ELLC - a fast, flexible light curve model for detached eclipsing binary stars and transiting exoplanets

Very high quality light curves are now available for thousands of detached eclipsing binary stars and transiting exoplanet systems as a result of surveys for transiting exoplanets and other large-scale photometric surveys. I have developed a binary star model (ELLC) that can be used to analyse the light curves of detached eclipsing binary stars and transiting exoplanet systems that is fast and accurate, and that can include the effects of star spots, Doppler boosting and light-travel time within binaries with eccentric orbits. The model represents the stars as triaxial ellipsoids. The apparent flux from the binary is calculated using Gauss-Legendre integration over the ellipses that are the projection of these ellipsoids on the sky. The model can also be used to calculate the flux-weighted radial velocity of the stars during an eclipse (Rossiter-McLaughlin effect). The main features of the model have been tested by comparison to observed data and other light curve models. The model is found to be accurate enough to analyse the very high quality photometry that is now available from space-spaced instruments, flexible enough to model a wide range of eclipsing binary stars and extrasolar planetary systems, and fast enough to enable the use of modern Monte Carlo methods for data analysis and model testing.Comment: Accepted for publication in A&A. Source code available from pypi.python.org/pypi/ellc. Definition of "third-light" changed from version ellc-1.0.0 to ellc-1.1.0 - this preprint describes the definition used in the later versio

Rational invariants of even ternary forms under the orthogonal group

In this article we determine a generating set of rational invariants of minimal cardinality for the action of the orthogonal group $\mathrm{O}_3$ on the space $\mathbb{R}[x,y,z]_{2d}$ of ternary forms of even degree $2d$. The construction relies on two key ingredients: On one hand, the Slice Lemma allows us to reduce the problem to dermining the invariants for the action on a subspace of the finite subgroup $\mathrm{B}_3$ of signed permutations. On the other hand, our construction relies in a fundamental way on specific bases of harmonic polynomials. These bases provide maps with prescribed $\mathrm{B}_3$-equivariance properties. Our explicit construction of these bases should be relevant well beyond the scope of this paper. The expression of the $\mathrm{B}_3$-invariants can then be given in a compact form as the composition of two equivariant maps. Instead of providing (cumbersome) explicit expressions for the $\mathrm{O}_3$-invariants, we provide efficient algorithms for their evaluation and rewriting. We also use the constructed $\mathrm{B}_3$-invariants to determine the $\mathrm{O}_3$-orbit locus and provide an algorithm for the inverse problem of finding an element in $\mathbb{R}[x,y,z]_{2d}$ with prescribed values for its invariants. These are the computational issues relevant in brain imaging.Comment: v3 Changes: Reworked presentation of Neuroimaging application, refinement of Definition 3.1. To appear in "Foundations of Computational Mathematics

112 lines on smooth quartic surfaces (characteristic 3)

Over a field k of characteristic 3, we prove that there are no smooth quartic surfaces S in IP^3 with more than 112 lines. Moreover, the surface with 112 lines is projectively equivalent over k-bar to the Fermat quartic. As a key ingredient, we derive a characteristic free upper bound for the number of lines met by a quadric on a smooth quartic surface.Comment: 11 pages; v2: minor edits following referee's comment

Revisiting Vinti theory : generalized equinoctial elements and applications to spacecraft relative motion

Early phases of complex astrodynamics applications often require broad searches of large solution spaces. For these studies, mission complexity generally motivates the use of the coarsest dynamical models with analytical solutions because of the implied lightening of the computational load. In this context, two-body dynamics are typically employed in practice, but higher-fidelity models with analytical solutions exist, an attractive prospect for modern applications that may require or benefit from greater accuracy. Vinti theory, which prescribes one of the many alternative described models known as intermediaries, is revisited because it leads to a direct generalization of two-body dynamics, naturally incorporating the dominant effect of oblateness and optionally the top/bottom-heavy characteristic of a celestial body without recourse to perturbation methods. Prior to the innovations introduced in this dissertation, Vinti theory and associated solutions possessed many singularities in popular orbital regimes. The theory has received limited use. The goals of this dissertation are to assess Vinti theory's effectiveness in a modern application and remove its long-standing disincentives. These objectives inform the two main contributions, respectively: 1) Vinti theory is applied to the relative motion problem through the development of a state transition matrix (STM), enabled by improvements to the existing theory; 2) a new nonsingular element set is introduced. The relative motion application leverages Vinti's approximate analytical solution with J₃. An analytical relative motion model is derived and subsequently reformulated so that Vinti's solution is piecewise differentiable, developed alongside boosts in accuracy and removal of singularities in polar and nearly circular or equatorial orbits. Some of these singularities reside in the solution, others in the partials. Solving the problem in oblate spheroidal elements leads to large linear regions of validity. The new STM is compared with side-by-side simulations of a benchmark STM obtained from perturbation methods and is shown to offer improved accuracy over a broad design space. To defray the costs of software development, robust code is provided online. The second major thrust area is the introduction of a nonsingular element set that is at once novel and familiar. Vinti theory suffers from other well-known singularities, strictly artifacts of classical elements that are detrimental to many applications. To mitigate these singularities, the standard (spherical) equinoctial elements are chosen to inform in a natural way their generalization to a new nonsingular element set: the oblate spheroidal equinoctial orbital elements. The new elements are derived without J₃ and concise algorithms presented for common coordinate transformations. The transformations are valid away from the nearly rectilinear orbital regime and are exact except near the poles. When near the poles, the transformations match the accuracy of the approximate analytical solution. As a result, the singularity on the poles is completely eliminated for the first time. Analytical state propagation of the new elements in time for bounded orbits completes their formal introduction. Benefits of the new elements are identified. The dissertation is organized as follows. To convey Vinti theory's broader context, extensive background on intermediaries and related topics is provided in Chapter 1. General enhancements that grew out of the main efforts, including the removal of some singularities, are consolidated in Chapter 2 along with mathematical preliminaries. Relative motion is explored as the selected application in Chapter 3 and the major deficiencies of Vinti theory are removed in Chapter 4 with the introduction of the new element set. Analytical orbit propagation in the new set is developed in Chapter 5.Aerospace Engineerin

