Matchings on infinite graphs

Elek and Lippner (2010) showed that the convergence of a sequence of bounded-degree graphs implies the existence of a limit for the proportion of vertices covered by a maximum matching. We provide a characterization of the limiting parameter via a local recursion defined directly on the limit of the graph sequence. Interestingly, the recursion may admit multiple solutions, implying non-trivial long-range dependencies between the covered vertices. We overcome this lack of correlation decay by introducing a perturbative parameter (temperature), which we let progressively go to zero. This allows us to uniquely identify the correct solution. In the important case where the graph limit is a unimodular Galton-Watson tree, the recursion simplifies into a distributional equation that can be solved explicitly, leading to a new asymptotic formula that considerably extends the well-known one by Karp and Sipser for Erd\"os-R\'enyi random graphs.Comment: 23 page

The 35Cl/37Cl isotopic ratio in dense molecular clouds: HIFI observations of hydrogen chloride towards W3A

We report on the detection with the HIFI instrument on board the Herschel satellite of the two hydrogen chloride isotopologues, H35Cl and H37Cl, towards the massive star-forming region W3A. The J=1-0 line of both species was observed with receiver 1b of the HIFI instrument at 625.9 and 624.9 GHz. The different hyperfine components were resolved. The observations were modeled with a non-local, non-LTE radiative transfer model that includes hyperfine line overlap and radiative pumping by dust. Both effects are found to play an important role in the emerging intensity from the different hyperfine components. The inferred H35Cl column density (a few times 1e14 cm^-2), and fractional abundance relative to H nuclei (~7.5e^-10), supports an upper limit to the gas phase chlorine depletion of ~200. Our best-fit model estimate of the H35Cl/H37Cl abundance ratio is ~2.1+/-0.5, slightly lower, but still compatible with the solar isotopic abundance ratio (~3.1). Since both species were observed simultaneously, this is the first accurate estimation of the [35Cl]/[37Cl] isotopic ratio in molecular clouds. Our models indicate that even for large line opacities and possible hyperfine intensity anomalies, the H35Cl and H37Cl J=1-0 integrated line-intensity ratio provides a good estimate of the 35Cl/37Cl isotopic abundance ratio.Comment: Accepted for publication in Astronomy and Astrophysics (Herschel special issue

Detection of hydrogen fluoride absorption in diffuse molecular clouds with Herschel/HIFI: a ubiquitous tracer of molecular gas

We discuss the detection of absorption by interstellar hydrogen fluoride (HF) along the sight line to the submillimeter continuum sources W49N and W51. We have used Herschel's HIFI instrument in dual beam switch mode to observe the 1232.4762 GHz J = 1 - 0 HF transition in the upper sideband of the band 5a receiver. We detected foreground absorption by HF toward both sources over a wide range of velocities. Optically thin absorption components were detected on both sight lines, allowing us to measure - as opposed to obtain a lower limit on - the column density of HF for the first time. As in previous observations of HF toward the source G10.6-0.4, the derived HF column density is typically comparable to that of water vapor, even though the elemental abundance of oxygen is greater than that of fluorine by four orders of magnitude. We used the rather uncertain N(CH)-N(H2) relationship derived previously toward diffuse molecular clouds to infer the molecular hydrogen column density in the clouds exhibiting HF absorption. Within the uncertainties, we find that the abundance of HF with respect to H2 is consistent with the theoretical prediction that HF is the main reservoir of gas-phase fluorine for these clouds. Thus, hydrogen fluoride has the potential to become an excellent tracer of molecular hydrogen, and provides a sensitive probe of clouds of small H2 column density. Indeed, the observations of hydrogen fluoride reported here reveal the presence of a low column density diffuse molecular cloud along the W51 sight line, at an LSR velocity of ~ 24kms-1, that had not been identified in molecular absorption line studies prior to the launch of Herschel.Comment: 4 pages, 3 figures, A&A Letter special issue, accepted on 07/13/201

Strong absorption by interstellar hydrogen fluoride: Herschel/HIFI observations of the sight-line to G10.6-0.4 (W31C)

