[ { "id": "authors:g8gjj-f9t19", "collection": "authors", "collection_id": "g8gjj-f9t19", "cite_using_url": "https://authors.library.caltech.edu/records/g8gjj-f9t19", "type": "article", "title": "Enceladus's Limit Cycle", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Lithwick", "given_name": "Yoram", "orcid": "0000-0003-4450-0528" }, { "family_name": "Luan", "given_name": "Jing", "orcid": "0000-0001-6048-1959" } ], "abstract": "

Enceladus exhibits some remarkable phenomena, including water geysers spraying through surface cracks, a global ice shell that is librating atop an ocean, a large luminosity, and rapid outward orbital migration. Here, we model the coupled evolution of Enceladus's orbit and interior structure. We find that Enceladus is driven into a periodic state—a limit cycle. Many of Enceladus's observed phenomena emerge from the model, and the predicted values for the orbital eccentricity, libration amplitude, shell thickness, and luminosity agree with observations. A single limit cycle lasts around 10 million years, and has three distinct stages: (1) freezing, (2) melting, and (3) resonant libration. In our model, Enceladus is currently in the freezing stage, meaning that its ice shell is getting thicker. That pressurizes the ocean, which in turn cracks the shell and pushes water up through the cracks. In this stage, the orbital eccentricity increases, as Saturn pushes Enceladus deeper into resonance with Dione. Once the eccentricity is sufficiently high, tidal heating begins to melt the shell, which is the second stage of the cycle. In the third stage, the shell remains close to 3 km thick. At that thickness, the shell's natural libration frequency is resonant with the orbital frequency. The shell's librations are consequently driven to large amplitude, for millions of years. Most of the tidal heating of Enceladus occurs during this stage, and the observed luminosity is a relic from the last episode of resonant libration, while the present-day heat production is small (∼1 GW).

", "doi": "10.3847/1538-4357/adf8e7", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2025-10-10", "series_number": "1", "volume": "992", "issue": "1", "pages": "28" }, { "id": "authors:bqw9p-8k138", "collection": "authors", "collection_id": "bqw9p-8k138", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180813-133513736", "type": "article", "title": "DAVs: Red Edge and Outbursts", "author": [ { "family_name": "Luan", "given_name": "Jing", "orcid": "0000-0001-6048-1959", "clpid": "Luan-Jing" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "As established by photometric surveys, white dwarfs with hydrogen atmospheres and surface gravity, log(g) \u2248 8.0 pulsate as they cool across the temperature range of 12,500 K \u2273 T_(eff) \u2273 10,800 K. Known as DAVs or ZZ Ceti stars, their oscillations are attributed to gravity modes excited by convective driving. Overstability requires convective driving to exceed radiative damping. Previous works have demonstrated that \u03c9 \u2273 max(\u03c4_c^( \u22121), L_(\u2113,b)) is a necessary and sufficient condition for overstability. Here \u03c4_c and L \u2113,b are the effective thermal timescale and Lamb frequency at the base of the surface convection zone. Below the observational red edge, L(\u2113,b) \u00bb \u03c4_c^( \u22121), so overstable modes all have \u03c9\u03c4_c \u00bb 1. Consequently, their photometric amplitudes are reduced by that large factor rendering them difficult to detect. Although proposed previously, the condition \u03c9 \u2273 L_(\u2113,b) has not been clearly interpreted. We show that modes with \u03c9 < L_(\u2113,b) suffer enhanced radiative damping that exceeds convective driving rendering them damped. A quasi-adiabatic analysis is adequate to account for this enhancement. Although this approximation is only marginally valid at the red edge, it becomes increasingly accurate toward both higher and lower T_(eff). Recently, Kepler discovered a number of cool DAVs that exhibit sporadic flux outbursts. Typical outbursts last several hours, are separated by days, and release ~10^(33) \u2013 10^(34) erg. We attribute outbursts to limit cycles arising from sufficiently resonant 3-mode couplings between overstable parent modes and pairs of radiatively damped daughter modes. Limit cycles account for the durations and energies of outbursts and their prevalence near the red edge of the DAV instability strip.", "doi": "10.3847/1538-4357/aad0f4", "issn": "1538-4357", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2018-08-10", "series_number": "1", "volume": "863", "issue": "1", "pages": "Art. No. 82" }, { "id": "authors:zdn81-j8648", "collection": "authors", "collection_id": "zdn81-j8648", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180215-155848486", "type": "article", "title": "Dense Regions in Supersonic Isothermal Turbulence", "author": [ { "family_name": "Robertson", "given_name": "Brant", "orcid": "0000-0002-4271-0364", "clpid": "Robertson-B-E" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "The properties of supersonic isothermal turbulence influence a variety of astrophysical phenomena, including the structure and evolution of star-forming clouds. This work presents a simple model for the structure of dense regions in turbulence in which the density distribution behind isothermal shocks originates from rough hydrostatic balance between the pressure gradient behind the shock and its deceleration from ram pressure applied by the background fluid. Using simulations of supersonic isothermal turbulence and idealized waves moving through a background medium, we show that the structural properties of dense, shocked regions broadly agree with our analytical model. Our work provides a new conceptual picture for describing the dense regions, which complements theoretical efforts to understand the bulk statistical properties of turbulence and attempts to model the more complex features of star-forming clouds like magnetic fields, self-gravity, or radiative properties.", "doi": "10.3847/1538-4357/aaa89e", "issn": "1538-4357", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2018-02-20", "series_number": "2", "volume": "854", "issue": "2", "pages": "Art. No. 88" }, { "id": "authors:jep02-rp336", "collection": "authors", "collection_id": "jep02-rp336", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170213-152600459", "type": "article", "title": "Classification of Satellite Resonances in the Solar System", "author": [ { "family_name": "Luan", "given_name": "Jing", "orcid": "0000-0001-6048-1959", "clpid": "Luan-Jing" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Several pairs of solar system satellites occupy mean motion resonances (MMRs). We divide these into two groups according to their proximity to exact resonance. Proximity is measured by the existence of a separatrix in phase space. MMRs between Io\u2013Europa, Europa\u2013Ganymede, and Enceladus\u2013Dione are too distant from exact resonance for a separatrix to appear. A separatrix is present only in the phase spaces of the Mimas\u2013Tethys and Titan\u2013Hyperion MMRs, and their resonant arguments are the only ones to exhibit substantial librations. Could there be a causal connection between the libration amplitude and the presence of a separatrix? Our suspicions were aroused by Goldreich & Schlichting, who demonstrate that sufficiently deep in a MMR, eccentricity damping could destabilize librations. However, our investigation reveals that libration amplitudes in both the Mimas\u2013Tethys and Titan\u2013Hyperion MMRs are fossils. Although the Mimas\u2013Tethys MMR is overstable, its libration amplitude grows on the tidal damping timescale of Mimas's inclination, which is considerably longer than a Hubble time. On the other hand, the Titan\u2013Hyperion MMR is stable, but tidal damping of Hyperion's eccentricity is too weak to have affected the amplitude of its libration.", "doi": "10.3847/1538-3881/153/1/17", "issn": "1538-3881", "publisher": "American Astronomical Society", "publication": "Astronomical Journal", "publication_date": "2017-01", "series_number": "1", "volume": "153", "issue": "1", "pages": "Art. No. 17" }, { "id": "authors:pc2cy-mcy62", "collection": "authors", "collection_id": "pc2cy-mcy62", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20160106-112101147", "type": "article", "title": "Thermal Conductivity Of Rubble Piles", "author": [ { "family_name": "Luan", "given_name": "Jing", "orcid": "0000-0001-6048-1959", "clpid": "Luan-Jing" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Rubble piles are a common feature of solar system bodies. They are composed of monolithic elements of ice or rock bound by gravity. Voids occupy a significant fraction of the volume of a rubble pile. They can exist up to pressure P \u2248 \u0454_y\u00b5, where \u0454_y is the monolithic material's yield strain and \u03bc its rigidity. At low P, contacts between neighboring elements are confined to a small fraction of their surface areas. As a result, the effective thermal conductivity of a rubble pile, k_(con)\u2248 k(\u0454_y\u00b5)^(1/2), can be orders of magnitude smaller than the thermal conductivity of its monolithic elements, k. In a fluid-free environment, only radiation can transfer energy across voids. It contributes an additional component, k_(rad)=16\u2113\u03c3T^3/3, to the total effective conductivity, k_(eff) = k_(con)+ k_(rad). Here \u2113, the inverse of the opacity per unit volume, is of the order of the size of the elements, and voids. An important distinction between k_(con) and k_(rad) is that the former is independent of the size of the elements, whereas the latter is proportional to it. Our expression for k_(eff) provides a good fit to the depth dependence of thermal conductivity in the top 140 cm of the lunar regolith. It also offers a good starting point for detailed modeling of thermal inertias for asteroids and satellites. Measurement of the response of surface temperature to variable insolation is a valuable diagnostic of a regolith. There is an opportunity for careful experiments under controlled laboratory conditions to test models of thermal conductivity such as the one we outline.", "doi": "10.1088/0004-637X/814/1/36", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2015-11-20", "series_number": "1", "volume": "814", "issue": "1", "pages": "Art. No. 36" }, { "id": "authors:eca2a-zt081", "collection": "authors", "collection_id": "eca2a-zt081", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140815-140932022", "type": "article", "title": "Secular Evolution of the Pulsar Triple System J0337+1715", "author": [ { "family_name": "Luan", "given_name": "Jing", "clpid": "Luan-J" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "The pulsar triple system J0337+1715 is remarkably regular and highly hierarchical. Secular interactions transfer angular momentum between inner and outer orbits unless their apsidal lines are parallel or anti-parallel. These choices correspond to orthogonal eigenmodes p and are characterized by e_(p,1)/e_(p,2) ~ a_(1)/a_(2) and e_(a,1)/e_(a,2) ~ (a_(1)/a_(2))^(\u22123/2)(m_(2)/m_(1)). Mode p dominates the current state so e1/e2 remains close to e_(p,1)/e_(p,2). A small contribution by Mode a causes e1 and e_(2) to oscillate with a period of ~10^(3)\u2009\u2009yr which should be apparent in a few years. These will reveal the effects of general relativity, and possibly the distortion of the inner white dwarf (WD). Phinney proposes that the epicyclic energy of a WD-pulsar binary reaches equipartition with the kinetic energy of a single convective eddy when the WD's progenitor fills its Roche lobe. We extend Phinney's theory to apply to modes rather than individual orbits. Thus we predict that Mode p and Mode a achieved equipartition with eddies in the giant envelopes of the progenitors of the outer and inner WD, respectively. The most effective eddies are those with lifetimes closest to the orbit period. These were far more energetic in the progenitor of the outer WD. This explains why Mode p overwhelms Mode a, and also why the inner binary's orbit is far more eccentric than orbits of other WD-pulsar binaries with similar orbit periods. Mode a's small but finite amplitude places a lower bound of Q ~ 10^(6) on the tidal quality parameter of the inner WD.", "doi": "10.1088/0004-637X/790/1/82", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2014-07-20", "series_number": "1", "volume": "790", "issue": "1", "pages": "Art. No. 82" }, { "id": "authors:3gd76-jxj22", "collection": "authors", "collection_id": "3gd76-jxj22", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140528-152806090", "type": "article", "title": "Physical Constraints on Fast Radio Bursts", "author": [ { "family_name": "Luan", "given_name": "Jing", "clpid": "Luan-J" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Fast radio bursts (FRBs) are isolated, ms radio pulses with dispersion measure (DM) of order 10^3 pc cm^(\u20133). Galactic candidates for the DM of high latitude bursts detected at GHz frequencies are easily dismissed. DM from bursts emitted in stellar coronas are limited by free-free absorption and those from H II regions are bounded by the nondetection of associated free-free emission at radio wavelengths. Thus, if astronomical, FRBs are probably extragalactic. FRB 110220 has a scattering tail of ~5.6 \u00b1 0.1 ms. If the electron density fluctuations arise from a turbulent cascade, the scattering is unlikely to be due to propagation through the diffuse intergalactic plasma. A more plausible explanation is that this burst sits in the central region of its host galaxy. Pulse durations of order ms constrain the sizes of FRB sources implying high brightness temperatures that indicates coherent emission. Electric fields near FRBs at cosmological distances would be so strong that they could accelerate free electrons from rest to relativistic energies in a single wave period.", "doi": "10.1088/2041-8205/785/2/L26", "issn": "2041-8205", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal Letters", "publication_date": "2014-04-20", "series_number": "2", "volume": "785", "issue": "2", "pages": "Art. No. L26" }, { "id": "authors:4x08d-vs261", "collection": "authors", "collection_id": "4x08d-vs261", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20140219-102144736", "type": "article", "title": "Overstable Librations can Account for the Paucity of Mean Motion Resonances among Exoplanet Pairs", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Schlichting", "given_name": "Hilke E.", "clpid": "Schlichting-H-E" } ], "abstract": "We assess the multi-planet systems discovered by the Kepler satellite in terms of current ideas about orbital migration and eccentricity damping due to planet-disk interactions. Our primary focus is on first order mean motion resonances, which we investigate analytically to lowest order in eccentricity. Only a few percent of planet pairs are in close proximity to a resonance. However, predicted migration rates (parameterized by \u03c4_n = n/|\u1e45|) imply that during convergent migration most planets would have been captured into first order resonances. Eccentricity damping (parameterized by \u03c4_e = e/|\u0117|) offers a plausible resolution. Estimates suggest \u03c4_e /\u03c4_n ~ (h/\u0251)^2 ~ 10^(\u20132), where h/\u0251 is the ratio of disk thickness to radius. Together, eccentricity damping and orbital migration give rise to an equilibrium eccentricity, e_(eq) ~ (\u03c4_e /\u03c4_n )^(1/2). Capture is permanent provided e_(eq) \u227e \u03bc^(1/3), where \u03bc denotes the planet to star mass ratio. But for e_(eq) \u2273 \u03bc^(1/3), capture is only temporary because librations around equilibrium are overstable and lead to passage through resonance on timescale \u03c4_e . Most Kepler planet pairs have e_(eq) > \u03bc^(1/3). Since \u03c4_n \u00bb \u03c4_e is the timescale for migration between neighboring resonances, only a modest percentage of pairs end up trapped in resonances after the disk disappears. Thus the paucity of resonances among Kepler pairs should not be taken as evidence for in situ planet formation or the disruptive effects of disk turbulence. Planet pairs close to a mean motion resonance typically exhibit period ratios 1%-2% larger than those for exact resonance. The direction of this shift undoubtedly reflects the same asymmetry that requires convergent migration for resonance capture. Permanent resonance capture at these separations from exact resonance would demand \u03bc(\u03c4_n /\u03c4_e )^(1/2) \u2273 0.01, a value that estimates of \u03bc from transit data and (\u03c4_e /\u03c4_n )^(1/2) from theory are insufficient to match. Plausible alternatives involve eccentricity damping during or after disk dispersal. The overstability referred to above has applications beyond those considered in this investigation. It was discovered numerically by Meyer & Wisdom in their study of the tidal evolution of Saturn's satellites.", "doi": "10.1088/0004-6256/147/2/32", "issn": "0004-6256", "publisher": "American Astronomical Society.", "publication": "Astronomical Journal", "publication_date": "2014-02", "series_number": "2", "volume": "147", "issue": "2", "pages": "Art. No. 32" }, { "id": "authors:zc1p5-zgp76", "collection": "authors", "collection_id": "zc1p5-zgp76", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120525-132231424", "type": "article", "title": "Adiabatic Heating of Contracting Turbulent Fluids", "author": [ { "family_name": "Robertson", "given_name": "Brant", "clpid": "Robertson-B" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Turbulence influences the behavior of many astrophysical systems, frequently by providing non-thermal pressure support through random bulk motions. Although turbulence is commonly studied in systems with constant volume and mean density, turbulent astrophysical gases often expand or contract under the influence of pressure or gravity. Here, we examine the behavior of turbulence in contracting volumes using idealized models of compressed gases. Employing numerical simulations and an analytical model, we identify a simple mechanism by which the turbulent motions of contracting gases \"adiabatically heat,\" experiencing an increase in their random bulk velocities until the largest eddies in the gas circulate over a Hubble time of the contraction. Adiabatic heating provides a mechanism for sustaining turbulence in gases where no large-scale driving exists. We describe this mechanism in detail and discuss some potential applications to turbulence in astrophysical settings.", "doi": "10.1088/2041-8205/750/2/L31", "issn": "2041-8205", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal Letters", "publication_date": "2012-05-10", "series_number": "2", "volume": "750", "issue": "2", "pages": "L31" }, { "id": "authors:c0zt4-fx549", "collection": "authors", "collection_id": "c0zt4-fx549", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120817-105459244", "type": "article", "title": "Why do we see the man in the Moon?", "author": [ { "family_name": "Aharonson", "given_name": "Oded", "orcid": "0000-0001-9930-2495", "clpid": "Aharonson-O" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Sari", "given_name": "Re'em", "orcid": "0000-0002-1084-3656", "clpid": "Sari-R" } ], "abstract": "Numerical simulations and analysis show that the Moon locks into resonance with a statistical preference of facing either the current near-side or far-side toward Earth. The near-side is largely covered by dense, topographically low, dark mare basalts, the pattern of which to some, resembles the image of a man's face. Although the Moon is locked in this configuration at present, the opposite one, with the current far-side facing Earth, is of lower potential energy and hence might be naively expected. Instead, we find that the probability of selecting each configuration depends upon the ratio of the asymmetry of the potential energy maxima, dominated by the octupole moment of the Moon, to the energy dissipated per tidal cycle within the Moon. If this ratio is small, the two configurations are equally likely. Otherwise, interesting dynamical behavior ensues. In the Moon's present orbit, with the best-estimated geophysical parameters and dissipation parameter Q = 35, trapping into the current higher-energy configuration is preferred. With Q = 100 in analogy with the solid Earth, the current configuration is nearly certain. The ratio of energies and corresponding probabilities were different in the past. Relative crater counts on the leading and trailing faces indicate an impact may have unlocked the Moon before it settled into the present configuration. Our analysis constrains the geophysical parameters at the time of the last such event.", "doi": "10.1016/j.icarus.2012.02.019", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "2012-05", "series_number": "1", "volume": "219", "issue": "1", "pages": "241-243" }, { "id": "authors:mh5q1-7nh06", "collection": "authors", "collection_id": "mh5q1-7nh06", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20101025-133610473", "type": "article", "title": "Elastic ice shells of synchronous moons: Implications for cracks on Europa and non-synchronous rotation of Titan", "author": [ { "family_name": "Goldreich", "given_name": "Peter M.", "clpid": "Goldreich-P-M" }, { "family_name": "Mitchell", "given_name": "Jonathan L.", "clpid": "Mitchell-J-L" } ], "abstract": "A number of synchronous moons are thought to harbor water oceans beneath their outer ice shells. A subsurface ocean frictionally decouples the shell from the interior. This has led to proposals that a weak tidal or atmospheric torque might cause the shell to rotate differentially with respect to the synchronously rotating interior. Applications along these lines have been made to Europa and Titan. However, the shell is coupled to the ocean by an elastic torque. As a result of centrifugal and tidal forces, the ocean would assume an ellipsoidal shape with its long axis aligned toward the parent planet. Any displacement of the shell away from its equilibrium position would induce strains thereby increasing its elastic energy and giving rise to an elastic restoring torque. In the investigation reported on here, the elastic torque is compared with the tidal torque acting on Europa and the atmospheric torque acting on Titan.\nRegarding Europa, it is shown that the tidal torque is far too weak to produce stresses that could fracture the ice shell, thus refuting an idea that has been widely advocated. Instead, it is suggested that the cracks arise from time-dependent stresses due to non-hydrostatic gravity anomalies from tidally driven, episodic convection in the satellite's interior.