On optimal truncation of divergent series solutions of nonlinear differential systems; Berry smoothing

We prove that for divergent series solutions of nonlinear (or linear) differential systems near a generic irregular singularity, the common prescription of summation to the least term is, if properly interpreted, meaningful and correct, and we extend this method to transseries solutions. In every direction in the complex plane at the singularity (Stokes directions {\em not} excepted) there exists a nonempty set of solutions whose difference from the ``optimally'' (i.e., near the least term) truncated asymptotic series is of the same (exponentially small) order of magnitude as the least term of the series. There is a family of generalized Borel summation formulas $\mathcal{B}$ which commute with the usual algebraic and analytic operations (addition, multiplication, differentiation, etc). We show that there is exactly one of them, $\mathcal{B}_0$, such that for any formal series solution $\tilde{f}$, $\mathcal{B}_0(\tilde{f})$ differs from the optimal truncation of $\tilde{f}$ by at most the order of the least term of $\tilde{f}$. We show in addition that the Berry (1989) smoothing phenomenon is universal within this class of differential systems. Whenever the terms ``beyond all orders'' {\em change} in crossing a Stokes line, these terms vary smoothly on the Berry scale $\arg(x)\sim |x|^{-1/2}$ and the transition is always given by the error function; under the same conditions we show that Dingle's rule of signs for Stokes transitions holds

The Arecibo HII Region Discovery Survey

We report the detection of radio recombination line emission (RRL) using the Arecibo Observatory at X-band (9GHz, 3cm) from 37 previously unknown HII regions in the Galactic zone 66 deg. > l > 31 deg. and |b| < 1 deg. This Arecibo HII Region Discovery Survey (Arecibo HRDS) is a continuation of the Green Bank Telescope (GBT) HRDS. The targets for the Arecibo HRDS have spatially coincident 24 micron and 20 cm emission of a similar angular morphology and extent. To take advantage of Arecibo's sensitivity and small beam size, sources in this sample are fainter, smaller in angle, or in more crowded fields compared to those of the GBT HRDS. These Arecibo nebulae are some of the faintest HII regions ever detected in RRL emission. Our detection rate is 58%, which is low compared to the 95% detection rate for GBT HRDS targets. We derive kinematic distances to 23 of the Arecibo HRDS detections. Four nebulae have negative LSR velocities and are thus unambiguously in the outer Galaxy. The remaining sources are at the tangent point distance or farther. We identify a large, diffuse HII region complex that has an associated HI and 13CO shell. The ~90 pc diameter of the G52L nebula in this complex may be the largest Galactic HII region known, and yet it has escaped previous detection.Comment: Accepted to ApJ Data can be found here: http://go.nrao.edu/hrd

The Green Bank Telescope H II Region Discovery Survey: IV. Helium and Carbon Recombination Lines

The Green Bank Telescope H II Region Discovery Survey (GBT HRDS) found hundreds of previously unknown Galactic regions of massive star formation by detecting hydrogen radio recombination line (RRL) emission from candidate H II region targets. Since the HRDS nebulae lie at large distances from the Sun, they are located in previously unprobed zones of the Galactic disk. Here we derive the properties of helium and carbon RRL emission from HRDS nebulae. Our target sample is the subset of the HRDS that has visible helium or carbon RRLs. This criterion gives a total of 84 velocity components (14% of the HRDS) with helium emission and 52 (9%) with carbon emission. For our highest quality sources, the average ionic He-4+/H+ abundance ratio by number, , is 0.068 +/- 0.023 (1-sigma). This is the same ratio as that measured for the sample of previously known Galactic H II regions. Nebulae without detected helium emission give robust y+ upper limits. There are 5 RRL emission components with y+ less than 0.04 and another 12 with upper limits below this value. These H II regions must have either a very low He-4 abundance or contain a significant amount of neutral helium. The HRDS has 20 nebulae with carbon RRL emission but no helium emission at its sensitivity level. There is no correlation between the carbon RRL parameters and the 8 microns mid-infrared morphology of these nebulae.Comment: Accepted to ApJ. The survey website can be found here: http://go.nrao.edu/hrd

Radio continuum observations of local star-forming galaxies using the Caltech Continuum Backend on the Green Bank Telescope

