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5.1. Summary

We reviewed the methods to observe and determine rotation curves of the Milky Way and spiral galaxies. Rotation characteristics of spiral galaxies and mass distributions were shown to be similar among the various types of galaxies. The dynamical structures are, thus, universal from Sa to Sc galaxies. The Milky Way exhibits representative universal characteristics from the central black hole to the dark halo.

Major methods to derive the dynamical mass distribution in disk galaxies were described, and some results were presented. The direct method does not employ any galactic models or functional forms, but straightly compute the mass distribution. On the other hand, the decomposition method yields fundamental parameters representing the bulge, disk and dark halo individually. The decomposition method is thus convenient to discuss the fundamental structures composed of bulge, disk and dark halo. The fitted dynamical parameters were also useful for statistical correlation analyses such as the size-mass and mass-mass relations, which provides with fundamental information for formation and evolution scenarios of galaxies in the universe.

5.2. Achievements in the Decades and the Future

In our previous review (Sofue and Rubin 2001) various advances have been expected for the future rotation curve studies thanks principally to instrumental developments. We summarize and evaluate the progress in the decades.

5.2.1. Extinction-free rotation kinematics in the central regions

This has been achieved extensively using CO line spectroscopy and imaging, and will be achieved in an ultimate way using the ALMA (Atacama Large mm and submm Array) with higher spatial (0.01 arcsec) resolution. Spectroscoy with eight- and ten-meter class optical/infrared telescopes using infrared spectral lines such as Br γ was anticipated, while the progress seems not fast, and no outstanding advances have been achieved.

5.2.2. VLBI astrometry

The VERA (VLBI Experiments for Radio Astrometry) observations have successfully achieved high-resolution astrometric measurements of paralactic distances, positions, proper motions, and radial velocities for a number of maser sources in the Milky Way. The observations resulted in a series of special volume in PASJ (Honma et al. 2015, and the literature therein). Many new aspects have been learnt in details of the three-dimensional kinematics of the Galaxy, which include the new values of (R0, V0), non-circular motions related to the density waves and bars. The SKA (Square-Kilometer Array) will be a future, probably more powerful tool for trigonometric determination of the Galactic rotation curve and dynamics even by simply extending the method employed with VERA.

5.2.3. High-redshift rotation curves

Thanks to the increase of aperture and sensitivity of the telescopes, particularly using the Hubble Space Telescope and ground-based 8-10 meter telescopes, galaxies at cosmological distances are becoming routine targets for rotation curve observations. Tens of galaxies at high redshifts have their rotation velocities measured. The data are still not sensitive enough to be compared with those for the nearby galaxies. The global rotation curve shapes seem to be similar already at z ∼ 3−5. Combined with cosmological simulations of structural formation and evolution, this would become one of the most popular fields in rotation curve study.

However, the inner rotation curves representing the bulge and central massive objects are still not resolved at high redshifts. As mentioned already, an angular resolution of 0″.01 corresponds only to ∼ 200 pc at z > ∼ 2. Higher resolution instruments are desired for more detailed study of the bulge and black hole formation, which are one of the major events in the primeval galaxies.

5.2.4. Method of analysis

Rotation curves presented in this review have been basically derived by the popular methods like PV tracing and tilted ring methods. The PV iteration method was also applied, while only in a few cases.

Development of sophisticated iteration methods of 2D and 3D velocity fields to produce more accurate rotation curves has been anticipated. However, not much advance was obtained in the decades. These will lead to more tightly constrained mass deconvolution, and precise distributions of the stellar and dark matter masses.

5.2.5. Dark halos

Dark halos are one of the major topics in the field not only of galaxy dynamics and kinematics but also of the structure formation and evolution in the expanding universe. As to the Milky Way and M31, it has been shown that their rotation curves, and hence the dark matter halos, seem to be merged at a half distance to each other. Both galaxies showed declining rotation at radii larger than R ∼ 50 kpc. Their outermost rotation curves are not flat anymore and is not well fitted by the current pseudo-isothermal models, but they seem to be better approximated by the NFW model. However, it may be mentioned that the isothermal model is simple and is enough to represent dark halos up to ∼ 30−50 kpc, beyond which it is still difficult to obtain rotation curves in usual galaxies.