\nTwo years of Cassini RADAR observations of Titan's surface have been interpreted as implying an angular displacement of ~0.24\u00b0 relative to synchronous rotation. Compatibility of the amplitude and phase of the observed non-synchronous rotation with estimates of the atmospheric torque requires that Titan's shell be decoupled from its interior. We find that the elastic torque balances the seasonal atmospheric torque at an angular displacement \u227e0.05\u00b0, effectively coupling the shell to the interior. Moreover, if Titan's surface were spinning faster than synchronous, the tidal torque tending to restore synchronous rotation would almost certainly be larger than the atmospheric torque. There must either be a problem with the interpretation of the radar observations, or with our basic understanding of Titan's atmosphere and/or interior.", "doi": "10.1016/j.icarus.2010.04.013", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "2010-10", "series_number": "2", "volume": "209", "issue": "2", "pages": "631-638" }, { "id": "authors:g58j7-9k734", "collection": "authors", "collection_id": "g58j7-9k734", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:GOLapj09", "type": "article", "title": "Tidal Evolution of Rubble Piles", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Sari", "given_name": "Re'em", "orcid": "0000-0002-1084-3656", "clpid": "Sari-R" } ], "abstract": "Many small bodies in the solar system are believed to be rubble piles, a collection of smaller elements separated by voids.We propose a model for the structure of a self-gravitating rubble pile. Static friction prevents its elements from sliding relative to each other. Stresses are concentrated around points of contact between individual elements. The effective dimensionless rigidity, \u03bc\u02dc rubble, is related to that of a monolithic body of similar composition and size, \u03bc\u02dc by \u03bc\u02dc rubble \u223c \u03bc\u02dc^1/2 \u03b5Y^\u22121/2, where \u03b5Y \u223c 10^\u22122 is the yield strain. This represents a reduction in effective rigidity below the maximum radius, Rmax \u223c [\u03bc\u03b5Y /(G\u03c1^2)]^1/2 \u223c 10^3 km, at which a rubble pile can exist. Our model for the rigidity of rubble piles is compatible with laboratory experiments on the speed of shear waves in sand. Densities derived for binary asteroids imply that they are rubble piles. Thus their tidal evolution proceeds faster than it would if they were monoliths. Binary orbit evolution is also driven by torques resulting from the asymmetrical scattering and reradiation of sunlight (YORP and BYORP effects). The tidal torque probably overcomes the radiative (YORP) torque and synchronizes the spins of secondaries in near-Earth binary asteroids and it definitely does so for secondaries of main-belt binary asteroids. Synchronization is a requirement for the radiative (BYORP) torque to act on the binary orbit. This torque clearly dominates the tidal torque for all near-Earth binary asteroids and for some binaries in the main belt. For other main-belt binaries, the tidal torque appears to be at least comparable in strength to the BYORP torque. An exciting possibility is that in these systems the angular momentum added to the orbit by the tidal torque might be removed by the radiative torque.", "doi": "10.1088/0004-637X/691/1/54", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2009-01-20", "series_number": "1", "volume": "691", "issue": "1", "pages": "54-60" }, { "id": "authors:dxvt2-5wn08", "collection": "authors", "collection_id": "dxvt2-5wn08", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20100708-092110052", "type": "article", "title": "Constraints on deep-seated zonal winds inside Jupiter and Saturn", "author": [ { "family_name": "Liu", "given_name": "Junjun", "clpid": "Liu-Junjun" }, { "family_name": "Goldreich", "given_name": "Peter M.", "clpid": "Goldreich-P-M" }, { "family_name": "Stevenson", "given_name": "David J.", "orcid": "0000-0001-9432-7159", "clpid": "Stevenson-D-J" } ], "abstract": "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 planet's magnetic field cannot exceed the planet's 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\u2013metallic 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 planet's convective envelope would involve breaking the Taylor\u2013Proudman 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.", "doi": "10.1016/j.icarus.2007.11.036", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "2008-08", "series_number": "2", "volume": "196", "issue": "2", "pages": "653-664" }, { "id": "authors:dy5bx-2z311", "collection": "authors", "collection_id": "dy5bx-2z311", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130228-102458528", "type": "article", "title": "Spontaneous axisymmetry breaking of the external magnetic field at Saturn", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Farmer", "given_name": "Alison J.", "clpid": "Farmer-A-J" } ], "abstract": "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 cos\u00f8 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 \u227e \u1e40 \u227e 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.", "doi": "10.1029/2006JA012163", "issn": "0148-0227", "publisher": "American Geophysical Union", "publication": "Journal of Geophysical Research A", "publication_date": "2007-05-26", "series_number": "A5", "volume": "112", "issue": "A5", "pages": "Art. No. A05225" }, { "id": "authors:zznf2-rep90", "collection": "authors", "collection_id": "zznf2-rep90", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130228-103329811", "type": "article", "title": "Imbalanced Strong MHD Turbulence", "author": [ { "family_name": "Lithwick", "given_name": "Y.", "clpid": "Lithwick-Y" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" }, { "family_name": "Sridhar", "given_name": "S.", "clpid": "Sridhar-S" } ], "abstract": "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 Els\u00e4sser 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 Els\u00e4sser 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 Els\u00e4sser field with the larger amplitude. Instead, the more general criterion that always applies to both Els\u00e4sser 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.", "doi": "10.1086/509884", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2007-01-20", "series_number": "1", "volume": "655", "issue": "1", "pages": "269-274" }, { "id": "authors:4s4yj-v5h96", "collection": "authors", "collection_id": "4s4yj-v5h96", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:SARapj06", "type": "article", "title": "Spherical accretion", "author": [ { "family_name": "Sari", "given_name": "Re'em", "orcid": "0000-0002-1084-3656", "clpid": "Sari-R" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "We compare different examples of spherical accretion onto a gravitating mass. Limiting cases include the accretion of a collisionally dominated fluid and the accretion of collisionless particles. We derive expressions for the accretion rate and density profile for semicollisional accretion, which bridges the gap between these limiting cases. Particle crossing of the Hill sphere during the formation of the outer planets is likely to have taken place in the semicollisional regime.", "issn": "0004-637X", "publisher": "Astrophysical Journal", "publication": "Astrophysical Journal", "publication_date": "2006-05-01", "series_number": "1", "volume": "642", "issue": "1", "pages": "L65-L67" }, { "id": "authors:j8qg8-0h587", "collection": "authors", "collection_id": "j8qg8-0h587", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130228-103838219", "type": "article", "title": "Folded Fields as the Source of Extreme Radio-Wave Scattering in the Galactic Center", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Sridhar", "given_name": "S.", "clpid": "Sridhar-S" } ], "abstract": "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.", "doi": "10.1086/503668", "issn": "2041-8205", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal Letters", "publication_date": "2006-04-01", "series_number": "2", "volume": "640", "issue": "2", "pages": "L159-L162" }, { "id": "authors:hedhr-wqc60", "collection": "authors", "collection_id": "hedhr-wqc60", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:FARicarus06", "type": "article", "title": "Understanding the behavior of Prometheus and Pandora", "author": [ { "family_name": "Farmer", "given_name": "Alison J.", "clpid": "Farmer-A-J" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "We revisit the dynamics of Prometheus and Pandora, two small moons flanking Saturn's 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\u2013Pandora 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 potential's 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\u2013Pandora 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.", "doi": "10.1016/j.icarus.2005.10.005", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "2006-02", "series_number": "2", "volume": "180", "issue": "2", "pages": "403-411" }, { "id": "authors:y4081-v7c79", "collection": "authors", "collection_id": "y4081-v7c79", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130225-090018853", "type": "article", "title": "Spoke formation under moving plasma clouds", "author": [ { "family_name": "Farmer", "given_name": "Alison J.", "clpid": "Farmer-A-J" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Goertz and Morfill [Goertz, C.K., Morfill, G., 1988. Icarus 53, 219\u2013229] propose that spokes on Saturn's rings form under radially moving plasma clouds produced by meteoroid impacts. We demonstrate that the speed at which a plasma cloud can move relative to the ring material is bounded from above by the difference between the Keplerian and corotation velocities. The radial orientation of new spokes requires radial speeds that are at least an order of magnitude larger than this upper limit, thus the model advanced by Goertz and Morfill fails to make radial spokes.", "doi": "10.1016/j.icarus.2005.07.025", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "2005-12-15", "series_number": "2", "volume": "179", "issue": "2", "pages": "535-538" }, { "id": "authors:k87aw-zkv06", "collection": "authors", "collection_id": "k87aw-zkv06", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-102316325", "type": "article", "title": "Magnetospheric Eclipses in the Double-Pulsar System PSR J0737\u20133039", "author": [ { "family_name": "Rafikov", "given_name": "Roman R.", "clpid": "Rafikov-R-R" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "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 A's radio beam only illuminates pulsar B's magnetosphere for about 10 minutes surrounding the time of eclipse. During this time it heats particles at r \u2273 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 B\u03b3 pair creation by energetic photons produced in B's outer gap. Relativistic pairs cool by synchrotron radiation close to where they are born. Reexcitation of their gyrational motions by cyclotron absorption of A's radio beam can result in their becoming trapped between conjugate mirror points in B's 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 A's 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 magnetosphere's evolution toward a steady state when illuminated by A's radio beam. We predict that A's linear polarization varies with both eclipse phase and B's rotational phase.", "doi": "10.1086/432248", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2005-09-20", "series_number": "1", "volume": "631", "issue": "1", "pages": "488-494" }, { "id": "authors:5v57v-mzs33", "collection": "authors", "collection_id": "5v57v-mzs33", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130225-101423808", "type": "article", "title": "Final Stages of Planet Formation", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Lithwick", "given_name": "Yoram", "clpid": "Lithwick-Y" }, { "family_name": "Sari", "given_name": "Re'em", "orcid": "0000-0002-1084-3656", "clpid": "Sari-R" } ], "abstract": "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.", "doi": "10.1086/423612", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2004-10-10", "series_number": "1", "volume": "614", "issue": "1", "pages": "497-507" }, { "id": "authors:1wa7r-bmr53", "collection": "authors", "collection_id": "1wa7r-bmr53", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:GOLaraa04", "type": "article", "title": "Planet formation by coagulation: A focus on Uranus and Neptune", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Lithwick", "given_name": "Yoram", "clpid": "Lithwick-Y" }, { "family_name": "Sari", "given_name": "Re'em", "orcid": "0000-0002-1084-3656", "clpid": "Sari-R" } ], "abstract": "Planets form in the circumstellar disks of young stars. We review the basic physical processes by which solid bodies accrete each other and alter each others' random velocities, and we provide order-of-magnitude derivations for the rates of these processes. We discuss and exercise the two-groups approximation, a simple yet powerful technique for solving the evolution equations for protoplanet growth. We describe orderly, runaway, neutral, and oligarchic growth. We also delineate die conditions under which each occurs. We refute a popular misconception by showing that the outer planets formed quickly by accreting small bodies. Then we address the final stages of planet formation. Oligarchy ends when the surface density of the oligarchs becomes comparable to that of the small bodies. Dynamical friction is no longer able to balance viscous stirring and the oligarchs' random velocities increase. In the inner-planet system, oligarchs collide and coalesce. In the outer-planet system, some of the oligarchs are ejected. In both the inner- and outer-planet systems, this stage ends once the number of big bodies has been reduced to the point that their mutual interactions no longer produce large-scale chaos. Subsequently, dynamical friction by the residual small bodies circularizes and flattens their orbits. The final stage of planet formation involves the clean up of the residual small bodies. Clean up has been poorly explored.", "doi": "10.1146/annurev.astro.42.053102.134004", "issn": "0066-4146", "publisher": "Annual Reviews", "publication": "Annual Review of Astronomy and Astrophysics", "publication_date": "2004-09-22", "volume": "42", "pages": "549-601" }, { "id": "authors:08bvn-zrf21", "collection": "authors", "collection_id": "08bvn-zrf21", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130228-110021913", "type": "article", "title": "Planet-Disk Symbiosis", "author": [ { "family_name": "Sari", "given_name": "Re'em", "orcid": "0000-0002-1084-3656", "clpid": "Sari-R" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Planets form in disks around young stars. Interactions with these disks cause them to migrate and thus affect their final orbital periods. We suggest that the connection between planets and disks may be deeper and involve a symbiotic evolution. By contributing to the outward transport of angular momentum, planets promote disk accretion. Here we demonstrate that planets sufficiently massive to open gaps could be the primary agents driving disk accretion. Those having masses below the gap opening threshold drift inward more rapidly than the disk material and can only play a minor role in its accretion. An even more intimate symbiosis involving gap opening planets may result if they acquire most of their mass prior to gap formation. Given a small initial eccentricity, just a fraction of a percent, the orbital eccentricity of a massive planet may grow rapidly once a mass in excess of the planet's mass has been repelled to form a gap around the planet's orbit. Then, as the planet's radial excursions approach the gap's width, subsequent eccentricity growth slows so that the planet's orbit continues to be confined within the gap.", "doi": "10.1086/421080", "issn": "2041-8205", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal Letters", "publication_date": "2004-05-01", "series_number": "1", "volume": "606", "issue": "1", "pages": "L77-L80" }, { "id": "authors:hsc9m-bmn65", "collection": "authors", "collection_id": "hsc9m-bmn65", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130225-091451386", "type": "article", "title": "Wave Damping by Magnetohydrodynamic Turbulence and Its Effect on Cosmic-Ray Propagation in the Interstellar Medium", "author": [ { "family_name": "Farmer", "given_name": "Alison J.", "clpid": "Farmer-A-J" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Cosmic rays scatter off magnetic irregularities (Alfv\u00e9n 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 Alfv\u00e9n 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 Alfv\u00e9n speed but 10^6 GeV cosmic rays would stream unimpeded by self-generated waves, unless the coronal gas were remarkably turbulence-free.", "doi": "10.1086/382040", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2004-04-01", "series_number": "2", "volume": "604", "issue": "2", "pages": "671-674" }, { "id": "authors:gefa7-j9n50", "collection": "authors", "collection_id": "gefa7-j9n50", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130226-132906363", "type": "article", "title": "Gauge freedom in the N-body problem of celestial mechanics", "author": [ { "family_name": "Efroimsky", "given_name": "M.", "clpid": "Efroimsky-M" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" } ], "abstract": "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.", "doi": "10.1051/0004-6361:20034058", "issn": "0004-6361", "publisher": "EDP Sciences", "publication": "Astronomy and Astrophysics", "publication_date": "2004-03", "series_number": "3", "volume": "415", "issue": "3", "pages": "1187-1199" }, { "id": "authors:0qneq-ssn87", "collection": "authors", "collection_id": "0qneq-ssn87", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:GOLicarus03b", "type": "article", "title": "Origin of chaos in the Prometheus\u2013Pandora system", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Rappaport", "given_name": "Nicole", "clpid": "Rappaport-N" } ], "abstract": "We demonstrate that the chaotic orbits of Prometheus and Pandora are due to interactions associated with the 121:118 mean motion resonance. Differential precession splits this resonance into a quartet of components equally spaced in frequency. Libration widths of the individual components exceed the splitting resulting in resonance overlap which causes the chaos. A single degree of freedom model captures the essential features of the chaotic dynamics. Mean motions of Prometheus and Pandora wander chaotically in zones of width 1.8 deg yr^\u22121 and 3.1 deg yr^\u22121, respectively.", "doi": "10.1016/j.icarus.2003.09.002", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "2003-12", "series_number": "2", "volume": "166", "issue": "2", "pages": "320-327" }, { "id": "authors:xz7dg-1ff18", "collection": "authors", "collection_id": "xz7dg-1ff18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:EFRjmp05", "type": "article", "title": "Gauge symmetry of the N-body problem in the Hamilton\u2013Jacobi approach", "author": [ { "family_name": "Efroimsky", "given_name": "Michael", "clpid": "Efroimsky-M" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "In most books the Delaunay and Lagrange equations for the orbital elements are derived by the Hamilton\u2013Jacobi method: one begins with the two-body Hamilton equations in spherical coordinates, performs a canonical transformation to the orbital elements, and obtains the Delaunay system. A standard trick is then used to generalize the approach to the N-body case. We reexamine this step and demonstrate that it contains an implicit condition which restricts the dynamics to a 9(N\u20131)-dimensional submanifold of the 12(N\u20131)-dimensional space spanned by the elements and their time derivatives. The tacit condition is equivalent to the constraint that Lagrange imposed \"by hand\" to remove the excessive freedom, when he was deriving his system of equations by variation of parameters. It is the condition of the orbital elements being osculating, i.e., of the instantaneous ellipse (or hyperbola) being always tangential to the physical velocity. Imposure of any supplementary condition different from the Lagrange constraint (but compatible with the equations of motion) is legitimate and will not alter the physical trajectory or velocity (though will alter the mathematical form of the planetary equations). This freedom of nomination of the supplementary constraint reveals a gauge-type internal symmetry instilled into the equations of celestial mechanics. Existence of this internal symmetry has consequences for the stability of numerical integrators. Another important aspect of this freedom is that any gauge different from that of Lagrange makes the Delaunay system noncanonical. In a more general setting, when the disturbance depends not only upon positions but also upon velocities, there is a \"generalized Lagrange gauge\" wherein the Delaunay system is symplectic. This special gauge renders orbital elements that are osculating in the phase space. It coincides with the regular Lagrange gauge when the perturbation is velocity independent.", "doi": "10.1063/1.1622447", "issn": "0022-2488", "publisher": "Journal of Mathematical Physics", "publication": "Journal of Mathematical Physics", "publication_date": "2003-12", "series_number": "12", "volume": "44", "issue": "12", "pages": "5958-5977" }, { "id": "authors:hr1wm-prd39", "collection": "authors", "collection_id": "hr1wm-prd39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:GOLicarus03a", "type": "article", "title": "Chaotic motions of Prometheus and Pandora", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Rappaport", "given_name": "Nicole", "clpid": "Rappaport-N" } ], "abstract": "Recent HST images of the Saturnian satellites Prometheus and Pandora show that their longitudes deviate from predictions of ephemerides based on Voyager images. Currently Prometheus is lagging and Pandora leading these predictions by somewhat more than 20\u25e6. We show that these discrepancies are fully accounted for by gravitational interactions between the two satellites. These peak every 24.8 d at conjunctions and excite chaotic perturbations. The Lyapunov exponent for the Prometheus-Pandora system is of order 0.35 yr^\u22121 for satellite masses based on a nominal density of 1.3 g cm^\u22123. Interactions are strongest when the orbits come closest together. This happens at intervals of 6.2 yr when their apses are anti-aligned. In this context we note the sudden changes of opposite signs in the mean motions of Prometheus and Pandora at the end of 2000 occured shortly after their apsidal lines were anti-aligned.", "doi": "10.1016/S0019-1035(02)00080-5", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "2003-04", "series_number": "2", "volume": "162", "issue": "2", "pages": "391-399" }, { "id": "authors:yaz1a-11451", "collection": "authors", "collection_id": "yaz1a-11451", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130225-103259425", "type": "article", "title": "Eccentricity Evolution for Planets in Gaseous Disks", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Sari", "given_name": "Re'em", "orcid": "0000-0002-1084-3656", "clpid": "Sari-R" } ], "abstract": "At least several percent of solar-type stars possess giant planets. Surprisingly, most move on orbits of substantial eccentricity. We investigate the hypothesis that interactions between a giant planet and the disk from which it forms promote eccentricity growth. These interactions are concentrated at discrete Lindblad and corotation resonances. Interactions at principal Lindblad resonances cause the planet's orbit to migrate and open a gap in the disk if the planet is sufficiently massive. Those at first-order Lindblad and corotation resonances change the planet's orbital eccentricity. Eccentricity is excited by interactions at external Lindblad resonances that are located on the opposite side of corotation from the planet, and damped by co-orbital Lindblad resonances that overlap the planet's orbit. If the planet clears a gap in the disk, the rate of eccentricity damping by co-orbital Lindblad resonances is reduced. Density gradients associated with the gap activate eccentricity damping by corotation resonances at a rate that initially marginally exceeds that of eccentricity excitation by external Lindblad resonances. But the corotation torque may be reduced as the result of the trapping of fluid in libration around potential maxima. This nonlinear saturation can tip the balance in favor of eccentricity excitation. A minimal initial eccentricity of the order of 1% is required to overcome viscous diffusion, which acts to unsaturate corotation resonances by reestablishing the large-scale density gradient. Thus, eccentricity growth is a finite-amplitude instability. Formally, the apsidal resonance, which is a special kind of co-orbital Lindblad resonance that exists in pressure-dominated disks, appears to damp eccentricity faster than external Lindblad resonances can excite it. However, the wavelength of the apsidal wave in a pressure-dominated disk is so long that it does not propagate. A self-gravity-dominated disk does not have an apsidal resonance. Nevertheless, apsidal waves are excited at gap edges. Although these propagate, their long wavelengths suggest that they are likely to be reflected at disk edges to form standing waves. Viscous damping of standing waves results in eccentricity damping, but at level far below that which traveling waves would produce. Although the level of eccentricity damping due to apsidal waves is reduced to a modest level in both pressure- and self-gravity-dominated disks, whether it drops well below that of Lindblad resonances depends sensitively on the disk's thickness and planet's mass. However, our analysis shows that with reasonable parameters, planet-disk interactions can promote eccentricity growth.", "doi": "10.1086/346202", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2003-03-10", "series_number": "2", "volume": "585", "issue": "2", "pages": "1024-1037" }, { "id": "authors:nf2sj-2es37", "collection": "authors", "collection_id": "nf2sj-2es37", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130225-092533716", "type": "article", "title": "Imbalanced Weak Magnetohydrodynamic Turbulence", "author": [ { "family_name": "Lithwick", "given_name": "Yoram", "clpid": "Lithwick-Y" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Weak MHD turbulence consists of waves that propagate along magnetic field lines, in both directions. When two oppositely directed waves collide, they distort each other, without changing their respective energies. Each wave suffers many collisions before cascading; by contrast, in strong MHD turbulence, waves cascade on the same timescale at which they collide. \"Imbalance\" means that more energy is going in one direction than the other. In general, MHD turbulence is imbalanced. Yet imbalanced MHD cascades are not understood. For example, turbulence in the solar wind is observed to be imbalanced, so solar wind turbulence will not be understood until a theory of the imbalanced cascade is developed. We solve weak MHD turbulence that is imbalanced. Of crucial importance is that the energies going in both directions are forced to equalize at the dissipation scale. This \"pinning\" of the energy spectra was discovered by Grappin and coworkers. It affects the entire inertial range. Weak MHD turbulence is particularly interesting because perturbation theory is applicable. Hence, it can be described with a simple kinetic equation. Galtier and coworkers derived this kinetic equation. We present a simpler, more physical derivation, based on the picture of colliding wavepackets. In the process, we clarify the role of the zero-frequency mode. We also explain why Goldreich & Sridhar claimed that perturbation theory is inapplicable, and why this claim is wrong. (Our \"weak\" is equivalent to Goldreich & Sridhar's \"intermediate.