Textbooks: [Gravitational Physics of Stellar and Galactic Systems (W.C. Saslaw, ed.), Cambridge, Cambridge University Press (1985)] and [Galactic Dynamics (J. Binney & S. Tremaine), Princeton, Princeton University Press (1987). Structure and Dynamics of Elliptical Galaxies (T. de Zeeuw, ed.), Int. Astron. Union Symp. 127, Dordrecht, Kluwer (1987)].

Reviews: Rotation in Normal Galaxies.

1. Burbidge, E.M., Burbidge, G.R. in Galaxies and the Universe (Sandage, A.R., Sandage, M. & Kristian, J., eds.) = Stars and stellar Systems Vol. IX, Univ. Chicago Press (1975). p. 81.
2. Bulges of Galaxies (B.J. Jarvis & D.M. Terndrup, eds.), ESO Conf. and Workshop Proc. No. 35, Garching, ESO (1990).
3. Structure and Dynamics of Elliptical Galaxies (T. de Zeeuw, ed.), Int. Astron. Union Symp. 127, Dordrecht, Kluwer (1987).
4. Dynamics of Galaxies and their Molecular Cloud Distributions (F. Combes & F. Casoli, eds.), Int. Astron. Union Symp. 146, Dordrecht: Kluwer (1991).
5. The Evolution of Galaxies and their Environment. Proceedings of the Third Teton Summer School of Astrophysics (D. Hollenbach, H.A. Thronson, & J.M. Shull, eds.), Washington, NASA Conference Publication 3190 (1993).

Conference Proceedings on this topic are to be found in:

1. Structure and Dynamics of Elliptical Galaxies (T. de Zeeuw, ed.), Int. Astron. Union Symp. 127, Dordrecht, Kluwer (1987).
2. Dynamics and interactions of galaxies (R. Wielen, ed.), Berlin: Springer (1990).
3. Windows on Galaxies (G. Fabbiano, J.S. Gallagher & A. Renzini, eds.), ASSI 160, Dordrecht, Kluwer (1990).
4. Dynamics of Galaxies and their Molecular Cloud Distributions (F. Combes & F. Casoli, eds.), Int. Astron. Union Symp. 146, Dordrecht: Kluwer (1991).

Warps: [Bicknell, G.V., Astrophys. J. 399 (1992) 1].

Gravitational Interaction: [Barnaby, D., & Thronson, H.A. Jr., Astron. J. 103 (1992) 41].

Kinematics of Elliptical and S0 systems. [Capaccioli, M. in Photometry, Kinematics and Dynamics of Galaxies (Evans, D.S., ed.), University of Texas (1979). p. 165].

Stellar Dynamics and Structure of Galaxies. [Freeman, K.C. in Galaxies and the Universe (Sandage, A., Sandage, M.; Kristian, J., eds.) = Stars and stellar Systems Vol. IX, Univ. Chicago Press (1975). p. 409].

Kinematics of Spiral and Irregular Galaxies. [van der Kruit, P.C., Allen, R.J. Annu. Rev. Astron. Astrophys. 16 (1978) 103].

Dynamics of Elliptical Galaxies:

1. Berkhuijsen, E.M., Bajaja, E., & Beck, R., Astron. Astrophys. 279 (1993) 359.
2. Maia, M.A.G., Da Costa, L.N., Giovanelli, R., & Haynes, M.P., Astron. J. 105 (1993) 2107.

and of Spiral Galaxies:

1. Baan, W.A., Rhoads, J., & Haschick, A.D., Astrophys. J. 401 (1992) 508.
2. Ball, R., Astrophys. J. 395 (1992) 418.
3. Heller, P., & Rohlfs, K., Astron. Astrophys. 291 (1994) 743.

Evidence for Counter-Rotation:

1. Dahlem, M., Golla, G., Whiteoak, J.B., Wielebinski, R., Huttemeister, S., & Henkel, C., Astron. Astrophys. 270 (1993) 29.
2. Martimbeau, N., Carignan, C., & Roy, J.R., Astron. J. 107 (1994) 543.

Early-Type Galaxies: [Capaccioli, M., & Longo, G., Astron. Astrophys. Rev. 4 (1994) 293].

Observations of Rotation Curves

Rotation curves are derived from radial-velocity measurements of emission lines in the optical region and from the 21-cm line of neutral hydrogen. Figure 10 shows typical rotation curves for 25 galaxies including our own Galaxy.

