# 3D motions in the Sculptor dwarf galaxy as a glimpse of a new era

D. Massari1 1 1 , M. A. Breddels1 , A. Helmi1 , L. Posti1 , A. G. A. Brown1 , E. Tolstoy1 1Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands 1Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands
1affiliationmark:

# Methods Section

D. Massari11affiliationmark: 22affiliationmark: affiliationmark: , M. A. Breddels11affiliationmark: , A. Helmi11affiliationmark: , L. Posti11affiliationmark: , A. G. A. Brown22affiliationmark: , E. Tolstoy11affiliationmark: 11affiliationmark: Kapteyn Astronomical Institute, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands 22affiliationmark: Leiden Observatory, Leiden University, P.O. Box 9513, 2300 RA Leiden, The Netherlands
${}^{\ast}$${}^{\ast}$affiliationtext: Corresponding author: massariATastro.rug.nl

The 3D motions of stars in small galaxies beyond our own are minute and yet they are crucial for our understanding of the nature of gravity and dark matter. Even for the dwarf galaxy Sculptor which is one of the best studied systems and inferred to be strongly dark matter dominated, there are conflicting reports on its mean motion around the Milky Way and the 3D internal motions of its stars have never been measured. Here we report, based on data from the Gaia space mission and the Hubble Space Telescope, a new precise measurement of Sculptor’s mean proper motion. From this we deduce that Sculptor is currently at its closest approach to the Milky Way and moving on an elongated high-inclination orbit that takes it much farther away than previously thought. For the first time we are also able to measure the internal motions of stars in Sculptor. We find    and    along the projected radial and tangential directions, implying that the stars in our sample move preferentially on radial orbits as quantified by the anisotropy parameter, which we find to be at a location beyond the core radius. Taken at face value such a high radial anisotropy requires abandoning conventional models for the mass distribution in Sculptor. Our sample is dominated by metal-rich stars and for these we find , a value consistent with multi-component models where Sculptor is embedded in a cuspy dark halo as expected for cold dark matter.

To measure the proper motions (PMs) of individual stars in Sculptor we used data taken years apart. The first epoch was acquired with the Advanced Camera for Surveys on board HST. The data set consists of two overlapping pointings separated by about 2′ ( pc, see Fig. 1), each split in several 400 sec exposures in the F775W filter. The overlapping field-of-view has been observed 11 times. We obtained a catalog of positions, instrumental magnitudes and Point Spread Function (PSF) fitting-quality parameters by treating each chip of each exposure independently. Stellar positions were corrected for filter-dependent geometric distortions. We then cross-matched the single catalogs to compute 3-clipped average positions, magnitudes and corresponding uncertainties. We built the complete HST catalog after excluding all the saturated sources and those that were measured less than 4 times. The second epoch is provided by the Gaia first data release. We extracted from the Gaia archive all sources in the direction of Sculptor.

We transformed the HST positions to the equatorial reference frame defined by the Gaia data (right ascension, RA, and declination, DEC), using a six-parameter linear transformation. We found stars in common and their PMs were computed as the difference between the Gaia and HST positions, divided by the temporal baseline. The uncertainties on the PMs were computed as the sum in quadrature between the Gaia and HST positional errors, divided by the temporal baseline, also taking into account the non-negligible correlations between Gaia’s RA and DEC uncertainties. After this first iteration, we repeated the procedure several times to compute the frame transformations using only likely members of Sculptor. These were selected using their location in the (G, G-m) color-magnitude diagram (Fig. 2a) and their previous PM determination. After three iterations, the number of selected stars stabilized at .

Our final catalog is shown in Fig. 2. Very distant objects such as background galaxies and quasars do not move and thus if present will have an apparent non-zero proper motion as a result of our procedure that sets Sculptor at rest. Although there are no known quasars in our field of view, we were able to identify two background galaxies using the Gaia astrometric excess noise parameter, and confirmed by eye (see Figs. 1b and 1c). Even though these are extended sources, their cores are well fit by a point source-like PSF, making them reliable for defining the absolute reference frame. The relative PMs measured for these two galaxies are red crosses in Fig. 2b. The fact that they both lie in the same region of this PM diagram supports our analysis. We adopted their weighted mean relative proper motion (blue cross in Fig. 2b) as the zero-point, thus the absolute PM for Sculptor is (, )=(, which corresponds to (  assuming a distance of  kpc to Sculptor. Fig. 2c shows that the motions of the stars in the field are coherent. Finally, Fig. 3 compares our PM measurement to previous estimates111during the publication process of this paper, a new estimate has been provided by Sohn et al.2017. More details and a thorough description of the extensive tests we have performed are reported in the Methods section.

