Understanding the formation of giant planets with substantial gaseous envelopes forces us to confront once again the physics of the gas within the protoplanetary disk. Unlike the case of terrestrial planet formation, two qualitatively different theories have been proposed to account for the formation of massive planets. In the core accretion theory of giant planet formation, the acquisition of a massive envelope of gas is the final act of a story that begins with the formation of a core of rock and ice via the identical processes that we discussed in the context of terrestrial planet formation. The time scale for giant planet formation in this model – and to a large extent its viability – hinges on how quickly the core can be assembled and on how rapidly the gas in the envelope can cool and accrete on to the core. In the competing disk instability theory, giant planets form promptly via the gravitational fragmentation of an unstable protoplanetary disk – a purely gaseous analog of the Goldreich–Ward mechanism for planetesimal formation that we discussed in Chapter 4. Fragmentation turns out to require that the disk be able to cool on a relatively short time scale that is comparable to the orbital time scale, and whether these conditions are realized within disks is the main theoretical issue that remains unresolved. Drawing on our prior results on gravitational instabilities in disks and on terrestrial planet formation, the goal in this chapter is to describe the physical principles behind both models and to provide a summary of some of the relevant observational constraints.

The classical theory of giant planet formation described in the preceding chapter predicts that massive planets ought to form on approximately circular orbits, with a strong preference for formation in the outer disk at a few AU or beyond. Most currently known extrasolar planets have orbits that are grossly inconsistent with these predictions and, irrespective of the still open question of what the typical planetary system looks like, their existence demands an explanation. Even within the Solar System the existence of a large resonant population of Kuiper Belt Objects, and the time scale problem for the formation of Uranus and Neptune, suggest that the classical theory is at best incomplete.

In this chapter we describe a set of physical mechanisms – gas disk migration, planetesimal scattering, and planet–planet scattering – that promise to reconcile the observed properties of extrasolar planetary systems with theory. The common feature of all of these mechanisms is that they result in energy and angular momentum exchange either among newly formed planets, or between planets and leftover solid or gaseous debris in the system. The exchange of energy and angular momentum drives evolution of the planetary semi-major axis and eccentricity, which can be substantial enough to make the final architecture of the system unrecognizable from its state immediately after planet formation.

This chapter is devoted to population orbit determination, that is not just computing the orbit for a single object, but compiling a catalog of orbits given a large number of observations. A survey is a project aiming at collecting observations of the largest and most representative sample of objects possible. We deal here only with the case in which the target population belongs to the Solar System; of course an astronomical survey may target simultaneously extrasolar populations. We deal with Earth satellites in Section 8.7. This chapter is based on our papers (Milani et al. 2005a, Milani et al. 2008, Milani et al. 2006) and ongoing research, in particular that in preparation for Pan-STARRS, a next-generation survey.

Operational constraints of Solar System surveys

The following three arguments should be taken into account in the definition of an identification/orbit determination procedure for a modern sky survey.

First, Moore's law tells us that the number of elements in an electronic chip grows exponentially with time; the doubling time has been around 18 months for more than 30 years. There is no indication that this trend might slow down; although in the last few years it has no longer been possible to increase the clock frequency, the increase in the complexity of the chips is now used to produce “multicore” CPUs. Assuming the multicores are used in an efficient parallelization procedure, the practical performance of computers continues to increase by a factor of 4 every three years.

One of the main assumptions used in Chapter 1 is that the dynamical model is deterministic. This assumption can be too optimistic for celestial bodies small enough to be significantly affected by complex non-gravitational interactions. Both drag and radiation pressure can be so poorly known that the errors in the dynamical model can affect the predictions by amounts exceeding, by orders of magnitude, the measurement accuracy.

When this is the case, there are three possible ways out, including the multi-arc strategy presented in this chapter. The others are the use of on-board accelerometers, see Chapters 16, 17, and the empirical parameterization of the unknown effects, see Section 14.5.

The multi-arc approach gives up the attempt to model the orbit of the spacecraft, over the entire time span of the observations, in a deterministic way with a single set of initial conditions. The time span of the observations is decomposed into shorter intervals and the set of observations belonging to each interval is called an observed arc, or just an arc. Each arc has its own set of initial conditions, as if there were a new spacecraft for each one of them. This results in over-parameterization, with the additional initial conditions absorbing the dynamical model uncertainties. Other parameters, e.g., in the dynamic model, can also be local to a single arc.