Hunting for intermediate-mass black holes in globular clusters: an astrometric study of NGC 6441
Maximilian Häberle, Mattia Libralato, Andrea Bellini, Laura L. Watkins, Jörg-Uwe Pott, Nadine Neumayer, Roeland P. van der Marel, Giampaolo Piotto, Domenico Nardiello
MMNRAS , 1–15 (2020) Preprint 17 February 2021 Compiled using MNRAS L A TEX style file v3.0
Hunting for intermediate-mass black holes in globular clusters:an astrometric study of NGC 6441
Maximilian Häberle ★ , Mattia Libralato , Andrea Bellini , Laura L. Watkins , Jörg-Uwe Pott ,Nadine Neumayer , Roeland P. van der Marel , , Giampaolo Piotto , , and Domenico Nardiello , Max-Planck-Institut für Astronomie, Königstuhl 17, 69117 Heidelberg, Germany AURA for the European Space Agency (ESA), ESA Office, Space Telescope Science Institute, 3700 San Martin Drive, Baltimore MD 21218, USA Space Telescope Science Institute, 3700 San Martin Drive, Baltimore MD 21218, USA Center for Astrophysical Sciences, Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD 21218, USA Dipartimento di Fisica e Astronomia “Galileo Galilei”, Università degli Studi di Padova, Vicolo dell’Osservatorio 3, I-35122 Padova, Italy Istituto Nazionale di Astrofisica (INAF), Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, I-35122 Padova, Italy Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We present an astrometric study of the proper motions (PMs) in the core of the globular cluster NGC 6441. The core of thiscluster has a high density and observations with current instrumentation are very challenging. We combine ground-based,high-angular-resolution NACO@VLT images with
Hubble Space Telescope
ACS/HRC data and measure PMs with a temporalbaseline of 15 yr for about 1400 stars in the centermost 15 arcseconds of the cluster. We reach a PM precision of ∼ µ as yr − for bright, well-measured stars.Our results for the velocity dispersion are in good agreement with other studies and extend already-existing analyses of thestellar kinematics of NGC 6441 to its centermost region never probed before. In the innermost arcsecond of the cluster, wemeasure a velocity dispersion of (19.1 ± − for evolved stars. Because of its high mass, NGC 6441 is a promisingcandidate for harbouring an intermediate-mass black hole (IMBH). We combine our measurements with additional data from theliterature and compute dynamical models of the cluster. We find an upper limit of 𝑀 IMBH < . × M (cid:12) but we can neitherconfirm nor rule out its presence. We also refine the dynamical distance of the cluster to 12 . + . − . kpc.Although the hunt for an IMBH in NGC 6441 is not yet concluded, our results show how future observations with extremely-large telescopes will benefit from the long temporal baseline offered by existing high-angular-resolution data. Key words: globular clusters: individual (NGC 6441) – astrometry – proper motions – stars: kinematics and dynamics
Globular clusters are the oldest surviving stellar systems in theGalaxy. Because of their long dynamical timescales, they are wit-nesses of the early history of the Milky Way. The study of the internaldynamics of globular clusters is an active field of research and canreveal information about the formation and evolution of the clustersthemselves, their interaction with the Galactic potential, but also thepossible existence of intermediate-mass black holes (IMBHs) in theircore.IMBHs have been predicted in the centres of globular clusters(Portegies Zwart & McMillan 2002; Miller & Hamilton 2002), buta definitive detection is still lacking (see Greene et al. 2019 for arecent review). As the sphere of influence of a hypothetical IMBHis limited to the very centre of the cluster, dynamical studies of thiscentral region are necessary to infer the presence of an IMBH.There are two observational methods to study the kinematic sig-nature of individual stars in a globular cluster: spectroscopic line-of- ★ E-mail: [email protected] sight (LOS) velocity measurements and astrometric measurementsof proper motions (PMs). While in the early days of the field theline-of-sight velocities of only small numbers of stars were mea-sured, in recent years significant progresses have been made both inspectroscopy and astrometry. The MUSE integral field spectrographallowed LOS velocity measurements of up to 20,000 stars within thehalf-light radius of a large sample of globular clusters (Kamann et al.2018).On the astrometry side, the best tool to study the crowded cores ofthe clusters is the
Hubble Space Telescope (HST) and there are cat-alogues with high precision PM measurements for up to a hundredthousand stars (Bellini et al. 2014; Libralato et al. 2018) in a sin-gle cluster. Additionally, the
Gaia satellite (Gaia Collaboration et al.2016) allows the study of stellar motions in the outskirts of globu-lar clusters. However, its completeness and precision are limited incluster centres, where the stellar density is high.Observations of the crowded cores of some globular clusters stillremain very challenging with current instrumentation. Crowding ef-fects limit the usability of spectroscopic facilities, therefore, astrom-etry with high resolution imagers is the method of choice. One such © a r X i v : . [ a s t r o - ph . GA ] F e b Häberle et al. example is the globular cluster NGC 6441: its core is extremelycrowded and neither of the current
HST imagers provides the nec-essary resolution for astrometric studies of the core. The AdvancedCamera for Surveys Wide Field Channel (ACS/WFC) has a pixelscale of 50 mas pixel − and the Wide Field Camera 3 (WFC3) UVIShas a pixel scale of 40 mas pixel − .The High Resolution Channel of the Advanced Camera for Sur-veys (ACS/HRC) had a smaller pixel scale of 25 mas pixel − and,while still affected by the high stellar density, could measure stellarposition in the core of NGC 6441 with high precision. A first epochof observations was taken in 2003, but the failure of this instrumentin 2006 made it impossible to observe the core of NGC 6441 again.Since there are no other suitable epochs of HST observations, were-observed the core of the cluster using the near infrared instrumentNACO at the Very Large Telescope (VLT) of the European SouthernObservatory (ESO). The adaptive-optics (AO)-assisted observationsclose to the diffraction limit with an 8-m class telescope in the 𝐾 S band have a point spread function (PSF) full width half maximum(FWHM) of about 73 mas, similar to that of the HST at visiblewavelengths, and with 13.2 mas pixel − the pixel scale is twice assmall as the one of ACS/HRC.NGC 6441 is a Galactic bulge cluster. With a mass of 1 . × M (cid:12) (Baumgardt & Hilker 2018), it is one of the most massive clustersin the Galaxy. Furthermore, it is metal rich ([Fe/H] = − .
55, Harris2010) and hosts at least two main stellar populations (Bellini et al.2014).The dynamics of the cluster have been studied in several papers,both with PMs (Watkins et al. 2015a) and LOS velocities (Kamannet al. 2018), but due to its high density, accurate measurements ofthe velocity dispersion of the stars in the very centre of the clusterare still lacking. Our study probes the velocity dispersion to unprece-dentedly small radii. While we cannot put strong constraints on thepresence of an IMBH, we add more than 20 PM measurements in theinnermost arcsecond. The combination of space-based and ground-based AO-assisted astrometric imaging with a long time baseline isa successful pilot study to showcase what will be possible with newinstrumentation expected in the next decade, such as the MICADOimager at the Extremely Large Telescope (Davies et al. 2016).This paper is divided in the following sections: first we describeour observations (Section 2), after which we present the data analysisin Section 3. The determination of the PMs is described in Section 4.We use our results to create dynamical models of the globular clusterin Section 5, and discuss and conclude our work in Section 6.
HST
ACS/HRC Observations
The
HST instrument ACS/HRC was in operation from 2002 to 2006.It featured a 1024 × ×
25 mas pixel − , giving it a field of view of ∼ ×
26 arcsec .The core of NGC 6441 was observed with the ACS/HRC in twobroadband filters (F555W and F814W) during Program GO-9835(PI: G. Drukier). The 36 exposures in the F555W filter have all thesame exposure time of 240 s. The F814W band exposures compriseof 5 short and 12 long exposures of 40 s and 440 s, respectively. NACO (short for NAOS-CONICA) was a near-infrared, adaptive-optics-assisted imager and spectrograph at the Very Large Telescope(VLT). A full description of the instrument and its performance canbe found in Lenzen et al. (2003) and Rousset et al. (2003). The in-strument was mounted on the Nasmyth B focus of UT4 from 2001to 2013, and then moved to UT1 in 2014, where it continued itsoperations until its recent decommissioning in October 2019. Theadaptive optics front end (NAOS) was equipped with a 185 actuatordeformable mirror, a tip/tilt mirror and two wavefront sensors (op-erating in the visual and the IR range). Using bright natural guidestars, the best Strehl ratio obtainable in the 𝐾 S band was around 50%in typical observing conditions, similar to what we achieved in ourobserving run.The camera CONICA was equipped with an Aladdin2 detector(1026 x 1024 pixels, InSb) that replaced the Aladdin3 detector thatwas in use from 2004 to 2013. The detector was affected by severalartefacts that are described in detail later. CONICA offered cam-eras with three different pixel scales (S13: 13.22 mas pixel − , S27:27.06 mas pixel − , S54: 54.3 mas pixel − ). For our observations ofthe core of NGC 6441, we made use of the S13 mode to obtain thehighest possible resolution. The second epoch of observations of the core of NGC 6441 wereobtained with the ESO Program ID 0101.D-0385 (PI: M. Libralato).We only made use of images taken during the August run becauseof their overall better quality. In these nights, there were 23 usableexposures of 200 s each, and 27 short observations of 30 s each. Theobservations were executed with a dither pattern that covered the fieldof the first-epoch
HST observation almost completely (see Figure 1).The observing strategy was designed to solve for the geometric dis-tortion of the detector using an auto-calibration approach (see, e.g.,Libralato et al. 2014) and to achieve an astrometric precision highenough to allow for the kinematic analysis of NGC 6441.
