Extragalactic

Last time I left off with the remark that while most of the sample of disk galaxies clearly exhibits a tight relationship between circular velocity and stellar mass, there are some apparent outliers as well. While some “cosmic variance” is expected most of the apparent outliers are due to model failures, which have several possible causes:

1. Violation of the physical assumptions of the model, namely that the stars and gas are rotating (together) in the plane of a thin disk that’s moderately inclined to our line of sight (see my original post on this topic).
2. Errors in the photometry. I use two photometric quantities (specifically nsa_elpetro_ba and nsa_elpetro_phi) from the MaNGA DRPALL catalog to set priors for the kinematic parameters cos_i and phi(cosine of the disk inclination and angle to receding side) and also to initialize the sampler. Since proper and in practice fairly informative priors are required for these parameters errors here will propagate into the models, sometimes in ways that are fatal. I’ll look in more detail at some examples below.
3. Bad velocity data.
4. Not enough data.
5. Sampler convergence failures with no obvious cause.

The first two bullet points are closely related: most of the failures to satisfy the physical assumptions are directly related to errors in the photometric decompositions. One fairly common failure was galaxies that were too nearly face on to obtain reliable rotation curves. As an example here is the galaxy with the lowest estimated rotation velocity in the sample, mangaid 1-135054 (plateifu 8550-12703):

Besides showing no sign of rotation the velocity field hints at possible large scale, low velocity outflow in the central region. There are also a few apparent outliers, although these had little effect on model results. Fortunately the model output gives us plenty of clues that the results shouldn’t be trusted. The median circular velocity estimate is unrealistically low with very large posterior uncertainty (above right), while the posterior marginal density for cos_ihas a mode near 1 and also very large uncertainty (below).

Zooming out a bit on SDSS imaging gives a likely explanation for the peculiar velocity field. The elliptical galaxy just to the NW has nearly the same redshift (the velocity difference is ∼75 km/sec) and is almost certainly interacting with our target.

A key assumption in using photometric properties as proxies for kinematic quantities is that disk galaxies have intrinsically circular surface brightness profiles. This is never quite the case in practice and sometimes morphological features like strong bars can make this assumption catastrophically wrong. Here was perhaps the most extreme example in DR14:

The photometric major axis angle was estimated to be 98.4^o, that is just south of east, while the position angle of the maximum recession velocity is close to due south. When I first examined this velocity field I had a hard time reconciling it with rotation in a thin disk. This was before I learned how to do Voronoi binning though. The image below shows the binned velocity field (with a target S/N of 6.25). This shows that the relative velocity does increase on a roughly north to south line, indicating that this is indeed a rotating disk galaxy.

I mentioned in an earlier post that without a proper prior on the kinematic position angle phi these models are inherently multi-modal (in fact they are infinitely modal and therefore would have improper posteriors). The solution to that of course is to have a proper prior. But if the prior is seriously in error the posterior estimates for the components of the velocity will end up scrambled as can be seen in the top row of the graph below, which shows the posterior distributions of the circular and “expansion” velocities¹Remember that the photometric position angle is determined modulo π while the direction to the maximum recession velocity is measured modulo 2π. We don’t care about a π radian error in the prior though because that just flips the signs of the velocity components, which causes no sampling issues and is trivially fixable in the generated quantities block. It’s smaller errors that cause problems.

The obvious solution to a bad prior is to correct it²Using the data you’re trying to model to establish a prior is, technically, cheating (the polite term is “Empirical Bayes”), but this seems a relatively benign use of the data., which is easy enough. The bottom row shows the results of re-running the model with a prior phicentered on 180^o and the same data. Now both the circular and expansion velocity curves are at least plausible. The posterior mean of phiis ≈185^o, which is very close to correct as can be seen in the binned velocity field shown above.

One more example. Bars were the most common cause of misleading photometric decompositions, but not the only one. Some galaxies are just asymmetrical. Here is the one that had the largest offset between the photometric and kinematic position angles:

And the velocity field (again this agrees well with the Marvin measurements of the data cube):

This time the velocity field looks unremarkable, but again because of the prior the estimated circular and expansion velocities are scrambled together. And once again also, changing the prior to be centered on the approximate actual position angle of the receding side produces reasonable estimates for both:

Next up, I’ll take a more holistic look at final sample selection, and maybe get to results.

As I mentioned in the previous two posts SDSS Data Release 15 went public back in December and a query for “normal” disk galaxies as judged by Galaxy Zoo 2 classifiers returned 588 hits. I’ve finally run the GP velocity models on all of the new data and made a second run on around 40 that were contaminated by foreground stars or neighboring galaxies. So far I haven’t found an alternative to selecting these by eye and doing the masking manually, so that’s an error prone process. The month+ long gap between postings by the way was due to travel — my computer wasn’t grinding away on these models for all that time. As I mentioned last post the sampling properties including execution time of the GP model with arctangent mean function are usually quite favorable using Stan. The median wall time for these runs was about a minute, with a range from 25 to 1600 seconds. All model runs used 500 warmup iterations and 500 post-warmup with 4 chains run in parallel. This is more than enough for inference.

Before I discuss the results I’ll show them. As I did for the first pass at this way back in July I retrieved stellar mass and uncertainty estimates made by the MPA-JHU group from CasJobs; all but a handful also have mass estimates from the Wisconsin group. I may look at those later but don’t anticipate any very significant differences.

