Peter Goldreich

Final Stages of Planet Formation

We address three questions regarding solar system planets: What determined their number? Why are their orbits nearly circular and coplanar? How long did they take to form? n nRunaway accretion in a disk of small bodies resulted in a tiny fraction of the bodies growing much larger than all the others. These big bodies dominated the viscous stirring of all bodies. Dynamical friction by small bodies cooled the random velocities of the big ones. Random velocities of small bodies were cooled by mutual collisions and/or gas drag. Runaway accretion terminated when the orbital separations of the big bodies became as wide as their feeding zones. This was followed by oligarchic growth during which the big bodies maintained similar masses and uniformly spaced semimajor axes. As the oligarchs grew, their number density decreased, but their surface mass density increased. We depart from standard treatments of planet formation by assuming that as the big bodies got bigger, the small ones got smaller as the result of undergoing a collisional fragmentation cascade. It follows that oligarchy was a brief stage in solar system evolution. n nWhen the oligarchs surface mass density matched that of the small bodies, dynamical friction was no longer able to balance viscous stirring, so their velocity dispersion increased to the extent that their orbits crossed. This marked the end of oligarchy. What happened next differed in the inner and outer parts of the planetary system. In the inner part, where the ratios of the escape velocities from the surfaces of the planets to the escape velocities from their orbits are smaller than unity, big bodies collided and coalesced after their random velocities became comparable to their escape velocities. In the outer part, where these ratios are larger than unity, the random velocities of some of the big bodies continued to rise until they were ejected. In both parts, the number density of the big bodies eventually decreased to the extent that gravitational interactions among them no longer produced large-scale chaos. After that their orbital eccentricities and inclinations were damped by dynamical friction from the remaining small bodies. n nThe last and longest stage in planet formation was the cleanup of small bodies. Our understanding of this stage is fraught with uncertainty. The surviving protoplanets cleared wide gaps around their orbits that inhibited their ability to accrete small bodies. Nevertheless, in the inner planet system, all of the material in the small bodies ended up inside planets. Small bodies in the outer planet system probably could not have been accreted in the age of the solar system. A second generation of planetesimals may have formed in the disk of small bodies, by either collisional coagulation or gravitational instability. In the outer planet system, bodies of kilometer size or larger would have had their random velocities excited until their orbits crossed those of neighboring protoplanets. Ultimately they would have either escaped from the Sun or become residents of the Oort Cloud. An important distinction is that growth of the inner planets continued through cleanup, whereas assembly of the outer planets was essentially complete by the end of oligarchy. These conclusions imply that the surface density of the protoplanetary disk was that of the minimum solar mass nebula in the inner planet region but a few times larger in the outer planet region. The timescale through cleanup was set by the accretion rate at the geometrical cross section in the inner planet region and by the ejection rate at the gravitationally enhanced cross section in the outer planet region. It was a few hundred million years in the former and a few billion years in the latter. However, since Uranus and Neptune acquired most of their mass by the end of oligarchy, they may have formed before Earth! n nA few implications of the above scenario are worth noting. Impacts among protoplanets of comparable size were common in the inner planet system but not in the outer. Ejections from the outer planet system included several bodies with masses in excess of Earth after oligarchy and an adequate number of kilometer-size bodies to populate the Oort comet cloud during cleanup. Except at the very end of cleanup, collisions prevented Uranus and Neptune from ejecting kilometer-size objects. Only Jupiter and, to a much lesser extent, Saturn were capable of populating the Oort Cloud with comets of kilometer size.

Constraints on deep-seated zonal winds inside Jupiter and Saturn

The atmospheres of Jupiter and Saturn exhibit strong and stable zonal winds. How deep the winds penetrate unabated into each planet is unknown. Our investigation favors shallow winds. It consists of two parts. The first part makes use of an Ohmic constraint; Ohmic dissipation associated with the planets magnetic field cannot exceed the planets net luminosity. Application to Jupiter (J) and Saturn (S) shows that the observed zonal winds cannot penetrate below a depth at which the electrical conductivity is about six orders of magnitude smaller than its value at the molecular–metallic transition. Measured values of the electrical conductivity of molecular hydrogen yield radii of maximum penetration of 0.96R_J and 0.86R_S, with uncertainties of a few percent of R. At these radii, the magnetic Reynolds number based on the zonal wind velocity and the scale height of the magnetic diffusivity is of order unity. These limits are insensitive to difficulties in modeling turbulent convection. They permit complete penetration along cylinders of the equatorial jets observed in the atmospheres of Jupiter and Saturn. The second part investigates how deep the observed zonal winds actually do penetrate. As it applies heuristic models of turbulent convection, its conclusions must be regarded as tentative. Truncation of the winds in the planets convective envelope would involve breaking the Taylor–Proudman constraint on cylindrical flow. This would require a suitable nonpotential acceleration which none of the obvious candidates appears able to provide. Accelerations arising from entropy gradients, magnetic stresses, and Reynolds stresses appear to be much too weak. These considerations suggest that strong zonal winds are confined to shallow, stably stratified layers, with equatorial jets being the possible exception.