\") We perform numerical simulations of the kinetic equation to verify our claims. We construct simplified model equations that illustrate the main effects. Finally, we show that a large magnetic Prandtl number does not have a significant effect, and that hyperviscosity leads to a pronounced bottleneck effect.", "doi": "10.1086/344676", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2003-01-10", "series_number": "2", "volume": "582", "issue": "2", "pages": "1220-1240" }, { "id": "authors:287jq-e0y11", "collection": "authors", "collection_id": "287jq-e0y11", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130305-110107206", "type": "article", "title": "Formation of Kuiper-belt binaries by dynamical friction and three-body encounters", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Lithwick", "given_name": "Yoram", "clpid": "Lithwick-Y" }, { "family_name": "Sari", "given_name": "Re'em", "orcid": "0000-0002-1084-3656", "clpid": "Sari-R" } ], "abstract": "The Kuiper belt is a disk of icy bodies that orbit the Sun beyond Neptune; the largest known members are Pluto and its companion Charon. A few per cent of Kuiper-belt bodies have recently been found to be binaries with wide separations and mass ratios of the order of unity. Collisions were too infrequent to account for the observed number of binaries, implying that these binaries formed through collisionless interactions mediated by gravity. These interactions are likely to have been most effective during the period of runaway accretion, early in the Solar System's history. Here we show that a transient binary forms when two large bodies penetrate one another's Hill sphere (the region where their mutual forces are larger than the tidal force of the Sun). The loss of energy needed to stabilize the binary orbit can then occur either through dynamical friction from surrounding small bodies, or through the gravitational scattering of a third large body. Our estimates slightly favour the former mechanism. We predict that five per cent of Kuiper-belt objects are binaries with apparent separations greater than 0.2 arcsec, and that most are in tighter binaries or systems of higher multiplicity.", "doi": "10.1038/nature01227", "issn": "0028-0836", "publisher": "Nature Publishing Group", "publication": "Nature", "publication_date": "2002-12-12", "series_number": "6916", "volume": "420", "issue": "6916", "pages": "643-646" }, { "id": "authors:h8qew-ymc75", "collection": "authors", "collection_id": "h8qew-ymc75", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-101423922", "type": "article", "title": "Tidal Evolution of the Planetary System around HD 83443", "author": [ { "family_name": "Wu", "given_name": "Yanqin", "clpid": "Wu-Y" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Two planets with an orbital period ratio of approximately 10:1 have been discovered around the star HD 83443. The inner and more massive planet, HD 83443b, has the smallest semimajor axis among all currently known exoplanets. Unlike other short-period exoplanets, it maintains a substantial orbital eccentricity, e_1 = 0.079 \u00b1 0.008, in spite of efficient tidal damping. This is a consequence of its secular interactions with HD 83443_c, whose orbital eccentricity e_2 = 0.42 \u00b1 0.06. Dissipation, associated with tides the star raises in the inner planet, removes energy but not angular momentum from its orbit, while secular interactions transfer angular momentum but not energy from the inner to the outer planet's orbit. The outward transfer of angular momentum decreases the tidal decay rate of the inner planet's orbital eccentricity while increasing that of the outer planet. The alignment of the apsides of the planets' orbits is another consequence of tidal and secular interactions. In this state the ratio of their orbital eccentricities, e_1/e_2, depends on the secular perturbations the planets exert on each other and on additional perturbations that enhance the inner planet's precession rate. Tidal and rotational distortions of the inner planet along with general relativity provide the most important of these extra precessional perturbations, each of which acts to reduce e_1/e_2. Provided the planets' orbits are coplanar, the observed eccentricity ratio uniquely relates sin i and C \u2261 (k_2/k_(2J))(R_1/R_J)^5, where the tidal Love number, k_2, and radius, R_1, of the inner planet are scaled by their Jovian equivalents.", "doi": "10.1086/324193", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2002-01-10", "series_number": "2", "volume": "564", "issue": "2", "pages": "1024-1027" }, { "id": "authors:27jfp-92302", "collection": "authors", "collection_id": "27jfp-92302", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130225-113105156", "type": "article", "title": "Compressible Magnetohydrodynamic Turbulence in Interstellar Plasmas", "author": [ { "family_name": "Lithwick", "given_name": "Yoram", "clpid": "Lithwick-Y" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Radio wave scintillation observations reveal a nearly Kolmogorov spectrum of density fluctuations in the ionized interstellar medium. Although this density spectrum is suggestive of turbulence, no theory relevant to its interpretation exists. We calculate the density spectrum in turbulent magnetized plasmas by extending the theory of incompressible magnetohydrodynamic (MHD) turbulence given by Goldreich & Sridhar to include the effects of compressibility and particle transport. Our most important results are as follows:\n\n1. Density fluctuations are due to the slow mode and the entropy mode. Both modes are passively\nmixed by the cascade of shear Alfv\u00e9n waves. Since the shear Alfv\u00e9n waves have a Kolmogorov spectrum,\nso do the density fluctuations.\n2. Observed density fluctuation amplitudes constrain the nature of MHD turbulence in the interstellar\nmedium. Slow mode density fluctuations are suppressed when the magnetic pressure is less than the gas\npressure. Entropy mode density fluctuations are suppressed by cooling when the cascade timescale is\nlonger than the cooling timescale. These constraints imply either that the magnetic and gas pressures are\ncomparable or that the outer scale of the turbulence is very small.\n3. A high degree of ionization is required for the cascade to survive damping by neutrals and thereby\nto extend to small length scales. Regions that are insufficiently ionized produce density fluctuations only\non length scales larger than the neutral damping scale. These regions may account for the excess of\npower that is found on large scales.\n4. Provided that the thermal pressure exceeds the magnetic pressure, both the entropy mode and the\nslow mode are damped on length scales below that at which protons can diffuse across an eddy during\nthe eddy's turnover time. Consequently, eddies whose extents along the magnetic field are smaller than\nthe proton collisional mean free path do not contribute to the density spectrum. However, in MHD\nturbulence eddies are highly elongated along the magnetic field. From an observational perspective, the\nrelevant length scale is that transverse to the magnetic field. Thus, the cutoff length scale for density\nfluctuations is significantly smaller than the proton mean free path.\n5. The Alfv\u00e9n mode is critically damped at the transverse length scale of the proton gyroradius and\nthus cascades to smaller length scales than either the slow mode or the entropy mode.", "doi": "10.1086/323470", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2001-11-20", "series_number": "1", "volume": "562", "issue": "1", "pages": "279-296" }, { "id": "authors:s8hca-fk474", "collection": "authors", "collection_id": "s8hca-fk474", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130418-104452858", "type": "article", "title": "Incompressible MHD Turbulence", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "The inertial range of incompressible MHD turbulence is most conveniently described in terms of counter propagating waves. Shear Alfv\u00e9n waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfv\u00e9n waves. Cascades composed entirely of shear Alfv\u00e9n waves do not generate a significant measure of slow waves. MHD turbulence is anisotropic with energy cascading more rapidly along k \u22a5 than along k \u2225. Anisotropy increases with k \u22a5 such that the excited modes are confined inside a cone bounded by k \u2225\u221d k^(2/3)_(\u22a5). The opening angle of the cone, \u03b8(k \u22a5)\u221d k^(-1/3)_(\u22a5) , defines the scale dependent anisotropy. MHD turbulence is generically strong in the sense that the waves which comprise it are critically damped. Nevertheless, deep inside the inertial range, turbulent fluctuations are small. Their energy density is less than that of the background field by a factor \u03b82(k \u22a5)\u226a 1. MHD cascades are best understood geometrically. Wave packets suffer distortions as they move along magnetic field lines perturbed by counter propagating wave packets. Field lines perturbed by unidirectional waves map planes perpendicular to the local field into each other. Shear Alfv\u00e9n waves are responsible for the mapping's shear and slow waves for its dilatation. The former exceeds the latter by \u03b8(-1)(k \u22a5)\u226b 1 which accounts for dominance of the shear Alfv\u00e9n waves in controlling the cascade dynamics.", "doi": "10.1023/A:1013165303280", "issn": "0004-640X", "publisher": "Springer", "publication": "Astrophysics and Space Science", "publication_date": "2001-10", "series_number": "1-2", "volume": "278", "issue": "1-2", "pages": "17-23" }, { "id": "authors:8a5pz-c2t18", "collection": "authors", "collection_id": "8a5pz-c2t18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-094949679", "type": "article", "title": "Simulations of Incompressible Magnetohydrodynamic Turbulence", "author": [ { "family_name": "Maron", "given_name": "Jason", "clpid": "Maron-J" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "We simulate incompressible MHD turbulence using a pseudospectral code. Our major conclusions are: (1) MHD turbulence is most conveniently described in terms of counterpropagating shear Alfv\u00e9n and slow waves. Shear Alfv\u00e9n waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfv\u00e9n waves. Cascades composed entirely of shear Alfv\u00e9n waves do not generate a significant measure of slow waves. (2) MHD turbulence is anisotropic, with energy cascading more rapidly along k_\u22a5 than along k_\u2225, where k_\u22a5 and k_\u2225 refer to wavevector components perpendicular and parallel to the local magnetic field, respectively. Anisotropy increases with increasing k_\u22a5 such that excited modes are confined inside a cone bounded by k_\u2225 \u221d k^y_\u22a5, where \u03b3 < 1. The opening angle of the cone, \u0398(k_\u22a5) \u221d k^\u22a5^(-(1-y)), defines the scale-dependent anisotropy. (3) The one-dimensional inertial range energy spectrum is well fitted by a power law, E(k_\u22a5) \u221d k_\u22a5^(-\u0251), with \u03b1 > 1. (4) MHD turbulence is generically strong in the sense that the waves that comprise it suffer order unity distortions on timescales comparable to their periods. Nevertheless, turbulent fluctuations are small deep inside the inertial range. Their energy density is less than that of the background field by a factor of \u0398^((\u03b1-1)/(1-\u03b3)) \u00ab 1. (5) MHD cascades are best understood geometrically. Wave packets suffer distortions as they move along magnetic field lines perturbed by counterpropagating waves. Field lines perturbed by unidirectional waves map planes perpendicular to the local field into each other. Shear Alfv\u00e9n waves are responsible for the mapping's shear and slow waves for its dilatation. The amplitude of the former exceeds that of the latter by 1/\u0398(k_\u22a5), which accounts for dominance of the shear Alfv\u00e9n waves in controlling the cascade dynamics. (6) Passive scalars mixed by MHD turbulence adopt the same power spectrum as the velocity and magnetic field perturbations. (7) Decaying MHD turbulence is unstable to an increase of the imbalance between the fluxes of waves propagating in opposite directions along the magnetic field. Forced MHD turbulence displays order unity fluctuations with respect to the balanced state if excited at low k_\u22a5 by \u03b4(t)-correlated forcing. It appears to be statistically stable to the unlimited growth of imbalance. (8) Gradients of the dynamic variables are focused into sheets aligned with the magnetic field whose thickness is comparable to the dissipation scale. Sheets formed by oppositely directed waves are uncorrelated. We suspect that these are vortex sheets, which the mean magnetic field prevents from rolling up. (9) Items 1-6 lend support to the model of strong MHD turbulence put forth by Goldreich & Sridhar (GS). Results from our simulations are also consistent with the GS prediction \u03b3 = 2/3, as are those obtained previously by Cho & Vishniac. The sole notable discrepancy is that one-dimensional energy spectra determined from our simulations exhibit \u03b1 \u2248 3/2, whereas the GS model predicts \u03b1 = 5/3. Further investigation is needed to resolve this issue.", "doi": "10.1086/321413", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2001-06-29", "series_number": "2", "volume": "554", "issue": "2", "pages": "1175-1196" }, { "id": "authors:7k2nx-ms604", "collection": "authors", "collection_id": "7k2nx-ms604", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-093646534", "type": "article", "title": "Gravity Modes in ZZ Ceti Stars. IV. Amplitude Saturation by Parametric Instability", "author": [ { "family_name": "Wu", "given_name": "Yanqin", "clpid": "Wu-Y" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "ZZ Ceti stars (also known as DAV stars) exhibit small-amplitude photometric pulsations in multiple gravity modes. As the stars cool, their dominant modes shift to longer periods. We demonstrate that parametric instability limits overstable modes to amplitudes similar to those observed. In particular, it reproduces the trend that longer period modes have larger amplitudes. Parametric instability is a form of resonant three-mode coupling. It involves the destabilization of a pair of stable daughter modes by an overstable parent mode. The three modes must satisfy exact angular selection rules and approximate frequency resonance. The lowest instability threshold for each parent mode is provided by the daughter pair that minimizes (\u03b4\u03c9^2 + \u03b3^2_d)/\u03ba^2, where \u03ba is the nonlinear coupling constant, \u03b4\u03c9 is the frequency mismatch, and \u03b3_d is the energy damping rate of the daughter modes. Parametric instability leads to a steady state if |\u03b4\u03c9| > \u03b3_d and to limit cycles if |\u03b4\u03c9| < \u03b3_d. The former behavior characterizes low radial order (n \u2264 3) parent modes, and the latter those with n \u2265 5. In either case, the overstable mode's amplitude is maintained at close to the instability threshold value. Although parametric instability defines an upper envelope for the amplitudes of overstable modes in ZZ Ceti stars, other nonlinear mechanisms are required to account for the irregular distribution of amplitudes of similar modes and the nondetection of modes with periods longer than 1200 s. Resonant three-mode interactions involving more than one excited mode may account for the former and Kelvin-Helmholtz instability of the mode-driven shear layer below the convection zone for the latter.", "doi": "10.1086/318234", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2001-01-01", "series_number": "1", "volume": "546", "issue": "1", "pages": "469-483" }, { "id": "authors:2er6z-5e080", "collection": "authors", "collection_id": "2er6z-5e080", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130225-095427902", "type": "article", "title": "Apse Alignment of Narrow Eccentric Planetary Rings", "author": [ { "family_name": "Chiang", "given_name": "E. I.", "clpid": "Chiang-E-I" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" } ], "abstract": "The boundaries of the Uranian \u2208, \u03b1, and \u03b2 rings can be fitted by Keplerian ellipses. The pair of ellipses that outline a given ring share a common line of apsides. Apse alignment is surprising because the quadrupole moment of Uranus induces differential precession. We propose that rigid precession is maintained by a balance of forces due to ring self-gravity, planetary oblateness, and interparticle collisions. Collisional impulses play an especially dramatic role near ring edges. Pressure-induced accelerations are maximal near edges because there (1) velocity dispersions are enhanced by resonant satellite perturbations and (2) the surface density declines steeply. Remarkably, collisional forces felt by material in the last ~100 m of a ~10 km wide ring can increase equilibrium masses up to a factor of ~100. New ring surface densities are derived that accord with Voyager radio measurements. In contrast to previous models, collisionally modified self-gravity appears to allow for both negative and positive eccentricity gradients; why all narrow planetary rings exhibit positive eccentricity gradients remains an open question.", "doi": "10.1086/309372", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2000-09-10", "series_number": "2", "volume": "540", "issue": "2", "pages": "1084-1090" }, { "id": "authors:kf6pv-s3x84", "collection": "authors", "collection_id": "kf6pv-s3x84", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-091606977", "type": "article", "title": "Growth of Perturbations in Gravitational Collapse and Accretion", "author": [ { "family_name": "Lai", "given_name": "Dong", "clpid": "Lai-D" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "When a self-gravitating spherical gas cloud collapses or accretes onto a central mass, the inner region of the cloud develops a density profile \u03c1 \u221d r^(-3/2) and the velocity approaches free fall. We show that in this region nonspherical perturbations grow with decreasing radius. In the linear regime, the tangential velocity perturbation increases as r^(-1), while the Lagrangian density perturbation, \u0394\u03c1/\u03c1, grows as r^(-1/2). Faster growth occurs if the central collapsed object maintains a finite multiple moment, in which case \u0394\u03c1/\u03c1 increases as r^(-l), where l specifies the angular degree of the perturbation. These scaling relations are different from those obtained for the collapse of a homogeneous cloud. Our numerical calculations indicate that nonspherical perturbations are damped in the subsonic region and that they grow and approach the asymptotic scalings in the supersonic region. The implications of our results to asymmetric supernova collapse and to black hole accretion are briefly discussed.", "doi": "10.1086/308821", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "2000-05-20", "series_number": "1", "volume": "535", "issue": "1", "pages": "402-411" }, { "id": "authors:ymk0x-qgd46", "collection": "authors", "collection_id": "ymk0x-qgd46", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-090802378", "type": "article", "title": "Gravity Modes in ZZ Ceti Stars. III. Effects of Turbulent Dissipation", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Wu", "given_name": "Yanqin", "clpid": "Wu-Y" } ], "abstract": "We investigate dynamical interactions between turbulent convection and g-mode pulsations in ZZ Ceti variables. Since our understanding of turbulence is rudimentary, we are compelled to settle for order-of-magnitude results. A key feature of these interactions is that convective response times are much shorter than pulsation periods. Thus the dynamical interactions enforce near uniform horizontal velocity inside the convection zone. They also give rise to a narrow shear layer in the region of convective overshoot at the top of the radiative interior. Turbulent damping inside the convection zone is negligible for all modes, but that in the region of convective overshoot may be significant for a few long-period modes near the red edge of the instability strip. These conclusions are in accord with those reached earlier by Brickhill. Our major new result concerns nonlinear damping arising from the Kelvin-Helmholtz instability of the aforementioned shear layer. Amplitudes of overstable modes saturate where dissipation due to this instability balances excitation by convective driving. This mechanism of amplitude saturation is most effective for long-period modes, and it may play an important role in defining the red edge of the instability strip.", "doi": "10.1086/307752", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1999-10-01", "series_number": "2", "volume": "523", "issue": "2", "pages": "805-811" }, { "id": "authors:vncm5-dak37", "collection": "authors", "collection_id": "vncm5-dak37", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-085742039", "type": "article", "title": "Gravity Modes in ZZ Ceti Stars. II. Eigenvalues and Eigenfunctions", "author": [ { "family_name": "Wu", "given_name": "Yanqin", "clpid": "Wu-Y" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "We report on numerical calculations of nonadiabatic eigenvalues and eigenfunctions for g-modes in ZZ Ceti variables. The spectrum of overstable l = 1 modes delineates the instability strip. Its blue edge occurs where \u03c9\u03c4_c \u2248 1 for the n = 1 mode. Here \u03c9 is radian frequency and \u03c4_c is about 4 times the thermal time at the bottom of the surface convection zone. As a ZZ Ceti cools, its convection zone deepens, longer period modes become overstable, but the critical value of \u03c9\u03c4_c separating overstable and damped modes rises. The latter is a consequence of enhanced radiative damping for modes that propagate immediately below the convection zone. The critical value of \u03c9\u03c4_c is of observational significance, because modes with the smallest value of \u03c9\u03c4_c are most observable photometrically. Maximum periods for overstable modes predicted for our cooler model envelopes are about a factor of 2 longer than the observational upper limit of 1200 s. We assess a number of plausible resolutions for this discrepancy among which convective overshoot and nonlinear saturation look promising. The nonadiabatic eigenfunctions enable us to predict relative amplitudes and phases of photospheric variations of flux and velocity, quantities made accessible by recent observations. We also present asymptotic formula for damping rates of high-order modes, a result of consequence for future investigations of nonlinear saturation of the amplitudes of overstable modes.", "doi": "10.1086/307412", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1999-07-10", "series_number": "2", "volume": "519", "issue": "2", "pages": "783-792" }, { "id": "authors:ba207-hyh93", "collection": "authors", "collection_id": "ba207-hyh93", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-084127575", "type": "article", "title": "Spectral Energy Distributions of Passive T Tauri Disks: Inclination", "author": [ { "family_name": "Chiang", "given_name": "E. I.", "clpid": "Chiang-E-I" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "We compute spectral energy distributions (SEDs) for passive T Tauri disks viewed at arbitrary inclinations. Semianalytic models of disks in radiative and hydrostatic equilibrium are employed. Over viewing angles for which the flared disk does not occult the central star, the SED varies negligibly with inclination. For such aspects, the SED shortward of ~80 \u03bcm is particularly insensitive to orientation, since short wavelength disk emission is dominated by superheated surface layers, which are optically thin. The SED of a nearly edge-on disk is that of a class I source. The outer disk occults inner disk regions, and emission shortward of ~30 \u03bcm is dramatically extinguished. Spectral features from dust grains may appear in absorption. However, millimeter-wavelength fluxes decrease by at most a factor of 2 from face-on to edge-on orientations. We present illustrative applications of our SED models. The class I source 04108+2803B is considered a T Tauri star hidden from view by an inclined circumstellar disk. Fits to its observed SED yield model-dependent values for the disk mass of ~0.015 M_\u2609 and a disk inclination of ~65\u00b0 relative to face-on. The class II source GM Aur represents a T Tauri star only slightly obscured by its circumstellar disk. Fitted parameters include a disk mass of ~0.050 M_\u2609 and an inclination of ~60\u00b0, where the viewing angle is chosen to reproduce the observed visual extinction of A_V = 0.5 