A multi-mass velocity dispersion model of 47 Tucanae indicates no evidence for an intermediate mass black hole

A multi-mass velocity dispersion model of 47 Tucanae indicates no evidence for an intermediate mass black hole

Christopher R. Mann Department of Physics and Astronomy
 University of British Columbia
 Vancouver, BC V6T 1Z1
 Canada
Harvey Richer Department of Physics and Astronomy
 University of British Columbia
 Vancouver, BC V6T 1Z1
 Canada
Jeremy Heyl Department of Physics and Astronomy
 University of British Columbia
 Vancouver, BC V6T 1Z1
 Canada
Jay Anderson Space Telescope Science Institute
 3700 San Martin Drive
 Baltimore, MD 21218
 USA
Jason Kalirai Space Telescope Science Institute
 3700 San Martin Drive
 Baltimore, MD 21218
 USA
Ilaria Caiazzo Department of Physics and Astronomy
 University of British Columbia
 Vancouver, BC V6T 1Z1
 Canada
Swantje D. Möhle Department of Physics and Astronomy
 University of British Columbia
 Vancouver, BC V6T 1Z1
 Canada
Alan Knee Department of Physics and Astronomy
 University of British Columbia
 Vancouver, BC V6T 1Z1
 Canada
Holger Baumgardt School of Mathematics and Physics
 University of Queensland
 St. Lucia, QLD 4068
 Australia
Abstract

In this paper we analyze stellar proper motions in the core of the globular cluster 47 Tucanae to explore the possibility of an intermediate mass black hole (IMBH) influence on the stellar dynamics. Our use of short wavelength photometry affords us an unprecedentedly clear view of stellar motions into the very centre of the crowded core, yielding proper motions for 50,000 stars in the central 2 arcminutes. We model the velocity dispersion profile of the cluster using an isotropic Jeans model. The density distribution is taken as a central IMBH point mass added to a combination of King profiles. We individually model the general low-mass cluster objects (main sequence/giant stars), as well as the concentrated populations of heavy binary systems and dark stellar remnants. Using un-binned likelihood model-fitting, we find that the inclusion of the concentrated populations in our model plays a crucial role in fitting for an IMBH mass. Taking into account all of these cluster sub-populations our model predicts an IMBH to cluster mass ratio of . The concentrated binaries and stellar-mass black holes produce a sufficient velocity dispersion signal in the core as to make an IMBH unnecessary to fit the observations. We additionally determine that a stellar-mass black hole retention fraction of becomes incompatible with our observations for 47 Tuc.

1 Introduction

There are two classes of black holes (BHs) for which we have strong observational evidence. Stellar-mass BHs are of the order of a few solar masses, while supermassive BHs in galactic cores are found to be in the range of . Bridging the gap between these two extremes is the regime of the intermediate-mass BH (IMBH). Though IMBHs are required by most theories of supermassive BH evolution, they have proven very difficult to confirm observationally. There are many proposed formation mechanisms (Miller & Hamilton, 2002; Sigurdsson & Hernquist, 1993; Fryer, 1999; Zwart et al., 2004; Quinlan & Shapiro, 1990; Gurkan et al., 2004; Freitag et al., 2006), but most require a dense stellar environment of the type that is typically found only in the cores of globular clusters. As such, globular clusters tend to be the primary targets for IMBH searches.

One of the many open questions surrounding IMBHs is whether or not they follow the same empirical bulge-BH relations observed for supermassive BHs in galaxies. Correlations are present between the mass of the central supermassive BH and total bulge mass, total bulge luminosity, and central bulge velocity dispersion (Lützgendorf et al., 2013). Such correlations suggest commensurate evolution and/or feedback between the BH and its surroundings. IMBHs may follow the same or similar relations with their host globular clusters. Discrepancies between the bulge-BH relations of supermassive BHs versus IMBHs may shed light on how IMBHs themselves form, and how they are involved in the growth of supermassive BHs (Mezcua, 2017).

There are several methods available to detect IMBHs. Systems actively accreting gas may show a corresponding x-ray and radio emission. The detected flux combined with accretion models place an upper limit on the mass of the accreting BH. This method is frequently used to constrain stellar-mass BHs in observed x-ray binaries as well as for galactic supermassive BHs. The technique can be applied to IMBHs using the BH accretion fundamental plane, an empirical correlation between a BH’s radio flux, x-ray flux, and mass (Merloni et al., 2003). However, this relation has not yet been explicitly confirmed to continue into the IMBH mass regime. Our ability to look for radio signatures of IMBHs in globular clusters will be dramatically expanded when the Next Generation Very Large Array comes online in roughly a decade. A proposal by Wrobel et al. (2018) outlines a plan to measure radio fluxes of many hundreds of globular clusters out to a distance of 25 Mpc. Reaching large numbers of globular clusters is important as it may be the case that only a small fraction of clusters are likely to keep an IMBH even if they are commonly born with one (Fragione et al., 2018).