To compute the orbit of Sculptor around the Milky Way and also to quantify the effect of “apparent rotation”, we combine our absolute PM measurement with literature values of the line of sight velocity , distance, and sky position of Sculptor. We use these as initial conditions (and also consider PMs within 1 of the measured values) for the integration of orbits in a multi-component Galactic potential. These show that Sculptor moves on a relatively high inclination orbit and that it is currently close to its minimum distance to the Milky Way, as we find its peri- and apocenter radii are  kpc and  kpc. The values of these orbital parameters depend on the assumed mass for the Milky Way halo, but variations of 30% lead to estimates within the quoted uncertainties (see the Methods section for more details).

Finally, we deduce the maximum apparent rotation for this orbit to be  at a position angle  . Therefore if we correct the velocity gradient along the major axis previously measured in Sculptor for this apparent rotation, we find an intrinsic rotation signal along this axis of amplitude  . This implies that at its half-light radius, , for a line-of-sight velocity dispersion  . Given the large pericentric distance and the small amount of rotation we have inferred, this implies that Sculptor did not originate in a disky dwarf that was tidally perturbed by the Milky Way.

We determined the internal transverse motions of the stars in Sculptor using a sub-sample selected such that: () mag, to avoid stars in the HST non-linear regime and those where the Gaia positional errors are more uncertain; () the errors on each of the PM components are smaller than  (corresponding to    at the distance of Sculptor); () the total PM vector is smaller than 0.23 (i.e.  , this limit is set by the apparent PM of the background galaxies). There are 15 stars that satisfy these criteria and hence have the best PM measurements.

We model the velocity dispersion of this sample using a multivariate Gaussian. The parameters of this distribution are the mean velocities in the radial and tangential directions on the plane of the sky (), the dispersions () and their correlation coefficient . We use Bayes theorem to derive the posterior distribution for these parameters (assuming a Gaussian-like prior on the dispersions) from a Markov Chain Monte Carlo (MCMC) algorithm. We find    and  , as shown in Fig. 4a.

If we assume spherical symmetry and neglect rotation (see the Methods section for details), we can use the Jeans equations to find a relation between the velocity dispersions measured at (the location of our fields) and the value of the anisotropy where :

 ^β=1−σ2Tσ2los+σ2R−σ2T. (1)

We determine  for 10 stars in common with a spectroscopic catalog. Using the MCMC chain samples, we obtain the probability distribution for shown in Fig. 4b. The two other histograms in this panel depict the results obtained assuming a flat-prior (dashed) or the more often quoted value  (dotted). In all cases, radial anisotropy is clearly favored, with a median value and the maximum a posteriori .

This is the first ever determination of the value of the anisotropy in an external galaxy. The anisotropy is the key missing ingredient to robustly establish the distribution of matter in Sculptor, reflected in a longstanding unresolved debate, as to whether or not this galaxy has the cuspy profile predicted by the concordance cosmological model in which dark matter is cold, constituted by weakly interacting particles.

The value of we have measured is surprising. A review of the literature indicates that most previous works have assumed spherical symmetry and derived, for a variety of mass models of Sculptor, for constant with radius. However, no physical system can have a constant anisotropy and with a light density profile that has a central slope , since has to satisfy in the spherically symmetric limit. Therefore, in this context, our result shows that the anisotropy in Sculptor cannot be constant with radius. Our measurement also rules out the simplest predictions for Sculptor’s anisotropy based on the alternative gravity model known as MOND.

Our results highlight the necessity to go beyond the standard assumptions. We may need to consider that Sculptor’s dark halo may be axisymmetric or even triaxial. Alternatively and quite plausibly our measurement may be biased towards the colder, more centrally concentrated, metal-rich(er) subcomponent of Sculptor. Of the 15 stars in our best PM sample, 9 have a metallicity measurement (see Methods section for details) and 6 of these have [Fe/H] dex, indicating that about half could belong to this subcomponent of Sculptor. From the 11 stars in our sample with [Fe/H] dex, , and that satisfy also the quality criteria, we determine the anisotropy to be clearly radial with and a median at a distance , as shown in Fig. 4c. This value is in excellent agreement with predictions if Sculptor’s metal-rich component is embedded in a cuspy dark halo profile. It remains to be seen if such a high value can also be consistent with cored models, since those published typically predict lower, though still radial, anisotropy. Another intriguing question is what formation mechanism produces a population of stars moving on such very elongated orbits.

