Lambert W Function for Applications in Physics

The Lambert W(x) function and its possible applications in physics are presented. The actual numerical implementation in C++ consists of Halley's and Fritsch's iterations with initial approximations based on branch-point expansion, asymptotic series, rational fits, and continued-logarithm recursion.Comment: 9 pages, 12 figures. Extended version of arXiv:1003.1628, updated link to source

Some Special Cases in the Stability Analysis of Multi-Dimensional Time-Delay Systems Using The Matrix Lambert W function

This paper revisits a recently developed methodology based on the matrix Lambert W function for the stability analysis of linear time invariant, time delay systems. By studying a particular, yet common, second order system, we show that in general there is no one to one correspondence between the branches of the matrix Lambert W function and the characteristic roots of the system. Furthermore, it is shown that under mild conditions only two branches suffice to find the complete spectrum of the system, and that the principal branch can be used to find several roots, and not the dominant root only, as stated in previous works. The results are first presented analytically, and then verified by numerical experiments

"Retired" Planet Hosts: Not So Massive, Maybe Just Portly After Lunch

Studies of the planet abundance as a function of stellar mass have suggested a strong increase in the frequency of planet occurrence around stars more massive than $1.5 M_\odot$, and that such stars are deficit in short period planets. These planet searches have relied on giant stars for a sample of high mass stars, which are hostile to precision Doppler measurements due to rotation and activity while on the main sequence. This paper considers the observed $v\sin i$ and observationally inferred mass for exoplanet hosting giants with the $v\sin i$ of distribution of field stars, which show discrepancies that can be explained by erroneous mass determinations of some exoplanet host stars. By comparison with an expected mass distribution constructed from integrating isochrones, it is shown that the exoplanet hosts are inconsistent with a population of massive stars. These stars are more likely to have originated from a main sequence population of late F/early G dwarfs with mass 1.0--$1.2 M_\odot$, only slightly more more massive than the typical FGK dwarfs with Doppler detected planets. The deficit of short period planets is most likely explained by tidal capture. The planet abundance difference requires either a steeper increase in planet frequency with mass than previously thought or a high rate of false positives due to signals of stellar origin. The measurement of photospheric Carbon isotope ratios is suggested as a method to discriminate whether this sample of giant stars is significantly more massive than the population of FGK dwarfs with Doppler detected planets.Comment: Accepted for publication in Astrophysical Journal Letter

Carbon and Oxygen abundances across the Hertzsprung gap

We derived atmospheric parameters and spectroscopic abundances for C and O for a large sample of stars located in the Hertzsprung gap in the Hertzsprung-Russell Diagram in order to detect chemical peculiarities and get a comprehensive overview of the population of stars in this evolutionary state. We have observed and analyzed high resolution spectra (R = 60 000) of 188 stars in the mass range 2 - 5 Msun with the 2.7 m Harlan J. Smith Telescope at the McDonald Observatory including 28 stars previously identified as Am/Ap stars. We find that the C and O abundances of the majority of stars in the Hertzsprung gap are in accordance with abundances derived for local lower mass dwarfs but detect expected peculiarities for the Am/Ap stars. The C and O abundances of stars with Teff < 6500 K are slightly lower than for the hotter objects but the C/O ratio is constant in the analyzed temperature domain. No indication of an alteration of the C and O abundances of the stars by mixing during the evolution across the Hertzsprung gap could be found before the homogenization of their atmospheres by the first dredge-up.Comment: 13 pages, 12 figures, accepted for publication in the Astrophysical Journa

Compulsory Deep Mixing of 3He and CNO Isotopes in the Envelopes of low-mass Red Giants

Three-dimensional stellar modeling has enabled us to identify a deep-mixing mechanism that must operate in all low mass giants. This mixing process is not optional, and is driven by a molecular weight inversion created by the 3He(3He,2p)4He reaction. In this paper we characterize the behavior of this mixing, and study its impact on the envelope abundances. It not only eliminates the problem of 3He overproduction, reconciling stellar and big bang nucleosynthesis with observations, but solves the discrepancy between observed and calculated CNO isotope ratios in low mass giants, a problem of more than 3 decades' standing. This mixing mechanism, which we call `$\delta\mu$-mixing', operates rapidly (relative to the nuclear timescale of overall evolution, ~ 10^8 yrs) once the hydrogen burning shell approaches the material homogenized by the surface convection zone. In agreement with observations, Pop I stars between 0.8 and 2.0\Msun develop 12C/13C ratios of 14.5 +/- 1.5, while Pop II stars process the carbon to ratios of 4.0 +/- 0.5. In stars less than 1.25\Msun, this mechanism also destroys 90% to 95% of the 3He produced on the main sequence.Comment: Final accepted version (submitted to Astrophys J in Jan 2007...

