We present an analysis of the stability spectrum for all stationary periodic solutions to the sine-Gordon equation. An analytical expression for the spectrum is given. From this expression, various quantitative and qualitative results about the spectrum are derived. Specifically, the solution parameter space is shown to be split into regions of distinct qualitative behavior of the spectrum, in one of which the solutions are stable. Additional results on the stability of solutions with respect to perturbations of an integer multiple of the solution period are given.
We used the Tycho-Gaia Astrometric Solution catalogue, part of Gaia Data Release 1, to search for candidate astrometric microlensing events expected to occur within the remaining lifetime of the Gaia satellite. Our search yielded one promising candidate. We predict that the nearby DQ type white dwarf LAWD 37 (WD 1142-645) will lens a background star and will reach closest approach on November 11th 2019 (± 4 days) with impact parameter 380 ± 10 mas. This will produce an apparent maximum deviation of the source position of 2.8 ± 0.1 mas. In the most propitious circumstance, Gaia will be able to determine the mass of LAWD 37 to ∼ 3%. This mass determination will provide an independent check on atmospheric models of white dwarfs with helium rich atmospheres, as well as tests of white dwarf mass radius relationships and evolutionary theory.
We present extensive optical photometry of the afterglow of GRB 221009A. Our data cover 0.9–59.9 days from the time of Swift and Fermi gamma-ray burst (GRB) detections. Photometry in rizy-band filters was collected primarily with Pan-STARRS and supplemented by multiple 1–4 m imaging facilities. We analyzed the Swift X-ray data of the afterglow and found a single decline rate power law f(t) ∝ t −1.556±0.002 best describes the light curve. In addition to the high foreground Milky Way dust extinction along this line of sight, the data favor additional extinction to consistently model the optical to X-ray flux with optically thin synchrotron emission. We fit the X-ray-derived power law to the optical light curve and find good agreement with the measured data up to 5−6 days. Thereafter we find a flux excess in the riy bands that peaks in the observer frame at ∼20 days. This excess shares similar light-curve profiles to the Type Ic broad-lined supernovae SN 2016jca and SN 2017iuk once corrected for the GRB redshift of z = 0.151 and arbitrarily scaled. This may be representative of an SN emerging from the declining afterglow. We measure rest-frame absolute peak AB magnitudes of M g = −19.8 ± 0.6 and M r = − 19.4 ± 0.3 and M z = −20.1 ± 0.3. If this is an SN component, then Bayesian modeling of the excess flux would imply explosion parameters of M ej = 7.1 − 1.7 + 2.4 M ⊙, M Ni = 1.0 − 0.4 + 0.6 M ⊙, and v ej = 33,900 − 5700 + 5900 km s−1, for the ejecta mass, nickel mass, and ejecta velocity respectively, inferring an explosion energy of E kin ≃ 2.6–9.0 × 1052 erg.
We used Gaia Data Release 2 to search for upcoming photometric microlensing events, identifying two candidates with high amplification. In the case of candidate 1, a spectrum of the lens (l1) confirms it is a usdM3 subdwarf with mass ≈ 0.11M , while the event reaches maximum amplification of 20 +20 −10 mmag on November 3rd 2019 (±1d). For candidate 2, the lens (l2) is a metal-poor M dwarf with mass ≈ 0.38M derived from SED fitting, and maximum amplification of 10 +40 −10 mmag occurs on June 3rd 2019 (±4d). This permits a new algorithm for mass inference on the microlens. Given the predicted time, the photometric lightcurve of these events can be densely sampled by ground-based telescopes. The lightcurve is a function of the unknown lens mass, together with 8 other parameters for all of which Gaia provides measurements and uncertainties. Leveraging this prior information on the source and lens provided by Gaia's astrometric solution, and assuming that a ground-based campaign can provide 50 measurements at mmag precision, we show for example that the mass of l1 can be recovered to within 20 per cent (68 per cent confidence limit).
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