Volume 490, Number 2, November I 2008
|Page(s)||477 - 486|
|Published online||18 August 2008|
A new equation for the mid-plane potential of power law discs
II. Exact solutions and approximate formulae
Université de Bordeaux, LAB, 351 cours de la Libération, Talence 33405, France e-mail: [jean-marc.hure;jean-pierre.busset]@obs.u-bordeaux1.fr
2 CNRS/INSU, UMR 5804/LAB, 2 rue de l'Observatoire, BP 89, 33271 Floirac Cedex, France e-mail: firstname.lastname@example.org
3 La Maurellerie, 37290 Bossay-sur-Claise, France e-mail: email@example.com
Accepted: 31 May 2008
Aims. The first-order ordinary differential equation (ODE) that describes the mid-plane gravitational potential in flat finite size discs of surface density (Huré & Hersant 2007, A&A, 467, 907) is solved exactly in terms of infinite series.
Methods. 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.
Results. Inside the disc, the gravitational potential consists of three terms: a power law of radius R with index , and two infinite series of the variables R and . 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.
Conclusions. Applications concern all theoretical models and numerical simulations where the influence of disc gravity must be checked and/or reliably taken into account.
Key words: gravitation / methods: analytical / accretion, accretion disks
© ESO, 2008
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.