Local Statistics of Realizable Vertex Models

We study planar "vertex" models, which are probability measures on edge subsets of a planar graph, satisfying certain constraints at each vertex, examples including dimer model, and 1-2 model, which we will define. We express the local statistics of a large class of vertex models on a finite hexagonal lattice as a linear combination of the local statistics of dimers on the corresponding Fisher graph, with the help of a generalized holographic algorithm. Using an $n\times n$ torus to approximate the periodic infinite graph, we give an explicit integral formula for the free energy and local statistics for configurations of the vertex model on an infinite bi-periodic graph. As an example, we simulate the 1-2 model by the technique of Glauber dynamics

Flight craft Patent

Designing spacecraft for flight into space, atmospheric reentry, and landing at selected site

Trees and Matchings

In this article, Temperley's bijection between spanning trees of the square grid on the one hand, and perfect matchings (also known as dimer coverings) of the square grid on the other, is extended to the setting of general planar directed (and undirected) graphs, where edges carry nonnegative weights that induce a weighting on the set of spanning trees. We show that the weighted, directed spanning trees (often called arborescences) of any planar graph G can be put into a one-to-one weight-preserving correspondence with the perfect matchings of a related planar graph H. One special case of this result is a bijection between perfect matchings of the hexagonal honeycomb lattice and directed spanning trees of a triangular lattice. Another special case gives a correspondence between perfect matchings of the ``square-octagon'' lattice and directed weighted spanning trees on a directed weighted version of the cartesian lattice. In conjunction with results of Kenyon, our main theorem allows us to compute the measures of all cylinder events for random spanning trees on any (directed, weighted) planar graph. Conversely, in cases where the perfect matching model arises from a tree model, Wilson's algorithm allows us to quickly generate random samples of perfect matchings.Comment: 32 pages, 19 figures (minor revisions from version 1

Quadri-tilings of the plane

We introduce {\em quadri-tilings} and show that they are in bijection with dimer models on a {\em family} of graphs $\{R^*\}$ arising from rhombus tilings. Using two height functions, we interpret a sub-family of all quadri-tilings, called {\em triangular quadri-tilings}, as an interface model in dimension 2+2. Assigning "critical" weights to edges of $R^*$, we prove an explicit expression, only depending on the local geometry of the graph $R^*$, for the minimal free energy per fundamental domain Gibbs measure; this solves a conjecture of \cite{Kenyon1}. We also show that when edges of $R^*$ are asymptotically far apart, the probability of their occurrence only depends on this set of edges. Finally, we give an expression for a Gibbs measure on the set of {\em all} triangular quadri-tilings whose marginals are the above Gibbs measures, and conjecture it to be that of minimal free energy per fundamental domain.Comment: Revised version, minor changes. 30 pages, 13 figure

The spectral energy distribution of self-gravitating protostellar disks

The long wavelength emission of protostellar objects is commonly attributed to a disk of gas and dust around the central protostar. In the first stages of disk accretion or in the case of high mass protostars, the disk mass is likely to be sufficiently large, so that the disk self-gravity may have an impact on the dynamics and the emission properties of the disk. In this paper we describe the spectral energy distribution (SED) produced by a simple, non-flaring, self-gravitating accretion disk model. Self-gravity is included in the calculation of the rotation curve of the disk and in the energy balance equation, as a term of effective heating related to Jeans instability. In order to quantify in detail the requirements on the mass of the disk and on the accretion rate posed on the models by realistic situations, we compare the SEDs produced by these models with the observed SEDs of a small sample of well-studied protostellar objects. We find that relatively modest disks - even lighter than the central star - can lead to an interesting fit to the infrared SED of the FU Orionis objects considered, while in the case of T Tauri stars the required parameters fall outside the range suggested as acceptable by the general theoretical and observational scenario. On the basis of the present results, we may conclude that the contribution of a self-gravitating disk is plausible in several cases (in particular, for FU Orionis objects) and that, in the standard irradiation dominated disk scenario, it would help softening the requirements encountered by Keplerian accretion models.Comment: 26 pages, 7 figures, accepted by A&

Terrestrial Planet Formation I. The Transition from Oligarchic Growth to Chaotic Growth

