# How reliable is it to use globular cluster lumniosity function as a distance indicator?

I have seen many research suggesting that GCLF can be used as a secondary distance indicator for extragalactic objects. However, this method seems to be only reliable when the GCs in use come from say, a single galaxy, and it can be used as a secondary distance indicator for the home galaxy.

My question right now is that I have large all sky survey data with extracted potential GCs candidates ranging from 1 to 20 Mpc. These can be considered as "rogue" GCs that do not necessarily belong to any galaxy. Can the GCLF be used to obtain at least some very crude estimates of the distance of individual GC? What I mean by that is we will obtain some probability distribution of the distance through GCLF and how reliable is the distance information provided by the previously obtained distribution? Thank you!

## How reliable is it to use globular cluster lumniosity function as a distance indicator? - Astronomy

In this section, we will review the globular cluster luminosity function (GCLF) as a distance indicator. The method is currently "unfashionable" in the literature mainly because some previous results seem to be in contradiction with other distance indicators (e.g. Ferrarese et al. 1999). We will try to shade some light on the discrepancies, and show that, if the proper corrections are applied, the GCLF competes well with other extragalactic distance indicators.

A nice overview of the method is given in Harris (2000), including some historical remarks and a detailed description of the method. A further review on the GCLF method was written by Whitmore (1997), who addressed in particular the errors accompanying the method. We will only briefly summarize the method here.

The basics of the method are to measure in a given filter (most often V) the apparent magnitudes of a large number of globular clusters in the system. The so constructed magnitude distribution, or luminosity function, peaks at a characteristic (turn-over) magnitude. The absolute value for this characteristic magnitude is derived from local or secondary distance calibrators, allowing to derive a distance modulus from the observed turn-over magnitude. Figure 8 shows a typical globular cluster luminosity function with its clear peak (taken from Della Valle et al. 1998).

The justification of the method is mainly empirical. Apparent turn-overs for galaxies at the same distance (e.g. in the same galaxy cluster) can be compared and a scatter around 0.15 mag is then obtained, without correcting for any external error. Similarly, a number of well observed apparent turn-over magnitudes can be transformed into absolute ones using distances from other distance indicators (Cepheids where possible, or a mean of Cepheids, surface brightness fluctuations, planetary luminosity function, . ) and a similar small scatter is found (see Harris 2000 for a recent compilation). Taking into account the errors in the photometry, the fitting of the GCLF, the assumed distances, etc. this hintsat an internal dispersion of the turn-over magnitude of < 0.1, making it a good standard candle. From a theoretical point of view, this constancy of the turn-over magnitude translates into a "universal" characteristic mass in the globular cluster mass distributions in all galaxies. Whether this is a relict of a characteristic mass in the mass function of the molecular clouds at the origin of the globular clusters, or whether it was implemented during the formation process of the globular clusters is still unclear.

The absolute turn-over magnitude lies around VTO

-7.5, and the determination of the visual turn-over is only accurate if the peak of the GCLF is reached by the observations. From an observational point of view, this means that the data must reach e.g. V

25 to determine distances in the Fornax or Virgo galaxy clusters (D

20 Mpc), and that with HST or 10m-class telescopes reaching typically V

28, the method could be applied as far out as 120 Mpc (including the Coma galaxy cluster).

The observational advantages of this method over others are that globular clusters are brighter than other standard candles (except for supernovae), and do not vary, i.e. no repeated observations are necessary. Further, they are usually measured at large radii or in the halo of (mostly elliptical) galaxies where reddening is not a concern.

A large number of distance determinations from the GCLF were only by-products of globular cluster system studies, and often suffered from purely practical problems of data taken for different purposes.

First, a good estimation of the background contamination is necessary to clean the globular cluster luminosity function from the luminosity function of background galaxies which tends to mimic a fainter turn-over magnitude. Next, the finding incompleteness for the globular clusters needs to be determined, in particular as a function of radius since the photon noise is changing dramatically with galactocentric radius. Proper reddening corrections need to be applied and might differ whether one uses the "classical" reddening maps of Burstein & Heiles (1984) or the newer maps from Schlegel et al. (1998). When necessary, proper aperture correction for slightly extended clusters on WFPC2/HST images has to be made (e.g. Puzia et al. 1999). Finally, several different ways of fitting the GCLF are used: from fitting a histogram, over more sophisticated maximum-likelihood fits taking into account background contamination and incompleteness. The functions fitted vary from Gaussians to Student (t5) functions, with or without their dispersion as a free parameter in addition to the peak value.

In addition to these, errors in the absolute calibration will be added (see below). Furthermore, dependences on galaxy type and environment were claimed, although the former is probably due to the mean metallicity of globular clusters differing in early- and late-type galaxies, while the latter was never demonstrated with a reliable set of data.

All the above details can introduce errors in the analysis that might sum up to several tenths of a magnitude. The fact that distance determinations using the GCLF are often a by-product of studies aiming at understanding globular cluster or galaxy formation and evolution, did not help in constructing a very homogeneous sample in the past. The result is a very inhomogeneous database (e.g. Ferrarese et al. 1999) dominated by large random scatter introduced in the analysis, as well as systematic errors introduced by the choice of calibration and the complex nature of globular cluster systems (see below). Nevertheless, most of these problems were recognized and are overcome by better methods and data in the recent GCLF distance determinations.

Harris (2000, see also Kavelaars et al. 2000) outline what we will call the classical way of measuring distances with the GCLF. This method implies that the GCLF is measured from all globular clusters in a system. In addition, it uses the GCLF as a "secondary" distance indicator, basing its calibration on distances derived by Cepheids an other distance indicators. The method compares the peak of the observed GCLF with the peak of a compilation of GCLFs from mostly Virgo and Fornax ellipticals, adopting from the literature a distance to these calibrators. This allowed, among others, Harris' group to determine a distance to Coma ellipticals and to construct the first Hubble diagram from GCLFs in order to derive a value for H0 (Harris 2000, Kavelaars et al. 2000).

