Uniquely and 2-Uniquely Hamiltonian Graphs

In graph theory a Hamilton cycle is a walk around the vertices of a graph in which each vertex is visited exactly once, and then it returns to the starting vertex. The problem of determining whether a graph contains a Hamilton cycle has been studied extensively and is determined to belong to the so-called NP-complete family of problems for arbitrary graphs. Due to the difficulty in solving such a problem for an arbitrary graph, we set our sights on a family of graphs described by graph theorist John Sheehan. A maximum uniquely Hamiltonian graph contains the greatest number of edges possible while maintaining a single Hamilton cycle. Sheehan shows that for a graph with n nodes (for n \u3e= 4), the maximum number of edges it can contain is equal to (n^2/4) + 1. We will describe an algorithm that finds the Hamilton cycle for any such graph or any of its subgraphs in polynomial time. This algorithm shows that these graphs do not suffer the same complexity issues as do arbitrary graphs for a Hamilton cycle problem. For any graph containing a single Hamilton cycle, that cycle can be revealed in polynomial time

Analysis of Neptune's 2017 Bright Equatorial Storm

We report the discovery of a large ($\sim$8500 km diameter) infrared-bright storm at Neptune's equator in June 2017. We tracked the storm over a period of 7 months with high-cadence infrared snapshot imaging, carried out on 14 nights at the 10 meter Keck II telescope and 17 nights at the Shane 120 inch reflector at Lick Observatory. The cloud feature was larger and more persistent than any equatorial clouds seen before on Neptune, remaining intermittently active from at least 10 June to 31 December 2017. Our Keck and Lick observations were augmented by very high-cadence images from the amateur community, which permitted the determination of accurate drift rates for the cloud feature. Its zonal drift speed was variable from 10 June to at least 25 July, but remained a constant $237.4 \pm 0.2$ m s$^{-1}$ from 30 September until at least 15 November. The pressure of the cloud top was determined from radiative transfer calculations to be 0.3-0.6 bar; this value remained constant over the course of the observations. Multiple cloud break-up events, in which a bright cloud band wrapped around Neptune's equator, were observed over the course of our observations. No "dark spot" vortices were seen near the equator in HST imaging on 6 and 7 October. The size and pressure of the storm are consistent with moist convection or a planetary-scale wave as the energy source of convective upwelling, but more modeling is required to determine the driver of this equatorial disturbance as well as the triggers for and dynamics of the observed cloud break-up events.Comment: 42 pages, 14 figures, 6 tables; Accepted to Icaru

Performance of the Gemini Planet Imager Non-Redundant Mask and spectroscopy of two close-separation binaries HR 2690 and HD 142527

The Gemini Planet Imager (GPI) contains a 10-hole non-redundant mask (NRM), enabling interferometric resolution in complement to its coronagraphic capabilities. The NRM operates both in spectroscopic (integral field spectrograph, henceforth IFS) and polarimetric configurations. NRM observations were taken between 2013 and 2016 to characterize its performance. Most observations were taken in spectroscopic mode with the goal of obtaining precise astrometry and spectroscopy of faint companions to bright stars. We find a clear correlation between residual wavefront error measured by the AO system and the contrast sensitivity by comparing phase errors in observations of the same source, taken on different dates. We find a typical 5-$\sigma$ contrast sensitivity of $2-3~\times~10^{-3}$ at $\sim\lambda/D$. We explore the accuracy of spectral extraction of secondary components of binary systems by recovering the signal from a simulated source injected into several datasets. We outline data reduction procedures unique to GPI's IFS and describe a newly public data pipeline used for the presented analyses. We demonstrate recovery of astrometry and spectroscopy of two known companions to HR 2690 and HD 142527. NRM+polarimetry observations achieve differential visibility precision of $\sigma\sim0.4\%$ in the best case. We discuss its limitations on Gemini-S/GPI for resolving inner regions of protoplanetary disks and prospects for future upgrades. We summarize lessons learned in observing with NRM in spectroscopic and polarimetric modes.Comment: Accepted to AJ, 22 pages, 14 figure

Performance of the Gemini Planet Imager Non-Redundant Mask and Spectroscopy of Two Close-Separation Binaries: HR 2690 and HD 142527

