13 Folding of simulated rotation curves

So far, the efficacy of the developed auto-folder has been judged by means of a comparison of the PS folding solution and the auto-folder's solution defined on the same sample; the assumption is, of course, that the PS solution over the sample can be considered "correct'' in an overall statistical sense. However, in the absence of objective certainties, it is useful to see how the auto-folder deals with the folding of simulated rotation curves for which the folded solution is known a priori.

The following describes the results of applying the auto-folder to three sets of five simulated rotation curves per set, where each set is chosen to represent a particular sampling regime - low-density sampling, medium-density sampling and high-density sampling.

13.1 Simulation details

The considerations of Roscoe (1999A) show that, to an extremely high level of statistical precision, optical rotation curves conform to a power-law, $V\,=\,A\,R^\alpha$ for $R\,>\,R_{{\rm min}}$ where $R_{{\rm min}}$ defines the exterior limit of the dynamical effects of the bulge on the disc.

Given this, and in order to preserve maximum authenticity otherwise, we used the existing PS sample as the basis for generating simulated unfolded rotation curves as follows:

• fold the PS sample using the auto-folder and record the set of parameters, $(\Delta R,\, V_{{\rm sys}})$, which define each folding solution. Here, $\Delta R$ is the offset between MFB's estimated dynamical centre and the auto-folder's estimated dynamical centre, whilst $V_{{\rm sys}}$ is the auto-folder's estimated systematic velocity for the galaxy concerned;
• record the sampling radii, $R_i,\,i\,=\,1,..,N$ on each rotation curve, and note which is on the approaching arm and which is on the receding arm;
• at each radial position, $R_i,\,i\,=\,1,..,N$, on any give rotation curve, record the value of the cross-correlation coefficient, $\rho_i$, used by MFB to indicate the quality of velocity measurement at R_i - see Sect. 2.1;
• calculate A, $\alpha$ and $R_{{\rm min}}$ for each rotation curve;
• for $R_i\,>\,R_{{\rm min}}$, form $V(R_i)\,=\,A\,R_i^\alpha \,+ \, \epsilon_i,~i=1,..,N$ where $\epsilon_i$ is a normal random variable defined so that 95% of its values lie within the interval [-a,+a] km s^-1 when $\rho_i\, = \,0.4$, and within the interval [-a/6,+a/6] km s^-1 when $\rho_i\,=\,0.9$. Here, a is a positive parameter with a typical value of 20 km s^-1;
• in the absence of any quantitative knowledge in the region $R\,<\,R_{{\rm min}}$, we simply form $V(R_i)\,=\,B\,R_i^\beta \,+\,\epsilon_i$ in this region, where B is chosen to ensure continuity of velocity across $R\,=\,R_{{\rm min}}$, and $\beta$ is a normal random variable chosen so that 95% of its values lie within the interval [0,1];
• unfold the rotation curve by assigning to each velocity, $V(R_i),\,i=1,..,N$, a positive or negative signature according to whether the sampling radius, R_i, is on an approaching or receding arm;
• make a random selection of $(\Delta R,\, V_{{\rm sys}})$ from the previously computed list, and transform the unfolded rotation curve according to $R_i\,\rightarrow\,R_i\,+\,\Delta R$and $V(R_i)\,\rightarrow\,V(R_i)\,+\,V_{{\rm sys}},~i=1,..,N$;
• apply the auto-folder to the folding of these simulated rotation curves.

13.2 Simulation results

When the velocity noise parameter, $\epsilon_i$, is set to zero, then the auto-folder folds all of the simulated rotation curves (864 of them) exactly, to within rounding errors. This demonstrates that the general logic of the underlying algorithm is working correctly.

Typically realistic simulated rotation curves resulted when the velocity noise parameter, $\epsilon_i$, was defined by setting a = 20 km s^-1 in the foregoing prescription. The results of applying the auto-folder to the folding of a sample of these noisy curves are shown in Figs. 9, 10 and 11 respectively. In all cases, it is seen that the auto-folder solution has the appearance of being reasonable in the sense that it is not obvious that the exact solution is necessarily better. However, a detailed comparison of the exact solutions with the auto-folder's solutions shows that, in general, the auto-folder's solutions improve as the sampling rate increases; this is a reasonable and expected result.