Kinematic study of stellar clusters is a common approach for central IMBH detection. One way to probe the gravitational environment is to examine pulsar accelerations using measurements of their various spin derivatives (Kızıltan et al., 2017; Gieles et al., 2018). There are, however, confounding factors such as the intrinsic spin-down of the pulsars. Of particular interest to the study at hand are the results for 47 Tuc by Kızıltan et al. (2017). They determined accelerations of millisecond pulsars in 47 Tuc and compare them against N-body simulations with different masses of central IMBHs. Their analysis suggests that an IMBH of mass ( of their total cluster mass) may be present in order to explain their observed pulsar accelerations. We would like to note a major deficiency with certain aspects of their method. The simulations used for comparison with their pulsar data contained no significant mass in binaries. They included no primordial binary systems, leaving only those systems that formed dynamically. In a cluster with a short central relaxation time such as 47 Tuc, the binaries and other massive objects will be centrally concentrated. As we demonstrate below, the presence of a concentrated and substantial mass distribution alone can mimic the dynamical effect of an IMBH, to say nothing of the dynamical heat source hardening binaries also provide to the cluster in general. Including binaries is crucial for a cluster like 47 Tuc whose binary mass fraction exceeds a few percent of the entire cluster mass. Kızıltan et al. (2017) also underestimated the neutron star masses in their simulation, which they took as the canonical (Baumgardt, private correspondence). The millisecond pulsars of 47 Tuc are almost certainly part of binary systems (Camilo et al., 2000) and have total dynamical mass . This mass difference will affect their distribution in the cluster and must be taken into account.

Another kinematic approach to detecting an IMBH is to measure and model the velocity dispersion of stars in the core of the globular cluster. This technique is commonly used in measuring galactic supermassive BHs (Chatzopoulos et al., 2015; Pagotto et al., 2017; Ahn et al., 2017; Walsh et al., 2013). For galactic studies individual stars generally cannot be resolved and so gas-dynamical models and spectral line widths are used to infer velocity dispersions. Similar techniques have produced upper limits for IMBHs in globular clusters where individual stars can be measured either in line-of-sight velocity or proper motion. However, individual stellar measurements are typically hindered by the crowding in the core, which is highly problematic as the core is the region where an IMBH’s gravitational influence will be most prevalent. Using integrated-light spectroscopy, one can try to work around the visibility issue in the crowded globular cluster cores. This method can be very useful, but also comes comes with its own limitations (see e.g. de Vita et al., 2017). Table 1 displays a selection of recent IMBH mass estimates and limits placed on various clusters through both emission and dynamical methods. It is clear that there is a lot of disagreement, even for studies with the same target. For a comprehensive review of the observational evidence for IMBHs, see Mezcua (2017).

This paper documents our use of proper motions in the core of the globular cluster 47 Tucanae to conduct analysis on the velocity dispersion profile and investigate whether an IMBH is required to explain the observed stellar motions. Section 2 outlines how we use UV images to overcome the visibility issues in the core when determining proper motions and how we characterize a population of dynamically heavy binary systems using an optical/infrared data set. It also outlines the process of estimating the populations of dark stellar remnants. In Section 3 we build the analytic dispersion model that is fit to the observations, and discuss our analytic methods while explaining and justifying our use of an un-binned likelihood maximization technique to fit the dispersion model. In Section 4 we report our results and uncertainties, verifying the validity of our velocity dispersion model with a re-sampling technique. Finally, Section 5 provides a summary and discussion of our findings.

Target Method [] ref.
Cen kinematic 12,000 (1)
Cen kinematic 40,000 (2)
Cen kinematic lower than ref. (2) (3)
NGC 6388 kinematic 2000 (4)
NGC 6388 kinematic 28,000 (5)
NGC 6388 x-ray 1500 (6)
NGC 6388 x-ray 600* 1200* (7)
47 Tuc kinematic 2300 (8)
NGC 6535 comparative simulation presence (9)
ULX-7 (M51) x-ray 1600* 35,000* (10)
M15 Radio VLBI 500 (11)
Molec. cloud CO-0.40-0.22 Radio, gas dynamics 100,000 (12)
NGC 6624 Pulsar timing 7500 (13)
NGC 6624 Pulsar timing   0 (14)
G1 (M31) kinematic 17,000 (15)
G1 (M31) comparative simulation   0 (16)

Note. – Sources: (1) van der Marel & Anderson (2010), (2) Noyola et al. (2008), (3) Zocchi et al. (2017), (4) Lanzoni et al. (2013), (5) Lützgendorf et al. (2015), (6) Cseh et al. (2010), (7) Bozzo et al. (2011), (8) Kızıltan et al. (2017), (9) Askar et al. (2017), (10) Earnshaw et al. (2016), (11) Kirsten & Vlemmings (2012), (12) Oka et al. (2017), (13) Perera et al. (2017), (14) Gieles et al. (2018), (15) Gebhardt et al. (2005), (16) Baumgardt et al. (2003).
Anisotropy considerations are unquantified, but suggest reference (2) estimate is too high.
* These estimates reflect different techniques and assumptions within the referenced study.
Comparison with simulations suggest the presence of an IMBH, but there are no reported constraints on mass.

Table 1: Recent IMBH findings/limits

2 Data

2.1 Proper Motions

Our proper motion data come from two epochs of Hubble Space Telescope (HST) observations. HST Program GO-12971 (PI: Richer) observed 47 Tuc over 10 orbits early in 2013, imaging the core in the F225W and F336W filters using Wide Field Camera 3. These data were used in conjunction with those of an earlier epoch; GO-9443 (PI: King) imaged the core in 2002 using the ACS High-Resolution Channel in the F475W filter. We match the stars between the F336W and F475W images, providing a baseline of years over which to calculate proper motions.

