The Galactic Centre
To determine the location of the galactic
centre relative to our Sun through analysis of the distribution of globular
clusters
Graeme Ing, HET603A, 2004
Project 4: Galactic Centre
Project Supervisor: Professor Duncan Forbes
Introduction
The purpose of the
experiment outlined in this paper is to reproduce the work carried out by
Harlow Shapley in 1920, in which he determined that the galactic centre was not
co-existent with the Sun, but lay approximately 18,000 parsecs (pc) away in the
constellation of Sagittarius. His conclusion came from analyzing the
distribution of globular clusters throughout the Milky Way.
Similarly, we intend
to construct a distribution model from a sample of globular clusters, and from
that model identify the centre of the distribution and thus the distance and
direction of the galactic centre. Rather than use variable stars as calibrators
of intra-galactic distance, we intend to use the principle adopted by Trumpler
in 1930, that “smaller is further”, and compare the angular diameter of
clusters to determine their distance.
Historical Perspective
Prior to 1900, it
was widely believed amongst astronomers that the Sun lay at the centre of our
galaxy, the Milky Way. Notable astronomers including William Herschel had made
detailed analyses of stellar density in all areas of the sky and determined
that the galaxy was a flattened disk. Because stars appeared in equal numbers
in all directions across this disk, Herschel concluded that our Sun must be
located at the centre of the galaxy.
In the late 19th
century, Jacobus Kapteyn devoted his career to finding the distance to more
distant stars using stellar parallax. He developed a technique called “mean
parallax” that allowed astronomers to more accurately measure distances up to
3000 light years; much greater than the 300 light years that traditional
trigonometric parallax allowed (Belkora, L. 2003). In his later years, Kapteyn,
working with Pieter van Rhijn, refined his model of our galaxy, known as the
“Kapteyn Universe”. In 1920, he estimated the diameter of our galaxy at 17
kiloparsecs (kpc), and placed our Sun a mere 650 parsecs (pc) from the centre
(Belkora, L. 2003). His distances were substantially less than reality, in part
because Kapteyn underestimated the effects of interstellar extinction, though
he was aware of the phenomena and reported to be quite concerned about its
effects in the 1900’s. Interstellar extinction refers to the absorption of
light from distant objects by gas and dust particles in the interstellar
medium, causing a reduction in the brightness of those objects. To a non-cautious
astronomer the objects appear more distant than they really are.
Interstellar
extinction is more prevalent in the plane (or disk) of the galaxy due to the high
quantities of gas and dust found there. Any further attempts to measure the
size of our galaxy would require distant, bright objects that lay outside of
the plane of the galaxy. Such objects are less influenced by the dimming effect
of interstellar extinction and so their distance can be measured more
accurately. Globular star clusters proved to be the perfect choice of object
since many lay scattered throughout the galactic halo, the sphere that
encompasses our galaxy. In the early 20th century, it was believed
that globular clusters lay equally distributed throughout the halo, many lying
high above or below the plane of the galaxy. It was proposed that if their
locations were accurately measured and plotted, the centre of this distribution
map would pinpoint the centre of the galaxy. The distance to the furthest
globulars in the halo would also provide a more accurate measurement of the
size of our galaxy (Freedman, R. & Kaufmann, W. 2001).
Harlow Shapley took
on the task, observing globular clusters between 1914 and 1917 from the Mount
Wilson Observatory. His first challenge was how to obtain accurate distances to
each globular cluster. Before the 20th century, the standard means
of calculating interstellar distances was via a variety of parallax methods.
Despite Kapteyn’s development of “mean parallax”, parallax proved unsuitable
for the enormous distances involved, since many globular clusters lay between
10 kpc and 30 kpc away.
The answer lay in
the research performed in 1912 by Henrietta Leavitt. Leavitt had determined
that Cepheid variable stars have an extremely predictable period to luminosity
ratio. The luminosity of a variable star could be determined by observing the
period of its brightness fluctuations. The distances to nearby Cepheids were calculated
using parallax and these formed “standard candles” to which other Cepheid
variables could be compared. Using the luminosity together with the observed
apparent brightness of a Cepheid, its distance could be calculated using the
inverse square law. Because the period-luminosity relationship is valid
whatever the distance, astronomers now possessed a method to calculate the distance
to Cepheids throughout and beyond the galaxy.
