The method of the renormalization group was originally introduced by Gell-Mann and Low as a means of dealing with the failure of perturbation theory at very high energies in quantum electrodynamics. An n-loop contribution to an amplitude involving momenta of order q, such as the vacuum polarization Πμν(q), is found to contain up to n factors of In() as well as a factor αn, so perturbation theory will break down when is large, even though the fine structure constant a is small. Even in a massless theory like a non-Abelian gauge theory we must introduce some scale μ to specify a renormalization point at which the renormalized coupling constants are to be defined, and in this case we encounter logarithms In(E/μ), so that perturbation theory may break down if E ≫ μ or E ≪ μ, even if the coupling constant is small.

Fortunately, there is a modified version of perturbation theory that can often be used in such cases. The key idea of this approach consists in the introduction of coupling constants gμ defined at a sliding renormalization scale μ — that is, a scale that is not related to particle masses in any fixed way. By then choosing μ to be of the same order of magnitude as the energy E that is typical of the process in question, the factors In(E/μ) are rendered harmless. We can then do perturbation theory as long as gμ remains small.

There are subtleties in the implications of symmetries in quantum field theory that have no counterpart in classical theories. Even in renormalizable theories, the infinities in quantum field theory require that some sort of regulator or cut-off be used in actual calculations. The regulator may violate symmetries of the theory, and even when this regulator is removed at the end of the calculation it may leave traces of this symmetry violation. This problem first emerged in trying to understand the decay rate of the neutral pion, in the form of an anomaly that violates a global symmetry of the strong interactions. Anomalies can also violate gauge symmetries, but in this case the theory becomes inconsistent, so that the condition of anomaly cancellation may be used as a constraint on physical gauge theories. The importance of anomalies will become even more apparent in the next chapter, where we shall study the non-perturbative effects of anomalies in the presence of topologically non-trivial field configurations.

The π° Decay Problem

By the mid-1960s the picture of the pion as a Goldstone boson associated with a spontaneously broken SU(2)⊗ SU(2) symmetry of the strong interactions had scored a number of successes, outlined here in Chapter 19. However, this picture also had a few outstanding failures. The most disturbing had to do with the rate of the dominant decay mode of the neutral pion, π0→ 2γ.

Most of this book has been devoted to applications of quantum field theory that can at least be described in perturbation theory, whether or not the perturbation series actually works well numerically. In using perturbation theory, we expand the action around the usual spacetime-independent vacuum values of the fields, keeping the leading quadratic term in the exponential exp(iI), and treating all terms of higher order in the fields as small corrections. Starting in the mid-1970s, there has been a growing interest in effects that arise because there are extended spacetime-dependent field configurations, such as those known as instantons, that are also stationary ‘points’ of the action. In principle, we must include these configurations in path integrals and sum over fluctuations around them. (In Section 20.7 we have already seen an example of an instanton configuration, applied in a different context.) Although such non-perturbative contributions are often highly suppressed, they are large in quantum chromodynamics, and produce interesting exotic effects in the standard electroweak theory.

There are also extended field configurations that occur, not only as correction terms in path integrals for processes involving ordinary particles, but also as possible components of actual physical states. These configurations include some that are particle-like, such as magnetic monopoles and skyrmions, which are concentrated around a point in space or, equivalently, around a world line in spacetime. There are also string-like configurations, similar to the vortex lines in superconductors discussed in Section 21.6, which are concentrated around a line in space or, equivalently, around a world sheet in spacetime.

During the first few hours after each impact, numerous phenomena were observed with telescopes on Earth, in orbit, and in space. The primary events in that time were: impacts themselves, rise and fall of large plumes of ejected material, and atmospheric waves; also of interest were the characteristic morphologies of fresh sites. Based on timing from Galileo instruments and ground-based observations, the Hubble Space Telescope (HST) recorded actual impact phenomena for fragments G and W, with the A and E impacts occurring just prior to the HST observation window. For these four events, plumes were directly imaged; plume development and collapse correlated with strong infrared emission near the jovian limb, supporting the interpretation that the IR brightness was created by the fall-back of plume material from high altitude (see chapter by Nicholson). For medium-to-large fresh impact sites imaged by HST within a few hours of impact, expanding rings were detected, caused by horizontal propagation of atmospheric waves (see chapters by Ingersoll and Zahnle). Initial site morphology at visible wavelengths was similar for all medium-to-large impacts: a dark streak surrounded by dark material, dominated by a large crescent-shaped ejecta to the southeast. Smaller impact sites typically only showed a dark patch (no ejecta) which dissipated quickly. This chapter summarizes the most recent measurements and interpretations of plumes and fresh impact sites as observed by HST.

