Quantum inequalities and `quantum interest' as eigenvalue problems

Quantum inequalities (QI's) provide lower bounds on the averaged energy density of a quantum field. We show how the QI's for massless scalar fields in even dimensional Minkowski space may be reformulated in terms of the positivity of a certain self-adjoint operator - a generalised Schroedinger operator with the energy density as the potential - and hence as an eigenvalue problem. We use this idea to verify that the energy density produced by a moving mirror in two dimensions is compatible with the QI's for a large class of mirror trajectories. In addition, we apply this viewpoint to the `quantum interest conjecture' of Ford and Roman, which asserts that the positive part of an energy density always overcompensates for any negative components. For various simple models in two and four dimensions we obtain the best possible bounds on the `quantum interest rate' and on the maximum delay between a negative pulse and a compensating positive pulse. Perhaps surprisingly, we find that - in four dimensions - it is impossible for a positive delta-function pulse of any magnitude to compensate for a negative delta-function pulse, no matter how close together they occur.Comment: 18 pages, RevTeX. One new result added; typos fixed. To appear in Phys. Rev.

Formation of '3D' multiplanet systems by dynamical disruption of multiple-resonance configurations

Assuming that giant planets are formed in thin protoplanetary discs, a '3D' system can form, provided that the mutual inclination is excited by some dynamical mechanism. Resonant interactions and close planetary encounters are thought to be the primary inclination-excitation mechanisms, resulting in a resonant and non-resonant system, respectively. Here we propose an alternative formation scenario, starting from a system composed of three giant planets in a nearly coplanar configuration. As was recently shown for the case of the Solar system, planetary migration in the gas disc (Type II migration) can force the planets to become trapped in a multiply resonant state. We simulate this process, assuming different values for the planetary masses and mass ratios. We show that such a triple resonance generally becomes unstable as the resonance excites the eccentricities of all planets and planet-planet scattering sets in. One of the three planets is typically ejected from the system, leaving behind a dynamically 'hot' (but stable) two-planet configuration. The resulting two-planet systems typically have large values of semimajor axial ratios (a1/a2 < 0.3), while the mutual inclination can be as high as 70{\deg}, with a median of \sim30{\deg}. A small fraction of our two-planet systems (\sim5 per cent) ends up in the stability zone of the Kozai resonance. In a few cases, the triple resonance can remain stable for long times and a '3D' system can form by resonant excitation of the orbital inclinations; such a three-planet system could be stable if enough eccentricity damping is exerted on the planets. Finally, in the single-planet resulting systems, which are formed when two planets are ejected from the system, the inclination of the planet's orbital plane with respect to the initial invariant plane -presumably the plane perpendicular to the star's spin axis- can be as large as \sim40{\deg}.Comment: 9 pages, 5 figures, published in MNRA

Simon Says (Fall 2014)

In this issue: New Electronic Databases Einstein\u27s Now Open - More Renovations in the Works New Music Acquisitions New Archival Acquisitions Research Clinic CSU ePress New Science Librarian - Paul Luft Upcoming Library Forum Events Upcoming Library Exhibits CSU Libraries Connectedhttps://csuepress.columbusstate.edu/library_newsletters/1015/thumbnail.jp

Formation, Survival, and Detectability of Planets Beyond 100 AU

Direct imaging searches have begun to detect planetary and brown dwarf companions and to place constraints on the presence of giant planets at large separations from their host star. This work helps to motivate such planet searches by predicting a population of young giant planets that could be detectable by direct imaging campaigns. Both the classical core accretion and the gravitational instability model for planet formation are hard-pressed to form long-period planets in situ. Here, we show that dynamical instabilities among planetary systems that originally formed multiple giant planets much closer to the host star could produce a population of giant planets at large (~100 AU - 100000 AU) separations. We estimate the limits within which these planets may survive, quantify the efficiency of gravitational scattering into both stable and unstable wide orbits, and demonstrate that population analyses must take into account the age of the system. We predict that planet scattering creates a population of detectable giant planets on wide orbits that decreases in number on timescales of ~10 Myr. We demonstrate that several members of such populations should be detectable with current technology, quantify the prospects for future instruments, and suggest how they could place interesting constraints on planet formation models.Comment: 13 pages (emulateapj format), 10 figures, accepted for publication in Ap

Secular Evolution of HD 12661: A System Caught at an Unlikely Time

The eccentricity evolution of multiple planet systems can provide valuable constraints on planet formation models. Unfortunately, the inevitable uncertainties in the current orbital elements can lead to significant ambiguities in the nature of the secular evolution. Integrating any single set of orbital elements inadequately describes the full range of secular evolutions consistent with current observations. Thus, we combine radial velocity observations of HD 12661 with Markov Chain Monte Carlo sampling to generate ensembles of initial conditions for direct n-body integrations. We find that any mean motion resonances are quite weak and do not significantly impact the secular evolution, and that current observations indicate circulation or large amplitude libration of the periapses. The eccentricity of the outer planet undergoes large oscillations for nearly all of the allowed two-planet orbital solutions. This type of secular evolution would arise if planet c had been impulsively perturbed, perhaps due to strong scattering of an additional planet that was subsequently accreted onto the star. Finally, we note that the secular evolution implied by the current orbital configuration implies that planet c spends ~96% of the time following an orbit more eccentric than that presently observed. Either this system is being observed during a relatively rare state, or additional planets are affecting the observed radial velocities and/or the system's secular eccentricity evolution.Comment: 5 pages, 2 figures, 1 table, accepted for publication in ApJ