It is worth taking a moment to note the benefits of using such blue/ultraviolet filters. 47 Tuc has historically been imaged at longer wavelengths and, in such filters, visibility in the core is hampered both by crowding and by the dominating brightness of the many red giant stars located there. These giants saturate a portion of the detector in any moderately deep image, bleeding outwards and reducing the ability to accurately measure their stellar neighbours. Our short wavelength filters act to suppress the light from the cool red giants and allow for an unimpeded deep exposure right in the cluster’s core. The shorter wavelengths also improve the diffraction limit of every star in the frame, reducing the degree of crowding. For a detailed description of the observations and completeness correction process, see Goldsbury et al. (2016).

As a result of this extremely clear view into the core, we achieve quality proper motion measurements of over 50,000 stars including within the central arcsecond, and within the central three arcseconds. Measurements this close to the cluster core have been extremely difficult in the past and yet these are exactly the stars that probe the region of greatest potential IMBH influence. A dynamical estimate (Peebles, 1972) sets the BH’s radius of influence () to be

(1)

where is the central velocity dispersion of the cluster. Adopting a distance to 47 Tuc of 4.69 kpc (Woodley et al., 2012) and measuring the central velocity dispersion to be kms, we find the sphere of influence of a BH in 47 Tuc to have a radius of . It is the central few arcseconds where an IMBH has its most pronounced effect, so visibility in this region is critical.

2.2 Binary Population

Throughout this paper we will be referring to different components and sub-populations of 47 Tuc with the following abbreviations in the text and quantity subscripts: intermediate-mass black hole (IMBH), low-mass cluster objects (Cl), binaries (bin), white dwarfs (WD), neutron stars (NS), and stellar-mass black holes (sBH).

When looking for a dispersion signature in the core, it is important to separately account for populations of dynamically massive objects. Relaxation effects cause more massive objects to concentrate in the core while the less massive ones diffuse outward. This segregation will prove to be important in the context of velocity dispersion and will be discussed in later sections. For the purpose of our analysis, we make the distinction between high-mass () and low-mass () cluster objects. For the low-mass category we group together the low-mass binaries, low-mass WDs, main sequence stars, and giants as a single composite population. In the high-mass category we individually model the populations of heavy binary systems, heavy WDs, NSs, and sBHs. These higher-mass objects are more concentrated than the typical cluster object and have the potential to affect the velocity dispersion deep into the core, mimicking the effects of an IMBH.

Figure 1: A colour-magnitude diagram of 47 Tuc stars in visual/infrared filters. In these filters the binary population is adequately separated from the main sequence. We use a fiducial main sequence line and isochrones to estimate where the binary sequence would lie for different mass ratio () values. Binaries with are distinct from the main sequence. To measure the binary fraction in each range, we draw a box bounded by these isochrones and compare binary counts against main sequence stars within a certain colour range.

First we make an observationally motivated estimate of the binary population in 47 Tuc. On a colour-magnitude diagram (CMD) the binary sequence lies above the main sequence (brighter and redder), and the extent of this offset depends on the mass ratio () of the binary pair. A system with mass ratio of has two identical stars and simply twice the flux in every filter, leading to a vertical rise of 0.75 magnitudes and no shift in colour. Systems with will be brightened by a lesser amount and reddened to a small degree depending on the specific value. While the UV filters mentioned previously provide excellent positional information and allow for high quality proper motion determination, they unfortunately are not ideal for identifying binary systems. In the UV CMD the binary population is largely blended with the thick main sequence, likely due to multiple populations with differing compositions (Richer et al., 2013). Only those binaries with a mass ratio close to unity stand significantly apart from the main sequence, and even those have substantial contamination. Figure 1 shows a CMD of 47 Tuc from another data set using the redder filters F606W and F814W (Sarajedini et al., 2007, GO-10775). These optical/infrared data cover a similar field of view as the UV data (central ) and provide a clearer distinction between the main sequence and the binary sequence, allowing us to distinguish systems down to . Of the hidden binaries that have , the large majority of them will have masses . Their masses will cause them to follow the distributions of similar mass main sequence stars and are thus left to be included as part of the general low-mass population.

For a more refined characterization of the binaries, we look at the binary fraction in three different ranges. These are displayed in Figure 1, and the specific values will be used in computing the binary mass distribution in Section 3.3.

We create the isochrones used to calculate stellar masses using MESA stellar evolution models (Modules for Experiments in Stellar Astrophysics, Paxton et al., 2011, 2013, 2015). The fluxes in each filter were determined using Phoenix atmospheres111https://phoenix.ens-lyon.fr/Grids/BT-Settl/CIFIST2011_2015/, by Baraffe et al. (2015) and Allard (2016).

2.3 Stellar Remnants

In addition to the concentrated binary systems, 47 Tuc will harbour a collection of dark stellar remnants in the form of heavy WDs, NSs, and sBHs. Most of these objects are virtually impossible to directly detect, but we can infer their parameters based on an initial mass function (IMF) for the cluster.