Figure 10. Rotation curves of 25 galaxies of various types [Bosma, A. Ph. Thesis, Univ. Groningen (1978)]. The parameters Vmaxrot (maximum rotational velocity) = Vmax) and Rmax (the corresponding distance from the center) are shown for M81 on the first picture left above.

No significant decrease in rotational velocities is observed in outer parts as would be expected in case of Keplerian rotation. (Exceptions like M81 are galaxies disturbed gravitationally by close companions). See Fig. 11.

Rotation Curves in the Outer Parts by HI Observations:

1. Salpeter, E.E. in Structure and Properties of Nearby Galaxies (Berkhuijsen, E.M., & Wielebinski, R., eds.), Int. Astron. Union Symp. 77, Reidel, Dordrecht (1978). p. 23.
2. Roberts, M.S. The Rotation Curves of Galaxies, in Dynamics of Stellar Systems. IAU Symp. 69 (Hayli, A., ed.), Reidel, Dordrecht (1975) p. 331.
3. Huchtmeier, W.K. Astron. Astrophys. 45 (1975) 259.

Extended Rotation Curve of the Sab galaxy NGC 7217. [Peterson, C.J., Rubin, V.C., Ford, E.K., Jr., & Roberts, M.S. Astrophys. J. 226 (1978) 770].

Extended Rotation Curves of 10 high-luminosity spirals. [Rubin, V.C., Ford, W.K., Jr., & Thonnard, N. Astrophys. J. 225 (1978) L107].

Rubin et al. [Rubin, V.C., Ford, W.K., Jr., & Thonnard, N. Astrophys. J. 238 (1980) 471] investigate the rotational properties of 21 Sc galaxies (in some cases up to more than 100 kpc distance from the center). In no case did they find a significant decrease but mostly increasing rotation curves. NGC 0801 for instance shows a constant V_rot up to 50 kpc. They suggest that Sc spirals of all luminosity classes have significant masses beyond the optical image. Also the rotation curve of our own Galaxy is believed to be flat perhaps out to 16 kpc [Blitz, L. Astrophys. J. 231 (1979) L115; erratum in 234 (1979) L172] contrary to Schmidt's model (Keplerian in the outer parts). See also:

1. Gunn, J.E., Knapp, G.R., & Tremaine, S.D. Astron. J. 84 (1979) 1181.
2. Blitz, L. Astrophys. J. 231 (1979) L115; erratum in 234 (1979) L172.

Altogether there is some evidence for massive coronae of galaxies.

1. Ostriker, J.P., Peebles, P.J.E., & Yahil, A. Astrophys. J. 193 (1974) L1.
2. Einasto, J., Kaasik, A., & Saar, E. Nature 250 (1974) 309.

But see also the critical comments in

1. Burbidge, G. Astrophys. J. 196 (1975) L7.
2. van den Bergh, S. in Structure and Properties of Nearby Galaxies (Berkhuijsen, E.M., & Wielebinski, R., eds.), Int. Astron. Union Symp. 77, Reidel, Dordrecht (1978). p. 247.

Van den Bergh (1978) points to the facts (1) that motions in the outer parts might not be circular, (b) that there might happen some "spillover", i.e., emission from the bright central area received a side lobe of the antenna.

The barred spiral NGC 7723 (type SBb) shows a constant angular velocity of 63 km s^-1 kpc^-1 (i.e. solid rotation) in the bar region and a constant linear velocity of 210 km s^-1 in the spiral region [Chevalier, R.A., & Flurenlid, I. Astrophys. J. 225 (1978) 67]. Large deviations from circular orbits in the SBbII galaxy NGC 5383 are reported in [Peterson, C.J., Rubin, V.C., Ford, K.W., Jr., & Thonnard, N. Astrophys. J. 219 (1978) 31].

Rotation and mass distribution in pairs: NGC 935 and IC 1801 (close pair of spirals) [Blackman, C.P. Mon. Not. R. Astron. Soc. 178 (1977) 15]; NGC 672 and IC 1727 (interacting barred spirals) [Carozzi-Meyssonnier, N. Astron. Astrophys. Suppl. 47 (1982) 237]; rotation curves for two other pairs are shown in Fig. 11. These pairs illustrate the lack of the Tully-Fisher relation.

