Saving Planetary Systems: Dead Zones & Planetary Migration

The tidal interaction between a disk and a planet leads to the planet's migration. A long-standing question regarding this mechanism is how to stop the migration before planets plunge into their central stars. In this paper, we propose a new, simple mechanism to significantly slow down planet migration, and test the possibility by using a hybrid numerical integrator to simulate the disk-planet interaction. The key component of the scenario is the role of low viscosity regions in protostellar disks known as dead zones, which affect planetary migration in two ways. First of all, it allows a smaller-mass planet to open a gap, and hence switch the faster type I migration to the slower type II migration. Secondly, a low viscosity slows down type II migration itself, because type II migration is directly proportional to the viscosity. We present numerical simulations of planetary migration by using a hybrid symplectic integrator-gas dynamics code. Assuming that the disk viscosity parameter inside the dead zone is (alpha=1e-4-1e-5), we find that, when a low-mass planet (e.g. 1-10 Earth masses) migrates from outside the dead zone, its migration is stopped due to the mass accumulation inside the dead zone. When a low-mass planet migrates from inside the dead zone, it opens a gap and slows down its migration. A massive planet like Jupiter, on the other hand, opens a gap and slows down inside the dead zone, independent of its initial orbital radius. The final orbital radius of a Jupiter mass planet depends on the dead zone's viscosity. For the range of alpha's noted above, this can vary anywhere from 7 AU, to an orbital radius of 0.1 AU that is characteristic of the hot Jupiters.Comment: 38 pages, 14 figures, some changes in text and figures, accepted for publication in Ap

From mean-motion resonances to scattered planets: Producing the Solar System, eccentric exoplanets and Late Heavy Bombardments

We show that interaction with a gas disk may produce young planetary systems with closely-spaced orbits, stabilized by mean-motion resonances between neighbors. On longer timescales, after the gas is gone, interaction with a remnant planetesimal disk tends to pull these configurations apart, eventually inducing dynamical instability. We show that this can lead to a variety of outcomes; some cases resemble the Solar System, while others end up with high-eccentricity orbits reminiscent of the observed exoplanets. A similar mechanism has been previously suggested as the cause of the lunar Late Heavy Bombardment. Thus, it may be that a large-scale dynamical instability, with more or less cataclysmic results, is an evolutionary step common to many planetary systems, including our own.Comment: 12 pages, 7 figures, submitted to Ap

Overcoming migration during giant planet formation

In the core accretion model, gas giant formation is a race between growth and migration; for a core to become a jovian planet, it must accrete its envelope before it spirals into the host star. We use a multizone numerical model to extend our previous investigation of the "window of opportunity" for gas giant formation within a disk. When the collision cross-section enhancement due to core atmospheres is taken into account, we find that a broad range of protoplanetary disks posses such a window.Comment: 4 pages, 3 figs, accepted to ApJ

A safety net for fast migrators: Interactions between gap-opening and sub-gap-opening bodies in a protoplanetary disk

Young planets interact with their parent gas disks through tidal torques. An imbalance between inner and outer torques causes bodies of mass \ga 0.1 Earth masses to lose angular momentum and migrate inward rapidly relative to the disk; this is known as ``Type I'' migration. However, protoplanets that grow to gas giant mass, O($10^2) M_\oplus$, open a gap in the disk and are subsequently constrained to migrate more slowly, locked into the disk's viscous evolution in what is called "Type II" migration. In a young planetary system, both Type I and Type II bodies likely coexist; if so, differential migration ought to result in close encounters when the former originate on orbits exterior to the latter. We investigate the resulting dynamics, using two different numerical approaches: an N-body code with dissipative forces added to simulate the effect of the gas disk, and a hybrid code which combines an N-body component with a 1-dimensional viscous disk model, treating planet-disk interactions in a more self-consistent manner. In both cases, we find that sub-gap-opening bodies have a high likelihood of being resonantly captured when they encounter a gap-opening body. A giant planet thus tends to act as a barrier in a protoplanetary disk, collecting smaller protoplanets outside of its orbit. Such behavior has two important implications for giant planet formation: First, for captured protoplanets it mitigates the problem of the migration timescale becoming shorter than the growth timescale. Secondly, it suggests one path to forming systems with multiple giant planets: Once the first has formed, it traps/accretes the future solid core of the second in an exterior mean-motion resonance, and so on. The most critical step in giant planet formation may thus be the formation of the very first one.Comment: Accepted for publication in Ap

Giant Planet Accretion and Migration: Surviving the Type I Regime

In the standard model of gas giant planet formation, a large solid core (~ 10 times the Earth's mass) forms first, then accretes its massive envelope (100 or more Earth masses) of gas. However, inward planet migration due to gravitational interaction with the proto-stellar gas disk poses a difficulty in this model. Core-sized bodies undergo rapid "Type I" migration; for typical parameters their migration timescale is much shorter than their accretion timescale. How, then, do growing cores avoid spiraling into the central star before they ever get the chance to become gas giants? Here, we present a simple model of core formation in a gas disk which is viscously evolving. As the disk dissipates, accretion and migration timescales eventually become comparable. If this happens while there is still enough gas left in the disk to supply a jovian atmosphere, then a window of opportunity for gas giant formation opens. We examine under what circumstances this happens, and thus, what predictions our model makes about the link between proto-stellar disk properties and the likelihood of forming giant planets.Comment: To appear in ApJ. 644. 12 pages, 9 figures. Figures degraded for size; see http://www.cita.utoronto.ca/~thommes/ for original

