arXiv subjects

Peter Goldreich

Publications and source records attributed to Peter Goldreich.

35 records · Page 2Linked to original sources

arXiv2004

Final Stages of Planet Formation

We address three questions regarding solar system planets. What determined their number? Why are their orbits nearly circular and coplanar? How long did they take to form? Runaway accretion in a disk of small bodies resulted in a tiny fraction of the bodies growing much larger than all the others. This was followed by oligarchic growth during which the big bodies maintained similar masses and uniformly spaced semi-major axes. 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. When 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. The last and longest stage in planet formation was the clean up of small bodies.

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arXiv2003

Gauge Symmetry of the N-body Problem in the Hamilton-Jacobi Approach

In most books the Delaunay and Lagrange equations for the orbital elements are derived by the Hamilton-Jacobi method: one begins with the 2-body Hamilton equations, performs a canonical transformation to the orbital elements, and obtains the Delaunay system. A standard trick is then used to generalise the approach to the N-body case. We re-examine this step and demonstrate that it contains an implicit condition which restricts the dynamics to a 9(N-1)-dimensional submanifold of the 12(N-1)-dimensional space spanned by the elements and their time derivatives. The tacit condition is equivalent to the so-called Lagrange constraint. 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 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 of the celestial-mechanics equations and 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 non-canonical. In a more general setting, when the disturbance depends not only upon positions but also upon velocities, there is a "generalised Lagrange gauge" wherein the Delaunay system is symplectic. This gauge renders orbital elements that are osculating in the phase space. It coincides with the regular Lagrange gauge when the perturbation is velocity-independent.

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arXiv2003

Wave damping by MHD turbulence and its effect upon cosmic ray propagation in the ISM

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

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arXiv2003

Origin of Chaos in the Prometheus-Pandora System

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 and 3.1 deg/yr, respectively.

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arXiv2003

Gauge Freedom in the N-body problem of Celestial Mechanics

We summarise research reported in (Efroimsky 2002, 2003; Efroimsky and Goldreich 2003a,b) and develop its application to planetary equations in non-inertial frames. We provide a practical example illustrating how the gauge formalism considerably simplifies the calculation of satellite motion about an oblate precessing planet.

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arXiv2003

Planet - Disk Symbiosis

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. Eccentricity growth during gap formation may involve an even more intimate symbiosis. 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.

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arXiv2002

Formation of Kuiper Belt Binaries

It appears that at least several percent of large Kuiper belt objects are members of wide binaries. Physical collisions are too infrequent to account for their formation. Collisionless gravitational interactions are more promising. These provide two channels for binary formation. In each, the initial step is the formation of a transient binary when two large bodies penetrate each other's Hill spheres. Stabilization of a transient binary requires that it lose energy. Either dynamical friction due to small bodies or the scattering of a third large body can be responsible. Our estimates favor the former, albeit by a small margin. We predict that most objects of size comparable to those currently observed in the Kuiper belt are members of multiple systems. More specifically, we derive the probability that a large body is a member of a binary with semi-major axis of order a. The probability depends upon sigma, the total surface density, Sigma, the surface density of large bodies having radius R, and theta=10^-4, the angle subtended by the solar radius as seen from the Kuiper belt. For (sigma/Sigma)R<a< R/theta, the probability is just (Sigma/rho R)theta^-2, the optical depth of the large bodies divided by the solid angle subtended by the Sun. For R<a<r_u=(sigma/Sigma)R, it varies inversely with semimajor axis and reaches (sigma/rho R)theta^-2 at a=R. Based on current surveys of the Kuiper belt, we estimate Sigma/rho=3 10^-4cm and R=100km. We obtain sigma/rho=0.3cm by extrapolating the surface density deduced for the minimum mass solar nebula. Rough predictions are: outside of the critical separation r_u/a_odot=3'', the binary probability is 0.3%; at separations of 0.2'', comparable to current resolving capabilities, it reaches 5%, in agreement with results from the HST binary survey by Brown.