PM measurements require multiple epochs of precise stellar positionmeasurements. In this section we describe how we extracted stellarpositions from the raw exposures of our two datasets.
HST
ACS/HRC 2003)
The data reduction of the ACS/HRC exposures was performed on _flt -type images by closely following the prescriptions given inBellini et al. (2014, 2017a).In brief, we started by deriving state-of-art, spatially-variable PSFmodels for each exposure by perturbing the library PSF models cre-ated by Jay Anderson . Our improved PSF models take into accounttelescope breathing effects (di Nino et al. 2008, see also Bellini et al.2017a; Bellini et al. 2018a), which can significantly change the shape These images are bias, dark and flat-field corrected by the standard
HST pipeline
CALACS , but are not resampled, so they retain the full signal of theastrometric scene. MNRAS , 1–15 (2020) unting for intermediate-mass black holes in globular clusters: an astrometric study of NGC 6441 x [pixel]37504000425045004750500052505500575060006250 y [ p i x e l ] -- HRC field of view+ NACO pointings x [pixel]Stacked NACO image . . . . . . . . . . Right ascension [degree] 37.05637.05537.05437.05337.05237.05137.05037.04937.04837.047 D e c li n a t i o n [ d e g r ee ] . . . . . . . . . . Right ascension [degree]
Figure 1.
The two panels show the field of the NACO observations in equatorial coordinates and in our pixel-based coordinate system. In the left plot, thecentres of all long NACO pointings are marked with a black cross and the footprint of each exposure is marked in red. The missing lower-left quadrant is alreadyremoved from the footprints. In the central regions, a maximum of 17 exposures overlap, while at the edges of the observed field the depth of coverage is muchsmaller. The right plot shows a stacked image of the NACO observations in an inverted grey scale.Epoch Telescope Instrument Program ID Filter 𝑡 exp 𝑁 images HST
ACS/HRC GO-9835 F555W 200 s 36F814W 440 s 12F814W 40 s 52018.61 VLT NACO 0101.D-0385 𝐾 S
200 s 2330 s 27
Table 1.
List of used observations. of the
HST
PSFs from one exposure to the next even within the sametelescope orbit.Preliminary stellar positions and fluxes of bright sources wereobtained through PSF fitting using the
FORTRAN code hst1pass (Anderson in preparation, see Bellini et al. 2018c for details). Pho-tometry of saturated stars includes all the relevant flux that has bledinto adjacent pixels following the prescriptions given in Gillilandet al. (2010). Stellar positions were corrected for the effects of ge-ometric distortion using the state-of-the-art solutions provided byAnderson & King (2004).Next, we made use of
Gaia
DR2 positions (Gaia Collaborationet al. 2016, 2018) to define a reference-frame system with Northup, East to the left, and with the same pixel scale of NACO of13.2 mas pixel − . We transformed stellar positions of each single-exposure catalogue on to the reference frame by means of general, six-parameter linear transformations. Our best estimate of positions andfluxes for all possible sources in the ACS/HRC field is obtained usingthe FORTRAN code
KS2 (Anderson in preparation, see Bellini et al. 2017a for details).
KS2 starts from the image-tailored PSF models,the lists of bright stars and their transformations on to the referenceframe, and goes through several waves of source finding, measuringand subtraction using all the exposures simultaneously. Our final first-epoch catalogue contains around 44 000 sources measured in bothF555W and F814W filters. In addition to the photometry results, theoutput contains several quality parameters such as the radial excessvalue (
RADXS ). If it is positive, the profile of a single star containsexcessive flux outside of the fit radius with respect to the PSF, if it islower than 0, this flux is lower than expected.To calibrate the photometry of the ACS/HRC data, we performedaperture photometry on the corresponding _drz images that areresampled and normalised to 1 s exposure time, corrected the resultsfor finite aperture (using the corrections from Bohlin 2016) andthen brought them onto the VegaMAG system using the zeropoints
MNRAS000
MNRAS000 , 1–15 (2020)
Häberle et al. available at the STScI website . Then we determined the zeropointbetween the calibrated aperture photometry and our PSF photometryby taking the 3 𝜎 -clipped median of the magnitude difference forbright isolated stars (we chose stars that had no brighter neighbourswithin a 18 pixel radius). A similar calibration process for HST photometry has been described e.g. in Bellini et al. (2017a).
We downloaded the raw NACO exposures and the correspondingcalibration frames (dark frames, flat-fields, bad-pixel maps) from theESO Science archive. The dark frames were not usable because thebackground showed patterns that varied over the course of the night.We therefore only divided the images by the flat-fields. Furthermore,we flagged all saturated pixels and added them to the bad-pixel map.The saturation threshold is set at 10 000 analogue-digital converterunits (ADUs) to avoid non-linearity effects following the recommen-dations of the NACO User manual (Schmidtobreick et al. 2018).The bottom-left quadrant of NACO presents a high number of badcolumns in a regular pattern (2 good, 2 bad, 3 good, 1 bad), so wechose not to use it in our analysis.
Insufficient thermal shielding within the instrument created a dif-fuse, non-static background pattern that cannot be corrected in thepre-reduction phase. The time-dependent nature of the backgroundpattern and the high density of stars in the cluster centre made it chal-lenging to correct this pattern. The correction we applied is the resultof two iterations. In each iteration, we used the median of multipleimages in which the background pattern did not change as our back-ground model. To remove the influence of stars, we flagged pixelsfrom an image if they were significantly brighter than in the other im-ages. Furthermore, we flagged a circular area with 𝑟 =
45 NACO pix-els around saturated stars to remove the influence of their bright andextended halos. Flagged pixels were excluded from the median cal-culation. In the first iteration this clipping was performed directly onthe raw images. In the second iteration we improved the model bysubtracting the modelled stellar fluxes from the raw images (based onthe PSF photometry results from the next paragraph) before runninga jackknife algorithm to find additional outliers. For some images itwas beneficial to shift the model in the 𝑦 -direction before subtractingit. The different steps are visualised in Figure 2. An accurate PSF model is a crucial ingredient for high-precisionastrometry. We created a 3 × https://acszeropoints.stsci.edu/ This effect is known since 2015, see ESO instrument history:
Figure 2.
The top-left panel (a) shows a typical raw NACO image ofNGC 6441. The thermal-related background pattern and the bad columnsin the bottom-left quadrant are clearly visible. In the top-right panel (b), wepresent the corresponding image model. Each star in the model image isobtained by rescaling the PSF by the stellar flux. The white circles have aradius of 45 NACO pixels and mark the region around saturated stars that weflagged and excluded from the model creation. The bottom-left image (panelc) is the resulting background model obtained after our iterative procedure.Finally, in the bottom-right panel (d), we show the image from panel (a) afterthe removal of the background pattern. refer to these papers for a detailed description of the PSF modelling.Here we provide only a brief overview of the method and discuss themajor differences with respect to the original papers.Because of the small field of view of NACO, we used a regular3 × µ m wavelength. The Strehl ratios we obtained ranged between 0.2and 0.49, with a median value of 0.37. This is compatible with thetypical Strehl ratios stated in the NACO User manual. The medianFWHM of our PSF was 73 mas. Figure 3 shows the 8 PSF modelsdetermined for an individual frame and how the local models differfrom a spatially constant PSF model. After the PSF model has been determined for each image, we used it tofit the raw position (x,y) in pixels and the instrumental magnitude of
MNRAS , 1–15 (2020) unting for intermediate-mass black holes in globular clusters: an astrometric study of NGC 6441 (1,3) (2,3) (3,3)(1,2) (2,2) (3,2)(1,1) (2,1) (3,1) Figure 3. (Upper panel:) Each of the nine squares is a representation of thedetector. Each black point shows the location of a local PSF model that isdetermined from stars within the green rectangle. (Lower-left panel:) The 3 × each star. To be able to select well measured stars, we also calculatedthe so called quality-of-fit ( QFIT ) value, which is a normalised sumof the fit residuals within the fitting radius (Anderson et al. 2006).The closer to 0 the
QFIT , the better is the PSF fit.
To achieve a sub-pixel astrometric precision, we have to correct thegeometric distortion present in the NACO images, which reaches upto 2 NACO pixels in the corners of the detector. We redeterminedthe GDC using our
HST catalogue as distortion-free reference, theprocess is described in detail in the Appendix A. We corrected thegeometric distortion to a level of ≤ .
