An elementary approach to toy models for D. H. Lehmer's conjecture

In 1947, Lehmer conjectured that the Ramanujan's tau function $\tau (m)$ never vanishes for all positive integers $m$, where $\tau (m)$ is the $m$-th Fourier coefficient of the cusp form $\Delta_{24}$ of weight 12. The theory of spherical $t$-design is closely related to Lehmer's conjecture because it is shown, by Venkov, de la Harpe, and Pache, that $\tau (m)=0$ is equivalent to the fact that the shell of norm $2m$ of the $E_{8}$-lattice is a spherical 8-design. So, Lehmer's conjecture is reformulated in terms of spherical $t$-design. Lehmer's conjecture is difficult to prove, and still remains open. However, Bannai-Miezaki showed that none of the nonempty shells of the integer lattice \ZZ^2 in \RR^2 is a spherical 4-design, and that none of the nonempty shells of the hexagonal lattice $A_2$ is a spherical 6-design. Moreover, none of the nonempty shells of the integer lattices associated to the algebraic integers of imaginary quadratic fields whose class number is either 1 or 2, except for \QQ(\sqrt{-1}) and \QQ(\sqrt{-3}) is a spherical 2-design. In the proof, the theory of modular forms played an important role. Recently, Yudin found an elementary proof for the case of \ZZ^{2}-lattice which does not use the theory of modular forms but uses the recent results of Calcut. In this paper, we give the elementary (i.e., modular form free) proof and discuss the relation between Calcut's results and the theory of imaginary quadratic fields.Comment: 18 page

Imaging Polarimetric Observations of a New Circumstellar Disk System

Few circumstellar disks have been directly observed. Here we use sensitive differential polarimetric techniques to overcome atmospheric speckle noise in order to image the circumstellar material around HD 169142. The detected envelope or disk is considerably smaller than expectations based on the measured strength of the far-IR excess from this system

UBVJHKLM photometry and modeling of R Coronae Borealis

We present the results of UBVJHKLM photometry of R CrB spanning the period from 1976 to 2001. Studies of the optical light curve have shown no evidence of any stable harmonics in the variations of the stellar emission. In the L band we found semi-regular oscillations with the two main periods of ~3.3 yr and 11.9 yr and the full amplitude of ~0.8 mag and ~0.6 mag, respectively. The colors of the warm dust shell (resolved by Ohnaka et al. 2001) are found to be remarkably stable in contrast to its brightness. This indicates that the inner radius is a constant, time-independent characteristic of the dust shell. The observed behavior of the IR light curve is mainly caused by the variation of the optical thickness of the dust shell within the interval \tau(V)= 0.2-0.4. Anticorrelated changes of the optical brightness (in particular with P ~ 3.3 yr) have not been found. Their absence suggests that the stellar wind of R CrB deviates from spherical symmetry. The light curves suggest that the stellar wind is variable. The variability of the stellar wind and the creation of dust clouds may be caused by some kind of activity on the stellar surface. With some time lag, periods of increased mass-loss cause an increase in the dust formation rate at the inner boundary of the extended dust shell and an increase in its IR brightness. We have derived the following parameters of the dust shell (at mean brightness) by radiative transfer modeling: inner dust shell radius r_in ~ 110 R_*, temperature T_dust(r_in) ~ 860 K, dust density \rho_dust(r_in) ~ 1.1x10^{-20} g cm^-3, optical depth \tau(V) ~ 0.32 at 0.55 micron, mean dust formation rate [dM/dt]_dust ~ 3.1x10^-9 M_sun / yr, mass-loss rate [dM/dt]_gas ~ 2.1x10^-7 M_sun / yr, size of the amorphous carbon grains <(~) 0.01 micron, and B-V ~ -0.28.Comment: 9 pages, 6 figures, accepted for publication in A&

Correlation dynamics between electrons and ions in the fragmentation of D2_2 molecules by short laser pulses

We studied the recollision dynamics between the electrons and D$_2^+$ ions following the tunneling ionization of D$_2$ molecules in an intense short pulse laser field. The returning electron collisionally excites the D$_2^+$ ion to excited electronic states from there D$_2^+$ can dissociate or be further ionized by the laser field, resulting in D$^+$ + D or D$^+$ + D$^+$, respectively. We modeled the fragmentation dynamics and calculated the resulting kinetic energy spectrum of D$^+$ to compare with recent experiments. Since the recollision time is locked to the tunneling ionization time which occurs only within fraction of an optical cycle, the peaks in the D$^+$ kinetic energy spectra provides a measure of the time when the recollision occurs. This collision dynamics forms the basis of the molecular clock where the clock can be read with attosecond precision, as first proposed by Corkum and coworkers. By analyzing each of the elementary processes leading to the fragmentation quantitatively, we identified how the molecular clock is to be read from the measured kinetic energy spectra of D$^+$ and what laser parameters be used in order to measure the clock more accurately.Comment: 13 pages with 14 figure

