Dynamics of prolate spheroidal mass distributions with varying
  eccentricity
  In this paper we calculate the potential for a prolate spheroidal
distribution as in a dark matter halo with a radially varying eccentricity.
This is obtained by summing up the shell-by-shell contributions of isodensity
surfaces, which are taken to be concentric and with a common polar axis and
with an axis ratio that varies with radius. Interestingly, the constancy of
potential inside a shell is shown to be a good approximation even when the
isodensity contours are dissimilar spheroids, as long as the radial variation
in eccentricity is small as seen in realistic systems. We consider three cases
where the isodensity contours are more prolate at large radii, or are less
prolate, or have a constant eccentricity. Other relevant physical quantities
like the rotation velocity, the net orbital and vertical frequency due to the
halo and an exponential disc of finite thickness embedded in it are obtained.
We apply this to the kinematical origin of Galactic warp, and show that a
prolate shaped halo is not conducive to making long-lived warps - contrary to
what has been proposed in the literature. The results for a prolate mass
distribution with a variable axis ratio obtained are general, and can be
applied to other astrophysical systems such as prolate bars, for a more
realistic treatment.
