Optimal Grid Drawings of Complete Multipartite Graphs and an Integer Variant of the Algebraic Connectivity

How to draw the vertices of a complete multipartite graph $G$ on different points of a bounded $d$-dimensional integer grid, such that the sum of squared distances between vertices of $G$ is (i) minimized or (ii) maximized? For both problems we provide a characterization of the solutions. For the particular case $d=1$, our solution for (i) also settles the minimum-2-sum problem for complete bipartite graphs; the minimum-2-sum problem was defined by Juvan and Mohar in 1992. Weighted centroidal Voronoi tessellations are the solution for (ii). Such drawings are related with Laplacian eigenvalues of graphs. This motivates us to study which properties of the algebraic connectivity of graphs carry over to the restricted setting of drawings of graphs with integer coordinates.Comment: Appears in the Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018

Signs of strong Na and K absorption in the transmission spectrum of WASP-103b

Context: Transmission spectroscopy has become a prominent tool for characterizing the atmospheric properties on close-in transiting planets. Recent observations have revealed a remarkable diversity in exoplanet spectra, which show absorption signatures of Na, K and $\mathrm{H_2O}$, in some cases partially or fully attenuated by atmospheric aerosols. Aerosols (clouds and hazes) themselves have been detected in the transmission spectra of several planets thanks to wavelength-dependent slopes caused by the particles' scattering properties. Aims: We present an optical 550 - 960 nm transmission spectrum of the extremely irradiated hot Jupiter WASP-103b, one of the hottest (2500 K) and most massive (1.5 $M_J$) planets yet to be studied with this technique. WASP-103b orbits its star at a separation of less than 1.2 times the Roche limit and is predicted to be strongly tidally distorted. Methods: We have used Gemini/GMOS to obtain multi-object spectroscopy hroughout three transits of WASP-103b. We used relative spectrophotometry and bin sizes between 20 and 2 nm to infer the planet's transmission spectrum. Results: We find that WASP-103b shows increased absorption in the cores of the alkali (Na, K) line features. We do not confirm the presence of any strong scattering slope as previously suggested, pointing towards a clear atmosphere for the highly irradiated, massive exoplanet WASP-103b. We constrain the upper boundary of any potential cloud deck to reside at pressure levels above 0.01 bar. This finding is in line with previous studies on cloud occurrence on exoplanets which find that clouds dominate the transmission spectra of cool, low surface gravity planets while hot, high surface gravity planets are either cloud-free, or possess clouds located below the altitudes probed by transmission spectra.Comment: Accepted for publication in A&

Topological Designs

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is large, both bounds are the best known for large genus. In genus one and two, we solve the problem exactly. Our methods generalize to variants in which the allowed number of pairwise intersections is odd, even, or bounded, and to surfaces with boundary components.Comment: 14 p., 4 Figures. To appear in Geometriae Dedicat

PyTranSpot\texttt{PyTranSpot} - A tool for multiband light curve modeling of planetary transits and stellar spots

Several studies have shown that stellar activity features, such as occulted and non-occulted starspots, can affect the measurement of transit parameters biasing studies of transit timing variations and transmission spectra. We present $\texttt{PyTranSpot}$, which we designed to model multiband transit light curves showing starspot anomalies, inferring both transit and spot parameters. The code follows a pixellation approach to model the star with its corresponding limb darkening, spots, and transiting planet on a two dimensional Cartesian coordinate grid. We combine $\texttt{PyTranSpot}$ with an MCMC framework to study and derive exoplanet transmission spectra, which provides statistically robust values for the physical properties and uncertainties of a transiting star-planet system. We validate $\texttt{PyTranSpot}$'s performance by analyzing eleven synthetic light curves of four different star-planet systems and 20 transit light curves of the well-studied WASP-41b system. We also investigate the impact of starspots on transit parameters and derive wavelength dependent transit depth values for WASP-41b covering a range of 6200-9200 $\AA$, indicating a flat transmission spectrum.Comment: 17 pages, 22 figures; accepted for publication in Astronomy & Astrophysic

GJ1214: Rotation period, starspots, and uncertainty on the optical slope of the transmission spectrum

Brightness inhomogeneities in the stellar photosphere (dark spots or bright regions) affect the measurements of the planetary transmission spectrum. To investigate the star spots of the M dwarf GJ 1214, we conducted a multicolor photometric monitoring from 2012 to 2016. The measured variability shows a periodicity of 125 +- 5 days, which we interpret as the signature of the stellar rotation period. This value overrules previous suggestions of a significantly shorter stellar rotation period. A light curve inversion of the monitoring data yields an estimation of the flux dimming of a permanent spot filling factor not contributing to the photometric variability, a temperature contrast of the spots of about 370 K and persistent active longitudes. The derived surface maps over all five seasons were used to estimate the influence of the star spots on the transmission spectrum of the planet from 400 nm to 2000 nm. The monitoring data presented here do not support a recent interpretation of a measured transmission spectrum of GJ 1214b as to be caused by bright regions in the stellar photosphere. Instead, we list arguments as to why the effect of dark spots likely dominated over bright regions in the period of our monitoring. Furthermore, our photometry proves an increase in variability over at least four years, indicative for a cyclic activity behavior. The age of GJ 1214 is likely between 6 and 10 Gyr. The long-term photometry allows for a correction of unocculted spots. For an active star such as GJ 1214, there remains a degeneracy between occulted spots and the transit parameters used to build the transmission spectrum. This degeneracy can only be broken by high-precision transit photometry resolving the spot crossing signature in the transit light curve.Comment: 11 pages, 8 figures, accepted for publication in Astronomy & Astrophysic

Irreducible triangulations of surfaces with boundary

A triangulation of a surface is irreducible if no edge can be contracted to produce a triangulation of the same surface. In this paper, we investigate irreducible triangulations of surfaces with boundary. We prove that the number of vertices of an irreducible triangulation of a (possibly non-orientable) surface of genus g>=0 with b>=0 boundaries is O(g+b). So far, the result was known only for surfaces without boundary (b=0). While our technique yields a worse constant in the O(.) notation, the present proof is elementary, and simpler than the previous ones in the case of surfaces without boundary

Crossing and weighted crossing number of near-planar graphs

A nonplanar graph G is near-planar if it contains an edge e such that G − e is planar. The problem of determining the crossing number of a near-planar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop min-max formulas involving efficiently computable lower and upper bounds. These min-max results are the first of their kind in the study of crossing numbers and improve the approximation factor for the approximation algorithm given by Hliněny and Salazar (Graph Drawing GD’06). On the other hand, we show that it is NP-hard to compute a weighted version of the crossing number for near-planar graphs

Physical properties and transmission spectrum of the WASP-74 planetary system from multi-band photometry

We present broad-band photometry of eleven planetary transits of the hot Jupiter WASP-74 b, using three medium-class telescopes and employing the telescope-defocussing technique. Most of the transits were monitored through I filters and one was simultaneously observed in five optical (U,g′,r′,i′,z′) and three near infrared (J, H, K) passbands, for a total of 18 light curves. We also obtained new high-resolution spectra of the host star. We used these new data to review the orbital and physical properties of the WASP-74 planetary system. We were able to better constrain the main system characteristics, measuring smaller radius and mass for both the hot Jupiter and its host star than previously reported in the literature. Joining our optical data with those taken with the HST in the near infrared, we built up an observational transmission spectrum of the planet, which suggests the presence of strong optical absorbers, as TiO and VO gases, in its atmosphere