circumstances [8]. Formation of giant planets by spontaneous collapse does not suffer the timescale problem of the nucleated instability model (because there is no need to wait for a nucleus to form), but there are other problems related to the long-term stability of the collapsing planet. Investigators differ on this issue. The differences are not fully understood, but might relate to the accuracy with which cooling processes are represented [14].

Neither core accretion nor nebula collapse predicted the over-abundance of heavy elements measured in Jupiter by the Galileo entry probe ( [120], see

Fig. 6. Metal abundances in Jupiter relative to those in the Sun, as measured by the Galileo entry probe. Helium and Neon are low in abundance because they are partly dissolved in the metallic hydrogen core. Oxygen is low, probably because the probe entered Jupiter's atmosphere at an (unrepresentative) hot-spot location, where conditions were atypically dry. The other measured elements are over-abundant relative to their Solar proportions. From [120]

Fig. 6. Metal abundances in Jupiter relative to those in the Sun, as measured by the Galileo entry probe. Helium and Neon are low in abundance because they are partly dissolved in the metallic hydrogen core. Oxygen is low, probably because the probe entered Jupiter's atmosphere at an (unrepresentative) hot-spot location, where conditions were atypically dry. The other measured elements are over-abundant relative to their Solar proportions. From [120]

Fig. 6). In fact, pure collapse models implicitly contradict it because gravitational instabilities provide no way to selectively accrete elements according to their molecular weight. Pressure gradient forces might help to concentrate solids near growing planets [56], and one might conjecture that Jupiter's heavy elements were accreted by the capture of ice-rich planetesimals in the extended atmosphere of the newly formed planet. There are problems with providing enough planetesimals to deliver the mass of Jupiter's metal excess above Solar composition. This process further fails to explain N and Ar, which are overabundant in Jupiter by factors of 3 or 4 (Fig. 6) but which are too volatile to be carried by asteroids or the known comets in any appreciable abundance. The suggestion advanced by Owen et al. [120] is that Jupiter's core grew by the accretion of ultra-cold (~30K) planetesimals, in which N, Ar, and other volatiles were efficiently trapped (probably by adsorption within amorphous water ice). But 30K is too cold to fit the protoplanetary disk at 5AU (c.f. Equation 2, which gives T = 125K at this distance). A convincing resolution of this puzzle has yet to be identified.

Fig. 7. Ice giant Uranus from the Voyager 2 spacecraft. Courtesy NASA Ice Giants

Compared to Jupiter and Saturn, Uranus (Fig. 7) and Neptune (Fig. 8) are an order of magnitude less massive and also compositionally distinct, being depleted in H2 and He. The bulk of their mass is contained in heavier elements that form ices at low temperatures, such as C, N, and O. Uranus and Neptune are known as "ice giants" for this reason. The difficulty in forming Uranus and Neptune on any reasonable timescale has motivated a number of novel, alternative suggestions. For example, in one well-publicized model, Uranus and Neptune are envisioned to have formed between Jupiter and Saturn, were then scattered outwards by mutual perturbations, and, finally, their orbits

Fig. 7. Ice giant Uranus from the Voyager 2 spacecraft. Courtesy NASA Ice Giants

Compared to Jupiter and Saturn, Uranus (Fig. 7) and Neptune (Fig. 8) are an order of magnitude less massive and also compositionally distinct, being depleted in H2 and He. The bulk of their mass is contained in heavier elements that form ices at low temperatures, such as C, N, and O. Uranus and Neptune are known as "ice giants" for this reason. The difficulty in forming Uranus and Neptune on any reasonable timescale has motivated a number of novel, alternative suggestions. For example, in one well-publicized model, Uranus and Neptune are envisioned to have formed between Jupiter and Saturn, were then scattered outwards by mutual perturbations, and, finally, their orbits

Fig. 8. Ice giant Neptune from the Voyager 2 spacecraft. Courtesy NASA

were circularized by friction with an assumed massive disk [149]. To make all this happen, the authors placed the giant planets initially at 6.0, 7.4, 9.0, and 11.1 AU and assumed that they were initially each of 10 M®, with an additional 95 M® of planetesimals between 12 AU and the assumed edge of the protoplanetary disk at 60 AU. In common with almost all other N-body Solar system simulations, they neglected collective interactions in the 95 M® disk (these might be expected to generate waves that could be important in the redistribution of angular momentum in the disk [155]). Dynamical effects of the few x 104 M® of nebular gas (which must also have been present in order to keep the overall disk composition in approximately cosmic proportions) were also neglected, except that some of this gas was used to feed the runaway growth of the gas giants. The authors assert that their scenario for Uranus and Neptune formation is insensitive to the above assumptions, and, indeed, it is easy to imagine that the first core to experience runaway mass growth should exert a strong gravitational influence on other cores nearby, perhaps scattering them outwards. On the other hand, the initial conditions may have been very different from the ones envisioned in [149]. Worst of all, it is not clear to me what new observations can be taken to test it.

