Measuring non-axisymmetry in spiral galaxies

We present a method for measuring small deviations from axisymmetry of the potential of a filled gas disc. The method is based on a higher order harmonic expansion of the full velocity field of the disc. This expansion is made by first fitting a tilted-ring model to the velocity field of the gas disc and subsequently expanding the velocity field along each ring into its harmonic terms. We use epicycle theory to derive equations for the harmonic terms in a distorted potential. The phase of each component of the distortion can vary with radius. We show that if the potential has a distortion of harmonic number m, the velocity field as seen on the sky exhibits an m − 1 and m + 1 distortion. As is to be expected, the effects of a global elongation of the halo are similar to an m = 2 spiral arm. The main difference is that the phase of the spiral arm can vary with radius. Our method allows a measurement of ϵ pot sin 2 ϕ2, where épot is the elongation of the potential and ϕ2 is one of the viewing angles. The advantage of this method over previous approaches to measure the elongations of disc galaxies is that, by using Hi data, one can probe the potential at radii beyond the stellar disc, into the regime where dark matter is thought to be the dominant dynamical component. The method is applied to the spiral galaxies NGC 2403 and 3198 and the harmonic terms are measured up to ninth order. The residual velocity field of NGC 2403 shows some spiral-like structures. The harmonic analysis indicates that the m = 3 term is dominant, with an average value of ∼0.02 υ c. This is consistent with an average ellipticity of the potential of ϵpot sin 2 ϕ2 = 0.064 ± 0.003, but spiral arms may couple significantly to this result. In the harmonic analysis of the kinematics of NGC 3198 the m= 2 and m= 3 terms are strongest (∼0.01υc). The inferred average elongation of the potential is 0.019 ± 0.003. Since the amplitude of the elongation is coupled to the viewing angles and may be influenced by spiral arms, more galaxies should be examined to separate these effects from true elongation in a statistical way.