Erratum: Galactic potential constraints from clustering in action space of combined stellar stream data

Monthly Notices of the Royal Astronomical Society, vol. 512, pp. 4455–4463

Abstract

This is an erratum to the paper ‘Galactic potential constraints from clustering in action space of combined stellar stream data’ (Reino et al. 2021). In the original paper, there were errors in our action-space probability density estimates when considering multiple streams in combination. The errors do not stem from our overall procedure but from a technical limitation of our density estimator, that was not known to us, or the author of the density estimator, a priori. When estimating the probability densities to calculate the weighted Kullback–Leibler Divergence we artificially separated the streams from each other by shifting them in Lz to heighten the sensitivity of our method. We deemed this to be equivalent to estimating the probability density of each stream separately after accounting for normalization. We have since reviewed this assumption and found that in the case of the EnLink algorithm which we used to calculate our densities, this assumption is not correct. In particular, EnLink induces a shear-like effect on to the densities of clusters with large separation. This effect is stronger the fewer points there are in a particular cluster. The shift that we applied to Lz therefore caused the necessary condition for the shear-effect to distort the estimated densities. As a consequence all our weighted combined-stream results were incorrect. All individual stream results, and all unweighted combined-stream results, on the other hand, were correct and remain unchanged. Illustration of shearing behaviour in EnLink density estimation (with method being ‘sbr’ and number of nearest neighbors being 10) for well-separated clusters of points in two dimensions. Upper left: estimated density ρ obtained by running EnLink on each cluster individually, normalized to integrate to one over the entire distribution (equation 2). Upper right: fractional deviation in density Δρ/ρ, relative to the individually computed densities, when EnLink is run on both clusters simultaneously at the locations shown. Lower left: fractional deviation in estimated density relative to individual-cluster estimates when the cluster with larger σ and more points is shifted by 40σ horizontally. Lower right: fractional deviation in density relative to individual estimates when a cluster with larger σ and fewer points shifted by 40σ horizontally. By construction the clusters are separated by a few times |$\boldsymbol{\Sigma }_2$| (the size of the larger cluster). Applying the density estimator to the entire distribution as constructed, including both clusters, produces estimates that differ only fractionally from those estimated for each cluster individually (Fig. 1, upper right-hand panel). However, if we artificially shift cluster 2 by 40σ2 in one component of |$\boldsymbol{J}$|⁠, then the clusters are now separated by many times their size. This shift is similar in magnitude, relative to the size of the clusters, to the shift applied in our paper to guarantee that all clusters were distinct in action space. Using EnLink on this density distribution produces density estimates that are severely sheared for the cluster with fewer points, as shown in the lower left-hand panel of Fig. 1. The shear is aligned with the direction toward the location to which the cluster with more points was shifted for the density estimation: densities in the cluster with fewer points are over-estimated along this line and under-estimated away from it. Finally, we can verify that the effect is due mainly to the number of points in the cluster, and not to its spread or compactness, by setting N1 = 500 and N2 = 100 for fixed |$\boldsymbol{\mu }_j$| and |$\boldsymbol{\Sigma }_j$|⁠, shifting cluster 2, and recalculating the densities with EnLink. In this case the shear is induced in cluster 2, which is now the one with fewer points (Fig. 1, lower right-hand panel). This correction has a minor effect on our results. Here, we present the rectified versions of the affected figures and tables. Figs 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 and Tables 1, 2, 3 are the correct versions of figs 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, B1 and tables 1, 2, 3, respectively, in the original paper. The new enclosed mass constraints we find are: |$M(\lt 20\ \mathrm{kpc}) = 2.22^{+0.10}_{-0.08}$| from the GD-1, Pal 5 and Orphan streams and |$4.10^{+2.05}_{-1.25} \times 10^{11} \ M_{\odot }$| when also including the Helmi stream. These values replace those originally reported in Reino et al. (2021). As Figs 10 and 11 show, our corrected three-stream results are in much better agreement with other recent measurements of the Galactic profile. All conclusions made in the original paper still apply. Weighted combined data results for the single-component potential. The best-fitting point is marked with a pink cross, the grey points represent potentials that resulted in unbound stars and were therefore discarded, and other points are colour-coded according to their KLD2 value. The orange region shows 1σ contours. Top: results when combining all four streams. Bottom: results for the combination of GD-1, Orphan, and Pal 5. The weighted combined data results for a single-component potential: marginalized single parameter distributions. The top panel shows the results from the combination of all four streams, and the bottom panel shows the results of the combination of GD-1, Orphan, and Pal 5 streams. The green points show the parameter values against the wKLD2 of the potential they belong to. The values of the parameters in the best-fitting potentials are marked with a pink cross. The black lines are drawn at wKLD2 = 0.5 that signifies the 1σ confidence interval. The light grey bars show the range of values that are accepted with 1σ confidence. Comparison of best-fitting parameter values for the single-component potential with their 1σ confidence intervals for enclosed mass (left), scale length (middle), and flattening (right). Results for individual streams are labelled with the stream name. Combined results are labelled with ‘GD-1/Orphan/Pal 5’ for the three-stream combination and ‘All’ for the four-stream combination. In addition, the combined stream labels end with an ‘s’ or a ‘w’ for the standard and weighted analyses, respectively. We remind the reader that e > 1 corresponds