In practice, an accurate GCLF turn-over is determined (see above) and calibrated without any further corrections using MV(TO) = -7.33 ± 0.04 (Harris 2000) or MV(TO) = -7.26 ± 0.06 using Virgo alone (Kavelaars et al. 2000).

The advantages of this approach are the following. Using all globular clusters (instead of a limited sub-population) often avoids problems with small number statistics. This is also the idea behind using Virgo GCLFs instead of the spars Milky Way GCLF as calibrators. The Virgo GCLFs, derived from giant elliptical galaxies rich in globular clusters, are well sampled and do not suffer from small number statistics. Further, since most newly derived GCLFs come from cluster ellipticals, one might be more confident to calibrate these using Virgo (i.e. cluster) ellipticals, in order to avoid any potential dependence on galaxy type and/or environment.

However, the method has a number of caveats. The main one is that giant ellipticals are known to have globular cluster sub-populations with different ages and metallicities. This automatically implies that the different sub-populations around a given galaxy will have different turn-over magnitudes. By using the whole globular cluster systems, one is using a mix of turn-over magnitudes. One could in principal try to correct e.g. for a mean metallicity (as proposed by Ashman, Conti & Zepf 1995), but this correction depends on the population synthesis model adopted (see Puzia et al. 1999) and implies that the mix of metal-poor to metal-rich globular clusters is known. This mix does not only vary from galaxy to galaxy (e.g. Gebhardt & Kissler-Patig 1999), but also varies with galactocentric radius (e.g. Geisler et al. 1996, Kissler-Patig et al. 1997). It results in a displacement of the turn-over peak and the broadening of the observed GCLF of the whole system. The Virgo ellipticals are therefore only valid calibrators for other giant ellipticals with a similar ratio of metal-poor to metal-rich globular clusters and for which the observations cover similar radii. This is potentially a problem when comparing ground-based (wide-field) studies with HST studies focusing on the inner regions of a galaxy. Or when comparing nearby galaxies where the center is well sampled to very distant galaxies for which mostly halo globulars are observed. In the worse case, ignoring the presence of different sub-populations and comparing very different galaxies in this respect, can introduce errors a several tenths of magnitudes.

Another caveat of the classical way, is that relative distances to Virgo can be derived, but absolute magnitudes (and e.g. values of H0) will still dependent on other methods such as Cepheids, surface brightness fluctuations (SBF), Planetary Nebulae luminosity functions (PNLF), and tip of red-giant branches (TRGB), i.e. the method will never overcome these other methods in accuracy and carry along any of their potential systematic errors.

As an alternative to the classical way, one can focus on the metal-poor clusters only. The idea is to isolate the metal-poor globular clusters of a system and to determine their GCLF. As a calibrator, one can use the GCLF of the metal-poor globular clusters in the Milky Way, which avoids any assumption on the distance of the LMC and will be independent of any other extragalactic distance indicator. For the Milky Way GCLF, the idea is to re-derive an absolute distance to each individual cluster, resulting in individual absolute magnitudes and allowing to derive an absolute luminosity function. Individual distances to the clusters are derived using the known apparent magnitudes of their horizontal branches and a relation between the absolute magnitude of the horizontal branch and the metallicity (e.g. Gratton et al. 1997). The latter is based on HIPPARCOS distances to sub-dwarfs fitted to the lower main sequence of chosen clusters. This methods follows a completely different path than methods based at some stage on Cepheids. In particular, the method is completely independently from the distance to the LMC.

In practice, an accurate GCLF turn-over (see above) for the metal-poor clusters in the target galaxy is derived and calibrated, without any further corrections, using MV(TO) = -7.62 ± 0.06 derived from the metal-poor clusters of the Milky Way (see Della Valle et al. 1998, Drenkhahn & Richtler 1999 note that the error is statistical only and does not include any potential systematic error associated with the distance to Galatic globular clusters, currently under debate).

The advantages of this method are the following. This method takes into account the known sub-structures of globular cluster systems. Using the metal-poor globular clusters is motivated by several facts. First, they appear to have a true universal origin (see Burgarella et al. 2000), and their properties seem to be relatively independent of galaxy type, environment, size and metallicity. Thus, to first order they can be used in all galaxies without applying any corrections. In addition, the Milky Way is justified as calibrator even for GCLFs derived from elliptical galaxies. Further, they appear to be "halo" objects, i.e. little affected by destruction processes that might have shaped the GCLF in the inner few kpc of large galaxies, or that affect objects on radial orbits. They will certainly form a much more homogeneous populations than the total globular cluster system (see previous sections). Using the Milky Way as calibrator allows this method to be completely independent on other distance indicators and to check independently derived distances and value of H0.

The method is not free from disadvantages. First, selecting metal-poor globular clusters requires better data than are currently used in most GCLF studies, implying more complicated and time-consuming observations. Second, even with excellent data a perfect separation of metal-poor and metal-rich clusters will not be possible and the sample will be somewhat contaminated by metal-rich clusters. Worse, the sample size will be roughly halved (for a typical ratio of blue to red clusters around one). This might mean that in some galaxies less than hundred clusters will be available to construct the luminosity function, inducing error > 0.1 on the peak determination due to sample size alone. Finally, the same concerns applies as for the whole sample: how universal is the GCLF peak of metal-poor globular clusters? This remains to be checked, but since variations of the order of < 0.1 seem to be the rule for whole samples, there is no reason to expect a much larger scatter for metal-poor clusters alone.