03 NACO pixel ( ≈ . + rotation pattern, if such a calibration is required. Most stars in our dataset have been measured on multiple NACOexposures. To combine the distortion-corrected single-image cata-logues, we matched them on the same reference system as definedfor the first epoch (see 3.1). We transformed the stellar positionsfrom each distortion-corrected single-image catalogue on to the ref-erence system using six-parameter linear transformations. Stars thathave been measured in at least three images were then added toour NACO master frame. Our best estimate for their position is theaveraged position from the individual transformed positions. The scatter of the single measurements, quantified by their rootmean square (rms) deviation from the master frame position, is ameasure of the astrometric precision we can reach for a given signalto noise ratio (S/N). In Figure 4, this positional rms is plotted asa function of the 𝑚 𝐾 S magnitude (see subsection 3.2.7 for our fluxcalibration). As expected, we see a decrease in precision for faintstars, i.e., those with a lower S/N ratio. For stars brighter than 𝑚 𝐾 S < .
5, the trend flattens out at a level of approximately 0.03 NACOpixel ( ≈ .
40 mas), while theoretical limits on centroiding precision(e.g. Lindegren 1978) predict: 𝜎 = 𝑘 · FWHM
PSF
SNR (1)(with 𝑘 close to unity).This fundamental limit of 0.03 pixel is worse than the 0.01 pixelachieved for astrometry with HST instruments, but comparable toother ground-based IR studies (Libralato et al. 2014, 2015; Kerberet al. 2019). It is most likely caused by systematic effects, such asresidual uncorrected distortion, imperfections in our PSF modelsand/or systematic effects related to the thermal background patternand its correction.
We brought our NACO PSF photometry on to the Two Micron All-Sky Survey (2MASS, Skrutskie et al. 2006) photometric system bycross-matching it with the NGC 6441 catalogue published by Valentiet al. (2010) (in the following Val10). Due to the depth of the NACOimages, most of the stars found in Val10 are saturated in our longexposures. Therefore, we used a two step-process. First, we createda catalogue based on the short NACO exposures whose photometryhas been zeropointed on to our long-exposure master-frame. Then,we determined the magnitude difference between the 63 stars incommon between our master frame and the Val10 catalogue. Ourbest estimate of the zeropoint is the sigma clipped median of thedifference between the magnitudes in both catalogues. Since we onlyhave one filter, we added this simple zeropoint without accountingfor possible colour effects. After obtaining calibrated photometry forboth the
HST and the NACO dataset, we created ( 𝑚 F555W − 𝑚 F814W )and ( 𝑚 F555W − 𝑚 𝐾 S ) colour-magnitudes diagrams, which can be seenin Figure 5. To measure PMs one has to determine by how much the positions ofindividual stars have changed over time.Compared to other studies, where multiple different epochs andfields of views are combined, the situation in our study is rathersimple: we only combine two single epochs with a large time span inbetween. For this reason, we do not use a linear fit through multipledatapoints for each star, but we determine the position differencebetween two single epoch master frames. Due to the more reliablydetermined PSF and the better general astrometric precision, wedecided to use only the long NACO exposures for this part of theanalysis, at the cost of being unable to determine the PMs for around20 stars, which were saturated in the long exposures.
The goal of this step is to create new, improved master frames by usingadditional clipping procedures and local corrections. The following
MNRAS000
MNRAS000 , 1–15 (2020)
Häberle et al. m K s P o s i t i o n a l r m s [ N A C O p i x e l ] RMS of saturated stars with multiple measurementsRMS of unsaturated stars with multiple measurementsEmpirical model of uncertaintiesFWHM / SNRPrecision Limit due to systematics (~0.03 pixel = 0.40 mas)Binned RMS P o s i t i o n a l r m s [ m a s ] Figure 4.
This plot shows the magnitude dependence of the astrometric precision reached with single, long (200 s) NACO exposures. We determine the positionalscatter of stars that have been measured in at least 3 images by determining the rms of single-image position measurements when transformed in our referencesystem. Overall, faint stars follow the theoretical prediction for centroiding accuracy, while the measurements of bright stars are limited by a noise floor of ∼ steps were performed on both the HST
ACS/HRC and the NACOdata.1. Selection of well-measured stars: we applied several selectionsbased on quality criteria (
QFIT value,
RADXS value) to restrict oursample to well-measured stars. Those well-measured stars were thenused to determine the parameters of the following transformations.2. Global transformations: for each image, we use all stars flaggedas “well-measured” to determine the optimal six-parameter lineartransformations to transform the single-image catalogues on to themaster frame. The residuals between the single-image catalogues andthe master frame are used in the next steps for a local correction.3. Local corrections: after the parameters of the global transfor-mations have been determined, we used local corrections to removeresidual local effects such as uncorrected distortion. For each star,we determined the clipped mean of the transformation residuals ofthe 50 closest neighbours in both the 𝑥 and the 𝑦 directions. Thesemean residuals are subtracted from the mean coordinates of the star.This procedure is called “boresight” correction (van der Marel &Anderson 2010).4. Error-based clipping and determination of mean positions: thelast steps gave us a list of multiple position measurements foreach star, all transformed in the same reference frame. We definedas improved master-frame positions the mean positions of theselocally-corrected measurements. Outliers were removed using ajackknife approach: for each star we excluded one measurement ata time and checked whether the excluded measurement deviatesmore than 10 standard deviations from the mean value of theremaining measurements. To determine the expected standarddeviation, we employed the empirical error model based on thetypical positional rms at a given magnitude (see Figure 4). After the clipping process, we calculated the mean of the positions based onthe remaining measurements. As the single-epoch position error,we employed the standard error of the mean for both coordinates: Δ 𝑥, 𝑦 master frame = 𝜎 𝑥,𝑦 √ 𝑁 The procedures used to match the two single-epoch master framesare similar to the methods used for the master frame creation. Thedifference is that, instead of matching single-image catalogues fromthe same epoch, we now directly match the master frame positions ofthe two different epochs. Since we want to measure PMs relative tothe bulk motion of the cluster, we use only bona-fide cluster membersto determine the parameters of the linear transformations between themaster frames.1. Selection of well-measured cluster members : we use the samequality selection as described above. In addition, we included a se-lection of bona-fide cluster members based on their location on theACS/HRC based ( 𝑚 F555W − 𝑚 F814W ) CMD (see Figure 5). OncePMs have been determined, we also restricted the sample of bona-fide cluster members on the basis of their location on the vector-pointdiagram.2. Global transformations: for each star, we use the positions ofall well-measured stars within a radius of 1000 NACO pixels todetermine the optimal six-parameter transformation between the twomaster frames. Six-parameter linear transformations also solve for the rotation betweenframes. By using cluster stars to compute the coefficients of such transfor-MNRAS , 1–15 (2020) unting for intermediate-mass black holes in globular clusters: an astrometric study of NGC 6441 m F555W m F814W m F W All Stars with PMSelection forkinematic studies 1 2 3 4 5 6 m F555W m K s m K s Figure 5.
This plot shows two colour-magnitude diagrams of our PM samplebased on the photometry of the ACS/HRC (F555W, F814W) and the NACO( 𝐾 S ) measurements. Measurements, that pass the quality selection for thekinematic analysis (Section 5) are marked in red.
3. Calculation of PM and PM error: the stellar PMs are obtained asthe difference between the transformed positions of the two masterframes, divided by the temporal baseline. The PM errors are cal-culated by quadratically adding the positional errors of the singleepochs and then dividing this result by the temporal baseline.
By construction, the mean motion of cluster members should beat location (0,0) on the vector-point diagram regardless of stellarpositions on the master frame. Local deviations of the mean motionare caused by small, uncorrected-for systematic effects, which wemitigated using an a-posteriori local correction as follows.For each cluster star in our PM catalogue, we chose the 50 closestneighbouring cluster stars and calculated their mean motion. Thisvalue is subtracted from the measured PM of the star. This stepleads to an additional statistical PM error of ∼ − percoordinate. Therefore, we kept both the uncorrected and the correctedposition residuals for the further analysis. The vector-point diagram and the PM uncertainties in both coordi-nates are shown in Figure 6. For bright stars with 𝑚 𝐾 S < . 𝑚 F555W < .
