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The rotation-anisotropy diagram below uses the oblate-rotator flattening proxy for $\beta =0,0.2,0.4$. Each curve starts at $e=\beta$ and rises as $e$ increases. The plotted domains cover typical spheroid values, roughly $0\le e\lesssim 0.7$ and $0\le v/\sigma \lesssim 1.5$, with the upper axis extended to show the reference curve's rise. The shaded regions are schematic overlapping populations, not measurements or sharp classification boundaries.

Giant elliptical galaxies commonly have low ordered rotation for their observed flattening, lying below the oblate isotropic rotator reference. Anisotropic velocity dispersions and, in many cases, triaxial structure provide support. Lower-luminosity elliptical galaxies are more commonly rotating, disky systems near the isotropic reference. Classical galactic bulges of spiral galaxies often overlap this rotating-spheroid locus; disk-like bulges can be more strongly rotation supported. These are population trends with substantial overlap, not universal rules for individual galaxies.

Observations usually use projected ellipticity and a line-of-sight rotation measure such as peak speed divided by a central velocity dispersion. The intrinsic $e$ and global dispersions used in the derivation are different quantities. Inclination, apertures, and unresolved streaming can change both coordinates. Consequently a point below the reference suggests extra random-motion support, but does not uniquely measure $\beta$.

For a singular isothermal sphere, spherical integration of the mass density givesThe spherical shell theorem makes the radial attraction $GM(<R)/R^2$. Balancing it with circular centripetal acceleration yieldsThus the model has a flat galaxy rotation curve, resembling the outer galaxy rotation curves of many spiral galaxies. Central rotation curves usually rise, and not all real curves are flat throughout. A pressure-supported elliptical galaxy does not itself rotate at this test-particle circular speed.

The model has divergent central mass density, although finite enclosed mass at each finite radius, and an infinite total mass because $M(<R)\propto R$. Its potential is logarithmic, requiring an outer cutoff for a finite escape energy. A globally constant isotropic velocity dispersion, spherical symmetry, and one common profile for all gravitating components are further idealizations. Stellar light commonly has a quite different radial profile from the dark matter halo; real disks are flattened, and halos are neither exact isothermal spheres nor infinite. Tidal truncation also changes the distribution near the satellite boundary.

Typical present elliptical galaxies are dominated by old stellar populations: their brightest short-lived stars have disappeared, leaving faint long-lived stars and stellar remnants, so luminosity per unit stellar mass is low. Ongoing star formation in many spiral galaxies and irregular galaxies supplies luminous young stars, giving a smaller stellar mass-to-light ratio in a fixed band; dark matter, the initial mass function and aperture must also be specified if the quoted ratio is dynamical.

A spiral galaxy has a galactic disk with spiral structure, commonly cold interstellar gas and ongoing star formation as well as an older stellar population.