Thus, the measurements of the dynamical mass of dark halos in most galaxies are limited to their outermost radii of measured rotation curves only to ∼ 20−30 kpc, where it is difficult to discriminate the halo models. Measurements to rato larger radii, up to ∼ 100 kpc, for a more number of galaxies are crucial for conclusive comparison with the cosmological scenarios of structural formation.

Dark matter density in the solar neighborhood has been determined in the range around ∼ 0.2−0.4 GeV cm−3. The physical property of the dark matter in view of elementary particle physics would be clarified when direct detection in the laboratory is achieved. Also indirect detection toward the Sun and the Galactic Center are proposed and partly performed. Although, at the moment, there appears no report of firm detection, this would be the fundamental field in the experimental physics in the near future.

5.2.6. Massive black holes

Coevolution of the spheroidal component and central supermassive black hole is a standard scenario in the structural evolution of galaxies based on the observational facts that black hole mass is positively correlated with the spheroidal mass in measured galaxies. This topic was, however, not reviewed in this paper, mainly because of insufficient sample galaxies with both the black hole mass and detailed rotation curves around the nuclei. Such study would be a promising subject for high-resolution submillimeter spectroscopy using ALMA and infrared spectroscopy with large-aperture telescopes including TMT.

5.2.7. Activities

Many of the galaxies with well defined rotation curves as discussed in this review are known as galaxies possessing various activities such as starburst, outflow, jets, and/or active galactic nuclei (AGN). However, there has been no clear correlation analysis among these activities and the galactic mass distribution. It could be a subject for a more detailed comparison study with careful inspection of the individual curves and mass structures, categorizing the curves into those with and without activities. It may be related to central bars by which the gas is accumulated to the nuclei, and hence the correlation of rotation curves of bar and non-barred galaxies.

5.2.8. Star formation

Correlation analysis of general star forming activity in the disk with the rotation curves and mass structure would be a fundamental subject related to ISM physics in galaxies. A relation is known that low surface brightness galaxies are generally slowly rotating and less massive. On the other hand, high velocity rotators, usually of late types as Sa and S0, show that their major star formation activity was over in the far past. In Sb and Sc galaxies, the star formation rate is not necessarily controlled by the mass structure, but rather related to spiral arms and/or bars, or to environmental effect such as interaction with other galaxies.

The relation between mass structure and star forming activity is, therefore, not well studied from observations of galaxies at the present epoch (redshift z ≃ 0). This is readily shown by the observed constant M/L ratio among galaxies, where M/L ratios do not represent star forming activity. Direct relation between the mass structure and star formation would be studied by rotation curve analysis of high-redshift galaxies during their major star formation.

5.2.9. Bars and spiral arms

In the current rotation curve studies the bar/non-bar discrimination has not necessarily been the major subject for the reason raised in chapter 2. In this article we reviewed on rotation curves based on the assumption that the galactic rotation is circular and axisymmetric. Therefore, we did not touch upon spiral arms and bars in detail.

The separation of arm and bar related parameters is still difficult even though the data get wider, deeper, and more accurate. The number of parameters, hence the number of freedom, is still large to reach a unique result. The difficulty has not been solved even by numerical simulations.

Some new technique to directly analyze the mass distribution and motion in arms and bars are desired. More detailed inspection, probably in a more sophisticated way, of spectroscopic 2D and 3D data cubes of a larger number of galaxies, particularly data with such facilities like 2D Fabry-Perot instruments, velocity cubes in HI and CO using more advanced interferometers like SKA and ALMA, would lead to deeper insight into mass structures of galaxies including bars and spiral arms.

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