5) we reach uncertainties of around 0.03 mas yr − .Similar precisions are reached in pure HST based studies (see Belliniet al. 2014) while the
Gaia
DR2 PMs have a higher uncertainty of mations, we are implicitly absorbing any potential systemic-rotation signalof the cluster and, as such, the rotation in the plane of the sky of NGC 6441cannot be directly detected with our PMs (see the discussion in, e.g., Belliniet al. 2017b and Libralato et al. 2018). around 0.1 mas yr − for stars with G=17 mag even in less crowdedfields (Lindegren et al. 2018). Only cluster stars (within 1.5 mas yr − from the origin of the vector-point diagram) that are well-measured in the HST and NACO datawere used in the analysis of the internal kinematics. For the
HST data,we defined as “well-measured” stars that have: (i) magnitude rms inboth filters lower than 0.025 mag; (ii)
QFIT parameter larger than0.985; (iii) the absolute value of the shape parameter
RADXS lowerthan 0.03. For the NACO data, we only required the 1D positionalerrors to be lower than 0.4 NACO pixel. Stars with a PM error ineither direction greater than 0.15 mas yr − or larger than half thelocal velocity dispersion 𝜎 𝜇 of the closest 50 cluster stars were ex-cluded from the analysis. We tested different quality selections, andfind negligible differences in the resulting velocity dispersions. Theparameters described above provide a good compromise betweenincluding as many stars as possible (for better statistics in the kine-matic analysis) and excluding poorly-measured objects. Out of thearound 1400 stars with a PM measurement, ∼ 𝜎 𝜇 ), radial( 𝜎 R ), and tangential ( 𝜎 T ) velocity dispersions as a function of dis-tance from the cluster’s centre in: i) one radial bin with all stars within1 arcsec from the cluster’s centre (23 stars); ii) four equally-populatedradial bins between 1 and 5 arcsec (107 stars each); iii) eight radialbins using all stars outside the centermost 5 arcsec (seven groups of97 stars and one with 87).The results are shown in Figure 7 and in Table B1 in the Ap-pendix. Our data are very consistent with the PM results of Watkinset al. (2015a) and the MUSE LOS measurements of Kamann et al.(2018), which are also shown in Figure 7. However, we cannot re-produce the dip in velocity dispersion in the innermost MUSE datapoint, that may be caused by crowding effects (Alfaro-Cuello et al.2020). In the innermost arcsecond we measure a combined velocitydispersion of (0.316 ± − . If we employ the dynamicaldistance estimate from Section 5 ( 𝐷 = .
74 kpc) this correspondsto (19.1 ± − . Fast moving stars in the centre of NGC 6441 could be a possiblesignature of an IMBH (Drukier & Bailyn 2003). The lack of LOSvelocities does not allow us to obtain a 3D kinematic picture of thecore of this cluster. However, we can still investigate the presenceand nature of fast-moving objects in the plane of the sky thanks toour high-precision PMs.We initially searched for stars with a PM that indicates a veloc-ity higher than the escape velocity of the cluster as a result of theinteraction with a potential IMBH. We considered fast-moving starsthose with total PM between 1.26 and 1.5 mas yr − . The lower limitcorresponds to the escape velocity of 𝑣 esc =
76 km s − reported byBaumgardt & Hilker (2018) for NGC 6441. We find 4 stars in thisvelocity range. Stars with a PM > − are not considered inour analysis because they are either mismatches between the NACO MNRAS000
76 km s − reported byBaumgardt & Hilker (2018) for NGC 6441. We find 4 stars in thisvelocity range. Stars with a PM > − are not considered inour analysis because they are either mismatches between the NACO MNRAS000 , 1–15 (2020)
Häberle et al. ]1.51.00.50.00.51.01.5 [ m a s y r ] Vector Point Diagram c o s [ m a s y r ] Proper motion errors
Median PM error of bright stars(0.032 mas yr )
10 11 12 13 14 15 16 17 m K s [ m a s y r ] Median PM error of bright stars(0.031 mas yr ) Figure 6.
The left panel shows the vector-point diagram with our measured PMs for NGC 6441. The sample is restricted to stars that have been used for thekinematic analysis. The two right panels show the magnitude dependence of the PM errors for the stars in the sample. The red line marks the median PM errorfor stars brighter than 𝑚 K S = . ∼ 𝑚 F555W = . − per PM component is reached. and HST catalogues, or are located in the outskirts of our field ofview, far away from the cluster centre.For each high-PM star, we investigated if its PM vector suggestsa past passage within 2.5 arcsec from the centre of the cluster (for asimilar procedure see Libralato et al. 2021). This specific radius is theinfluence radius of an IMBH with a mass of 𝑀 BH = . × M (cid:12) ,i.e., the upper limit of the IMBH mass we computed in Sect. 5.3.2.We find only one high-PM star with a PM vector consistent with anejection caused by the interaction with an IMBH in the core of thecluster.We also performed a statistical analysis of the total PMs as done byAnderson & van der Marel (2010) for the globular cluster 𝜔 Centauri.We divided our sample in 12 radial bins of 100 stars each, anddetermined various percentiles of the total PM distribution in eachsuch bin (see Figure 8). If an IMBH is harboured in the core ofNGC 6441, we would expect a higher number of high-velocity starscloser to the centre. However, we do not detect a significant differencebetween the distribution within the innermost 2 arcseconds and thedistributions in the outer bins at
𝑅 >
Kinematics tell us how fast the stars are moving inside the cluster.Dynamical models offer us a way to determine the underlying physicsthat make the stars move as they do. We ran dynamical models ofNGC 6441 to determine its mass profile, anisotropy profile, and dynamical distance. We also used our models to assess whether wecould determine the existence and properties of a possible IMBH atthe centre of the cluster.
We used a compilation of 4 kinematic datasets. We started withthe NACO PM catalogue of well-measured stars that was used todetermine the velocity dispersion profile of the cluster in Section 5.1.We augmented this with the catalogue of
HST
PMs from Watkinset al. (2015b), which probe further out in the cluster than the NACOdataset alone. To this, we added the LOS velocity dispersion profilesfrom Kamann et al. (2018) and Baumgardt & Hilker (2018). We usedthe surface brightness profile from Trager et al. (1995) as a proxy forthe surface number density profile from which the kinematic sampleswere drawn.For the dynamical models themselves, we used the spherical JeansAnisotropic Multi-Gaussian-Expansion (JAM, MGE) models (Cap-pellari 2008, 2015). The methodology broadly follows that describedin Section 5.2 and Appendix C of Hénault-Brunet et al. (2019), andwe refer to that paper for details. Here we briefly summarise theanalysis.In the JAM models, the stellar and mass densities are treated asMGEs. Each component of the MGEs has a width and a weight. Weassumed that the widths were the same for the stellar and mass MGEsbut allowed the weights to vary independently so as to fit a variablemass-to-light ratio (M/L). By giving each component of the stellarMGE an anisotropy, we were also able to fit for a variable anisotropyprofile. We used 5 Gaussian components to fit the cluster. We triedinitially with 6 and found that this was too many, so reduced to 5and found that this gave much improved and non-degenerate results.Altogether, this gave us 20 free parameters: 5 widths 𝑠 𝑖 , 5 stellar MNRAS , 1–15 (2020) unting for intermediate-mass black holes in globular clusters: an astrometric study of NGC 6441 [ m a s y r ] T , R [ m a s y r ] Tangential Velocity Dispersion T (this work)Radial Velocity Dispersion R (this work) T / R Anisotropy (this work)Anisotropy (Watkins et al. 2015) T , R [ k m s ] [ k m s ] Combined Velocity Dispersion (this work)Combined Velocity Dispersion (Watkins et al. 2015)MUSE LOS Velocity Dispersion z (Kamann et al. 2018) Figure 7.
This plot shows the results of the determination of the velocity dispersion both in units of mas yr − and km s − (converted using our best estimate forthe dynamical distance of 12.74 kpc). The upper panel shows the combined velocity dispersion (calculated by treating the tangential and radial proper motionas individual measurements). The middle panel shows the velocity dispersion calculated separately for the two components. The lower panel shows the ratiobetween the two, i.e., the anisotropy of the cluster. t o t a l [ m a s y r ] t o t a l [ m a s y r ]
25 %50 %75 %90 %95 %99 %Percentiles
Figure 8.
The upper panel shows the total proper motion (i.e. the quadraticsum of both components) of all stars in our measured sample and their distancefrom the centre. We separate the sample in radial bins of 100 stars each. Thelower plot shows different percentiles of the PM in each bin. density weights 𝜈 𝑖 , 5 mass density weights 𝜌 𝑖 , and 5 anisotropies 𝛽 (cid:48) 𝑖 .There were two further parameters – the distance 𝐷 , and the IMBHmass 𝑀 BH – taking the total number of free parameters to 22.In the spherical JAM models, the mean velocities are everywherezero, and the velocity distributions are characterised by the disper-sions in the projected-radial, projected-tangential, and LOS direc-tions. The likelihood of the data given the model was calculateddifferently for the PMs and the LOS velocities due to the differentdatasets available. We treated the PMs discretely, that is we did notbin the stars, but instead for each star calculated the likelihood ofobserving a star with the measured velocity and uncertainty giventhe mean velocity and velocity dispersion predicted by the model atthe position of the star, assuming Gaussian velocity distributions anduncertainties. For the LOS velocities, we had only binned velocitydispersion profiles, so we calculated the likelihood of measuring agiven velocity dispersion and uncertainty given the velocity disper-sion predicted by the model at the position of the bin.The Gaussian widths, Gaussian weights and the IMBH mass wereall fitted in log-space, as they had the potential to span many orders ofmagnitude. We used priors to restrict the range of certain parametersfor both physical and computational reasons, but otherwise used flatpriors within the allowed ranges.To avoid degeneracies between different components and to en-sure that the models fit for 5 separate components, we insisted thatlog 𝑠 𝑖 + − log 𝑠 𝑖 > .