mag.", "doi": "10.1086/307351", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1999-07-01", "series_number": "1", "volume": "519", "issue": "1", "pages": "279-284" }, { "id": "authors:9pm7t-1mj98", "collection": "authors", "collection_id": "9pm7t-1mj98", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-082524842", "type": "article", "title": "Gravity Modes in ZZ Ceti Stars. I. Quasi-adiabatic Analysis of Overstability", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Wu", "given_name": "Yanqin", "clpid": "Wu-Y" } ], "abstract": "We analyze the stability of g-modes in white dwarfs with hydrogen envelopes. All relevant physical processes take place in the outer layer of hydrogen-rich material, which consists of a radiative layer overlaid by a convective envelope. The radiative layer contributes to mode damping, because its opacity decreases upon compression and the amplitude of the Lagrangian pressure perturbation increases outward. The convective envelope is the seat of mode excitation, because it acts as an insulating blanket with respect to the perturbed flux that enters it from below. A crucial point is that the convective motions respond to the instantaneous pulsational state. Driving exceeds damping by as much as a factor of 2 provided \u03c9\u03c4_c\u22651, where \u03c9 is the radian frequency of the mode and \u03c4_c\u22484\u03c4_(th), with \u03c4_(th) being the thermal time constant evaluated at the base of the convective envelope. As a white dwarf cools, its convection zone deepens, and lower frequency modes become overstable. However, the deeper convection zone impedes the passage of flux perturbations from the base of the convection zone to the photosphere. Thus the photometric variation of a mode with constant velocity amplitude decreases. These factors account for the observed trend that longer period modes are found in cooler DA variables. Overstable modes have growth rates of order \u03b3~1/(n\u03c4_\u03c9), where n is the mode's radial order and \u03c4_\u03c9 is the thermal timescale evaluated at the top of the mode's cavity. The growth time, \u03b3^(\u22121), ranges from hours for the longest period observed modes (P\u224820 minutes) to thousands of years for those of shortest period (P\u22482 minutes). The linear growth time probably sets the timescale for variations of mode amplitude and phase. This is consistent with observations showing that longer period modes are more variable than shorter period ones. Our investigation confirms many results obtained by Brickhill in his pioneering studies of ZZ Cetis. However, it suffers from two serious shortcomings. It is based on the quasiadiabatic approximation that strictly applies only in the limit \u03c9\u03c4_c \u00bb 1, and it ignores damping associated with turbulent viscosity in the convection zone. We will remove these shortcomings in future papers.", "doi": "10.1086/306705", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1999-02-01", "series_number": "2", "volume": "511", "issue": "2", "pages": "904-915" }, { "id": "authors:hc37a-tec34", "collection": "authors", "collection_id": "hc37a-tec34", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-075841374", "type": "article", "title": "Spectral Energy Distributions of T Tauri Stars with Passive Circumstellar Disks", "author": [ { "family_name": "Chiang", "given_name": "E. I.", "clpid": "Chiang-E-I" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" } ], "abstract": "We derive hydrostatic, radiative equilibrium models for passive disks surrounding T Tauri stars. Each disk is encased by an optically thin layer of superheated dust grains. This layer reemits directly to space about half the stellar energy it absorbs. The other half is emitted inward and regulates the interior temperature of the disk. The heated disk flares. As a consequence, it absorbs more stellar radiation, especially at large radii, than a flat disk would. The portion of the spectral energy distribution contributed by the disk is fairly flat throughout the thermal infrared. At fixed frequency, the contribution from the surface layer exceeds that from the interior by about a factor 3 and is emitted at more than an order of magnitude greater radius. Spectral features from dust grains in the superheated layer appear in emission if the disk is viewed nearly face-on.", "doi": "10.1086/304869", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1997-11-20", "series_number": "1", "volume": "490", "issue": "1", "pages": "368-376" }, { "id": "authors:ywgwb-6k265", "collection": "authors", "collection_id": "ywgwb-6k265", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-073328900", "type": "article", "title": "Magnetohydrodynamic Turbulence Revisited", "author": [ { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" }, { "family_name": "Sridhar", "given_name": "S.", "clpid": "Sridhar-S" } ], "abstract": "In 1965, Kraichnan proposed that MHD turbulence occurs as a result of collisions between oppositely directed Alfv\u00e9n wave packets. Recent work has generated some controversy over the nature of nonlinear couplings between colliding Alfv\u00e9n waves. We find that the resolution to much of the confusion lies in the existence of a new type of turbulence, intermediate turbulence, in which the cascade of energy in the inertial range exhibits properties intermediate between those of weak and strong turbulent cascades. Some properties of intermediate MHD turbulence are the following: (1) in common with weak turbulent cascades, wave packets belonging to the inertial range are long-lived; (2) however, components of the strain tensor are so large that, similar to the situation in strong turbulence, perturbation theory is not applicable; (3) the breakdown of perturbation theory results from the divergence of neighboring field lines due to wave packets whose perturbations in velocity and magnetic fields are localized, but whose perturbations in displacement are not; (4) three-wave interactions dominate individual collisions between wave packets, but interactions of all orders n \u2265 3 make comparable contributions to the intermediate turbulent energy cascade; (5) successive collisions are correlated since wave packets are distorted as they follow diverging field lines; (6) in common with the weak MHD cascade, there is no parallel cascade of energy, and the cascade to small perpendicular scales strengthens as it reaches higher wavenumbers; (7) for an appropriate weak excitation, there is a natural progression from a weak, through an intermediate, to a strong cascade.", "doi": "10.1086/304442", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1997-08-20", "series_number": "2", "volume": "485", "issue": "2", "pages": "680-688" }, { "id": "authors:bdj4k-z3b34", "collection": "authors", "collection_id": "bdj4k-z3b34", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130227-071422504", "type": "article", "title": "Single-Sided Shepherding", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Rappaport", "given_name": "Nicole", "clpid": "Rappaport-N" }, { "family_name": "Sicardy", "given_name": "Bruno", "clpid": "Sicardy-B" } ], "abstract": "Narrow ringlets are observed to form at isolated Lindblad resonances in recent simulations (J. H\u00e4nninen and H. Salo, Icarus 108, 325-346 (1994) and 117, 435-438 (1995)). Our remarks are directed toward the interpretation of this phenomenon. Ringlet formation is a consequence of the negative angular momentum luminosity promoted by satellite perturbations of the streamlines of particle flow. Contraction halts once the surface density reaches a value such that the angular momentum luminosity vanishes. Our estimate for the formation time, t_f \u223c (M_p/M_s)^1/2\u03a9,^(-1), for low order resonances in optically thin rings is consistent with the results of the simulations. Ringlets drift across the width of resonance, W \u223c (M_s/M_p)^(1/2)\u0251, as a result of the unbalanced satellite torque. This occurs on the much longer timescale t_d \u223c (M_p/M_s)\u03a9^(-1) and cannot be observed in current simulations.", "doi": "10.1006/icar.1995.1200", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "1995-12", "series_number": "2", "volume": "118", "issue": "2", "pages": "414-417" }, { "id": "authors:0e07m-rph17", "collection": "authors", "collection_id": "0e07m-rph17", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130418-071010312", "type": "article", "title": "Toward a theory of interstellar turbulence. II. Strong Alfv\u00e9nic turbulence", "author": [ { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" }, { "family_name": "Sridhar", "given_name": "S.", "clpid": "Sridhar-S" } ], "abstract": "We continue to investigate the possibility that interstellar turbulence is caused by nonlinear interactions\namong shear Alfv\u00e9n waves. Here, as in Paper I, we restrict attention to the symmetric case where the oppositely directed waves carry equal energy fluxes. This precludes application to the solar wind in which the outward flux significantly exceeds the ingoing one. All our detailed calculations are carried out for an incompressible\nmagnetized fluid. In incompressible magnetohydrodynamics (MHD), nonlinear interactions only occur between oppositely direct waves. Paper I contains a detailed derivation of the inertial range spectrum for the weak turbulence of shear Alfv\u00e9n waves. As energy cascades to high perpendicular wavenumbers, interactions become so strong that the assumption of weakness is no longer valid. Here, we present a theory for the strong turbulence of shear Alfv\u00e9n waves. It has the following main characteristics. (1) The inertial-range energy spectrum exhibits a critical balance beween linear wave periods and nonlinear turnover timescales. (2) The \"eddies\" are elongated in the direction of the field on small spatial scales; the parallel and perpendicular components of the wave vector, k_z and k_\u22a5, are related by k_z \u2248 k^(2/3) _\u22a5L^(-1/3), where L is the outer scale of the turbulence. (3) The \"one-dimensional\" energy spectrum is proportional to k^(-5/3) _\u22a5-an anisotropic Kolmogorov energy spectrum. Shear Alfv\u00e9nic turbulence mixes specific entropy as a passive contaminant. This gives rise to an electron density power spectrum whose form mimics the energy spectrum of the turbulence. Radio, wave scattering by these electron density fluctuations produces anisotropic scatter-broadened images. Damping by ion-neutral collisions restricts Alfv\u00e9nic turbulence to highly ionized regions of the interstellar medium. We expect negligible generation of compressive MHD waves by shear Alfv\u00e9n waves belonging to the critically balanced cascade. Viscous and collisionless damping are also unimportant in the interstellar medium (ISM). Our calculations support the general picture of interstellar turbulence advanced by Higdon.", "doi": "10.1086/175121", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1995-01-10", "series_number": "2", "volume": "438", "issue": "2", "pages": "763-775" }, { "id": "authors:0kq0q-6fz02", "collection": "authors", "collection_id": "0kq0q-6fz02", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130417-153456891", "type": "article", "title": "Toward a theory of interstellar turbulence. I: Weak Alfv\u00e9nic turbulence", "author": [ { "family_name": "Sridhar", "given_name": "S.", "clpid": "Sridhar-S" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" } ], "abstract": "We study weak Alfv\u00e9nic turbulence of an incompressible, magnetized fluid in some detail, with a view to developing a firm theoretical basis for the dynamics of small-scale turbulence in the interstellar medium. We prove that resonant 3-wave interactions are absent. We also show that the Iroshnikov-Kraichnan theory of incompressible, magnetohydrodynamic turbulence-which is widely accepted-describes weak 3-wave turbulence; consequently, it is incorrect. Physical arguments, as well as detailed calculations of the coupling coefficients are used to demonstrate that these interactions are empty. We then examine resonant 4-wave interactions, and show that the resonance relations forbid energy transport to small spatial scales along the direction of the mean magnetic field, for both the shear Alfv\u00e9n wave and the pseudo Alfv\u00e9n wave. The threedimensional inertial-range energy spectrum of 4-wave shear Alfv\u00e9n turbulence guessed from physical arguments reads E(k_z,k_\u22a5) ~ V_Av_LL^(-1/3) k^(-10/3) _\u22a5, where V_A is the Alfv\u00e9n speed, and v_L is the velocity difference across the outer scale L. Given this spectrum, the velocity difference across \u03bb_\u22a5 ~ k^(-1) _\u22a5 is V_(\u03bb\u22a5) ~ v_L(\u03bb_\u22a5/L)^(2/3). We derive a kinetic equation, and prove that this energy spectrum is a stationary solution and that it implies a positive flux of energy in k-space, along directions perpendicular to the mean magnetic field. Using this energy spectrum, we deduce that 4-wave interactions strengthen as the energy cascades to small, perpendicular spatial scales; beyond an upper bound in perpendicular wavenumber, k_\u22a5L ~ (V_A/v_L)^(3/2), weak turbulence theory ceases to be valid. Energy excitation amplitudes must be very small for the 4-wave inertial-range to be substantial.\nWhen the excitation is strong, the width of the 4-wave inertial-range shrinks to zero. This seems likely to be\nthe case in the interstellar medium. The physics of strong turbulence is explored in Paper II.", "doi": "10.1086/174600", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1994-09-10", "series_number": "2", "volume": "432", "issue": "2", "pages": "612-621" }, { "id": "authors:x6t8d-f0z10", "collection": "authors", "collection_id": "x6t8d-f0z10", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130417-151224265", "type": "article", "title": "Effect of nonlinear interactions on p-mode frequencies and line widths", "author": [ { "family_name": "Kumar", "given_name": "Pawan", "clpid": "Kumar-P" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Kerswell", "given_name": "Richard", "clpid": "Kerswell-R" } ], "abstract": "We calculate the effect of nonlinear interactions among solar acoustic modes upon the modal frequencies and energy loss rates (or line widths). The frequency shift for a radial p-mode of frequency 3 mHz is found to be about -0.5 \u00b5Hz. The magnitude of nonlinear frequency shift increases more rapidly with frequency than the inverse mode mass (mode mass is defined as the ratio of energy in the mode to its surface velocity amplitude squared). This frequency shift is primarily due to nonresonant three-mode interactions and is dominated by high l surface gravity waves (\u0192-modes) and p-modes. The line width of a radial p-mode of frequency 3 mHz, due to resonant nonlinear interactions, is about 0.3 \u00b5Hz. This result is consistent with that of Kumar & Goldreich (1989). We also find, in agreement with these authors, that the most important nonlinear interactions of trapped p-modes involve \u0192-modes and high-frequency p-modes (frequency greater than about 5 mHz) which propagate in the solar photosphere. Thus, using the arguments advanced by Kumar & Goldreich (1989), we\nconclude that nonlinear couplings cannot saturate the overstable solar p-modes at their small observed amplitudes.\nBoth the nonlinear frequency shifts and line widths, at a fixed frequency, are proportional to the inverse of mode mass which for modes of degree greater than about 100 is ~ l^(0.8). Therefore, the frequency of an \u0192-mode of l = 1000, due to nonlinear interactions, is decreased by approximately 0.4%.", "doi": "10.1086/174159", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1994-05-20", "series_number": "1", "volume": "427", "issue": "1", "pages": "483-496" }, { "id": "authors:j7n3t-zzx83", "collection": "authors", "collection_id": "j7n3t-zzx83", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130417-134602700", "type": "article", "title": "Excitation of neutron star normal modes during binary inspiral", "author": [ { "family_name": "Reisenegger", "given_name": "Andreas", "clpid": "Reisenegger-A" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "As a compact binary inspirals due to the emission of gravitational waves, its orbital period decreases continuously down to approximately 1 ms, its value at coalescence. During the last part of the inspiral, the two stars are close together, and their tidal interactions become strong. Neutron stars have many normal modes (core g-modes, crustal discontinuity modes, shear modes, etc.) whose periods lie in the range (approximately several ms) swept by the orbital period. Some of these modes are resonantly excited by the tidal force. The amount of energy a mode absorbs is proportional to the square of the overlap integral between its displacement field and the tidal force field. For all modes of interest, this overlap is poor, resulting in relatively weak excitation. For the best case, the absorbed energy is only a small fraction (approximately 10^(-6)) of the orbital energy, so the orbital phase shift is too weak to be detected by observations of the gravitational wave signal emitted by the inspiraling binary. However, with displacement amplitudes of excited quadrupole modes ranging up to 0.5% of the stellar radius, the possibility of a detectable electromagnetic signature cannot be dismissed. Both the periods of the modes and the energy they absorb depend quite strongly on the internal structure of the star. Their observation could shed light on the correct high-density equation of state.", "doi": "10.1086/174105", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1994-05-10", "series_number": "2", "volume": "426", "issue": "2", "pages": "688-691" }, { "id": "authors:yamw4-t5p22", "collection": "authors", "collection_id": "yamw4-t5p22", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130417-140841036", "type": "article", "title": "Excitation of solar p-modes", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Murray", "given_name": "Norman", "clpid": "Murray-N" }, { "family_name": "Kumar", "given_name": "Pawan", "clpid": "Kumar-P" } ], "abstract": "We investigate the rates at which energy is supplied to individual p-modes as a function of their frequencies\nv and angular degrees \u2113. The observationally determined rates are compared with those calculated on the hypothesis that the modes are stochastically excited by turbulent convection. The observationally determined excitation rate is assumed to be equal to the product of the mode's energy\nE and its (radian) line width \u0413. We obtain E from the mode's mean square surface velocity with the aid of its\nvelocity eigenfunction. We assume that \u0413 measures the mode's energy decay rate, even though quasi-elastic\nscattering may dominate true absorption. At fixed \u2113, E\u0413 rises as v^7 at low v, reaches a peak at v \u2248 3.5 mHz,\nand then declines as v^(-4\u20224) at higher v. At fixed v, E\u0413 exhibits a slow decline with increasing \u2113. To calculate energy input rates, P_ \u03b1, we rely on the mixing-length model of turbulent convection. We find entropy fluctuations to be about an order of magnitude more effective than the Reynolds stress in exciting p-modes. The calculated P_ \u03b1 mimic the v^7 dependence of E\u0413 at low v and the v^(-4\u20224) dependence at high v. The break of 11.4 powers in the v-dependence of E\u0413 across its peak is attributed to a combination of (1) the reflection of high-frequency acoustic waves just below the photosphere where the scale height drops precipitously and (2) the absence of energy-bearing eddies with short enough correlation times to excite high-frequency modes. Two parameters associated with the eddy correlation time are required to match the\nlocation and shape of the break. The appropriate values of these parameters, while not unnatural, are poorly\nconstrained by theory. The calculated P_ \u03b1 can also be made to fit the magnitude of E\u0413 with a reasonable value\nfor the eddy aspect ratio. Our results suggest a possible explanation for the decline of mode energy with increasing \u2113 at fixed v. Entropy fluctuations couple to changes in volume associated with the oscillation mode. These decrease with decreasing n at fixed v, becoming almost zero for the \u0192-mode.", "doi": "10.1086/173904", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1994-03-20", "series_number": "1", "volume": "424", "issue": "1", "pages": "466-479" }, { "id": "authors:dxv7h-9pb51", "collection": "authors", "collection_id": "dxv7h-9pb51", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130417-143450002", "type": "article", "title": "The effects of scattering on solar oscillations", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Murray", "given_name": "Norman", "clpid": "Murray-N" } ], "abstract": "Acoustic modes are scattered by turbulent velocity fluctuations in the solar convection zone. The strongest\nscattering occurs near the top of the acoustic cavity where the mode changes character from propagating to evanescent. This layer is located at depth z_1 ~ g/\u03c9^2 below the photosphere. The scattering optical depth \u03c4_s of order M^2_1 where M_1 is the Mach number of the energy-bearing eddies at z_1. The corresponding contribution to the line width is y^s ~ \u03c9M^2_1/\u03c0(n + 1), where n is the mode's radial order. At the top of the acoustic cavity the correlation time of energy-bearing eddies is much longer than \u03c9^(-1). Also, the pressure scale height H and the eddy correlation length \u039b are comparable to \u03c9/c, where c is the sound speed. Thus scattering couples modes of similar \u03c9 and all \u2113 and has little effect on the sum of their energies. Observations show that mode energies decline with decreasing n (increasing \u2113) at fixed \u03c9. Consequently, scattering damps p-modes and excites \u0192-modes.", "doi": "10.1086/173905", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1994-03-20", "series_number": "1", "volume": "424", "issue": "1", "pages": "480-490" }, { "id": "authors:294qf-6f967", "collection": "authors", "collection_id": "294qf-6f967", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-154541025", "type": "article", "title": "Magnetic field decay in isolated neutron stars", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Reisenegger", "given_name": "Andreas", "clpid": "Reisenegger-A" } ], "abstract": "We investigate three mechanisms that promote the loss of magnetic flux from an isolated neutron star. Ohmic decay produces a diffusion of the magnetic field with respect to the charged particles. It proceeds at a rate that is inversely proportional to the electric conductivity and independent of the magnetic field strength. Ohmic decay occurs in both the fluid core and solid crust of a neutron star, but it is too slow to directly affect magnetic fields of stellar scale. Ambipolar diffusion involves a drift of the magnetic field and charged particles relative to the neutrons. The drift speed is proportional to the second power of the magnetic field strength if the protons form a normal fluid. Variants of ambipolar diffusion include both the buoyant rise and the dragging by superfluid neutron\nvortices of magnetic flux tubes. Ambipolar diffusion operates in the outer part of the fluid core where the\ncharged particle composition is homogeneous, protons and electrons being the only species. The charged particle\nflux associated with ambipolar diffusion decomposes into a solenoidal and an irrotational component. Both components are opposed by frictional drag. The irrotational component perturbs the chemical equilibrium between neutrons, protons, and electrons, thus generating pressure gradients that effectively choke it. The solenoidal component is capable of transporting magnetic flux from the outer core to the crust on a short time scale. Magnetic flux that threads the inner core, where the charged particle composition is inhomogeneous, would be permanently trapped unless particle interactions could rapidly smooth departures from chemical equilibrium. Magnetic fields undergo a Hall drift related to the Hall component of the electric field. The drift speed is proportional to the magnetic field strength. Hall drift occurs throughout a neutron star. Unlike ohmic decay and ambipolar diffusion which are dissipative, Hall drift conserves magnetic energy. Thus, it cannot by itself be responsible for magnetic field decay. However, it can enhance the rate of ohmic dissipation. In the solid crust, only the electrons are mobile and the tangent of the Hall angle is large. There, the evolution of the magnetic field resembles that of vorticity in an incompressible fluid at large Reynolds number. This leads us to speculate that the magnetic field undergoes a turbulent cascade terminated by ohmic dissipation at small scales. The small-scale components of the magnetic field are also transported by Hall drift waves from the inner crust where ohmic dissipation is slow to the outer crust where it is rapid. The diffusion of magnetic flux through the crust takes ~ 5 x 10^8/B_(12) yr, where B_(12) is the crustal magnetic field strength measured in units of 