The Chemical Compositions of the SRd Variable Stars-- II. WY Andromedae, VW Eridani, and UW Librae

Chemical compositions are derived from high-resolution spectra for three stars classed as SRd variables in the General Catalogue of Variable Stars. These stars are shown to be metal-poor supergiants: WY And with [Fe/H] = -1.0, VW Eri with [Fe/H] = -1.8, and UW Lib with [Fe/H] = -1.2. Their compositions are identical to within the measurement errors with the compositions of subdwarfs, subgiants, and less evolved giants of the same FeH. The stars are at the tip of the first giant branch or in the early stages of evolution along the asymptotic giant branch (AGB). There is no convincing evidence that these SRd variables are experiencing thermal pulsing and the third dredge-up on the AGB. The SRds appear to be the cool limit of the sequence of RV Tauri variables.Comment: 14 pages, 1 figure, 4 table

Oscillator Strengths for B-X, C-X, and E-X Transitions in Carbon Monoxide

Band oscillator strengths for electronic transitions in CO were obtained at the Synchrotron Radiation Center of the University of Wisconsin-Madison. Our focus was on transitions that are observed in interstellar spectra with the Far Ultraviolet Spectroscopic Explorer; these transitions are also important in studies of selective isotope photodissociation where fractionation among isotopomers can occur. Absorption from the ground state (X ^1Sigma^+ v'' = 0) to A ^1Pi (v'= 5), B ^1Sigma^+ (v' = 0, 1), C ^1Sigma^+ (v' = 0, 1), and E ^1Pi (v' = 0) was measured. Fits to the A - X (5, 0) band, whose oscillator strength is well known, yielded the necessary column density and excitation temperature. These parameters were used in a least-squares fit of the observed profiles for the transitions of interest to extract their band oscillator strengths. Our oscillator strengths are in excellent agreement with results from recent experiments using a variety of techniques. This agreement provides the basis for a self-consistent set of f-values at far ultraviolet wavelengths for studies of interstellar (and stellar) CO.Comment: 22 pages, 3 figures, ApJS (in press

Rubidium in Metal-Deficient Disk and Halo Stars

We report the first extensive study of stellar Rb abundances. High-resolution spectra have been used to determine, or set upper limits on, the abundances of this heavy element and the associated elements Y, Zr, and Ba in 44 dwarfs and giants with metallicities spanning the range -2.0 <[Fe/H] < 0.0. In metal-deficient stars Rb is systematically overabundant relative to Fe; we find an average [Rb/Fe] of +0.21 for the 32 stars with [Fe/H] < -0.5 and measured Rb. This behavior contrasts with that of Y, Zr, and Ba, which, with the exception of three new CH stars (HD 23439A and B and BD +5 3640), are consistently slightly deficient relative to Fe in the same stars; excluding the three CH stars, we find the stars with [Fe/H] < -0.5 have average [Y/Fe], [Zr/Fe], and [Ba/Fe] of --0.19 (24 stars), --0.12 (28 stars), and --0.06 (29 stars), respectively. The different behavior of Rb on the one hand and Y, Zr, and Ba on the other can be attributed in part to the fact that in the Sun and in these stars Rb has a large r-process component while Y, Zr, and Ba are mostly s-process elements with only small r-process components. In addition, the Rb s-process abundance is dependent on the neutron density at the s-processing site. Published observations of Rb in s-process enriched red giants indicate a higher neutron density in the metal-poor giants. These observations imply a higher s-process abundance for Rb in metal-poor stars. The calculated combination of the Rb r-process abundance, as estimated for the stellar Eu abundances, and the s-process abundance as estimated for red giants accounts satisfactorily for the observed run of [Rb/Fe] with [Fe/H].Comment: 23 pages, 5 tables, 7 figure

Spectroscopic Study of IRAS 19285+0517(PDS 100): A Rapidly Rotating Li-Rich K Giant

We report on photometry and high-resolution spectroscopy for IRAS 19285+0517. The spectral energy distribution based on visible and near-IR photometry and far-IR fluxes shows that the star is surrounded by dust at a temperature of $T_{\rm {d}}$ $\sim$ 250 K. Spectral line analysis shows that the star is a K giant with a projected rotational velocity $v sin i$ = 9 $\pm$ 2 km s$^{-1}$. We determined the atmospheric parameters: $T_{\rm {eff}}$ = 4500 K, log $g$ = 2.5, $\xi_{t}$ = 1.5 km s$^{-1}$, and [Fe/H] = 0.14 dex. The LTE abundance analysis shows that the star is Li-rich (log $\epsilon$(Li) = 2.5$\pm$0.15), but with essentially normal C, N, and O, and metal abundances. Spectral synthesis of molecular CN lines yields the carbon isotopic ratio $^{12}$C/$^{13}$C = 9 $\pm$3, a signature of post-main sequence evolution and dredge-up on the RGB. Analysis of the Li resonance line at 6707 \AA for different ratios $^{6}$Li/$^{7}$Li shows that the Li profile can be fitted best with a predicted profile for pure $^{7}$Li. Far-IR excess, large Li abundance, and rapid rotation suggest that a planet has been swallowed or, perhaps, that an instability in the RGB outer layers triggered a sudden enrichment of Li and caused mass-loss.Comment: To appear in AJ; 40 pages, 9 figure