The dark clouds that were easily seen in small telescopes after the comet impacts were caused by small particles which were deposited in Jupiter's stratosphere. Observations from the Hubble Space Telescope and from ground-based instruments at visible and infrared wavelengths indicate that the mean radius of the particles is in the range 0.1 to 0.3 μm, and the total volume of particles is approximately the same as that for a 1-km diameter sphere. In the dark core regions of freshly-formed impacts, the particles are distributed over a large vertical extent, between about 1 mb and 200 mb or deeper. The diffuse outlying haze is confined to the high-altitude end of the range. Such a distribution probably reflects different methods of emplacement of the debris as a function of distance from the impact. The color of the particles, and their volatility as required to make waves visible, suggest an organic material as the main constituent. These relatively volatile materials are thought to have condensed onto more refractory grains after the plume material cooled, some 30 minutes or more after impact. The most refractory materials expected to condense from an evolving fireball are Al2O3, magnesium and iron silicates, and soot, depending on the C/O ratio. A silicate spectral feature was observed, confirming that cometary material was incorporated into the grains, although silicate grains make up only 10–20% of the particle volume.

Galileo observations in the UV, visible, and infrared uniquely characterize the luminous phenomena associated primarily with the early stages of the impacts of SL9 fragments—the bolide and fireball phases—because of the spacecraft's direct view of the impact sites. The single luminous events, typically 1 min in duration at near-IR wavelengths, are interpreted as initial bolide flashes in the stratosphere followed immediately by development of a fireball above the ammonia clouds, which subsequently rises, expands, and cools from ∼ 8000 K to ∼ 1000 K over the first minute. The brightnesses of the bolide phases were remarkably similar for disparate events, including L and N, which were among the biggest and smallest of the impacts as classified by Earth-based phenomena. Subsequent fireball brightnesses differ much more, suggesting that the similar-sized fragments were near the threshold for creating fireballs and large dark features on Jupiter's face. Both bolides and fireballs were much dimmer than had been predicted before the impacts, implying that impactor masses were small (∼0.5 km diameter). Galileo data clarify the physical interpretation of the “first precursor,” as observed from Earth: it probably represents a massive meteor storm accompanying the main fragment, peaking ∼10s before the fragment penetrates to the tropopause; hints of behind-the-limb luminous phenomena, recorded from Earth immediately following the peak of the first precursor, may be due to reflection of the late bolide/early fireball stages from comet debris very high in Jupiter's atmosphere.

Measurements of thermal emission in spectral regions, ranging from the near-infrared to mm wavelengths provide information on the atmospheric thermal structure over impact sites from μbar levels in the upper stratosphere down to the upper troposphere. Systematic time series of observations relevant to this entire height range over individual spots do not exist. However, by piecing together information at different times from various spots, it is possible to obtain a provisional, semi-quantitative picture of the behavior of the thermal structure over a typical impact site. Immediately after fall-back of the ejecta plume, the upper stratosphere is heated to ∼ 600–1300 K above ambient temperature. The amplitude of the temperature perturbation diminishes with increasing depth in the atmosphere, but even in the upper troposphere a temperature increase of a few kelvins is observed. Initially, the upper stratosphere cools very rapidly with time scales of tens of minutes, presumably the result of strong radiative cooling associated with the high temperatures. After the initial cooling, all levels continue to cool at slower rates with time scales of a few days; however, this is still very rapid compared to radiative cooling of the ambient atmosphere. Enhancements in infrared opacity necessary to produce the cooling radiatively do not appear to be viable, suggesting that dynamical effects may play a dominant role. Possible mechanisms include horizontal mixing with the ambient atmosphere and adiabatic cooling produced by upward motion associated with an anticyclonic vortex.

Two months after the discovery of comet Shoemaker-Levy 9 came the astonishing announcement that the comet would impact Jupiter in July 1994. Computing the orbital motion of this remarkable comet presented several unusual challenges. We review the pre-impact orbit computations and impact predictions for SL9, from the preliminary orbit solutions shortly after discovery to the final set of predictions before the impacts. The final set of predicted impact times were systematically early by an average of 7 minutes, probably due to systematic errors in the reference star catalogs used in the reduction of the fragments' astrometric positions. The actual impact times were inferred from the times of observed phenomena for 16 of the impacts. Orbit solutions for the fragments were refined by using the actual impact times as additional data, and by estimating and removing measurement biases from the astrometric observations. The final orbit solutions for 21 fragments are tabulated, along with final estimates of the impact times and locations. The pre-breakup orbital history of the comet was investigated statistically, via a Monte Carlo analysis. The progenitor nucleus of SL9 was most likely captured by Jupiter around 1929 ± 9 years. Prior to capture, the comet was in a low-eccentricity, low-inclination heliocentric orbit entirely inside Jupiter's orbit, or, less likely, entirely outside. The ensemble of possible pre-capture orbits is consistent with a group of Jupiter family comets known as the quasi-Hildas.