Effects of Turbulence, Eccentricity Damping, and Migration Rate on the Capture of Planets into Mean Motion Resonance

Pairs of migrating extrasolar planets often lock into mean motion resonance as they drift inward. This paper studies the convergent migration of giant planets (driven by a circumstellar disk) and determines the probability that they are captured into mean motion resonance. The probability that such planets enter resonance depends on the type of resonance, the migration rate, the eccentricity damping rate, and the amplitude of the turbulent fluctuations. This problem is studied both through direct integrations of the full 3-body problem, and via semi-analytic model equations. In general, the probability of resonance decreases with increasing migration rate, and with increasing levels of turbulence, but increases with eccentricity damping. Previous work has shown that the distributions of orbital elements (eccentricity and semimajor axis) for observed extrasolar planets can be reproduced by migration models with multiple planets. However, these results depend on resonance locking, and this study shows that entry into -- and maintenance of -- mean motion resonance depends sensitively on migration rate, eccentricity damping, and turbulence.Comment: 43 pages including 14 figures; accepted for publication in The Astrophysical Journa

Simon Says (Spring 2011)

In this issue: Library 2.0 & CSU Libraries Tenth Annual Faculty Research Forums Information Commons Move Department Spotlight: Reference Blog Highlights Films on Demand National Library Week Interlibrary Loan Electronic Article Delivery CSU Libraries Offer Two New JSTOR Collections Rite of Passage Convocations Return New Employees Join the CSU Libraries On the Road with CSU Libraries African American Read-In Faculty Emeritus Designationshttps://csuepress.columbusstate.edu/library_newsletters/1014/thumbnail.jp

Simon Says (Spring 2008)

Inside this issue: CSU History Showcased in Digital Collections Expanded LIBR 1105: In Full Swing African American Read-In 2008-2013 CSU Libraries Strategic Plan Columbus State University Archives Receives Conservation Bookshelf Library Staff Development Day: Road Trip! Libraries Host Rite of Passage Convocations Georgia Depository Meeting Midland Middle School Tour Collection Analysis for a Trimmer Figure New Faculty/Staff WilsonWeb OmniFile Full Text Faculty Research Forum Series 2008: Seven Faculty Present their Research Findingshttps://csuepress.columbusstate.edu/library_newsletters/1010/thumbnail.jp

The effects of fly-bys on planetary systems

Most of the observed extrasolar planets are found on tight and often eccentric orbits. The high eccentricities are not easily explained by planet-formation models, which predict that planets should be on rather circular orbits. Here we explore whether fly-bys involving planetary systems with properties similar to those of the gas giants in the solar system, can produce planets with properties similar to the observed planets. Using numerical simulations, we show that fly-bys can cause the immediate ejection of planets, and sometimes also lead to the capture of one or more planets by the intruder. More common, however, is that fly-bys only perturb the orbits of planets, sometimes leaving the system in an unstable state. Over time-scales of a few million to several hundred million years after the fly-by, this perturbation can trigger planet-planet scatterings, leading to the ejection of one or more planets. For example, in the case of the four gas giants of the solar system, the fraction of systems from which at least one planet is ejected more than doubles in 10^8 years after the fly-by. The remaining planets are often left on more eccentric orbits, similar to the eccentricities of the observed extrasolar planets. We combine our results of how fly-bys effect solar-system-like planetary systems, with the rate at which encounters in young stellar clusters occur. For example, we measure the effects of fly-bys on the four gas giants in the solar system. We find, that for such systems, between 5 and 15 per cent suffer ejections of planets in 10^8 years after fly-bys in typical open clusters. Thus, encounters in young stellar clusters can significantly alter the properties of any planets orbiting stars in clusters. As a large fraction of stars which populate the solar neighbourhood form in stellar clusters, encounters can significantly affect the properties of the observed extrasolar planets.Comment: 22 pages, 15 figures, 5 tables. Accepted for publication in MNRA

Long period planets from dynamical relaxation

Recent imaging campaigns indicate the likely existence of massive planets (~ 1-10 MJ) on ~1000 year orbits about a few percent of stars. Such objects are not easily explained in most current planet formation models. In this Letter we use ensembles of 100 N-body simulations to evaluate the potential for planet scattering during relaxation of dynamically active systems to explain the population of giant planets with projected separations up to a few 100 AU. We find that such a mechanism could indeed be at play, and that statistical samples of long period planets could place interesting constraints on early stage planet formation scenarios. Results from direct imaging and microlensing surveys are complementary probes of this dynamical relaxation process.Comment: Accepted for publication in ApJ Letters, this is final versio