Yoram Lithwick

Lower Limits on Lorentz Factors in Gamma-Ray Bursts

As is well known, the requirement that gamma-ray bursts be optically thin to high-energy photons yields a lower limit on the Lorentz factor (γ) of the expansion. In this paper, we provide a simple derivation of the lower limit on γ due to the annihilation of photon pairs and correct the errors in some of the previous calculations of this limit. We also derive a second limit on γ due to scattering of photons by pair-created electrons and positrons. For some bursts, this limit is the more stringent. In addition, we show that a third limit on γ, obtained by considering the scattering of photons by electrons that accompany baryons, is nearly always less important than the second limit. Finally, we evaluate these limits for a number of bursts.

Hot Jupiters from secular planet-planet interactions

About 25 per cent of ‘hot Jupiters’ (extrasolar Jovian-mass planets with close-in orbits) are actually orbiting counter to the spin direction of the star. Perturbations from a distant binary star companion can produce high inclinations, but cannot explain orbits that are retrograde with respect to the total angular momentum of the system. Such orbits in a stellar context can be produced through secular (that is, long term) perturbations in hierarchical triple-star systems. Here we report a similar analysis of planetary bodies, including both octupole-order effects and tidal friction, and find that we can produce hot Jupiters in orbits that are retrograde with respect to the total angular momentum. With distant stellar mass perturbers, such an outcome is not possible. With planetary perturbers, the inner orbits angular momentum component parallel to the total angular momentum need not be constant. In fact, as we show here, it can even change sign, leading to a retrograde orbit. A brief excursion to very high eccentricity during the chaotic evolution of the inner orbit allows planet–star tidal interactions to rapidly circularize that orbit, decoupling the planets and forming a retrograde hot Jupiter.

Compressible Magnetohydrodynamic 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 magnetohydrodynamic (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 constrain the nature of MHD turbulence in the interstellar medium. Slow mode density fluctuations are suppressed when the magnetic pressure is less than the gas pressure. Entropy mode density fluctuations are suppressed by cooling when the cascade timescale is longer than the cooling timescale. These constraints 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 length scales. Regions that are insufficiently ionized produce density fluctuations only on length scales larger than the neutral damping scale. These regions may account for the excess of power that is found on large scales. 4. Provided that the thermal pressure exceeds the magnetic pressure, both the entropy mode and the slow mode are damped on length scales below that at which protons can diffuse across an eddy during the eddys 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 length scale is that transverse to the magnetic field. Thus, the cutoff length scale for density fluctuations is significantly smaller than the proton mean free path. 5. The Alfven mode is critically damped at the transverse length scale of the proton gyroradius and thus cascades to smaller length scales than either the slow mode or the entropy mode.

Particle stirring in turbulent gas disks: Including orbital oscillations

We describe the diusion and random velocities of solid particles due to stochastic forcing by turbulent gas. We include the orbital dynamics of Keplerian disks, both in-plane epicycles and vertical oscillations. We obtain a new result for the diusion of solids. The Schmidt number (ratio of gas to particle diusivity) is Sc 1 + (t stop) 2 , in terms of the particle stopping time tstop and the orbital frequency . The standard result, Sc = 1 +tstop=teddy, in terms of the eddy turnover time, teddy, is shown to be incorrect. The main dierence is that Sc rises quadratically, not linearly, with stopping time. Consequently, particles larger than 10 cm in protoplanetary disks will suer less radial diusion and will settle closer to the midplane. Such a layer of boulders would be more prone to gravitational collapse. Our predictions of RMS speeds, vertical scale height and diusion coecients will help interpret numerical simulations. We conrm previous results for the vertical stirring of particles (scale heights and random velocities), and add a correction for arbitrary ratios of eddy to orbital times. The particle layer becomes thinner for teddy > 1= with the strength of turbulent diusion held xed. We use two analytic techniques { the Hinze-Tchen formalism and the FokkerPlanck equation with velocity diusion { with identical results when the regimes of validity overlap. We include simple physical arguments for the scaling of our results. Subject headings: Disks, Planetary Formation, Solar Nebula