In Figure 2 we have plotted the observed stellar mass function of the visual/infrared data as blue points. Overlaid as a dashed red line is a Kroupa (2001) IMF normalized to the heaviest observed objects. The Kroupa IMF follows a broken power law, , where in the mass ranges , and , respectively. The deficit of low-mass stars between the data and IMF is presumably due to the preferential loss of low-mass stars from our field of view (segregation) and from the cluster as a whole (evaporation). Baumgardt & Sollima (2017) verify this strong depletion of low-mass stars, and yet the present-day mass function agrees well with an evolved Kroupa IMF (Baumgardt & Hilker, 2018). We make the assumption that the highest mass stars still observed in the cluster () remain at their natal abundance and use this IMF to estimate how many progenitor objects there were at different masses. This is, in several ways, a conservative approach to estimating the number of stellar remnants. The Kroupa IMF is among the steepest mass functions, and assuming some fraction of the stars have been lost would only serve to raise the IMF and increase the remnant estimates.

Figure 2: A Kroupa initial mass function is used to estimate the number of progenitor objects that form remnants of various masses. The red, blue, green, and purple regions respectively count progenitor objects that would create low-mass WDs (), high-mass WDs (), NSs, and sBHs. For the species above we use these counts to infer their population parameters and include them in the velocity dispersion model. The large number of predicted low-mass WD progenitors does agree with observed WD counts when one considers the WD birth rate, time a WD takes to cool below detection thresholds, and the age of the cluster (Goldsbury et al., 2016).

As mentioned previously, we are interested in objects that have a mass above , and would therefore be distributed more centrally than main sequence stars. The red region of Figure 2 marks progenitor stars that would form WDs of . These WDs would relax into a distribution comparable to the main sequence stars. The blue region marks progenitors that produced WDs of mass (Cummings et al., 2016). The green region indicates the more massive progenitors that form NSs, and the purple region marks the progenitors that produce sBHs of up to (Spera & Mapelli, 2017). The number of objects in each coloured mass range is determined by integrating the mass function. We cut off the IMF beyond because the BHs above tend to be completely lost during cluster evolution (Morscher et al., 2015).

We have to consider that natal kicks imparted upon formation as well as gravitational encounters within the cluster will cause the loss of NSs and sBHs. The retention fraction in globular clusters is not a well constrained quantity. Simulations tend to produce values anywhere in the range of about (Baumgardt & Sollima, 2017; Morscher et al., 2015; Moody & Sigurdsson, 2009; Mackey et al., 2008). We adopt a fairly conservative estimate of 10% retention for NSs and sBHs, but we demonstrate later that this choice is well motivated.

3 An Analytic Model for Dispersion

3.1 Jeans Model

In order to determine whether or not our observations are consistent with a central IMBH, we create a theoretical velocity dispersion model that includes the contributions of all the non-IMBH cluster objects as well as an IMBH mass parameter that affects the dispersion signal in the core. We use a 3D version of the King density model (King, 1962) as a starting point in determining the shape of the dispersion profile. The King density profile,

(2)

describes how the density () falls off with 3D radius () as we move away from the core. The density drops to zero at a prescribed tidal radius () which is set by the Galactic potential. The profile scales with , a constant that encodes the total mass contained in the distribution and the parameter describes an effective core radius. For radii interior to the distribution is largely flat.

The density profile and the associated enclosed mass as a function of radius, , can be used in the isotropic Jeans equation,

(3)

to isolate the square of the velocity dispersion . Here and are the gravitational potential and constant, respectively. We use the isotropic version as the level of anisotropy is low in the core region of 47 Tuc where our proper motion data occur (Heyl et al., 2017; Watkins et al., 2015; Bellini et al., 2017). The squared velocity dispersion then takes the form,

(4)

This, of course, is the dispersion profile for 3D radius . Once is obtained, it is integrated against the density along the line of sight to produce the projected squared velocity dispersion,

(5)

where is the on-sky projected radius from the centre. In the second line we have simply parcelled the equation into two parts: a function that holds the dependence, and a scaling term out front. will have units of [(velocity)/mass] and describe how the dispersion falls off with . The term is the total mass of the distribution in question and scales the magnitude of the dispersion profile. It arises from in Equation 4 which involves an integral over the density profile and a coefficient that can be written in terms of . We adopt this format for ease of writing separate component contributions later on in Section 3.4.

The operations between Equations 35 are fully generic manipulations of the Jeans equation and make no specifications about the density profile. In our full density distribution, we combine different King models with individual total masses and core radii to capture the unique dynamical contributions of the various cluster populations. For an analogous procedure using Gaussian distributions and with detailed formalism, see Emsellem et al. (1994) and Cappellari (2008). Before we apply the Jeans equation directly to our cluster model, there are a few things to consider.

3.2 Treatment of Core Parameter

The parameter in Equation 2 acts as a characteristic core radius. This is not necessarily a global value as objects of different mass will be distributed with different core radii. To correct for the inevitability that our tracer stars (those for which we have proper motions) are not perfectly representative of the true mass distribution, we make a distinction between two different core radius parameters. In the right-hand side of Equation 3, the factor generates the force acting on each particle. This must therefore reflect the true underlying mass distribution of the cluster, whether those objects are detected in the data or not. Conversely, the terms of Equation 3 describe the distribution of tracer particles whose motions we are observing. These tracers need not be the same objects that are generating the potential. We therefore calculate using the measured core radius of the stars for which we have proper motions, , and calculate the potential generating term using a more globally representative cluster parameter, , which is left as a fitting parameter. We fit a King distribution to our tracer stars and find . Because our tracer stars (with completeness corrections) are reasonably representative of the total cluster population we expect to be similar to , but allow them the freedom to differ. The binary and stellar remnant populations that we discuss next will also have their own distribution parameters. Due to the differing concentrations, the functional form of the velocity dispersion in Equation 5 now becomes where specifies the mass distribution in that particular population.