Two examples of distance determinations from metal-poor clusters were given in Della Valle et al. (1998), and Puzia et al. (1999).

The first study derived a distance modulus for NGC 1380 in the Fornax cluster of (m - M) = 31.35 ± 0.09 (not including a potential systematic error of up to 0.2). In this case, the GCLF of the metal-poor and the metal-rich clusters peaked at the same value, i.e. the higher metallicity was compensated by a younger age (few Gyr) of the red globular cluster population, so that it would not make a difference whether one uses the metal-poor clusters alone or the whole system. As a comparison, values derived from Cepheids and a mean of Cepheids/SBF/PNLF to Fornax are (m - M) = 31.54 ± 0.14 (Ferrarese et al. 1999) and (m - M) = 31.30 ± 0.04 (from Kavelaars et al. 2000).

In the case of NGC 4472 in the Virgo galaxy cluster, Puzia et al. (1999) derived turn-overs from the metal-poor and metal-rich clusters of 23.67 ± 0.09 and 24.13 ± 0.11 respectively. Using the metal-poor clusters alone, their derived distance is then (m - M) = 30.99 ± 0.11. This compares with the Cepheid distance to Virgo from 6 galaxies of (m - M) = 31.01 ± 0.07 and to the mean of Cepheids/SBF/TRGB/PNLF of (m - M) = 30.99 ± 0.04 (from Kavelaars et al. 2000). Both cases show clearly the excellent agreement of the GCLF method with other popular methods, despite the completely different and independent calibrators used. The accuracy of the GCLF method will always be limited by the sample size and lies around

A nice example of the "classical way" is the recent determination of the distance to Coma. At the distance of

100 Mpc the separation of metal-poor and metal-rich globular clusters is barely feasible anymore, and using the full globular cluster systems is necessary. Kavelaars et al. (2000) derived turn-over values of MV(TO) = 27.82 ± 0.13 and MV(TO) = 27.72 ± 0.20 for the two galaxies NGC 4874 and IC 4051 in Coma, respectively. Using Virgo ellipticals as calibrators and assuming a distance to Virgo of (m - M) = 30.99 ± 0.04, they derive a distance to Coma of 102 ± 6 Mpc. Adding several turn-over values for distant galaxies (taken from Lauer et al. 1998), they construct a Hubble diagram for the GCLF technique and derive a Hubble constant of H0 = 69 ± 9 km s -1 Mpc -1 . This example demonstrates the reach in distance of the method.

In summary, we think that the method is mature now and that most errors in the analysis can be avoided, as well as good choices for the calibration made. In the future, with HST and 10m-class telescope data, a number of determinations in the 100 Mpc range will emerge, and eventually, using metal-poor globular clusters only, this will give us a grasp on the distance scale completely independent from distances based at some stage on the LMC distance or Cepheids. *****

## Globular cluster luminosity function as distance indicator

Globular clusters are among the first objects used to establish the distance scale of the Universe. In the 1970-ies it has been recognized that the differential magnitude distribution of old globular clusters is very similar in different galaxies presenting a peak at M V∼−7.5. This peak magnitude of the so-called Globular Cluster Luminosity Function has been then established as a secondary distance indicator. The intrinsic accuracy of the method has been estimated to be of the order of ∼0.2 mag, competitive with other distance determination methods. Lately the study of the Globular Cluster Systems has been used more as a tool for galaxy formation and evolution, and less so for distance determinations. Nevertheless, the collection of homogeneous and large datasets with the ACS on board HST presented new insights on the usefulness of the Globular Cluster Luminosity Function as distance indicator. I discuss here recent results based on observational and theoretical studies, which show that this distance indicator depends on complex physics of the cluster formation and dynamical evolution, and thus can have dependencies on Hubble type, environment and dynamical history of the host galaxy. While the corrections are often relatively small, they can amount to important systematic differences that make the Globular Cluster Luminosity Function a less accurate distance indicator with respect to some other standard candles.

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## Title: THE ACS FORNAX CLUSTER SURVEY. VIII. THE LUMINOSITY FUNCTION OF GLOBULAR CLUSTERS IN VIRGO AND FORNAX EARLY-TYPE GALAXIES AND ITS USE AS A DISTANCE INDICATOR

We use a highly homogeneous set of data from 132 early-type galaxies in the Virgo and Fornax clusters in order to study the properties of the globular cluster luminosity function (GCLF). The globular cluster system of each galaxy was studied using a maximum likelihood approach to model the intrinsic GCLF after accounting for contamination and completeness effects. The results presented here update our Virgo measurements and confirm our previous results showing a tight correlation between the dispersion of the GCLF and the absolute magnitude of the parent galaxy. Regarding the use of the GCLF as a standard candle, we have found that the relative distance modulus between the Virgo and Fornax clusters is systematically lower than the one derived by other distance estimators, and in particular, it is 0.22 mag lower than the value derived from surface brightness fluctuation measurements performed on the same data. From numerical simulations aimed at reproducing the observed dispersion of the value of the turnover magnitude in each galaxy cluster we estimate an intrinsic dispersion on this parameter of 0.21 mag and 0.15 mag for Virgo and Fornax, respectively. All in all, our study shows that the GCLF properties vary systematically with galaxy mass showingmore » no evidence for a dichotomy between giant and dwarf early-type galaxies. These properties may be influenced by the cluster environment as suggested by cosmological simulations. « less

## 2. DATA AND GCLF INGREDIENTS

Each one of the 132 galaxies included in this study was observed with the Advanced Camera for Surveys (ACS) during a single HST orbit, as part of the ACS Virgo Cluster Survey (ACSVCS) and the ACS Fornax Cluster Survey (ACSFCS). The goals and main observational features of these two surveys are extensively discussed in Côté et al. (2004) and Jordán et al. (2007a), respectively. We refer the interested reader to these publications for further details.