Secular dynamics in hierarchical three-body systems

The secular approximation for the evolution of hierarchical triple configurations has proven to be very useful in many astrophysical contexts, from planetary to triple-star systems. In this approximation the orbits may change shape and orientation, on time scales longer than the orbital time scales, but the semi major axes are constant. For example, for highly inclined triple systems, the Kozai-Lidov mechanism can produce large-amplitude oscillations of the eccentricities and inclinations. Here we revisit the secular dynamics of hierarchical triple systems. We derive the secular evolution equations to octupole order in Hamiltonian perturbation theory. Our derivation corrects an error in some previous treatments of the problem that implicitly assumed a conservation of the z-component of the angular momentum of the inner orbit (i.e., parallel to the total angular momentum of the system). Already to quadrupole order, our results show new behaviors including the possibility for a system to oscillate from prograde to retrograde orbits. At the octupole order, for an eccentric outer orbit, the inner orbit can reach extremely high eccentricities and undergo chaotic flips in its orientation. We discuss applications to a variety of astrophysical systems, from stellar triples to merging compact binaries and planetary systems. Our results agree with those of previous studies done to quadrupole order only in the limit in which one of the inner two bodies is a massless test particle and the outer orbit is circular;our results agree with previous studies at octupole order for the eccentricity evolution, but not for the inclination evolution.

Formation of Kuiper-belt binaries by dynamical friction and three-body encounters

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 Systems history. Here we show that a transient binary forms when two large bodies penetrate one anothers 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.

RESONANT REPULSION OF KEPLER PLANET PAIRS

Planetary systems discovered by the Kepler space telescope exhibit an intriguing feature. While the period ratios of adjacent low-mass planets appear largely random, there is a significant excess of pairs that lie just wide of resonances and a deficit on the near side. We demonstrate that this feature naturally arises when two near-resonant planets interact in the presence of weak dissipation that damps eccentricities. The two planets repel each other as orbital energy is lost to heat. This moves near-resonant pairs just beyond resonance, by a distance that reflects the integrated dissipation they experienced over their lifetimes. We find that the observed distances may be explained by tides if tidal dissipation is unexpectedly efficient (tidal quality factor ~10). Once the effect of resonant repulsion is accounted for, the initial orbits of these low mass planets show little preference for resonances. This could constrain their origin.

The Eccentric Kozai Mechanism for a Test Particle

We study the dynamical evolution of a test particle that orbits a star in the presence of an exterior massive planet, considering octupole-order secular interactions. In the standard Kozai mechanism (SKM), the planets orbit is circular and so the particle conserves vertical angular momentum. As a result, the particles orbit oscillates periodically, exchanging eccentricity for inclination. However, when the planets orbit is eccentric, the particles vertical angular momentum varies and its Kozai oscillations are modulated on longer timescales—we call this the eccentric Kozai mechanism (EKM). The EKM can lead to behavior that is dramatically different from the SKM. In particular, the particles orbit can flip from prograde to retrograde and back again, and it can reach arbitrarily high eccentricities given enough time. We map out the conditions under which this dramatic behavior (flipping and extreme eccentricities) occurs and show that when the planets eccentricity is sufficiently high, it occurs quite generically. For example, when the planets eccentricity exceeds a few percent of the ratio of semimajor axes (outer to inner), around half of randomly oriented test particle orbits will flip and reach extreme eccentricities. The SKM has often been invoked for bringing pairs of astronomical bodies (star-star, planet-star, compact-object pairs) close together. Including the effect of the EKM will enhance the rate at which such matchmaking occurs.

Imbalanced Strong MHD Turbulence

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

Densities and eccentricities of 139 Kepler planets from transit time variations

We extract densities and eccentricities of 139 sub-Jovian planets by analyzing transit time variations (TTVs) obtained by the Kepler mission through Quarter 12. We partially circumvent the degeneracies that plague TTV inversion with the help of an analytical formula for the TTV. From the observed TTV phases, we find that most of these planets have eccentricities of order a few percent. More precisely, the r.m.s. eccentricity is 0.018^{+0.005}_{-0.004}, and planets smaller than 2.5R_\earth are around twice as eccentric as those bigger than 2.5R_\earth. We also find a best-fit density-radius relationship \rho ~3 g/cm^3 \times (R/3R_\earth)^{-2.3} for the 56 planets that likely have small eccentricity and hence small statistical correction to their masses. Many planets larger than 2.5R_\earth are less dense than water, implying that their radii are largely set by a massive hydrogen atmosphere.