Rotation of spiral galaxies is measured by spectroscopic observations of emission lines such as Hα, HI and CO lines from disk objects, namely population I objects and interstellar gases. In these lines the velocity dispersion is negligibly small compared to rotational velocity, which implies that the pressure term is negligible in the Virial theorem, so that the dynamical balance between the gravitational and centrifugal forces may be used to calculate the mass in sufficient accuracy. Absorption lines are also used for bulge's kinematics using velocity dispersion and rotation. The dynamical approach is essential particularly for measurement of the mass of the dark matter and black holes, which are not measurable in surface photometry assuming a mass-to-luminosity ratio.
In sections 2 and 3 we review the various methods to derive rotation curves of the Milky Way and spiral galaxies, respectively, and describe the general characteristics of observed rotation curves. The progress in the rotation curve studies will be also reviewed briefly. In section 4 we review the methods to determine the mass distributions in disk galaxies using the rotation curves, and describe their dynamical mass structures.
By definition, a rotation curve is the mean circular velocity around the nucleus as a single function of radius. Non-circular streaming motions such as due to spiral arms, bars, and/or expansion/contraction motions are not considered here. Limitation of the current rotation curve analyses is discussed in section 2. Elliptical galaxies, for which rotation curve analysis is not applicable, are beyond the scope of this review. Considerations that employ unconventional physical laws such as MOND (modified Newtonian dynamics) are also out of the scope of this review.
There are a number of articles and reviews on rotation curves and mass determination of galaxies, that include Sofue and Rubin (2001) and Sofue (2013a), and the literature therein. Individual references will be given in the related sections in this review.