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Figure 2.13 A possible time line for the formation of the giant planets. The times are far more uncertain than the sequence of events.

Quite why this resonance explains the large inclination of Saturn is beyond our scope. The inclination of Uranus is remarkable (97.8°), though for this less massive planet this can be explained by the accumulated effects of icy-rocky planetesimals that just happened to nudge the rotation axis predominantly in one direction. Alternatively, an impact with a large embryo could account for it.

This is the core-accretion model of giant planet formation.

The time line in Figure 2.13 does present us with some difficulties. The first is that Neptune's kernel might not have formed before the T Tauri phase had swept far too much of the gas away for Neptune to acquire anywhere near the amount of hydrogen and helium that it contains. The second is that the T Tauri phase could have been earlier, which gives us the same difficulty with Uranus as we have with Neptune.

Both of these difficulties can be overcome if we include giant planet migration in our models. These models also account neatly for other features of the Solar System.

Giant planet migration in the Solar System

If Uranus and Neptune formed closer to the Sun than we find them today (but not as close as Jupiter or Saturn), their kernels could have formed rapidly enough so that by the time the gas disc was removed by the T Tauri wind of the Sun they had acquired their modest hydrogen-helium envelopes. In this case they must have since migrated outwards.

The evidence for giant planet migration in exoplanetary systems (Section 2.1.2) lends credibility to the view that migration has indeed occurred in the Solar System. Note that the migration of terrestrial planets is slight - their masses are too small to excite appreciable spiral density waves in the disc. By contrast, the giant planets could well have migrated significant distances. Indeed, it is possible that one or more (growing) giants were consumed by the proto-Sun. But clearly four survived, possibly through gravitational interactions between them, via the spiral density waves they each induced in the disc.

But if Uranus and Neptune formed closer to the Sun than they are today, and remained there until the solar nebula had largely dissipated, how could they have moved outwards? The answer is that there is another way for migration to occur. This is via the scattering of planetesimals from one giant to another.

A recent computer simulation starts, after clearance of the gas disc, with Jupiter, Saturn, Uranus, and Neptune in circular orbits at the respective solar distances 5.5 AU, 8.2 AU, 14.2 AU, and 11.5 AU (yes, Neptune is placed closer to the Sun than Uranus). □ So, which way does each giant need to move?

Jupiter needs to move inwards to 5.2 AU, Saturn outwards to 9.6 AU, Uranus outwards to 19.2 AU, and Neptune outwards to 30.1 AU. The model crucially includes a large population of planetesimals in the range 15-35 AU, which constituted the Edgeworth-Kuiper (E-K) belt at this time, though more distant bodies are not excluded. From the details of the simulation it emerges that Uranus and Neptune scattered planetesimals predominantly inwards, and as a result these two giants gained angular momentum, and thus moved outwards. Saturn is also a net inward scatterer, so also moves outwards, though not by much owing to its large mass. Jupiter is a net outward scatterer, so moves inwards, again not by much. Many of the outward-scattered planetesimals escape into interstellar space; those that do not quite make it contribute to the Oort cloud. Uranus and Neptune move into the E-K belt, ejecting many objects, some of which make a major contribution to a late heavy bombardment in the inner Solar System, others of which become the major component of the Oort cloud. The migration of Jupiter and Saturn disturbs the asteroid belt, and this also contributes to the heavy bombardment.

The simulated migration of Jupiter and Saturn causes them to pass through their mutual 2:1 mean motion resonance. This causes the eccentricity of the orbits of Uranus and Neptune to increase to the extent that they interact, with the outcome that they exchange orbits. Their orbits are reduced in eccentricity by further interaction with planetesimals. After several million years, with the planetesimal population depleted, we end up with the four giant planets in their present orbits. This kind of simulation can also explain the high eccentricity of Pluto, via the outward migration of Neptune, and its capture into its 3:2 mmr with Neptune.

Formation of giant planets by gravitational instability in the disc

In the mid 1990s, before the problem of the slow formation of the Uranus and Neptune kernels had been solved by models that included migration, a radical alternative model was put forward, which solved the problem, and also tackled some other difficulties. In this alternative model, the gas in the outer nebula becomes gravitationally unstable, and fragments of higher density form. Each of these contracts to create, in at least some cases, what is called a protoplanet, rather in the manner that the Sun formed at the centre of the nebula. The fragment further contracts to form the giant. This one stage process is distinctly different from the core-accretion model elaborated above. Note that the gas in the inner nebula is too hot to become gravitationally unstable, so the model does not change the mode of formation of the terrestrial planets.