3.3 Inclusion of Binary Populations

If there is no IMBH in the mass distribution, we expect the velocity dispersion to flatten off in the core as . However, a central BH point mass causes to remain positive for any and produces a rising slope into the cluster core. A concentrated population of objects in the cluster can also cause the dispersion profile to rise farther into the core before flattening out, mimicking the central effects of a BH. When testing for an IMBH it is therefore very important to take careful consideration of populations more centrally concentrated than the typical main sequence star, which may cause the dispersion to rise deeper into the core.

For this reason we model the binary population separately from the general lower-mass cluster stars. However, the binary systems are not a single homogeneous population in terms of their mass. For a given primary mass (i.e. position along the main sequence) systems with high mass ratios are more massive than those with low -values, and for a given mass ratio, binaries at the top of the main sequence are more massive than those below them. The higher mass systems will have correspondingly small distribution parameters. We therefore break the binary contribution into components to account for binary sub-populations with different distributions. The expression for the squared velocity dispersion of the binary stars becomes,

(6)

The index signifies different mass bins down the main sequence in the range of . The index specifies one of the three mass ratio ranges seen in Figure 1: lower (), moderate (), or high (). The mass of binaries in each -bin and -range () is calculated as,

(7)

The first item, , is the observed binary fraction for a given mass ratio range (). These values are determined by counting stars in a section of the CMD as seen in Figure 1. We assume that these fractional values hold true for the entire main sequence. The factors are the proportion of the population that is visible within our field of view according to the distribution parameter for that population. Therefore, the term modifies the observed binary fraction to describe the entire cluster, including objects beyond our image field. It is necessary because the binary fraction will drop away from the core since the binary systems are more concentrated than the main sequence stars. The factor accounts for the mass of the secondary star in the binary system where is the midpoint of the mass ratio range . At the end of Equation 7, the quantity is the observed mass in main sequence bin , obtained by counting stars and consulting the isochrone. It is divided by , the total observed mass of the main sequence, to determine the mass proportion of the entire cluster our selection corresponds to. Finally, multiplying by scales this proportion to the fitted dynamical cluster mass.

Because of the shape and the width of the main sequence and giant branch in Figure 1 the binary stars cannot be identified for all masses and thus, their distributions cannot be measured directly. We make use of the results of Goldsbury et al. (2013) to determine the parameter of Equation 6. In order to quantify the mass segregation in 54 globular clusters (including 47 Tuc), Goldsbury et al. (2013) measured the distributions of various stellar groups, determining a power-law relation between the objects’ mass and their distribution parameter ( in their paper works out to be equivalent to in ours). We use this power-law to determine given that the mass of the binary system is the mass of the primary times the factor .

3.4 Inclusion of Dark Stellar Remnants

The dark stellar remnants are included in a much simpler manner. Each of the heavy WD, NS, and sBH populations are given a single distribution specified by their total mass and core radius parameter. The heavy WDs are assumed to have a mean mass of , the mid-point of the considered mass range ( to the Chandrasekhar limit ). The NSs are all conservatively estimated to have masses of , and the sBHs are all assumed to have masses of (Morscher et al., 2015; Cummings et al., 2016; Spera & Mapelli, 2017). The populations’ overall masses are simply given by the predicted number of objects multiplied by their mass. The number of objects in each of Figure 2’s coloured mass ranges is determined by integrating the mass function. This value is then corrected to include everything beyond the data’s field of view. This correction process is analogous to the treatment used for binaries within Equation 7. With estimates of the remnant masses we can infer their distributions and include them in our velocity dispersion model using the Goldsbury et al. (2013) mass segregation power law as was done for the binary systems. However, the sBHs and NSs will suffer additional losses, largely due to natal kicks from their progenitor supernovae. We assume a moderate value of 10% retention for NSs and sBHs. The projected squared velocity dispersion contribution from the dark stellar remnant populations becomes

(8)

where is an index that refers to the dark stellar remnant in question (WD, NS, sBH). The full and final expression for the squared velocity dispersion comes together as:

(9)

We have now built a distribution up from 47 Tuc’s different mass components. Using measured binary fractions and inferred distributions we account for the individual contributions of binary sub-populations that have differing distributions and total masses. An initial mass function normalized to observed cluster stars allows us to estimate how many dark stellar remnants remain in the cluster. The fit parameters and describe the gross shape of the dispersion curve (its vertical scale and the radius at which it falls off), while the concentrated binaries and remnants bring up the dispersion value towards the core. Any central rise observed in the data that is not due to the concentrated populations is taken up by the IMBH fit parameter .

All of this put together produces a radial profile for the projected velocity dispersion that depends on three parameters: central IMBH mass, total cluster mass, and cluster core radius. The rest of the inputs are measured or inferred quantities. The projected dispersion profile, , is now in a form that can be fit to the data and the optimal values for the 3 input parameters found.