Resonant inclination excitation of migrating giant planets

The observed orbits of extrasolar planets suggest that many giant planets migrate a considerable distance towards their parent star as a result of interactions with the protoplanetary disk, and that some of these planets become trapped in eccentricity-exciting mean motion resonances with one another during this migration. Using three-dimensional numerical simulations, we find that as long as the timescale for damping of the planets' eccentricities by the disk is close to or longer than the disk-induced migration timescale, and the outer planet is more than half the mass of the inner, resonant inclination excitation will also occur. Neither the addition of a (simple, fixed) disk potential, nor the introduction of a massive inner planetary system, inhibit entry into the inclination resonance. Therefore, such a mechanism may not be uncommon in the early evolution of a planetary system, and a significant fraction of exoplanetary systems may turn out to be non-coplanar.Comment: to appear in ApJ 597, November 1, 200

Unstable Planetary Systems Emerging Out Of Gas Disks

The discovery of over 400 extrasolar planets allows us to statistically test our understanding of formation and dynamics of planetary systems via numerical simulations. Traditional N-body simulations of multiple-planet systems without gas disks have successfully reproduced the eccentricity (e) distribution of the observed systems, by assuming that the planetary systems are relatively closely packed when the gas disk dissipates, so that they become dynamically unstable within the stellar lifetime. However, such studies cannot explain the small semi-major axes (a) of extrasolar planetary systems, if planets are formed, as the standard planet formation theory suggests, beyond the ice line. In this paper, we numerically study the evolution of three-planet systems in dissipating gas disks, and constrain the initial conditions that reproduce the observed semi-major axis and eccentricity distributions simultaneously. We adopt the initial conditions that are motivated by the standard planet formation theory, and self-consistently simulate the disk evolution, and planet migration by using a hybrid N-body and 1D gas disk code. We also take account of eccentricity damping, and investigate the effect of saturation of corotation resonances on the evolution of planetary systems. We find that the semi-major axis distribution is largely determined in a gas disk, while the eccentricity distribution is determined after the disk dissipation. We also find that there may be an optimum disk mass which leads to the observed a-e distribution. Our simulations generate a larger fraction of planetary systems trapped in mean-motion resonances (MMRs) than the observations, indicating that the disk's perturbation to the planetary orbits may be important to explain the observed rate of MMRs. We also find much lower occurrence of planets on retrograde orbits than the current observations of close-in planets suggest.Comment: 12 pages, 9 figures, accepted for publication in Ap

Cost-Effectiveness Evaluation Of Quadrivalent Influenza Vaccines For Seasonal Influenza Prevention: A Dynamic Modeling Study Of Canada And The United Kingdom

Background: The adoption of quadrivalent influenza vaccine (QIV) to replace trivalent influenza vaccine (TIV) in immunization programs is growing worldwide, thus helping to address the problem of influenza B lineage mismatch. However, the price per dose of QIV is higher than that of TIV. In such circumstances, cost-effectiveness analyses provide important and relevant information to inform national health recommendations and implementation decisions. This analysis assessed potential vaccine impacts and cost-effectiveness of a country-wide switch from TIV to QIV, in Canada and the UK, from a third-party payer perspective. Methods: An age-stratified, dynamic four-strain transmission model which incorporates strain interaction, transmission-rate seasonality and age-specific mixing in the population was used. Model input data were obtained from published literature and online databases. In Canada, we evaluated a switch from TIV to QIV in the entire population. For the UK, we considered two strategies: Children aged 2-17 years who receive the live-attenuated influenza vaccine (LAIV) switch to the quadrivalent formulation (QLAIV), while individuals aged > 18 years switch from TIV to QIV. Two different vaccination uptake scenarios in children (UK1 and UK2, which differ in the vaccine uptake level) were considered. Health and cost outcomes for both vaccination strategies, and the cost-effectiveness of switching from TIV/LAIV to QIV/QLAIV, were estimated from the payer perspective. For Canada and the UK, cost and outcomes were discounted using 5 % and 3.5 % per year, respectively. Results: Overall, in an average influenza season, our model predicts that a nationwide switch from TIV to QIV would prevent 4.6 % influenza cases, 4.9 % general practitioner (GP) visits, 5.7 % each of emergency room (ER) visits and hospitalizations, and 6.8 % deaths in Canada. In the UK (UK1/UK2), implementing QIV would prevent 1.4 %/1.8 % of influenza cases, 1.6 %/2.0 % each of GP and ER visits, 1.5 %/1.9 % of hospitalizations and 4.3 %/4.9 % of deaths. Discounted incremental cost-utility ratios of $7,961 and 7,989 pound/7,234 pound per quality-adjusted life-year (QALY) gained are estimated for Canada and the UK (UK1/UK2), both of which are well within their respective cost-effectiveness threshold values. Conclusions: Switching from TIV to QIV is expected to be a cost-effective strategy to further reduce the burden of influenza in both countries.GlaxoSmithKline Biologicals S