Acknowledgements: We have made use of data from the European Space Agency mission Gaia (http://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement. This work is also based on observations made with the NASA/ESA Hubble Space Telescope, obtained from the Data Archive at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. A.H. and L.P. acknowledge financial support from a Vici grant from the Netherlands Organisation for Scientific Research. M.B. and A.H. are grateful to NOVA for financial support.

Authors’ contributions: D.M. performed the data analysis and the proper motion measurements, M.B. developed the statistical tools, A.H. derived the relations between observables and orbital anisotropy, coordinated the work and led the scientific interpretation, L.P. performed the orbit computation, A.B. and E.T. contributed to the presentation of the paper. All the authors critically contributed to the work presented here. Correspondence and requests for materials should be addressed to massari@astro.rug.nl”

## References

• Lin & Faber (1983) 1. Lin, D. N. C., & Faber, S. M.  Some implications of nonluminous matter in dwarf spheroidal galaxies, Astrophys. J. Lett., 266, 21-25 (1983)
• Strigari (2013) 2. Strigari, L. E. Galactic searches for dark matter, Phys. Rep., 531, 1-88 (2013)
• Walker et al. (2007) 3. Walker, M. G., Mateo, M., Olszewski, E. W., et al.  Velocity Dispersion Profiles of Seven Dwarf Spheroidal Galaxies, Astrophys. J. Lett. , 667, 53-56 (2007)
• Battaglia et al. (2008) 4. Battaglia, G., Helmi, A., Tolstoy, E., et al. The Kinematic Status and Mass Content of the Sculptor Dwarf Spheroidal Galaxy, Astrophys. J. Lett., 681, L13 (2008)
• Schweitzer et al. (1995) 5. Schweitzer, A. E., Cudworth, K. M., Majewski, S. R., & Suntzeff, N. B. The Absolute Proper Motion and a Membership Survey of the Sculptor Dwarf Spheroidal Galaxy, Astron. J., 110, 2747-2757 (1995)
• Piatek et al. (2006) 6. Piatek, S., Pryor, C., Bristow, P., et al. Proper Motions of Dwarf Spheroidal Galaxies from Hubble Space Telescope Imaging. IV. Measurement for Sculptor, Astron. J., 131, 1445-1460 (2006)
• Walker et al. (2008) 7. Walker, M. G., Mateo, M., & Olszewski, E. W. Systemic Proper Motions of Milky Way Satellites from Stellar Redshifts: The Carina, Fornax, Sculptor, and Sextans Dwarf Spheroidals, Astrophys. J. Lett., 688, L75 (2008)
• Gaia Collaboration et al. (2016a) 8. Gaia Collaboration, Prusti, T., de Bruijne, J. H. J., et al. The Gaia mission, Astron. Astrophys., 595, A1 (2016)
• Battaglia et al. (2013) 9. as reviewed in Battaglia, G., Helmi, A., & Breddels, M. Internal kinematics and dynamical models of dwarf spheroidal galaxies around the Milky Way, New Astron. Rev., 57, 52-79 (2013)
• Strigari et al. (2017) 10. Strigari, L. E., Frenk, C. S., & White, S. D. M.  Dynamical Models for the Sculptor Dwarf Spheroidal in a ΛCDM Universe, Astrophys. J., 838, 123-132 (2017)
• Anderson (2007) 11. Anderson, J. Variation of the Distortion Solution, Inst. Sci. Rep. ACS 2007-08, 12 pages, 8 (2007)
• Gaia Collaboration et al. (2016b) 12. Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. Gaia Data Release 1. Summary of the astrometric, photometric, and survey properties, Astron. Astrophys., 595, A2 (2016)
• Irwin & Hatzidimitriou (1995) 13. Irwin, M., & Hatzidimitriou, D. Structural parameters for the Galactic dwarf spheroidals, Mon. Not. R. Astron. Soc., 277, 1354-1378 (1995)
• Massari et al. (2017) 14. Massari, D., Posti, L., Helmi, A., Fiorentino, G., & Tolstoy, E. The power of teaming up HST and Gaia: the first proper motion measurement of the distant cluster NGC 2419, Astron. Astrophys., 598, L9 (2017)
