How Far is the Galactic Center?
This exercise is based on the ground-breaking work of Harlow Shapley, who began his study of the distances to globular clusters in 1914.
To the measurer of the sidereal universe, star
clusters are beacon lights. They point the way to the center of our
Galaxy and to its edges... . The globular clusters are a sort of framework—a
vague skeleton of the whole Galaxy—the first and still the best
indicators of its extent and orientation.
These remarks, written around 1920, are mostly true to this day. At the time of its writing, however, the Milky Way was generally believed to be 20,000 light-years across and about 6,000 light-years in thickness. Our Sun was thought to be somewhere near the center of this assemblage of stars.
In the early 1800's, John Herschel noted that almost all of the globular clusters discovered were found in the southern hemisphere with nearly a third of them confined to the vicinity of the constellation Sagittarius. The northern hemisphere, meanwhile, had virtually no globular clusters. Many were also found to be far from the luminous band of the Milky Way. Shapley reasoned that the globular clusters were oriented around the center of our galaxy and that an accurate determination of the distances to the clusters would give a measure of the Sun's distance from the galactic center. In this activity we'll recreate the basic work of Shapley and use modern corrections to arrive at reasonable estimates of the size of our Milky and our location in it.
Procedure
Figure 1.
In this exercise we will determine the distance to 20 globular clusters by measuring the RR Lyrae gap of each cluster. Actually, the location of those "standard candle" variable stars are NOT plotted because their magnitudes vary over a period of several hours' time (see figure at left). There is no point plotting the position of a star on the H-R diagram when it will move from that position rapidly.
By determining the position of "blue" edge of the gap, however, it is possible to determine the apparent magnitude m of the stars that would fill the gap if they had been plotted on the cluster's diagram. Using the example of M72, we find that the magnitude m of the gap is m = 17.0 . It is important to note that the magnitudes decrease as you move up the y-axis. It is unnecessary to record the values along the x-axis.
Once the apparent magnitude of the RR Lyrae gap is determined, the distance to the cluster can be ascertained using the extinction of starlight in the direction of the cluster. The extinction A is simply the amount by which the magnitude of starlight is dimmed by its passage through dust and gas along the line of sight. Sometimes extinction is referred to as absorption (hence the variable A) and other times it is referred to as interstellar reddening due to the fact that more starlight from the blue end of the spectrum is scattered while traveling towards Earth than is light from the red end of the the stars' spectra. Therefore, the light that eventually reaches observers on Earth is not only dimmer than if it had not passed through gas and dust, but it is also artificially reddened (this phenomenon is well known and explains why it is more difficult to determine a star's surface temperature based on its color than on a careful analysis of its spectrum).
Figure 2.
Using the extinction of starlight per 1,000 light-years from Earth (found in Table I: Cluster Parameters), the distance to each cluster is calculated with the aid of the nomogram. For the example of M72, the extinction A = 0.10 and the distance is simply the value on the center scale (see Figure 2).
Using a straight edge extending from the HB Apparent Magnitude of 17.0 and the Extinction in Magnitudes of 0.10, we find that the measured distance to M72 is 59 (thousand light-years).
Example Calculation
The coordinates of the cluster can be found using the calculated distance
and the given values of galactic latitude b and galactic longitude l. It
is necessary to perform these calculations because the distance calculated
does not give any indication of the direction the cluster lies from Earth.
The distance and location of the galactic center can only be ascertained
after converting the distance to the clusters to two-dimensional coordinates.
The equations needed to determined the x-coordinate (along the galactic
plane) and z-coordinate (above or below the galactic plane) are
X = Dcosl cosb
and
Z = Dsinb .
For our example cluster M72, we find that
X = 59 x cos 35°.2 x cos (–32°.7) = 59 x 0.8171 x 0.7965 = 40.6 (thousand light-years)
Z = 59 x sin (–32°.7) = 59 x (–0.5402) = –31.9 (thousand light-years)
If you would like to create your data table and have the calculations done using Excel, here are examples of the syntax for determining the value of X and Z:
Data Presentation
After completing the data collection for all 20 globular
clusters and finding the average values of X and Z, the results
can be plotted as a scatter plot (no lines) using Excel's Chart Wizard.
An example is shown in below.
When submitting your results, include a data table, average value of X and Z, and a plot of the cluster distribution.