Shapley was very
familiar with variable stars, having done considerable research into Cepheids
at
In 1920, Shapley
went on to create a distribution map of globular clusters and discovered that
their density was far from equal in all directions. There was a noticeable concentration
of clusters toward the constellation of Sagittarius. If the assumption that
globular clusters orbited the centre of the galaxy were valid, this would
indicate that the Sun was not at the centre of our galaxy. Using his model,
Shapley determined that the centre of our galaxy lay 18 kpc from the Sun, in
Sagittarius. He also dramatically increased the known size of our galaxy to a
diameter of 100 kpc. (Belkora, L. 2003). At the same time, his rival Curtis had
declared a galactic diameter of 10 kpc, considerably less, with the Sun 3 kpc
from the center (Trimble, V. 1995).
We now know that
Shapley’s distances had become greatly exaggerated by his own determination
that interstellar extinction played no effect in his measurements of globular
clusters. He had erred in the opposite direction to Kapteyn. During the mid 20th
century, several other astronomers published their own values for the Sun’s galactocentric
distance (known as Ro). In the 1920’s, Sir Harold Spencer Jones and Sir James
Jeans calculated Ro as 20 kpc; in 1930, Robert Trumpler declared Ro as 18 kpc;
Jan Oort reported Ro as 10 kpc in 1932. Ironically, Oort’s original estimate in
1927 was 6.3 kpc +/- 2kpc, very close to the accepted modern value of Ro.
Walter Baade revisited Shapley’s work with RR Lyrae stars in globular clusters,
in 1953, and arrived at a much more accurate value of Ro of 8.2 kpc (Trimble,
V. 1995).
Eisenhauer et al
(2003) have determined the distance of the Sun from the galactic centre to be 8
kpc +/- 0.4 kpc, using spectroscopic observations of the star S2 observed to
orbit Sagittarius A*, the supermassive
black hole believed to lie at the galactic centre. This is the currently
accepted value for Ro.
The globular cluster distribution model
In our attempt to
replicate Shapley’s research, the chief objective of our experiment is to
produce a distribution model of globular clusters within the Milky Way. Like
Shapley’s model, our model should clearly demonstrate an asymmetric
distribution of globular clusters, with the majority of globular clusters lying
in one particular sector of the model. We will estimate the centre of the
distribution, which will mark the location of the galactic centre. The distance
of this location from the origin of our model, (The Sun) will allow us to
calculate the Sun’s galactocentric distance.
Our model will
consist of a circular grid representing polar coordinates. To plot the location
of globular clusters within the model we require a set of polar coordinates for
each cluster in our data set. Polar coordinates consist of a radial coordinate
representing distance from the centre, and an angular coordinate relative to
zero degrees (Wolfram Research). We will obtain the Right Ascension (R.A.) of
each globular cluster from an ephemeris, and the R.A. converted to degrees will
form the angular coordinate. We must calculate the radial coordinate, or
distance to the cluster.
Calculating the cluster distance
For our experiment,
we decided to use a simple distance estimation method developed by Trumpler in
1930, which he called “Diameter Distance” (
(Equation 1) 
(Equation 2)
Diameter of calibrator cluster, 
known distance to calibrator cluster
Diameter of sample cluster, 
distance to sample cluster
Equation 1 takes the
form of an inverted ratio because distance increases as diameter decreases.
Measuring the cluster diameter
We will physically
measure each cluster diameter from optical images. A typical globular cluster
consists of a densely packed core in which individual stars cannot be resolved,
surrounded by a looser sphere of easily resolved stars (Figure 1). There are
several ways to measure the diameter of the cluster from its image:
- Estimate the furthest boundaries of the
cluster. This involves determining where the loose edges of the cluster
end and the background star field begins. In many cases, particularly the
looser globulars, this is impossible to do accurately. Additionally, many
images had too small a field of view to show the entirety of the cluster.
