Fitting Rotation Curves: Mass Models Kinematics

Long before dark matter in galaxies became the focus of two or three international conferences a year, optical emission line studies were used to measure the kinematics, and, indirectly, the mass distribution in spirals. In those days, spiral galaxies were assumed to be as flat as they looked, and it was natural to devise a method of using the measured kinematics of galaxies to infer the surface mass density of the luminous disk. First attempts assumed that the mass of a galaxy could be reasonably well approximated by either a superposition of oblate spheroidal shells of a given flattening (Burbidge, Burbidge & Prendergast 1959, Brandt 1960) or an infinitely thin disk (Toomre 1963, Nordsieck 1973, Bosma 1981). The rotation curve was then inverted via Poisson's equation to produce the cylindrical mass interior to a given radius.

Later, when more sensitive optical emission spectroscopy and especially HI synthesis observations consistently indicated that rotation speeds remained nearly constant to large galactocentric radius, dark matter was invoked and modeled as a spherical, isothermal halo with , which produces absolutely flat rotation curves at all radii. In order to account for the slower speeds in the very inner regions of galaxies and the contribution to the rotation curve provided by the luminous mass, the spherical dark halo was then modified to have a a finite core. The mass density of the luminous matter was assumed to be proportional to the density of the light, scaled by a radially-constant mass-to-light ratio M/L.

Interestingly, much of the original observational motivation for spherical, isothermal dark halos has now been lost, especially over radial range (r < 40 kpc) of spiral rotation curves (Sackett 1995). One of the strongest constraints beyond this radius relies on statistical arguments based on the orbits of satellites (Zaritsky & White 1994) that do seem to suggest that halos may be isothermal at > 100 kpc. On the other hand, the only known measurement from a cold tracer at these radii, the giant HI ring in Leo (kpc), seems to indicate that the halo has already truncated by 60 kpc (Schneider 1985). The kinematic advantage of HI rings like that in Leo is the simple orbit structure of cold gas; unfortunately only one such ring is known. The spatial resolution and sensitivity of the Parkes MultiBeam should make it an ideal instrument to search for more.

With this separation of galactic mass into luminous and dark components, the philosophy changed from rotation curve inversion to rotation curve fitting. Rotation curve fitting generally proceeds by comparing kinematic observations with model rotation curves resulting from multi-component mass models built from at least three of the following components:

• Stellar Bulge: assume M follows L, and fit for
• Stellar Disk: assume M follows L, and fit for
• Gaseous Disk: assume M follows known 21cm emission
• Dark Halo: assume a parameterized form, typically fitting two free parameters that control the central dark density and halo core radius

Scale lengths of the disk and bulge are typically measured from photometry, with assumptions for the flattening of the bulge and the scale height of the stellar disk. The flattening of the dark halo is also fixed by assumption (generally to be spherical). Such a procedure produces ``best-fit'' halo parameters, or -- if the mass-to-light ratios are fixed by the maximum disk hypothesis (van Albada & Sancisi 1986) -- ``maximum disk'' halo parameters.

Given the large number of implicit assumptions, the complications of non-circular motions, the degeneracy of the fitting parameters, and the fact that rotation curves generally exhibit only two or three distinct features (turn-over radius, peak speed, and outer slope), one might be reasonably concerned that rotation curve fitting introduces spurious, model-dependent trends in inferred dark matter properties. We are therefore led to begin again, reducing the number of model assumptions and asking, ``In what way is the radial distribution of the total mass (luminous + dark) of a galaxy constrained by its rotation curve?''

© Copyright Astronomical Society of Australia 1997