A study of the orbits of the logarithmic potential for galaxies

Valluri, S. R.; Wiegert, Paul; Drozd, John; Silva, M. Da

doi:10.1111/j.1365-2966.2012.22071.x

Cited by 19 publications

(18 citation statements)

References 34 publications

(55 reference statements)

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“…Systems with −2 < α < 1 are of relevant interest in quantum mechanics and astrophysics, for instance in describing the galactic dynamics in presence of power law densities or massive black-holes and in modelling the gravitational lensing, see [1,2]. Also, the logarithmic potential appears in particle physics [3,4] and in a model for the dynamics of vortex filaments in ideal fluid, see e.g.…”

Section: Introductionmentioning

confidence: 99%

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A study of the apsidal angle and a proof of monotonicity in the logarithmic potential case

Castelli¹

2014

Journal of Mathematical Analysis and Applications

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This paper concerns the behaviour of the apsidal angle for orbits of central force system with homogeneous potential of degree −2 ≤ α ≤ 1 and logarithmic potential. We derive a formula for the apsidal angle as a fixed end-points integral and we study the derivative of the apsidal angle with respect to the angular momentum . The monotonicity of the apsidal angle as function of is discussed and it is proved in the logarithmic potential case.

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Section: Introductionmentioning

confidence: 99%

“…A study for the logarithmic case using the p − ellipse approximation and the Lambert W function is presented in [2]. Touma and Tremaine in [1], by means of the Mellin transform, provide an asymptotic series on for any α ∈ [−2, 1].…”

Section: Introductionmentioning

confidence: 99%

A study of the apsidal angle and a proof of monotonicity in the logarithmic potential case

Castelli¹

2014

Journal of Mathematical Analysis and Applications

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“…It has since then been applied to numerous scientific problems, with physics leading the way. Valluri, Jeffrey & Corless (2000) and Caillol (2003) review some applications in physics; later applications have ranged from projectile range (Morales 2005) and capillary rise (Fries & Dreyer 2008) to solar winds (Cranmer 2004) and the dynamics of galaxies (Valluri et al 2012).…”

Section: Introductionmentioning

confidence: 99%

The Lambert W function in ecological and evolutionary models

Lehtonen

2016

Methods Ecol Evol

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Summary The Lambert W function is a mathematical function with a long history, but which was named and rigorously defined relatively recently. It is closely related to the logarithmic function and arises from many models in the natural sciences, including a surprising number of problems in ecology and evolution. I describe the basic properties of the function and present examples of its application to models of ecological and evolutionary processes. The Lambert W function makes it possible to solve explicitly several models where this is not possible with elementary functions. I present examples of such models from existing literature, as well as novel models. Solving models explicitly with the Lambert W function can provide deeper insight and a new point of view on a biological problem. Explicit solutions with the Lambert W function are easily amenable to further mathematical operations, such as differentiation and integration. These advantages apply to a wide range of models, from the marginal value theorem to population growth rates and disease epidemics.

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“…Analytic approximation have been employed to study the radial equation and the apsidal angle for small eccentricity. In the logarithmic potential case Touma and Tremaine [23] used the epyciclic approximation and found the value ∆ 0 θ = π/ √ 2 in the limit of e → 0, Struck in [25] and Valluri et al in [16] used p-ellipses curves and the Lambert W function to approximate the radial solution up to order e 2 and the apsidal angle to high accuracy.…”

Section: Introductionmentioning

confidence: 99%

The monotonicity of the apsidal angle in power-law potential systems

Castelli

2015

Journal of Mathematical Analysis and Applications

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In a central force system the apsidal angle is the angle at the centre of force between two consecutive apsis and measures the precession rate of the line of apsis. The apsidal angle has applications in different fields and the Newton's apsidal precession theorem has been extensively studied by astronomers, physicist and mathematicians. The perihelion precession of Mercury, the dynamics of galaxies, the vortex dynamics, the JWKB quantisation condition are some examples where the apsidal angle is of interest.In case of eccentric orbits and forces far from inverse square, numerical investigations provide evidence of the monotonicity of the apsidal angle with respect to the orbit parameters, such as the orbit eccentricity. However, no proof of this statement is available.In this paper central force systems with f (r) ∼ µr −(α+1) are considered. We prove that for any −2 < α < 1 the apsidal angle is a monotonic function of the orbital eccentricity, or equivalently of the angular momentum. As a corollary, the conjecture stating the absence of isolated cases of zero precession is proved.

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A study of the orbits of the logarithmic potential for galaxies

Cited by 19 publications

References 34 publications

A study of the apsidal angle and a proof of monotonicity in the logarithmic potential case

A study of the apsidal angle and a proof of monotonicity in the logarithmic potential case

The Lambert W function in ecological and evolutionary models

The monotonicity of the apsidal angle in power-law potential systems

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