Changing Views on Curves and Surfaces

Visual events in computer vision are studied from the perspective of algebraic geometry. Given a sufficiently general curve or surface in 3-space, we consider the image or contour curve that arises by projecting from a viewpoint. Qualitative changes in that curve occur when the viewpoint crosses the visual event surface. We examine the components of this ruled surface, and observe that these coincide with the iterated singular loci of the coisotropic hypersurfaces associated with the original curve or surface. We derive formulas, due to Salmon and Petitjean, for the degrees of these surfaces, and show how to compute exact representations for all visual event surfaces using algebraic methods.Comment: 31 page

Reflection groups and cones of sums of squares

We consider cones of real forms which are sums of squares and invariant under a (finite) reflection group. Using the representation theory of these groups we are able to use the symmetry inherent in these cones to give more efficient descriptions. We focus especially on the An, Bn and Dn case where we use so-called higher Specht polynomials to give a uniform description of these cones. These descriptions allow us, to deduce that the description of the cones of sums of squares of fixed degree 2d stabilizes with. Furthermore, in cases of small degree, we are able to analyze these cones more explicitly and compare them to the cones of non-negative forms

Separable triaxial potential-density pairs in MOND

We study mass models that correspond to MOND (triaxial) potentials for which the Hamilton-Jacobi equation separates in ellipsoidal coordinates. The problem is first discussed in the simpler case of deep-MOND systems, and then generalized to the full MOND regime. We prove that the Kuzmin property for Newtonian gravity still holds, i.e., that the density distribution of separable potentials is fully determined once the density profile along the minor axis is assigned. At variance with the Newtonian case, the fact that a positive density along the minor axis leads to a positive density everywhere remains unproven. We also prove that (i) all regular separable models in MOND have a vanishing density at the origin, so that they would correspond to centrally dark-matter dominated systems in Newtonian gravity; (ii) triaxial separable potentials regular at large radii and associated with finite total mass leads to density distributions that at large radii are not spherical and decline as ln(r)/r^5; (iii) when the triaxial potentials admit a genuine Frobenius expansion with exponent 0<epsilon<1, the density distributions become spherical at large radii, with the profile ln(r)/r^(3+2epsilon). After presenting a suite of positive density distributions associated with MOND separable potentials, we also consider the important family of (non-separable) triaxial potentials V_1 introduced by de Zeeuw and Pfenniger, and we show that, as already known for Newtonian gravity, they obey the Kuzmin property also in MOND. The ordinary differential equation relating their potential and density along the z-axis is an Abel equation of the second kind that, in the oblate case, can be explicitly reduced to canonical form.Comment: 17 pages, 4 figures (low resolution), accepted by MNRA

Quartic Metamaterials: The Inverse Method, Perturbations, and Bulk Optical Neutrality

A primary goal of photonics is designing material structures that support predetermined electromagnetic field distributions. We have developed an inverse method to determine material parameters for a quartic metamaterial from six desired plane waves. This work inspired us to study how perturbations to the parameters can result in optical neutrality

4th Order Symmetric Tensors and Positive ADC Modelling

International audienceHigh Order Cartesian Tensors (HOTs) were introduced in Generalized DTI (GDTI) to overcome the limitations of DTI. HOTs can model the apparent diffusion coefficient (ADC) with greater accuracy than DTI in regions with fiber heterogeneity. Although GDTI HOTs were designed to model positive diffusion, the straightforward least square (LS) estimation of HOTs doesn't guarantee positivity. In this chapter we address the problem of estimating 4th order tensors with positive diffusion profiles. Two known methods exist that broach this problem, namely a Riemannian approach based on the algebra of 4th order tensors, and a polynomial approach based on Hilbert's theorem on non-negative ternary quartics. In this chapter, we review the technicalities of these two approaches, compare them theoretically to show their pros and cons, and compare them against the Euclidean LS estimation on synthetic, phantom and real data to motivate the relevance of the positive diffusion profile constraint