2-INTEGRAL MODELS FOR OBLATE ELLIPTIC GALAXIES WITH CUSPS

Self-consistent two-integral distribution functions (DFs) f(E, L(z)) h ave been numerically constructed for a set of oblate density distribut ions, whose isocontours are oblate spheroids with various axial ratios , and whose radial profiles are proportional to r-4 at large radii and to r(-gamma) in the centre with 0 less-than-or-equal-to gamma less-th an-or-equal-to 2. From these models the line-of-sight velocity profile s (VPs) on the projected minor and major axes have been computed for v arious inclinations. The main results are as follows. (i) DFs f(E, L(z )) are strongly increasing functions of angular momentum even for mode rate flattening. This implies that the kinematics of E greater than or similar to E2 galaxies cannot be modelled with spherical models. (ii) Isotropic rotator models for E greater than or similar to E3 develop a secondary peak on the retrograde circular orbits; they are therefore unlikely to exist in nature. (iii) Non-rotating two-integral models l ead to flat-topped or double-peaked velocity profiles (VPs), which are not observed among elliptical galaxies. Non-rotating galaxies must th erefore have three-integral DFs. Maximally rotating DFs f (E, L(z)) al so lead to VPs not seen in galaxies. (iv) The VPs of the isotropic rot ator models are always asymmetric in the sense that the wing of the VP extending to retrograde orbits is shallower than that on the prograde side. This property is in accordance with observations, and is preser ved when the secondary peak in the DF on retrograde circular orbits is removed. (v) Fitting of a Gaussian to such asymmetric VPs leads to an overestimate of upsilon by about 15 per cent, and an underestimate of sigma by 0-10 per cent. In terms of Gaussian fit velocities, isotropi c rotator models have (upsilon/sigma) almost-equal-to 1.2 when seen e dge-on, and (upsilon/sigma) almost-equal-to 1.4 when seen at 60-degre es inclination. This suggests that many ellipticals near the classical isotropic line in the upsilon/sigma - epsilon diagram are not in fact consistent with isotropic rotator models.