- Estimate the distance from the centre of
the cluster at which the star density appears reduced to an arbitrary
threshold, e.g. the distance at which the density appears to be about half
that of the core. After analyzing images it was determined this is not
constant for all clusters. Some have large, dense cores and a thin halo of
looser stars, whilst others have the appearance of an open cluster with an
even density of stars throughout the image.
- Measure the diameter of the core, i.e.
the point at which the “solid” white core breaks up into resolvable stars.
We chose method #3
as the most consistent and accurate, providing that the images meet the
following requirements:
- Images are obtained from the same source
library and are all black and white optical images.
- Images are all to the same scale and
square.
- The “solid” white core of the cluster
must be fully contained within the image.
Because equation 1
requires the ratio of diameters, the units of our measurements are unimportant;
all that is required is that the diameter of each cluster be measured in an
identical manner. For this experiment, we have chosen to load each image into
image-manipulation software and measure the number of pixels from one side of
the “solid” core to the other. Noticing that such cores are not perfectly
circular, we decided to derive an average diameter from two distinct
measurements, one vertical and one horizontal (Figure 1).
Figure 1: NGC 6205, showing
lines of measurement.
(Image copyright Digitized
Sky Survey, DSS)
The Selection of a cluster data set
We decided to use Harris, W. E. (2003) as a recent catalog of known globular clusters. Harris provides New General Catalog identification, R.A., Declination and distance from the Sun, using the J2000 coordinate system, for 150 clusters within the Milky Way. We determined that a 20% sample would form a reasonable quantity to provide a sufficient number of objects to model, yet few enough to perform calculations in a reasonable time. Harris’ database is ordered by R.A., allowing us to use a simple scatter selection, picking every fourth or fifth cluster, so as not to favour any segment of the sky.
We obtained all images from the Digitized Sky Survey (DSS), via the Space Telescope Science Institute (STSCI) web site, downloading an optical image of each cluster in our data set using the default image size of 15x15 arcminutes. All images were sourced from the POSS2/UKSTU survey, which DSS defines as “Red, all sky, 1.0 arcsec/pixel”. We chose this survey for two reasons:
- The image resolution is identical in both the northern and southern hemispheres, whereas some sky surveys were not.
- Conveniently, the measurement of cluster diameter in pixels would double as a measurement of the angular diameter in arcsecs.
After an initial analysis of the images in our data set, we rejected 14 candidates on the basis that they had no measurable “solid” centre (Figure 2). We selected and downloaded alternatives from the Harris catalog until we had obtained a suitable data set of 34 images.
Figure 2:
Unsuitable Images
Left: NGC 288
Right: NGC 6809
(Images copyright Digitized
Sky Survey, DSS)
We chose NGC 6205 as the calibrator cluster for our data set. We selected this cluster visually because the diameter of its “solid” core was of average size compared to the smallest and largest clusters in the data set.
Experimental assumptions and approximations
The experimental procedure used in this project makes a
number of assumptions and each step is subject to some error and approximation.
The intent of the experiment is to arrive at a result for the Sun’s
galactocentric distance that is within 100% error of the accepted distance of 8 kpc +/- 0.4 kpc. Shapley himself was in
error by greater than 100%, largely due to his decision not to allow for
interstellar extinction.
Approximations and
assumptions in this experiment include:
The major assumption is the one made by Trumpler in 1930, by
adopting the rule “smaller is further”. This assumes that all globular clusters
are the same size, when in reality they differ widely, from less than 10
arcminutes across to greater than 50 arcminutes (Wilson, B. 2003). If our data
set contains smaller than average clusters that are close to the Sun, or very
large clusters that are very distant, then our model will be inaccurate. We
propose that multiple excessive differences from the norm would be required to alter
significantly the accuracy of this experiment. Trumpler minimized this error by
dividing clusters into several groups of similar density and core size (
A further assumption made is that the central “solid” core of a globular cluster is representative of its size. If tiny clusters exist with huge and very dense cores, or large, disperse clusters possess negligible cores, then our core diameter measurements would be atypical and our distance calculations would be dramatically skewed.