2. Additionally, we limited the widths of theinnermost and outermost MGE components such that 𝑠 > 𝑅 and 𝑠 < 𝑅 max /√
3, where 𝑅 and 𝑅 max are the projected radial po- MNRAS000
3, where 𝑅 and 𝑅 max are the projected radial po- MNRAS000 , 1–15 (2020) Häberle et al. sitions of the 25-th star in the PM dataset and outermost point insurface brightness profile, respectively. The first condition ensuredthat there were at least 25 stars inside of 𝑠 to constrain the innercluster properties, and the second condition ensured that the outerslope of the density profiles was at least 3, which is required for afinite system.The light and mass weights were assumed to be positive. Wefurther added a lower limit on the M/L of each component such that 𝜌 𝑖 / 𝜈 𝑖 > .
1, as values lower than this would be unrealistic physically. 𝛽 (cid:48) is a modified anisotropy 𝛽 (cid:48) = 𝜎 𝑟 − 𝜎 𝑡 𝜎 𝑟 + 𝜎 𝑡 (2)where 𝜎 𝑟 and 𝜎 𝑡 are the velocity dispersions in the radial and tan-gential directions in a spherical coordinate system. This modifiedanisotropy is defined to be both symmetric about isotropy ( 𝛽 (cid:48) = 𝛽 (cid:48) = ± 𝛽 (cid:48) ≈ ± . to avoid extreme anisotropies that are computationallyintensive but not seen in real clusters.Finally, we restricted how much the density and anisotropy profilescould change from component to component by insisting that − < Δ 𝜌,𝜈 / Δ 𝑠 < − < Δ 𝛽 (cid:48) / Δ 𝑠 <
2, where Δ 𝜌 = 𝜌 𝑖 + − 𝜌 𝑖 andsimilarly for Δ 𝜈 and Δ 𝑠 and Δ 𝛽 (cid:48) . In practice, very few componentshit the limits of these ranges.We set a lower limit of the central black hole mass of 0.1 M (cid:12) ;the BH mass was fit in log space and very small masses are indistin-guishable in the model so this parameter has the potential to go to −∞ if not limited. No other restrictions were set. The distance wasfit in linear space, and was assumed to be positive with no furtherrestrictions. We explored the available parameter space using the affine-invariantMarkov Chain Monte Carlo software emcee (Foreman-Mackey et al.2013). We used 100 walkers and ran for 7 500 steps to be sure that thechains had fully converged. The resulting fits to the cluster are shownin Figure 9. In each panel, the coloured data points show the datawith their uncertainties (in cases where there are no points, data iseither not available or not measurable). The black solid lines show themedian of the fits for the profile, the darker shaded region spans the15.9 to 84.1 percentiles (equivalent to the 1-sigma confidence regionfor a Gaussian distribution) and the light shaded region spans the 2.5to 97.5 percentiles (equivalent to the 2-sigma confidence region fora Gaussian distribution).The left panels show from top to bottom the projected radial dis-persion profile 𝜎 R , the projected tangential dispersion profile 𝜎 T , andthe LOS velocity dispersion profile 𝜎 z . In the top and middle panels,the blue points show the profiles calculated from NACO PMs and theorange points show the profiles calculated from the HST
PMs, how-ever we stress that the fits were not done to these profiles but to theindividual measurements. The profiles are shown here for visualisa-tion purposes. In the bottom panel, the green points show the MUSEdispersion profile from Kamann et al. (2018) and the purple pointsthe dispersion profile from Baumgardt & Hilker (2018); for these wedid fit directly to the binned dispersion profiles. In general, these fitsare very good. The models struggle in the outer regions where thereare only LOS constraints but not PMs, and show some broadening The actual limits correspond to cases where one dispersion is 4 or timesthe other. in the centre where there are very few stars. The only poorly-fit pointis the central point in the MUSE LOS dispersion profile, which themodels were not able to constrain. Kamann et al. (2018) also pointout this central dip in their dispersion profile; they note that it couldpotentially be due to crowding but argue that this is likely not the caseas the cluster does not have a steep central surface brightness profile.That the models are unable to reconcile the PM and LOS dispersionprofile may suggest otherwise and that the central MUSE data pointis more affected by crowding than previously believed. This effectdue to crowding is also clearly seen in the MUSE velocity dispersionprofile of the cluster M54 (Alfaro-Cuello et al. 2019, 2020) and canbe overcome by higher spatial resolution data (Alfaro-Cuello et al.in prep.).The panels in the central column show the anisotropies, from topto bottom they are tangential over radial, radial PM over LOS, andtangential PM over LOS. For the top panel, we are able to showdata (NACO PMs in blue and HST
PMs in orange as for the PMdispersion panels) as we have the same set of bins for both the radialPM and tangential PM datasets. For the middle and lower panels, weshow only the model fits, where we have used the distance of eachmodel to convert the PMs from mas yr − to km s − so that isotropyis at 1; as we do not have the same data coverage (and, hence, bins)for the LOS and PM samples we cannot calculate these anisotropiesdirectly, but the models given us an insight into what we cannotmeasure easily from the existing data. All panels are not constant,indicating that there is some anisotropy in the system and it variesthrough the cluster.The right column shows, from top to bottom, the surface brightnessprofile, the projected M/L, and a proxy for the projected sphericalanisotropy profile. The surface brightness profile is a key part of themodel and so we are able to compare the model fit to the data, and itis a very good fit overall.We cannot measure the mass directly – indeed, this is one ofthe main motivators for carrying out these models – so the M/Lprofile has no data points against which to compare. The M/L isclearly not constant through the cluster. It is ∼ . (cid:12) / L (cid:12) at thecentre, falls slightly in the intermediate regions and then increasesto ∼
10 M (cid:12) / L (cid:12) in the outer parts. This is consistent with a clusterthat has some mass segregation, whereby high-mass stars tend tobe more centrally concentrated than low-mass stars. The differencein mass between high-mass stars and low-mass stars is only arounda factor of a few, 10 at most, but the low-mass stars are far morenumerous than the high-mass stars, so it is the low-mass stars thatdominate the mass budget. However, the high-mass stars are manyhundreds or thousands times brighter than the low-mass stars, so itis the high-mass stars that dominate the light budget, despite theirlower numbers. This interplay typically gives an M/L profile muchlike what we see here. The dashed line shows the global M/L of ∼ . + . − . × M (cid:12) . This value is higher than the values reported by other au-thors. McLaughlin & van der Marel (2005) determined a mass inthe range of (1 . + . − . − . + . − . ) × M (cid:12) based on M/L valuesof stellar population fits. Baumgardt & Hilker (2018) report a massof 1 . × M (cid:12) based on a comparison between isotropic N-bodymodels of globular clusters with observed LOS velocity measure-ments. We advise caution in interpreting these global measurements.The profiles are best constrained where there is a lot of kinematic MNRAS , 1–15 (2020) unting for intermediate-mass black holes in globular clusters: an astrometric study of NGC 6441 R [arcsec] . . . . σ R [ m a s y r − ] HST Watkins+15NACO Haeberle+0 . . . . σ T [ m a s y r − ] R [arcsec] σ z [ k m s − ] Baumgardt+18Kamann+18 10 R [arcsec] . . . . σ T / σ R . . σ R / σ z R [arcsec] . . . σ T / σ z R [arcsec] Σ L [ L (cid:12) p c − ] Trager+9510 M / L fi t [ M (cid:12) L − (cid:12) ] R [arcsec] − . . . β Σ L / Σ L Figure 9.
Best-fitting model profiles. From left to right then top to bottom: radial PM dispersion, tangential PM dispersion, LOS velocity dispersion, tangentialPM / radial PM anisotropy, radial PM / LOS velocity anisotropy, tangential PM / LOS velocity anisotropy; surface brightness, projected M/L, proxy for projectedspherical anisotropy. In all panels, the solid black line is the median of the fits, the darker shaded regions span the 15.9 to 84.1 percentiles (approximately the1-sigma confidence region) and the lighter shaded regions span the 2.5 to 97.5 percentiles (approximately the 2-sigma confidence region). The coloured datapoints show data from this work and from literature sources to which the fits were performed as shown. In the anisotropy panels, the dotted lines highlightisotropy. In the mass-to-light panel, the dashed line shows the global M/L for the cluster calculated as the total mass over the total light. Overall, the fits to thedata points are good. The cluster is best fit by a variable M/L, consistent with a cluster with some mass segregation, and a variable anisotropy profile, consistentwith a cluster that is more relaxed in the central regions than the outer parts due to their different relaxation times. data (the intermediate radii) and less well constrained where the datais sparse (the innermost and outermost regions). In particular, theouter regions of the mass profile are driven by the outermost data-point in the (Baumgardt & Hilker 2018) LOS profile, where there isno corresponding PM data at all. This is reflected in the large scatterwe see in the outer regions of the fits.The middle panels of Figure 9 show how the dispersions measuredin two orthogonal directions compare in a 3-dimensional system. Thelower panel in the right column effectively shows the 3-dimensional(spherical) anisotropy. Recall, in the JAM models, each componentof the stellar density MGE is assigned an anisotropy 𝛽 (cid:48) . We calculatea luminosity-weighted anisotropy profile by multiplying the stellardensity weights by the anisotropy and calculating the correspondingMGE profile. We divide this by the stellar density (surface bright-ness) MGE to get a proxy for the anisotropy profile. Here isotropyis at 0. The slight negative bias at the centre indicates some mild(spherical) tangential anisotropy there, although the models are alsoconsistent with isotropy. The positive bias in the outer parts indi-cates (spherical) radial anisotropy there. This trend is consistent withtheoretical expectations for cluster evolution and with cluster simu-lations. Clusters are expected to develop with some radial anisotropy,but they become more isotropic as they relax (Baumgardt & Makino2003; Vesperini et al. 2014; Tiongco et al. 2016). As relaxation timesare shorter at the centres, the centres relax and reach isotropy first.For NGC 6441, Harris (2010) reports a half-mass relaxation time oflog ( 𝑡 hm / yr ) = .