3.5 Binned vs. Un-binned Analysis

Figure 3: A radial binning scheme is chosen that follows a logarithmic spacing out to the limit of the UV data (). When the number of bins (and thus, the size of the bins) is changed, the binned-fit parameter fluctuates dramatically and unpredictably. The other fit parameters (not shown) follow similar variations. We note this unreliability as a justification for using an un-binned likelihood analysis in determining best-fit model parameters. Note: A negative IMBH mass simply means the non-IMBH components of the model produce a higher dispersion in the core than the data indicate. The negative mass is an attempt by the optimizer to bring the dispersion value down to improve the fit.

With a theoretical velocity dispersion model now built, we are in a position to find the set of the 3 parameters (, , and ) that best fit the observations. This is most simply and commonly done by radially binning the data, finding the velocity dispersion in each bin, then finding a least-squares fit of the model to the data.

We must, however, be careful in trusting the results of a binned data set. Any binning scheme has some degree of arbitrariness that may influence the result in unpredictable ways. At the very least, introducing bins will smear out the radius-dependent signal over a larger range of radii. At worst, the choice of binning system may unintentionally amplify or suppress a particular result.

As a test of the influence of bin choice, we set up a logarithmically spaced binning scheme then steadily change the number (and thus the size) of the bins. Figure 3 shows how the binned-fit parameter fluctuates as we slowly scale the bin sizes. The fluctuations are both pronounced and unsystematic. To bypass the issue altogether, we use the following un-binned likelihood maximization technique to find the best-fit dispersion model.

Again, under the assumption of isotropy in the core for 47 Tuc, we assume the observed velocity components (, ) of each star are samples from Gaussian-distributed populations. The widths of these Gaussian populations are estimated by the model dispersion value at each star’s projected distance from the core, broadened by the uncertainty in the velocity measurements. We find the set of , , and parameters for that maximize the global log-likelihood function,

(10)

Here is the model dispersion value at the projected radius of star , and and are the measurement uncertainties in the velocity components of star that arises from our photometric data. This likelihood estimator is equivalent to Equation 8 of Walker et al. (2006), but has been adapted for 2D proper motions and a continuous rather than 1D radial velocities and binned dispersion values. Furthermore, we ignore the constant offset introduced to the value by Gaussian normalization. For a full derivation and explanation of the entire un-binned likelihood technique, see Goldsbury et al. (2016). We employ this un-binned system to avoid any unwanted bias introduced through arbitrarily choosing a binning system to apply to the data. The results of the fit are plotted in Figure 4.

4 Results

4.1 Model Fitting Results

Having built a velocity dispersion model that includes measured binary fractions, inferred dark stellar remnants, and a NS and sBH retention fraction of 10%, we obtain the following best-fit parameter values: , , (uncertainties are determined through bootstrapping). Of perhaps more value for comparison purposes is the ratio of . The ratio avoids scaling discrepancies in cluster distance and total mass. The dispersion model described by these parameters is shown in Figure 4, plotted over binned velocity dispersion data. The binned data are included for visualization purposes only and the fit was made using the un-binned likelihood maximization described in the previous section. Of concentrated heavy populations, the binaries and sBHs play the dominating roles on the central dispersion rise. The binary population contain a large overall mass and their distribution causes the dispersion to continue rising into . The sBHs, while comprising a lesser total mass, are much more concentrated and continue this dispersion rise into . The combined effect leaves little room for any substantial IMBH which would drive the dispersion beyond the data.

We note that throughout this paper the reported velocities (and thus the fitted masses that scale the dispersion curves) are dependent on the choice of distance to the cluster. We have adopted a distance of kpc from Woodley et al. (2012) who fit spectral energy distributions of 47 Tuc’s WDs. Additionally, our proper motion data only extends to , a small fraction of the tidal radius, (Harris, 1996, 2010 edition). This means we are really only fitting the central region of the cluster. Our mass estimates involve integrals of the density distributions and therefore assume the density beyond follows a perfect King distribution out to the tidal radius. There may be substantial deviation in the tails which may contribute to our fairly large cluster mass determination. However, Giersz & Heggie (2011) model a cluster mass of and note that perhaps consists of stellar remnants. In light of the different masses between studies, the ratio may be a better way to compare results.

Figure 4: The solid red line shows the best-fitting (un-binned) velocity dispersion model. Model parameters are given in Section 4.1. Note that the black points display binned data and are included for visualization purposes only. The model was fit using an un-binned likelihood maximization. Dashed lines show how the different components (cluster stars, binaries, sBHs, WDs, and NSs) contribute to the overall velocity dispersion profile with the individual components being added in quadrature. We see here that the sBHs and binaries produce a dispersion rise in the core that is nearly adequate to explain the observations. The un-binned likelihood maximization requires only a very minor IMBH contribution to fit the proper motion data.

4.2 Re-sampling

We employ a system of statistical re-sampling in order to assess the efficacy of our modelling process. This re-sampling involves generating new and velocity components for each star. These velocities are drawn randomly from Gaussian distributions whose widths are determined by the dispersion value at the radius of each data point, broadened by a measurement error of the velocity. Keeping the original positions for each star ensures their radial distribution remains unchanged. Once all the stars have had their velocities re-sampled according to the dispersion model, a best-fit parameter set is determined for this generated data. The parameters are recorded along with the log-likelihood value (Equation 10) of the newly generated velocities measured against the original data’s best fit . This process is repeated many times over to build distributions of the model parameters and log-likelihood values.