Imbalanced Strong MHD Turbulence

We present a phenomenological model of imbalanced MHD turbulence in an incompressible magnetofluid. The steady state cascades, of waves traveling in opposite directions along the mean magnetic field, carry unequal energy fluxes to small length scales, where they decay as a result of viscous and resistive dissipation. The inertial range scalings are well understood when both cascades are weak. We study the case in which both cascades are, in a sense, strong. The inertial range of this imbalanced cascade has the following properties: (1) The ratio of the rms Elsasser amplitudes is independent of scale and is equal to the ratio of the corresponding energy fluxes. (2) In common with the balanced strong cascade, the energy spectra of both Elsasser waves are of the anisotropic Kolmogorov form, with their parallel correlation lengths equal to each other on all scales, and proportional to the two-thirds power of the transverse correlation length. (3) The equality of cascade time and wave period (critical balance) that characterizes the strong balanced cascade does not apply to the Elsasser field with the larger amplitude. Instead, the more general criterion that always applies to both Elsasser fields is that the cascade time is equal to the correlation time of the straining imposed by oppositely directed waves. (4) In the limit of equal energy fluxes, the turbulence corresponds to the balanced strong cascade. Our results are particularly relevant for turbulence in the solar wind. Spacecraft measurements have established that in the inertial range of solar wind turbulence, waves traveling away from the Sun have higher amplitudes than those traveling toward it. Result 1 allows us to infer the turbulent flux ratios from the amplitude ratios, thus providing insight into the origin of the turbulence.

Spontaneous axisymmetry breaking of the external magnetic field at Saturn

Saturn’s magnetic field is remarkably axisymmetric. Early evidence for nnonaxisymmetry came from the periodicity of Saturn’s kilometric radio bursts (SKR). nSubsequently, percent-level variations of the SKR period were found to occur on ntimescales of years. A recent breakthrough has been the direct detection of a nnonaxisymmetric component of the field that rotates with a period close to that of the SKR nand whose magnitude varies only weakly with distance from Saturn. The latter implies nthat it must be supported by currents external to the planet. We explore the hypothesis that ncentrifugally driven convection spontaneously breaks the axisymmetry of the external nmagnetic field at Saturn. The density of the outflowing plasma close to its source is nassumed to contain a substantial part that varies as coso and rotates uniformly. We ndemonstrate that the plasma stream must narrow with distance from the planet, while the nfield-aligned currents joining stream to ionosphere increase rapidly. These currents nproduce a nonaxisymmetric component of magnetic field whose magnitude varies ninversely with radial distance in the planet’s equatorial plane. For a rate of plasma outflow n10^4 ≾ Ṁ ≾ 10^5g s^(-1), this component’s strength is compatible with that observed. nAdditionally, we postulate that the SKR is associated with the narrow range of longitudes nover which large currents flow along magnetic field lines connecting the tip of the outflow nto the auroral ionosphere.

WAVE DAMPING BY MAGNETOHYDRODYNAMIC TURBULENCE AND ITS EFFECT ON COSMIC-RAY PROPAGATION IN THE INTERSTELLAR MEDIUM

Cosmic rays scatter off magnetic irregularities (Alfven waves) with which they are resonant, that is, waves of wavelength comparable to their gyroradii. These waves may be generated either by the cosmic rays themselves, if they stream faster than the Alfven speed, or by sources of MHD turbulence. Waves excited by streaming cosmic rays are ideally shaped for scattering, whereas the scattering efficiency of MHD turbulence is severely diminished by its anisotropy. We show that MHD turbulence has an indirect effect on cosmic-ray propagation by acting as a damping mechanism for cosmic-ray-generated waves. The hot (coronal) phase of the interstellar medium is the best candidate location for cosmic-ray confinement by scattering from self-generated waves. We relate the streaming velocity of cosmic rays to the rate of turbulent dissipation in this medium for the case in which turbulent damping is the dominant damping mechanism. We conclude that cosmic rays with up to 10^2 GeV could not stream much faster than the Alfven speed but 10^6 GeV cosmic rays would stream unimpeded by self-generated waves, unless the coronal gas were remarkably turbulence-free.