09, while the core relaxation time is significantlyshorter with log ( 𝑡 core / yr ) = . 𝑟 core = .
13 arcsec fromHarris (2010) (converted to 𝑟 core = .
48 pc using our dynamicaldistance estimate), and assumed an average stellar mass of M (cid:12) . Theother parameters are based on our model. With our total mass estimateof 2 . + . − . × M (cid:12) , we find a half-mass radius of 8 . + . − . pc. Wedetermine a core density of ( . + . − . )× M (cid:12) pc − . The resultingvalues for the relaxation times are log ( 𝑡 hm / yr ) = . + . − . in thecore, and log ( 𝑡 core / yr ) = . + . − . at the half-mass radius.While our values for the relaxation time in the core are similarto the values in the literature, our value for the half-mass relaxationtime is larger as the previously reported values mainly because ofthe larger estimate for the half-mass radius. Due to the variable M/Lratio and our high total mass estimate, this value is higher then thehalf-light radius, which was used as a proxy for the half-mass radiusin previous studies.Although we advise caution in interpreting these global values (seeabove), we can see that the relaxation time at the half-mass radiusis longer than in the core. This difference in relaxation times couldexplain why we observe isotropy in the centre, but anisotropy in theouter regions of the cluster. Similar astrometric findings for otherclusters are discussed in Watkins et al. (2015a).Figure 10 shows the distribution of distance (top panel) and IMBHmass (bottom panel) estimates at the end of the MCMC run. Toconstruct these plots, we selected 10 000 points in total, 100 walk-ers from each of 100 steps at 50 step intervals. The distance isnicely constrained, and approximately Gaussian overall. To deter-mine a distance estimate and uncertainty, we calculate the medianand 15.9 and 84.1 percentiles of the distribution to obtain distance 𝐷 = . + . − . kpc. This estimate is in reasonable agreement withthe estimate from Kamann et al. (2018) of 12 . ± . MNRAS000
48 pc using our dynamicaldistance estimate), and assumed an average stellar mass of M (cid:12) . Theother parameters are based on our model. With our total mass estimateof 2 . + . − . × M (cid:12) , we find a half-mass radius of 8 . + . − . pc. Wedetermine a core density of ( . + . − . )× M (cid:12) pc − . The resultingvalues for the relaxation times are log ( 𝑡 hm / yr ) = . + . − . in thecore, and log ( 𝑡 core / yr ) = . + . − . at the half-mass radius.While our values for the relaxation time in the core are similarto the values in the literature, our value for the half-mass relaxationtime is larger as the previously reported values mainly because ofthe larger estimate for the half-mass radius. Due to the variable M/Lratio and our high total mass estimate, this value is higher then thehalf-light radius, which was used as a proxy for the half-mass radiusin previous studies.Although we advise caution in interpreting these global values (seeabove), we can see that the relaxation time at the half-mass radiusis longer than in the core. This difference in relaxation times couldexplain why we observe isotropy in the centre, but anisotropy in theouter regions of the cluster. Similar astrometric findings for otherclusters are discussed in Watkins et al. (2015a).Figure 10 shows the distribution of distance (top panel) and IMBHmass (bottom panel) estimates at the end of the MCMC run. Toconstruct these plots, we selected 10 000 points in total, 100 walk-ers from each of 100 steps at 50 step intervals. The distance isnicely constrained, and approximately Gaussian overall. To deter-mine a distance estimate and uncertainty, we calculate the medianand 15.9 and 84.1 percentiles of the distribution to obtain distance 𝐷 = . + . − . kpc. This estimate is in reasonable agreement withthe estimate from Kamann et al. (2018) of 12 . ± . MNRAS000 , 1–15 (2020) Häberle et al. . . . . . D [kpc] P ( D ) . +0 . − . kpc − log M BH /M (cid:12) . . . . . P ( l og M B H ) M BH < . × M (cid:12) log M BH < . Figure 10.
The final distributions of the distance (top) and IMBH mass(bottom) parameters at the end of the MCMC chains. The distance is wellconstrained and the median and 15.9 and 84.1 percentiles were used to providea distance estimate as shown in the top right corner. The IMBH mass isunconstrained and we can neither confirm nor rule out the presence of anIMBH, but we are able to place an upper limit on a possible IMBH mass, asgiven in the top right corner. from the values of 11.6 kpc from Harris (2010) and of 11 . ± .
14 kpcfrom Baumgardt et al. (2019).The putative IMBH mass is not at all well constrained as evidencedby the flat distribution in the lower panel of Figure 10. With this datawe cannot conclusively state whether or not an IMBH is present asmodels with a fairly massive BH and models with sub-solar-mass(effectively no) BH are equally likely, although models with verymassive BHs are ruled out. The best we can do here is a place anupper limit on the mass of a possible IMBH at 𝑀 BH < . × M (cid:12) . There are several scaling relations for black-hole masses in galacticbulges that are valid over a wide mass range. We extrapolate them tothe mass of NGC 6441 and compare the predicted black hole masswith our upper limit.Using our estimate for the mass of the cluster of 2 . + . − . × M (cid:12) , the extrapolation of the relation between the Bulge and BHmasses of Schutte et al. (2019) gives an expected black hole mass of1 . + . − . × M (cid:12) , significantly smaller than our upper limit.If we use the relation between velocity dispersion and BHmass of Gebhardt et al. (2000), our central velocity dispersion of(19.1 ± − translates in an IMBH mass of 1 . + . − . × M (cid:12) which is quite similar to our upper limit.Although these estimates are consistent with our upper limit,they are extrapolated from measurements of black-hole masses ina different (higher) mass regime than that of IMBHs in GCs. Amore similar mass range is found in nuclear star clusters, in whichthe mass ratio between stellar cluster mass and black hole is typi-cally 𝑀 BH / 𝑀 NSC ≈ .
25 with a scatter of 2 (Nguyen et al. 2018; Greene et al. 2019). This is much higher than the upper limit of 𝑀 BH / 𝑀 NGC 6441 ≤ . BH < (cid:12) , based on the (missing) radio signature of an accretingIMBH in NGC 6441.We can conclude that a larger sample of stars with precise PMsin the central region is necessary to observationally reach the lowerend of the predicted black hole mass range and to put a final answeron the existence/non-existence of an IMBH in this cluster. We determined PMs of around 1400 stars in the inner 15 arcsecondsof the globular cluster NGC 6441 by combining space-based andground-based position measurements taken 15 years apart. Becauseof the high astrometric precision of both epochs and the long timebasebetween them, our PM measurements reach a precision of 30 µ as yr − for bright stars ( 𝑚 𝐾 S < . 𝑚 F555W < . HST -based studies (see Bellini et al. 2014) whilethe
Gaia
DR2 PMs have a higher uncertainty of around 0.1 mas yr − for stars with 𝐺 =
17 mag in less crowded fields (Lindegren et al.2018). This proves the potential of combined ground-based AO andspace-based astrometry with a long temporal baseline.With the PM data we were able to determine the velocity dispersionprofile of evolved stars in core of the cluster. In the innermost arcsec-ond we measure a velocity dispersion of (0.316 ± − which corresponds to (19.1 ± − assuming a distance of12.74 kpc.Using our PM measurements, we searched for signatures of apotential IMBH. Although we find one fast-moving object, whoseprojected trajectory is compatible with being ejected from the corebecause of the interaction with an IMBH, a statistical analysis of thePMs in our field does not show any signs of the presence of an IMBHin the centre of NGC 6441. A complete 5D picture (including radialvelocity measurements) and deeper observations of the core are stillneeded to clearly confirm or rule out any hypothesis.We used Jeans models to fit a combination of our newly-obtainedkinematic data and existing kinematic catalogues. From the best-fitmodels we could determine the underlying physical properties of thecluster: the global M/L of the cluster is ∼ . (cid:12) / L (cid:12) , but it variesfrom ∼ . (cid:12) / L (cid:12) in the core of the cluster to ∼ . (cid:12) / L (cid:12) inthe outskirts. This is consistent with mass segregation where bright,high-mass stars are more centrally concentrated than low-mass stars.In the core we do not observe significant anisotropy, however theouter parts of the cluster show some radial anisotropy. By combiningLOS velocity measurements with our PM measurements we obtaina dynamical distance of 𝐷 = . + . − . kpc. The models include apossible IMBH in the centre of the cluster. Our results are compatiblewith both the existence and non-existence of such a black hole, andwe can only place an upper limit of 𝑀 BH < . × M (cid:12) on themass of the black hole. This value is about one order of magnitudelarger than the mass predicted by extrapolating the relation betweenBH and bulge masses, but consistent with other predictions for GCs.In future studies, deeper observations of the cluster would bebeneficial as the number of stars in our kinematic sample is clearlylimited by the number of detectable stars in the NACO exposures.In the second half of this decade, instruments at extremely large MNRAS , 1–15 (2020) unting for intermediate-mass black holes in globular clusters: an astrometric study of NGC 6441 telescopes, such as ELT MICADO, will allow PM measurements withan even higher precision and be able to finally answer the questionof whether IMBHs are present in the cores of globular clusters. ACKNOWLEDGEMENTS
Based on observations with the NASA/ESA
Hubble Space Telescope ,obtained at the Space Telescope Science Institute,which is operatedby AURA, Inc., under NASA contract NAS5-26555.Based on observations collected at the European Southern Obser-vatory under ESO programme 0101.D-0385.This work has made use of data from the European SpaceAgency (ESA) mission
Gaia ( ), processed by the Gaia
Data Processing and Analysis Consor-tium (DPAC, ). Funding for the DPAC has been provided by nationalinstitutions, in particular the institutions participating in the
Gaia
Multilateral Agreement.MH acknowledges financial funding from the
Studienstiftung desdeutschen Volkes that supported his visit at STScI.Support for this work was provided by grants for
HST programsAR-14322 and AR-15055 provided by the Space Telescope ScienceInstitute, which is operated by AURA, Inc., under NASA contractNAS 5-26555.