• Dinescu et al. (1999) 15. Dinescu, D. I., Girard, T. M., & van Altena, W. F. Space Velocities of Globular Clusters. III. Cluster Orbits and Halo Substructure, Astron. J., 117, 1792-1815 (1999)
• Sohn et al. (2012) 16. Sohn, S. T., Anderson, J., & van der Marel, R. P. The M31 Velocity Vector. I. Hubble Space Telescope Proper-motion Measurements, Astrophys. J., 753, 7 (2012)
• Lindegren et al. (2016) 17. Lindegren, L., Lammers, U., Bastian, U., et al. Gaia Data Release 1. Astrometry: one billion positions, two million proper motions and parallaxes, Astron. Astrophys., 595, A4 (2016)
• Martínez-Vázquez et al. (2015) 18. Martínez-Vázquez, C. E., Monelli, M., Bono, G., et al. Variable stars in Local Group Galaxies - I. Tracing the early chemical enrichment and radial gradients in the Sculptor dSph with RR Lyrae stars, Mon. Not. R. Astron. Soc., 454, 1509-1516 (2015)
• Piffl et al. (2014) 19. Piffl, T., Binney, J., McMillan, P. J., et al. Constraining the Galaxy’s dark halo with RAVE stars, Mon. Not. R. Astron. Soc., 445, 3133-3151 (2014)
• Mayer et al. (2001) 20. Mayer, L., Governato, F., Colpi, M., et al. The Metamorphosis of Tidally Stirred Dwarf Galaxies, Astrophys. J., 559, 754-784 (2001)
• Michalik et al. (2015) 21. Michalik, D., Lindegren, L., Hobbs, D., & Butkevich, A. G. Gaia astrometry for stars with too few observations. A Bayesian approach, Astron. Astrophys., 583, A68 (2015)
• Foreman-Mackey et al. (2013) 22. Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. emcee: The MCMC Hammer, Public. Astron. Soc. Pac., 125, 306 (2013)
• Strigari et al. (2007) 23. Strigari, L. E., Bullock, J. S., & Kaplinghat, M. Determining the Nature of Dark Matter with Astrometry, Astrophys. J. Lett., 657, 1-4 (2007)
• Walker et al. (2009) 24. Walker, M. G., Mateo, M., & Olszewski, E. W. Stellar Velocities in the Carina, Fornax, Sculptor, and Sextans dSph Galaxies: Data From the Magellan/MMFS Survey, Astron. J., 137, 3100-3108 (2009)
• Walker & Peñarrubia (2011) 25. Walker, M. G., & Peñarrubia, J. A Method for Measuring (Slopes of) the Mass Profiles of Dwarf Spheroidal Galaxies, Astrophys. J., 742, 20 (2011)
• Amorisco & Evans (2012) 26. Amorisco, N. C., & Evans, N. W.  Dark matter cores and cusps: the case of multiple stellar populations in dwarf spheroidals, Mon. Not. R. Astron. Soc., 419, 184-196 (2012)
• Breddels et al. (2013) 27. Breddels, M. A., & Helmi, A. Model comparison of the dark matter profiles of Fornax, Sculptor, Carina and Sextans, Astron. Astrophys., 558, A35 (2013)
• Navarro et al. (1996) 28. Navarro, J. F., Frenk, C. S., & White, S. D. M. The Structure of Cold Dark Matter Halos, Astrophys. J., 462, 563-575 (1996)
• An & Evans (2006) 29. An, J. H., & Evans, N. W.  A Cusp Slope-Central Anisotropy Theorem, Astrophys. J., 642, 752-758 (2006)
• Angus (2008) 30. Angus, G. W.  Dwarf spheroidals in MOND, Mon. Not. R. Astron. Soc., 387, 1481-1488 (2008)

## 1 Description of the HST data and procedures

To measure the proper motions (PMs) of stars in the Sculptor dwarf spheroidal galaxy we used two epochs of data obtained with the two best astrometric space facilities available at the moment: the HST and the Gaia mission. The first epoch of observations was acquired with the Wide Field Channel (WFC) of the Advanced Camera for Survey (ACS) on board the HST. This camera is made up of two pixel detectors separated by a gap of about 50 pixels. Its pixel scale is pixel, for a total field of view (FoV) . The data set (GO-9480, PI: Rhodes), consists of two overlapping pointings separated by about 2′. In turn, the first pointing is split in five sec long exposure images in the F775W filter. The second pointing is made up of six exposures with the same characteristics. The overlapping FoV has thus been observed 11 times. This data set has been acquired on the 26th of September, 2002.

