Modular Solutions to Equations of Generalized Halphen Type

Solutions to a class of differential systems that generalize the Halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group. These functions all uniformize Riemann surfaces of genus zero and have $q$--series with integral coefficients. Rational maps relating these functions are derived, implying subgroup relations between their automorphism groups, as well as symmetrization maps relating the associated differential systems.Comment: PlainTeX 36gs. (Formula for Hecke operator corrected.

Filaments in Galactic Winds Driven by Young Stellar Clusters

The starburst galaxy M82 shows a system of H$\alpha$-emitting filaments which extend to each side of the galactic disk. We model these filaments as the result of the interaction between the winds from a distribution of Super Stellar Clusters (SSCs). We first derive the condition necessary for producing a radiative interaction between the cluster winds (a condition which is met by the SSC distribution of M82). We then compute 3D simulations for SSC wind distributions which satisfy the condition for a radiative interaction, and also for distributions which do not satisfy this condition. We find that the highly radiative models, that result from the interaction of high metallicity cluster winds, produce a structure of H$\alpha$ emitting filaments, which qualitatively agrees with the observations of the M82, while the non-radiative SSC wind interaction models do not produce filamentary structures. Therefore, our criterion for radiative interactions (which depends on the mass loss rate and the terminal velocity of the SSC winds, and the mean separation between SSCs) can be used to predict whether or not an observed galaxy should have associated H$\alpha$ emitting filaments.Comment: 10 pages, 6 Figures. ApJ Accepted, August 7, 200

Irregular conformal blocks, with an application to the fifth and fourth Painlev\'e equations

We develop the theory of irregular conformal blocks of the Virasoro algebra. In previous studies, expansions of irregular conformal blocks at regular singular points were obtained as degeneration limits of regular conformal blocks; however, such expansions at irregular singular points were not clearly understood. This is because precise definitions of irregular vertex operators had not been provided previously. In this paper, we present precise definitions of irregular vertex operators of two types and we prove that one of our vertex operators exists uniquely. Then, we define irregular conformal blocks with at most two irregular singular points as expectation values of given irregular vertex operators. Our definitions provide an understanding of expansions of irregular conformal blocks and enable us to obtain expansions at irregular singular points. As an application, we propose conjectural formulas of series expansions of the tau functions of the fifth and fourth Painlev\'e equations, using expansions of irregular conformal blocks at an irregular singular point.Comment: 26 page

What Controls the Star Formation in Luminous Starburst Mergers ?

In order to understand what controls the star formation process in luminous starburst mergers (e.g., NGC 6240, Arp 220, and so on), we investigate observational properties of two samples of high-luminosity starburst galaxies mapped in CO($J$=1--0) independently using both the Owens Valley Radio Observatory (Scoville et al. 1991) and the IRAM interferometer (Downes & Solomon 1998). We find that the surface density of far-infrared luminosity, $\Sigma$(FIR), is proportional linearly to the H$_2$ surface mass density, $\Sigma$(H$_2$), for the two samples; $\Sigma$(FIR) $\propto \Sigma$(H$_2$)$^{1.01\pm0.06}$ with a correlation coefficient of 0.96. It is often considered that $\Sigma$(FIR) provides a good measure of the star formation rate per unit area, $\Sigma$(SFR). It is also known that molecular gas is dominated in circumnuclear regions in the luminous starburst mergers; i.e., $\Sigma$(gas) $\simeq \Sigma$(H$_2$). Therefore, the above relationship suggests a star formation law; $\Sigma$(SFR) $\propto \Sigma$(gas). We suggest that this star formation law favors the gravitational instability scenario rather than the cloud-cloud collision one.Comment: 14 pages, 2 figures. The Astrophysical Journal (Letters), in pres

Large-scale distributions of mid- and far-infrared emission from the center to the halo of M82 revealed with AKARI

The edge-on starburst galaxy M82 exhibits complicated distributions of gaseous materials in its halo, which include ionized superwinds driven by nuclear starbursts, neutral materials entrained by the superwinds, and large-scale neutral streamers probably caused by a past tidal interaction with M81. We investigate detailed distributions of dust grains and polycyclic aromatic hydrocarbons (PAHs) around M82 to understand their interplay with the gaseous components. We performed mid- (MIR) and far-infrared (FIR) observations of M82 with the Infrared Camera and Far-Infrared Surveyor on board AKARI. We obtain new MIR and FIR images of M82, which reveal both faint extended emission in the halo and very bright emission in the center with signal dynamic ranges as large as five and three orders of magnitude for the MIR and FIR, respectively. We detect MIR and FIR emission in the regions far away from the disk of the galaxy, reflecting the presence of dust and PAHs in the halo of M82. We find that the dust and PAHs are contained in both ionized and neutral gas components, implying that they have been expelled into the halo of M82 by both starbursts and galaxy interaction. In particular, we obtain a tight correlation between the PAH and H$\alpha$ emission, which provides evidence that the PAHs are well mixed in the ionized superwind gas and outflowing from the disk.Comment: 12 pages, 8 figures, accepted for publication in A&

Connection Formulae for Asymptotics of Solutions of the Degenerate Third Painlev\'{e} Equation. I

The degenerate third Painlev\'{e} equation, $u^{\prime \prime} = \frac{(u^{\prime})^{2}}{u} - \frac{u^{\prime}}{\tau} + \frac{1}{\tau}(-8 \epsilon u^{2} + 2ab) + \frac{b^{2}}{u}$, where $\epsilon,b \in \mathbb{R}$, and $a \in \mathbb{C}$, and the associated tau-function are studied via the Isomonodromy Deformation Method. Connection formulae for asymptotics of the general as $\tau \to \pm 0$ and $\pm i0$ solution and general regular as $\tau \to \pm \infty$ and $\pm i \infty$ solution are obtained.Comment: 40 pages, LaTeX2