The surveys targeted a total of 100 galaxies in the Virgo cluster and 43 galaxies in Fornax, and included observations in the F475W (≈ Sloan g) and F850LP (≈ Sloan z) passbands, with exposure times of

1210 s, respectively. In what follows, we will refer to the F475W filter as "g" and to F850LP as "z" due to their close proximity to the corresponding Sloan passbands.

Jordán et al. (2004) describes the pipeline implemented to automate the reduction procedure and analysis of all images in both surveys. The final output from this pipeline is a preliminary catalog of GC candidates and expected contamination per galaxy, including photometric and morphological properties that are later used to evaluate the probability pGC that a given object is a GC (see Jordán et al. 2009 for details). For the purposes of this study, and as defined in previous ACSVCS and ACSFCS papers, we constructed the GC candidate samples by selecting all sources that have pGC𢙐.5.

Our catalog of GC candidates in a given galaxy differs from the intrinsic GC population due to two effects: the existence of contamination in the sample and the level of completeness of the observations.

In order to quantify the average number of contaminants per field of view we have used archival ACS imaging of 17 blank-high latitude fields that have been observed in both the g and z bands, to the same or deeper depth than our images. These control fields were processed using the same pipeline implemented for the science data, and were then used to build customized control fields, as if a given galaxy was in front of it (the details of this process are explained in Peng et al. 2006, where a full list of the control fields used is also available). For each of our target galaxies, the result is a catalog containing 17 different estimates of the expected foreground and background contamination. These are later used to obtain an average estimate of the contamination in the field of view of a given galaxy.

The completeness function needs to be built considering four parameters: the magnitude of the source (m), its size as measured by the projected half-light radius (rh), its color ((gz)0), and the surface brightness of the local background over which the object lies (Ib). The completeness function f(m, rh, (gz)0, Ib) was obtained by performing simulations that added model GCs of different sizes (rh = (1, 3, 6, 10) pc), colors ((gz)0 = (0.7, 1.1, 1.5, 1.9) mag), and with King (1966) concentration parameter of c = 1.5, to the images. Although the effect of the color of the clusters has not been considered in previous publications (e.g., Peng et al. 2006 Jordán et al. 2007b), we have now established that it also has a small but measurable effect over the expected completeness. Overall, roughly 6 million fake GCs were added for the completeness tests for each color, with equal fractions at each of the four sizes and avoiding physical overlaps with sources already present. These images were then reduced through exactly the same procedure used with the science data. The final output of the process is a four-dimensional table that is used to evaluate f given an arbitrary set of (m, rh, (gz)0, Ib). The random uncertainty in the mean completeness curve is essentially zero, so the completeness limits at 90% and 50% are robust and can be determined with negligible error for a given population of objects.

This paper focuses on the study of the 89 early-type galaxies discussed by Jordán et al. (2007b) and all 43 galaxies of the ACSFCS. Our analysis is restricted to those galaxies that have more than five GC candidates and for which we were able to usefully constrain the GCLF parameters. These restrictions exclude 11 galaxies in the Virgo sample but none in Fornax.

## Nearby galaxies: HR diagrams (again), SBF, and globular clusters

We need to find new methods to reach out father into space, well beyond the Milky Way Galaxy and outside the Local Group. If we wish to measure the distance to a galaxy in the Virgo Cluster, for example, we have to leave fundamental methods behind and start to adopt more indirect techniques.

1. first, use a "primary" technique like parallax to determine the distance to a nearby object of type X
2. then, ASSUME that the nearby object is identical to a distant object of type X
3. and use the inverse square law to find the relative distances of the two objects

These methods are not as secure as the direct ones --- they make our ladder begin to tilt and sway. But sometimes, these secondary methods are the best we can do.

#### HR diagrams again: Tip of the Red Giant Branch (TRGB)

• Y = 0.246, which means the ratio of helium to hydrogen by mass is 0.246
• Z = 0.001, meaning that the ratio of (everything heavier than helium) to hydrogen by mass is 0.001

This should be relatively typical of an old stellar population, like that in the halo of our Milky Way, or the halo of other galaxies.

Let's begin with very young stellar populations. The figure below shows the HR diagrams for clusters of stars with ages of 100 and 500 Myr.

Now, as time passes, the stars continue to evolve . but the rate of change decreases. This is simply a result of the longer lifetimes and slower evolution of the lower-mass stars which remain.

But not only does the pace of change slow down -- something rather peculiar, but very, very useful for astronomers, is happening at the tip of the red giant branch.

Let's zoom in on this region. One thing you'll notice is a set of parallel tracks for the stars of each age. In those cases, the lower, more solid track shows stars of slightly lower mass, which are just ascending the red giant branch (RGB) for the first time, and then starting to descend it the upper, more sparse track shows stars of slightly higher mass, which are making a second journey up the red giant branch before they leave for good (probably turning into planetary nebulae). The track of those making their second trip is sometimes called the Asymptotic Giant Branch (AGB).

If we look at even older groups of stars, we will see that this tip of the RGB becomes redder . but remains roughly the same luminosity.

The absolute magnitude of this Tip of the Red Giant Branch is nearly constant from 5, to 8, to 11 Gyr.

Now, I've cheated just a little bit. If you look closely, you'll see that on the y-axis of these graphs, I've put the "absolute magnitude in I-band," rather than the more common "absolute magnitude in V-band." It turns out that in the V-band, the absolute magnitude of this tip does change a bit more strongly, growing fainter as time passes:

Astronomers have examined the possibilities, and it seems that the I-band is just the wavelength at which these red giants of different ages appear to have about the same luminosity. I don't know exactly why that it, but it's certainly convenient for us ground-based observers!