We retrieved from the archive only _FLC images, which are corrected for charge transfer efficiency (CTE) losses by the pre-reduction pipeline adopting a pixel-based correction. The data-reduction was performed with the img2xymWFC.0910 program. We treated each chip of each exposure independently, and we obtained a catalog with positions, instrumental magnitudes and Point Spread Function (PSF) fitting-quality parameter for each of them. Stellar positions were corrected for filter-dependent geometric distortions. We then cross-matched the single catalogs to compute 3-clipped average positions, magnitudes and corresponding uncertainties (defined as the rms of the residuals around the mean value). We finally built the total HST catalog after excluding all the saturated sources and those that were measured less than 4 times.

## 2 Error analysis

Since for this study a good control of all the uncertainties is fundamental, in the following we summarize every source of measurement error in an attempt to find and correct possible unaccounted for terms.

### 2.1 Intrinsic errors

#### 2.1.1 Hst

From the analysis of many HST dithered images, a general trend for the behavior of ACS/WFC single exposures positional errors as a function of instrumental magnitude and adopted filter has been derived. This trend has been modeled for three filters (F435W, F606W and F814W), but very similar results were found for all of them, and especially for the two redder ones. Our exposures have been observed in the filter F775W (instrumental magnitudes were calibrated onto the VEGAmag system using publicly available aperture corrections and zeropoints), so that it is reasonable to compare the positional errors we obtained with the model describing F606W and F814W. Such a comparison is shown in Fig. 5. To compute our single-exposure positional errors, we multiplied the rms values in the HST global catalog obtained as described above, by sqrt(), where is the number of times each star has been measured.

Single-exposure errors computed in this way still contain another source of uncertainty given by possible residuals in the geometric distortion solution. It has been reported that the distortion solution for the F775W filter is slightly worse than e.g. that for the F606W filter because of the lower number of images available used for modeling. The expected residuals should be of the order of pixels. Indeed, this explains very well why our errors are located systematically above the expectation given by the red solid line in Fig. 5. By adding in quadrature an additional term of pixels, which mimics the effect of distortion residuals, the expected trend (dashed red line) matches well the median behavior obtained from our data. Therefore, we conclude that the estimated errors for the HST first-epoch position are reasonable and robust.

#### 2.1.2 Gaia

The Gaia positional uncertainties and correlations have been extensively analyzed and discussed in the recent literature. Their determination will certainly improve in the next data releases, but they are currently in the best shape allowed by the amount of data collected so far. We therefore take the errors at their face value.

### 2.2 Systematic uncertainties

PM measurements can be affected by several systematic uncertainties. In the following we test our measurements against a comprehensive list of systematic effects, based on the prescriptions described in previous work.

Chromatic effects. Differential chromatic refraction (DCR) is one of the most common sources of systematic uncertainties on astrometric measurements. This is due to the fact that DCR shifts the position of photons on the detector proportionally to their wavelength and to the zenithal angle of the observations. Since this effect is induced by the atmosphere, our data taken from space facilities should be unaffected, but possible chromatic effects could still play a role. We checked for this by looking for trends of our PMs as a function of color (G-m). As evident in the top panel of Fig. 6 (where the two PM components and are shown with black and red symbols, respectively) no such trends are apparent. In fact the best least squares linear fit, , has coefficients that are consistent with zero within 1 (e.g  and  ). We can therefore rule out the presence of systematic chromatic effects affecting our PMs.