10^(12) G.", "doi": "10.1086/171646", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1992-08-10", "series_number": "1", "volume": "395", "issue": "1", "pages": "250-258" }, { "id": "authors:appmc-70446", "collection": "authors", "collection_id": "appmc-70446", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-154510940", "type": "article", "title": "A new class of g-modes in neutron stars", "author": [ { "family_name": "Reisenegger", "given_name": "Andreas", "clpid": "Reisenegger-A" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Because a neutron star is born hot, its internal composition is close to chemical equilibrium. In the fluid\ncore, this implies that the ratio of the number densities of charged particles (protons and electrons) to neutrons,\nx \u2261 n_c/n_n, is an increasing function of the mass density. This composition gradient stably stratifies the matter giving rise to a Brunt-V\u00e4is\u00e4l\u00e4 frequency N ~ (xg/2H)^(1/2) ~ 500 s^(-1), where g is the gravitational acceleration, and H is the density scale height. Consequently, a neutron star core provides a cavity that supports gravity modes (g-modes). These g-modes are distinct from those previously identified with the thermal stratification of the surface layers and the chemical stratification of the crust. We compute the lowest-order, quadrupolar, g-modes for cold, Newtonian, neutron star models with M/M_\u2609 = 0.581 and M/M_\u2609 = 1.405 and show that the crustal and core g-modes have similar periods. We also discuss damping mechanisms and estimate damping rates for the core g-modes. Particular attention is paid to damping due to the emission of gravitational radiation.", "doi": "10.1086/171645", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1992-08-10", "series_number": "1", "volume": "395", "issue": "1", "pages": "240-249" }, { "id": "authors:x5gg9-qcq78", "collection": "authors", "collection_id": "x5gg9-qcq78", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130430-134847768", "type": "article", "title": "Puzzles and Prospects in Planetary Ring Dynamics", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "I outline some of the main processes that shape planetary rings. Then I focus on two\noutstanding issues, the role of self-gravity in the precession of narrow rings and the dynamics of\nNeptune's arcs. By airing these well-defined but unsolved problems, I hope to encourage others\nto join me in the quest for their solutions.", "issn": "0074-1809", "publisher": "Kluwer", "publication": "IAU Symposia", "publication_date": "1992", "volume": "152", "pages": "65-73" }, { "id": "authors:dq418-bz526", "collection": "authors", "collection_id": "dq418-bz526", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-154444141", "type": "article", "title": "Thermal and mechanical damping of solar p-modes", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Kumar", "given_name": "Pawan", "clpid": "Kumar-P" } ], "abstract": "Nonadiabatic effects associated with the transfer of energy and with turbulent stresses add small imaginary parts, \u03c9_i^(1) and \u03c9_i^(2), to solar p-mode eigenfrequencies. Numerical calculations have shown that these quite\ndifferent processes make comparable contributions to \u03c9i at frequencies well below the acoustic cutoff at \u03c9_(ac). We derive analytic expressions which reveal the connection between \u03c9_i^(1) and \u03c9_i^(2). Our estimates yield \u03c9_i \u221d \u03c9^8\nfor \u03c9 \u00ab \u03c9_(ac), in good agreement with the numerical calculations. However, the observed line width is proportional to \u03c9^(4.2) at low frequencies. We suspect that there is an unmodeled component of perturbed convective\nenergy transport or of turbulent viscosity that makes an important contribution to \u03c9_i at \u03c9 \u00ab \u03c9_(ac).", "doi": "10.1086/170126", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1991-06-10", "series_number": "1", "volume": "374", "issue": "1", "pages": "366-368" }, { "id": "authors:e1974-r1a83", "collection": "authors", "collection_id": "e1974-r1a83", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-114003503", "type": "article", "title": "Implications of solar p-mode frequency shifts", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Murray", "given_name": "Norman", "clpid": "Murray-N" }, { "family_name": "Willette", "given_name": "Gregory", "clpid": "Willette-G" }, { "family_name": "Kumar", "given_name": "Pawan", "clpid": "Kumar-P" } ], "abstract": "We relate entropy and magnetic field perturbations to variations of solar p-mode eigenfrequencies. The frequency\nvariations result from changes in path length and propagation speed. These produce shifts of opposite sign. Path length changes dominate for entropy perturbations, and propagation speed changes dominate for most types of magnetic field perturbations. The p-mode frequencies increased along with solar activity between 1986 and 1989. The frequency shifts exhibit a rapid rise with increasing frequency followed by a precipitous drop. The positive component signals a strengthening of the photospheric magnetic field to an rms value of order 200 G. The sudden drop at high frequency is due to a combination of a resonance and an increase in temperature in the chromospheric cavity. The magnetic stress perturbation decays above the top of the convection zone on a length scale comparable to the pressure scale height and grows gradually with depth below. The former characteristic implies that the stress is mainly due to small magnetic elements of the enhanced network, a conclusion supported by our analysis of Kitt Peak magnetograms. The latter property suggests that the flux tubes which pierce the photosphere\nstrengthen with depth, at least to a pressure level of 10^8 dynes cm^(-2). The presence of a resonance in the chromospheric cavity means that the transition layer maintains enough coherence to partially reflect acoustic waves even near cycle maximum. The fractional chromospheric temperature rise implies a much larger fractional increase in the rate of mechanical heating, as indicated by the\nvariation of the Ca II H and K lines.", "doi": "10.1086/169858", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1991-04-01", "series_number": "2", "volume": "370", "issue": "2", "pages": "752-762" }, { "id": "authors:et8gj-kfg69", "collection": "authors", "collection_id": "et8gj-kfg69", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-143153072", "type": "article", "title": "Wave generation by turbulent convection", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Kumar", "given_name": "Pawan", "clpid": "Kumar-P" } ], "abstract": "We consider wave generation by turbulent convection in a plane parallel, stratified atmosphere that sits in a\ngravitational field, g. The atmosphere consists of two semi-infinite layers, the lower adiabatic and polytropic\nand the upper isothermal. The adiabatic layer supports a convective energy flux given by mixing length theory; F_c ~ pv^3_H, where p is mass density and v_H is the velocity of the energy bearing turbulent eddies. Acoustic waves with \u03c9 > \u03c9_(\u0251c) and gravity waves with \u03c9 < 2k_h H_i\u03c9b propagate in the isothermal layer whose acoustic cutoff frequency, \u03c9_(ac), and Brunt-V\u00e4is\u00e4l\u00e4 frequency, \u03c9_b, satisfy \u03c9^2_(\u0251c) = yg/4H_i and \u03c9^2_b = (y-1)g/yH_i, where y and H_i denote the adiabatic index and scale height. The atmosphere traps acoustic waves in upper part of the adiabatic layer (p-modes) and gravity waves on the interface between the adiabatic and isothermal layers (f-modes). These modes obey the dispersion relation \u03c9^2\u22482/m gk_h(n + m/2), for \u03c9 < \u03c9_(\u0251c). Here, m is the polytropic index, k_h is the magnitude of the horizontal wave vector, and n is the\nnumber of nodes in the radial displacement eigenfunction; n = 0 for f-modes. Wave generation is concentrated at the top of the convection zone since the turbulent Mach number, M =\nv_H/c, peaks there; we assume M_t \u00ab 1. The dimensionless efficiency, \u03b7, for the conversion of the energy carried\nby convection into wave energy is calculated to be \u03b7~M_t^(5/12) for p-modes,f-modes, and propagating acoustic\nwaves, and \u03b7~M, for propagating gravity waves. Most of the energy going into p-modes, f-modes, and propagating\nacoustic waves is emitted by inertial range eddies of size h ~ M_t^(3/2)H_t, at \u03c9 ~ \u03c9_(\u0251c) and k_h ~ 1/H_t. The\nenergy emission into propagating gravity waves is dominated by energy bearing eddies of size ~ H_t and is concentrated at \u03c9 ~ v_t/H_t ~ M_t \u03c9_(\u0251c) and k_h ~ 1/H_t. We find the power input to individual p-modes, E_p, to vary as \u03c9<^(2m^2+7m-3)/(m+3) at frequencies \u03c9 \u00ab v_t/H_t. Libbrecht has shown that the amplitudes and linewidths of the solar p-modes imply E_p \u221d \u03c9^8 for \u03c9 \u00ab 2 x 10^(-2) s^(-1). The theoretical exponent matches the observational one for m \u2248 4, a value obtained from the density profile in the upper part of the solar convection zone. This agreement supports the hypothesis that the solar p-modes are stochastically excited by turbulent convection.", "doi": "10.1086/169376", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1990-11-10", "series_number": "2", "volume": "363", "issue": "2", "pages": "694-704" }, { "id": "authors:ssw41-vg836", "collection": "authors", "collection_id": "ssw41-vg836", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-132048465", "type": "article", "title": "Neutron starquake models for gamma-ray bursts", "author": [ { "family_name": "Blaes", "given_name": "O.", "clpid": "Blaes-O" }, { "family_name": "Blandford", "given_name": "R.", "orcid": "0000-0002-1854-5506", "clpid": "Blandford-R-D" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" }, { "family_name": "Madau", "given_name": "P.", "clpid": "Madau-P" } ], "abstract": "We assess neutron starquake models for \u03b3-ray bursts. The elastic energy the crust can store is sufficient to account for that radiated in a single burst, but it is insufficient to supply the \u2273 10^6 bursts each star produces over its lifetime, and so it must be replenished. Seismic waves are radiated if shear stress is relieved by brittle fracture. However they cannot propagate directly to the surface but are temporarily trapped below a reflecting layer. Between the reflecting layer and the surface the displacement amplitude of the wave is nearly constant and the strain is very small. At low frequencies, \u227e 10^4 Hz, the reflection is associated with an evanescent zone. At high frequencies, \u2273 10^4 Hz, the reflection occurs where the magnetic field stress starts to dominate the crustal rigidity. The shaking of the stellar surface couples the seismic waves to Alf\u00e9n waves which propagate out into the magnetosphere. At low frequencies, the coupling coefficient, T, is proportional to the square of the magnetic field, B, and increases with the seventh power of the wave frequency, v. At high frequencies, T is proportional to B^(4/7)v^(3/7). Alfv\u00e9n wave luminosities sufficient to power Galactic \u03b3-ray bursts are possible if magnetic fields \u2273 10^(11) G cover at least part of the stellar surface. The conversion of Alfv\u00e9n waves into \u03b3-rays may occur if the waves are charge-starved or if their amplitudes approach that of the background magnetic field.", "doi": "10.1086/167754", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1989-08-15", "series_number": "2", "volume": "343", "issue": "2", "pages": "839-848" }, { "id": "authors:9k9dj-pzg90", "collection": "authors", "collection_id": "9k9dj-pzg90", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130304-082325115", "type": "article", "title": "Neptune's Story", "author": [ { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" }, { "family_name": "Murray", "given_name": "N.", "clpid": "Murray-N" }, { "family_name": "Longaretti", "given_name": "P. Y.", "clpid": "Longaretti-P-Y" }, { "family_name": "Banfield", "given_name": "D.", "clpid": "Banfield-D" } ], "abstract": "It is conjectured that Triton was captured from a heliocentric\norbit as the result of a collision with what was then\none of Neptune's regular satellites. The immediate postcapture\norbit was highly eccentric with a semimajor axis a\n~ 10^3R_N and a periapse distance rp that oscillated periodically\nabove a minimum value of about 5R_N. Dissipation\ndue to tides raised by Neptune in Triton caused Triton's\norbit to evolve to its present state in \u227e10^9 years. For\nmuch of this time Triton was almost entirely molten.\nWhile its orbit was evolving, Triton cannibalized most of\nthe regular satellites of Neptune and also perturbed\nNereid, thus accounting for that satellite's highly eccentric\nand inclined orbit. The only regular satellites of\nNeptune that survived were those that formed well within\n5RN and they move on inclined orbits as the result of\nchaotic perturbations forced by Triton. Neptune's arcs are\nconfined around the corotation resonances of one of these\ninner satellites. The widths and lengths of the arcs imply\nthat the satellite's radius is at least 30/(sin i)^(2/3) kilometers\nfor i \u227e 1, where i is the angle of inclination.", "doi": "10.1126/science.245.4917.500", "issn": "0036-8075", "publisher": "American Association for the Advancement of Science", "publication": "Science", "publication_date": "1989-08-04", "series_number": "4917", "volume": "245", "issue": "4917", "pages": "500-504" }, { "id": "authors:2d3zm-r3673", "collection": "authors", "collection_id": "2d3zm-r3673", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130226-153610592", "type": "article", "title": "The Formation of Sharp Edges in Planetary Rings by Nearby Satellites", "author": [ { "family_name": "Borderies", "given_name": "Nicole", "clpid": "Borderies-N" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "Sharp edges, boundaries between regions of high and low optical depth in planetary rings, are maintained by shepherd satellites which transfer angular momentum to and from the ring particles. We derive equations that govern the shapes of the perturbed streamlines near such a boundary. These equations are solved for a simple numerical model whose parameters are chosen to resemble those of the Encke division and its associated satellite. The results of our calculation faithfully reproduce the sharp edges which bound the division and imply that the ring thickness in the unperturbed regions far from the edges is of order 10 m. In particular, the angle-averaged surface density is found to vary on a much shorter radial length scale than that over which the satellite torque is applied. We demonstrate that this striking feature is related to the local reversal of the viscous transport of angular momentum in the most strongly perturbed regions.", "doi": "10.1016/0019-1035(89)90145-0", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "1989-08", "series_number": "2", "volume": "80", "issue": "2", "pages": "344-360" }, { "id": "authors:ytmkh-p0496", "collection": "authors", "collection_id": "ytmkh-p0496", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-103448215", "type": "article", "title": "Tides in rotating fluids", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Nicholson", "given_name": "Philip D.", "orcid": "0000-0003-2275-4463", "clpid": "Nicholson-P-D" } ], "abstract": "We consider the tidal disturbance forced in a differentially rotating fluid by a rigidly rotating external\npotential. The fluid is assumed to be inviscid, insulated, and self-gravitating, and to have laminar unperturbed\nand perturbed velocity fields. The external potential may exert a steady torque on the fluid which is of second\norder in Its strength. However, to this order, we prove that there are no secular changes in the angular momenta of fluid particles, except possibly at corotation where the angular velocity, \u03a9(r,\u03b8), is equal to the pattern speed of the potential, \u03a9_p. A corollary of our theorem is that, except at corotation, all of the angular momentum transferred to the fluid by the external potential must be transported away by internal stresses. In the applications of which we are aware, these stresses are associated with waves.", "doi": "10.1086/167664", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1989-07-15", "series_number": "2", "volume": "342", "issue": "2", "pages": "1075-1078" }, { "id": "authors:3jfs9-sej77", "collection": "authors", "collection_id": "3jfs9-sej77", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-105241949", "type": "article", "title": "Tidal friction in early-type stars", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Nicholson", "given_name": "Philip D.", "orcid": "0000-0003-2275-4463", "clpid": "Nicholson-P-D" } ], "abstract": "The tidal torque on an early-type star is concentrated near the boundary between the convective core and radiative envelope and a train of gravity waves is excited there. The angular momentum which the torque removes from the fluid is transported outward by the gravity waves, which carry negative angular momentum. Before the surface layers are despun to synchronous rotation, the gravity waves propagate to just below the photosphere where they suffer radiative damping and are partially reflected. It is here that the negative angular momentum is deposited and the primary tidal despinning takes place. The surface layers cannot be\nspun down below synchronous rotation because as a train of gravity waves approaches a corotation resonance its group velocity and wavelength tend to zero, its amplitude diverges, and it is completely absorbed. Thus, tidal despinning to synchronous rotation proceeds from the outside toward the inside of the star. Our picture provides a neat explanation for the otherwise puzzling discovery by Giuricin, Mardirossian, and Mezzetti that Zahn's theory for tidal evolution in early-type close binaries seems to be compatible with the observed rates of orbit circularization while significantly underestimating the observed rates of spin synchronization.", "doi": "10.1086/167665", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1989-07-15", "series_number": "2", "volume": "342", "issue": "2", "pages": "1079-1084" }, { "id": "authors:r3egq-17312", "collection": "authors", "collection_id": "r3egq-17312", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130305-082856101", "type": "article", "title": "Nonlinear interactions among solar acoustic modes", "author": [ { "family_name": "Kumar", "given_name": "Pawan", "clpid": "Kumar-P" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "We evaluate the rates at which nonlinear interactions transfer energy among the normal modes of a plane-parallel,\nstratified atmosphere. The atmosphere resembles the outer part of the Sun including the convection\nzone and the optically thin region above the photosphere up to the temperature minimum. The acoustic\nmodes are assigned energies such that their photospheric velocities match those of the Sun's p-modes. The\nnonlinearity parameter is the acoustic Mach number, M, the ratio of the total acoustic velocity due to all of\nthe modes to the sound speed. For M^2 \u226a 1 the leading nonlinear interactions are those which couple three-modes.\nWe show that every p-mode in the 5 minute band is involved in many near-resonant triplets. As a\nconsequence, the energy transfer rates are independent of the mode line widths. Because M increases with\nheight, the dominant contributions to the three-mode coupling coefficients occur in the upper part of the convection\nzone and in the optically thin isothermal layer. Moreover, the coupling coefficients tend to increase\nwith \u03c9 and k_h.\n\nNonlinear interactions which couple two trapped modes and one propagating mode drain energy from the\ntrapped modes. They are far more effective than interactions among three trapped modes which drive the\nmodes toward equipartition of energy. Thus, every trapped mode suffers a net loss of energy due to its nonlinear\ninteractions. Estimates of the nonlinear energy transfer rates are plagued by two uncertainties. Some of the\ncoefficients which couple two trapped modes to a propagating mode formally diverge as the thickness of the\nisothermal layer is increased to infinity; physically, this reflects the exponential growth of the acoustic Mach\nnumber with height in the isothermal layer. Also, the energy transfer rates are sensitive to the unknown energies\nof the high-degree trapped modes. Plausible assumptions lead to energy transfer rates which are somewhat\nsmaller than the products of the mode energies and line widths. Thus, nonlinear mode coupling is\nprobably not the dominant damping process for the solar p-modes, at least for those with small l. However,\nthis cannot be regarded as a secure conclusion. The observational signature of damping due to nonlinear\nmode coupling would be a decrease in the energy per mode with increasing l at fixed \u03c9. In addition, it might\nbe responsible, at least in part, for the steep decline in the energy per mode at frequencies above 3 mHz which\nis usually attributed to radiative damping.\n\nOur investigation indirectly bears on the question of the stability of the p-modes. We find that nonlinear\nmode couplings cannot limit the growth of overstable p-modes. This favors the hypothesis that the Sun's\np-modes are stochastically excited by turbulent convection.", "doi": "10.1086/167616", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1989-07-01", "series_number": "1", "volume": "342", "issue": "1", "pages": "558-575" }, { "id": "authors:jmd58-vwa91", "collection": "authors", "collection_id": "jmd58-vwa91", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130312-145928531", "type": "article", "title": "Distribution functions for the time-averaged energies of stochastically excited solar p-modes", "author": [ { "family_name": "Kumar", "given_name": "Pawan", "clpid": "Kumar-P" }, { "family_name": "Franklin", "given_name": "Joel", "clpid": "Franklin-J-N" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "We study the excitation of a damped harmonic oscillator by a random force as a model for the stochastic\nexcitation of a solar p-mode by turbulent convection. An extended sequence of observations is required to\nseparate different p-modes and thus determine the energies of individual modes. Therefore, the observations\nyield time-averaged values of the energy. We apply the theory of random differential equations to calculate\ndistribution functions for the time-averaged energy of the oscillator. The instantaneous energy satisfies a\nBoltzmann distribution. With increasing averaging time the distribution function narrows, and its peak shifts\ntoward the mean energy. We also perform numerical integrations to generate finite sequences of time-averaged\nenergies. These are treated as simulated data from which we obtain approximate probability distributions for\nthe time-averaged energy. A comparison of our calculated distributions with those determined observationally\nshould help to resolve whether the solar p-modes are stochastically excited. If they are, modes of the same\nfrequency with degree l \u227e 200 should have identical values for the products of their mean energies, line-widths,\nand masses. If, in addition, turbulence or radiative dissipation provides the principal damping mechanism, the\nmean energies should be independent of angular order, l.", "doi": "10.1086/166345", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1988-05-15", "series_number": "2", "volume": "328", "issue": "2", "pages": "879-887" }, { "id": "authors:5gkvh-ezh41", "collection": "authors", "collection_id": "5gkvh-ezh41", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130311-145716444", "type": "article", "title": "The interaction of acoustic radiation with turbulence", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Kumar", "given_name": "Pawan", "clpid": "Kumar-P" } ], "abstract": "We derive expressions for the spectral emissivity and absorptivity of acoustic radiation by low Mach number\n(M \u226a 1) turbulent fluids. The emissivity and absorptivity depend on the manner in which the turbulence is excited. We consider three types of turbulence. The first is free turbulence, that is, turbulence which is not subject to external forces. The second and third examples are special cases of forced turbulence, turbulence maintained by stirring with spoons and turbulent pseudoconvection. Acoustic quadrupoles are the lowest order acoustic multipoles present in free turbulence, and they control both its emissivity and absorptivity. Acoustic dipoles are created in forced turbulence, and they enhance the acoustic emissivity by M^(-2) compared to that of free turbulence. The acoustic absorptivity of forced turbulence is quite subtle. The absorptivity of turbulence which is maintained by stirring is dominated by acoustic dipoles and exceeds that of free turbulence by M^(-2). The dipole absorptivity of turbulent pseudoconvection is reduced by M^2 below that of turbulence maintained by stirring. Thus, the absorptivity of turbulent pseudoconvection is no larger than that of free turbulence. We apply our results to estimate the equilibrium energies of the acoustic modes in a box filled with fluid some of which is turbulent. For both free turbulence and turbulence maintained by stirring, the most highly excited acoustic modes attain energies E ~ Mv^2, where M and v are the typical mass and velocity of an\nenergy bearing eddy. The quality factors, or Q's, of the modes are larger by M^(-2) in the former case than in\nthe latter. For turbulent pseudoconvection, the most energetic acoustic modes have equilibrium energies\nE ~ Mc^2 , where c is the sound speed. Their Q's are comparable to those of modes in equilibrium with free\nturbulence. We evaluate the scattering of acoustic radiation by turbulent fluids. For all types of turbulence, the scattering opacity is smaller by M^3 than the absorptive opacity for frequencies near the peak of the acoustic spectrum. Radiation scattered by free turbulence and turbulent pseudoconvection suffers frequency shifts\n\u0394\u03c9 ~ \u03c9. The frequency shifts are much smaller, \u0394\u03c9 ~ M\u03c9, for radiation scattered by turbulence maintained\nby stirring.