## 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

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
−4*πG*Σ(*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.

Key words: gravitation / accretion, accretion disks / methods: analytical

*© ESO, 2014*