Back to the Future: Inverted Kinematics Mass

The simplest case of rotation curve inversion is, of course, obtained under the (extreme) assumption of a spherical potential. In that case, the circular velocity is all that is required compute the mass interior to R via the usual ; the density of a shell of radius r is thus given by . Computation of the mass column within the cylindrical distance R, however, requires an integration along the cylindrical z coordinate, and thus knowledge of the kinematics to very large R. Specifically,

so that in general, the rotation curve at infinity must be known in order to compute the projected mass density . The other extreme is to assume that the galaxy can be described by an axisymmetric, infinitely thin disk; in this case the rotation is influenced by mass outside R as well. Using Laplace's equation, Gauss' Law and Bessel function identities, it is possible to show that the surface mass density of such a disk is related to the rotation curve and its derivate via (Binney & Tremaine 1987),

which again formally requires kinematic information at infinite distance. Integrating over the area of the disk within R then gives the enclosed mass.

Effects of Extrapolation and Noise

Note that inverting the rotation curve to obtain the surface mass column is not completely model-independent: assumptions of axisymmetry and the form of the vertical potential must be made. On the other hand, having made these assumption (which are also made in rotation curve fitting) the radial dependence of is then entirely fixed by the kinematics. Two concerns remain about application of this technique to real data: (1) the effect of missing kinematic information at very large R, and (2) the effects of noise in the rotation curve and its derivative. We now examine each of these in turn.

 
Figure: The input rotation curve for a thin exponential disk (upper left) is inverted to produce a surface mass column (upper right), enclosed mass M(<R) (lower left), and local mass-to-light ratio M/L (lower right). Two extrapolation schemes are used for beyond 10 scale lengths: flat (long dash) and Keplerian (dotted).

The effect of extrapolating the rotation curve beyond the measured kinematics is shown in Fig. 1. As a test, we begin with the theoretically-derived rotation curve of a thin exponential disk extrapolated beyond 10 disk scale lengths with one of two extreme schemes that can reasonably be expected to bracket reality: flat extrapolation (= constant) and Keplerian decline (.) For each extrapolation, the rotation curve is inverted to produce an inferred surface mass density for both the spherical (Eq. 1) and flat disk (Eq. 2) geometries. Note that in this case both the Keplerian and flat extrapolations deviate from the true surface mass density (shown as the solid straight line in the log plot) by overestimating at large R because the rotation of an exponential disk falls faster than Keplerian, an exact result that can be proven analytically. This departure does not begin at the end of the ``measured'' rotation curve, but at half the kinematical radius, in this case at 5 scale lengths.

The incorrect (in this example) assumption that the mass is spherically distributed leads to inferred surface mass densities that are too high in the inner regions and thus must be compensated for by outer shells with unphysical negative ; this results in an enclosed mass that actually declines with radius in the outer regions. The flat extrapolation scheme produces an overestimate of the enclosed mass interior to the last kinematical point where the departure has already reached 30%. This then has a very large effect on the local estimate for the mass-to-light ratio, which although taken to be constant in this example, is overestimated by more than a factor of 150 at the end of the rotation curve.

 
Figure: Same as Fig. 1, but now inversion is performed under the assumption of disk geometry for the HI rotation curve for NGC 2903 (solid squares) and the same data to which artificial Gaussian noise has been added (open squares). For each case both flat and Keplerian extrapolations are shown. Also shown is a maximum disk fit using a spherical, isothermal halo and a luminous disk with ; the and of the resulting model halo and disk are shown (solid lines) in the top panels.

The inversion procedure depends not only on the measured rotation curve, but also on its derivative. Binney & Tremaine (1987) have suggested that the noise in real data will make the technique unreliable, but as Fig. 2 demonstrates, this effect is much smaller than the uncertainties introduced by the extrapolation procedure. For this example, Gaussian noise has been added to the rotation curve of NGC 2903 (Begeman 1987) with the result that the derived is noisier, but suffers no systematic offset. As for the exponential disk, inversion of this more realistic rotation curve demonstrates that the flat extrapolation begins to differ noticeably from a Keplerian one at half the kinematic radius; by the end of the rotation curve, the two differ by a factor of 3 in , and by about 35% in the enclosed mass.

For comparison, a traditional maximum disk fit has been done for NGC 2903 and also displayed in Fig. 2. Not surprisingly, the mass column of resulting disk component, which has an , agrees well with that derived from the inversion technique in the inner regions. In the outer half of the galaxy, the model isothermal halo (with a core) has a that is indistinguishable from an exponential disk with scale length equal to that inferred from inversion with flat extrapolation. Since the model halo is assumed to be spherical, however, the mass normalization must be larger by the expected to produce the same rotation curve. (Note that because the enclosed mass is defined within a sphere of radius r rather than within a cylinder of radius R, the inverted rotation curve with flat extrapolation and the maximum disk + isothermal halo fit produce similar enclosed masses but quite different projected surface mass densities.) The Keplerian extrapolation produces a somewhat shorter outer scale length for the total mass column than either the isothermal halo fit or the flat extrapolation.

© Copyright Astronomical Society of Australia 1997