The breakup of Comet Shoemaker-Levy 9 is discussed both in the context of splitting as a cometary phenomenon, comparing this object with other split comets, and as an event with its own idiosyncrasies. The physical appearance of the comet is described, features diagnostic of the nature of tidal splitting are identified, and the implications for modelling the event are spelled out. Among the emphasized issues is the problem of secondary fragmentation, which documents the comet's continuing disintegration during 1992–94 and implies that in July 1992 the parent object split tidally near Jupiter into 10–12, not 21, major fragments. Also addressed are the controversies involving models of a strengthless agglomerate versus a discrete cohesive mass and estimates for the sizes of the progenitor and its fragments.

Introduction

Splitting is a relatively common phenomenon among comets, even though its detection is observationally difficult because companions are almost invariably very diffuse objects with considerable short-term brightness variations. Comet Shoemaker-Levy 9's behavior was generally less erratic than that of an average split comet, which may have in part been due to a major role of large-sized dust. The breakup products that contributed most significantly to the comet's total brightness are referred to below as components, or, because of their diffuse appearance, as condensations, both common terms of cometary phenomenology. The terms nuclei and fragments are instead reserved for genuine solid bodies of substantial dimensions (≳ 1 km across) that were “hidden” in the condensations.

A new analytical model that is calibrated against numerical simulations performed with the CTH shock physics code provides a useful description of the entry of Periodic Comet Shoemaker-Levy 9 into the Jovian atmosphere. Mass loss due to radiative heating of fragments larger than 100 m in diameter is insignificant because of energy conservation during the ablative process. Nevertheless, radiative ablation is a major contributor to atmospheric energy deposition at high altitude and plays an important role in early-time fireball evolution. The analytical model provides the initial conditions from which fireball and plume evolution can be calculated using CTH. The results from these simulations suggest that if the tops of the plumes originated from a specific level of the Jovian atmosphere then maximum plume heights are independent of fragment size provided the fragments penetrated at least 30 km below this level. If the tops of the plumes originated from the visible cloud tops, then fragment masses greater than 4 × 1012 g, corresponding to 200 m diameter fully dense water ice, are required to explain the observations. If the plumes originated from the NH4SH layer then masses greater than 3 × 1013 g (400 m water ice) are required. The lateral extent and mass of the observable plume are functions of fragment size and contribute to the lateral extent and albedo of the debris patterns after re-impact with the atmosphere.

The SL9 impacts are known by their plumes. Several of these were imaged by HST towering 3000 km above Jupiter's limb. The heat released when they fell produced the famous infrared main events. The reentry shocks must have been significantly hotter than the observed color temperature would imply, which indicates that the shocks were radiatively cooled, and that most of the energy released on reentry was radiated. This allows us to use the infrared luminosities of the main event to estimate the energy of the impacts; we find that the R impact released some 0.3 − 1 × 1027 ergs. Shock chemistry generates a suite of molecules not usually seen on Jupiter. The chemistry reflects a wide range of different shock temperatures, pressures, and gas compositions. The primary product, apart from H2, is CO, the yield of which depends only weakly on the comet's composition, and so can be used to weigh the comet. Abundant water and S2 are consistent with a somewhat oxidized gas (presumably the comet itself), but the absence of SO2 and CO2 shows that conditions were neither too oxidizing nor the shocks too hot. Meanwhile, production of CS, CS2, and HCN appears to require a source in dry jovian air; i.e., the airbursts occurred above the jovian water table. Tidal disruption calculations and models of the infrared light curves agree on an average fragment diameter of about half a kilometer.

Images of Jupiter taken by the Hubble Space Telescope (HST) reveal two concentric circular rings surrounding five of the impact sites from comet Shoemaker-Levy 9 (SL9). The rings are visible 1.0 to 2.5 hours after the impacts. The outer ring expands at a constant rate of 450 ms−1. The inner ring expands at about half that speed. The rings appear to be waves. Other features (diffuse rings and crescent) further out appear to be debris thrown out by the impact. Sound waves (p-modes), internal gravity waves (g-modes), surface gravity waves (f-modes), and rotational waves (r-modes) all are excited by the impacts. Most of these waves do not match the slow speed, relatively large amplitude, and narrow width of the observed rings. Ingersoll and Kanamori have argued that internal gravity waves trapped in a stable layer within the putative water cloud are the only waves that can match the observations. If they are correct, and if moist convection in the water cloud is producing the stable layer, then the O/H ratio on Jupiter is roughly ten times that on the Sun.

Introduction

Much of what we know about the interior of the Earth has come from the study of seismic waves—a branch of seismology. Recently, much has been learned about the interior of the Sun from helioseismology. Now, the SL9 impacts give us an opportunity to do jovian seismology. The waves probe Jupiter's atmosphere to depths that cannot be reached by remote-sensing instruments.

