Extremely Flat Haloes and the Shape of the Galaxy
  We present a set of highly flattened galaxy models with asymptotically
constant rotation curves. The mass density in the equatorial plane falls like
(distance)$^{-1}$ at large radii. Although the inner equidensity contours may
be spherical, oblate or prolate, the outer parts are always severely flattened.
The elongated shape is supported by rotation or tangential velocity anisotropy.
The models are thickened Mestel discs, and form a previously undiscovered part
of the Miyamoto & Nagai sequence of flattened galaxies. The properties of the
models -- axis ratios, velocity dispersions, streaming velocities and
distribution functions -- are all discussed in some detail. We pose the
question: are extremely flattened or disk-like haloes possible for the Milky
Way galaxy? This has never been examined before, as very flattened halo models
were not available. We fit the rotation curve and the vertical kinematics of
disc stars in the solar neighbourhood to constrain the overall shape of the
Galaxy. Denoting the ratio of polar axis to major axis by $q$, we show that
models with $q\lesssim 0.57$ cannot simultaneously reproduce the in-plane and
out-of-plane constraints. The kinematics of the Sagittarius galaxy also
strongly disfavour models with high flattening, as the orbital plane precession
is too great and the height reached above the Galactic plane is too small. At
least for our Galaxy, the dark halo cannot be flatter than E4 (or axis ratio $q
\sim 0.57$) at the Solar circle. Models in which the dark matter is accounted
for by a massive baryonic disc or by decaying neutrinos are therefore ruled out
by constraints from the rotation curve and the vertical kinematics.