to an oblate potential, while e < 1 corresponds to a prolate potential. As in Fig. 2, but showing the weighted combined data results for the two-component potential. Top: results for our four-stream data set. Bottom: results for the combination of GD-1, Orphan, and Pal 5. Comparison of the best-fitting parameter values for the two-component potential with their 1σ confidence intervals for enclosed mass (left), scale length (middle), and flattening (right). Results for individual streams are labelled with the stream name. Combined results are labelled with ‘GD-1/Orphan/Pal 5’ for the three-stream combination and ‘All’ for the four-stream combination. We remind the reader that e > 1 corresponds to an oblate potential in our convention. The orbits for the GD-1 (top), Pal 5 (middle), and Orphan (bottom) streams. These shaded regions correspond to the allowed orbits of the stars whose current position has been marked with a dot of the same colour. The edges of the shaded regions are defined by the potentials that produce the highest and the lowest enclosed mass within 20 kpc among those that are within 1σ of the best-fitting results of the combined analysis of GD-1, Orphan, and Pal 5 streams. Other stars in the stream are shown with purple dots. Axes are in the Galactocentric frame. Comparison of actions of GD-1 (orange points), Pal 5 (teal points), and Orphan (yellow points) produced by two different two-component potentials: the best-fitting potential of the three-stream combination and another potential. Left: KLD1 values of the three-stream combination as a function of enclosed mass and scale length of the outer component. The higher the KLD1 value, the more clustered the action space. Centre: action distribution of the best-fitting potential (black cross in left-hand panel). Right: action distribution of a different potential with KLD1 0.35 lower than the corresponding best fit (purple cross in left-hand panel). All Orphan stars have been shifted up by |$2500\ \rm kpc\ km\ s^{-1}$| in Jν for clarity on both panels. Histograms of the current λ for stars in GD-1 and Pal 5 in different two-component potentials, normalized by the pericentre and apocentre positions of each star’s orbit to serve as a proxy for the orbital phase, with pericentre at 0 and apocentre at 1. Top left-hand panel: the approximate orbital phase of GD-1 stars assuming the best-fitting GD-1 potential. Top right-hand panel: the position of GD-1 stars assuming the potential best fit to the combined GD-1, Orphan, and Pal 5 data. Bottom left-hand panel: the position of Pal 5 stars assuming the best-fitting Pal 5 potential. Bottom right-hand panel: the position of Pal 5 stars assuming the potential best fit to the combined GD-1, Orphan, and Pal 5 data. Comparison of our two-component potential results with previous measurements of the enclosed mass at different radii. The light blue shaded area represents the combined results for all four streams (left-hand panel) and for GD-1, Orphan, and Pal 5 (right-hand panel). The darker shaded regions show the subset of the potentials that are compatible with the current measurements of the local standard of rest velocity. The black markers signify individual stream results at their respective average radii. The coloured markers show measurements of the enclosed mass by other authors. The markers showing the results of Koposov, Rix & Hogg (2010) and Newberg et al. (2010) are slightly offset from 20 and 50 kpc, respectively, for clarity. Rotation curves corresponding to the weighted combined-stream results of the two-component potential model. The lighter shaded region shows the rotation curves for potentials within 1σ of the best fit for each data set. The purple, orange, teal, and yellow data points correspond to the results of the Helmi, GD-1, Pal 5, and Orphan streams, respectively: they show the rotation velocity of the best-fitting potential at the mean Galactocentric distance of the stars in that stream. The darker shaded region shows the subset of the rotation curves that are compatible with the current measurements of the local standard of rest velocity. For comparison, the dashed black and cyan lines are the rotation curves from the galpy MWP otential2014 (Bovy 2015) and McMillan (2017), respectively, and the grey dots represent the data from Eilers et al. (2019). This is the corrected version of the top panel of fig. 13 in the original paper. Comparison of action space of GD-1 (top panels), Pal 5 (middle panels), and Orphan (bottom panels) produced by two different two-component potentials. The action of stars are coloured based on their energies. Left: KLD1 values of corresponding individual stream analysis given as a function of enclosed mass and scale length of the outer component. The higher the KLD1 value, the more clustered the action space. Centre: action distribution of the best-fitting individual-stream potentials (black cross in left-hand panel). Right: action distribution of the same stream in the best-fitting three-stream potential (purple cross in left-hand panel). The KLD1 value of the purple cross potential in a single-stream analysis is reduced by 0.86 (for GD-1), 0.23 (for Pal 5), and 0.16 (for Orphan) compared to the KLD1 value of the black cross potential. The individual and combined stream results for a single-component potential. Best-fit parameters are given with their 1σ confidence intervals. We remind the reader that e > 1 corresponds to an oblate potential, while e < 1 corresponds to a prolate potential. The individual and combined stream results for a single-component potential. Best-fit parameters are given with their 1σ confidence intervals. We remind the reader that e > 1 corresponds to an oblate potential, while e < 1 corresponds to a prolate potential. The individual and combined stream results for a two-component potential. Best-fitting parameters are given with their 1σ confidence intervals. Note that e > 1 corresponds to an oblate potential in our convention. The individual and combined stream results for a two-component potential. Best-fitting parameters are given with their 1σ confidence intervals. Note that e > 1 corresponds to an oblate potential in our convention. KLD1 values for the best-fitting results from individual and combined streams for single-component (1-comp) and two-component (2-comp) potentials. KLD1 values for the best-fitting results from individual and combined streams for single-component (1-comp) and two-component (2-comp) potentials.

