Self-gravity in thin discs and edge effects: an extension of Paczynski’s approximation

¹ Univ. Bordeaux, LAB, UMR 5804, 33270 Floirac, France
² CNRS, LAB, UMR 5804, 33270 Floirac, France
e-mail: audrey.trova@obs.u-bordeaux1.fr; jean-marc.hure@obs.u-bordeaux1.fr; franck.hersant@obs.u-bordeaux1.fr

Received: 24 June 2013
Accepted: 9 January 2014

Abstract

Because hydrostatic equilibrium of gaseous discs is partly governed by the gravity field, we have estimated the component caused by a vertically homogeneous disc with particular attention to the outer regions where self-gravity appears most often. The accuracy of the integral formula is better than 1% regardless of the disc thickness, radial extension and radial density profile. At order zero, the field is even algebraic for thin discs and reads −4πGΣ(R) × f_edges(R) at disc surface, which means a correction of Paczynski’s formula by a multiplying factor f_edges ≲ ½, which depends on the relative distance to the edges and the local disc thickness. For very centrally condensed discs, however, this local contribution can be surpassed by the action of mass stored in the inner regions, possibly resulting in f_edges ≫ 1. A criterion setting the limit between these two regimes is derived. These results are robust in the sense that the details of vertical stratification are not critical. We briefly discuss how hydrostatic equilibrium is affected. In particular, the disc flaring probably does not reverse in the self-gravitating region, which contradicts what is usually obtained from Paczynski’s formula. This suggests that i) these outer regions are probably not fully shadowed by the inner ones (when the disc is illuminated by a central star); and ii) the flared shape of discs does not firmly prove the absence or weakness of self-gravity.