On the augmented density of a spherical anisotropic dynamic system

An, J.

doi:10.1111/j.1365-2966.2011.18324.x

Cited by 7 publications

(6 citation statements)

References 15 publications

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“…If the AD is multiplicatively separable into functions of the potential and the radius dependences like , this results in the necessary condition stated by An (2011b), that is, for the radius part of the AD. We have also discovered a few equivalent statements of this condition, notably the complete monotonicity of the function defined in as well as . The similar argument for the potential part of a separable AD on the other hand recovers the conditions derived by Van Hese et al (2011) and An (2011a), which are further generalized with fractional calculus to indicate that for all accessible Ψ is necessary if or there exists such that is well defined or such that is non‐zero and finite. The df of an escapable system with a separable AD may be inverted from the latter utilizing the inverse Laplace transform as in . The non‐negativity of the resulting df is guaranteed if its Laplace transformation is completely monotonic.…”

Section: Discussionmentioning

confidence: 62%

“…Ciotti & Morganti (2010b) have essentially hypothesized that the necessary conditions of Ciotti & Morganti (2010a), which concerns the behaviour of the potential‐dependent parts of augmented densities, may be applicable to any system for which the potential and radial dependences of the augmented density are multiplicatively separable. This has been subsequently proven by Van Hese, Baes & Dejonghe (2011) and An (2011a) whereas An (2011b) was able to find necessary conditions on the radius‐dependent parts of separable augmented densities, which results in the constraints on the behaviour of the anisotropy parameter that can be consistent with separable augmented densities.…”

Section: Introductionmentioning

confidence: 83%

“…An (2011a) has shown that the Abel transformation of the augmented moment function results in an integral transformation of the df similar to but with different powers on K and L 2 . This is generalized by means of fractional calculus (), that is, for any pair of non‐negative reals ξ≥μ≥ 0, where Γ( x ) is the gamma function and the operators a I x λ and a D x λ are as defined in .…”

Section: Models For Spherical Dynamical Systemsmentioning

confidence: 99%

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Phase-space consistency of stellar dynamical models determined by separable augmented densities

An¹,

Hese²,

Baes³

2012

Monthly Notices of the Royal Astronomical Society

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Assuming the separable augmented density, it is always possible to construct a distribution function of a spherical population with any given density and anisotropy. We consider under what conditions the distribution constructed as such is in fact non-negative everywhere in the accessible phase space. We first generalize the known necessary conditions on the augmented density using fractional calculus. The condition on the radius part R(r 2 ) (whose logarithmic derivative is the anisotropy parameter) is equivalent to the complete monotonicity of w −1 R(w −1 ). The condition on the potential part on the other hand is given by its derivative up to any order not greater than 3 2 − β 0 being non-negative where β 0 is the central anisotropy parameter. We also derive a specialized inversion formula for the distribution from the separable augmented density, which leads to sufficient conditions on separable augmented densities for the non-negativity of the distribution. These last conditions are generalizations of the similar condition derived earlier for the generalized Cuddeford system to arbitrary separable systems.

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Section: Discussionmentioning

confidence: 62%

Section: Introductionmentioning

confidence: 83%

Section: Models For Spherical Dynamical Systemsmentioning

confidence: 99%

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Phase-space consistency of stellar dynamical models determined by separable augmented densities

An¹,

Hese²,

Baes³

2012

Monthly Notices of the Royal Astronomical Society

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“…This is then placed into the practical context of a joint likelihood analysis of the variance and kurtosis in dwarf spheroidal galaxies. We evaluate the contribution by Lokas (2002) in establishing a model for the kurtosis that may be used to lift the degeneracy ( Lokas et al 2005) and extend the method to general anisotropy as proposed by An (2011b) with the separable augmented density system.…”

Section: Introductionmentioning

confidence: 99%

“…To simplify the mathematical description of the higher order Jeans equations there has been much success in the literature since the advent of the augmented density formalism by Dejonghe (1986). Whilst an application (Dejonghe 1987;Baes & van Hese 2007) of this method has generally been limited to models (Plummer 1915;Hernquist 1990) with particularly simple potential-density pairs, the recent work of An (2011b) demonstrates for generic density and anisotropy that a separable system (Ciotti & Morganti 2010) solves the Jeans degeneracy problem completely by specifying moments at all orders with the potential and anisotropy parameter alone. This is however by no means a general solution and without a strong physical motivation its practical use is difficult to evaluate.…”

Section: Introductionmentioning

confidence: 99%

Analytical solutions to the mass-anisotropy degeneracy with higher order Jeans analysis: a general method

Richardson¹,

Fairbairn²

2013

Monthly Notices of the Royal Astronomical Society

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The Jeans analysis is often used to infer the total density of a system by relating the velocity moments of an observable tracer population to the underlying gravitational potential. This technique has recently been applied in the search for Dark Matter in objects such as dwarf spheroidal galaxies where the presence of Dark Matter is inferred via stellar velocities. A precise account of the density is needed to constrain the expected gamma ray flux from DM self-annihilation and to distinguish between cold and warm dark matter models. Unfortunately the traditional method of fitting the second order Jeans equation to the tracer dispersion suffers from an unbreakable degeneracy of solutions due to the unknown velocity anisotropy of the projected system. To tackle this degeneracy one can appeal to higher moments of the Jeans equation. By introducing an analog to the Binney anisotropy parameter at fourth order, β ′ we create a framework that encompasses all solutions to the fourth order Jeans equations rather than the restricted range imposed by the separable augmented density. The condition β ′ = f (β) ensures that the degeneracy is lifted and we interpret the separable augmented density system as the order-independent case β ′ = β. For a generic choice of β ′ we present the line of sight projection of the fourth moment and how it could be incorporated into a joint likelihood analysis of the dispersion and kurtosis. The framework is then extended to all orders such that constraints may be placed to ensure a physically positive distribution function. Having presented the mathematical framework, we then use it to make preliminary analyses of simulated dwarf spheroidal data leading to interesting results which strongly motivate further study.

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Constraints on Velocity Anisotropy of Spherical Systems With Separable Augmented Densities

An¹

2011

ApJ

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If the augmented density of a spherical anisotropic system is assumed to be multiplicatively separable to functions of the potential and the radius, the radial function, which can be completely specified by the behavior of the anisotropy parameter alone, also fixes the anisotropic ratios of every higher-order velocity moment. It is inferred from this that the non-negativity of the distribution function necessarily limits the allowed behaviors of the radial function. This restriction is translated into the constraints on the behavior of the anisotropy parameter. We find that not all radial variations of the anisotropy parameter satisfy these constraints and thus that there exist anisotropy profiles that cannot be consistent with any separable augmented density.

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On the augmented density of a spherical anisotropic dynamic system

Cited by 7 publications

References 15 publications

Phase-space consistency of stellar dynamical models determined by separable augmented densities

Phase-space consistency of stellar dynamical models determined by separable augmented densities

Analytical solutions to the mass-anisotropy degeneracy with higher order Jeans analysis: a general method

Constraints on Velocity Anisotropy of Spherical Systems With Separable Augmented Densities

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