Growth rate of a migrating protoplanet

We investigated the collision rate (i.e., the growth rate) of a migrating protoplanet with planetesimals. The collision rate strongly depends on the orbital elements of planetesimals (e.g., their eccentricities and inclinations). Thus we calculated the orbital evolutions of 2000 planetesimals in the vicinity of the migrating protoplanet and obtained the collision rate by counting the number of collisions with the protoplanet. For slow migration, the protoplanet makes a gap around its orbit in the planetesimals disk. On the other hand, for rapid migration, the protoplanet cannot shepherd planetesimals and keeps catching planetesimals. The obtained collision rate becomes larger with an increase in the migration speed. The comparison of the obtained collision rates with that of the previous work with no migration shows that the rapid migration of a protoplanet can enhance the collision rate by more than the factor 10. Using the obtained collision rate, we examined the growth of a migrating protoplanet. Our results suggest that, due to the enhancement of the collision rate, planets can be formed before they fall to the sun.     

Grain sedimentation in a giant gaseous protoplanet

We present a calculation of the sedimentation of grains in a giant gaseous protoplanet such as that resulting from a disk instability of the type envisioned by Boss [Boss, A.P., 1998. Earth Moon Planets 81, 19-26]. Boss [Boss, A.P., 1998. Earth Moon Planets 81, 19-26] has suggested that such protoplanets would form cores through the settling of small grains. We have tested this suggestion by following the sedimentation of small silicate grains as the protoplanet contracts and evolves. We find that during the course of the initial contraction of the protoplanet, which lasts some 4×¹⁰⁵ years, even very small (>1 μm) silicate grains can sediment to create a core both for convective and non-convective envelopes, although the sedimentation time is substantially longer if the envelope is convective, and grains are allowed to be carried back up into the envelope by convection. Grains composed of organic material will mostly be evaporated before they get to the core region, while water ice grains will be completely evaporated. These results suggest that if giant planets are formed via the gravitational instability mechanism, a small heavy element core can be formed due to sedimentation of grains, but it will be composed almost entirely of refractory material. Including planetesimal capture, we find core masses between 1 and 10 M_⊕, and a total high-Z enhancement of ∼40 M_⊕. The refractories in the envelope will be mostly water vapor and organic residuals.     

The evolution of jupiter from a protoplanet

The evolution of an object of Jupiter mass contracting under gravity while being heated at its outer surface by the Sun is considered. Degenerate pressure in its interior is included. It is found that a good agreement with the present effective temperature and luminosity of Jupiter is predicted. The derived age of the evolved body is not in contradiction with the ages of other objects in the Solar System.     

An efficient, low-velocity, resonant mechanism for capture of satellites by a protoplanet

Numerical simulations of the gravitational scattering of planetesimals by a protoplanet reveal that a significant fraction of scattered planetesimals can become trapped as so-called quasi-satellites in heliocentric 1:1 co-orbital resonance with the protoplanet. While trapped, these resonant planetesimals can have deep low-velocity encounters with the protoplanet that result in temporary or permanent capture onto highly eccentric prograde or retrograde circumplanetary orbits. The simulations include solar nebula gas drag and use planetesimals with diameters ranging from ∼1 to ∼1000 km. Initial protoplanet eccentricities range from ep=0 to 0.15 and protoplanet masses range from 300 Earth-masses (M_⊕) down to 0.1M_⊕. This mass range effectively covers the final masses of all planets currently thought to be in possession of captured satellites—Jupiter, Saturn, Neptune, Uranus, and Mars. For protoplanets on moderately eccentric orbits (ep?0.1) most simulations show from 5-20% of all scattered planetesimals becoming temporarily trapped in the quasi-satellite co-orbital resonance. Typically, 20-30% of the temporarily trapped quasi-satellites of all sizes came within half the Hill radius of the protoplanet while trapped in the resonance. The efficiency of the resonance trapping combined with the subsequent low-velocity circumplanetary capture suggests that this trapped-to-captured transition may be important not only for the origin of captured satellites but also for continued growth of protoplanets.     