We observed radio continuum emission in 27 local (D < 70 Mpc) star-forming galaxies with the Robert C. Byrd Green Bank Telescope between 26 GHz and 40 GHz using the Caltech Continuum Backend. We obtained detections for 22 of these galaxies at all four sub-bands and four more marginal detections by taking the average flux across the entire bandwidth. This is the first detection (full or marginal) at these frequencies for 22 of these galaxies. We fit spectral energy distributions (SEDs) for all of the four-sub-band detections. For 14 of the galaxies, SEDs were best fit by a combination of thermal free-free and nonthermal synchrotron components. Eight galaxies with four-sub-band detections had steep spectra that were only fit by a single nonthermal component. Using these fits, we calculated supernova rates, total number of equivalent O stars, and star formation rates within each ~23 arcsecond beam. For unresolved galaxies, these physical properties characterize the galaxies' recent star formation on a global scale. We confirm that the radio-far-infrared correlation holds for the unresolved galaxies' total 33 GHz flux regardless of their thermal fractions, though the scatter on this correlation is larger than that at 1.4 GHz. In addition, we found that for the unresolved galaxies, there is an inverse relationship between the ratio of 33 GHz flux to total far-infrared flux and the steepness of the galaxy's spectral index between 1.4 GHz and 33 GHz. This relationship could be an indicator of the timescale of the observed episode of star formation.Comment: 36 pages, 9 figures; accepted for publication in ApJ. First and second author affiliation updated to reflect departmental name chang

The Electron Temperature Gradient in the Galactic Disk

We derive the electron temperature gradient in the Galactic disk using a sample of HII regions that spans Galactocentric distances 0--17 kpc. The electron temperature was calculated using high precision radio recombination line and continuum observations for more than 100 HII regions. Nebular Galactocentric distances were calculated in a consistent manner using the radial velocities measured by our radio recombination line survey. The large number of nebulae widely distributed over the Galactic disk together with the uniformity of our data provide a secure estimate of the present electron temperature gradient in the Milky Way. Because metals are the main coolants in the photoionized gas, the electron temperature along the Galactic disk should be directly related to the distribution of heavy elements in the Milky Way. Our best estimate of the electron temperature gradient is derived from a sample of 76 sources for which we have the highest quality data. The present gradient in electron temperature has a minimum at the Galactic Center and rises at a rate of 287 +/- 46 K/kpc. There are no significant variations in the value of the gradient as a function of Galactocentric radius or azimuth. The scatter we find in the HII region electron temperatures at a given Galactocentric radius is not due to observational error, but rather to intrinsic fluctuations in these temperatures which are almost certainly due to fluctuations in the nebular heavy element abundances. Comparing the HII region gradient with the much steeper gradient found for planetary nebulae suggests that the electron temperature gradient evolves with time, becoming flatter as a consequence of the chemical evolution of the Milky Way's disk.Comment: 43 pages, 9 figures (accepted for publication in the ApJ

HII Region Ionization of the Interstellar Medium: A Case Study of NGC 7538

Using data from the Green Bank Telescope, we analyze the radio continuum (free-free) and radio recombination line (RRL) emission of the compact HII region NGC 7538 (Sharpless 158). We detect extended radio continuum and hydrogen RRL emission beyond the photodissociation region (PDR) toward the north and east, but a sharp decrease in emission toward the south and west. This indicates that a non-uniform PDR morphology is affecting the amount of radiation "leaking" through the PDR. The strongest carbon RRL emission is found in the western PDR that appears to be dense. We compute a leaking fraction $f_R = 15 \pm 5$ % of the radio continuum emission measured in the plane of the sky which represents a lower limit when accounting for the three-dimensional geometry of the region. We detect an average $^4\textrm{He}^+/\textrm{H}^+$ abundance ratio by number of $0.088 \pm 0.003$ inside the HII region and a decrease in this ratio with increasing distance from the region beyond the PDR. Using Herschel Space Observatory data, we show that small dust temperature enhancements to the north and east of NGC 7538 coincide with extended radio emission, but that the dust temperature enhancements are mostly contained within a second PDR to the east. Unlike the giant HII region W43, the radiation leaking from NGC 7538 seems to only affect the local ambient medium. This suggests that giant HII regions may have a large effect in maintaining the ionization of the interstellar medium.Comment: Accepted for publication in ApJ (15 pages, 10 figures, 2 tables