An equally fascinating but rather different scenario for rapid ice giant formation assumes that these planets started out as gas giants and were then eroded down to their observed masses by intense fluxes of ionizing radiation from a nearby, massive star [9]. According to this model, the future ice giants are selectively depleted in mass relative to the surviving gas giants because they are more distant from the sun. Photoionized hydrogen (whose temperature is ~104K and thermal velocity ~10kms_1) escapes more rapidly from heliocentric orbit at the distances of Uranus and Neptune than at Jupiter and Saturn, leaving the former two planets unprotected from the radiation while the latter two are heavily shielded. Again, the authors do not suggest observational tests of this model, although non-thermal loss of gases from planetary atmospheres often leads to isotopic fractionation effects that might be expected in this extreme case.

The Domain of the Comets

There are several useful definitions of what it is to be a comet, not all of them mutually consistent. The different definitions are used concurrently, sometimes without a clear understanding of the differences between them. The three different classification schemes are idealized in Fig. 9.

Observationally, a comet is any object showing a gravitationally unbound atmosphere, known as a "coma" (from the Greek for "hair"). The coma is a low-surface brightness region surrounding the central, mass-dominant nucleus. It owes its brightness to a combination of sunlight resonantly scattered from molecules and molecular fragments (radicals) and light scattered from tiny dust particles entrained in the outflowing gas. The visibility of the coma depends on the instrumental sensitivity and angular resolution.

Fig. 9. Schematic diagram showing three different criteria for distinguishing comets from asteroids. Observationally, a comet is any body showing a coma (unbound atmosphere) at any point in its orbit. Dynamically, the distinction is made based on some model parameter, typically the Tisserand parameter, Tj. JFC, HFC, and LPC denote Jupiter-Family Comets, Halley-Family Comets, and Long-Period Comets. The Main-Belt Comets (MBCs) are located with the asteroids, in the middle panel of the figure. Compositionally, the distinction is based on the presence or absence of bulk ice in the body. The different definitions lead to the same classification in most cases, but there are growing numbers of bodies that are "cometary" by one definition but not the others

Fig. 9. Schematic diagram showing three different criteria for distinguishing comets from asteroids. Observationally, a comet is any body showing a coma (unbound atmosphere) at any point in its orbit. Dynamically, the distinction is made based on some model parameter, typically the Tisserand parameter, Tj. JFC, HFC, and LPC denote Jupiter-Family Comets, Halley-Family Comets, and Long-Period Comets. The Main-Belt Comets (MBCs) are located with the asteroids, in the middle panel of the figure. Compositionally, the distinction is based on the presence or absence of bulk ice in the body. The different definitions lead to the same classification in most cases, but there are growing numbers of bodies that are "cometary" by one definition but not the others

For this reason, objects that are discovered by survey telescopes as asteroids" (i.e., bodies having no atmospheres) are commonly reclassified as comets based on the subsequent detection of comae by observers using more sensitive telescopes. Moreover, the strength of the coma diminishes rapidly with heliocentric distance, falling to invisibility beyond the orbit of Jupiter except in a few unusual cases. On longer timescales, cometary activity can evolve in response to evolutionary process on the surface, in a crust or "mantle" that throttles the release of escaping gas. What appears as a comet now might look completely asteroidal to observers of the twenty second century. Obviously, this observational definition of comet-hood is not at all a perfect one.

Compositionally, a comet may be defined as a small body in which a substantial part of the mass is contained in ice. Practically, we may expect all objects that condensed beyond the "snow-line" to contain bulk water ice. The snow-line is now near the orbit of Jupiter; all small bodies from the Jovian Trojans outward are likely to be compositional comets by this reasoning, whether or not they show comae. In the past, the snow-line may have been closer to the sun, meaning that ice could be present in many of the main-belt asteroids. These bodies are compositionally comets. Unfortunately, we have no meaningful way to estimate the bulk composition of a body without drilling into it, and this definition of comet-hood is consequently hard to apply.

Dynamically, a comet is any body with a Tisserand parameter measured with respect to Jupiter, TJ < 3 (the main-belt asteroids have Tj > 3). The Tisserand parameter is a constant of the motion in the restricted, circular three-body approximation, defined by

where aJ is the semimajor axis of Jupiter's orbit (assumed circular); a, e, and i are the semimajor axis, eccentricity and inclination of the small body orbit. Bodies with TJ < 3 strongly interact with the planet, indicating a short dynamical lifetime and a source elsewhere. Those with TJ > 3 are effectively decoupled from the planet. This definition, although seemingly clean-cut, also suffers from ambiguity. Some main-belt asteroids can be scattered onto orbits with TJ < 3. A few comets (the most famous is 2P/Encke) have TJ > 3 (although only slightly so), making them dynamically asteroidal.