\nWe investigate the rate at which nonlinear interactions transfer energy among the acoustic modes. If all of\nthe fluid in the box is turbulent, this rate is slower, by M^3 for free turbulence, by M^5 for turbulence maintained\nby stirring, and by M for turbulent pseudoconvection, than the rate at which the individual acoustic\nmodes exchange energy with the turbulence. If only a small portion of the fluid is turbulent, the nonlinear mode interactions can be significant, especially for modes in equilibrium with turbulent pseudoconvection. Our results have potential applications to the acoustic radiation in regions of extended turbulence which often arise in nature. In particular, they should prove useful in understanding the excitation of solar oscillations.", "doi": "10.1086/166108", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1988-03-01", "series_number": "1", "volume": "326", "issue": "1", "pages": "462-478" }, { "id": "authors:ydjhm-fka85", "collection": "authors", "collection_id": "ydjhm-fka85", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-102358064", "type": "article", "title": "Physics of modes in a differentially rotating system -\n analysis of the shearing sheet", "author": [ { "family_name": "Narayan", "given_name": "Ramesh", "orcid": "0000-0002-1919-2730", "clpid": "Narayan-R" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Goodman", "given_name": "Jeremy", "clpid": "Goodman-J" } ], "abstract": "We analyse the linear non-vortical modes of the shearing sheet, a model compressible two-dimensional fluid system with constant density, constant shear, and Coriolis force. This model has several features found in differentially\nrotating systems of interest in astrophysics, such as disc galaxies, accretion tori, planetary rings, protostellar nebulae, and possibly even rotating stars.", "issn": "0035-8711", "publisher": "Royal Astronomical Society", "publication": "Monthly Notices of the Royal Astronomical Society", "publication_date": "1987-09-01", "series_number": "1", "volume": "228", "issue": "1", "pages": "1-41" }, { "id": "authors:a9mts-z1562", "collection": "authors", "collection_id": "a9mts-z1562", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-100854712", "type": "article", "title": "The stability of accretion tori. II. Non-linear evolution to discrete planets", "author": [ { "family_name": "Goodman", "given_name": "Jeremy", "clpid": "Goodman-J" }, { "family_name": "Narayan", "given_name": "Ramesh", "orcid": "0000-0002-1919-2730", "clpid": "Narayan-R" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Hawley has shown through two-dimensional computer simulations that a slender torus in which a linear Papaloizou-Pringle (PP) instability with azimuthal wavenumber m, is excited evolves non-linearly to a configuration with m nearly disconnected 'planets'. We present an analytical fluid equilibrium that we believe represents his numerical planets. The fluid has an ellipsoidal figure and is held together by the Corio lis force associated with the retrograde fluid motion. There is a bifurcation between the torus and planet configurations at precisely the vorticity below which the PP instability switches on. Although the solution is three-dimensional, there is perfect hydrostatic equilibrium and the motion is\nentirely two-dimensional. We analyse the linear modes of the analytical planet and find that there are numerous instabilities, though they are not as violent as the\nPP instability in the torus. We also discuss the energy and vorticity of neutral modes, and we argue that when the torus breaks up into planets, neutral modes with negative energy and non-zero vorticity are excited in order to conserve total energy and specific vorticity. We speculate that the fluid in Hawley's simulations may be approaching two-dimensional turbulence.", "issn": "0035-8711", "publisher": "Royal Astronomical Society", "publication": "Monthly Notices of the Royal Astronomical Society", "publication_date": "1987-04-01", "series_number": "3", "volume": "225", "issue": "3", "pages": "695-711" }, { "id": "authors:5jjs5-kwd73", "collection": "authors", "collection_id": "5jjs5-kwd73", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-085411501", "type": "article", "title": "Shepherding of the Uranian Rings. II. Dynamics", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Porco", "given_name": "Carolyn C.", "clpid": "Porco-C-C" } ], "abstract": "We explore the dynamical significance of the orbital resonances, identified in Paper I (Porco and Goldreich 1987), involving the satellites 1986U7 and 1986U8, and the \u0454, \u03b4, and y rings. We demonstrate that these satellites are capable of exerting torques on the \u0454 ring which supply at the inner edge, and remove at the outer edge, the unperturbed angular-momentum luminosity transported by viscous stresses outward across the ring, provided that the ring has a mass which is comparable to the predicted\nvalue M_\u0454 = 6.1 X 10^(18) g, and that it is not more than a few meters thick. Thus there is no compelling reason to question the applicability of the standard theory of shepherding to the \u0454 ring. However, the standard theory does place rather stringent requirements on the ring's properties, suggesting that confinement might be due to the reduction of the angular-momentum luminosity by flux reversal. Flux reversal could be associated either with the ring's eccentricity gradient or with nonlinear density waves\nin its interior. The drag due to the planet's extended neutral hydrogen atmosphere probably has only a minor effect on the dynamics of the \u0454, \u03b4, and y rings. However, it poses a severe problem for the shepherding of the \u0251 and \u03b2 rings unless their masses have been seriously underestimated. This problem, and the large s-band optical depths of these rings, lead us to question the proposal that self-gravity is responsible for enforcing rigid precession in narrow rings.", "doi": "10.1086/114355", "issn": "0004-6256", "publisher": "American Astronomical Society", "publication": "Astronomical Journal", "publication_date": "1987-03", "series_number": "3", "volume": "93", "issue": "3", "pages": "730-737" }, { "id": "authors:ffr8j-t0s61", "collection": "authors", "collection_id": "ffr8j-t0s61", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-075058984", "type": "article", "title": "Shepherding of the Uranian rings. I. Kinematics", "author": [ { "family_name": "Porco", "given_name": "Carolyn C.", "clpid": "Porco-C-C" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "We identify several orbital resonances involving the newly discovered satellites, 1986U7 and 1986U8, and the Uranian rings. The most important resonances in eccentric rings are known as eccentric reson\u00adances and are generalizations of the more familiar Lindblad resonances. In keeping with the notation established for Lindblad resonances, we distinguish inner and outer eccentric resonances by the symbols IER and OER. We show that by reducing the absolute radius scale of the Uranian ring system by 0.0124% the 24:25 OER of 1986U7 and the 14:13 IER of 1986U8 fall at the inner and outer edges of the \u0404 ring. The same scale change also brings the 23:22 IER of 1986U7 into coincidence with the outer edge of the \u03b4 ring and the 6:5 IER of 1986U8 close to the center of the \u03b3 ring. Furthermore, adopting the latest Voyager value of GM_u and our reduced radius scale, we find that the pattern speed of the m = 2 distortion in the \u03b4 ring corresponds to that expected for a normal mode excited either by an internal viscous overstability or parametrically by shepherd satellites. These kinematic results make a compel\u00ad ling case for our proposed reduction in the ring radius scale and also imply \nthat 1986U7 and 1986U8 are the inner and outer shepherds for the \u0404 ring, that 1986U7 is the outer shepherd for the \u03b4 ring, and that 1986U8 is an outer shepherd for the \u03b3 ring.", "doi": "10.1086/114354", "issn": "0004-6256", "publisher": "American Astronomical Society", "publication": "Astronomical Journal", "publication_date": "1987-03", "series_number": "3", "volume": "93", "issue": "3", "pages": "724-729" }, { "id": "authors:yfqy2-6z114", "collection": "authors", "collection_id": "yfqy2-6z114", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130301-104357332", "type": "article", "title": "Nonlinear Density Waves in Planetary Rings", "author": [ { "family_name": "Borderies", "given_name": "Nicole", "clpid": "Borderies-N" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "We discuss the steady-state structure of the nonlinear density waves generated in a planetary ring at the Lindblad resonances of a satellite. We show that strong density waves lead to an enhancement of the background surface density in the wave zone.", "doi": "10.1016/0019-1035(86)90054-0", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "1986-12", "series_number": "3", "volume": "68", "issue": "3", "pages": "522-533" }, { "id": "authors:0w8b8-49p07", "collection": "authors", "collection_id": "0w8b8-49p07", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130311-105632799", "type": "article", "title": "Towards a theory for Neptune's arc rings", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" }, { "family_name": "Borderies", "given_name": "Nicole", "clpid": "Borderies-N" } ], "abstract": "It is proposed that the incomplete rings of Neptune consist of a number of short arcs centered on the corotation resonances of a single satellite. The satellite must have a radius of the order of 100 km or more and move on an inclined orbit. Corotation resonances are located at potential maxima. Thus, mechanical energy dissipated by interparticle collisions must be continually replenished to prevent the arcs from spreading. It is shown that each corotation resonance is associated with a nearby Lindblad resonance, which excites the ring particles' orbital eccentricity, thus supplying the energy required to maintain the arc. The ultimate energy reservoir is the satellite's orbital energy. Therefore, interaction with the arcs damps the satellite's orbital inclination. The self-gravity of the arcs limits their contraction and enforces a relation between arc length and mass. The estimated arc masses are so small, of the order of 10^16 g, that the satellite's orbital inclination suffers negligible decay over the age of the solar system. The inferred surface mass densities are comparable to those found in the major rings of Saturn and Uranus.", "doi": "10.1086/114178", "issn": "0004-6256", "publisher": "American Astronomical Society", "publication": "Astronomical Journal", "publication_date": "1986-08", "series_number": "2", "volume": "92", "issue": "2", "pages": "490-494" }, { "id": "authors:fyrw5-6wy88", "collection": "authors", "collection_id": "fyrw5-6wy88", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-071925944", "type": "article", "title": "The stability of accretion tori. I. Long-wavelength modes of slender tori", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Goodman", "given_name": "Jeremy", "clpid": "Goodman-J" }, { "family_name": "Narayan", "given_name": "Ramesh", "orcid": "0000-0002-1919-2730", "clpid": "Narayan-R" } ], "abstract": "We elucidate the inviscid instabilities of an isentropic torus found previously by Papaloizou & Pringle. The torus is a polytrope of index, n, and has a small ratio of minor radius, \u0251, to orbital radius, r_0. In equilibrium it rotates on cylinders with angular velocity profile \u03a9,\u221dr^(-q). Linear modes are proportional to exp i(m\u00d8-wt). For small \u03b2\u2261m\u03b1/r_0 , we justify the use of height-averaged equations by appealing to approximate vertical hydrostatic equilibrium. The effective polytropic index for the resulting two-dimensional problem is N\u2261n+\u00bd; thus an \nincompressible torus in three dimensions behaves compressibly in two. Height averaged modes obey an ordinary differential equation, which we solve numerically to obtain the growth rate as a function of q, n, and \u03b2. The error made\nin predicting the growth rate of the actual three-dimensional system is small everywhere along the principal branch even for \u03b2~0.5, and is less than 1 per cent\nfor the fastest-growing mode. We analytically solve the artificial case n=-\u00bd, which is two-dimensionally incompressible, and show that it has all the qualitative\nfeatures of the general case, except that it does not have a resonance at corotation. In the general case, with n>-\u00bd and q<2, the corotation resonance absorbs energy and angular momentum, so the growing and decaying modes do not occur in complex-conjugate pairs. We solve a second special case, namely n=2-q=0, almost analytically in three dimensions, without height-averaging. Papaloizou & Pringle asserted that this system is stable but we show that there is an unstable mode for small \u03b2 just as in the other systems. In fact this principal unstable branch, with corotation at the pressure maximum, is qualitatively the same for all n and is essentially independent both of compressibility and of the gradient in vorticity per unit surface density. Thus the modes are not sonic, nor are they\nsimilar to those of the Kelvin-Helmholtz instability. Instead they are composed of two edge waves, akin to surface waves in water although modified by shear and\nrotation, coupled across a forbidden region around corotation.", "issn": "0035-8711", "publisher": "Royal Astronomical Society", "publication": "Monthly Notices of the Royal Astronomical Society", "publication_date": "1986-07-15", "series_number": "2", "volume": "221", "issue": "2", "pages": "339-364" }, { "id": "authors:yf72c-dvn55", "collection": "authors", "collection_id": "yf72c-dvn55", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130301-102826680", "type": "article", "title": "Evolution of the Janus-Epimetheus Coorbital Resonance Due to\n Torques from Saturn's Rings", "author": [ { "family_name": "Lissauer", "given_name": "Jack J.", "orcid": "0000-0001-6513-1659", "clpid": "Lissauer-J-J" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "We analyze the interactions between Saturn's coorbital satellites, Janus and Epimetheus, and the outer edge of the A ring, which is presumably maintained by these moons at their 7:6 resonance. Using two distinct but conceptually related methods, we show that ring torques are driving these satellites into a tighter lock. Unless there is a counterbalancing force which we have neglected, their orbital configuration will evolve from the current horseshoe-type lock to one of tadpole orbits around a single Lagrange point in \u223c20 myr. This finding adds an additional member to the list of short time scale problems associated with the interactions between Saturn's rings and its inner moons", "doi": "10.1016/0019-1035(85)90066-1", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "1985-12", "series_number": "3", "volume": "64", "issue": "3", "pages": "425-434" }, { "id": "authors:p3pwd-kjh74", "collection": "authors", "collection_id": "p3pwd-kjh74", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130301-083144733", "type": "article", "title": "A Granular Flow Model for Dense Planetary Rings", "author": [ { "family_name": "Borderies", "given_name": "Nicole", "clpid": "Borderies-N" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "We study the viscosity of a differentially rotating particle disk in the limiting case where the particles are densely packed and their collective behavior resembles that of a liquid. The pressure tensor is derived from the equations of hydrodynamics and from a simple kinetic model of collisions described by Haff (1983). We find that density waves and narrow circular rings are unstable if the liquid approximation applies. The resulting development of nonlinear perturbations may give rise to \"splashing\" of the ring material in the vertical direction. These results may help in understanding the origin of the ellipticities of ringlets, the nonaxisymmetric features near the outer edge of the Saturnian B ring, and the unexplained residuals in kinematic models of the Saturnian and Uranian rings.", "doi": "10.1016/0019-1035(85)90054-5", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "1985-09", "series_number": "3", "volume": "63", "issue": "3", "pages": "406-420" }, { "id": "authors:j3vp3-epd75", "collection": "authors", "collection_id": "j3vp3-epd75", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130312-113632727", "type": "article", "title": "Non-axisymmetric instability in thin discs", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Narayan", "given_name": "Ramesh", "orcid": "0000-0002-1919-2730", "clpid": "Narayan-R" } ], "abstract": "Thin discs of arbitrary specific angular momentum are shown to have unstable non-axisymmetric modes provided there is at least one good reflecting edge.", "issn": "0035-8711", "publisher": "Royal Astronomical Society", "publication": "Monthly Notices of the Royal Astronomical Society", "publication_date": "1985-03-01", "series_number": "1", "volume": "213", "issue": "1", "pages": "7-10" }, { "id": "authors:0btw7-zs347", "collection": "authors", "collection_id": "0btw7-zs347", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130301-080054810", "type": "article", "title": "The Eccentric Saturnian Ringlets at 1.29R_s and 1.45R_s", "author": [ { "family_name": "Porco", "given_name": "C.", "clpid": "Porco-C-C" }, { "family_name": "Nicholson", "given_name": "P. D.", "orcid": "0000-0003-2275-4463", "clpid": "Nicholson-P-D" }, { "family_name": "Borderies", "given_name": "N.", "clpid": "Borderies-N" }, { "family_name": "Danielson", "given_name": "G. E.", "clpid": "Danielson-G-E" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" }, { "family_name": "Holberg", "given_name": "J. B.", "clpid": "Holberg-J-B" }, { "family_name": "Lane", "given_name": "A. L.", "clpid": "Lane-A-L" } ], "abstract": "The shapes and kinematics of the two major eccentric ringlets in Saturn's C ring are studied in data acquired by four Voyager experiments: imaging science (ISS), radio science (RSS), ultraviolet spectrometer (UVS), and photopolarimeter (PPS). It is found that the ringlets have mean widths of \u223c25 km (Titan, 1.29 R_s) and \u223c64 km (Maxwell, 1.45R_s), eccentricities of order 10^(\u22124), sharp edges on a scale of \u223c1 km, normal optical depths \u03c4 \u223c 1\u22122, and are embedded in essentially empty gaps (\u03c4 < 0.05). In addition, they exhibit positive linear width-radius relations, suggesting that differential precession across the ringlets is being prevented by the self-gravity of the ring particles. The kinematics of the Maxwell ringlet are determined solely by Saturn's nonspherical gravity field; the kinematics of the Titan ringlet are apparently determined by its interaction with Titan. Masses, mean surface mass densities, and mass extinction coefficients have been calculated. The comparatively large optical depths and mass extinction coefficients in these features suggest an environment and particle size distribution different from the remainder of the C ring and presumably caused by the mechanism responsible for ring confinement.", "doi": "10.1016/0019-1035(84)90134-9", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "1984-10", "series_number": "1", "volume": "60", "issue": "1", "pages": "1-16" }, { "id": "authors:b49cr-n5a03", "collection": "authors", "collection_id": "b49cr-n5a03", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130301-081258771", "type": "article", "title": "Saturn's Nonaxisymmetric Ring Edges at 1.95R_s and 2.27R_s", "author": [ { "family_name": "Porco", "given_name": "C.", "clpid": "Porco-C-C" }, { "family_name": "Danielson", "given_name": "G. E.", "clpid": "Danielson-G-E" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" }, { "family_name": "Holberg", "given_name": "J. B.", "clpid": "Holberg-J-B" }, { "family_name": "Lane", "given_name": "A. L.", "clpid": "Lane-A-L" } ], "abstract": "The outer edges of Saturn's A and B rings, at 2.27 R_s and 1.95 R_s, have been examined using data acquired by four Voyager experiments. The shapes and kinematics of these features are influenced by their proximity to strong low-order Lindblad resonances. The data for the A-ring edge are consistent with a seven-lobed radial distortion of amplitude 6.7 \u00b1 1.5 km which rotates with the mass-weighted mean angular velocity of the coorbital satellite system. The B-ring edge has essentially a double-lobed figure of radial amplitude 74 \u00b1 9 km which rotates with the mean motion of Mimas, though there is an indication that it is not completely described with a simple Saturn-centered ellipse. An upper limit of 10 m has been placed on the vertical thickness in the unperturbed region of the B ring.", "doi": "10.1016/0019-1035(84)90135-0", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "1984-10", "series_number": "1", "volume": "60", "issue": "1", "pages": "17-28" }, { "id": "authors:e8sar-0x966", "collection": "authors", "collection_id": "e8sar-0x966", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130312-110957161", "type": "article", "title": "Excitation of inclinations in ring-satellite systems", "author": [ { "family_name": "Borderies", "given_name": "Nicole", "clpid": "Borderies-N" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "Resonant gravitational interactions between a ring and a satellite produce secular variations of their orbital\ninclinations. Interactions at vertical resonances, analogous to Lindblad resonances but involving inclinations\ninstead of eccentricities, excite inclinations. There is no inclination analog of the corotation resonance. An\nequatorial ring changes the inclination of a nearby satellite in qualitatively the same way that a satellite in an equatorial orbit changes the inclination of a nearby ring. Viscous dissipation in a ring leads to an equilibrium\nvalue of its inclination. These results provide a basis for discussing the origins of the inclinations of planetary\nrings.", "doi": "10.1086/162423", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1984-09-01", "series_number": "1", "volume": "284", "issue": "1", "pages": "429-434" }, { "id": "authors:bhe6f-m9p50", "collection": "authors", "collection_id": "bhe6f-m9p50", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130312-110011572", "type": "article", "title": "Self-similar spherical voids in an expanding universe", "author": [ { "family_name": "Fillmore", "given_name": "James A.", "clpid": "Fillmore-J-A" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "We derive similarity solutions which describe the evolution of spherically symmetric voids in a perturbed Einstein-de Sitter universe filled with cold, collisionless matter. The character of a solution depends upon the profile of the initial density deficit. Gradual perturbations give rise to holes within which the density rises smoothly to the background value. Steep perturbations result in voids bounded by overdense shells with sharp edges, i.e., collisionless gravitational shocks.", "doi": "10.1086/162071", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1984-06-01", "series_number": "1", "volume": "281", "issue": "1", "pages": "9-12" }, { "id": "authors:z1v82-5qh19", "collection": "authors", "collection_id": "z1v82-5qh19", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130312-104834500", "type": "article", "title": "Self-similar gravitational collapse in an expanding universe", "author": [ { "family_name": "Fillmore", "given_name": "James A.", "clpid": "Fillmore-J-A" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "We derive similarity solutions which describe the collapse of cold, collisionless matter in a perturbed Einstein-de Sitter universe. We obtain three classes of solutions, one each with planar, cylindrical, and spherical symmetry. Our solutions can be computed to arbitrary accuracy, and they follow the development of structure in both the linear and nonlinear regimes.", "doi": "10.1086/162070", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1984-06-01", "series_number": "1", "volume": "281", "issue": "1", "pages": "1-8" }, { "id": "authors:1y4j1-vac81", "collection": "authors", "collection_id": "1y4j1-vac81", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20131127-134520567", "type": "article", "title": "A