CTE losses. Defects in the silicon lattice of the ACS detector can lead to an inefficient read-out of the charge that causes deferred-charge trails developing from each source along the vertical direction. This effects tends to systematically move the centroid of sources in the same (vertical) direction, and more significantly affects faint objects. The images we used in this study have already been corrected for CTE losses, but we further checked for the existence of possible residuals by looking for trends among our measured PMs and magnitude (faint stars should be more affected) and positions (trends along the ACS Y-direction should be observed). The first of these tests is shown in the bottom panel of Fig. 6, where the two PM components are plotted against Gaia G-band magnitudes. As in the previous case, no trend is found (the slopes being  and  ). The second test is shown in Fig. 7. We rotated the PMs by degrees, such that their X- and Y- components correspond to the horizontal and vertical direction of the ACS detector. Again, in all cases the slopes of the best linear fit are fully consistent with zero, i.e. no trends are apparent. We can then conclude that residuals CTE effects are not affecting our measurements.

Other systematic effects. In general, other not well identified systematic effects could affect our PM measurements or their estimated uncertainties. We checked for their presence by looking for trends between PMs and all the other measured quantities (photometric parameters, positions, quality of the PSF fitting, astrometric excess noise), finding none. Therefore, after this analysis we can conclude that our PM measurements do not suffer from (the better) known systematic effects.

Possible global systemic motions of the dSph like expansion/contraction or rotation on the plane of the sky could translate into systematic uncertainties on our absolute PM estimate. However, given the large distance of Sculptor (we adopt throughout the paper a distance of kpc, obtained from the analysis of RR Lyrae variable stars), they are negligible compared to the uncertainty on the absolute zero-point. For example, if we assume that the total rotational signal of    reported in the literature corresponds to rotation on the plane of the sky, then the corresponding PM at the location of our HST FoV would be of only  .

## 3 The orbit of Sculptor around the Milky Way and its apparent rotation

We use the observed position on the sky, distance, heliocentric radial velocity and our newly obtained PM measurements of Sculptor to derive its orbit. In a right-handed Cartesian heliocentric reference frame, where points towards the Galactic center, in the direction of rotation and is positive towards the Galactic North pole, Sculptor lies at  kpc and moves with velocity . We then correct for the Sun’s position and velocity w.r.t. the Galactic center assuming  kpc, and  . We integrate these initial conditions, together with 100 random realizations assuming that the errors in the observables are Gaussian, in an axisymmetric Galactic potential for 4 Gyr forward and backward in time using an 8 order Runge-Kutta method. The Galactic potential has several components: a flattened bulge, a gaseous exponential disc, thin and thick stellar exponential discs and a flattened () dark matter halo. The total baryonic (stars and cold gas) mass of the model is , while the dark halo follows an NFW profile whose virial mass is and its concentration . As reported in the main part of the paper, we find that Sculptor has recently (approximately Myr ago) reached its minimum distance to the Milky Way, and is currently moving outwards. The peri- and apocenter radii are  kpc and  kpc, and the orbit has a relatively high inclination of deg. These values are, of course dependent on the characteristic parameters of the Galactic potential. To give a flavor of how they change we vary the mass of the Milky Way halo by 30%. We find that for , then  kpc and  kpc, while for ,  kpc and  kpc. As expected, only the apocentric distance varies strongly with . In fact, if we assume to be half of our fiducial value (i.e. ), then Sculptor would be unbound.

Now that we have determined the orbital motion of Sculptor, we may quantify the magnitude of the “apparent” rotation. The total apparent velocity field induced by the orbit is shown in Fig. 8, where the black ellipse corresponds to the tidal radius of Sculptor and the direction of its PM is indicated by the black arrow. The velocity field, color-coded in steps of , has a maximum magnitude of at , that is projected to an apparent velocity gradient of   along the major axis and    along the minor axis.

## 4 Velocity dispersion and anisotropy

In this section we describe the procedure for deriving the velocity dispersion on the plane of the sky as well as the velocity anisotropy of Sculptor.

We transform the PM components from the equatorial reference to radial and tangential components on the plane of the sky according to the equatorial-polar coordinates relation:

 [μRμT]=[cos(ϕ)sin(ϕ)−sin(ϕ)cos(ϕ)]×[μαcos(δ)μδ],

where , and and are the (local Cartesian) gnomonic projected coordinates. Uncertainties are fully propagated taking into account the correlation coefficient between Gaia ’s RA and DEC estimates. The projected velocities in the radial and tangential direction therefore are , with the distance to Sculptor.

