Detection of Extended Hot Water in the Outflow from NGC 2071

We report the results of spectroscopic mapping observations carried out toward a ~1 min x 1 min region within the northern lobe of the outflow from NGC 2071 using the Infrared Spectrograph (IRS) of the Spitzer Space Telescope. These observations covered the 5.2-37 um spectral region and have led to the detection of a number of ionic, atomic, and molecular lines, including fine-structure emission of Si+, Fe+, S++, S, the S(0)-S(7) pure rotational lines of H2, the R(3) and R(4) transitions of HD, and at least 11 transitions of H2O. In addition, the 6.2, 7.4, 7.6, 7.9, 8.6 and 11.3 um PAH emission bands were also observed and several transitions of OH were tentatively detected. Most of the detected line transitions were strong enough to map including, for the first time, three transitions of hot H2O. We find that: (1) the water emission is extended; (2) the extended emission is aligned with the outflow; and, (3) the spatial distribution of the water emission generally follows that observed for H2. Based on the measured line intensities, we derive an HD abundance relative to H2 of 1.1-1.8 10^-5 and an H2O number density of 12-2 cm^3. The H2 density in the water-emitting region is not well constrained by our observations, but is likely between 3 10^4 and 10^6 cm^3, yielding an H2O abundance relative to H2 of between 2 10^-5 and 6 10^-4. Future observations planned for the Herschel Space Observatory should greatly improve the density estimate, and thus our knowledge of the H2O abundance, for the water-emitting regions reported here. Finally, we note a possible departure from the H2O ortho-to-para ratio of 3:1 expected for water formed in hot post-shocked gas, suggesting that a significant fraction of the water vapor we detect may arise from H2O sputtered from cold dust grains.Comment: 35 pages, 15 figures, 4 tables, accepted for publication in Ap

A Reexamination of Phosphorus and Chlorine Depletions in the Diffuse Interstellar Medium

We present a comprehensive examination of interstellar P and Cl abundances based on an analysis of archival spectra acquired with the Space Telescope Imaging Spectrograph of the Hubble Space Telescope and the Far Ultraviolet Spectroscopic Explorer. Column densities of P II, Cl I, and Cl II are determined for a combined sample of 107 sight lines probing diffuse atomic and molecular gas in the local Galactic interstellar medium (ISM). We reevaluate the nearly linear relationship between the column densities of Cl I and H$_2$, which arises from the rapid conversion of Cl$^+$ to Cl$^0$ in regions where H$_2$ is abundant. Using the observed total gas-phase P and Cl abundances, we derive depletion parameters for these elements, adopting the methodology of Jenkins. We find that both P and Cl are essentially undepleted along sight lines showing the lowest overall depletions. Increasingly severe depletions of P are seen along molecule-rich sight lines. In contrast, gas-phase Cl abundances show no systematic variation with molecular hydrogen fraction. However, enhanced Cl (and P) depletion rates are found for a subset of sight lines showing elevated levels of Cl ionization. An analysis of neutral chlorine fractions yields estimates for the amount of atomic hydrogen associated with the H$_2$-bearing gas in each direction. These results indicate that the molecular fraction in the H$_2$-bearing gas is at least 10% for all sight lines with $\log N({\rm H}_2)\gtrsim18$ and that the gas is essentially fully molecular at $\log N({\rm H}_2)\approx21$.Comment: 44 pages, 21 figures, accepted for publication in Ap

Evaluation of binomial double sums involving absolute values

We show that double sums of the form $$\sum_{i,j=-n} ^{n} |i^sj^t(i^k-j^k)^\beta| \binom {2n} {n+i} \binom {2n} {n+j}$$ can always be expressed in terms of a linear combination of just four functions, namely $\binom {4n}{2n}$, ${\binom {2n}n}^2$, $4^n\binom {2n}n$, and $16^n$, with coefficients that are rational in $n$. We provide two different proofs: one is algorithmic and uses the second author's computer algebra package Sigma; the second is based on complex contour integrals. In many instances, these results are extended to double sums of the above form where $\binom {2n}{n+j}$ is replaced by $\binom {2m}{m+j}$ with independent parameter $m$.Comment: AmS-LaTeX, 42 pages; substantial revision: several additional and more general results, see Proposition 11 and Theorems 15-1

Effective equidistribution and the Sato-Tate law for families of elliptic curves

Extending recent work of others, we provide effective bounds on the family of all elliptic curves and one-parameter families of elliptic curves modulo p (for p prime tending to infinity) obeying the Sato-Tate Law. We present two methods of proof. Both use the framework of Murty-Sinha; the first involves only knowledge of the moments of the Fourier coefficients of the L-functions and combinatorics, and saves a logarithm, while the second requires a Sato-Tate law. Our purpose is to illustrate how the caliber of the result depends on the error terms of the inputs and what combinatorics must be done.Comment: Version 1.1, 24 pages: corrected the interpretation of Birch's moment calculations, added to the literature review of previous results

Dosimetric comparison of extended dose range film with ionization measurements in water and lung equivalent heterogeneous media exposed to megavoltage photons

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/135483/1/acm20025.pd