Of course, it's not perfect. If we look at models with different metallicities, we see that there IS some variation in the level of the TRGB, but only for very metal-poor populations.

#### Using the TRGB to measure distances

which I'll use as a source for the following material.

1. Take pictures of a galaxy, aiming your telescope away from spiral arms and young stars, so that you see mostly an old population
2. Measure the apparent magnitude of each individual, resolved star you can distinguish
3. If you consider only the very brightest stars, there will be only a few -- rare supergiants and some foreground stars in our own Milky Way
4. But as you count fainter and fainter stars, you will eventually reach stars on the galaxy's TRGB -- at which point suddenly lots of stars will appear (because these are the brightest members of the main population of the galaxy)
5. Measure that apparent magnitude of the TRGB
6. Using the known absolute magnitude of the TRGB, compute the distance to the galaxy

Here's an example. The galaxy NGC 1365 is relatively nearby, so we can resolve individual stars with HST. The purple squares mark the location of some images taken by HST in 2013.

Here's a copy of one of those HST images, which you can grab for yourself from the Hubble Legacy Archive. Let's zoom in one small region, away from the spiral arms:

If we set the image contrast so that it shows only the very brightest objects . we don't see very many.

But if we lower the threshold, a few more stars appear.

Lowering the threshold further makes a few more stars appear .

. but when we reach a certain point, suddenly a WHOLE BUNCH of stars appear, all at once. We have finally reached the level of the TRGB. (Click the image below to animate the sequence)

Now, let's look at how Jang et al., arXiv:1703.10616 put this into a quantitative form. They begin by making a color-magnitude diagram of stars in the images:

Figure 2 taken from Jang et al., arXiv:1703.10616 and slightly modified.

The authors make a histogram of number of stars as a function of magnitude, smooth that histogram, and finally apply a matched filter to find the precise value at which stars suddenly increase in number.

Figure 5 taken from Jang et al., arXiv:1703.10616

They include a correction for a small amount of extinction, and end up with a distance to NGC 1365 of 18.1 Mpc:

Table 2 taken from Jang et al., arXiv:1703.10616

uses TRGB to measure the distance to M60, a giant elliptical galaxy in the Virgo Cluster. In that case, the color-corrected F814W apparent magnitude of the TRGB is found to be m814W = 27.04, leading to a distance modulus of (m - M) = 31.05, or a distance of 16.2 Mpc.

In the past few years, we have managed to just barely reach the Virgo Cluster with TRGB.

#### Surface Brightness Fluctuations

The next technique is a little bit like TRGB, because it depends upon the fact that old stellar populations in different galaxies have similar properties. Instead of looking only at the very brightest red giants, however, this method includes all the bright red giants in a galaxy, together.

Let's consider a simple situation: a one-dimensional "galaxy" made up of bar-shaped "stars". If the galaxy is close enough, we can resolve the individual "stars", so that each one is separated from its neighbors by empty, black space.

Note the high contrast

If the galaxy is ten times farther away, then the stars begin to blend together. The average pixel now contains light from one star (appearing light grey), but due to the random location of stars, a few pixels are still empty (black), and a few pixels contain the combined light of several stars (bright white).

No longer pure black vs. pure white

If we move a single galaxy farther and farther away, the average pixel contains a blend of light due to a larger and larger number of stars as a result, the range of pixel values decreases, since the size of the random fluctuations decreases as we add together more and more stars. In the figure below, the gold bars show how the entire image at 1 Mpc is compressed into the first 10 pixels of the image at 10 Mpc, the entire image at 10 Mpc is compressed into the first 10 pixels of the image at 100 Mpc, and so forth.

• if a galaxy is close to us, we see large differences between adjacent pixels
• if a galaxy is far from us, we see only small differences between adjacent pixels

Let's quantify these statements. In the synthetic one-dimensional "galaxies", there is an average of 1 star for every 10 pixels when viewed from 1 Mpc. If every star has an identical intensity of 1 unit, then the average pixel value must be 0.1.

When we view this "galaxy" from a distance of 10 Mpc, each pixel now contains 10 times as many stars. I've scaled the intensities so that the average value is still 0.1, but you can see that the changes from one pixel to the next are now much smaller. If you click on the picture, you'll see the progression as we move the galaxy from 1 Mpc to 10000 Mpc.

We can be even more quantitative by computing the fractional difference from the average pixel value, on a pixel-by-pixel basis. If a galaxy is close to us, this fractional difference can be very large, but it shrinks rapidly with distance.

As one might expect, there's a simple relationship between the distance of the galaxy and the typical size of these deviations from the average pixel value.

• take an image of some nearby galaxy at a known distance
• compute f(near), the fractional deviation from the average pixel value
• take an image of some distant galaxy
• compute f(far), the fractional deviation from the average pixel value
• compare f(far) to f(near) to estimate the relative distances of the galaxies
• compute the distance to the distant galaxy

Now, in the real world, with its three-dimensional galaxies consisting of billions of stars drawn from heterogeneous populations, the problem is much more complicated but the basic idea is the same.

Let's look at a real example, taken from Blakeslee, Ap&SS 341, 179 (2012). HST observations of the galaxy IC 3032 in the Virgo Cluster show a pretty typical elliptical: bright at the center, fading away at larger distances:

Taken from Fig 7 of Blakeslee, Ap&SS 341, 179 (2012).

If one makes a model of the galaxy's brightness, fits it to the distribution of light in the image, and subtracts it from the original, one is left with . lumps. (Click on the image to activate animation)

Those lumps are NOT individual stars in the galaxy for one thing, they are much brighter than even the brightest red giants at this distance. Instead, they are due to Surface Brightness Fluctuations (SBF) in the number of bright giant stars falling within each resolution element of the image.