We model the velocity dispersion for the sample selected as described in the main body of the paper, by a multivariate Gaussian including a covariance term. This Gaussian is characterized by velocity dispersions in the (projected) radial and tangential directions (), their correlation coefficient and the mean velocities (). The posterior for these parameters , including the data , is given by Bayes theorem:

 p(σR,σT,v0,R,v0,T,ρR,T|D)=p(D|σR,σT,v0,R,v0,T,ρR,T)p(σR,σT,v0,R,v0,T,ρR,T)/p(D). (2)

The likelihood here, , is a product of Gaussians, , and the covariance matrix is the sum of the covariance matrices associated to the intrinsic kinematics of the population and to the measurement uncertainties (which is equivalent to a convolution of the two Gaussians representing these contributions), i.e.

 p(D|σR,σT,v0,R,v0,T,ρR,T) = ∏iN([vR,ivT,i],[v0,Rv0,T],Σi), (3) Σi = [σ2RρR,TσRσTρR,TσRσTσ2T]+[ϵ2σR,iρRi,TiϵσR,iϵσT,iρRi,TiϵσR,iϵσT,iϵ2σT,i.] (4)

Furthermore, for the prior in Eq. (2), we assume it has a weak Gaussian-like form for the correlation coefficient of the intrinsic kinematics of the population (with mean 0, and dispersion 0.8) while a flat prior is assumed for the mean velocities. We explore two different priors for the velocity dispersion: a Gaussian-like for the (logarithm of the) velocity dispersions (with mean [], and unity dispersion), and a uniform prior.

We have used a Markov Chain Monte Carlo (MCMC) algorithm to estimate the posterior all the parameters, but, except for and , we consider all as nuisance parameters. As reported in the main body of the paper, we find for our best PM sample    and    in the case of the Gaussian prior (similar values are obtained for the flat case).

In our analysis we have left the mean projected velocities and as free (nuisance) parameters, and find values entirely consistent with those determined in the main body of the paper. If Sculptor would rotate with an amplitude of  , this would induce a gradient in the field where our stars are found of order 0.5 , that therefore would be negligible.

The anisotropy provides a measure of the intrinsic orbital distribution of the system, and is defined as , where and are the intrinsic (3D) velocity dispersions in the tangential and radial directions, respectively. To obtain an estimate of the orbital anisotropy from the observables , and we use the spherical Jeans equations. These link the measured dispersions with intrinsic properties of the system, namely , , and the light density profile ( in 3D and projected ) as follows:

 σ2los(R) = 2I⋆(R)∫∞R(1−βR2r2)ν⋆σ2rrdr√r2−R2, (5) σ2R(R) = 2I⋆(R)∫∞R(1−β+βR2r2)ν⋆σ2rrdr√r2−R2, (6) σ2T(R) = 2I⋆(R)∫∞R(1−β)ν⋆σ2rrdr√r2−R2. (7)

If we define , and

 f1(R)=∫∞RQ(r)dr,f2(R)=∫∞Rβ(r)R2r2Q(r)dr,f3(R)=∫∞Rβ(r)Q(r)dr,

then

 σ2los(R) = 2I⋆(R)(f1(R)−f2(R)), σ2R(R) = 2I⋆(R)(f1(R)−f3(R)+f2(R)), σ2T(R) = 2I⋆(R)(f1(R)−f3(R)).

If we do not make any assumptions on , we may use the mean value theorem in the form

 ∫baf(x)g(x)dx=f(c)∫bag(x)dx,c∈[a,b],

which holds provided does not change sign in . In our case we could apply this theorem to say that such that , where is the location where we have measured the velocity dispersions and with our dataset. This means that

 ^β=β(^r)=1−σ2Tσ2\rm los+σ2R−σ2T,with^r∈[RHST,rmax), (8)

where we have used that in reality, Sculptor has a maximum (finite) radial extent which we denote by . Note that if is constant, then Eq. (8) holds at every radius.

As discussed in the main body of the paper, there are indications that our sample may be dominated by metal-rich stars, a component known to have its own characteristic spatial distribution and kinematics. To derive the metallicity of our stars, we took the measured iron spectral index Fe reported in the spectroscopic sample observed with the MIKE spectrograph at the Magellan 6.5m telescope, and applied the following relation

 [Fe/H]=(7.02±2.10)ΣFe−3.97±2.03 (9)

to calibrate it to [Fe/H]. Since the metal-rich and metal-poor populations of Sculptor have been clearly separated on the basis of their metallicity, we preferred to work with the iron spectral index rather than with the mean reduced Mg index used in other works.