Authors 7

1. Stella Reino corresponding Aachen Leiden Observatory

   Leiden University · Leiden Observatory

   Affiliation as printed

   Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, the Netherlands

2. Elena Maria Rossi Aachen Leiden Observatory

   Leiden University · Leiden Observatory

   Affiliation as printed

   Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, the Netherlands

3. Robyn E. Sanderson

   Flatiron Institute · University of Pennsylvania

   Affiliation as printed

   Center for Computational Astrophysics, Flatiron Institute, 162 5th Ave., New York, NY 10010, USA

   Department of Physics and Astronomy, University of Pennsylvania, 209 S 33rd St, Philadelphia, PA 19104, USA

4. Elena Sellentin Aachen Leiden Observatory

   Leiden University · Leiden Observatory

   Affiliation as printed

   Leiden Observatory, Leiden University, Niels Bohrweg 2, NL-2333 CA Leiden, the Netherlands

5. A. Helmi

   University of Groningen

   Affiliation as printed

   Kapteyn Astronomical Institute, University of Groningen, PO Box 800, NL-9700 AV Groningen, the Netherlands

6. Helmer H. Koppelman

   University of Groningen

   Affiliation as printed

   Kapteyn Astronomical Institute, University of Groningen, PO Box 800, NL-9700 AV Groningen, the Netherlands

7. Sanjib Sharma

   The University of Sydney

   Affiliation as printed

   Sydney Institute for Astronomy, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia