9 The Mk III auto-folder

The Mk III auto-folder is based on the working hypothesis that the folding process should only use the exterior parts of the rotation curve, so that hole-cutting becomes an active part of this process. However, there is one immediate practical difficulty: the hole-cutting algorithm requires rotation curves to be already folded. The resolution of this problem is embodied in the algorithmic definition of the Mk III auto-folder, described in Sect. 9.1, and the solution arising from it applied over the PS data is shown in Fig. 5 right.

Figure 5: Comparison of Mk II and Mk III auto-folder solutions. The vertical lines indicate the positions of the A, B, C, D and E peaks in the PS solution

   
9.1 Algorithmic definition of the Mk III auto-folder

The Mk III auto-folder is defined by the following algorithm:

• For any given rotation curve, fold using the Mk II auto-folder;
• record the number of Fourier modes used for this fold, say $N_{\rm f}$, and the estimates for $V_{{\rm sys}}$ and $O_{{\rm dyn}}$;
• apply the hole-cutting algorithm defined Roscoe 1999A to this folded rotation curve, and record the removed points;
• take the original unfolded rotation curve, and reduce it by removing these recorded points;
• fold this reduced rotation curve using the Fourier method already described, but taking care to redefine the Fourier coefficients in the manner described in the next subsection;
• for this second-stage folding process, use either the $N_{\rm f}$ modes recorded at the first-stage folding process, or the number of modes defined by the algorithm of Sect. 6.2, whichever is the least;
• use the recorded estimates for $V_{{\rm sys}}$ and $O_{{\rm dyn}}$ as the initial guess for the minimization process of this second-stage folding.

9.2 Partition of Fourier region

The application of the hole-cutting algorithm of Roscoe 1999A cuts a hole out of the rotation curve which is centred on the estimated dynamical centre of the galaxy; consequently, instead of representing a continuous curve, the reduced rotation curve data represents two separated continuous sections with a gap in between. This means that the Fourier cosine coefficients cannot be computed using (2), but must be computed by an integral of the form

 

$$A(m) = {1 \over 2\,X} \int_{-X}^{-X'} f(x) \cos (m \pi x/X) ~{\rm d}x ~+~$$

$${1 \over 2\, X} \int_{X'}^X f(x) \cos (m \pi x/X) ~{\rm d}x,$$ (3)

where X' is some positive number representing the boundaries of the cut-out hole. In practice, of course, the hole-cutting strategy will generally remove a section which is non-symmetric about the current estimate of $O_{{\rm dyn}}$, and so the data on one of the two sections of the reduced rotation curve must be interpolated/extrapolated to ensure that the hole which has been effectively cut out of the rotation curve is centred exactly on the current estimate of $O_{{\rm dyn}}$. Once the removed section is centred in this way, and the Fourier modes computed using (3), rather than (2), the process is as before.