A short proof of a result of Katz and West

We give a short proof of a result due to Katz and West: Let $R$ be a Noetherian ring and $I_1,\ldots,I_t$ ideals of $R$. Let $M$ and $N$ be finitely generated $R$-modules and $N' \subseteq N$ a submodule. For every fixed $i \ge 0$, the sets $\mathrm{Ass}_R\left( \mathrm{Ext}_R^i(M, N/I_1^{n_1}\cdots I_t^{n_t} N') \right)$ and $\mathrm{Ass}_R\left( \mathrm{Tor}_i^R(M, N/I_1^{n_1}\cdots I_t^{n_t} N') \right)$ are independent of $(n_1,\ldots,n_t)$ for all sufficiently large $n_1,\ldots,n_t$.Comment: 3 pages, revised versio

Suppression of gravitational instabilities by dominant dark matter halo in low surface brightness galaxies

The low surface brightness galaxies are gas-rich and yet have a low star formation rate, this is a well-known puzzle. The spiral features in these galaxies are weak and difficult to trace, although this aspect has not been studied much. These galaxies are known to be dominated by the dark matter halo from the innermost regions. Here we do a stability analysis for the galactic disc of UGC 7321, a low surface brightness, superthin galaxy, for which the various observational input parameters are available. We show that the disc is stable against local, linear axisymmetric and non-axisymmetric perturbations. The Toomre Q parameter values are found to be large (>> 1) mainly due to the low disc surface density and the high rotation velocity resulting due to the dominant dark matter halo, which could explain the observed low star formation rate. For the stars-alone case, the disc shows finite swing amplification but the addition of dark matter halo suppresses that amplification almost completely. Even the inclusion of the low-dispersion gas which constitutes a high disc mass fraction does not help in causing swing amplification. This can explain why these galaxies do not show strong spiral features. Thus the dynamical effect of a halo that is dominant from inner regions can naturally explain why star formation and spiral features are largely suppressed in low surface brightness galaxies, making these different from the high surface brightness galaxies.Comment: 7 pages, 2 figures, 1 table, accepted for publication in MNRA

Role of gas in supporting grand spiral structure

The density wave theory for the grand-design two-armed spiral pattern in galaxies is successful in explaining several observed features. However, the long-term persistence of this spiral structure is a serious problem since the group transport would destroy it within about a billion years as shown in a classic paper by Toomre. In this paper we include the low velocity dispersion component, namely gas, on an equal footing with stars in the formulation of the density wave theory, and obtain the dispersion relation for this coupled system. We show that the inclusion of gas makes the group transport slower by a factor of few, thus allowing the pattern to persist longer - for several billion years. Though still less than the Hubble time, this helps in making the spiral structure more long-lived. Further we show that addition of gas is essential to get a stable wave for the observed pattern speed for the Galaxy, which otherwise is not possible for a one-component stellar disc.Comment: 6 pages, 3 figures, 1 table, accepted for publication in MNRA

Dwarf irregular galaxies with extended HI gas disks: Suppression of small-scale spiral structure by dark matter halo

Dwarf irregular galaxies with extended HI disk distributions, such as DDO 154, allow measurement of rotation curves, hence deduction of dark matter halo properties to large radial distances, up to several times the optical radius. These galaxies contain a huge reservoir of dark matter halo, which dominates over most of disk. We study the effect of the dark matter halo on small-scale spiral features by carrying out the local, non-axisymmetric perturbation analysis in the disks of five such late-type, gas-rich dwarf irregular galaxies, namely, DDO 154, NGC 3741, DDO 43, NGC 2366, and DDO 168 which host a dense and compact dark matter halo. We show that when the gas disk is treated alone, it allows a finite swing amplification; which would result in small-scale spiral structure in the outer gas disk, but the addition of dark matter halo in the analysis results in a higher Toomre Q parameter which prevents the amplification almost completely. This trend is also seen to be true in regions inside the optical radius. This implies absence of strong small-scale spiral arms in these galaxies, which is in agreement with observations. Hence despite being gas-rich, and in fact having gas as the main baryonic component, these galaxies cannot support small-scale spiral structure which would otherwise have been expected in normal gas-rich galaxies.Comment: 30 pages, 6 figures, 1 table, accepted for publication in New Astronom

Electrical Conductivity at the Core of a Magnetar

An expression for the electrical conductivity at the core of a magnetar is derived using Boltzmann kinetic equation with the relaxation time approximation. The rates for the relevant scattering processes, e.g., electron-electron and electron-proton are evaluated in presence of strong quantizing magnetic fields using tree level diagrams. It is found that in presence of a strong quantizing magnetic field, electrical conductivity behaves like a second rank tensor. However, if the zeroth Landau levels are only occupied by the charged particles, it again behaves like a scaler of a one dimensional system.Comment: REVTEX File, 4 .eps figures (included