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Extremely low-mass white dwarf stars (ELMs) are M < 0.3 M-circle dot helium-core white dwarfs born either as a result of a common-envelope phase or after a stable Roche lobe overflow episode in a multiple system. The Universe is not old enough for ELMs to have formed through single-star evolution channels. As remnants of binary evolution, ELMs can shed light onto the poorly understood phase of common-envelope evolution and provide constraints to the physics of mass accretion. Most known ELMs will merge in less than a Hubble time, providing an important contribution to the signal to be detected by upcoming space-based gravitational wave detectors. There are currently less than 150 known ELMs; most were selected by colour, focusing on hot objects, in a magnitude-limited survey of the Northern hemisphere only. Recent theoretical models have predicted a much larger space density for ELMs than estimated observationally based on this limited sample. In order to perform meaningful comparisons with theoretical models and test their predictions, a larger well-defined sample is required. In this work, we present a catalogue of ELM candidates selected from the second data release of Gaia (DR2). We have used predictions from theoretical models and analysed the properties of the known sample to map the space spanned by ELMs in the Gaia Hertzsprung-Russell diagram. Defining a set of colour cuts and quality flags, we have obtained a final sample of 5762 ELM candidates down to T-eff approximate to 5000 K.
On 2019 April 25.346 and 26.640 UT the Laser Interferometer Gravitational-Wave Observatory (LIGO) and Virgo gravitational-wave (GW) observatory announced the detection of the first candidate events in Observing Run 3 that contained at least one neutron star (NS). S190425z is a likely binary neutron star (BNS) merger at d(L) = 156 +/- 41 Mpc, while S190426c is possibly the first NS-black hole (BH) merger ever detected, at d(L) = 377 +/- 100 Mpc, although with marginal statistical significance. Here we report our optical follow-up observations for both events using the MMT 6.5 m telescope, as well as our spectroscopic follow-up of candidate counterparts (which turned out to be unrelated) with the 4.1 m SOAR telescope. We compare to publicly reported searches, explore the overall areal coverage and depth, and evaluate those in relation to the optical/near-infrared (NIR) kilonova emission from the BNS merger GW170817, to theoretical kilonova models, and to short gamma-ray burst (SGRB) afterglows. We find that for a GW170817-like kilonova, the partial volume covered spans up to about 40% for S190425z and 60% for S190426c. For an on-axis jet typical of SGRBs, the search effective volume is larger, but such a configuration is expected in at most a few percent of mergers. We further find that wide-field gamma-ray and X-ray limits rule out luminous on-axis SGRBs, for a large fraction of the localization regions, although these searches are not sufficiently deep in the context of the gamma-ray emission from GW170817 or off-axis SGRB afterglows. The results indicate that some optical follow-up searches are sufficiently deep for counterpart identification to about 300 Mpc, but that localizations better than 1000 deg(2) are likely essential.