Collisional growth of planetesimals

Safronov's (1972) demonstration that relative velocities of planetesimals would be comparable to the dominant size bodies' escape velocities, combined with a plausible size distribution that has most mass in the largest bodies, yielded his evolution model with limited growth of the largest planetesimal with respect to its next largest neighbors. A numerical simulation of planetesimal accretion (Greenberget al., 1978) suggests that at least over one stage of collisional accretion, velocities were much lower than the escape velocity of the largest bodies, because the bulk of the mass still resided in km-scale bodies. The low velocities at this early stage may conceivably have permitted early runaway growth, which, in turn, would have kept the velocities low and permitted continued runaway growth of the largest bodies.Paper presented at the European Workshop on Planetary Sciences, organised by the Laboratorio di Astrofisica Spaziale di Frascati, and held between April 23–27, 1979, at the Accademia Nazionale del Lincei in Rome, Italy.     

Oligarchic growth of giant planets

Runaway growth ends when the largest protoplanets dominate the dynamics of the planetesimal disk; the subsequent self-limiting accretion mode is referred to as “oligarchic growth.” Here, we begin by expanding on the existing analytic model of the oligarchic growth regime. From this, we derive global estimates of the planet formation rate throughout a protoplanetary disk. We find that a relatively high-mass protoplanetary disk (∼10 × minimum-mass) is required to produce giant planet core-sized bodies (∼10 M_⊕) within the lifetime of the nebular gas (?10 million years). However, an implausibly massive disk is needed to produce even an Earth mass at the orbit of Uranus by 10 Myrs. Subsequent accretion without the dissipational effect of gas is even slower and less efficient. In the limit of noninteracting planetesimals, a reasonable-mass disk is unable to produce bodies the size of the Solar System’s two outer giant planets at their current locations on any timescale; if collisional damping of planetesimal random velocities is sufficiently effective, though, it may be possible for a Uranus/Neptune to form in situ in less than the age of the Solar System. We perform numerical simulations of oligarchic growth with gas and find that protoplanet growth rates agree reasonably well with the analytic model as long as protoplanet masses are well below their estimated final masses. However, accretion stalls earlier than predicted, so that the largest final protoplanet masses are smaller than those given by the model. Thus the oligarchic growth model, in the form developed here, appears to provide an upper limit for the efficiency of giant planet formation.     

Non-intrusive measurements of crater growth

An experimental technique to measure crater growth is presented whereby a high speed video captures profiles of a crater forming after impact obtained using a vertical laser sheet centered on the impact point. Unlike previous so called “quarter-space experiments,” where projectiles were launched along a transparent Plexiglas sheet so that growth of half a crater could be viewed, the use of the laser sheet permits viewing changes in crater shape without any physical interference to the cratering process. This technique indicates that for low velocity impacts (<300 m/s) into 220 μm glass beads that are without cohesion and where the projectile is not disrupted, craters initially grow somewhat proportionally, but that later their depths remain essentially constant while their diameters continue to expand. In addition, these experiments indicate that as the impact velocity increases, the rate of growth and the transient depth to diameter ratio at the end of ejecta excavation decreases. These last two observations are probably due to the large time of penetration of the projectile, which becomes a significant fraction of the time of crater formation. This is contrary to the expectations for the scaling rules, which assumes a point source. Very high curtain angles (>45°) are also seen, and could be due to the low friction angle of the target. Significant crater modification, which is rarely seen in “quarter-space experiments,” is also observed and appears to be controlled by the dynamic angle of repose of the target. These latter observations indicate that differences in target friction angles may need to be considered when determining near rim ejecta-mass distributions and large-scale crater modification processes on the planets.     

A semi-analytic model for oligarchic growth

A new semi-analytic model for the oligarchic growth phase of planetary accretion is developed. The model explicitly calculates damping and excitation of planetesimal eccentricities e and inclinations i due to gas drag and perturbations from embryos. The effects of planetesimal fragmentation, enhanced embryo capture cross sections due to atmospheres, inward planetesimal drift, and embryo-embryo collisions are also incorporated. In the early stages of oligarchic growth, embryos grow rapidly as e and i fall below their equilibrium values. The formation of planetesimal collision fragments also speeds up embryo growth as fragments have low-e, low-i orbits, thereby optimizing gravitational focussing. At later times, the presence of thick atmospheres captured from the nebula aids embryo growth by increasing their capture cross sections. Planetesimal drift due to gas drag can lead to substantial inward transport of solid material. However, inward drift is greatly reduced when embryo atmospheres are present, as the drift timescale is no longer short compared to the accretion timescale. Embryo-embryo collisions increase embryo growth rates by 50% compared to the case where growth is solely due to accretion of planetesimals. Formation of 0.1-Earth-mass protoplanets at 1 AU and 10-Earth-mass cores at 5 AU requires roughly 0.1 and 1 million years respectively, in a nebula where the local solid surface density is 7 g cm⁻² at each of these locations.     