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arXiv2002

Imbalanced Weak MHD Turbulence

MHD turbulence consists of waves that propagate along magnetic fieldlines, in both directions. When two oppositely directed waves collide, they distort each other, without changing their respective energies. In weak MHD turbulence, a given wave suffers many collisions before cascading. "Imbalance" means that more energy is going in one direction than the other. In general, MHD turbulence is imbalanced. A number of complications arise for the imbalanced cascade that are unimportant for the balanced one. 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. We call this the "pinning" of the energy spectra. 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 et al. (2000) 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.

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arXiv2002

Chaotic Motions of F-Ring Shepherds

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 degrees. We show that these discrepancies are fully accounted for by gravitational interactions between the two satellites. These peak every 24.8 days at conjunctions and excite chaotic perturbations. The Lyapunov exponent for the Prometheus-Pandora system is of order 0.35 inverse years for satellite masses based on a nominal density of 1.3 gm/cm^3. Interactions are strongest when the orbits come closest together. This happens at intervals of 6.2 years 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.

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arXiv2002

Eccentricity Evolution for Planets in Gaseous Disks

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 which are located on the opposite side of corotation from the planet, and damped by co-orbital Lindblad resonances which 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 which initially marginally exceeds that of eccentricity excitation by external Lindblad resonances. But the corotation torque drives a mass flux which reduces the density gradient near the resonance. Sufficient partial saturation of corotation resonances can tip the balance in favor of eccentricity excitation. A minimal initial eccentricity of a few percent 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, interactions at the apsidal resonance, which is a special kind of co-orbital Lindblad resonance, appears to damp eccentricity faster than external Lindblad resonances can excite it. However, apsidal waves have such long wavelengths that they do not propagate in protoplanetary disks. This reduces eccentricity damping by the apsidal resonance to a modest level.

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arXiv2001

Tidal Evolution Of the Planetary System around HD 83443

Two planets with orbital period ratio approximately 10:1 have been discovered around the star HD 83443. The inner and more massive planet, HD 83443b, has the smallest semi-major axis among all currently known exoplanets. Unlike other short period exoplanets, it maintains a substantial orbital eccentricity, e_1=0.079+/- 0.008, in spite of efficient tidal damping. This is a consequence of its secular interactions with HD 83443c whose orbital eccentricity e_2=0.42+/-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 upon 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=(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.

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arXiv2001

Compressible MHD Turbulence in Interstellar Plasmas

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 MHD turbulence given by Goldreich & Sridhar to include the effects of compressibility and particle transport. Our most important results are as follows. (1) Density fluctuations are due to the slow mode and the entropy mode. Both modes are passively mixed by the cascade of shear Alfven waves. Since the shear Alfven waves have a Kolmogorov spectrum, so do the density fluctuations. (2) Observed density fluctuation amplitudes imply either that the magnetic and gas pressures are comparable, or that the outer scale of the turbulence is very small. (3) A high degree of ionization is required for the cascade to survive damping by neutrals and thereby to extend to small lengthscales. Regions that are insufficiently ionized produce density fluctuations only on lengthscales larger than the neutral damping scale. These regions may account for the excess of power that is found on large scales. (4) Both the entropy mode and the slow mode are damped on lengthscales below that at which protons can diffuse across an eddy during the eddy's turnover time. Consequently, eddies whose extents along the magnetic field are smaller than the proton collisional mean free path do not contribute to the density spectrum. However, in MHD turbulence eddies are highly elongated along the magnetic field. From an observational perspective, the relevant lengthscale is that transverse to the magnetic field. Thus the cut-off lengthscale for density fluctuations is significantly smaller than the proton mean free path.

