3 The adopted measure of folding quality

The very noisy nature of rotation curve data means that any solution to the folding of any given rotation curve is likely to contain a large element of uncertainty; it follows that, given two reasonable folding techniques, there are likely to be many rotation curves which are best folded by one particular technique, and vice-versa for other rotation curves whilst, at the same time, any objective judgement about which are the best individual solutions is likely to be extremely difficult (if not impossible) at best. It follows that any judgement about the relative merits of any two folding methods must be made by some global statistical method applied to the folded solution of a large number of rotation curves. The method used for the present development is described in the following.

3.1 The ln A distribution

As we have indicated, the basic result (shorn of the fine details) of Roscoe 1999A was that optical rotation curve data is described, to a high statistical precision, by the power-law structure $V = A R^\alpha$ where $(A,\alpha)$ are constants which differ from galaxy to galaxy. Essentially, but with the "hole-cutting'' data-reduction process described in Roscoe 1999A, $\ln A$ and $\alpha$ for each rotation curve were determined by linear regression of $\ln V$ data on $\ln R$ data. Thus, for the PS sample of 900 good quality rotation curves we obtain 900 pairs $(\ln A,\alpha)$.

Figure 2: Comparison of MFB and PS solutions. The vertical lines in both diagrams indicate the positions of the A, B, C, D and E peaks in the PS solution

Figure 2 (left) shows the $\ln A$ distribution for the MFB solution (that is, when the rotation curves are folded using MFB's optical centre and their estimate of $V_{{\rm sys}}$), whilst Fig. 2 (right) shows the corresponding distribution for the PS solution. A comparison shows that the peaks labelled B, C and D in the MFB solution are reproduced - enhanced and with much reduced noise - in the PS solution. Since the MFB solutions were determined directly from observational estimates of optical centres and systematic redshifts whilst the PS solutions were determined using a "by-eye'' folding process without any reference to an underlying power-law structure (and hence without reference to any $\ln A$ distribution), then the foregoing comparison strongly suggests that the B, C, D peak structure reflects objective physical qualities in the underlying distribution of rotation curves. This conclusion was reached independently (with the addition of the A and E peaks), supported by very strong statistics, in Roscoe 1999B.

For these reasons, the $\ln A$ distribution is adopted here as a suitable statistical representation of a folding solution over a large number of rotation curves.