We use this process to address whether or not our model is faithfully describing the data. Being able to find a set of best-fit model parameters does not necessarily mean one has accurately modelled the data. For example, any set of curved data will have a straight line that fits it best, but that does not mean a straight line is a good description of those data. To determine the quality of our model, we re-sample stellar velocities based on the dispersion parameters that best fit our data. Figure 5 shows the distribution of log-likelihood values generated when we draw these samples many times. As can be seen, the line indicating the original data’s log-likelihood value falls very near the middle of the distribution. Such close agreement indicates our model is a good description of the data. We would expect to draw data from our best-fitting model that differs at least as much as the real data do from that model of the time. Artificial velocity data generated from our model really do look like the cluster’s velocity data.

We similarly verify that the distribution of values of the generated data are centred around the input model’s value. This indicates our model can extract the input IMBH mass without systematic bias in either direction.

Figure 5: Presented here is a histogram of the log-likelihood () values for the best-fit curves of re-sampled data sets. Here the re-sampled data are drawn from the original data best-fit dispersion curve. All fits used un-binned likelihood maximization. Fits to the re-sampled data are in close agreement with our fit to the original data, indicating the model is a good description. We would expect a deviation of at least this magnitude in 87.3% of cases.

5 Conclusion

5.1 Summary

Our use of UV imaging data has allowed us to glean velocity information from stars right into the very centre of the globular cluster 47 Tucanae. Historically, crowding has been a limiting factor in IMBH velocity dispersion searches as it severely reduces the numbers of stars measurable at very small projected radius. Resolution and accuracy in this region are of key importance in trying to estimate the mass of a central IMBH through this method. With visibility right in to the cluster core, we are able to probe the region of the cluster where an IMBH’s influence would be most pronounced.

Beginning with a King model, we used the isotropic Jeans equation to build a velocity dispersion profile with the added influence of a central IMBH. The dispersion profile was built with contributions from several sub-components: a central IMBH point mass, a collective group of low-mass cluster objects (), and individual contributions from concentrated populations such as the binary systems, heavy white dwarfs, neutron stars, and stellar-mass black holes.

To avoid the known and unknown biases of binning our data, we employ an un-binned likelihood analysis when fitting our model. This technique weighs the combined likelihoods of each individual star’s velocities being drawn from a given dispersion model. It prevents the radial smearing-out that occurs when data are binned together, and avoids the unpredictable fluctuations that result from the arbitrary choice of a binning scheme.

To determine how good a fit our dispersion model is to the data, we randomly re-sample the velocity components of our stars based on the best-fit dispersion curve. The log-likelihood of the real data compared to the best-fit model is in very good agreement with the distribution of re-sampled log-likelihood values, indicating our model does a good job of describing the data.

Our analysis produces a best-fit value of for a central IMBH in the core of 47 Tucanae, giving a ratio of . This value is consistent with zero, unlike the recent results of Kızıltan et al. (2017) who found a ratio of , however, our results do agree at the high and low ends of our respective uncertainties. We emphasize that the proper characterization of the binary population and sBHs can have a marked effect on the IMBH estimate and their omission will likely bias that estimate high. We explored a range of sBH retention rates to determine the effect it can have on the final fitted IMBH mass. We observe that in order to find the fraction concluded by Kızıltan et al. (2017) we would need to assume essentially all sBHs are ejected from the cluster. Retaining only raises the central dispersion enough for our model to require no IMBH at all, and pushing the retention beyond creates higher dispersion in the core than is observed, requiring a significant negative IMBH mass. Table 2 displays the final fit results under these different retention assumptions.

Retention (%) [] []
0 4348 1344642 0.0032
5 2587 1363820 0.0019
10 838 1384564 0.0006
12 134 1393112 0.0001
15 -893 1407125 -0.0006
20 -2602 1431780 -0.0018
25 -4285 1458876 -0.0029

Note. – Assuming different retention fractions of the sBH population produces a range of IMBH estimates. Typical error bars on the values are . Retention of is the approximate range around zero IMBH. A very low retention fraction is required to produce a significant IMBH detection. Considering that our mass function is already conservative in its sBH estimate, these results suggest that a retention may not be possible in 47 Tuc. The Kızıltan et al. (2017) mass fraction of requires a sBH retention very near zero.

Table 2: Exploring Retention Fraction
Figure 6: The cumulative distribution of stars (incompleteness corrected) as a function of distance from the core (). The dashed red line is an relation, indicative of a flat density distribution. The apparent deviation in the core from the line is not statistically significant as the deviation is small and the number of stars is low in this region. Significant deviation () only occurs beyond a radius of where the cluster no longer remains flat. We find similar results when we relocate the cluster’s centroid within the uncertainty range of Goldsbury et al. (2010).

5.2 Discussion

The velocity dispersion profile investigated in this study does not support an IMBH detection for 47 Tuc unless one posits a very low sBH retention (). The concentrated populations of binary stars and dark stellar remnants alone are enough to explain the central velocity dispersion.