One way to measure quantitatively the size of the fluctuations in a galaxy's light is to take the Fourier transform of the two-dimensional image of the residuals, after the smooth model of galaxy light has been subtracted. If the galaxy is nearby, then the power of those fluctuations (the amplitude of the curved line in the diagram below) will be large:

Figure taken from Tonry and Schneider, AJ 96, 807 (1988).

If the galaxy is distant, then the power of those fluctuations will be small:

Figure taken from Tonry and Schneider, AJ 96, 807 (1988).

• a real galaxy has stars with a wide range of different colors and luminosities
• even though the old populations in most galaxies are similar, they may have somewhat different
• ages
• mass functions
• metallicities

So the basic ASSUMPTION that the stellar census in old populations of all galaxies is identical has to be modified. Astronomers who use the SBF method must try to account for variations in the population of stars from one galaxy to the next. It's a very complicated business.

The literature on SBF often uses the notion of the "fluctuation magnitude", denoted by a magnitude with a horizontal bar over it. This "flucuation magnitude" depends on the mix of stars within the overall stellar population but in many cases, it is dominated by the light of giant stars. It is, alas, not exactly the same in all galaxies, because the stellar population isn't the same in all galaxies. Fortunately, it doesn't very VERY much if one chooses a set of galaxies with similar properties, so as long as one compares, say, giant ellipticals to giant ellipticals with similar colors, the method is pretty reliable. Look at these results for two sets of galaxies in the Virgo and Fornax clusters:

The SBF method appears to give relatively precise distances when used properly Blakeslee et al., ApJ 694, 556 (2009) claim an intrinsic scatter of only about 0.06 mag. We can check this by looking at a recent determination of the distances to galaxies in the Fornax cluster.

Figure 4 taken from Blakeslee, J. P, ApSS 341, 179 (2012)

This method works best at long wavelengths, in the optical I-band and in the near-infrared, because emission from a galaxy at those wavelengths is dominated by a relatively few, luminous, red giant stars. If we observe at shorter wavelengths, then we dilute the light of these few, luminous giants with the light from many more less-luminous subgiants and main-sequence stars. That means that the number of stars contributing to the light inside each pixel (or resolution element) increases, and the size of random fluctuations in that light will decrease.

it can be used both in elliptical galaxies and in some spiral galaxies -- in any spiral with a prominent bulge, like NGC 4459 shown below.

The reach of the SBF method is larger than that of other methods we have examined so far. Although the results are still somewhat preliminary, it appears that SBF can be used to measure distances to the Coma Cluster of galaxies -- which it finds to be about 95 Mpc away from us, roughly six times more distant the the main Virgo Cluster!

First, a near-IR image of the galaxy NGC 4874 taken with HST at 1.6 microns (in H-band):

Taken from Jensen et al., "The Surface Brightness Fluctuation Distance to the Coma Cluster" (2015)

Now, the same image after subtracting a model for the galaxy's light, leaving surface brightness fluctuations and a boatload of globular clusters.

Taken from Jensen et al., "The Surface Brightness Fluctuation Distance to the Coma Cluster" (2015)

So, there is a good chance that we can soon measure the distance to the Coma Cluster (and other objects at similar distances) via SBF.

#### A more "secondary" indicator: Globular Cluster Luminosity Function (GCLF)

TRGB We need to understand the evolution and appearance of individual stars through the red giant stage

But our final topic today is one which involves more assumptions, including some which are very poorly understood. When we try to apply the Globular Cluster Luminosity Function method, we will have to fall back upon a somewhat weak argument:

We will simple assume that if two groups of objects have a similar appearance, that their properties must be the same in detail. It's quite a step away from individual stars, for which we can work out the physics in detail, or even a population of many stars.

Okay, enough philosophy. Back to science.

Globular clusters are collections of 10,000 to 1,000,000 stars orbiting within a compact space of a few parsecs. The stars within them belong to very old populations in some cases, it appears that they may date from the time of our Galaxy's formation, or even earlier.

When we look at globular clusters within our own Milky Way, we can see very clearly the individual stars (except when they get in each other's way near the center, perhaps).

Image of M80 courtesy of NASA and Wikipedia

When we look at other galaxies, we can't resolve the individual stars instead, we just see a compact, bright ball of light. In this picture of the Sombrero Galaxy, for example, note the many, many little dots which appear blueish in color. (Click on the image below to enlarge)

Image of M104 from HST and Spitzer courtesy of NASA/JPL-Caltech/University of Arizona

Here, I'll zoom in to show them more clearly.

Now, globular clusters are NOT identical: some are much larger than others, some are much more luminous than others. If we count the number of clusters in the Milky Way as a function of their absolute magnitudes, we find a roughly gaussian distribution. Yes, there's a tail at the low-luminosity end no, for our purposes, that's not very important (why not?).

Figure based on data from http://physwww.mcmaster.ca/

If we look at the distribution of apparent magnitudes of globular clusters around other galaxies, we see something like a gaussian distribution (well, sometimes . more on that in a moment). In the figure below, the left-hand panels show histograms of the GCs in a pair of Virgo Cluster galaxies.

Figure taken from Jordan et al., ApJS 180, 54 (2009)

The right-hand panels in the figure above show a histogram of the COLORS of the GCs around those two Virgo galaxies. It seems that GCs come in two flavors, "red" and "blue" the difference has something to do with metallicity. Is the mixture of flavors the same in all galaxies? Does it matter for the use of GCLF as a distance indicator? Good questions. Figure taken from Secker and Harris, AJ 105, 1358 (1993) -->

Figure taken from Miller and Lotz, ApJ 670, 1074 (2007)

Let's put aside all these questions and just make the GIANT ASSUMPTION that the processes which formed globular clusters in the Milky Way also governed the formation of globular clusters around all other galaxies. In that case, the luminosity function for cluster around all galaxies should have the same absolute magnitude at its peak -- right? (Actually, in some cases, this might not be SO crazy -- see Harris et al., ApJ 797, 128 (2014))

If you have extra time, you might read about the dangers of incompleteness when one is comparing luminosity functions.

