Modal analysis of gravitational instabilities in nearly Keplerian, counter-rotating collisionless discs

We present a modal analysis of instabilities of counter-rotating, self-gravitating collisionless stellar discs, using the recently introduced modified WKB formulation of spiral density waves for collisionless systems (Gulati \& Saini). The discs are assumed to be axisymmetric and in coplanar orbits around a massive object at the common center of the discs. The mass in both discs is assumed to be much smaller than the mass of the central object. For each disc, the disc particles are assumed to be in near circular orbits. The two discs are coupled to each other gravitationally. The perturbed dynamics of the discs evolves on the order of the precession time scale of the discs, which is much longer than the Keplerian time scale. We present results for the azimuthal wave number $m=1$ and $m=2$, for the full range of disc mass ratio between the prograde and retrograde discs. The eigenspectra are in general complex, therefore all eigenmodes are unstable. Eigenfunctions are radially more compact for $m = 1$ as compared to $m = 2$. Pattern speed of eigenmodes is always prograde with respect to the more massive disc. The growth rate of unstable modes increases with increasing mass fraction in the retrograde disc, and decreases with $m$; therefore $m=1$ instability is likely to play the dominant role in the dynamics of such systems.Comment: 24 pages, 8 figures, 1 tabl

Slow pressure modes in thin accretion discs

Thin accretion discs around massive compact objects can support slow pressure modes of oscillations in the linear regime that have azimuthal wavenumber $m=1$. We consider finite, flat discs composed of barotropic fluid for various surface density profiles and demonstrate--through WKB analysis and numerical solution of the eigenvalue problem--that these modes are stable and have spatial scales comparable to the size of the disc. We show that the eigenvalue equation can be mapped to a Schr\"odinger-like equation. Analysis of this equation shows that all eigenmodes have discrete spectra. We find that all the models we have considered support negative frequency eigenmodes; however, the positive eigenfrequency modes are only present in power law discs, albeit for physically uninteresting values of the power law index $\beta$ and barotropic index $\gamma$.Comment: 9 pages, 7 figures, 1 table, accepted in MNRAS for pulicatio

MATHCAD Geometrie Benutzerhandbuch

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INTERNET ADDICTION AND LEVEL OF ANXIETY AMONG SCHOOL STUDENTS

Internet Addiction is a rising problem among the Indian school students. The purpose of this research was to find out the extent of internet addiction among higher secondary level school students and the anxiety of higher secondary level school students having different levels of internet addiction. The research was conducted on 600 students of CBSE schools in Bhopal. The information was collected by Internet Addiction Scale and General Anxiety Scale. The findings revealed that 10% had normal level of internet addiction, 56% had mild level of internet addiction, 33% had moderate level of internet addiction and just 1% had severe level of internet addiction, the lower the level of internet addiction the lower will be the anxiety of the higher secondary level school students and girls are found to have significantly higher anxiety than boys

Unstable m=1 modes of counter-rotating Keplerian discs

We study the linear m= 1 counter-rotating instability in a two-component, nearly Keplerian disc. Our goal is to understand these slow modes in discs orbiting massive black holes in galactic nuclei. They are of interest not only because they are of large spatial scale and can hence dominate observations but also because they can be growing modes that are readily excited by accretion events. Self-gravity being non-local, the eigenvalue problem results in a pair of coupled integral equations, which we derive for a two-component softened gravity disc. We solve this integral eigenvalue problem numerically for various values of mass fraction in the counter-rotating component. The eigenvalues are in general complex, being real only in the absence of the counter-rotating component, or imaginary when both components have identical surface density profiles. Our main results are as follows: (i) the pattern speed appears to be non-negative, with the growth (or damping) rate being larger for larger values of the pattern speed; (ii) for a given value of the pattern speed, the growth (or damping) rate increases as the mass in the counter-rotating component increases; (iii) the number of nodes of the eigenfunctions decreases with increasing pattern speed and growth rate. Observations of lopsided brightness distributions would then be dominated by modes with the least number of nodes, which also possess the largest pattern speeds and growth rates

Enhanced biomass and steviol glycosides in Stevia rebaudiana treated with phosphate-solubilizing bacteria and rock phosphate

Biofertilizers offer alternative means to promoting cultivation of medicinal plants less dependent on chemical fertilizers. Present study was aimed at evaluating the potential of phosphate-solubilizing bacteria (PSB) Burkholderia gladioli MTCC 10216, B. gladioli MTCC 10217, Enterobacter aerogenes MTCC 10208 and Serratia marcescens MTCC 10238 for utilizing Mussoorie rock phosphate (MRP) to enhance plant growth, and stevioside (ST) and rebaudioside-A (R-A) contents of Stevia rebaudiana. The solubilization of MRP by PSB strains varied from 1.4 to 15.2 lg ml-1, with the highest solubilization by Enterobacter aerogenes 10208. The PSB treatment increased the growth and ST and R-A contents of plants. Plant growth and stevioside contents were more pronounced with plants treated with a mixture of strains and grown in MRP amended soil compared to the unamended soil. The increment in shoot length (47.8%), root length (17.4%), leaf dry weight (164%), stem dry weight (116%), total shoot biomass (136%) resulted in enhanced productivity of ST (291%) and R-A (575%) in plants inoculated with mixture of PSB as compared to the uninoculated plants. The soils of PSB treated plants contained more available P than the soils of uninoculated plants (increase of 86–576%). PSB inoculated plants also recorded higher P content (64–273% increase) compared to uninoculated plants. The PSB strains differed in the extent of rhizosphere colonization, carbon source utilization pattern and whole cell fatty acids methyl esters compositio

Isolation and identification of phosphate solubilizing bacteria able to enhance the growth and aloin-A biosynthesis of Aloe barbadensis Miller

Aloe barbadensis in soil containing tricalcium phosphate (TCP). PSB were identified based on 16S rRNA gene sequencing as Pseudomonas synxantha, Burkholderia gladioli, Enterobacter hormaechei and Serratia marcescens. These PSB solubilized 25–340 μg ml−1 of TCP into the liquid phase. The treatment of plants with individual PSB or mixture of these increased soil available P, P uptake in plants and plant growth. The increase in aloin-A content due to higher plant biomass and unit biomass production was 673%, 294%, 276%, 119% and 108% in plants treated with a PSB consortium, P. synxantha, S. marcescens, B. gladioli, and E. hormaechei in TCP amended soil, respectivel