# 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.

As you will see, some of the methods described here are called "secondary distance indicators" because we

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)

Let's look at the HR diagram of a group of stars, all born at the same time, with identical chemical compositions. I'll choose a metal-poor population, with

• 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.

To make models of stars as they evolve, I'll use the

The site provides a set of stars with masses varying from 0.5 to 8.4 solar, and follows them as the evolve into the red giant branch.

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.

```

Q:  What changes do you see between 100 and 500 Myr?

Q:  What happens to the magnitude of the most
luminous red giants as the cluster ages?

```

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.

```

Q:  What happens to the tip of the red giant branch
between 1 Gyr and 2 Gyr?

```

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

This method was first described back in 1993

but it has been developed and extended over the past two decades. A good recent article is

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

The basic idea is pretty simple:

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.

```

Q: Based on this CMD, what is the apparent magnitude
of the TRGB?

Q: Using that apparent magnitude, and absolute magnitude
MI = -3.95  for the TRGB,
what is the distance to NGC 1365?

```

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

Another recent paper by this same group

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.

```

Q:  Using only these two galaxies,

M60 in the Virgo Cluster
NGC 1365 in the Fornax Cluster

what is the difference in distance modulus

Δ  =  (m-M)(Fornax)  -  (m-M)(Virgo)

```

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.

The pattern is clear:

• 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.

One could use this relationship to determine distances in the following way:

• 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)

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

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).

Now, real life is much more complicated than that simple example.

• 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:

Figure taken from Blakeslee et al., ApJ 694, 556 (2009)

```

Q:  What is the difference in distance modulus between
Fornax and Virgo, using these SBF measurements?

Q:  How does that compare to the difference in distance
modulus between NGC 1365 (Fornax) and M60 (Virgo),
using the TRGB method?

```

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.

Two reasons the SBF method is so promising are

• 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 newest camera aboard HST, the Wide Field Camera 3 (WFC3), can take images in the near-infrared, where SBF are particular prominent, and so detect them in very distant galaxies.

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!

The figures below are taken from a poster published in 2015, not from a refereed paper, so please consider them preliminary. Another poster was shown in Jan 2017, but the SBF group has still not published a paper on this object.

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

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

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)

The methods we've discussed so far today are not as mathematically sound as trigonometric parallax, but they don't rely on THAT many assumptions.

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

SBF
We need to understand the appearance of entire populations of stars, with a range of masses and metallicities and ages, all mixed together -- so this has more uncertainties than TRGB

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:

I don't quite understand how globular clusters form, or why they have a particular distribution of luminosities ... but I am going to ASSUME that the clusters around galaxy A have the same distribution as those around galaxy B

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/~harris/mwgc.dat

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 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))

```

Q:  Choose either the Virgo or the Fornax
distribution.  Compare it to the
distribution of Milky Way GC absolute
magnitudes.  Use it to estimate the
distance to the Virgo or Fornax galaxy
cluster.

```

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.