The globular cluster technique may be losing some popularity now, but it was one the popular methods for nearby galaxies in the past because ground-based telescopes could measure the properties of globular clusters out to large distances. With HST, it can be used out to Coma and beyond.

• the original paper, Tonry and Schneider, AJ 96, 807 (1988)
• a more recent summary of the technique, Blakeslee, Ap&SS 341, 179 (2012)
• a discussion including the calibration of population effects, Jensen et al., ApJ 808, 91 (2015)

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In: Astrophysical Journal , Vol. 419, No. 2, 20.12.1993, p. 479-484.

Research output : Contribution to journal › Review article › peer-review

T1 - A comparison of the planetary nebula luminosity function and surface brightness fluctuation distance scales

N2 - Two of the best techniques for measuring distances greater than ∼3 Mpc are the planetary nebula luminosity function (PNLF) and the surface brightness fluctuation (SBF) method. We compare the results of both methods and analyze the internal and external errors associated with the measurements. We find that the PNLF distances are systematically larger than the SBF distances by 0.07 ± 0.03 mag, but this error can be entirely attributed to uncertainties in the Local Group calibrations which both methods employ. After correcting for this effect, we find the random scatter in the difference between the PNLF and SBF distance determinations Δ = (m-M)SBF - (m-M)PNLF = 0.17 mag, is in exact agreement with that predicted from the internal uncertainties of the methods. We show that Δ is not measurably correlated with such parameters as galaxy color metallicity, specific PN density, and specific globular cluster frequency, but does correlate slightly with galactic absolute B magnitude. We discuss the reality of this correlation and show that the trend is not important for extragalactic distance applications.

AB - Two of the best techniques for measuring distances greater than ∼3 Mpc are the planetary nebula luminosity function (PNLF) and the surface brightness fluctuation (SBF) method. We compare the results of both methods and analyze the internal and external errors associated with the measurements. We find that the PNLF distances are systematically larger than the SBF distances by 0.07 ± 0.03 mag, but this error can be entirely attributed to uncertainties in the Local Group calibrations which both methods employ. After correcting for this effect, we find the random scatter in the difference between the PNLF and SBF distance determinations Δ = (m-M)SBF - (m-M)PNLF = 0.17 mag, is in exact agreement with that predicted from the internal uncertainties of the methods. We show that Δ is not measurably correlated with such parameters as galaxy color metallicity, specific PN density, and specific globular cluster frequency, but does correlate slightly with galactic absolute B magnitude. We discuss the reality of this correlation and show that the trend is not important for extragalactic distance applications.

## Morphology

In contrast to open clusters, most globular clusters remain gravitationally-bound for time periods comparable to the life spans of the majority of their stars. (A possible exception is when strong tidal interactions with other large masses result in the dispersal of the stars.)

At present the formation of globular clusters remains a poorly understood phenomenon. However, observations of globular clusters show that these stellar formations arise primarily in regions of efficient star formation, and where the interstellar medium is at a higher density than in normal star-forming regions. Globular cluster formation is prevalent in starburst regions and in interacting galaxies.

After they are formed, the stars in the globular cluster begin to gravitationally interact with each other. As a result the velocity vectors of the stars are steadily modified, and the stars lose any history of their original velocity. The characteristic interval for this to occur is the relaxation time. This is related to the characteristic length of time a star needs to cross the cluster as well as the number of stellar masses in the system. The value of the relaxation time varies by cluster, but the mean value is on the order of 10 9 years.

Ellipticity of Globulars
Galaxy Ellipticity
Milky Way 0.07±0.04
LMC 0.16±0.05
SMC 0.19±0.06
M31 0.09±0.04

Although globular clusters generally appear spherical in form, ellipticities can occur due to tidal interactions. Clusters within the Milky Way and the Andromeda Galaxy are typically oblate spheroids in shape, while those in the Large Magellanic Cloud are more elliptical.

Astronomers characterize the morphology of a globular cluster by means of standard radii. These are the core radius (rc), the half-light radius (rh) and the tidal radius (rt). The overall luminosity of the cluster steadily decreases with distance from the core, and the core radius is the distance at which the apparent surface luminosity has dropped by half. A comparable quantity is the half-light radius, or the distance from the core within which half the total luminosity from the cluster is received. This is typically larger than the core radius.

Note that the half-light radius includes stars in the outer part of the cluster that happen to lie along the line of sight, so theorists will also use the half-mass radius (rm)&mdashthe radius from the core that contains half the total mass of the cluster. When the half-mass radius of a cluster is small relative to the overall size, it has a dense core. An example of this is the Messier 3, which has an overall visible dimension of about 18 arc seconds, but a half-mass radius of only 1.12 arc seconds.

Finally the tidal radius is the distance from the centre of the globular cluster at which the external gravitation of the galaxy has more influence over the stars in the cluster than does the cluster itself. This is the distance at which the individual stars belonging to a cluster can be separated away by the galaxy. The tidal radius of M3 is about 38&Prime.

### Mass segregation and luminosity

In measuring the luminosity curve of a given globular cluster as a function of distance from the core, most clusters in the Milky Way steadily increase in luminosity as this distance decreases, up to a certain distance from the core, then the luminosity levels off. Typically this distance is about 1&ndash2 parsecs from the core. However about 20% of the globular clusters have undergone a process termed "core collapse". In this type of cluster, the luminosity continues to steadily increase all the way to the core region. An example of a core-collapsed globular is M15.