Our measurement of core diameters is also subject to approximation errors. When examined closely, the cores are imperfect circles with fuzzy edges, and determining the edge of the core will require careful analysis. We expect to encounter up to a 10% error in this stage of the experiment.
Other approximation errors lie in the physical construction of our model. Distances (radial coordinate) are rounded to the nearest 100 parsecs. The angular coordinate suffers two stages of compound error: The Right Ascension taken from the Harris catalog will be rounded to the nearest degree when converted to the angular coordinate. As will be shown later, the small size of our model introduces a +/- 3º error when plotting within 10 kpc, and a +/- 1º error plotting out to 30 kpc.
Experimental Procedure
The first step was a careful measurement of the core diameter of each cluster in our data set, measuring once vertically and once horizontally, and then averaging the values. This data is presented in Table 1 below.
The next step involved applying equation 2 to calculate the distance to each cluster in our data set. Our calibrator cluster, NGC 6205, was determined to have an average core diameter of 150 and a distance of 7700 pc taken from the Harris catalog.
Example 1: NGC 362 appears smaller than NGC 6205.
Its core diameter was measured as 88.
Using
equation 2: 
Because NGC 362 is smaller, it is considered further than NGC 6205.
Taken from Harris, the actual distance is 8500 pc, a 54% error.
Example 2: NGC 104 appears larger than NGC 6205.
Its core diameter was measured as 194.
Using
equation 2: 
Because NGC 104 is larger, it is considered closer than NGC 6205.
Taken from Harris, the actual distance is 4500 pc, a 33% error.
Table 1 lists the calculated distance to each cluster, the known distance according to Harris and our percentage error.
Table 1: - Core diameter measurements and galactocentric distances
|
Cluster |
X Diameter |
Y Diameter |
Avg. Diameter |
Calculated distance (pc) |
Known distance (pc) |
% error |
|
NGC 6205 |
152 |
148 |
150 |
- |
7700 |
- |
|
NGC 104 |
198 |
190 |
194 |
6000 |
4500 |
+ 33 |
|
NGC 362 |
84 |
92 |
88 |
13100 |
8500 |
+ 54 |
|
NGC 1851 |
86 |
88 |
87 |
13300 |
12100 |
+ 10 |
|
NGC 2298 |
28 |
32 |
30 |
38500 |
10700 |
+ 260 |
|
NGC 2808 |
152 |
162 |
157 |
7400 |
9600 |
- 23 |
|
NGC 4147 |
34 |
38 |
36 |
32100 |
19300 |
+ 66 |
|
NGC 4833 |
128 |
96 |
112 |
10300 |
6500 |
+ 58 |
|
NGC 5139 |
462 |
440 |
451 |
2600 |
5300 |
- 51 |
|
NGC 5634 |
62 |
62 |
62 |
18600 |
25200 |
- 26 |
|
NGC 5824 |
44 |
42 |
43 |
26900 |
32000 |
- 16 |
|
NGC 5904 |
142 |
150 |
146 |
7900 |
7500 |
+ 5 |
|
NGC 5946 |
36 |
36 |
36 |
32100 |
10600 |
+ 203 |
|
NGC 6139 |
54 |
56 |
55 |
21000 |
10100 |
+ 108 |
|
NGC 6254 |
140 |
140 |
140 |
8250 |
4400 |
+ 88 |
|
NGC 6273 |
96 |
112 |
104 |
11100 |
8600 |
+ 29 |
|
NGC 6304 |
46 |
48 |
47 |
24600 |
6000 |
+ 310 |
|
NGC 6333 |
70 |
82 |
76 |
15200 |
7900 |
+ 92 |
|
NGC 6356 |
80 |
72 |
76 |
15200 |
15200 |
0 |
|
NGC 6388 |
118 |
116 |
117 |
9800 |
10000 |
- 2 |
|
NGC 6402 |
134 |
126 |
130 |
8900 |
9300 |
- 4 |
|
NGC 6440 |
54 |
62 |
58 |
20000 |
8400 |
+ 138 |
|
NGC 6441 |
106 |
104 |
105 |
11000 |
11700 |
- 6 |
|
NGC 6517 |
48 |
52 |
50 |