Diffuse Ionized Gas in the Milky Way Disk

We analyze the diffuse ionized gas (DIG) in the first Galactic quadrant from l=18deg to 40deg using radio recombination line (RRL) data from the Green Bank Telescope. These data allow us to distinguish DIG emission from HII region emission and thus study the diffuse gas essentially unaffected by confusion from discrete sources. We find that the DIG has two dominant velocity components, one centered around 100km/s associated with the luminous HII region W43, and the other centered around 45km/s not associated with any large HII region. Our analysis suggests that the two velocity components near W43 may be caused by non-circular streaming motions originating near the end of the Galactic bar. At lower Galactic longitudes, the two velocities may instead arise from gas at two distinct distances from the Sun, with the most likely distances being ~6kpc for the 100km/s component and ~12kpc for the 45km/s component. We show that the intensity of diffuse Spitzer GLIMPSE 8.0um emission caused by excitation of polyaromatic hydrocarbons (PAHs) is correlated with both the locations of discrete HII regions and the intensity of the RRL emission from the DIG. This implies that the soft ultra-violet photons responsible for creating the infrared emission have a similar origin as the harder ultra-violet photons required for the RRL emission. The 8.0um emission increases with RRL intensity but flattens out for directions with the most intense RRL emission, suggesting that PAHs are partially destroyed by the energetic radiation field at these locations.Comment: Accepted for publication in ApJ (16 pages, 11 figures, 2 tables

Diffuse Ionized Gas in the Milky Way Disk

We analyze the diffuse ionized gas (DIG) in the first Galactic quadrant from l=18deg to 40deg using radio recombination line (RRL) data from the Green Bank Telescope. These data allow us to distinguish DIG emission from HII region emission and thus study the diffuse gas essentially unaffected by confusion from discrete sources. We find that the DIG has two dominant velocity components, one centered around 100km/s associated with the luminous HII region W43, and the other centered around 45km/s not associated with any large HII region. Our analysis suggests that the two velocity components near W43 may be caused by non-circular streaming motions originating near the end of the Galactic bar. At lower Galactic longitudes, the two velocities may instead arise from gas at two distinct distances from the Sun, with the most likely distances being ~6kpc for the 100km/s component and ~12kpc for the 45km/s component. We show that the intensity of diffuse Spitzer GLIMPSE 8.0um emission caused by excitation of polyaromatic hydrocarbons (PAHs) is correlated with both the locations of discrete HII regions and the intensity of the RRL emission from the DIG. This implies that the soft ultra-violet photons responsible for creating the infrared emission have a similar origin as the harder ultra-violet photons required for the RRL emission. The 8.0um emission increases with RRL intensity but flattens out for directions with the most intense RRL emission, suggesting that PAHs are partially destroyed by the energetic radiation field at these locations.Comment: Accepted for publication in ApJ (16 pages, 11 figures, 2 tables

Untangling the Recombination Line Emission from HII Regions with Multiple Velocity Components

HII regions are the ionized spheres surrounding high-mass stars. They are ideal targets for tracing Galactic structure because they are predominantly found in spiral arms and have high luminosities at infrared and radio wavelengths. In the Green Bank Telescope HII Region Discovery Survey (GBT HRDS) we found that >30% of first Galactic quadrant HII regions have multiple hydrogen radio recombination line (RRL) velocities, which makes determining their Galactic locations and physical properties impossible. Here we make additional GBT RRL observations to determine the discrete HII region velocity for all 117 multiple-velocity sources within 18deg. < l < 65deg. The multiple-velocity sources are concentrated in the zone 22deg. < l < 32deg., coinciding with the largest regions of massive star formation, which implies that the diffuse emission is caused by leaked ionizing photons. We combine our observations with analyses of the electron temperature, molecular gas, and carbon recombination lines to determine the source velocities for 103 discrete H II regions (88% of the sample). With the source velocities known, we resolve the kinematic distance ambiguity for 47 regions, and thus determine their heliocentric distances.Comment: 44 pages, 5 figures, 16 pages of tables; Accepted by ApJ