ZONAL TOROIDAL HARMONIC EXPANSIONS OF EXTERNAL GRAVITATIONAL FIELDS FOR RING-LIKE OBJECTS

Journal Article · Mon Aug 01 04:00:00 EDT 2016 · Astronomical Journal (Online)

DOI:https://doi.org/10.3847/0004-6256/152/2/35· OSTI ID:22662989

Fukushima, Toshio ^[1]

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1. National Astronomical Observatory, Ohsawa, Mitaka, Tokyo 181-8588 (Japan)

We present an expression of the external gravitational field of a general ring-like object with axial and plane symmetries such as oval toroids or annular disks with an arbitrary density distribution. The main term is the gravitational field of a uniform, infinitely thin ring representing the limit of zero radial width and zero vertical height of the object. The additional term is derived from a zonal toroidal harmonic expansion of a general solution of Laplace’s equation outside the Brillouin toroid of the object. The special functions required are the point value and the first-order derivative of the zonal toroidal harmonics of the first kind, namely, the Legendre function of the first kind of half integer degree and an argument that is not less than unity. We developed a recursive method to compute them from two pairs of seed values explicitly expressed by some complete elliptic integrals. Numerical experiments show that appropriately truncated expansions converge rapidly outside the Brillouin toroid. The truncated expansion can be evaluated so efficiently that, for an oval toroid with an exponentially damping density profile, it is 3000–10,000 times faster than the two-dimensional numerical quadrature. A group of the Fortran 90 programs required in the new method and their sample outputs are available electronically.

• OSTI ID: 22662989
• Journal Information: Astronomical Journal (Online), Journal Name: Astronomical Journal (Online) Journal Issue: 2 Vol. 152; ISSN 1538-3881
• Country of Publication: United States
• Language: English

Related Subjects

79 ASTRONOMY AND ASTROPHYSICS
ACCRETION DISKS
DAMPING
DENSITY
EXPANSION
GRAVITATIONAL FIELDS
HARMONICS
LEGENDRE POLYNOMIALS
MECHANICS
PROTOPLANETS
QUADRATURES
SYMMETRY
TWO-DIMENSIONAL CALCULATIONS