Gauge Freedom in the N-body Problem of Celestial Mechanics

The goal of this paper is to demonstrate how the internal symmetry of the N-body celestial-mechanics problem can be exploited in orbit calculation. n nWe start with summarising research reported in (Efroimsky [CITE], [CITE]; Newman & Efroimsky [CITE]; Efroimsky & Goldreich [CITE]) and develop its application to planetary equations in non-inertial frames. This class of problems is treated by the variation-of-constants method. As explained in the previous publications, whenever a standard system of six planetary equations (in the Lagrange, Delaunay, or other form) is employed for N objects, the trajectory resides on a 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the orbital elements and their time derivatives. The freedom in choosing this submanifold reveals an internal symmetry inherent in the description of the trajectory by orbital elements. This freedom is analogous to the gauge invariance of electrodynamics. In traditional derivations of the planetary equations this freedom is removed by hand through the introduction of the Lagrange constraint, either explicitly (in the variation-of-constants method) or implicitly (in the Hamilton-Jacobi approach). This constraint imposes the condition (called osculation condition) that both the instantaneous position and velocity be fit by a Keplerian ellipse (or hyperbola), i.e., that the instantaneous Keplerian ellipse (or hyperbola) be tangential to the trajectory. Imposition of any supplementary constraint different from that of Lagrange (but compatible with the equations of motion) would alter the mathematical form of the planetary equations without affecting the physical trajectory. n nHowever, for coordinate-dependent perturbations, any gauge different from that of Lagrange makes the Delaunay system non-canonical. Still, it turns out that in a more general case of disturbances dependent also upon velocities, there exists a generalised Lagrange gauge, i.e., a constraint under which the Delaunay system is canonical (and the orbital elements are osculating in the phase space). This gauge reduces to the regular Lagrange gauge for perturbations that are velocity-independent. n nFinally, we provide a practical example illustrating how the gauge formalism considerably simplifies the calculation of satellite motion about an oblate precessing planet.

Magnetospheric eclipses in the double-pulsar system PSR J0737-3039

We argue that eclipses of radio emission from the millisecond pulsar A in the double-pulsar system PSR J0737-3039 are due to synchrotron absorption by plasma in the closed field line region of the magnetosphere of its normal pulsar companion B. On the basis of a plausible geometric model, pulsar As radio beam only illuminates pulsar Bs magnetosphere for about 10 minutes surrounding the time of eclipse. During this time it heats particles at r ≳ 10^9 cm to relativistic energies and enables extra plasma, beyond that needed to maintain the corotation electric field, to be trapped by magnetic mirroring. An enhancement of the plasma density by a factor of ~10^2 is required to match the duration and optical depth of the observed eclipses. The extra plasma might be supplied by a source near B through Bγ pair creation by energetic photons produced in Bs outer gap. Relativistic pairs cool by synchrotron radiation close to where they are born. Reexcitation of their gyrational motions by cyclotron absorption of As radio beam can result in their becoming trapped between conjugate mirror points in Bs magnetosphere. Because the trapping efficiency decreases with increasing optical depth, the plasma density enhancement saturates even under steady state illumination. The result is an eclipse with finite, frequency-dependent optical depth. After illumination by As radio beam ceases, the trapped particles cool and are lost. The entire cycle repeats every orbital period. We speculate that the asymmetries between eclipse ingress and egress result in part from the magnetospheres evolution toward a steady state when illuminated by As radio beam. We predict that As linear polarization varies with both eclipse phase and Bs rotational phase.

Folded Fields as the Source of Extreme Radio-Wave Scattering in the Galactic Center

A strong case has been made that radio waves from sources within about half a degree of the Galactic center undergo extreme diffractive scattering. However, problems arise when standard (Kolmogorov) models of electron density fluctuations are employed to interpret the observations of scattering in conjunction with those of free-free radio emission. Specifically, the outer scale of a Kolmogorov spectrum of electron density fluctuations is constrained to be so small that it is difficult to identify an appropriate astronomical setting. Moreover, an unacceptably high turbulent heating rate results if the outer scale of the velocity field coincides with that of the density fluctuations. We propose an alternative model based on folded magnetic field structures that have been reported in numerical simulations of small-scale dynamos. Nearly isothermal density variations across thin current sheets suffice to account for the scattering. There is no problem of excess turbulent heating, because the outer scale for the velocity fluctuations is much larger than the widths of the current sheets. We speculate that interstellar magnetic fields could possess geometries that reflect their origins: fields maintained by the Galactic dynamo could have large correlation lengths, whereas those stirred by local energetic events might exhibit folded structures.

Understanding the behavior of Prometheus and Pandora

We revisit the dynamics of Prometheus and Pandora, two small moons flanking Saturns F ring. Departures of their orbits from freely precessing ellipses result from mutual interactions via their 121:118 mean motion resonance. Motions are chaotic because the resonance is split into four overlapping components. Orbital longitudes were observed to drift away from predictions based on Voyager ephemerides. A sudden jump in mean motions took place close to the time at which the orbits apses were antialigned in 2000. Numerical integrations reproduce both the longitude drifts and the jumps. The latter have been attributed to the greater strength of interactions near apse antialignment (every 6.2 yr), and it has been assumed that this drift-jump behavior will continue indefinitely. We re-examine the dynamics of the Prometheus–Pandora system by analogy with that of a nearly adiabatic, parametric pendulum. In terms of this analogy, the current value of the action of the satellite system is close to its maximum in the chaotic zone. Consequently, at present, the two separatrix crossings per precessional cycle occur close to apse antialignment. In this state libration only occurs when the potentials amplitude is nearly maximal, and the “jumps” in mean motion arise during the short intervals of libration that separate long stretches of circulation. Because chaotic systems explore the entire region of phase space available to them, we expect that at other times the Prometheus–Pandora system would be found in states of medium or low action. In a low action state it would spend most of the time in libration, and separatrix crossings would occur near apse alignment. We predict that transitions between these different states can happen in as little as a decade. Therefore, it is incorrect to assume that sudden changes in the orbits only happen near apse antialignment.