Distance to the Center of the Galaxy
Adapted from: University of Victoria, Astronomy 120Objective
In this exercise, we use the distance to the globular cluster M4 to calibrate the distances to a number of other globular clusters. By determining the center of their distribution, we get an estimate of the distance to the center of the Galaxy. [Note: It is assumed that the student has already completed the exercise "RR Lyrae Stars and the Distance to M4." If so, we must use the corrected magnitude that took into account blocking of light from the star by dust.]
Materials
- Browser needed for on-line measurements
- Distance to globular cluster M4, either from reference or as determined in the "RR Lyae and the Distance to M4" lab
- Table 1 in Excel (with formulas!)
- Scientific Calculator (If Excel spreadsheet not used. PC calculator will work.)
Introduction
In studying the distribution of globular star clusters, I was led some years ago to consider the circumstance that stars near the Sun fail to show the same galactic circle as that outlined by Milky Way star clouds. Further, when the globular star clusters appeared to show that the galactic center is at a great distance from the Sun in the direction of Sagittarius, and yet the evidence from star counts seemed to indicate that the Sun is near tthe center of the stellar system, the question arose as to whether the star distribution in the solar vicinity might not be only a local phenomenon; our Sun -- might it not be near the center of a subsystem, but remote from the real galactic center?
Harlow Shapley, 1920, Flights from Chaos, p. 123
The System of Globular Clusters
Image courtesy of A Galactic
Globular Cluster Database
In the years 1916 - 1917 Harlow Shapley had been taking photographs of globular clusters at Mt. Wilson (California) with the 60-inch reflecting telescope. He noticed the marked concentration of these clusters toward the region of the sky near Sagittarius, although there were none near the middle of the Milky Way band itself.
Shapley also noticed that the Hertzsprung-Russell diagrams of the globular clusters differed in many significant respects from that of stars in the solar neighborhood. In the figure at the left, we see the color-magnitude diagram (CMD) of M4. The CMD'S of all globular clusters are similar in that they exhibit little or no main sequence, a strong giant branch, and a so-called "horizontal branch" running to the left of the giant branch. Here is a clue, leading us to presume that globular clusters are among themselves composed of similar stars, while their stellar content seems to differ in some significant way from that of the solar neighborhood. We begin to question the validty of the assumption made by earlier observers that the number of stars in each brightness interval is the same in all parts of the Galaxy that could be observed.
Because of calibration of the Cepheid period-luminosity relation, Shapley was not in a good position to estimate the distance to those clusters in which he could detect Cepheids and measure their periods. This he did for the brighter clusters. In the fainter clusters, the Cepheids are below the threshold of detection. But these faint clusters also appeared smaller, thus supporting the hunch that they were of similar construction to the brighter ones but simply at greater distances.
The assumption of similar essence could be used to convert the brightness and size data into distance data. Furthermore, Shapley noticed that the brightest stars of the nearer clusters were about two magnitudes brighter than the RR Lyrae type of Cepheid variable, and thus of a known absolute magnitude. So in those clusters too faint to exhibit RR Lyrae variables, but bright enough for resolution of their brightest stars, a distance could be estimated. Finally, Shapley accumulated sufficient data to construct a picture of the distribution of the globular clusters with respect to the Milky Way band. The model comprised a more or less spherical distribution of clusters centered far off in the direction of Sagittarius, at a distance estimated by Shapley to be about 16 kpc. The fact that most clusters appear to be in one region of the sky demonstrates that we are outside most of the spherical distribution. In computing distances, Shapley assumed no absorption of the light of the RR Lyrae stars or the whole clusters. If absorbing material were present, making the stars appear too faint, it would have led Shapley to overestimate the distances. We may thus say that the upper limit of the distance to the center of the globular cluster system was 16 kpc.
This was a real revolution in conception of the size of our stellar system. Supposing the clusters to be symmetrically situated with respect to the Milky Way stars, we would be in a flat, circular system at least tens of thousands of parsecs in diameter!
Computation of the Distance to the Center of the Globular Cluster System
In this exercise we shall carry out a computation of the distance to the center of the globular cluster system very similar to that done by Shapley. Instead of the 86 clusters Shapley used, we shall use only 16, but that will be enough to give a reasonable estimate of the distance. Since, in Shapley's words, "it appears to be a tenable hypothesis that the supersystem of globular clusters is coextensive with the Galaxy itself," by finding the distance to the center of the cluster system, we shall also find the distance to the center of the Galaxy.
Milky Way region around the constellation of Sagittarius We have chosen a section of the sky near to the direction of the apparent center of the globular cluster system in Sagittarius. In this area of about 15o x 25o (centered on right ascension 18h 30m and declination -32o), there are over a dozen globular clusters. Look for the symbol on the map that represents globular clusters. We shall find the distance to the center of this group of clusters by averaging their distances. In doing so, we must make several assumptions:
- The angular size of a cluster is inversely proportional to its distance. This is tantamount to presuming that all clusters are of more or less the same intrinsic size.
- We see in this limited region a representative selection of globular clusters, both near and far.
