4.2 Kinematic modelling: general description
4.2.3 Fitting process
To fit the model to the data we use the KinMS_MCMC code, which utilises Gibbs sampling and adaptive stepping3 to explore the parameter space. To measure the
goodness-of-fit of the model to the data a chi-squared statistics is used: χ2 =X i datai−modeli σi !2 = 1 σ2 X i (datai−modeli)2 (4.4)
where the sum is performed over all the pixels within the region of the data cube that the model fits, and σ is the rms noise level measured in line-free channels of the data cube (see Table 3.4), assumed to be constant for all the pixels. The 3Gibbs sampling is an MCMC algorithm that produces samples from a posterior distribution conditioned on the observed data and thus allows to obtain final estimates of a model parameter. It is applicable when the joint distribution is not known explicitly or is difficult to sample from directly, but the conditional distribution of each variable is known and is easy to sample from. The Gibbs sampling algorithm generates an instance from the distribution of each variable in turn, conditional on the current values of the other variables. In the version used in this work it incorporates an adaptive stepping algorithm that allows us to explore the parameter space by progressively adapting the intervals within which the priors are left free to vary, in order to make the code converge faster.
Figure 4.2: Example of the arctangent models used to fit the CO velocity curves (adapted from Courteau 1997). The different curves are for different values of the truncation radius, rt, and the same circular velocity in the flat part of the curve
(here assumed to be vc = 200 km s−1).
posterior distribution of each model is then described by the log-likelihood function ln P = −χ2/2. As in Smith et al. (2019), we re-scale the uncertainty of the fitted
parameters by a factor of (2N)0.25, where N is the number of pixels with detected
emission in the mask, as defined in Section 3.3. This ensures that the uncertainty in the χ2 statistics for high N does not lead to unrealistically small uncertainties in
our final fit values.
To ensure our fitting process converges, we set reasonable priors for the physical parameters that are left free to vary in the fit. Specifically, the CO central velocity offset (i.e. the shift with respect to the kinematic centre) is allowed to vary within the channel widths of each data cube (listed in Table 3.2). The gas velocity dispersion (σgas) is constrained to be less than the maximum line-of-sight velocity widths
shown in Chapter 3. Within KinMS, the kinematic PA of the gas distribution on the sky is defined as the angle measured counterclockwise from the north to the receding side of the velocity field. We then made an initial estimate of the the kinematic PA by looking at the moment 1 maps of our CO detections, and then allowed it to vary by ±40 degrees around the estimated values. By definition, the disc inclination (θinc) is the angle between the observer line-of-sight and the
normal to the disc plane. In our modelling, we initially left it free to vary over the full physical range (0◦−90◦, where 0◦ and 90◦ mean “face-on” and “edge-on”,
Table 4.1: Best-fit model parameters.
Target PAkin θinc vflat rturn σgas voffset(vCO) Rhole χ2red
(deg) (deg) (km s−1) (arcsec) (km s−1) (km s−1) (km s−1) (arcsec)
(1) (2) (3) (4) (5) (6) (7) (8) (9) IC 1531 356±1.0 32±2.5 272±29 0.07±0.02 3.4±2.3 −25±1.0 (7677) − 1.2 NGC 612 183±0.05 81±0.01 453±0.04 1.2±0.05 20±0.02 −95±0.01 (8879) 0.3 1.2 IC 4296 240-220 68±1.5 404±1.5 0.01±0.002 64±1.5 −16 ± 2.0 (3691) − 1.0 NGC 7075 322±4.0 46±1.5 446±56 0.03±0.02 5.1±2.6 7.8±3.0 (5491) − 1.1
Notes. − Columns: (1) Target name. (2) Kinematic position angle of the CO disc (i.e. the PA measured counter-
clockwise from North to the receding side of the velocity field). It is given as a range where a position angle warping is modelled. (3) Inclination angle of the disc. (4) Asymptotic (or maximum) circular velocity in the flat part of the rotation curve. (5) Effective radius at which the rotation curve turns over. (6) Gas velocity dispersion. (7) Velocity offset of the kinematic centre, i.e. offset between the expected and observed velocity zero-point of the line emission. Specifically, the expected zero-point corresponds to the redshifted frequency of the CO(2-1) transition listed in column (3) of Table 3.2; the observed zero-point consists in the best estimate of the line systemic velocity as inferred from the observed spectral profiles. The corresponding CO central velocity is reported in parentheses. (8) Radius of the central surface brightness cut-off. (9) Reduced χ2of the best-fit model.
respectively); in subsequent iterations, it is constrained to vary within ±20◦ of the
value at which the first chain converges. The maximum circular velocity of CO (vflat) and the turnover radius (rturn) are constrained to lie within ±40 km s−1 and
two beam widths, respectively, around best guess values determined by visually inspecting the velocity curves (PVD) of the six CO discs. In particular, in those cases in which the turn over of the velocity curve is resolved (e.g. NGC 612), we initially set vflat as the velocity value associated with the flat part of the curve and
rturn as the radius where the curve starts to turn over; when the flattening of the
velocity curve is not resolved (e.g. NGC 7075), we assume as best initial guesses for vflat and rturn the maximum sampled circular velocity and the corresponding radius,
respectively.
Initially, the step size in each fit is manually scaled on a source-by-source basis to ensure a minimum acceptance fraction and the chain convergence. Once the MCMC chains converged, we re-ran the entire chain for an additional 105 steps to produce the full final posterior probability distribution. For each model parameter these probability surfaces are then marginalised over to produce a best-fit value (median of the marginalised posterior distribution) and associated 68 and 99% confidence levels (CL).
(a) (b)
(c)
(d)
Figure 4.3: IC 1531 observed, mock and residual (data-model) mean velocity maps (panels a, b and c, respectively). The black dashed line in panel a indicates the direction of the radio jet axis. The synthesised beam is shown in the bottom-left corner of each panel. The wedges to the right show the colour scale. East is to the left and North to the top. Velocities are measured in the source frame and the zero-point corresponds to the intensity-weighted centroid of the CO emission (vCO;
see Table 4.1). The maps are created with the masked moment technique described in Section 3.3 of Chapter 3, using a data cube with a channel width of 20 km s−1.
CO PVD (panel d) extracted within a rectangular slit which includes all of the CO emission along the major axis. The starting point of the slit (i.e. the smallest offset value) is oriented with respect to the kinematic position angle, indicated in the bottom-left corner of the panel. The contours of the best-fit model are overlaid in cyan. The x-axis indicates the position offset along the extraction axis. The y-axes indicate the velocities centred on the expected (right axis) and observed (left axis) velocity zero-point of the line emission, reported in column (7) of Table 4.1 and in column (3) of Table 3.2, respectively. The contour levels are drawn at 1, 3, 9...times