23100 |
10800 |
+ 114 |
|
NGC 6541 |
118 |
110 |
114 |
10100 |
7000 |
+ 44 |
|
NGC 6569 |
60 |
56 |
58 |
20000 |
10700 |
+ 87 |
|
NGC 6626 |
90 |
88 |
89 |
13000 |
5600 |
+ 132 |
|
NGC 6652 |
58 |
52 |
55 |
21000 |
10100 |
+ 108 |
|
NGC 6681 |
76 |
72 |
74 |
15600 |
9000 |
+ 73 |
|
NGC 6723 |
118 |
108 |
113 |
10200 |
8700 |
+ 17 |
|
NGC 6779 |
84 |
80 |
82 |
14000 |
10100 |
+ 39 |
|
NGC 6864 |
64 |
66 |
65 |
17800 |
20700 |
- 14 |
|
NGC 7078 |
152 |
152 |
152 |
7600 |
10300 |
- 26 |
|
NGC 7099 |
88 |
82 |
85 |
13600 |
8000 |
+ 70 |
For simplicity in presentation, we elected to construct a two-dimensional model. To achieve this it was necessary for us to reduce the straight-line distances listed in Table 1 into two dimensions (referred to hereafter as compensated distance); otherwise, our distances would be in error. As Figure 3 shows, the amount of error is proportional to the declination of the cluster.
d0 = straight-line
distance
d1 = compensated distance
Figure 3a: Small declination Figure
3b: Large declination
Using basic trigonometry: d1 = d0 cos dec. (Equation 3)
Example 1: NGC 362 d1 = 13100 cos 71
d1 = 4300 pc
Example 2: NGC 104 d1 = 6000 cos 72
d1 = 1900 pc
Table 2 lists the Right Ascension and Declination of each cluster, taken from Harris, and the compensated distances calculated via equation 3.
Table 2: - Polar coordinates and compensated distances
|
Cluster |
Straight-line distance (pc) |
R.A. (hr,
min) (nearest minute) |
Angular coord. (nearest degree) |
Declination (nearest degree) |
Compensated distance (pc) |
|
NGC 6205 |
7700 |
16 42 |
251 |
+36 |
6200 |
|
NGC 104 |
6000 |
00 24 |
6 |
-72 |
1900 |
|
NGC 362 |
13100 |
01 03 |
16 |
-71 |
4300 |
|
NGC 1851 |
13300 |
05 14 |
79 |
-40 |
10200 |
|
NGC 2298 |
38500 |
06 49 |
102 |
-36 |
31100 |
|
NGC 2808 |
7400 |
09 12 |
138 |
-65 |
3100 |
|
NGC 4147 |
32100 |
12 10 |
183 |
+19 |
30400 |
|
NGC 4833 |
10300 |
13 00 |
195 |
-71 |
3400 |
|
NGC 5139 |
2600 |
13 27 |
202 |
-47 |
1800 |
|
NGC 5634 |
18600 |
14 30 |
218 |
-6 |
18500 |
|
NGC 5824 |
26900 |
15 04 |
226 |
-33 |
22600 |
|
NGC 5904 |
7900 |
15 19 |
230 |
+2 |
7900 |
|
NGC 5946 |
21100 |
15 35 |
234 |
-51 |
24200 |
|
NGC 6139 |
21000 |
16 28 |
247 |
-39 |
16300 |
|
NGC 6254 |
8300 |
16 57 |
254 |
-4 |
8300 |
|
NGC 6273 |
11100 |
17 03 |
256 |
-26 |
10000 |
|
NGC 6304 |
24600 |
17 15 |
259 |
-29 |
21600 |
|
NGC 6333 |
15200 |
17 19 |
260 |
-19 |
14400 |
|
NGC 6356 |
15200 |
17 24 |
261 |
-18 |
14500 |
|
NGC 6388 |
9800 |
17 36 |
264 |
-45 |
7000 |
|
NGC 6402 |
8900 |
17 38 |
265 |
-3 |
8900 |
|
NGC 6440 |
20000 |
17 49 |
267 |
-20 |
18800 |
|
NGC 6441 |
11000 |
17 50 |
268 |
-37 |
8800 |
|
NGC 6517 |
23100 |
18 02 |
271 |
-9 |
22800 |
|
NGC 6541 |
10100 |
18 08 |
272 |
-44 |
7300 |
|
NGC 6569 |
20000 |
18 14 |
274 |
-32 |
17000 |
|
NGC 6626 |
13000 |
18 25 |
276 |
-25 |
11800 |
|
NGC 6652 |
21000 |
18 36 |
279 |
-33 |
17600 |
|
NGC 6681 |
15600 |
18 43 |
281 |
-32 |
13200 |
|
NGC 6723 |
10200 |
19 00 |
285 |
-37 |
8100 |
|
NGC 6779 |
14000 |
19 17 |
289 |
+30 |
12100 |
|
NGC 6864 |
17800 |
20 06 |
302 |
-22 |
16500 |
|
NGC 7078 |
7600 |
21 30 |
323 |
+12 |
7400 |
|
NGC 7099 |
13600 |
21 40 |
325 |
-23 |
12500 |