To explore further the possibility that our measurement of the anisotropy could be affected by the presence of the different populations in Sculptor, we repeat the procedure outlined above to determine the value of where again , now using sample that includes only stars with [Fe/H] dex. To enlarge the statistics, we also include 5 fainter, similarly metal-rich stars. For this sample we find , a value that is much more tightly constrained that the obtained using the best PM sample without a metallicity cut (compare Fig. 4b and 4c in the main body of the paper). The reason for this is not a decrease in the errors (the data satisfy the same quality criteria), nor different numbers of objects, but the heterogeneity present in our original best PM sample. That is, this sample contained stars drawn from the different components in Sculptor with their own, apparently rather different orbital structure. Unfortunately our sample of metal-poor stars with good PM measurements is too small to make a similar analysis and results in an anisotropy that is relatively unconstrained.

## References

• Anderson & Bedin (2010) 31. Anderson, J., & Bedin, L. R. An Empirical Pixel-Based Correction for Imperfect CTE. I. HST’s Advanced Camera for Surveys, Public. Astron. Soc. Pac., 122, 1035-1064 (2010)
• Ubeda & Anderson (2012) 32. Ubeda, L., Anderson, J., Study of the evolution of the ACS/WFC charge transfer efficiency, STScI Inst. Sci. Rep. ACS 2012-03 (2012)
• Anderson & King (2006) 33. Anderson, J., & King, I., PSFs, Photometry, and Astronomy for the ACS/WFC, STScI Inst. Sci. Rep. ACS 2006-01 (2006)
• Bellini et al. (2014) 34. Bellini, A., Anderson, J., van der Marel, R. P., et al. Hubble Space Telescope Proper Motion (HSTPROMO) Catalogs of Galactic Globular Clusters. I. Sample Selection, Data Reduction, and NGC 7078 Results, Astrophys. J., 797, 115 (2014) 35. http://www.stsci.edu/hst/acs/analysis/zeropoints.
• Anderson (2006) 36. Anderson, J. Empirical PSFs and Distortion in the WFC Camera, The 2005 HST Calibration Workshop: Hubble After the Transition to Two-Gyro Mode, 11 (2006)
• Arenou et al. (2017) 37. Arenou, F., Luri, X., Babusiaux, C., et al. Gaia Data Release 1. Catalogue validation, Astron. Astrophys., 599, A50 (2017)
• Trippe et al. (2010) 38. Trippe, S., Davies, R., Eisenhauer, F., et al. High-precision astrometry with MICADO at the European Extremely Large Telescope, Mon. Not. R. Astron. Soc., 402, 1126-1140 (2010)
• Chiaberge et al. (2006) 39. Chiaberge, M., Riess, A., Mutchler, M., Sirianni, M., & Mack, J. Mack, J. ACS Charge Transfer Efficiency. Results from Internal and External Tests, The 2005 HST Calibration Workshop: Hubble After the Transition to Two-Gyro Mode, 36 (2006)
• Schönrich et al. (2010) 40. Schönrich, R., Binney, J., & Dehnen, W. Local kinematics and the local standard of rest, Mon. Not. R. Astron. Soc., 403, 1829-1833 (2010)
• Binney & Tremaine (2008) 41. Binney, J., & Tremaine, S. 2008, Galactic Dynamics: Second Edition, by James Binney and Scott Tremaine. ISBN 978-0-691-13026-2 (HB). Published by Princeton University Press, Princeton, NJ USA (2008)
• Tolstoy et al. (2004) 42. Tolstoy, E., Irwin, M. J., Helmi, A., et al. Two Distinct Ancient Components in the Sculptor Dwarf Spheroidal Galaxy: First Results from the Dwarf Abundances and Radial Velocities Team, Astrophys. J. Lett., 617, 119-122 (2004)
• Zin & West (1984) 43. Zinn, R., & West, M. J.  The globular cluster system of the galaxy. III - Measurements of radial velocity and metallicity for 60 clusters and a compilation of metallicities for 121 clusters, Astrophys. J. Suppl., 55, 45-66 (1984)
• Sohn et al. (2017) 44. Sohn, S. T., Patel, E., Besla, G., et al. Space Motions of the Dwarf Spheroidal Galaxies Draco and Sculptor Based on HST Proper Motions with a ∼10 yr Time Baseline, Astrophys. J., 849 (2017)
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