simple derivation of capture probabilities for the J+1:J and J+2:J orbit-orbit resonance problems", "author": [ { "family_name": "Borderies", "given_name": "Nicole", "clpid": "Borderies-N" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "We present a simplified analytic derivation of the capture probabilities for the j+1\u2236j and j+2\u2236j orbital resonances. We apply Henrard's method which is based on an extension of the theory of adiabatic invariants and recover the results originally obtained by Yoder.", "doi": "10.1007/BF01231120", "issn": "0923-2958", "publisher": "Springer", "publication": "Celestial Mechanics", "publication_date": "1984-02", "series_number": "2", "volume": "32", "issue": "2", "pages": "127-136" }, { "id": "authors:jx9mt-gr134", "collection": "authors", "collection_id": "jx9mt-gr134", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130422-111540285", "type": "article", "title": "Eccentric Ringlet in the Maxwell Gap at 1.45 Saturn Radii: Multi-Instrument Voyager Observations", "author": [ { "family_name": "Esposito", "given_name": "L. W.", "clpid": "Exposito-L-W" }, { "family_name": "Borderies", "given_name": "N.", "clpid": "Borderies-N" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" }, { "family_name": "Cuzzi", "given_name": "J. N.", "clpid": "Cuzzi-J-N" }, { "family_name": "Holberg", "given_name": "J. B.", "clpid": "Holberg-J-B" }, { "family_name": "Lane", "given_name": "A. L.", "clpid": "Lane-A-L" }, { "family_name": "Pomphrey", "given_name": "R. B.", "clpid": "Pomphrey-R-B" }, { "family_name": "Terrile", "given_name": "R. J.", "clpid": "Terrile-R-J" }, { "family_name": "Lissauer", "given_name": "J. J.", "orcid": "0000-0001-6513-1659", "clpid": "Lissauer-J-J" }, { "family_name": "Marouf", "given_name": "E. A.", "clpid": "Marouf-E-A" }, { "family_name": "Tyler", "given_name": "G. L.", "clpid": "Tyler-G-L" } ], "abstract": "The Voyager spacecraft observed a narrow, eccentric ringlet in the Maxwell gap (1.45 Saturn radii) in Saturn's rings. Intercomparison of the Voyager imaging, photopolarimeter, ultraviolet spectrometer, and radio science observations yields results not available from individual observations. The width of the ringlet varies from about 30 to about 100 kilometers, its edges are sharp on a radial scale < 1 kilometer, and its opacity exhibits a double peak near the center. The shape and width of the ringlet are consistent with a set of uniformly precessing, confocal ellipses with foci at Saturn's center of mass. The ringlet precesses as a unit at a rate consistent with the known dynamical oblateness of Saturn; the lack of differential precession across the ringlet yields a ringlet mass of about 5 x 10^(18) grams. The ratio of surface mass density to particle cross-sectional area is about five times smaller than values obtained elsewhere in the Saturn ring system, indicating a relatively larger fraction of small particles. Also, comparison of the measured transmission of the ringlet at radio, visible, and ultraviolet wavelengths indicates that about half of the total extinction is due to particles smaller than 1 centimeter in radius, in contrast even with nearby regions of the C ring. However, the color and brightness of the ringlet material are not measurably different from those of nearby C ring particles. We find this ringlet is similar to several of the rings of Uranus.", "doi": "10.1126/science.222.4619.57", "issn": "0036-8075", "publisher": "American Association for the Advancement of Science", "publication": "Science", "publication_date": "1983-10-07", "series_number": "4619", "volume": "222", "issue": "4619", "pages": "57-60" }, { "id": "authors:pa4kh-fxj16", "collection": "authors", "collection_id": "pa4kh-fxj16", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130308-084452674", "type": "article", "title": "The dynamics of elliptical rings", "author": [ { "family_name": "Borderies", "given_name": "Nicole", "clpid": "Borderies-N" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "We investigate the evolution of eccentric rings under the influence of (1) differential precession due to the planetary quadrupole moment; (2) self-gravity; (3) viscous forces due to interparticle collisions; and (4) eccentricity excitation by shepherd satellites. The principal conclusions are that: (a) Uniform precession can be enforced by self-gravity (Goldreich and Tremaine 1979b);\nthe resulting configuration is both dynamically and secularly stable. (b) Due to viscous forces the line of apsides at the inner ring edge is not exactly aligned with the line of apsides at the outer edge; the apse shift may be detectable in the \u0251 and \u03b2 rings of Uranus. (c) The mean\neccentricity is determined by a balance between viscous damping and excitation by shepherds. (d) We expect the dimensionless eccentricity gradient \u0251\u0394e/\u0394\u0251 to be positive and of order unity in most eccentric rings, as observed.", "doi": "10.1086/113446", "issn": "0004-6256", "publisher": "American Astronomical Society", "publication": "Astronomical Journal", "publication_date": "1983-10", "series_number": "10", "volume": "88", "issue": "10", "pages": "1560-1568" }, { "id": "authors:7jes0-qm295", "collection": "authors", "collection_id": "7jes0-qm295", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130418-091213582", "type": "article", "title": "Magnetic focusing in the Sco X-1 radio source", "author": [ { "family_name": "Achterberg", "given_name": "A.", "clpid": "Achterberg-A" }, { "family_name": "Blandford", "given_name": "R. D.", "orcid": "0000-0002-1854-5506", "clpid": "Blandford-R-D" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" } ], "abstract": "There has been much theoretical discussion of the confinement of the radio jets associated with extragalactic radio sources. There are now several examples of sources where the minimum pressure inferred in the jets appears to exceed the external gas pressure, which suggests that magnetic pinching may be playing an important part in the confinement. This issue has been highlighted by recent remarkable observations, using the Very Large Array, of the radio lobes associated with the galactic X-ray source Sco X-1 by Fomalont et al. We argue here that these results necessitate magnetic focusing and that this also strengthens the case for magnetic focusing in the extragalactic sources. We show that a sufficient overpressure with respect to the ambient interstellar medium can be achieved if the radius of the jet is reduced by a factor 10\u2013100, and the converging flow becomes dissipative. The radio lobes which form at that point quickly die out again due to electron expansion losses in the now rapidly diverging jet. On the basis of this model it is predicted that the Faraday rotation changes sign across the radio lobes, and that the shape of the lobes should be conical with the apex pointing towards Sco X-1.", "doi": "10.1038/304607a0", "issn": "0028-0836", "publisher": "Nature Publishing Group", "publication": "Nature", "publication_date": "1983-08-18", "series_number": "5927", "volume": "304", "issue": "5927", "pages": "607-609" }, { "id": "authors:dykq9-t9136", "collection": "authors", "collection_id": "dykq9-t9136", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130228-101759239", "type": "article", "title": "Perturbed Particle Disks", "author": [ { "family_name": "Borderies", "given_name": "Nicole", "clpid": "Borderies-N" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "The Boltzmann moment equations are solved to determine the velocity ellipsoid in a particle disk near an isolated satellite resonance. In a coordinate frame which rotates with the pattern speed of the perturbation potential, the solutions are stationary functions of the azimuthal angle. From the velocity ellipsoid we obtain the stress tensor due to particle collisions and consequently, the viscous angular momentum flux. We show that the magnitude of the rate of deformation tensor in a perturbed particle disk is bounded from above by K\u03a9(1 + \u03c4^2)^\u00bd where \u03a9 is the orbital angular velocity, \u03c4 is the optical depth, and K is a dimensionless constant of order unity. It is also found that in sufficiently perturbed regions there are ranges of azimuthal angle within which the radial component of the angular momentum flux is negative. It is even possible for the angular momentum luminosity, the radial flux integrated over azimuth, to be negative. These results are important for understanding sharp edges and the decay of density waves in planetary rings. They are also relevant to the damping of differential precession and eccentricity in narrow ringlets.", "doi": "10.1016/0019-1035(83)90055-6", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "1983-07", "series_number": "1", "volume": "55", "issue": "1", "pages": "124-132" }, { "id": "authors:z9e8v-4kx39", "collection": "authors", "collection_id": "z9e8v-4kx39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130307-112553837", "type": "article", "title": "Precession of inclined rings", "author": [ { "family_name": "Borderies", "given_name": "Nicole", "clpid": "Borderies-N" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "Differential precession due to the planet's quadrupole moment tends to destroy the alignment of particles in inclined rings. We propose that alignment is maintained by the self-gravity of the ring. This hypothesis predicts that \u03b4i/\u03b4\u0251>0 across the ring. If \u03b4i/i0<(1, \u03b4e/e_0 <(1, \u0251\u03b4i/\u03b4\u0251<(1,\nand \u0251\u03b4e/\u03b4\u0251< 1, a further prediction is that \u03b4i/i_0 = \u03b4e/e_0. The \u0251 and \u03b2 rings of Uranus may be used to test these predictions.", "doi": "10.1086/113309", "issn": "0004-6256", "publisher": "American Astronomical Society", "publication": "Astronomical Journal", "publication_date": "1983-02", "series_number": "2", "volume": "88", "issue": "2", "pages": "226-228" }, { "id": "authors:5n27y-2ep90", "collection": "authors", "collection_id": "5n27y-2ep90", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130228-092555967", "type": "article", "title": "The Variations in Eccentricity and Apse Precession Rate of a Narrow Ring Perturbed by a Close Satellite", "author": [ { "family_name": "Borderies", "given_name": "Nicole", "clpid": "Borderies-N" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "We derive a Hamiltonian which describes the first-order perturbations of orbital eccentricity and apse precession rate of a narrow ring due to a close satellite whose orbit is also eccentric. Our treatment covers cases in which the satellite crosses the ring. The level curves of the Hamiltonian are displayed for several values of the parameters. We apply our results to the interaction of Saturn's F ring with its inner shepherd satellite.", "doi": "10.1016/0019-1035(83)90022-2", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "1983-01", "series_number": "1", "volume": "53", "issue": "1", "pages": "84-89" }, { "id": "authors:evp4w-7g572", "collection": "authors", "collection_id": "evp4w-7g572", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130418-084850346", "type": "article", "title": "Sharp edges of planetary rings", "author": [ { "family_name": "Borderies", "given_name": "Nicole", "clpid": "Borderies-N" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "The ring systems of Saturn and Uranus exhibit several sharp edges across which the optical depth drops from order unity to essentially zero. At least two and perhaps all of these features are associated with the location of orbital resonances between a satellite and the ring particles. It is remarkable that the optical depth varies on a distance scale which is much finer than that over which angular momentum can be transferred between a satellite and the ring material. The important features of this phenomenon are: (1) A perturbed band of width \u0394a/a \u2243 (M_s/M_p)^(\u00bd) adjacent to the edge within which the angular momentum transfer occurs. (2) Streamlines perturbed such that the angular momentum luminosity decreases smoothly across the band to zero at the edge even though the optical depth remains constant. (3) Dynamical equilibrium requires a relation between the random velocity, the rate of deformation and the optical depth.", "doi": "10.1038/299209a0", "issn": "0028-0836", "publisher": "Nature Publishing Group", "publication": "Nature", "publication_date": "1982-09-16", "series_number": "5880", "volume": "299", "issue": "5880", "pages": "209-211" }, { "id": "authors:1f7ta-01410", "collection": "authors", "collection_id": "1f7ta-01410", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130313-143651943", "type": "article", "title": "The Dynamics of Planetary Rings", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "The discovery of ring systems around Uranus and Jupiter, and the Pioneer and\nVoyager spacecraft observations of Saturn, have shown that planetary rings\nare both more common and more complex than previously suspected. These\nring systems, interesting in their own right, also serve as prototypes for more\nmassive disk systems such as accretion disks and spiral galaxies occurring\nelsewhere in astronomy.\nDisks and rings are a natural consequence of dissipation in rotating systems.\nA cloud of debris surrounding a spherical planet settles into a flat circular ring\nbecause interparticle collisions dissipate energy but conserve total angular\nmomentum. Since planets are oblate, only the component of angular momentum\nalong the spin axis is conserved, and the flat ring lies in the equatorial\nplane.\nCollisions redistribute angular momentum among the particles and the ring\nspreads, transferring mass inward and angular momentum outward (Lynden-Bell\n& Pringle 1974). However, the spreading process occurs on a much\nlonger timescale than the flattening process since the collision speeds in a flat\nring are much lower than the orbital speeds (see Sections 2.2 and 5. 3).\nSpreading can be slowed by gravitational interactions with satellites; nevertheless,\na ring cannot live forever and an important constraint on possible ring models is that they yield survival times at least comparable to the age of the\nsolar system (cf. Sections 5, 6).\nThis review was written in October 1981, shortly after the Voyager 2\nencounter with Saturn. Analysis of the data from the Voyager encounters is not\nyet complete. Therefore the emphasis in this review is on the basic physical\nprocesses that occur in planetary rings, rather than on a detailed confrontation\nof theoretical predictions with observation.\nTable 1 lists some of the important properties of the planets with known ring\nsystems.\nFor the sake of brevity we shall refer to a series of papers we have written\non rings (Goldreich & Tremaine 1978a, b, c, 1979a, b, c, 1980, 1981) as GT\n1, . . . , GT 8.", "doi": "10.1146/annurev.aa.20.090182.001341", "issn": "0066-4146", "publisher": "Annual Reviews", "publication": "Annual Review of Astronomy and Astrophysics", "publication_date": "1982-09", "volume": "20", "pages": "249-283" }, { "id": "authors:v5w28-24447", "collection": "authors", "collection_id": "v5w28-24447", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130307-111008312", "type": "article", "title": "Linear polarization of radio frequency lines in molecular clouds and circumstellar envelopes", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Kylafis", "given_name": "Nikolaos D.", "clpid": "Kylafis-N-D" } ], "abstract": "We predict that interstellar lines possess a few percent linear polarization provided that the optical depth in the source region is both anisotropic and of order unity and the radiative rates are at least comparable to the collision rates. These conditions are expected to be met in many sources which emit radio and far-infrared line radiation. Under circumstances in which the Zeeman splitting\nexceeds both the radiative and collisional rates the linear polarization is aligned either parallel or perpendicular to the projection of the magnetic field on the plane of the sky. This \"strong magnetic field\" limit is expected to apply to all radio frequency lines and to many of those far infrared lines which form between levels whose magnetic moments are comparable to the Bohr magneton. The \"weak magnetic field\" limit is relevant to most far-infrared lines formed between levels with magnetic moments of order the nuclear magneton. In this limit the polarization direction is determined by the orientation of the propagation direction with respect to the anisotropic optical depth.", "doi": "10.1086/159663", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1982-02-15", "series_number": "1", "volume": "253", "issue": "1", "pages": "606-621" }, { "id": "authors:yzw7z-w8b24", "collection": "authors", "collection_id": "yzw7z-w8b24", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130307-074422132", "type": "article", "title": "Radial widths, optical depths, and eccentricities of the Uranian rings", "author": [ { "family_name": "Nicholson", "given_name": "P. D.", "orcid": "0000-0003-2275-4463", "clpid": "Nicholson-P-D" }, { "family_name": "Matthews", "given_name": "K.", "clpid": "Matthews-Keith-Astronomy" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" } ], "abstract": "Observations of the stellar occultation by the Uranian rings of 15/16 August 1980 are used to estimate radial widths and normal optical depths for segments of rings 6, 5, 4, \u0251, \u03b2, \u03b7, y, and \u03b4. Synthetic occultation profiles are generated to match the observed light curves. A review of\npublished data confirms the existence of width-radius relations for rings \u0251 and \u03b2, and indicates that the optical depths of these two rings vary inversely with their radial widths. Masses are obtained for rings \u0251 and \u03b2, on the assumption that differential precession is prevented by their self-gravity. A quantitative comparison of seven \u0454-ring occultation profiles obtained over a period of 3.4 yr reveals a consistent structure, which may reflect the presence of unresolved gaps and subrings. Elliptical models for rings 6, 5, 4, \u0251, \u03b2, and \u0454 are presented for comparison\nwith the results of previous studies, particularly that of Elliot et al. (1981a).", "doi": "10.1086/113117", "issn": "0004-6256", "publisher": "American Astronomical Society", "publication": "Astronomical Journal", "publication_date": "1982-02", "series_number": "2", "volume": "87", "issue": "2", "pages": "433-447" }, { "id": "authors:1342e-mgq98", "collection": "authors", "collection_id": "1342e-mgq98", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130306-154922514", "type": "article", "title": "The Uranus Occultation of 10 June 1979. I. The Rings", "author": [ { "family_name": "Nicholson", "given_name": "P. D.", "orcid": "0000-0003-2275-4463", "clpid": "Nicholson-P-D" }, { "family_name": "Matthews", "given_name": "K.", "clpid": "Matthews-Keith-Astronomy" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" } ], "abstract": "Observations and analysis of a stellar occultation by the rings of Uranus on 10 June 1979 are presented. Occultations by rings 4, \u0251, \u03b2, y, \u03b4, and \u0454 are identified, and radii and azimuths of the occulting segments in the plane of the rings calculated. Results for rings y and \u03b4 are consistent\nwith the hypothesis (Elliot et al. 1978; Nicholson et \u0251l. 1978) that these two rings are circular and coplanar, and an approximate upper limit of 8 X 10^(-5) is placed on the eccentricity of either ring. Coplanar elliptical models are presented for rings \u0251 and \u03b2, with eccentricities of \n(6.0\u00b10.3) x 10^(-4) and (4.9\u00b10.5) x 10^(-4), respectively. For ring 4 two possible elliptical models are obtained, with eccentricities of (1.2\u00b10.4) x 10^(-3) and (6.0\u00b10.3) x 10^(-4), the former being preferred. The width-radius relation established previously for the \u0454 ring is confirmed, and the elliptical model for this ring is slightly revised. An improved estimate for Uranus's J_2 of\n(3.390 \u00b1 0.005) x 10^(-3), based on the apsidal precession of the \u0454 ring, and an upper limit for \u2502J_4\u2502 of ~1 x 10^(-4), based on the precession of rings 4 and \u03b2, are obtained.", "doi": "10.1086/112921", "issn": "0004-6256", "publisher": "American Astronomical Society", "publication": "Astronomical Journal", "publication_date": "1981-04", "series_number": "4", "volume": "86", "issue": "4", "pages": "596-606" }, { "id": "authors:v2z13-n1e06", "collection": "authors", "collection_id": "v2z13-n1e06", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130306-155257978", "type": "article", "title": "The origin of the eccentricities of the rings of Uranus", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "We consider the effect of gravitational perturbations from a nearby satellite on the eccentricity e of a narrow particulate ring. The perturbations near a resonance in an eccentric ring may be divided into corotation and Lindblad terms. For small e, the corotation terms damp e, whereas the Lindblad terms excite e. In the absence of saturation the corotation terms win by. a small margin, and e damps.\nHowever, if the perturbations open gaps at the strongest resonances, then the Lindblad terms win, and e grows. This result offers an explanation for the existence of both circular and eccentric rings around Uranus. We also show that eccentricity changes induced by circular rings on eccentric satellite orbits are similar to those induced by satellites with circular orbits on eccentric rings.", "doi": "10.1086/158671", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1981-02-01", "series_number": "1", "volume": "243", "issue": "1", "pages": "1062-1075" }, { "id": "authors:74sk2-twb94", "collection": "authors", "collection_id": "74sk2-twb94", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130307-073145033", "type": "article", "title": "On mapping the magnetic field direction in molecular clouds by polarization measurements", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Kylafis", "given_name": "Nikolaos D.", "clpid": "Kylafis-N-D" } ], "abstract": "We predict that interstellar radio-frequency lines possess a few percent linear polarization, provided that (1) the radiative transition rate is at least comparable to the collision rate, (2) the optical depth is moderate. and anisotropic, and (3) the number of extrema of the velocity component along the line of sight through the source is small. If the Zeeman splitting exceeds both the collisional frequency and the radiative transition rate, then the polarization is aligned either perpendicular to or parallel to the projection of the magnetic field on the plane of the sky.", "doi": "10.1086/183446", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1981-01-15", "series_number": "2", "volume": "243", "issue": "2", "pages": "L75-L78" }, { "id": "authors:zvdjc-40x55", "collection": "authors", "collection_id": "zvdjc-40x55", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130306-154423987", "type": "article", "title": "Disk-Satellite Interactions", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "We calculate the rate at which angular momentum and energy are transferred between a disk and a satellite which orbit the same central mass. A satellite which moves on a circular orbit exerts a torque on the disk only in the immediate vicinity of its Lindblad resonances. The direction of angular momentum transport is outward, from disk material inside the satellite's orbit to the\nsatellite and from the satellite to disk material outside its orbit. A satellite with an eccentric orbit exerts a torque on the disk at corotation resonances as well as at Lindblad resonances. The angular momentum and energy transfer at Lindblad resonances tends to increase the satellite's orbit eccentricity whereas the transfer at corotation resonances tends to decrease it. In a Keplerian disk, to lowest order in eccentricity and in the absence of nonlinear effects, the corotation resonances dominate by a slight margin and the eccentricity damps. However, if the strongest corotation resonances saturate due to particle trapping, then the eccentricity grows. We present an illustrative application of our results to the interaction between Jupiter and the protoplanetary disk. The angular momentum transfer is shown to be so rapid that substantial\nchanges in both the structure of the disk and the orbit of Jupiter must have taken place on a time scale of a few thousand years.", "doi": "10.1086/158356", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1980-10-01", "series_number": "1", "volume": "241", "issue": "1", "pages": "425-441" }, { "id": "authors:5150e-70q38", "collection": "authors", "collection_id": "5150e-70q38", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130306-154242198", "type": "article", "title": "Homologously collapsing stellar cores", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Weber", "given_name": "Stephen V.", "clpid": "Weber-S-V" } ], "abstract": "We investigate the collapse of nonrotating gas spheres with a polytropic equation of state: n = 3, corresponding to y = 4/3. Such polytropes provide a reasonable approximation to collapsing stellar cores during the early phase before nuclear density is reached. We find a family of exact\nhomologously collapsing configurations. Homologous collapse of the entire core is possible if the pressure at a given density is reduced by up to 3% from the value for a marginally stable static core. For a greater pressure reduction, an inner core can collapse homologously, the mass of which varies as the 3/2 power of the reduced pressure at the onset of collapse. Linear perturbations of\nthese homologously collapsing solutions are separable in space and time. Low order radial and nonradial modes are calculated, and it is found that all modes are essentially stable.", "doi": "10.1086/158065", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1980-06-15", "series_number": "1", "volume": "238", "issue": "1", "pages": "991-997" }, { "id": "authors:a3s0y-49b65", "collection": "authors", "collection_id": "a3s0y-49b65", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130306-153254452", "type": "article", "title": "The excitation of density waves at the Lindblad and corotation resonances by an external potential", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "We calculate the linear response of a differentially rotating two-dimensional gas disk to a rigidly rotating external potential. The main assumptions are that the sound speed is much smaller than the orbital velocity and that the external potential varies on the scale of the disk\nradius. We investigate disks both with and without self-gravity. The external potential exerts torques on the disk only at the Lindblad and corotation resonances. The torque is positive at the outer Lindblad resonance and negative at the inner Lindblad resonance; at corotation the torque has the sign of the radial gradient of vorticity\nper unit surface density. The torques are of the same order of magnitude at both types of resonance and are independent of the sound speed in the disk. The external potential also excites density waves in the vicinity of the Lindblad and corotation resonances. The long trailing wave is excited at a Lindblad resonance. It transports away from the resonance all of the angular momentum which is deposited there by the external torque. Short trailing waves are excited at the corotation resonance. The amplitudes of the excited waves are the same on both sides of the resonance and are small unless the disk is almost gravitationally unstable. No net angular momentum is transported away from the corotation\nregion by the waves. Thus the angular momentum deposited there by the external torque accumulates in the gas.