Initially, this gravitational instability model seemed promising, particularly because fragments appear in the models when the nebula is only a few hundred years old. But further modelling has revealed huge difficulties. First, with a solar nebula twice the minimum mass solar nebula, the gas becomes gravitationally unstable only beyond about 10 AU. Even at 14 times the minimum, instability extends inwards to only about 7 AU, still beyond the orbit of Jupiter. Worse, the fragments are themselves unstable, and usually do not form protoplanets. At best, the fragments might promote kernel formation. It is also difficult to see how Uranus and Neptune can be so different in composition from Jupiter and Saturn. Therefore, the core-accretion model is secure, for the time being.

Question 2.8

If the proto-Sun went through its T Tauri phase much earlier than in Figure 2.13, what might the planets in the outer Solar System be like today?

2.2.6 The Origin of the Oort Cloud, the E-K Belt, and Pluto

Regardless of the model used, the Oort cloud consists of icy-rocky planetesimals that were flung out by the giant planets, but not fast enough to escape from the Solar System. The remaining question is how they became confined to a thick shell rather than retrace the orbit of ejection. This is because their perihelion distances were increased, and the orbital eccentricities consequently reduced, by the overall gravitational force of the stars and interstellar matter that constitute our Galaxy. The force has this effect because it varies across a planetesimal orbit, i.e. it is a differential force.

□ What is a suitable name for this force?

A differential force is a tidal force (Section 1.4.5), and so a suitable name is the Galactic tidal force, though it is usually called the Galactic tide. The planetesimal orbits were subsequently randomised in orientation by this tidal force and also by passing stars and giant molecular clouds, yielding a spherical shell of 1012-1013 bodies greater than a kilometre across, 103-105 AU from the Sun. The Oort members were thus emplaced (Section 1.2.3), perhaps on a time scale as short as 103Ma. In Section 3.2.6 you will see how the Oort cloud can account for some of the comets observed today in the inner Solar System.

In the migration model outlined in the previous section, Uranus and Neptune generated the majority of Oort cloud members, with Jupiter making a smaller contribution.

This model, and others like it, also show that few planetesimals within about 40 AU of the Sun survived the migration of Neptune to its final orbit, with a semimajor axis of 30.1 AU. Kepler's third law shows that an object with a semimajor axis of about 40 AU will orbit the Sun three times for every two orbits of Neptune - a 3:2 mmr. As Neptune migrated outwards it captured bodies into this resonance and swept them before it, thus clearing the space. The bodies beyond 40 AU are a mixture of planetesimals, even embryos, that formed from the solar nebula and have always resided there, and those scattered by the giant planets to modest distances. This mixture constitutes the E-K belt of icy-rocky bodies, with known sizes up to about 1500 km radius, and clustering around the mid plane of the erstwhile solar nebula.

That there are no giant planets beyond Neptune is readily explained by the low spatial density of objects in the E-K belt and their slow orbital motion, resulting in a very low collision rate, and the lack of sufficient nebular gas to reduce their eccentricities and hence their collision speeds - Figure 2.11 illustrates this in a different context. In Section 3.2.6 you will see that the E-K belt makes a further contribution to the observed comets.

In models that do not involve giant planet migration, the great majority of E-K objects (EKOs) are icy-rocky planetesimals that formed more or less where the E-K belt resides today.

The best-known member of the E-K belt is the planet Pluto, an icy-rocky body with a radius of 1153 km. With a semimajor axis of 39.8 AU it is in the 3:2 mmr with Neptune. The migration model shows that as Neptune migrated outwards it would have captured Pluto into this resonance when Neptune was at about 25 AU.

□ What would have been the semimajor axis of Pluto's orbit at this time? From Kepler's third law (equation (1.3)) this would have been 25 x (3/2)2/3 = 33AU (to two significant figures). As Pluto was pushed outwards in this resonance a secular resonance would have increased the orbital inclination of Pluto to about that observed, 17.1 °. Its orbital eccentricity would also have increased, again to about that observed, 0.25. This high eccentricity means that Pluto comes closer to the Sun than Neptune - see Figure 1.5. The orbits do not intersect because of their different orbital inclinations. Moreover, owing to the 3:2 resonance, Pluto is always near aphelion at the times the orbits are close, so Pluto and Neptune avoid the close approaches that would otherwise destabilise Pluto's orbit. As mentioned in Section 1.2.3, Pluto is not the largest EKO.

Interactions between large EKOs can account for Pluto's satellites, via capture, and the strange orbit of Triton (Section 2.3.1). Collisions of large EKOs with Neptune can explain its large axial inclination, 28.3°.

With Pluto and the E-K belt in place, there was ejection of some of the remaining objects in the giant region, and the collisional evolution of smaller objects everywhere, including the generation of dust. Thus, we have the Solar System as we see it today.

In Sections 3.2.6 and 3.2.7 we shall return to the Oort cloud and the E-K belt, and, in the case of the latter, explore its populations in more detail.

Question 2.9

In a few sentences, discuss whether the E-K belt could blend into the Oort cloud.

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