There are now at least two plausible choices for reference circular velocities: the velocity at a fiducial radius, and again I choose 1.5 effective radii since the MaNGA IFUs are meant to cover out to that radius in the primary sample. The other obvious choice is the asymptotic velocity v_c in the arctangent mean function. This seems in principle to be the better choice since it estimates the circular velocity in the flat part of the rotation curve, but it might be a considerable extrapolation from the actual data in some cases.

Both sets of results are shown below for all model runs that ran to completion (N=582). Plotted are the median, 2.5, and 97.5 percentiles (≈ ± 2σ) of posterior predictions for the log-circular velocity at 1.5r_eff (top graph) and the same quantiles of the posteriors of the parameter v_c (bottom graph). These are plotted against the median, 16, and 84 percentiles (≈ ± 1σ) of the total stellar mass estimates per the MPA group.

Evidently most of the sample follows a tight linear relationship with either measure of circular velocity, but there are some apparent outliers as well. I’m feeling a bit blocked right now, so I’ll end the post here. Next time I’ll look at some of the causes of model failure, what to do about them, and get to the results.

I haven’t had a lot of time for this hobby lately and won’t for another month or so, but I haven’t been completely inactive. When I first started modeling disk galaxy rotation curves with Gaussian processes I used a low order polynomial for the mean function. This was an admitted hack, and not a very good one at that. It’s not uncommon in the rotation curve modeling industry to adopt a functional form for the circular velocity. Sometimes it’s physically motivated (usually based on a bulge/disk/dark matter halo model), sometimes it’s purely phenomenological. With a bit of digging in the literature I found a particularly simple form in Courteau (1997):

\(v(r) = v_{0} + \frac{2}{\pi} v_{c} \arctan(r/r_{t})\)

Making this the mean function for the GP model involves just a few simple changes to the Stan code I outlined in that post. We only need two parameters for the curve, and the parameter block now looks like:

parameters {       
                
  real phi;       
                
  real<lower=0., upper=1.> cos_i;  // cosine disk inclination       
                
  real x_c;  //kinematic centers       
                
  real y_c;       
                
  real v_sys;     //system velocity offset (should be small)       
                
  real v_c;       
                
  real<lower=0> r_t;       
                
  real<lower=0.> sigma_los;       
                
  real<lower=0.> alpha;       
                
  real<lower=0.> rho;       
                
}       
                 

The key line in the model block is:

  v ~ multi_normal_cholesky(v_sys + sin_i * (2./pi() * v_c * atan(r/r_t) .* yhat ./ r), L_cov);       

And then in the generated quantities block are posterior predictions for new data at the observed points and in concentric rings:

  // model and residuals at the original points       
                
         
                
  v_rot = 2./pi() * v_c * atan(r/r_t);       
                
  vf_model = gp_pred_rng(xyhat, v-v_sys-sin_i * (v_rot .* yhat ./ r), xyhat, alpha, rho, sigma_los);       
                
  vf_model = vf_model + v_sys + sin_i * (v_rot .* yhat ./r);       
                
  vf_res = v - vf_model;       
                
         
                
  // model in rings       
                
         
                
  v_gp = gp_pred_rng(xy_pred, v-v_sys-sin_i * (v_rot .* yhat ./ r), xyhat, alpha, rho, sigma_los);       
                
  v_pred = (v_gp + sin_i * 2./pi() * v_c * atan(r_pred/r_t) .* y_pred ./ r_pred) *v_norm/sin_i;       
                
  vrot_pred[1] = 0.;       
                
  vexp_pred[1] = 0.;       
                

The full code is available in my github repository at https://github.com/mlpeck/vrot_stanmodels.

This model turned out to have very nice sampling properties using Stan’s version of NUTS, and convergence is fast enough in terms of both number of iterations and walltime to make it easily feasible to model a large sample.

Since I’ve been using this as a running example here are posterior circular and expansion velocity estimates for plateifu 8942-12702 (mangaid 1-218280):

These curves are shaped just slightly differently from the ones I posted in July, but the confidence bounds are somewhat tighter. Whether this is actually due to better inference remains to be seen.

Second topic: SDSS Data release 15 went public in December as promised. I haven’t had time to dig into it much, but I did rerun the query I used to get a sample of high confidence disk galaxies based on GZ2 classifications. Run in the DR15 context that query got 588 hits, an increase of 229 from DR14 and still about 13% of the entire MaNGA galaxy sample excluding ancillary targets that aren’t also part of the main samples.

I downloaded the RSS files and have done a first pass at rotation curve modeling of the 229 new galaxies. Unfortunately I won’t have time to actually analyze what I have until, most likely, March. To conclude for now here is the kinematically most interesting example I saw from the new to DR15 sample. GZ2 classifiers called this a normal disk galaxy by a large majority

and were about evenly divided on whether it has a bar. And here is its velocity field measured from the stacked RSS file on the left, with the modeled velocity field on the right. The left graph is interpolated to the same resolution as the data cubes.

This is an incomplete map because of the signal/noise threshold I set, but it’s apparent that the outer disk is counter-rotating relative to the inner disk. This is confirmed for the stellar velocity field in the SDSS DAP, while the ionized gas rotates in the same sense as the outer disk throughout. Keep in mind that I measure blended velocities that in any given case could be dominated either by stellar light or ionized gas — in this galaxy the emission lines are very weak, so I measured the stellar velocity field. The GP based model does a good job of fitting the data, but the rotation curve isn’t exactly the standard shape:

One thing this demonstrates is the versatility of the gaussian process model with a simple mean function (even if it’s wrong).