Oligarchic growth with migration and fragmentation

In the core-accretion model, giant-planet cores form by oligarchic growth from a population of planetesimals prior to the dispersal of the disk gas. Once a core reaches a critical mass of roughly 10 Earth masses, it begins to accrete a gaseous envelope, forming a giant planet. Collisions between planetesimals cause fragmentation. Planetesimal fragments are more easily captured by cores, speeding up growth, but fragments are also lost by radial drift, reducing the total solid mass in the disk. Interaction with the gas causes cores to undergo inward type-I migration. Migration allows a core to accrete planetesimals from a larger region, but migrating cores may be lost if they reach the star. Thus, migration and fragmentation have both a positive and a negative impact on core formation. Here we describe results of new simulations of oligarchic growth that include fragmentation and/or migration. In the absence of migration, cores grow until they reach their isolation mass, which increases with distance from the star, or until the disk gas disperses. Fragmentation increases the maximum core mass by increasing growth rates in the outer disk, allowing objects to reach their isolation mass during the disk lifetime. When migration is present, cores migrate inwards rapidly when they approach 1 Earth mass. Most migrating cores are lost. Migrating cores gain little extra mass since they are passing through regions that have been depleted by earlier generations of cores. For a disk viscosity parameter alpha=1e−3 and planetesimal radius = 10 km, the maximum core mass is roughly 4 and 0.5 Earth masses with/without fragmentation, respectively, with little dependence on the disk mass. Formation and survival of 10-Earth-mass cores, in the presence of migration, requires large alpha (1e−2) and a massive disk (0.1 solar masses). When alpha is large, type-I migration rates decrease rapidly with time, allowing large, late-forming cores to survive. The addition of a stochastic (random-walk) migration component makes little difference to the outcome, provided that stochastic migration affects only cores larger than 0.01 Earth masses. Stochastic migration becomes increasingly important if it also affects lower-mass objects.     

Cosmic structure growth and dark energy

Dark energy has a dramatic effect on the dynamics of the Universe, causing the recently discovered acceleration of the expansion. The dynamics are also central to the behaviour of the growth of large-scale structure, offering the possibility that observations of structure formation provide a sensitive probe of the cosmology and dark energy characteristics. In particular, dark energy with a time-varying equation of state can have an influence on structure formation stretching back well into the matter-dominated epoch. We analyse this impact, first calculating the linear perturbation results, including those for weak gravitational lensing. These dynamical models possess definite observable differences from constant equation of state models. Then we present a large-scale numerical simulation of structure formation, including the largest volume to date involving a time-varying equation of state. We find the halo mass function is well described by the Jenkins et al. mass function formula. We also show how to interpret modifications of the Friedmann equation in terms of a time-variable equation of state. The results presented here provide steps toward realistic computation of the effect of dark energy in cosmological probes involving large-scale structure, such as cluster counts, the Sunyaev–Zel'dovich effect or weak gravitational lensing.     

Maximum growth rates of magnetoatmospheric instabilities

Simple bounds on the growth rates of magneto-atmospheric instabilities are derived, both for perturbations along and perpendicular to a horizontal magnetic field. The physical significance of a singular level for stable modes is briefly discussed.     