Core-collapse is thought to occur when the more massive stars in a globular encounter their less massive companions. As a result of the encounters the larger stars tend to lose kinetic energy and start to settle toward the core. Over a lengthy period of time this leads to a concentration of massive stars near the core.

The Hubble Space Telescope has been used to provide convincing observational evidence of this stellar mass-sorting process in globular clusters. Heavier stars slow down and crowd at the cluster's core, while lighter stars pick up speed and tend to spend more time at the cluster's periphery. The globular star cluster 47 Tucanae, which is made up of about 1 million stars, is one of the densest globular clusters in the Southern Hemisphere. This cluster was subjected to an intensive photographic survey, which allowed astronomers to track the motion of its stars. Precise velocities were obtained for nearly 15,000 stars in this cluster.

The overall luminosities of the globular clusters within the Milky Way and M31 can be modeled by means of a gaussian curve. This gaussian can be represented by means of an average magnitude Mv and a variance &sigma 2 . This distribution of globular cluster luminosities is called the Globular Cluster Luminosity Function (GCLF). (For the Milky Way, Mv = &minus7.20±0.13, &sigma=1.1±0.1 magnitudes.) The GCLF has also been used as a " standard candle" for measuring the distance to other galaxies, under the assumption that the globular clusters in remote galaxies follow the same principles as they do in the Milky Way.

### N-body simulations

Computing the interactions between the stars within a globular cluster requires solving what is termed the N-body problem. That is, each of the stars within the cluster continually interacts with the other N&minus1 stars, where N is the total number of stars in the cluster. The naive CPU computational "cost" for a dynamic simulation increases in proportion to N 3 , so the potential computing requirements to accurately simulate such a cluster can be enormous. An efficient method of mathematically simulating the N-body dynamics of a globular cluster is done by sub-dividing into small volumes and velocity ranges, and using probabilities to describe the locations of the stars. The motions are then described by means of a formula called the Fokker-Planck equation. This can be solved by a simplified form of the equation, or by running Monte Carlo simulations and using random values. However the simulation becomes more difficult when the effects of binaries and the interaction with external gravitation forces (such as from the Milky Way galaxy) must also be included.

The results of N-body simulations have shown that the stars can follow unusual paths through the cluster, often forming loops and often falling more directly toward the core than would a single star orbiting a central mass. In addition, due to interactions with other stars that result in an increase in velocity, some of the stars gain sufficient energy to escape the cluster. Over long periods of time this will result in a dissipation of the cluster, a process termed evaporation. The typical time scale for the evaporation of a globular cluster is 10 10 years.

Binary stars form a significant portion of the total population of stellar systems, with up to half of all stars occurring in binary systems. Numerical simulations of globular clusters have demonstrated that binaries can hinder and even reverse the process of core collapse in globular clusters. When a star in a cluster has a gravitational encounter with a binary system, a possible result is that the binary becomes more tightly bound and kinetic energy is added to the solitary star. When the massive stars in the cluster are sped up by this process, it reduces the contraction at the core and limits core collapse.

## How reliable is it to use globular cluster lumniosity function as a distance indicator? - Astronomy

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>endstream endobj 112 0 obj > stream J.+n6!J#]7":u%c_?1q!)K>tti36 _A+VI)8o"5l18 endstream endobj 113 0 obj > stream J../V!J#]3":[email protected])S7'P.XiHow reliable is it to use globular cluster lumniosity function as a distance indicator? - Astronomy,[nobr][H1toH2]

## How reliable is it to use globular cluster lumniosity function as a distance indicator? - Astronomy

We test whether the peak absolute magnitude Mv(TO) of the Globular Cluster Luminosity Function (GCLF) can be used for reliable extragalactic distance determinations. Starting with the luminosity function of the Galactic Globular Clusters listed in Harris catalog, we determine Mv(TO) either using current calibrations of the absolute magnitude Mv(RR) of RR Lyrae stars as a function of the cluster metal content [Fe/H] and adopting selected cluster samples. We show that the peak magnitude is slightly affected by the adopted Mv(RR)-[Fe/H] relation, while it depends on the criteria to select the cluster sample. As for the GCLFs in other external galaxies, using Surface Brightness Fluctuations (SBF) measurements we give evidence that the luminosity functions of the blue (metal-poor) Globular Clusters peak at the same luminosity within

0.2 mag, whereas for the red (metal-rich) samples the agreement is within

0.5 mag even accounting for the theoretical metallicity correction expected for clusters with similar ages and mass distributions. Then, using the SBF absolute magnitudes provided by a Cepheid distance scale calibrated on a fiducial distance to LMC (m(LMC)=18.50 mag), we show that the Mv(TO) value of the metal-poor clusters in external galaxies(-7.67+/-0.23 mag) is in excellent agreement with the value of both Galactic (-7.66+/-0.11 mag) and M31 (-7.65+/-0.19 mag)ones

## Extragalactic Globular Clusters and Galaxy Formation

AbstractGlobular cluster (GC) systems have now been studied in galaxies ranging from dwarfs to giants and spanning the full Hubble sequence of morphological types. Imaging and spectroscopy with the Hubble Space Telescope and large ground-based telescopes have together established that most galaxies have bimodal color distributions that reflect two subpopulations of old GCs: metal-poor and metal-rich. The characteristics of both subpopulations are correlated with those of their parent galaxies. We argue that metal-poor GCs formed in low-mass dark matter halos in the early universe and that their properties reflect biased galaxy assembly. The metal-rich GCs were born in the subsequent dissipational buildup of their parent galaxies and their ages and abundances indicate that most massive early-type galaxies formed the bulk of their stars at early times. Detailed studies of both subpopulations offer some of the strongest constraints on hierarchical galaxy formation that can be obtained in the near-field.