The timescale for the loss of volatiles from a body is just Tdv ~ M/ (dM/dt)), where M is the mass and dM/dt the rate of loss of mass. Whipple and authors since have assumed that mass loss is predominantly by sublimation [?], at a rate that can be calculated from the assumption of radiative equilibrium on the nucleus. There is growing evidence that the mass loss in at least some comets may be dominated by disintegration of the nucleus, in which mass is shed in macroscopic blocks or chunks rather than molecule-by-molecule as in the process of sublimation. Neglecting this possibility for the moment, we write the energy balance equation for a sublimating ice patch as a

Here, Lq is the luminosity of the Sun, R is the heliocentric distance, A and e are the albedo and the emissivity of the surface, 0 is the angle between the direction to the Sun and the surface normal, L(T) is the latent heat of sublimation of the ice at temperature T, dm/dt is the mass loss rate per unit area and fc represents the conducted energy flux from the surface while fg is the flux of energy carried by gas flow into the nucleus. A few things should be noted. The quantity Lq/(4nR2) is the flux of sunlight falling on the projected surface. When evaluated at R =1 AU, this quantity is called the Solar Constant, Fq, and has the value Fq = 1360 Wm-2. The first term on the right-hand side represents the power per unit area lost by radiation into space. The second term is the power per unit area consumed by sublimation. Physically this power is used to break the bonds connecting molecules together in the solid phase. The last term in the equation accounts for thermal conduction and can be either positive or negative, depending on the temperature gradient in the upper layers of the nucleus.

For a non-volatile (L ^ to) black-body (A = 0, e =1) material oriented perpendicular to the Sun (0 = 0) and neglecting thermal conduction, the temperature is just

1/4 393

This corresponds to the temperature at the sub-Solar point on a perfectly absorbing body. The average temperature on a spherical isothermal object will be reduced by a factor 41/4, because the average value of cos(0) over the sunlit hemisphere is 1/4, giving T — 278/rAUj .

For a sublimating surface, (3) cannot be solved without prior knowledge of the temperature dependence of the latent heat. The Clausius-Clapeyron equation (for the slope of the solid-gas phase boundary in pressure vs. temperature space) can be used or, more directly, measurements of the thermal pressure exerted by sublimating water ice as a function of temperature can be employed. For illustrative purposes, we here consider an extreme approximation.

When close to the Sun (say for Rau < 1 AU) water ice, the dominant cometary volatile, uses so much energy to sublimate that we may write

4nR2 K ' dt as a rough approximation to (4). Then, we see that the characteristic mass loss rate per unit area (again with 0 = 0) is just dt L(T)R2Al] 1 J

and we have assumed for simplicity that the surface is perfectly absorbing, A =0. Substituting Fq = 1360Wm~2 and L(T) = 2x106 Jkg^1 (for water ice), we have dm/dt — 7x10-4/RAU [kgs-1 m-2].

The rate at which the sublimation surface recedes into the body of the nucleus is just di -i dm . .

where p is the bulk density. With p ~ 1000 kg m-3, we estimate di/dt ~ 0.7 ^ms-1 at Raj = 1 AU. The sublimation lifetime of a nucleus of radius rn is then r n pr n

and, with the standard values as above, we obtain

In this equation, the unit of time is denoted yr1 to emphasize that it is the number of years of equivalent exposure to sunlight at 1 AU.

Of course, no real comets circle the Sun in the orbit of the Earth. Instead, they follow eccentric orbits with larger semimajor axes and are hot enough to sublimate only when they dip in to perihelion. Still, the approximation described above nicely illustrates the fact that sublimation can potentially limit the active lifetimes of the comets to very small values, certainly values that are tiny compared with the 4.6 Gyr age of the Solar system.

Less approximate solutions of the energy balance equation are plotted in Fig. 10. There I show the average value of dm/dt computed around the orbits of comets having eccentricities e = 0, 0.5 and 0.9, as a function of the semimajor axis. At a given semimajor axis, the net effect of non-zero eccentricity is to increase the orbitally averaged mass loss rate relative to the circular orbit approximation, because sublimation grows fast enough near perihelion to overwhelm the long period of inactivity as the comet sails out to and back from aphelion. Figure 10 shows that, for a typical short-period comet having a = 4 AU and e = 0.3, the orbitally averaged mass loss rate is dm/dt ~ 10-7 [kgs-1 m-2], giving di/dt ~ 10-10 ms-1 and Tdv ~ 3x10s yr.

All of the above is simplistic and intended merely to make a point, namely that sublimation can destroy nuclei quickly. We will have more to say about this later. For now, we use it to assert that the active comets must be derived from inactive source regions, if they are (as we believe) as old as the Solar system.

Source Regions

Three distinct source regions of the comets are now recognized. One, the Oort Cloud, was identified half a century ago [119] and is well known as the source of the long-period comets. The second, the Kuiper belt, was discovered in 1992 [73] and has played a major role in revamping our understanding of the Solar system. It is the source of the Jupiter Family Comets. The third, the

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