Figure 11. Extended rotation curves for two pairs of galaxies [Rubin, V.C., Ford, W.K., Jr., & Thonnard, N. Astrophys. J. 225 (1978) L107].

A correlation between maximum rotational velocity V^max_rot and morphological type is reported in [Brosche, P. Astron. Astrophys. 13 (1971) 293] , see Fig. 12.

Figure 12. Maximum rotational velocity vs. morphological type [Brosche, P. Astron. Astrophys. 13 (1971) 293]. Open circles: optical observations: squares: 21-cm observations. Numbers are NGC numbers.

with
V_rot(R) = circular velocity at distance R from the center,
n = shape parameter, numerical index.
A = 3^3/2n V^max_rot / R_max; B = 2^1/n / R_max,
V^max_rot = maximum rotational velocity - see Figure 10
R_max = corresponding distance from the axis of rotation - see Figure 10

From (1) follows the gravitational attraction at distance R

Thus
for small R / R_max : F(R) R (constant angular velocity, solid rotation).
for large R / R_max : F(R) 1 / R² (Keplerian motion).

With n = 3, Eq. (2) yields the Bottlinger-Lohmann formula.

Dynamical considerations indicate that the larger the degree of mass concentration toward the center of the galaxy, the greater the n value.

The function R_max / R_Ho (R_Ho = Holmberg radius) increases with the morphological type, from 0.3 for spirals of type Sab to 2 for irregular systems [Huchtmeier, W.K. Astron. Astrophys. 45 (1975) 259].

Further dynamical parameters

The above method has been extended and modified by Takase and Kinoshita [Takase, B., & Kinoshita, H. Publ. Astron. Soc. Japan 19 (1967) 409] to calculate the projected mass, the angular momentum and the rotational energy:

Projected mass	M(R) = mass contained in the cylinder of radius R whose axis coincides with the rotational axis = 0R d M' (R)	(3)
Angular momentum	Q(R) = 0R R Vrot d M' (R)	(4)
Rotational energy	T(R) = 1/2 0R V2rot d M' (R)	(5)

The relationships between angular momentum, rotational energy and total mass are shown in Fig. 13. A similar investigation of statistical characteristics of the dynamical structure of 21 flat galaxies [Miyamoto, M., Satoh, C., & Ohashi, M. Astrophys. Space Science 67 (1980) 147] yields the relation

with angular momentum Q in [< G(10¹¹M)³ kpc >^1/2] (G = gravitational constant) and mass M in [10¹¹ M]. Tables of functions for use in calculating mass distribution, density and mass-surface density are given in [Brandt, J.C., & Scheer, L.S. Astron. J. 70 (1965) 471]. The mass-angular momentum relation is also reflected in a relation between optical luminosity and angular momentum density (angular momentum per unit mass). Vettolani et al. [Vettolani, G., Marano, B., Zamorani, G., & Bergamini, R. Mon. Not. R. Astron. Soc. 193 (1980) 269] found the following relations:

with a correlation coefficient r = 0.93 for a sample of 89 late-type galaxies, studied by [Shostak, G.S. Astron. Astrophys. 68 (1978) 321] and

with r = 0.88 for a sample of 38 bright galaxies (M < -17) out of low surface-brightness galaxies studied by [Fisher, J.R., & Tully, R.B. Astron. Astrophys. 44 (1975) 151].

Here A = angular momentum density in [km kpc s^-1], M₂₀ = M + 20 (M = absolute magnitude).

Figure 13. Statistical relationship of total angular momentum Q in [g cm2 s-1] and of the total kinetic energy T in [g cm2 s-2] to the total mass M in [Photometry, Kinematics and Dynamics of Galaxies (Evans, D.S., ed.), University of Texas (1979)] for galaxies of different type [Takase, B., & Kinoshita, H. Publ. Astron. Soc. Japan 19 (1967) 409].

Dynamical parameters of about 40 galaxies with known rotation curves are studied in [Brosche, P. Astron. Astrophys. 23 (1973) 259]. Figure 14 shows the relation between D_v / R_max (D_v = optical diameter) and morphological type (Fig. 14).

Figure 14. The ratio of optical diameter Dv to Rmax vs. morphological type [Brosche, P. Astron. Astrophys. 23 (1973) 259].