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arXiv2001

Simulations of Incompressible MHD Turbulence

We simulate incompressible MHD turbulence in the presence of a strong background magnetic field. Our major conclusions are: 1) MHD turbulence is most conveniently described in terms of counter propagating shear Alfven and slow waves. Shear Alfven waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfven waves, as does a passive scalar. 2) MHD turbulence is anisotropic with energy cascading more rapidly along k_perp than along k_parallel, where k_perp and k_parallel refer to wavevector components perpendicular and parallel to the local magnetic field. Anisotropy increases with increasing k_perp. 3) MHD turbulence is generically strong in the sense that the waves which comprise it suffer order unity distortions on timescales comparable to their periods. Nevertheless, turbulent fluctuations are small deep inside the inertial range compared to the background field. 4) Decaying MHD turbulence is unstable to an increase of the imbalance between the flux of waves propagating in opposite directions along the magnetic field. 5) Items 1-4 lend support to the model of strong MHD turbulence by Goldreich & Sridhar (GS). Results from our simulations are also consistent with the GS prediction gamma=2/3. The sole notable discrepancy is that 1D power law spectra, E(k_perp) ~ k_perp^{-alpha}, determined from our simulations exhibit alpha ~ 3/2, whereas the GS model predicts alpha = 5/3.

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arXiv2000

Gravity-Modes in ZZ Ceti Stars: IV. Amplitude Saturation by Parametric Instability

ZZ Ceti stars exhibit small amplitude photometric pulsations in multiple gravity-modes. We demonstrate that parametric instability, a form of resonant 3-mode coupling, limits overstable modes to amplitudes similar to those observed. In particular, it reproduces the observed trend that longer period modes have larger amplitudes. Parametric instability involves the destabilization of a pair of stable daughter modes by an overstable parent mode. The 3-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 $(δω^2+γ_d^2)/κ^2$, where $κ$ is the nonlinear coupling constant, $δω$ is the frequency mismatch, and $γ_d$ is the energy damping rate of the daughter modes. 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 non-detection of modes with periods longer than $1,200\s$. Resonant 3-mode interactions involving more than one excited mode may account for the former. Our leading candidate for the latter is Kelvin-Helmholtz instability of the mode-driven shear layer below the convection zone.

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arXiv1999

Growth of Perturbation in Gravitational Collapse and Accretion

When a self-gravitating spherical gas cloud collapses or accretes onto a central mass, the inner region of the cloud develops a density profile $ρ\propto r^{-3/2}$ 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, $Δρ/ρ$, grows as $r^{-1/2}$. Faster growth occurs if the central collapsed object maintains a finite multiple moment, in which case $Δρ/ρ$ 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.

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arXiv1998

Gravity-Modes in ZZ Ceti Stars III. Eigenvalues and Eigenfuctions

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 $ωτ_c \approx 1$ for the $n=1$ mode. Here $ω$ is radian frequency and $τ_c$ is about four times the thermal timescale 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 $ωτ_c$ separating overstable and damped modes rises. The latter is a consequence of enhanced radiative damping for modes which propagate immediately below the convection zone. The critical value of $ωτ_c$ is of observational significance because modes with the smallest value of $ωτ_c$ are most observable photometrically. Maximum periods for overstable modes predicted for our cooler model envelopes are about a factor two longer than the observational upper limit of $1,200\s$. 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 amplidues of overstable modes.

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arXiv1998

Gravity-Modes in ZZ Ceti Stars: I.Quasiadiabatic Analysis of Overstability

We analyze the stability of g-modes in variable white dwarfs with hydrogen envelopes. In these stars, the radiative layer contributes to mode damping because its opacity decreases upon compression and the amplitude of the Lagrangian pressure perturbation increases outward. The overlying 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 two provided $ωτ_c\geq 1$, where $ω$ is the radian frequency of the mode and $τ_c\approx 4τ_{th}$ with $τ_{th}$ being the thermal time constant evaluated at the base of the convective envelope. As a white dwarf cools, its convection zone deepens, and modes of lower frequency 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 DAVs. The linear growth time, ranging from hours for the longest period observed modes ($P\approx 20$ minutes) to thousands of years for those of shortest period ($P\approx 2 $ minutes), probably sets the time-scale 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.

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