The Gemini Planet Imager (GPI) contains a 10-hole non-redundant mask (NRM), enabling interferometric resolution in complement to its coronagraphic capabilities. The NRM operates both in spectroscopic (integral field spectrograph, henceforth IFS) and polarimetric configurations. NRM observations were taken between 2013 and 2016 to characterize its performance. Most observations were taken in spectroscopic mode, with the goal of obtaining precise astrometry and spectroscopy of faint companions to bright stars. We find a clear correlation between residual wavefront error measured by the adaptive optic system and the contrast sensitivity by comparing phase errors in observations of the same source, taken on different dates. We find a typical 5σ contrast sensitivity of (2-3) × 10-3 at ∼λ/D. We explore the accuracy of spectral extraction of secondary components of binary systems by recovering the signal from a simulated source injected into several data sets. We outline data reduction procedures unique to GPI\u27s IFS and describe a newly public data pipeline used for the presented analyses. We demonstrate recovery of astrometry and spectroscopy of two known companions to HR 2690 and HD 142527. NRM+polarimetry observations achieve differential visibility precision of σ ∼ 0.4% in the best case. We discuss its limitations on Gemini-S/GPI for resolving inner regions of protoplanetary disks and prospects for future upgrades. We summarize lessons learned in observing with NRM in spectroscopic and polarimetric modes

Analysis of Neptune’s 2017 Bright Equatorial Storm

We report the discovery of a large ( ∼ 8500 km diameter) infrared-bright storm at Neptune’s equator in June 2017. We tracked the storm over a period of 7 months with high-cadence infrared snapshot imaging, carried out on 14 nights at the 10 m Keck II telescope and 17 nights at the Shane 120 inch reflector at Lick Observatory. The cloud feature was larger and more persistent than any equatorial clouds seen before on Neptune, remaining intermittently active from at least 10 June to 31 December 2017. Our Keck and Lick observations were augmented by very high-cadence images from the amateur community, which permitted the determination of accurate drift rates for the cloud feature. Its zonal drift speed was variable from 10 June to at least 25 July, but remained a constant 237.4 ± 0.2 m s^(−1) from 30 September until at least 15 November. The pressure of the cloud top was determined from radiative transfer calculations to be 0.3-0.6 bar; this value remained constant over the course of the observations. Multiple cloud break-up events, in which a bright cloud band wrapped around Neptune’s equator, were observed over the course of our observations. No “dark spot” vortices were seen near the equator in HST imaging on 6 and 7 October. The size and pressure of the storm are consistent with moist convection or a planetary-scale wave as the energy source of convective upwelling, but more modeling is required to determine the driver of this equatorial disturbance as well as the triggers for and dynamics of the observed cloud break-up events

Gemini Planet Imager Observations of the AU Microscopii Debris Disk: Asymmetries within One Arcsecond

We present Gemini Planet Imager (GPI) observations of AU Microscopii, a young M dwarf with an edge-on, dusty debris disk. Integral field spectroscopy and broadband imaging polarimetry were obtained during the commissioning of GPI. In our broadband imaging polarimetry observations, we detect the disk only in total intensity and find asymmetries in the morphology of the disk between the southeast and northwest sides. The southeast side of the disk exhibits a bump at 1$''$ (10 AU projected separation) that is three times more vertically extended and three times fainter in peak surface brightness than the northwest side at similar separations. This part of the disk is also vertically offset by 69$\pm$30 mas to the northeast at 1$''$ when compared to the established disk mid-plane and consistent with prior ALMA and Hubble Space Telescope/STIS observations. We see hints that the southeast bump might be a result of detecting a horizontal sliver feature above the main disk that could be the disk backside. Alternatively when including the morphology of the northwest side, where the disk mid-plane is offset in the opposite direction $\sim$50 mas between 0$.''$4 and 1$.''$2, the asymmetries suggest a warp-like feature. Using our integral field spectroscopy data to search for planets, we are 50% complete for $\sim$4 $M_\mathrm{Jup}$ planets at 4 AU. We detect a source, resolved only along the disk plane, that could either be a candidate planetary mass companion or a compact clump in the disk.Comment: 14 pages, 4 figures, accepted by ApJ Letter