Response of a galactic disc to vertical perturbations : Strong
  dependence on density distribution
  We study the self-consistent, linear response of a galactic disc to
non-axisymmetric perturbations in the vertical direction as due to a tidal
encounter, and show that the density distribution near the disc mid-plane has a
strong impact on the radius beyond which distortions like warps develop. The
self-gravity of the disc resists distortion in the inner parts. Applying this
approach to a galactic disc with an exponential vertical profile, Saha & Jog
showed that warps develop beyond 4-6 disc scalelengths, which could hence be
only seen in HI. The real galactic discs, however, have less steep vertical
density distributions that lie between a sech and an exponential profile. Here
we calculate the disc response for such a general sech^(2/n) density
distribution, and show that the warps develop from a smaller radius of 2-4 disc
scalelengths. This naturally explains why most galaxies show stellar warps that
start within the optical radius. Thus a qualitatively different picture of
ubiquitous optical warps emerges for the observed less-steep density profiles.
The surprisingly strong dependence on the density profile is due to the fact
that the disc self-gravity depends crucially on its mass distribution close to
the mid-plane. General results for the radius of onset of warps, obtained as a
function of the disc scalelength and the vertical scaleheight, are presented as
contour plots which can be applied to any galaxy.