We use a hybrid, multiannulus, n-body-coagulation code to investigate the growth of km-sized planetesimals at 0.4-2 AU around a solar-type star. After a short runaway growth phase, protoplanets with masses of roughly 10^26 g and larger form throughout the grid. When (i) the mass in these `oligarchs' is roughly comparable to the mass in planetesimals and (ii) the surface density in oligarchs exceeds 2-3 g/sq cm at 1 AU, strong dynamical interactions among oligarchs produce a high merger rate which leads to the formation of several terrestrial planets. In disks with lower surface density, milder interactions produce several lower mass planets. In all disks, the planet formation timescale is roughly 10-100 Myr, similar to estimates derived from the cratering record and radiometric data.Comment: Astronomical Journal, accepted; 22 pages + 15 figures in ps format; eps figures at http://cfa-www.harvard.edu/~kenyon/dl/ revised version clarifies evolution and justifies choice of promotion masse

Azimuthally Symmetric Theory of Gravitation (I)

From a purely none-general relativistic standpoint, we solve the empty space Poisson equation ($\nabla^{2}\Phi=0$) for an azimuthally symmetric setting, i.e., for a spinning gravitational system like the Sun. We seek the general solution of the form $\Phi=\Phi(r,\theta)$. This general solution is constrained such that in the zeroth order approximation it reduces to Newton's well known inverse square law of gravitation. For this general solution, it is seen that it has implications on the orbits of test bodies in the gravitational field of this spinning body. We show that to second order approximation, this azimuthally symmetric gravitational field is capable of explaining at least two things (1) the observed perihelion shift of solar planets (2) that the mean Earth-Sun distance must be increasing -- this resonates with the observations of two independent groups of astronomers (Krasinsky & Brumberg 2004; Standish 2005) who have measured that the mean Earth-Sun distance must be increasing at a rate of about $7.0\pm0.2 m/cy$ (Standish 2005) to $15.0\pm0.3 m/cy$ (Krasinsky & Brumberg 2004). In-principle, we are able to explain this result as a consequence of loss of orbital angular momentum -- this loss of orbital angular momentum is a direct prediction of the theory. Further, we show that the theory is able to explain at a satisfactory level the observed secular increase Earth Year ($1.70\pm0.05 ms/yr$; Miura et al. 2009}). Furthermore, we show that the theory makes a significant and testable prediction to the effect that the period of the solar spin must be decreasing at a rate of at least $8.00\pm2.00 s/cy$.Comment: 2 figures, 2 tables, 13 pages. Accepted to MNRAS 2009 December 9. Received 2009 December 9; in original form 2009 September 5: ref. MN-09-1767-MJ.R3

The Nature of Hypervelocity Stars and the Time between Their Formation and Ejection

We obtain Keck HIRES spectroscopy of HVS5, one of the fastest unbound stars in the Milky Way halo. We show that HVS5 is a 3.62 ± 0.11 M_☉ main-sequence B star at a distance of 50 ± 5 kpc. The difference between its age and its flight time from the Galactic center is 105 ± 18 (stat) ±30 (sys) Myr; flight times from locations elsewhere in the Galactic disk are similar. This 10^8 yr "arrival time" between formation and ejection is difficult to reconcile with any ejection scenario involving massive stars that live for only 10^7 yr. For comparison, we derive arrival times of 10^7 yr for two unbound runaway B stars, consistent with their disk origin where ejection results from a supernova in a binary system or dynamical interactions between massive stars in a dense star cluster. For HVS5, ejection during the first 10^7 yr of its lifetime is ruled out at the 3σ level. Together with the 10^8 yr arrival times inferred for three other well-studied hypervelocity stars (HVSs), these results are consistent with a Galactic center origin for the HVSs. If the HVSs were indeed ejected by the central black hole, then the Galactic center was forming stars ≃200 Myr ago, and the progenitors of the HVSs took ≃100 Myr to enter the black hole's loss cone

Thermonuclear runaways in thick hydrogen rich envelopes of neutron stars

A Lagrangian, fully implicit, one dimensional hydrodynamic computer code was used to evolve thermonuclear runaways in the accreted hydrogen rich envelopes of 1.0 Msub solar neutron stars with radii of 10 km and 20 km. Simulations produce outbursts which last from about 750 seconds to about one week. Peak effective temeratures and luninosities were 26 million K and 80 thousand Lsub solar for the 10 km study and 5.3 millison and 600 Lsub solar for the 20 km study. Hydrodynamic expansion on the 10 km neutron star produced a precursor lasting about one ten thousandth seconds