- There is not such a large amount of absorption of light between us and the clusters that it would obscure the most distant ones, nor seriously affect their apparent sizes. [Note: This is why one use the "dust-corrected" distance for M4 and not the one calculated from the data, which would be overstated.]
Caveat
When we take a photograph of a section of sky, the volume of space recorded in the picture increases very rapidly with distance. In the first kiloparsec of distance from us, a narrow wedge of space (volume A) is imaged. At a great distance away, however, an extra kiloparsec of distance includes a much larger slice of space (Volume B). Clearly we should expect to see more clusters in volume B than in volume A just because of its greater size. That means we shall see an anomalously greater number of distant clusters and an anomalously smaller number of nearby clusters for any given region of sky.
We must account for this effect before averaging the cluster distances, or else the preponderance of distant clusters will yield a lopsided result. [Thought question: Will our calculated distances be overstated or understated?] The area of the beam covered by the photograph increases as the square of the distance, and this is the correction factor needed. Essentially, we weight the closer clusters more than those farther away. Table 1, given at the end of this exercise and which guides your calculations, takes care of this when it asks you to square the angular sizes. The average distance, Rave, once you calculate it using the method outlined, takes this into account.
If this is all we did, then we would have the relative distances between these globular clusters, but NOT their actual distances! Why? Well we really don't know the relationsip between angular size and distance yet. We could calculate this if we knew the ACTUAL size of a globular cluster (and, recall, we are assuming they are all the same actual size), but we do not know that. We can solve this conundrum by remembering that we determined the distance to the globular cluster M4 (see the RR Lyrae and Distance to M4 lab, or we could look it up). All we need to do is measure its angular size (shown in this equation by the Greek letter theta 
on the same scale as the rest of the clusters) and we have the relationship:
where RM4 is the distance to M4 in parsecs or kiloparsecs. It may not be immediately obvious, but we are using the small-angle formula: angular size = actual size divided by distance. The link to the University of Victoria web page contains the mathematical logic. It is a bit hard to follow, and don't be too concerned if you do not understand it all. We are doing something called "normalizing" where we need to "weight" some of our data to make things come out right. This process is very much like "weighting" one of your scores on an exam or a homework assignment or something similar.
Procedure
The first thing we must do is measure the angular size of the globular cluster M4. NGC 6121 IS the same thing as M4, just listed with its "New General Catalog" number rather than its Messier designation. NGC 6121 is the first cluster shown in the table of clusters:
Web page containing the links to the 16 Clusters.
Click on its image and follow the directions given there. Enter your data in Table 1 below.Follow the same process for the remaining 15 clusters:
- Measure the apparent diameter, in pixels, of each cluster. Follow the directions given on the "clickable images" web pages, and enter your data in the correct columns in Table 1.
- Make sure you choose a suitable criterion for measuring size, such as the diameter of the darkest part of each image. Use the same criterion when measuring all of the clusters.
- Measure each cluster twice: once horizontally (X) and once vertically (Y).
- Use the scale factor of 1 pixel = 0.05 arc minutes--if measuring using the on-line script--to determine the width of each cluster in arc minutes. That is, multiply column 4 by column 5 and put the answer in column 6. [Determine your own scale factor if measuring from the screen or from a print out of each image (instructions given on the web page for each image).]
- Sum the "angular-sizes" column (6) and the "angular-sizes-squared" column (7). Divide as given by the formula. The formula looks really, really stinky? Then, what it is saying is divide column 6 by column 7. Then, multiply that number by the distance to M4 and your measured angular size of M4. Your answer will be Rave -- the distance to the center of the distribution, and theoretically the center of the Galaxy. [Caution: watch your units for the distance. Pick parsecs or kiloparsecs and then stick with it.]
Questions
The approximate distance to the center of the Milky Way is 8.5 kpc (kiloparsecs) or 8,500 parsecs. Compare your results by finding the percentage difference:
[(your value - actual value) / actual value] Would Harlow Shapley be proud of you? Is your answer within about 25% of the real answer? Comment very briefly on your results, analyzing why you are way off, if you are. [NOTE: it is okay to be way off here. Your instructor was about 40% off, deriving a much smaller distance.] In commenting, think about what we are assuming--as the next question hints.Astronomers (indeed, most scientists) MUST make assumptions in order to move forward in their work. The primary assumption made here is that these globular clusters are all actually the same size. Well, even a superficial look at the images reveals that that cannot be the case, as some clusters definitely have more stars than others, and they just LOOK different. What affect does this assumption (all being the same size) have on your results?
We used just one cluster, M4, to calibrate our results. Plus, we used just one star in that cluster for its distance determination.
- What level of confidence arises when we use just one object to calibrate many? Explain your answer.
- Suppose M4 were an unusually large cluster. What effect would this have on your calculated result for Rave?
- How about if M4 were unusually small?
[To answer these two questions, you need to think abstractly. If M4 is unusually large, what does this mean for the actual sizes of the other clusters? If M4 is unusually small, what does this mean for the actual sizes of the other clusters? Our assumed size directly affects our interpretation of the distances to these clusters: If we think a cluster is large, we will interpret its "small angular size" as meaning it is very far away. Think about this for a while.]
In 4 - 5 sentences, summarize your results for this exercise.