DATA AVAILABILITY
The data underlying this article were accessed from the ESO ScienceArchive and the Mikulski Archive for Space Telescopes (MAST) .The numerical values of the velocity dispersion determined in thiswork are provided in the article (Table B1) and its online supplemen-tary material. REFERENCES
Alfaro-Cuello M., et al., 2019, The Astrophysical Journal, 886, 57Alfaro-Cuello M., et al., 2020, The Astrophysical Journal, 892, 20Anderson J., King I., 2003, Publications of the Astronomical Society of thePacific, 115, 113Anderson J., King I. R., 2004, Multi-filter PSFs and Distortion Correctionsfor the HRC, Instrument Science Report ACS 2004-15Anderson J., van der Marel R. P., 2010, The Astrophysical Journal, 710, 1032Anderson J., Bedin L. R., Piotto G., Yadav R. S., Bellini A., 2006, Astronomy& Astrophysics, 454, 1029Baumgardt H., Hilker M., 2018, Monthly Notices of the Royal AstronomicalSociety, 478, 1520Baumgardt H., Makino J., 2003, Monthly Notices of the Royal AstronomicalSociety, 340, 227Baumgardt H., Hilker M., Sollima A., Bellini A., 2019, Monthly Notices ofthe Royal Astronomical Society, 482, 5138Bellini A., Bedin L. R., 2009, Publications of the Astronomical Society ofthe Pacific, 121, 1419Bellini A., Bedin L. R., 2010, Astronomy and Astrophysics, 517, A34Bellini A., et al., 2014, The Astrophysical Journal, 797, 115Bellini A., Anderson J., Bedin L. R., King I. R., van der Marel R. P., PiottoG., Cool A., 2017a, The Astrophysical Journal, 842, 6 http://archive.eso.org https://archive.stsci.edu/ Bellini A., Bianchini P., Varri A. L., Anderson J., Piotto G., van der MarelR. P., Vesperini E., Watkins L. L., 2017b, The Astrophysical Journal,844, 167Bellini A., Anderson J., Grogin N. A., 2018a, Focus-diverse, empirical PSFmodels for the ACS/WFC, Instrument Science Report ACS 2018Bellini A., et al., 2018b, The Astrophysical Journal, 853, 86Bellini A., et al., 2018c, ApJ, 853, 86Bohlin R. C., 2016, The Astronomical Journal, 152, 60Cappellari M., 2008, MNRAS, 390, 71Cappellari M., 2015, arXiv e-prints, p. arXiv:1504.05533Davies R., et al., 2016, in Evans C. J., Simard L., Takami H., eds, Society ofPhoto-Optical Instrumentation Engineers (SPIE) Conference Series Vol.9908, Ground-based and Airborne Instrumentation for Astronomy VI. p.99081Z ( arXiv:1607.01954 ), doi:10.1117/12.2233047Djorgovski S., 1993, in Djorgovski S. G., Meylan G., eds, AstronomicalSociety of the Pacific Conference Series Vol. 50, Structure and Dynamicsof Globular Clusters. p. 373Drukier G. A., Bailyn C. D., 2003, The Astrophysical Journal, 597, L125Foreman-Mackey D., Hogg D. W., Lang D., Goodman J., 2013, PASP, 125,306Gaia Collaboration et al., 2016, A&A, 595, A2Gaia Collaboration et al., 2018, A&A, 616, A1Gebhardt K., et al., 2000, The Astrophysical Journal Letters, 539, L13Giersz M., Leigh N., Hypki A., Lützgendorf N., Askar A., 2015, MonthlyNotices of the Royal Astronomical Society, 454, 3150Gilliland R. L., Rajan A., Deustua S., 2010, WFC3 UVIS Full Well Depths,and Linearity Near and Beyond Saturation, Space Telescope WFC In-strument Science ReportGreene J. E., Strader J., Ho L. C., 2019, arXiv e-prints, 1911,arXiv:1911.09678Harris W. E., 2010, arXiv:1012.3224 [astro-ph]Hénault-Brunet V., Gieles M., Sollima A., Watkins L. L., Zocchi A., ClaydonI., Pancino E., Baumgardt H., 2019, MNRAS, 483, 1400Kamann S., et al., 2018, Monthly Notices of the Royal Astronomical Society,473, 5591Kerber L. O., et al., 2019, Monthly Notices of the Royal Astronomical Society,484, 5530Lenzen R., et al., 2003, in Instrument Design and Perfor-mance for Optical/Infrared Ground-based Telescopes. In-ternational Society for Optics and Photonics, pp 944–952,doi:10.1117/12.460044,
Libralato M., Bellini A., Bedin L. R., Piotto G., Platais I., Kissler-Patig M.,Milone A. P., 2014, Astronomy & Astrophysics, 563, A80Libralato M., et al., 2015, Monthly Notices of the Royal Astronomical Society,450, 1664Libralato M., et al., 2018, The Astrophysical Journal, 861, 99Libralato M., et al., 2021, MNRAS, 500, 3213Lindegren L., 1978, International Astronomical Union Colloquium, -1, 197Lindegren L., et al., 2018, Astronomy & Astrophysics, 616, A2McLaughlin D. E., van der Marel R. P., 2005, The Astrophysical JournalSupplement Series, 161, 304Miller M. C., Hamilton D. P., 2002, Monthly Notices of the Royal Astronom-ical Society, 330, 232Nguyen D. D., et al., 2018, The Astrophysical Journal, 858, 118Plewa P. M., et al., 2015, Monthly Notices of the Royal Astronomical Society,453, 3234Plewa P. M., et al., 2018, Research Notes of the American AstronomicalSociety, 2, 35Portegies Zwart S. F., McMillan S. L. W., 2002, The Astrophysical Journal,576, 899Rousset G., et al., 2003, in Adaptive Optical System TechnologiesII. International Society for Optics and Photonics, pp 140–149,doi:10.1117/12.459332,
MNRAS000
MNRAS000 , 1–15 (2020) Häberle et al.
Schmidtobreick L., Pompei E., Smoker J., 2018, NACO VLT User Manualv103Schutte Z., Reines A. E., Greene J. E., 2019, The Astrophysical Journal, 887,245Skrutskie M. F., et al., 2006, The Astronomical Journal, 131, 1163Tiongco M. A., Vesperini E., Varri A. L., 2016, Monthly Notices of the RoyalAstronomical Society, 455, 3693Trager S. C., King I. R., Djorgovski S., 1995, AJ, 109, 218Tremou E., et al., 2018, The Astrophysical Journal, 862, 16Valenti E., Ferraro F. R., Origlia L., 2010, Monthly Notices of the RoyalAstronomical Society, 402, 1729Vesperini E., Varri A. L., McMillan S. L. W., Zepf S. E., 2014, MonthlyNotices of the Royal Astronomical Society, 443, L79Watkins L. L., van der Marel R. P., Bellini A., Anderson J., 2015a, TheAstrophysical Journal, 803, 29Watkins L. L., van der Marel R. P., Bellini A., Anderson J., 2015b, ApJ, 812,149di Nino D., Makidon R. B., Lallo M., Sahu K. C., Sirianni M., CasertanoS., 2008, HST Focus Variations with Temperature, Instrument ScienceReport ACS 2008-03van der Marel R. P., Anderson J., 2010, The Astrophysical Journal, 710, 1063
APPENDIX A: GEOMETRIC DISTORTION CORRECTION
A set of geometric distortion corrections (GDCs) for NACO hasalready been published by Plewa et al. (2015, 2018). However, theauthors themselves noted that the distortion of the NACO detectoris not stable over time, but shows abrupt changes most likely linkedto instrument interventions. Furthermore, we do not know the exactdefinition of the PSF centre used to determine the literature GDC andthere is a degeneracy between the PSF definition and the GDC. Forthese reasons, we decided to independently solve for the geometricdistortion (GD) of the NACO detector using our data in order toachieve the best astrometric precision possible.To determine the GDC, we tried both an autocalibration approachand the use of the ACS/HRC catalogue as a distortion-free reference.For the scientific analysis, we relied on the calibration based on theexternal ACS/HRC catalogue, as the resulting geometric distortionmodel led to a much better correction with fewer residual distortions.
