A family of anisotropic integral operators and behaviour of its maximal eigenvalue

We study the family of compact integral operators $\mathbf K_\beta$ in $L^2(\mathbb R)$ with the kernel K_\beta(x, y) = \frac{1}{\pi}\frac{1}{1 + (x-y)^2 + \beta^2\Theta(x, y)}, depending on the parameter $\beta >0$, where $\Theta(x, y)$ is a symmetric non-negative homogeneous function of degree $\gamma\ge 1$. The main result is the following asymptotic formula for the maximal eigenvalue $M_\beta$ of $\mathbf K_\beta$: M_\beta = 1 - \lambda_1 \beta^{\frac{2}{\gamma+1}} + o(\beta^{\frac{2}{\gamma+1}}), \beta\to 0, where $\lambda_1$ is the lowest eigenvalue of the operator $\mathbf A = |d/dx| + \Theta(x, x)/2$. A central role in the proof is played by the fact that $\mathbf K_\beta, \beta>0,$ is positivity improving. The case $\Theta(x, y) = (x^2 + y^2)^2$ has been studied earlier in the literature as a simplified model of high-temperature superconductivity.Comment: 16 page

How do methanol masers manage to appear in the youngest star vicinities and isolated molecular clumps?

General characteristics of methanol (CH3OH) maser emission are summarized. It is shown that methanol maser sources are concentrated in the spiral arms. Most of the methanol maser sources from the Perseus arm are associated with embedded stellar clusters and a considerable portion is situated close to compact HII regions. Almost 1/3 of the Perseus Arm sources lie at the edges of optically identified HII regions which means that massive star formation in the Perseus Arm is to a great extent triggered by local phenomena. A multiline analysis of the methanol masers allows us to determine the physical parameters in the regions of maser formation. Maser modelling shows that class II methanol masers can be pumped by the radiation of the warm dust as well as by free-free emission of a hypercompact region hcHII with a turnover frequency exceeding 100 GHz. Methanol masers of both classes can reside in the vicinity of hcHIIs. Modelling shows that periodic changes of maser fluxes can be reproduced by variations of the dust temperature by a few percent which may be caused by variations in the brightness of the central young stellar object reflecting the character of the accretion process. Sensitive observations have shown that the masers with low flux densities can still have considerable amplification factors. The analysis of class I maser surveys allows us to identify four distinct regimes that differ by the series of their brightest lines.Comment: 8 pages, 4 figures, invited presentation at IAU242 "Astrophysical Masers and their environments

Sources of Radiation in the Early Universe: The Equation of Radiative Transfer and Optical Distances

We have derived the radiative-transfer equation for a point source with a specified intensity and spectrum, originating in the early Universe between the epochs of annihilation and recombination, at redshifts z_\s =10^8\div 10^4. The direct radiation of the source is separated from the diffuse radiation it produces. Optical distances from the source for Thomson scattering and bremsstrahlung absorption at the maximum of the thermal background radiation are calculated as a function of the redshift z.The distances grow sharply with decreasing z, approaching asymptotic values, the absorption distance increasing more slowly and reaching their limiting values at lower z. For the adopted z values, the optical parameters of the Universe can be described in a flat model with dusty material and radiation, and radiative transfer can be treated in a grey approximation.Comment: 14 pages, 2 figure

Methanol in W3(H2O) and Surrounding Regions

We present the results of an interferometric study of 38 millimeter-wave lines of CH3OH in the region around the water maser source W3(H2O) and a region extending about 30" to the south and west of the hydroxyl maser source W3(OH). The methanol emitting region around W3(H2O) has an extent of 2.0" x 1.2" (4400 x 2600 AU). The density is of order 1.e7 cm-3, sufficient to thermalize most of the methanol lines. The kinetic temperature is approximately 140 K and the methanol fractional abundance greater than 1.e-6, indicative of a high degree of grain mantle evaporation. The W3(H2O) source contains sub-structure, with peaks corresponding to the TW source and Wyrowski's B/C, separated by 2500 AU in projection. The kinematics are consistent with these being distinct protostellar cores in a wide binary orbit and a dynamical mass for the region of a few tens of Mo. The extended methanol emission to the southwest of W3(OH) is seen strongly only from the lowest excitation lines and from lines known elsewhere to be class I methanol masers, namely the 84.5 GHz 5(-1)-4(0)E and 95.2 GHz 8(0)-7(1)A+ lines. Within this region there are two compact clumps, which we denote as swA and swB, each about 15" (0.16 pc projected distance) away from W3(OH). Excitation analysis of these clumps indicates the presence of lines with inverted populations but only weak amplification. The sources swA and swB appear to have kinetic temperatures of order 50-100 K and densities of order 1.e5 - 1.e6 cm-3. The methanol fractional abundance for the warmer clump is of order 1.e-7, suggestive of partial grain mantle evaporation. The clumping occurs on mass scales of order 1 Mo.Comment: 28 pages including 6 figures and 4 tables, accepted by Ap

Non-equilibrium excitation of methanol in Galactic molecular clouds: multi-transitional observations at 2 mm

We observed 14 methanol transitions near lambda=2 mm in Galactic star-forming regions. Broad, quasi-thermal J(0)-J(-1)E methanol lines near 157 GHz were detected toward 73 sources. Together with the 6(-1)-5(0)E and 5(-2)-6(-1)E lines at 133 GHz and the 7(1)-7(0)E line at 165 GHz, they were used to study the methanol excitation. In the majority of the observed objects, the Class I 6(-1)-5(0)E transition is inverted, and the Class II 5(-2)-6(-1)E and 6(0)-6(-1)E transitions are overcooled. This is exactly as predicted by models of low gain Class I masers. The absence of the inversion of Class II transitions 5(-2)-6(-1)E and 6(0)-6(-1)E means that quasi-thermal methanol emission in all objects arises in areas without a strong radiation field, which is required for the inversion.Comment: 23 pages paper (uses aasms4.sty), 12 pages tables (uses apjpt4.sty), 10 Jpeg figures, submitted to the ApJ

Molecular gas in high-mass filament WB673

We studied the distribution of dense gas in a filamentary molecular cloud containing several dense clumps. The center of the filament is given by the dense clump WB673. The clumps are high-mass and intermediate-mass star-forming regions. We observed CS(2-1), 13CO(1-0), C18O(1-0) and methanol lines at 96GHz toward WB673 with the Onsala Space Observatory 20-m telescope. We found CS(2-1) emission in the inter-clump medium so the clumps are physically connected and the whole cloud is indeed a filament. Its total mass is $10^4$ M$_{\odot}$ and mass-to-length ratio is 360 M$_{\odot}$pc$^{-1}$ from 13CO(1-0) data. Mass-to-length ratio for the dense gas is $3.4-34$ M$_{\odot}$pc$^{-1}$ from CS(2-1) data. The PV-diagram of the filament is V-shaped. We estimated physical conditions in the molecular gas using methanol lines. Location of the filament on the sky between extended shells suggests that it could be a good example to test theoretical models of formation of the filaments via multiple compression of interstellar gas by supersonic waves