We report the detection of strong absorption by interstellar hydrogen fluoride along the sight-line to the submillimeter continuum source G10.6-0.4 (W31C). We have used Herschel's HIFI instrument, in dual beam switch mode, to observe the 1232.4763 GHz J=1-0 HF transition in the upper sideband of the Band 5a receiver. The resultant spectrum shows weak HF emission from G10.6-0.4 at LSR velocities in the range -10 to -3 km/s, accompanied by strong absorption by foreground material at LSR velocities in the range 15 to 50 km/s. The spectrum is similar to that of the 1113.3430 GHz 1(11)-0(00) transition of para-water, although at some frequencies the HF (hydrogen fluoride) optical depth clearly exceeds that of para-H2O. The optically-thick HF absorption that we have observed places a conservative lower limit of 1.6E+14 cm-2 on the HF column density along the sight-line to G10.6-0.4. Our lower limit on the HF abundance, 6E-9 relative to hydrogen nuclei, implies that hydrogen fluoride accounts for between ~ 30 and 100% of the fluorine nuclei in the gas phase along this sight-line. This observation corroborates theoretical predictions that - because the unique thermochemistry of fluorine permits the exothermic reaction of F atoms with molecular hydrogen - HF will be the dominant reservoir of interstellar fluorine under a wide range of conditions.Comment: Accepted for publication in Astronomy and Astrophysics (Herschel special issue). This revised version corrects a typographic error in the HTML abstract, in which the lower limit on the HF abundance (should be 6E-9) was previously misstated. The abstract in the PDF version is correct and the latter has not been modifie

Large atom number dual-species magneto-optical trap for fermionic 6Li and 40K atoms

We present the design, implementation and characterization of a dual-species magneto-optical trap (MOT) for fermionic 6Li and 40K atoms with large atom numbers. The MOT simultaneously contains 5.2x10^9 6Li-atoms and 8.0x10^9 40K-atoms, which are continuously loaded by a Zeeman slower for 6Li and a 2D-MOT for 40K. The atom sources induce capture rates of 1.2x10^9 6Li-atoms/s and 1.4x10^9 40K-atoms/s. Trap losses due to light-induced interspecies collisions of ~65% were observed and could be minimized to ~10% by using low magnetic field gradients and low light powers in the repumping light of both atomic species. The described system represents the starting point for the production of a large-atom number quantum degenerate Fermi-Fermi mixture

Interstellar OH+, H2O+ and H3O+ along the sight-line to G10.6-0.4

We report the detection of absorption lines by the reactive ions OH+, H2O+ and H3O+ along the line of sight to the submillimeter continuum source G10.6$-$0.4 (W31C). We used the Herschel HIFI instrument in dual beam switch mode to observe the ground state rotational transitions of OH+ at 971 GHz, H2O+ at 1115 and 607 GHz, and H3O+ at 984 GHz. The resultant spectra show deep absorption over a broad velocity range that originates in the interstellar matter along the line of sight to G10.6$-$0.4 as well as in the molecular gas directly associated with that source. The OH+ spectrum reaches saturation over most velocities corresponding to the foreground gas, while the opacity of the H2O+ lines remains lower than 1 in the same velocity range, and the H3O+ line shows only weak absorption. For LSR velocities between 7 and 50 kms$^{-1}$ we estimate total column densities of $N$(OH+) $> 2.5 \times 10^{14}$ cm$^{-2}$, $N$(H2O+) $\sim 6 \times 10^{13}$ cm$^{-2}$ and $N$(H3O+) $\sim 4.0 \times 10^{13}$ cm$^{-2}$. These detections confirm the role of O$^+$ and OH$^+$ in initiating the oxygen chemistry in diffuse molecular gas and strengthen our understanding of the gas phase production of water. The high ratio of the OH+ by the H2O+ column density implies that these species predominantly trace low-density gas with a small fraction of hydrogen in molecular form

Herschel/HIFI observations of interstellar OH+ and H2O+ towards W49N: a probe of diffuse clouds with a small molecular fraction