A1 Determination of the GDC using an external referencecatalogue
A1.1 The reference catalogue
The
HST
ACS/HRC observations of NGC 6441 have a very highprecision and a very reliable distortion correction that reaches anaccuracy < 0.01 HRC pixel (see the Instrument Science Report An-derson & King 2004). In comparison with the uncorrected NACOcatalogues, they can be considered effectively distortion free. How-ever, we have to take into account the 15-year long time baseline be-tween the
HST and the NACO observations. The velocity dispersionin the cluster centre of around 18 km s − at a distance of 12.74 kpc(our dynamical distance estimate) leads to an rms displacement ofaround 0.4 NACO pixel. While this effect is purely statistical and isaveraged out when measurements of multiple stars are combined, itstill leads to a decreased precision of the GDC and can mask smallerGD effects.To overcome the limitations caused by the stellar motions, wemade use of a PM catalogue for the core of NGC 6441 createdusing the HRC exposures from 2003 and WFC3/UVIS observationsfrom 2014 and that will be the subject of a future paper (Bellini et al., in preparation). Suffice here to say that the data reductionand proper-motion computation of this catalogue closely followedthe prescriptions given in great details in Bellini et al. (2014) andBellini et al. (2018b). The number of well-measured stars in the coreof the cluster is relatively small due to the larger pixel scale of theWFC3/UVIS channel (40 mas pixel − ). On the other hand, we onlyuse bright, well-measured stars to determine the GDC anyway.The PMs in this catalogue were used to propagate the HRC 2003positions to the epoch of the NACO observations. We only in-cluded stars in the reference catalogue with total PM error 𝜎 PM < .
07 mas yr − , which corresponds to an error in the displacementof 0.08 NACO pixel. Furthermore, we restricted the selection of ref-erence stars using different quality criteria. In the end, around 1600stars were available as reference stars. A1.2 Matching individual frames on reference catalogue
As a first step, we matched the astrometric catalogues containing un-corrected NACO positions onto the reference catalogue using lineartransformations. The parameters of the linear transformations weredetermined using a least-squares fit. The cut-off radius for match-ing stars was initially 3.5 NACO pixels and it was progressivelydecreased down to 1 NACO pixel during the iterative process.To avoid absorbing the linear distortion terms (the so-called skew)already at this level, we used 4-parameter transformations, whichcontain a shift in x and y direction, a rotation and a change of scale.The residuals of the linear-transformation fit now contained boththe individual measurement errors, and also the geometric distortionof the NACO images. We divided the detector in 10 ×
10 quadraticbins and collected the residuals from all matched NACO catalogues.We then calculated the 3 𝜎 -clipped median of the residuals in eachbin containing at least 20 residuals. The maps of the binned residualscan be seen in Figure A1 and provide a first indicator of how thegeometric distortion correction will look like. A1.3 Fit of a 2D polynomial model
We modelled the binned residuals using the following 2D third orderstandard polynomials: 𝛿𝑥 = 𝑎 𝑥 + 𝑎 𝑦 + 𝑎 𝑥 + 𝑎 𝑥𝑦 + 𝑎 𝑦 + 𝑎 𝑥 + 𝑎 𝑥 𝑦 + 𝑎 𝑥𝑦 + 𝑎 𝑦 𝛿𝑦 = 𝑏 𝑥 + 𝑏 𝑦 + 𝑏 𝑥 + 𝑏 𝑥𝑦 + 𝑏 𝑦 + 𝑏 𝑥 + 𝑏 𝑥 𝑦 + 𝑏 𝑥𝑦 + 𝑏 𝑦 (A1)We used a least-square fit to determine the coefficients. The coeffi-cients 𝑎 and 𝑎 were set to zero to lower the degrees of freedomof the fit and to enforce that, at the centre of the detector, our GDCwill lead to the same 𝑥 -scale as the detector and the corrected andraw 𝑦 -axis will be aligned. However, as we cannot assume that 𝑥 and 𝑦 axis have the same scale/are perpendicular, we can not set theparameters 𝑏 and 𝑏 to zero.A higher order of the polynomials did not lead to a better fit ofthe data. Also, the use of a different polynomial base (e.g. Zernikepolynomials) had no significant influence on the result. MNRAS , 1–15 (2020) unting for intermediate-mass black holes in globular clusters: an astrometric study of NGC 6441 A1.4 Iterative Process
To avoid that the linear transformations between the single NACOcatalogues and the reference frame are biased by uncorrected geo-metric distortion, we repeated the procedures described in an iterativeprocess. After the polynomial coefficients have been determined, weapplied 75% of the corrections to the raw NACO catalogues andrepeated the full process (described in A1.2 and A1.3) with the newcorrected coordinates until the polynomial coefficients converge (inour case after 200 iterations).
A2 Determination of the GDC with an autocalibrationapproach
Initially, we planned to solve for the GDC using an autocalibrationapproach in which the distortion-free reference frame is obtainedby combining multiple different pointings. As the position measure-ments of each star are based on measurements at different parts of thedetector, the effect of the GD randomises and therefore is averagedout. This is a well-proven technique and has been used to calibratethe GD of the
HST (Anderson & King 2003; Bellini & Bedin 2009),but also for various ground-based studies (Anderson et al. 2006;Bellini & Bedin 2010; Libralato et al. 2014, 2015). We refer to thesepublications for a detailed description of the iterative process.In comparison to the GDC obtained with an external reference(see section above) our autocalibration result showed significant dif-ferences (see Figure A2): we were unable to determine the lineardistortion (the so-called skew) terms, which caused a rms differenceof 0.68 NACO pixel, but also the nonlinear differences had an rmsof 0.28 NACO pixel.By comparing the NACO master frames with independent
HST catalogues of NGC 6441 (based on WFC3/UVIS and ACS/WFCobservations), we could verify that the differences between the dis-tortion corrections indeed are caused by uncorrected GD in the au-tocalibration reference frame.It can be easily understood why we were not able to determinethe linear distortion terms using the autocalibration: the linear skewterms are the same over the whole field of the detector. Even if starsare measured on different detector positions, their position measure-ments are affected by the same skew and therefore the reference framealso has the same skew. This degeneracy is lifted if an instrumentwith multiple detectors is used, as is the case for the papers citedabove, or if there are pointings with different orientation on sky.The nonlinear differences are most likely caused by a relativelylow number of stars in comparison with e.g., the autocalibration ofthe
HST or ground-based wide-field instruments.The failure of the autocalibration approach with our 2018 NACOdataset highlights the importance of the choice of the dither + rotationpattern. Especially if no external reference with sufficient astrometricprecision is available (as will possibly be the case for future instru-ment at ELTs), a particular attention on this is required during thepreparation of future observations. APPENDIX B: VELOCITY DISPERSION DATA
Table B1 contains the values of the PM-based velocity dispersionscomputed in our work.
This paper has been typeset from a TEX/L A TEX file prepared by the author. MNRAS000
This paper has been typeset from a TEX/L A TEX file prepared by the author. MNRAS000 , 1–15 (2020) Häberle et al. (a) Distortion map before the correction.rms of residuals: 0.321 NACO pixel (b) Distortion map after the correction.rms of residuals: 0.032 NACO pixel
Figure A1.
The panels (a) and (b) show the distortion map of the NACO detector before (a) and after (b) the polynomial geometric distortion correction has beenapplied. The large central field shows a map of the detector with the grid of 10 ×
10 binned residuals between the NACO measurements and the
HST reference.The distortion vectors are only plotted when more than 20 measurements fell in the respective bin. The length of the vectors is enlarged by a factor of 250. Theside plots show all combinations of 𝑥 and 𝑦 residuals as a function 𝑥 and 𝑦 positions. 𝑅 [arcsec] 𝜎 [mas yr − ] 𝜎 R [mas yr − ] 𝜎 T [mas yr − ]0.76 0.316 ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± ± Table B1.
Results for the proper motion dispersion profiles in the inner region of NGC 6441.MNRAS , 1–15 (2020) unting for intermediate-mass black holes in globular clusters: an astrometric study of NGC 6441 y [ N A C O p i x e l ] HST Calibration y [ N A C O p i x e l ] Autocalibration y [ N A C O p i x e l ] Linear Differences (RMS = 0.65 pixel) y [ N A C O p i x e l ] Non linear differences (RMS = 0.28 pixel)
Figure A2.
This figure shows a comparison of the two distortion corrections we obtained with different methods: the external calibration (upper left) versusthe autocalibration (upper right). Significant differences (total differences between models: 0.68 pixels) can be observed between the maps. The deviations aredominated by a linear part (lower left, rms = 0.65 NACO pixel), however, also the nonlinear differences are significant (lower right, rms = 0.28 NACO pixel).MNRAS000