Scenarios for the Origin of the Orbits of the Trans-Neptunian Objects 2000 CR105 and 2003 VB12

Explaining the origin of the orbit of 2000 CR105 (a ~ 230AU, q ~ 45AU) is a major test for our understanding of the primordial evolution of the outer Solar System. Gladman et al. (2001) showed that this objects could not have been a normal member of the scattered disk that had its perihelion distance increased by chaotic diffusion. In this paper we explore four seemingly promising mechanisms for explaining the origin of the orbit of this peculiar object: (i) the passage of Neptune through a high-eccentricity phase, (ii) the past existence of massive planetary embryos in the Kuiper belt or the scattered disk, (iii) the presence of a massive trans-Neptunian disk at early epochs which exerted tides on scattered disk objects, and (iv) encounters with other stars. Of all these mechanisms, the only one giving satisfactory results is the passage of a star. Indeed, our simulations show that the passage of a solar mass star at about 800 AU only perturbs objects with semi-major axes larger than roughly 200 AU to large perihelion distances. This is in good agreement with the fact that 2000 CR105 has a semi-major axis of 230AU and no other bodies with similar perihelion distances but smaller semi-major axes have yet been discovered. The discovery of 2003 VB12, (a=450AU, q=75AU) announced a few days before the submission of this paper, strengthen our conclusions.Comment: AJ submitted. 27 pages, 6 figure

A Unified Theory for the Effects of Stellar Perturbations and Galactic Tides on Oort Cloud Comets

We examine the effects of passing field stars on the angular momentum of a nearly radial orbit of an Oort cloud comet bound to the Sun. We derive the probability density function (PDF) of the change in angular momentum from one stellar encounter, assuming a uniform and isotropic field of perturbers. We show that the total angular momentum follows a Levy flight, and determine its distribution function. If there is an asymmetry in the directional distribution of perturber velocities, the marginal probability distribution of each component of the angular momentum vector can be different. The constant torque attributed to Galactic tides arises from a non-cancellation of perturbations with an impact parameter of order the semimajor axis of the comet. When the close encounters are rare, the angular momentum is best modeled by the stochastic growth of stellar encounters. If trajectories passing between the comet and sun occur frequently, the angular momentum exhibits the coherent growth attributed to the Galactic tides.Comment: 8 pages, 2 figures; accepted to A

Dynamic walking stability of the TUlip robot by means of the extrapolated center of mass

The TUlip robot was created to participate in the teensize league of Robocup. The TUlip robot is a bipedal robot intended for dynamic walking. It has six degrees of freedom for each leg: three for the hip, one for the knee and two for the ankle. This paper elaborates on the algorithm for the sideways control during gait. The algorithm uses the extrapolated center of mass (XcoM) to achieve limit cycle stability. The algorithm is tested in simulation using a linear inverted pendulum and, then, experimentally applied to the TUlip robot. The result is an adaptive behavior of the TUlip robot, promising for future application to legged robot stability

Anisotropic interactions of a single spin and dark-spin spectroscopy in diamond

The nitrogen-vacancy (N-V) center in diamond is a promising atomic-scale system for solid-state quantum information processing. Its spin-dependent photoluminescence has enabled sensitive measurements on single N-V centers, such as: electron spin resonance, Rabi oscillations, single-shot spin readout and two-qubit operations with a nearby 13C nuclear spin. Furthermore, room temperature spin coherence times as long as 58 microseconds have been reported for N-V center ensembles. Here, we have developed an angle-resolved magneto-photoluminescence microscopy apparatus to investigate the anisotropic electron spin interactions of single N-V centers at room temperature. We observe negative peaks in the photoluminescence as a function of both magnetic field magnitude and angle that are explained by coherent spin precession and anisotropic relaxation at spin level anti-crossings. In addition, precise field alignment unmasks the resonant coupling to neighboring dark nitrogen spins that are not otherwise detected by photoluminescence. The latter results demonstrate a means of investigating small numbers of dark spins via a single bright spin under ambient conditions.Comment: 13 pages, 4 figure

Modeling galactic halos with predominantly quintessential matter

This paper discusses a new model for galactic dark matter by combining an anisotropic pressure field corresponding to normal matter and a quintessence dark energy field having a characteristic parameter $\omega_q$ such that $-1<\omega_q< -\frac{1}{3}$. Stable stellar orbits together with an attractive gravity exist only if $\omega_q$ is extremely close to $-\frac{1}{3}$, a result consistent with the special case studied by Guzman et al. (2003). Less exceptional forms of quintessence dark energy do not yield the desired stable orbits and are therefore unsuitable for modeling dark matter.Comment: 12 pages, 1 figur

Open problems in algebraic geometry

The open problems presented here were collected on the occasion of a workshop on Arithmetic Geometry at the University ofUtrecht, 26{30 June, 2000. This workshop was organized by the editors of the present article, and was made possible by support of: | NWO, the Netherlands Organization for Scientic Research, | KNAW, the Royal Netherlands Academy of Arts and Sciences, | MRI, the Mathematical Research Institute, | the Faculty of Mathematics and Computer Science of the University of Utrecht, and | the Department of Mathematics of the University of Utrecht. We thank these organizations heartily for their support. All problems in this list have been reviewed by at least one referee. In this process many useful suggestions and new references have come up. We thank all referees for their valuable comments

Dark matter effects in vacuum spacetime

We analyze a toy model describing an empty spacetime in which the motion of a test mass (and the trajectories of photons) evidence the presence of a continuous and homogeneous distribution of matter; however, since the energy-momentum tensor vanishes, no real matter or energy distribution is present at all. Thus, a hypothetical observer will conclude that he is immersed in some sort of dark matter, even though he has no chance to directly detect it. This suggests yet another possibility of explaining the elusive dark matter as a purely dynamical effect due to the curvature of spacetime.Comment: 5 pages, 2 figures, expanded with comments about the exact motion and curvature invariant

The a-number of hyperelliptic curves

It is known that for a smooth hyperelliptic curve to have a large $a$-number, the genus must be small relative to the characteristic of the field, $p>0$, over which the curve is defined. It was proven by Elkin that for a genus $g$ hyperelliptic curve $C$ to have $a_C=g-1$, the genus is bounded by $g<\frac{3p}{2}$. In this paper, we show that this bound can be lowered to $g <p$. The method of proof is to force the Cartier-Manin matrix to have rank one and examine what restrictions that places on the affine equation defining the hyperelliptic curve. We then use this bound to summarize what is known about the existence of such curves when $p=3,5$ and $7$.Comment: 7 pages. v2: revised and improved the proof of the main theorem based on suggestions from the referee. To appear in the proceedings volume of Women in Numbers Europe-