Two-integral models for oblate elliptical galaxies with cusps

Self-consistent two-integral distribution functions (DFs) f(E, L_z) have been numerically constructed for a set of oblate density distributions, whose isocontours arc oblate spheroids with various axial ratios, and whose radial profiles are proportional to r^-4 at large radii and to r^-γ in the centre with 0 ≤ γ ≤ 2. From these models the line- of-sight velocity profiles (VPs) on the projected minor and major axes have been computed for various inclinations. The main results are as follows. (i) DFs f(E, L_z) are strongly increasing functions of angular momentum, even for moderate flattening. This implies that the kinematics of E≿E2 galaxies cannot be modelled with spherical models. (ii) Isotropic rotator models for E≿E3 develop a secondary peak on the retrograde circular orbits; they are therefore unlikely to exist in nature. (iii) Non-rotating two-integral models lead to flat-topped or double-peaked velocity profiles (VPs), which are not observed among elliptical galaxies. Non-rotating galaxies must therefore have three-integral DFs. Maximally rotating DFs f(E, L_z) also lead to VPs not seen in galaxies. (iv) The VPs of the isotropic rotator 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 preserved 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 v by about 15 per cent, and an underestimate of sigma by 0–10 per cent. In terms of Gaussian fit velocities, isotropic rotator models have $$\left( v/\sigma\right)^\ast\approx1.2$$ when seen edge-on, and $$\left( v/\sigma\right)^\ast\approx1.4$$ when seen at 60° inclination. This suggests that many ellipticals near the classical isotropic line in the v/σ − ε diagram are not in fact consistent with isotropic rotator models.