The gravitational field of a disk

Krough, F. T.; Ng, Edward W.; Snyder, W. Van

doi:10.1007/bf01230419

Cited by 23 publications

(19 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, all the analytical estimates do not properly match the numerical solutions close to the star. Binney & Tremaine (2008), needs to be computed in a different way, for example, in that suggested in Krough et al (1982). Unfortunately, the equation for the force given in Krough et al (1982) only works for a disk with constant density.…”

Section: Comparison Between the Different Analytical Equationsmentioning

confidence: 99%

“…Binney & Tremaine (2008), needs to be computed in a different way, for example, in that suggested in Krough et al (1982). Unfortunately, the equation for the force given in Krough et al (1982) only works for a disk with constant density. This formalism could be integrated to obtain an equation for the force outside a disk with variable density, but it would probably be a more efficient approach to start from the integral of Eq.…”

Section: Comparison Between the Different Analytical Equationsmentioning

confidence: 99%

See 1 more Smart Citation

Pericenter precession induced by a circumstellar disk on the orbit of massive bodies: comparison between analytical predictions and numerical results

Fontana¹,

Marzari²

2016

A&A

View full text Add to dashboard Cite

Context. Planetesimals and planets embedded in a circumstellar disk are dynamically perturbed by the disk gravity. It causes an apsidal line precession at a rate that depends on the disk density profile and on the distance of the massive body from the star. Aims. Different analytical models are exploited to compute the precession rate of the perihelion˙ . We compare them to verify their equivalence, in particular after analytical manipulations performed to derive handy formulas, and test their predictions against numerical models in some selected cases. Methods. The theoretical precession rates were computed with analytical algorithms found in the literature using the Mathematica symbolic code, while the numerical simulations were performed with the hydrodynamical code FARGO. Results. For low-mass bodies (planetesimals) the analytical approaches described in Binney & Tremaine (2008, Galactic Dynamics, p. 96), Ward (1981, and Silsbee & Rafikov (2015a, ApJ, 798, 71) are equivalent under the same initial conditions for the disk in terms of mass, density profile, and inner and outer borders. They also match the numerical values computed with FARGO away from the outer border of the disk reasonably well. On the other hand, the predictions of the classical Mestel disk (Mestel 1963, MNRAS, 126, 553) for disks with p = 1 significantly depart from the numerical solution for radial distances beyond one-third of the disk extension because of the underlying assumption of the Mestel disk is that the outer disk border is equal to infinity. For massive bodies such as terrestrial and giant planets, the agreement of the analytical approaches is progressively poorer because of the changes in the disk structure that are induced by the planet gravity. For giant planets the precession rate changes sign and is higher than the modulus of the theoretical value by a factor ranging from 1.5 to 1.8. In this case, the correction of the formula proposed by Ward (1981) to account for the presence of a gap is a better approximation, at least in predicting a positive precession rate. Conclusions. Analytical modeling of the precession rate of massive bodies embedded in a circumstellar disk is accurate for planetesimals and terrestrial planets, but it becomes inaccurate when Jupiter-sized planets are considered. The changes they induce in the disk reverse the precession rate and increase it by more than 50%. This discrepancy can be partly solved by including the effects of a gap in the disk density profile in the analytical equations, as done in Ward (1981).

show abstract

Section: Comparison Between the Different Analytical Equationsmentioning

confidence: 99%

Section: Comparison Between the Different Analytical Equationsmentioning

confidence: 99%

Pericenter precession induced by a circumstellar disk on the orbit of massive bodies: comparison between analytical predictions and numerical results

Fontana¹,

Marzari²

2016

A&A

View full text Add to dashboard Cite

show abstract

“…This potential has been already derived by Krough et al (1982) and by Lass and Blitzer (1983). Nonetheless, the formula given there does not represent the potential function at every point in the space for which the potential function has a real finite value.…”

Section: The Annular Disk and Its Potential Functionmentioning

confidence: 96%

“…Some works in literature deal with the computation of the gravitational potential of a massive disk, like Krough et al (1982) or Lass and Blitzer (1983). They provide expressions that are mathematically correct, but not appropriated for numeric evaluation and do not cover the whole space.…”

mentioning

confidence: 99%

Computation of families of periodic orbits and bifurcations around a massive annulus

Tresaco

Elipe

Riaguas

2011

Astrophys Space Sci

View full text Add to dashboard Cite

This paper studies the main features of the dynamics around a planar annular disk. It is addressed an appropriated closed expression of the gravitational potential of a massive disk, which overcomes the difficulties found in previous works in this matter concerning its numerical treatment. This allows us to define the differential equations of motion that describes the motion of a massless particle orbiting the annulus. We describe the computation methods proposed for the continuation of uni-parametric families of periodic orbits, these algorithms have been applied to analyze the dynamics around a massive annulus by means of a description of the main families of periodic orbits found, their bifurcations and linear stability.

show abstract

“…Again, there are not many options because we lack formulae for the potential of the hollow cylinder, for the cone, and for the piece of spherical shell (see Table 1). Fortunately, there is the formula for the potential of the circular disk, i.e., a closed-form for 4 δ K(k)R dR (Durand 1953;Krough et al 1982;Lass & Blitzer 1983;Huré 2012). We can therefore deduce an axially symmetrical hyperkernel κ R by exchanging the role of P and P (see also Appendix C).…”

Section: Axial Symmetrymentioning

confidence: 99%

Exact, singularity-free recasting of the Newtonian potential in continuous media

Huré¹

2013

A&A

View full text Add to dashboard Cite

The gravitational potential is a key function involved in many astrophysical problems. Its evaluation inside continuous media from Newton's law is known to be challenging because of the diverging kernel 1/|r − r |. This difficulty is generally treated with avoidance techniques (e.g. multipole expansions, softening length), which are themselves not without drawbacks. In this article, we present a new path that basically fixes the point-mass singularity problem in systems with at least two dimensions. It consists of recasting the gravitational potential ψ in an equivalent integro-differential form,where (q 1 , q 2 ) is a pair of independent spatial variables (linear and/or angular), f is a known function, and H is an auxiliary scalar function. In contrast with ψ, this "hyperpotential" H is the convolution of the mass density with a finite amplitude kernel κ. We show that closed-form expressions for κ can be directly deduced from the potential of homogeneous sheets. We then give a few formulae appropriate to the Cartesian, cylindrical and spherical coordinate systems, including axial symmetry. The method is essentially not limited, either on the geometry of the source or on the distribution, and its implementation is straightforward. Several tests based upon simple quadrature/differentiation schemes are presented (the homogeneous rectangular sheet, cuboid and disk, the Maclaurin disk and a truncated Lane-Emden solution). Compared with a direct summation, the extra computational cost is low and the gain is real: no truncated series, no free parameter, and a relative accuracy better than 1% for typically 16 nodes per spatial direction using the most basic numerical schemes.

show abstract

The gravitational field of a disk

Cited by 23 publications

References 8 publications

Pericenter precession induced by a circumstellar disk on the orbit of massive bodies: comparison between analytical predictions and numerical results

Pericenter precession induced by a circumstellar disk on the orbit of massive bodies: comparison between analytical predictions and numerical results

Computation of families of periodic orbits and bifurcations around a massive annulus

Exact, singularity-free recasting of the Newtonian potential in continuous media

Contact Info

Product

Resources

About