Kinemetry: a generalization of photometry to the higher moments of the line-of-sight velocity distribution

We present a generalization of surface photometry to the higher-order moments of the line-of-sight velocity distribution of galaxies observed with integral-field spectrographs. The generalization follows the approach of surface photometry by determining the best-fitting ellipses along which the profiles of the moments can be extracted and analysed by means of harmonic expansion. The assumption for the odd moments (e.g. mean velocity) is that the profile along an ellipse satisfies a simple cosine law. The assumption for the even moments (e.g. velocity dispersion) is that the profile is constant, as it is used in surface photometry. We test the method on a number of model maps and discuss the meaning of the resulting harmonic terms. We apply the method to the kinematic moments of an axisymmetric model elliptical galaxy and probe the influence of noise on the harmonic terms. We also apply the method to SAURON observations of NGC 2549, NGC 2974, NGC 4459 and NGC 4473 where we detect multiple co- and counter-rotating (NGC 2549 and NGC 4473, respectively) components. We find that velocity profiles extracted along ellipses of early-type galaxies are well represented by the simple cosine law (with 2 per cent accuracy), while possible deviations are carried in the fifth harmonic term which is sensitive to the existence of multiple kinematic components, and has some analogy to the shape parameter of photometry. We compare the properties of the kinematic and photometric ellipses and find that they are often very similar, but a study on a larger sample is necessary. Finally, we offer a characterization of the main velocity structures based only on the kinemetric parameters which can be used to quantify the features in velocity maps.