Three generalised negative Binomial Distributions

We extend the velocity channel analysis (VCA), introduced by Lazarian & Pogosyan, of the intensity fluctuations in the velocity slices of position-position-velocity (PPV) spectroscopic data from Doppler broadened lines to study statistical anisotropy of the underlying velocity and density that arises in a turbulent medium from the presence of magnetic field. In particular, we study analytically how the anisotropy of the intensity correlation in the channel maps changes with the thickness of velocity channels. In agreement with the earlier VCA studies we find that the anisotropy in the thick channels reflects the anisotropy of the density field, while the relative contribution of density and velocity fluctuations to the thin velocity channels depends on the density spectral slope. We show that the anisotropies arising from Alfvén, slow and fast magnetohydrodynamical modes are different, in particular, the anisotropy in PPV created by fast modes is opposite to that created by Alfvén and slow modes, and this can be used to separate their contributions. We successfully compare our results with the recent numerical study of the PPV anisotropies measured with synthetic observations. We also extend our study to the medium with self-absorption as well as to the case of absorption lines. In addition, we demonstrate how the studies of anisotropy can be performed using interferometers.

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“…To evaluate this integral, we will use generalised Gegenbauer polynomial expansion (Plunkett & Jain 1975) defined as…”

Section: Appendix A: Turbulence Statistics In Ppv Spacementioning

confidence: 99%

Extending velocity channel analysis for studying turbulence anisotropies

Kandel

Lazarian²,

Pogosyan

2016

Mon. Not. R. Astron. Soc.

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“…The GGD can be defined as a mix-parameter transformation allowing the mean number of clusters in a generalized Hermite set-up (the Poisson part of the Poisson-Binomial law) to vary according to a gamma distribution in the population (Plunkett & Jain, 1975). The Gegenbauer distribution is obtained with m = 2, and corresponds to the parameter-mix of a Hermite distribution with an extra, gamma-distributed, parameter.…”

Section: 3mentioning

confidence: 99%

“…Plunkett & Jain (1975) obtained estimators for the parameters of the Gegenbauer distribution based on the method of moments. Medhi & Borah (1984) provided estimation procedures for the parameters of the GGD based on its first three and two moments.…”

Section: 3mentioning

confidence: 99%

“…The generalized Hermite (GHD) and generalized Gegenbauer (GGD) distributions can be constructed as functions of sums of correlated Poisson random variables (rv) and have been used to model epidemiological and linguistic data (Gupta & Jain (1974), Plunkett & Jain (1975), Medhi & Borah (1984), Pustet & Altmann (2005)). Both distributions owe their names to the orthogonal polynomials related to their probability generating functions (pgf) and belong to the class of extended generalized hypergeometric probability distributions (EGHPD) proposed by Kumar (2002).…”

Section: Introductionmentioning

confidence: 99%

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Some remarks on the generalized Hermite and generalized Gegenbauer probability distributions and their applications

Cortina‐Borja¹

2007

Exact Methods in the Study of Language and Text

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“…Otherwise, use the transformation of the form (3.7) based on eigenvalues and eigenvectors of the adjusted matrix. Plunkett and Jain (1975) developed the Gegenbauer distribution which has the p.g.f.…”

Section: Applications Of the Transformation To Other Data Setsmentioning

confidence: 99%