A new equation for the mid-plane potential of power law discs. II. Exact solutions and approximate formulae

Details A new equation for the mid-plane potential of power law discs. II. Exact solutions and approximate formulae A new equation for the mid-plane potential of power law discs. II. Exact solutions and approximate formulae

2008-08-12

arxiv.org/abs/0808.1634v1

Astronomy and Astrophysics

Astrophysics

Accepted for publication in A&A, 13 pages, 8 figures

Scientific paper

10.1051/0004-6361:200809682

The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs in which the surface density is a power-law function of the radius R with exponent s (Hur\'e & Hersant 2007) is solved exactly in terms of infinite series. The formal solution of the ODE is derived and then converted into a series representation by expanding the elliptic integral of the first kind over its modulus before analytical integration. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index 1+s, and two infinite series of the variables R and 1/R. The convergence of the series can be accelerated, enabling the construction of reliable approximations. At the lowest-order, the potential inside large astrophysical discs (s ~ -1.5 +/- 1) is described by a very simple formula whose accuracy (a few percent typically) is easily increased by considering successive orders through a recurrence. A basic algorithm is given. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account.

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