Beyond studying the velocity dispersion, there are several independent avenues of kinematic analysis available to collectively probe for the presence of an IMBH. In some cases they provide contradictory results, but should be collectively considered. As mentioned previously, the Kızıltan et al. (2017) study produced a positive IMBH result based on accelerations of millisecond pulsars in 47 Tuc. There is numerical evidence that the presence of an IMBH should cause a weak central cusp in the density/surface brightness profile (Baumgardt et al., 2005; Noyola & Baumgardt, 2011), however other work has shown that an observed cusp may also be caused by a state change in the degree of core collapse (Trenti et al., 2010; Vesperini & Trenti, 2010), and is not sufficient to infer an IMBH. Either way, we find no evidence for a central cusp in the density distribution. Figure 6 shows the cumulative distribution of objects (corrected for incompleteness) as a function of projected distance from the core. This distribution is compared against an relation, which would indicate a perfectly flat density profile. By eye there appears to be a mild deviation from in the centre, but the difference is very small (low end of the logarithmic scale) and includes very few objects (). The deviation is not statistically significant and we cannot reject the null hypothesis (i.e. that the data are drawn from an distribution) to any reasonable significance until we look beyond where the cluster density begins to fall off. Additionally, we find no sign of a small population of high velocity stars. If the IMBH were part of a binary system, 3-body interactions should periodically eject high velocity stars into the cluster on largely radial orbits. Our data do not show any stars with velocities beyond kms while N-body simulations including central IMBHs show a small population of stars with velocities up to kms (Baumgardt, private correspondence).

There are many contrary findings concerning IMBHs in globular clusters. Of particular note are the conflicting results of Lützgendorf et al. (2015) and Lanzoni et al. (2013) on NGC 6388, van der Marel & Anderson (2010) and Noyola et al. (2008) on Centauri, and Perera et al. (2017) and Gieles et al. (2018) on NGC 6624. There is clearly more work to be done in characterizing and evaluating the shortcomings and strengths of different IMBH detection methods and we could benefit from more studies that evaluate the accuracy and biases of these techniques. For example, de Vita et al. (2017) provide insight on how using integrated light spectroscopy might affect results versus using velocity measurements of individual objects. Simulations can be extremely useful with their ability to explicitly incorporate a central IMBH of known mass to determine its observable effects, but the expensive computation limitations require simplified physics and assumptions about initial conditions and short-timescale dynamics. Including an appropriate binary population is one aspect that suffers in this regard.

In the course of our study, we have found that building a careful mass model of the cluster that includes concentrated heavy populations has an important bearing on an inferred IMBH presence. Due to their concentrated distribution and substantial total mass, binary systems and sBHs cause the velocity dispersion to continue rising deeper into the core than would be expected from lower mass cluster stars, mimicking the central slope produced by an IMBH. As a result, we find no need to invoke an IMBH to explained the observed velocity dispersion unless the core has a sBH retention fraction near zero.

Consideration of the sBH retention fraction leads to an unanticipated additional result of this study. We employ a conservative initial mass function to estimate the number of sBHs, and still find that a retention of becomes incompatible with our dispersion observations. Above this fraction our model requires a significantly negative IMBH mass in an attempt to drop the dispersion values in the core, which is clearly unphysical. For 47 Tuc it seems that we can place a constraint on the sBH retention to be .

It should be noted that a King density distribution that has isotropic stellar velocities is not a self-consistent dynamical model once an IMBH point mass is added in the core. We use an isotropic model because there is no evidence for anisotropy in the core, and we use a King model because there is no evidence for a central density cusp. The IMBH parameter allows us to capture any additional rise in the central dispersion that is observed. Limiting the input profiles in such a way makes it easier to detect an IMBH influence by keeping the number of fit parameters low. Even so, the model requires very little “extra” dispersion in the core to fit the observations.

We have shown here that proper motion dispersion analysis can prove to be a useful tool in the search for IMBHs, even in the crowded environments of a globular cluster core. With the right choice of photometric filters we can overcome the crowding issues and glean a multitude of data in the region most sensitive to an IMBH’s influence. Our inclusion of concentrated populations in our mass model is sufficient to explain the observed velocity dispersion in 47 Tuc’s core. That being said, the dispersion signature of an IMBH and a tight population of sBHs is very similar. A trade-off in mass between them would not be very evident in the velocity data. Independent estimates of the sBH retention and progenitor IMF are likely the best way to disentangle this degeneracy.

Acknowledgements

We would like to extend special thanks and acknowledgement to Laura Watkins whose helpful suggestions set us on the track of looking at independent mass populations. We also greatly appreciate the consultation and advice of Vincent Hénault-Brunet with regards to our dynamical formulation and mass modelling.

The presented research used NASA/ESA Hubble Space Telescope observations obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy Inc. under NASA contract NAS5-26555. These observations are associated with proposals GO-12971 (PI: H. Richer), GO-9443 (PI: I. King), and GO-10775 (PI: A. Sarajedini). This work was supported by NASA/HST grants GO-12971, the Natural Sciences and Engineering Research Council of Canada, the Canadian Foundation for Innovation, and the British Columbia Knowledge Development Fund. We have made use of the NASA ADS, arXiv.org, and the Mikulski Archive for Space Telescopes.

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