Technique for Modeling the Gravitational Field of a Galactic Disk

Eckhardt, Donald H.; Pestaña, José Luis G.

doi:10.1086/341745

Cited by 17 publications

(13 citation statements)

References 15 publications

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“…Such singularities have been considered "inconvenient from the point of view of numerical work" by Binney & Tremaine (1987) and "bothersome" by Eckhardt & Pestaña (2002). Methods were suggested to circumvent such singularities by evaluating the radial gradient of potential at a vertical distance z slightly away from z = 0 (cf Binney & Tremaine 1987;Eckhardt & Pestaña 2002), which seem to be somewhat ad hoc by nature and lack of desirable mathematical elegance. On the other hand, it is the axisymmetric mass distribution within an idealized rotating infinitesmally thin disk that has often been of practical interest especially for rotation curve analysis (Toomre 1963).…”

Section: Introductionmentioning

confidence: 99%

“…The elliptic integrals of the first kind and second kind that appear in the radial gradient of potential can become mathematically singular at the midplane (when z = 0) where the radius of the source approaches that of the observation point. Such singularities have been considered "inconvenient from the point of view of numerical work" by Binney & Tremaine (1987) and "bothersome" by Eckhardt & Pestaña (2002). Methods were suggested to circumvent such singularities by evaluating the radial gradient of potential at a vertical distance z slightly away from z = 0 (cf Binney & Tremaine 1987;Eckhardt & Pestaña 2002), which seem to be somewhat ad hoc by nature and lack of desirable mathematical elegance.…”

Section: Introductionmentioning

confidence: 99%

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Modeling the Newtonian dynamics for rotation curve analysis of thin-disk galaxies

Feng

Gallo

2011

Res. Astron. Astrophys.

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We present an efficient, robust computational method for modeling the Newtonian dynamics for rotation curve analysis of thin-disk galaxies. With appropriate mathematical treatments, the apparent numerical difficulties associated with singularities in computing elliptic integrals are completely removed. Using a boundary element discretization procedure, the governing equations are transformed into a linear algebra matrix equation that can be solved by straightforward Gauss elimination in one step without further iterations. The numerical code implemented according to our algorithm can accurately determine the surface mass density distribution in a disk galaxy from a measured rotation curve (or vice versa). For a disk galaxy with a typical flat rotation curve, our modeling results show that the surface mass density monotonically decreases from the galactic center toward periphery, according to Newtonian dynamics. In a large portion of the galaxy, the surface mass density follows an approximately exponential law of decay with respect to the galactic radial coordinate. Yet the radial scale length for the surface mass density seems to be generally larger than that of the measured brightness distribution, suggesting an increasing mass-to-light ratio with the radial distance in a disk galaxy. In a nondimensionalized form, our mathematical system contains a dimensionless parameter which we call the "galactic rotation number" that represents the gross ratio of centrifugal force and gravitational force. The value of this galactic rotation number is determined as part of the numerical solution. Through a systematic computational analysis, we have illustrated that the galactic rotation number remains within ±10% of 1.70 for a wide variety of rotation curves. This implies that the total mass in a disk galaxy is proportional to V 2 0 R g , with V 0 denoting the characteristic rotation velocity (such as the "flat" value in a typical rotation curve) and R g the radius of the galactic disk. The predicted total galactic mass of the Milky Way is in good agreement with the star-count data.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Modeling the Newtonian dynamics for rotation curve analysis of thin-disk galaxies

Feng

Gallo

2011

Res. Astron. Astrophys.

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“…The calculation of the Newtonian gravitational acceleration vector due to a uniform ring or disk is one of fundamental but complicated tasks in the Celestial Mechanics and Dynamical Astronomy (Eckhardt and Pestana 2002;Broucke and Elipe 2005). One example of the problems requiring such calculation is the orbit integration of a test particle moving around a celestial object with a ring structure such as Saturn.…”

Section: Introductionmentioning

confidence: 99%

Precise computation of acceleration due to uniform ring or disk

Fukushima

2010

Celest Mech Dyn Astr

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We describe a precise numerical method to evaluate the acceleration vector of an inverse-square force law caused by a uniform ring or disk. The method consists of two parts. One is rewriting of the analytic expressions of the vector to reduce the loss of information in its computation. The other is a procedure to compute a special linear combination of the complete elliptic integrals of the first and the second kind appeared in the rewritten expressions.

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“…Actually, Gauss had already solved the problem and he used it to state his famous averaging theorem: replacing a perturbing body by an equivalent continuous mass spread over its orbit does not change the secular effects while it removes the periodic terms of the perturbation [67]. See also [68].[69] analytically treated the equivalent problem of calculating the off-axis electric field of a ring of charge. Numerical computation of the acceleration due to a uniform ring can be found in [70].…”

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confidence: 99%

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confidence: 99%

Orbital Perturbations Due to Massive Rings

Iorio

2012

Earth Moon Planets

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We analytically work out the long-term orbital perturbations induced by a homogeneous circular ring of radius R r and mass m r on the motion of a test particle in the cases (I): r > R r and (II): r < R r . In order to extend the validity of our analysis to the orbital configurations of, e.g., some proposed spacecraft-based mission for fundamental physics like LISA and ASTROD, of possible annuli around the supermassive black hole in Sgr A * coming from tidal disruptions of incoming gas clouds, and to the effect of artificial space debris belts around the Earth, we do not restrict ourselves to the case in which the ring and the orbit of the perturbed particle lie just in the same plane. From the corrections ∆̟ (meas) to the standard secular perihelion precessions, recently determined by a team of astronomers for some planets of the Solar System, we infer upper bounds on m r for various putative and known annular matter distributions of natural origin (close circumsolar ring with R r = 0.02 − 0.13 au, dust ring with R r = 1 au, minor asteroids, Trans-Neptunian Objects). We find m r ≤ 1.4 × 10 −4 m ⊕ (circumsolar ring with R r = 0.02 au), m r ≤ 2.6 × 10 −6 m ⊕ (circumsolar ring with R r = 0.13 au), m r ≤ 8.8 × 10 −7 m ⊕ (ring with R r = 1 au), m r ≤ 7.3 × 10 −12 M ⊙ (asteroidal ring with R r = 2.80 au), m r ≤ 1.1×10 −11 M ⊙ (asteroidal ring with R r = 3.14 au), m r ≤ 2.0×10 −8 M ⊙ (TNOs ring with R r = 43 au). In principle, our analysis is valid both for baryonic and nonbaryonic Dark Matter distributions.

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Technique for Modeling the Gravitational Field of a Galactic Disk

Cited by 17 publications

References 15 publications

Modeling the Newtonian dynamics for rotation curve analysis of thin-disk galaxies

Modeling the Newtonian dynamics for rotation curve analysis of thin-disk galaxies

Precise computation of acceleration due to uniform ring or disk

Orbital Perturbations Due to Massive Rings

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