The growth and decay of sunspot groups

Digitized Mount Wilson sunspot data from 1917 to 1985 are analyzed to examine the growth and decay rates of sunspot group umbral areas. These rates are distributed roughly symmetrically about a median rate of decay of a few hemisphere day^-1. Percentage area change rates average 502% day^-1 for growing groups and -45% day^-1 for decaying groups. These values are significantly higher than the comparable rates for plage magnetic fields because spot groups have shorter lifetimes than do plages. The distribution of percentage decay rates also differs from that of plage magnetic fields. Small spot groups grow at faster rates on average than they decay, and large spot groups decay on average at faster rates than they grow. Near solar minimum there is a marked decrease in daily percentage spot area growth rates. This decrease is not related to group area, nor is it due to latitude effects. Sunspot groups with rotation rates close to the average (for each latitude) have markedly slower average rates of daily group growth and decay than do those groups with rotation rates faster or slower than the average. Similarly, sunspot groups with latitude drift rates near zero have markedly slower average rates of daily group growth and decay than do groups with significant latitude drifts in either direction. Both of these findings are similar to results for plage magnetic fields. These various correlations are discussed in the light of our views of the connection of the magnetic fields of spot groups to subsurface magnetic flux tubes. It is suggested that a factor in the rates of growth or decay of spot groups and plages may be the inclination angle to the vertical of the magnetic fields of the spots or plages. Larger inclination angles may result in faster growth and decay rates.Operated by the Association of Universities for Research in Astronomy, Inc., under Cooperative Agreement with the National Science Foundation.     

Planet formation: Mechanism of early growth

Experiments in vacuum (approx. 0.5 to 1 mbar) and in air quantify mechanics of collisions, rebound, and fragmentation at low velocities (1–50 m/sec), under the conditions usually postulated for the preplanetary environment in the primitive solar nebula. Such collisions have been little studied experimentally. Contrary to widespread assumptions, accretionary growth of the largest meteoroid- and asteroid-sized bodies in a given swarm results spontaneously from the simple mechanics of these collisions, without other ad hoc sticking mechanisms. The smaller bodies in the swarm are less likely to grow. Granular surfaces form, either by gravitational collapse of dust swarms or by rapid formation of regolith surfaces on solid planetesimals; these surfaces strongly promote further growth by retarding rebound. Growth of large bodies increases modal collision velocities, causing fragmentation of smaller bodies and eventual production of interstellar dust as a by-product planetesimal interactions.     

Black hole growth in hierarchical galaxy formation

We incorporate a model for black hole growth during galaxy mergers into the semi-analytical galaxy formation model based on ΛCDM proposed by Baugh et al. Our black hole model has one free parameter, which we set by matching the observed zero-point of the local correlation between black hole mass and bulge luminosity. We present predictions for the evolution with redshift of the relationships between black hole mass and bulge properties. Our simulations reproduce the evolution of the optical luminosity function of quasars. We study the demographics of the black hole population and address the issue of how black holes acquire their mass. We find that the direct accretion of cold gas during starbursts is an important growth mechanism for lower mass black holes and at high redshift. On the other hand, the re-assembly of pre-existing black hole mass into larger units via merging dominates the growth of more massive black holes at low redshift. This prediction could be tested by future gravitational wave experiments. As redshift decreases, progressively less massive black holes have the highest fractional growth rates, in line with recent claims of 'downsizing' in quasar activity.     

Grain growth by radiation pressure induced coagulation

A theoretical treatment is given of the growth of grains as a consequence of their mutual coagulation brought about by relative motions induced by radiation pressure. Analytical and numerical techniques are employed to tackle the relevant coagulation equation. The results are of particular astrophysical significance in the context of forming very small grains following a nucleation process, in the production of grains large enough to allow condensation of volatiles onto their surface, and in any situation where the supply of volatiles has been exhausted. It was found that in interstellar clouds, grains composed of iron, graphite and glassy carbon, being typical examples of three basic types of material, could grow to a size where condensation of the volatiles was possible. On the other hand, olivine, a typical silicate, could not. If a source of radiation existed at the centre of the cloud, then growth could occur if the cloud was turbulent or if the density was high enough; otherwise the grains were driven out of the regions of interest at high velocity. In the latter case, with a high cloud density, re-radiation has to be taken into account.     

Source mergers and bubble growth during reionization

The recently introduced models of reionization bubbles based on extended Press–Schechter theory (by Furlanetto, Hernquist & Zaldarriaga) are generalized to include mergers of ionization sources. Sources with a recent major merger are taken to have enhanced photon production due to star formation, and accretion on to a central black hole if a black hole is present. This produces a scatter in the number of ionized photons corresponding to a halo of a given mass and a change in photon production over time for any given halo mass. By extending previous methods, photon production histories, bubble distributions and ionization histories are computed for several different parameter and recombination assumptions. The resulting distributions interpolate between previously calculated limiting cases.