From circular paths to elliptic orbits: A geometric approach to Kepler's motion

The hodograph, i.e. the path traced by a body in velocity space, was introduced by Hamilton in 1846 as an alternative for studying certain dynamical problems. The hodograph of the Kepler problem was then investigated and shown to be a circle, it was next used to investigate some other properties of the motion. We here propose a new method for tracing the hodograph and the corresponding configuration space orbit in Kepler's problem starting from the initial conditions given and trying to use no more than the methods of synthetic geometry in a sort of Newtonian approach. All of our geometric constructions require straight edge and compass only.Comment: 9 pages, 4 figure

Non-detection of L-band Line Emission from the Exoplanet HD189733b

We attempt to confirm bright non-local thermodynamic equilibrium (non-LTE) emission from the exoplanet HD 189733b at 3.25 μm, as recently reported by Swain et al. based on observations at low spectral resolving power (λ/δλ ≈ 30). Non-LTE emission lines from gas in an exoplanet atmosphere will not be significantly broadened by collisions, so the measured emission intensity per resolution element must be substantially brighter when observed at high spectral resolving power. We observed the planet before, during, and after a secondary eclipse event at a resolving power λ/δλ = 27, 000 using the NIRSPEC spectrometer on the Keck II telescope. Our spectra cover a spectral window near the peak found by Swain et al., and we compare emission cases that could account for the magnitude and wavelength dependence of the Swain et al. result with our final spectral residuals. To model the expected line emission, we use a general non-equilibrium formulation to synthesize emission features from all plausible molecules that emit in this spectral region. In every case, we detect no line emission to a high degree of confidence. After considering possible explanations for the Swain et al. results and the disparity with our own data, we conclude that an astrophysical source for the putative non-LTE emission is unlikely. We note that the wavelength dependence of the signal seen by Swain et al. closely matches the 2ν_2 band of water vapor at 300 K, and we suggest that an imperfect correction for telluric water is the source of the feature claimed by Swain et al

A Measurement of Water Vapour amid a Largely Quiescent Environment on Europa

Previous investigations proved the existence of local density enhancements in Europas atmosphere, advancing the idea of a possible origination from water plumes. These measurement strategies, however, were sensitive either to total absorption or atomic emissions, which limited the ability to assess the water content. Here we present direct searches for water vapour on Europa spanning dates from February 2016 to May 2017 with the Keck Observatory. Our global survey at infrared wavelengths resulted in non-detections on 16 out of 17 dates, with upper limits below the water abundances inferred from previous estimates. On one date (26 April 2016) we measured 2,095 658 tonnes of water vapour at Europas leading hemisphere. We suggest that the outgassing ls than previously estimated, with only rare localized events of stronger activity

The formation heritage of Jupiter Family Comet 10P/Tempel 2 as revealed by infrared spectroscopy

We present spectral and spatial information for major volatile species in Comet 10P/Tempel 2, based on high-dispersion infrared spectra acquired on UT 2010 July 26 (heliocentric distance Rh = 1.44 AU) and September 18 (Rh = 1.62 AU), following the comet's perihelion passage on UT 2010 July 04. The total production rate for water on July 26 was (1.90 +/- 0.12) x 10^28 molecules s-1, and abundances of six trace gases (relative to water) were: CH3OH (1.58% +/- 0.23), C2H6 (0.39% +/- 0.04), NH3 (0.83% +/- 0.20), and HCN (0.13% +/- 0.02). A detailed analysis of intensities for water emission lines provided a rotational temperature of 35 +/- 3 K. The mean OPR is consistent with nuclear spin populations in statistical equilibrium (OPR = 3.01 +/- 0.18), and the (1-sigma) lower bound corresponds to a spin temperature > 38 K. Our measurements were contemporaneous with a jet-like feature observed at optical wavelengths. The spatial profiles of four primary volatiles display strong enhancements in the jet direction, which favors release from a localized vent on the nucleus. The measured IR continuum is much more sharply peaked and is consistent with a dominant contribution from the nucleus itself. The peak intensities for H2O, CH3OH, and C2H6 are offset by ~200 km in the jet direction, suggesting the possible existence of a distributed source, such as the release of icy grains that subsequently sublimed in the coma. On UT September 18, no obvious emission lines were present in our spectra, nevertheless we obtained a 3-sigma upper limit Q(H2O) < 2.86 x 10^27 molecules s-1

The Volatile Composition of the Split Ecliptic comet 73P/Schwassmann-Wachmann 3: A Comparison of Fragments C and B

The composition of fragments C and B of the Jupiter-family comet 73P/Schwassmann-Wachmann 3 (SW3) was investigated in early April of 2006 at IR wavelengths using high-dispersion echelle spectroscopy. Both fragments were depleted in ethane, and C was depleted in most forms of volatile carbon. In particular, fragment C shows a severe depletion of CH_(3)OH but a "normal" abundance of HCN (which has a similar volatility). Thermal processing is a possible explanation, but since fragment B is perhaps sublimating fresher material because of the frequent outbursts and fragmentation, the observed depletions might have cosmogonic implications. The chemistry of the volatile ices in SW3, like in the Oort Cloud comet C/1999 S4 (LINEAR), may be associated with sublimation of icy mantles from precometary grains followed by subsequent gas-phase chemistry and recondensation

Measuring molecular abundances in comet C/2014 Q2 (Lovejoy) using the APEX telescope

Comet composition provides critical information on the chemical and physical processes that took place during the formation of the Solar system. We report here on millimetre spectroscopic observations of the long-period bright comet C/2014 Q2 (Lovejoy) using the Atacama Pathfinder Experiment (APEX) band 1 receiver between 2015 January UT 16.948 to 18.120, when the comet was at heliocentric distance of 1.30 AU and geocentric distance of 0.53 AU. Bright comets allow for sensitive observations of gaseous volatiles that sublimate in their coma. These observations allowed us to detect HCN, CH3OH (multiple transitions), H2CO and CO, and to measure precise molecular production rates. Additionally, sensitive upper limits were derived on the complex molecules acetaldehyde (CH3CHO) and formamide (NH2CHO) based on the average of the strongest lines in the targeted spectral range to improve the signal-to-noise ratio. Gas production rates are derived using a non-LTE molecular excitation calculation involving collisions with H2O and radiative pumping that becomes important in the outer coma due to solar radiation. We find a depletion of CO in C/2014 Q2 (Lovejoy) with a production rate relative to water of 2 per cent, and relatively low abundances of Q(HCN)/Q(H2O), 0.1 per cent, and Q(H2CO)/Q(H2O), 0.2 per cent. In contrast the CH3OH relative abundance Q(CH3OH)/Q(H2O), 2.2 per cent, is close to the mean value observed in other comets. The measured production rates are consistent with values derived for this object from other facilities at similar wavelengths taking into account the difference in the fields of view. Based on the observed mixing ratios of organic molecules in four bright comets including C/2014 Q2, we find some support for atom addition reactions on cold dust being the origin of some of the molecules.Comment: 10 pages, 7 figures, to be published in MNRA