\nWe briefly discuss the behavior of particle disks and prove that the external torques on particle disks are identical to those on gas disks.", "doi": "10.1086/157448", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1979-11-01", "series_number": "3", "volume": "233", "issue": "3", "pages": "857-871" }, { "id": "authors:9vwzf-r5564", "collection": "authors", "collection_id": "9vwzf-r5564", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130306-144033573", "type": "article", "title": "Precession of the \u0454 ring of Uranus", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "The boundaries of the \u0454 ring can be fit by aligned Keplerian ellipses. Differential precession due to the quadrupole moment of Uranus tends to destroy the apse alignment. We propose that apse alignment is maintained by the self-gravity of the ring. The required ring mass\n~5 x 10^(18) g which corresponds to a surface density at quadrature ~25 g cm^(-2).", "doi": "10.1086/112587", "issn": "0004-6256", "publisher": "American Astronomical Society", "publication": "Astronomical Journal", "publication_date": "1979-10", "series_number": "10", "volume": "84", "issue": "10", "pages": "1638-1641" }, { "id": "authors:fx953-sp222", "collection": "authors", "collection_id": "fx953-sp222", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130304-081212930", "type": "article", "title": "Towards a theory for the uranian rings", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "Interparticle collisions, radiation drag and differential precession all tend to disrupt the rings of Uranus. The first two effects lead to radial spreading which would disrupt a free ring in \u2272 10^8 yr. We propose that the rings are confined in radius by gravitational torques from a series of small satellites that orbit within the ring system. Differential precession tends to destroy the apse alignment of the elliptical \u03b5 ring. We suggest that apse alignment is maintained by the self-gravity of the ring. The resulting mass of the \u03b5 ring is ~5 \u00d7 10^(18) g. Its radial confinement requires (for example) a pair of satellites of mass ~10^(19) g, in circular orbits roughly 500 km away on either side of the ring.", "doi": "10.1038/277097a0", "issn": "0028-0836", "publisher": "Nature Publishing Group", "publication": "Nature", "publication_date": "1979-01-11", "series_number": "5692", "volume": "277", "issue": "5692", "pages": "97-99" }, { "id": "authors:fh8n9-b7686", "collection": "authors", "collection_id": "fh8n9-b7686", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130306-152013976", "type": "article", "title": "The Absorption of Trapped Line Photons by Dust", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Kwan", "given_name": "John", "clpid": "Kwan-J" } ], "abstract": "We derive the rate at which photons in an optically thick line are absorbed by cold dust. This rate is approximately equal to the dust optical depth to the cloud center times the rate at which the photons escape from the cloud. Our derivation is in response to a recent article by Strel'nitskii in which he incorrectly criticized our previous application of this result to models for the pumping of cosmic masers. Strel'nitskii now agrees that his criticism was unjustified.", "doi": "10.1086/156713", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1979-01", "series_number": "1", "volume": "227", "issue": "1", "pages": "150-151" }, { "id": "authors:de6j6-gsf57", "collection": "authors", "collection_id": "de6j6-gsf57", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130306-134542488", "type": "article", "title": "The rings of Uranus: Results of the 10 April 1978 occultation", "author": [ { "family_name": "Nicholson", "given_name": "P. D.", "orcid": "0000-0003-2275-4463", "clpid": "Nicholson-P-D" }, { "family_name": "Persson", "given_name": "S. E.", "clpid": "Persson-S-E" }, { "family_name": "Matthews", "given_name": "K.", "clpid": "Matthews-Keith-Astronomy" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" }, { "family_name": "Neugebauer", "given_name": "G.", "clpid": "Neugebauer-G" } ], "abstract": "Observations of the 10 April 1978 stellar occultation by the rings of Uranus are presented. Nine rings were observed and their radii and widths are calculated. Rings \u03b7,y, and \u03b4 are found to be most likely circular and coplanar, in agreement with previous analyses; the remaining rings are either noncircular or slightly inclined. The width of the \u0454 ring is a linear function of its radius from the center of Uranus, projected onto the satellites' orbital plane; this suggests that it forms one continuous noncircular ring. The optical depth profile of the \u0454 ring has not changed\nsignificantly since March 1977. A model of this ring which fits all available observations adequately is that of a uniformly precessing Keplerian ellipse coplanar with the satellites' orbits. This model permits predictions of the radius and width of the \u0454 ring for future occultations.\nThe precession rate is used to determine J_2 for Uranus, on the assumption that precession is caused solely by the planetary oblateness and not by satellite-ring interactions.", "doi": "10.1086/112318", "issn": "0004-6256", "publisher": "American Astronomical Society", "publication": "Astronomical Journal", "publication_date": "1978-10", "series_number": "10", "volume": "83", "issue": "10", "pages": "1240-1248" }, { "id": "authors:ef5pg-y1e07", "collection": "authors", "collection_id": "ef5pg-y1e07", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130306-141115931", "type": "article", "title": "The excitation and evolution of density waves", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "We study the linear oscillations of a thin self-gravitating gas sheet. The unperturbed velocity field of the sheet is a parallel shear flow. A Coriolis acceleration is included to simulate the effects of rotation. The sheet exhibits Lindblad resonances, and it can sustain both short and long\nwavelength density waves. We derive equations which govern the excitation and evolution of density waves in all regions\nof space, including the Lindblad resonances and the forbidden region around corotation. These equations are solved in the tight winding limit. An initial disturbance in the form of a wave packet of short leading waves evolves as follows. The packet propagates toward corotation, is reflected at the boundary of the forbidden region, and\nbecomes a packet of long leading waves. It then travels back to the Lindblad resonance, where it is reflected and becomes a packet of long trailing waves. Subsequently, this packet moves toward corotation and is reflected again at the boundary of the forbidden region. The packet is now made\nup of short trailing waves and propagates away from corotation indefinitely. For sufficiently stable disks, the forbidden region around corotation is wide and density waves\nare almost completely reflected at its boundaries. For marginally stable disks, some of the incident wave tunnels through the forbidden region and the reflected wave is amplified. The excitation of density waves by an arbitrary external potential is considered. In our model sheet, the sole effect of a barlike potential is to excite the long trailing wave at the Lindblad resonances. The amplitude of the excited wave is calculated.", "doi": "10.1086/156203", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1978-06-15", "series_number": "1", "volume": "222", "issue": "1", "pages": "850-858" }, { "id": "authors:g13a2-hnn05", "collection": "authors", "collection_id": "g13a2-hnn05", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130305-112750779", "type": "article", "title": "The formation of the Cassini division in Saturn's rings", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Tremaine", "given_name": "Scott", "clpid": "Tremaine-S" } ], "abstract": "The satellite Mimas excites a trailing spiral density wave in Saturn's rings at the position of the 2:1 resonance. The density wave carries negative angular momentum and propagates outward. The wave is damped by a combination of nonlinear and viscous effects, and its negative angular momentum is transferred to the ring particles. Consequently, the particles just outside the 2:1 resonance spiral inward, opening a gap. The inner edge of the gap is close to the resonance position in agreement with the location of the inner edge of the Cassini division. Despite its tiny mass, Minas is able to clear a gap as wide as the Cassini division. We estimate the ability of Saturn's satellites to open other gaps in the rings. The upper limit to the width of Encke's division implies that the velocity dispersion of the ring particles is <10^(-2) cm sec^(-1).", "doi": "10.1016/0019-1035(78)90165-3", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "1978-05", "series_number": "2", "volume": "34", "issue": "2", "pages": "240-253" }, { "id": "authors:wng3y-6hs60", "collection": "authors", "collection_id": "wng3y-6hs60", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130305-151706819", "type": "article", "title": "On the Radiative Acceleration of Quasar Absorption Line Clouds", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" } ], "abstract": "Radiation may expel dense gas clouds from quasar emission line regions. Small clouds can be accelerated to velocities approaching c if they are confined so as to maintain high densities.\nThe kinetic energies, inferred from the spectra, of low to medium excitation quasar absorption clouds are too large for radiation pressure to have played a dominant role in their acceleration. If these clouds are physically associated with the quasars, they are probably accelerated by the same stresses that are responsible for their confinement.\nModels for absorption clouds based on radiatively driven instabilities in expanding quasar envelopes predict column densities that are much larger than those deduced from observation.", "doi": "10.1088/0031-8949/17/3/015", "issn": "0031-8949", "publisher": "IOP", "publication": "Physica Scripta", "publication_date": "1978", "series_number": "3", "volume": "17", "issue": "3", "pages": "225-228" }, { "id": "authors:cbtkd-eha65", "collection": "authors", "collection_id": "cbtkd-eha65", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130418-074237036", "type": "article", "title": "Revenge of tiny Miranda", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Nicholson", "given_name": "Philip D.", "orcid": "0000-0003-2275-4463", "clpid": "Nicholson-P-D" } ], "abstract": "DERMOTT and Gold have proposed a resonance model for the rings of Uranus. They assume the rings are, in fact, arcs composed of small particles librating about stable resonances determined by pairs of satellites, either Ariel and Titania or Ariel and Oberon. Dermott and Gold dismiss as insignificant resonances involving tiny Miranda. We report here that, by a wide margin, the strongest resonances are all associated with Miranda. Furthermore, we show that the hypothesis that the rings are made up of librating particles, while original and ingenious, is incorrect.", "doi": "10.1038/269783b0", "issn": "0028-0836", "publisher": "Nature Publishing Group", "publication": "Nature", "publication_date": "1977-10-27", "series_number": "5631", "volume": "269", "issue": "5631", "pages": "783-785" }, { "id": "authors:vs5em-gcj36", "collection": "authors", "collection_id": "vs5em-gcj36", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130306-075048182", "type": "article", "title": "Solar seismology. I. The stability of the solar p-modes", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Keeley", "given_name": "Douglas A.", "clpid": "Keeley-D-A" } ], "abstract": "We investigate the stability of the radial p-modes of the Sun by computing nonadiabatic eigen-values and e1genfunctions for a solar envelope model which extends from an inner radius r \u2248 0.3 R_\u2609 out to an optical depth r \u2248 3 x 10^(-4). Our calculations take into account in a crude\nfashion the response of the convective flux to the oscillation. The dynamical effect of turbulence in the convection zone is parametrized in terms of a turbulent shear viscosity. The results of our calculations are as follows. If damping by turbulent viscosity is neglected,\nall modes with penriods longer than 6 minutes are unstable. The familiar K-mechanism, which operates in the H ionization-H^- opacity region, is the dominant source of driving of the oscillations. Modes with periods shorter than 6 minutes are stabilized by radiative damping in the solar atmosphere. When turbulent dissipation of pulsational energy is included, all modes are predicted to be stable. However, the margin of stability is very small. In view of the large uncertainty that must be assigned to our estimate of turbulent damping, we conclude that theoretical calculations cannot unequivocally resolve the question of the stability of the solar p-modes.", "doi": "10.1086/155005", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1977-02-01", "series_number": "1", "volume": "211", "issue": "1", "pages": "934-942" }, { "id": "authors:vg4g4-hyx38", "collection": "authors", "collection_id": "vg4g4-hyx38", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130305-111808710", "type": "article", "title": "Turbulent Viscosity and Jupiter's Tidal Q", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Nicholson", "given_name": "Philip D.", "orcid": "0000-0003-2275-4463", "clpid": "Nicholson-P-D" } ], "abstract": "A recent estimate of tidal dissipation by turbulent viscosity in Jupiter's convective interior predicts that the current value of the planet's tidal Q \u223c 5 \u00d7 10^6. We point out a fundamental error in this calculation, and show that turbulent dissipation alone implies that at present Q \u223c 5 \u00d7 10^(13). Our reduced estimate for the rate of tidal dissipation shows conclusively that tidal torques have produced only negligible modifications of the orbits of the Galilean satellites over the age of the solar system.", "doi": "10.1016/0019-1035(77)90163-4", "issn": "0019-1035", "publisher": "Elsevier", "publication": "Icarus", "publication_date": "1977-02", "series_number": "2", "volume": "30", "issue": "2", "pages": "301-304" }, { "id": "authors:w48h8-c1c82", "collection": "authors", "collection_id": "w48h8-c1c82", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130418-072712096", "type": "article", "title": "Spiral structure as an explanation for the asymmetric brightness of Saturn's A ring", "author": [ { "family_name": "Colombo", "given_name": "G.", "clpid": "Colombo-G" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" }, { "family_name": "Harris", "given_name": "A. W.", "orcid": "0000-0001-7431-2013", "clpid": "Harris-A-W" } ], "abstract": "REITSEMA et al. and Lumme and Irvine have recently confirmed earlier observations which suggested that the A ring of Saturn is fainter in the quadrants following conjunctions of the particles with the Earth\u2013Saturn line and brighter in the quadrants preceding conjunctions. Surprisingly, no intrinsic azimuthal brightness variation is found in the B ring. The brightness variation has been linked to the presence of synchronously rotating particles in the A ring, the effect caused by either a systematic variation in albedo over their surfaces or variations in their geometric projections. Of these two possibilities, the latter seems more promising since the asymmetry of the brightness variation with respect to the Earth\u2013Saturn line is difficult to account for by means of an albedo variation. Furthermore, an albedo variation would yield the same brightness pattern for the B ring if it contained synchronously rotating particles. In this paper, however, we propose yet another mechanism to explain the phenomenon\u2014spiral wakes in the A ring.", "doi": "10.1038/264344a0", "issn": "0028-0836", "publisher": "Nature Publishing Group", "publication": "Nature", "publication_date": "1976-11-25", "series_number": "5584", "volume": "264", "issue": "5584", "pages": "344-345" }, { "id": "authors:hfp1e-5qz37", "collection": "authors", "collection_id": "hfp1e-5qz37", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20190403-153008011", "type": "article", "title": "OH-IR stars. II. A model for the 1612 MHz masers", "author": [ { "family_name": "Elitzur", "given_name": "Moshe", "orcid": "0000-0001-8143-3550", "clpid": "Elitzur-M" }, { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Scoville", "given_name": "Nick", "orcid": "0000-0002-0438-3323", "clpid": "Scoville-N-Z" } ], "abstract": "A model is presented for the 1612 MHz OH masers associated with infrared stars. Its principal conclusions are as follows. The central stars are losing ~3 x 10^(-5) M_\u2299 yr^(-1), and the masers operate in the outer regions (r > 10^(16) cm) of the circumstellar envelopes. The maser radiation is narrowly beamed in the radial direction, both inward and outward. Thus the two maser emission features originate in the near and the far sides of the expanding circumstellar gas. The 1612 MHz maser is powered by the absorption of 35 \u00b5 photons which excite the OH molecules from the ^2\u03a0_(3/2), J = 3/2 ground state to the ^2\u03a0_(1/2), J = 5/2 state. The excited OH molecules return to the ground state by a series of radiative decays. In most cases, the radiative cascade proceeds directly down the ^2\u03a0_(1/2) ladder. The final and the most important step in the pump cycle is the radiative decay from the ^2\u03a0_(1/2), J = 1/2 state to the ^2\u03a0_(3/2), J = 3/2 state. If the transitions which link these two states are optically thick, a strong inversion of the F = 1 \u2192 F = 2 1612 MHz transition is produced.", "doi": "10.1086/154289", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1976-04-15", "volume": "205", "pages": "384-396" }, { "id": "authors:fp1y8-0b361", "collection": "authors", "collection_id": "fp1y8-0b361", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130306-085559284", "type": "article", "title": "OH-IR stars. I. Physical properties of circumstellar envelopes", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Scoville", "given_name": "Nick", "orcid": "0000-0002-0438-3323", "clpid": "Scoville-N-Z" } ], "abstract": "A theoretical model of the circumstellar envelope which surrounds a OH-IR star is developed. The circumstellar gas is ejected by radiation pressure which acts on dust grains that condense in the atmosphere of the central star. The dust grains transfer momentum to the gas by collisions\nwith the gas molecules. These collisions are the dominant source of heat input to the circumstellar gas. The major sources of cooling are the emission of radiation by H_2O molecules and adiabatic expansion. The gas temperature decreases from T \u2248 2 x 10^3 K near the stellar surface at r \u2248 6 x 10^(13) cm, to T \u2248 8 x 10^2 K at r = 10^(15) cm and to T \u2248 10^2 K at r = 10^(16) cm. The OH molecule abundance in the circumstellar envelope is controlled by chemical exchange reactions and by the dissociation of H^2O molecules. The reaction OH + H_2 \u2194 H_2O + H + 0.69 eV,\nwhich has an activation energy of 0.3 eV, rapidly converts OH molecules into H_2O molecules in the warm (T \u2273 5 x 10^2 K) inner (r \u227e 2 x 10^(15) cm) region of the circumstellar envelope. Beyond r \u2248 2 x 10^(15) cm, T is so low that the exchange reaction is very slow and the mean lifetime of an\nOH molecule is greater than the expansion time scale for the circumstellar envelope. In the outer region of the circumstellar envelope, OH molecules are produced from the photodissociation of H_2O molecules by the interstellar ultraviolet radiation and from the dissociation of H_2O molecules by collisions with dust grains. These processes are capable of producing OH number densities greater than 1 cm^(-3) at r \u2248 10^(16) cm. The predicted values of the gas temperature, T, and the OH abundance, n_(OH), depend upon the rate of mass loss from the central star, \u0424. The results quoted above are based on a calculation with \u0424 = 3 x 10^(-5) M_\u2609 yr^(-1). In general, T varies inversely and n_(OH) varies directly with \u0424.", "doi": "10.1086/154257", "issn": "0004-637X", "publisher": "American Astronomical Society", "publication": "Astrophysical Journal", "publication_date": "1976-04-01", "volume": "205", "pages": "144-154" }, { "id": "authors:r3bxd-fdx81", "collection": "authors", "collection_id": "r3bxd-fdx81", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20151208-113004039", "type": "article", "title": "Rotation of the Sun", "author": [ { "family_name": "Goldreich", "given_name": "Peter", "clpid": "Goldreich-P-M" }, { "family_name": "Schubert", "given_name": "Gerald", "clpid": "Schubert-G" } ], "abstract": "Dicke has interpreted his recent measurement of the sun's oblateness as implying a fast (1.8-day period) rotation of the solar radiative interior. We find that differentially rotating solar models, such as the one proposed by Dicke, are unstable. The rate of turbulent diffusion in the unstable regions of these models is so rapid that it appears to preclude a fast spinning solar interior. As a corollary of the stability analysis, we conclude that the loss of a significant fraction of a star's angular momentum must be accompanied by the mixing of material below its convective zone. Such mixing inevitably leads to the depletion of lithium in the star's photosphere.", "doi": "10.1126/science.156.3778.1101", "issn": "0036-8075", "publisher": "American Association for the Advancement of Science", "publication": "Science", "publication_date": "1967-05-26", "series_number": "3778", "volume": "156", "issue": "3778", "pages": "1101-1102" }, { "id": "authors:srxgn-nes81", "collection": "authors", "collection_id": "srxgn-nes81", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:ABEpasp66", "type": "article", "title": "On the origin of planetary nebulae", "author": [ { "family_name": "Abell", "given_name": "G. O.", "clpid": "Abell-G-O" }, { "family_name": "Goldreich", "given_name": "P.", "clpid": "Goldreich-P-M" } ], "abstract": "[No abstract]", "doi": "10.1086/128336", "issn": "0004-6280", "publisher": "Astronomical Society of the Pacific", "publication": "Publications of the Astronomical Society of the Pacific", "publication_date": "1966-06", "series_number": "463", "volume": "78", "issue": "463", "pages": "232-241" } ]