Unified ab initio treatment of attosecond photoionization and Compton scattering

We present a new theoretical approach to attosecond laser-assisted photo- and Compton ionization. Attosecond x-ray absorption and scattering are described by \hat{\mathrsfs{S}}^{(1,2)}-matrices, which are coherent superpositions of "monochromatic" $\hat{S}^{(1,2)}$-matrices in a laser-modified Furry representation. Besides refining the existing theory of the soft x-ray photoelectron attosecond streak camera and spectral phase interferometry (ASC and ASPI), we formulate a theory of hard x-ray photoelectron and Compton ASC and ASPI. The resulting scheme has a simple structure and leads to closed-form expressions for ionization amplitudes. We investigate Compton electron interference in the separable Coulomb-Volkov continuum with both Coulomb and laser fields treated non-perturbatively. We find that at laser-field intensities below 10$^{13}$ Wcm$^{-2}$ normalized Compton lines almost coincide with the lines obtained in the laser-free regime. At higher intensities, attosecond interferences survive integration over electron momenta, and feature prominently in the Compton lines themselves. We define a regime where the electron ground-state density can be measured with controllable accuracy in an attosecond time interval. The new theory provides a firm basis for extracting photo- and Compton electron phases and atomic and molecular wavefunctions from experimental data.Comment: 15 pages, 5 figure

Lithium in the Symbiotic Mira V407 Cyg

We report an identification of the lithium resonance doublet LiI 6708A in the spectrum of V407 Cyg, a symbiotic Mira with a pulsation period of about 745 days. The resolution of the spectra used was R~18500 and the measured equivalent width of the line is ~0.34A. It is suggested that the lithium enrichment is due to hot bottom burning in the intermediate mass AGB variable, although other possible origins cannot be totally ruled out. In contrast to lithium-rich AGB stars in the Magellanic clouds, ZrO 5551A, 6474A absorption bands were not found in the spectrum of V407Cyg. These are the bands used to classify the S-type stars at low-resolution. Although we identified weak ZrO 5718A, 6412A these are not visible in the low-resolution spectra, and we therefore classify the Mira in V407 Cyg as an M type. This, together with other published work, suggests lithium enrichment can precede the third dredge up of s-process enriched material in galactic AGB stars.Comment: 4 pages, 2 figures, to be published in MNRA

Speckle interferometry and radiative transfer modelling of the Wolf-Rayet star WR 118

WR 118 is a highly evolved Wolf-Rayet star of the WC10 subtype surrounded by a permanent dust shell absorbing and re-emitting in the infrared a considerable fraction of the stellar luminosity. We present the first diffraction-limited 2.13micron speckle interferometric observations of WR 118 with 73 mas resolution. The speckle interferograms were obtained with the 6m telescope at the Special Astrophysical Observatory. The two-dimensional visibility function of the object does not show any significant deviation from circular symmetry. The visibility curve declines towards the diffraction cut-off frequency to 0.66 and can be approximated by a linear function. Radiative transfer calculations have been carried out to model the spectral energy distribution, given in the range of 0.5-25micron, and our 2.13micron visibility function, assuming spherical symmetry of the dust shell. Both can be fitted with a model containing double-sized grains (``small'' and ``large'') with the radii of a = 0.05micron and 0.38micron, and a mass fraction of the large grains greater than 65%. Alternatively, a good match can be obtained with the grain size distribution function n(a)~a^-3, with a ranging between 0.005micron and 0.6micron. At the inner boundary of the modelled dust shell (angular diameter (17 +/- 1)mas), the temperature of the smallest grains and the dust shell density are 1750K +/- 100K and (1 +/- 0.2)x10^-19 g/cm^3, respectively. The dust formation rate is found to be (1.3 +/- 0.5)x10^-7 Msol/yr assuming Vwind = 1200 km/s.Comment: 6 pages including 4 PostScript figures, also available from http://www.mpifr-bonn.mpg.de/div/ir-interferometry/publications.html; accepted for publication in Astronomy & Astrophysic