We report the detection of absorption by interstellar hydroxyl cations and water cations, along the sight-line to the bright continuum source W49N. We have used Herschel's HIFI instrument, in dual beam switch mode, to observe the 972 GHz N = 1 - 0 transition of OH+ and the 1115 GHz 1(11) - 0(00) transition of ortho-H2O+. The resultant spectra show absorption by ortho-H2O+, and strong absorption by OH+, in foreground material at velocities in the range 0 to 70 km/s with respect to the local standard of rest. The inferred OH+/H2O+ abundance ratio ranges from ~ 3 to ~ 15, implying that the observed OH+ arises in clouds of small molecular fraction, in the 2 - 8% range. This conclusion is confirmed by the distribution of OH+ and H2O+ in Doppler velocity space, which is similar to that of atomic hydrogen, as observed by means of 21 cm absorption measurements, and dissimilar from that typical of other molecular tracers. The observed OH+/H abundance ratio of a few E-8 suggests a cosmic ray ionization rate for atomic hydrogen of (0.6 - 2.4) E-16 s-1, in good agreement with estimates inferred previously for diffuse clouds in the Galactic disk from observations of interstellar H3+ and other species.Comment: Accepted for publication in A&A Letter

Excitation and Abundance of C3 in star forming cores:Herschel/HIFI observations of the sight-lines to W31C and W49N

We present spectrally resolved observations of triatomic carbon (C3) in several ro-vibrational transitions between the vibrational ground state and the low-energy nu2 bending mode at frequencies between 1654-1897 GHz along the sight-lines to the submillimeter continuum sources W31C and W49N, using Herschel's HIFI instrument. We detect C3 in absorption arising from the warm envelope surrounding the hot core, as indicated by the velocity peak position and shape of the line profile. The sensitivity does not allow to detect C3 absorption due to diffuse foreground clouds. From the column densities of the rotational levels in the vibrational ground state probed by the absorption we derive a rotation temperature (T_rot) of ~50--70 K, which is a good measure of the kinetic temperature of the absorbing gas, as radiative transitions within the vibrational ground state are forbidden. It is also in good agreement with the dust temperatures for W31C and W49N. Applying the partition function correction based on the derived T_rot, we get column densities N(C3) ~7-9x10^{14} cm^{-2} and abundance x(C3)~10^{-8} with respect to H2. For W31C, using a radiative transfer model including far-infrared pumping by the dust continuum and a temperature gradient within the source along the line of sight we find that a model with x(C3)=10^{-8}, T_kin=30-50 K, N(C3)=1.5 10^{15} cm^{-2} fits the observations reasonably well and provides parameters in very good agreement with the simple excitation analysis.Comment: Accepted for publication in Astronomy and Astrophysics (HIFI first results issue

Nitrogen hydrides in interstellar gas: Herschel/HIFI observations towards G10.6-0.4 (W31C)

The HIFI instrument on board the Herschel Space Observatory has been used to observe interstellar nitrogen hydrides along the sight-line towards G10.6-0.4 in order to improve our understanding of the interstellar chemistry of nitrogen. We report observations of absorption in NH N=1-0, J=2-1 and ortho-NH2 1_1,1-0_0,0. We also observed ortho-NH3 1_0-0_0, and 2_0-1_0, para-NH3 2_1-1_1, and searched unsuccessfully for NH+. All detections show emission and absorption associated directly with the hot-core source itself as well as absorption by foreground material over a wide range of velocities. All spectra show similar, non-saturated, absorption features, which we attribute to diffuse molecular gas. Total column densities over the velocity range 11-54 km/s are estimated. The similar profiles suggest fairly uniform abundances relative to hydrogen, approximately 6*10^-9, 3*10^-9, and 3*10^-9 for NH, NH2, and NH3, respectively. These abundances are discussed with reference to models of gas-phase and surface chemistry.Comment: 5 pages, 3 figures, 2 online pages with 2 figures. Accepted for publication in A&A July 6 (Herschel/HIFI special issue