One-dimensional photochemical models are used to provide an assessment of the chemical composition of the Shoemaker-Levy 9 impact sites soon after the impacts, and over time, as the impact-derived molecular species evolve due to photochemical processes. Photochemical model predictions are compared with the observed temporal variation of the impact-derived molecules in order to place constraints on the initial composition at the impact sites and on the amount of aerosol debris deposited in the stratosphere. The time variation of NH3, HCN, OCS, and H2S in the photochemical models roughly parallels that of the observations. S2 persists too long in the photochemical models, suggesting that some of the estimated chemical rates constants and/or initial conditions (e.g., the assumed altitude distribution or abundance of S2) are incorrect. Models predict that CS and CO persist for months or years in the jovian stratosphere. Observations indicate that the model results with regard to CS are qualitatively correct (although the measured CS abundance demonstrates the need for a larger assumed initial abundance of CS in the models), but that CO appears to be more stable in the models than is indicated by observations. The reason for this discrepancy is unknown. We use model-data comparisons to learn more about the unique photochemical processes occurring after the impacts.

The Hubble Space Telescope Wide Field Planetary Camera 2 imaging data provide the highest spatial resolution of individual Shoemaker-Levy 9 impact sites. Analysis of images obtained with the F410M filter yielded horizontal translation rates of tropospheric cloud structures and the east-west components have been interpreted as zonal winds which vary with latitude. When the tropospheric zonal winds between −60° and −30°, which were derived from the SL9 images, are compared with Voyager data there are no discernible changes in the magnitude or latitudinal positions of wind minima and maxima. This result provides additional evidence of the long-term stability of the zonal winds. Changes in individual sites during a two week period in July 1994 have been mapped. Their evolution is consistent with zonal winds decreasing with height and it provides evidence that local circulation associated with isolated weather systems perturbs the lower stratosphere.

Introduction

On July 16, 1994 at 21h30–51m the first multicolor images revealed the site of the A fragment impact of Comet P/Shoemaker-Levy 9 (SL9) as it rotated into view about 1.5 hours after it formed. The lack of color dependence and the resulting orientation and morphology of the ejecta blanket had not been anticipated. The blowout region was located more to the east than expected and dark rings and crescent-shaped structures centered on the impact site were observed, but the most obvious aspect of site A was the dark core (see the chapter by Hammel).

Earth-based observations at near- and mid-infrared wavelengths were obtained for at least 15 of the SL9 impacts, ranging from the spectacular G, K and L events to the barely-detected N and V impacts. Although there were a few exceptions, most of the IR lightcurves fit a common pattern of one or two relatively faint precursor flashes, followed several minutes later by the main infrared event as the explosively-ejected plume crashed down onto the jovian atmosphere. Correlations with the impact times recorded by the Galileo spacecraft and plumes imaged by the Hubble Space Telescope lead to an interpretation of the twin precursors in terms of (i) the entry of the bolide into the upper atmosphere, and (ii) the re-appearance of the rising fireball above Jupiter's limb. Positive correlations are observed between the peak IR flux observed during the splashback phase and both pre-impact size estimates for the individual SL9 fragments and the scale of the resulting ejecta deposits. None of the fragments observed to have moved off the main train of the comet by May 1994 produced a significant impact signature. Earth-based fireball temperature estimates are on the order of 750 K, 30–60 sec after impact. For the larger impacts, the unexpectedly protracted fireball emission at 2.3 μm remains unexplained. A wide range of temperatures has been inferred for the splashback phase, where shocks are expected to have heated the re-entering plume material at least briefly to several thousand K, and further modelling is required to reconcile these data.

This review attempts to give a coherent explanation of the main observations of the entry Comet Shoemaker-Levy 9 and the aftermath of the resulting explosions by using models of the tidal breakup of the comet, the entry of individual fragments into the jovian atmosphere, and the resulting fireballs and plumes. A critical review shows that the models appear reasonably well understood. The biggest theoretical uncertainties currently concern how to best tie models of the entry to models of the resulting fireballs. The key unknown before the impact was the size and kinetic energy of the comet fragments. The evidence now available includes the behavior of the chain of fragments, the luminosity of the observed visible fireballs and later infrared emission, the chemistry of the spots, and the lack of seismic waves or perturbations at the water cloud pressure level. These observations point to the fragments having diameters under a kilometer, densities of order 0.5 g cm−3, and kinetic energies of order 1027 erg.

Introduction

In this review and in the review by Zahnle (this volume; hereafter “the plume review”), we make the argument that the fragments of Comet Shoemaker-Levy 9 that hit Jupiter were quite small, with diameters of under a kilometer and densities of order 0.5 g cm−3. The largest fragments probably had kinetic energies of order 1027 ergs.