The final step was to construct the distribution model. To do so we plotted compensated distance against angular coordinate for each cluster, on a polar coordinate grid (Figure 4). The centre of the grid represents the Sun and each circle represents a compensated distance of 10 kpc, 20 kpc and 30 kpc respectively.
Figure 4: Distribution model
Final Analysis
From our distribution model (Figure 4), all that remained
was to estimate the centre of the plotted clusters. To do so we visually
identified a circle that encompassed the majority of the clusters, allowing us
to identify the centre of the distribution and hence the location of the galactic
centre. We determined that the galactic centre lay approximately 10.5 kpc from
the Sun with an angular coordinate of 272º, corresponding to a Right Ascension
of 18h 02m. This compares favourably with the accepted value of Ro of 8 kpc (Eisenhauer et al 2003), giving an
error of 31%. Using the object Sagittarius A*, the RA of the galactic centre is
believed to be 17h 45m 40s (DSS).
It is visually obvious
from our model that there is a bias of clusters in the range of R.A. 17h to
19h, or the constellation Sagittarius; a pattern seen by the pioneers of
galactic modeling and that used by Shapley to determine that the galactic
centre did not coincide with the position of the Sun.
Experimental errors
and challenges
As an indicator of the accuracy of our initial distance calculations, we calculated the margin of error, presented in the rightmost column of Table 1. The average error was 72%, within our target of 100% and only 25% of our values exceeded 100% error. Given the broad assumptions and estimations involved in the experiment, we believe these errors are acceptable.
After completion of the experiment, we attempted to determine the cause of our over-estimation of Ro. We believe the major factor lay in the selection of our initial data set. The majority of clusters in our data set were smaller than our calibrator (NGC 6205). Since we based our experiment upon the principle of “smaller is further”, a predominance of smaller clusters could lead to over-estimation of the distance to most clusters in the data set, a hypothesis validated by the data in Table 1. This could cause the large value for Ro that we arrived at. We should have been able to obtain a more accurate result had we ensured that the core diameter of our calibrator cluster represented the mean of all the clusters in our data set, rather than simply the smallest and largest.
Other ways that we could have increased the accuracy of our model include using a significantly larger grid to plot upon, reducing the error of +/- 3º plotting angular coordinates at the 10 kpc range; as well as a finer set of distance rings, representing each 5 kpc, or even each 1 kpc. Whilst this would have little effect on our final value of Ro, it might have produced a more accurate R.A. value.
Another alternative would have been to replace the physical model with a mathematical one. This would have obviated the need to collapse our straight-line distances into two-dimensions and would have allowed our entire model to be accurate to 100 pc and 1º of angular coordinate. Estimating the centre of a 3-dimensional mathematical model would have required detailed computer modeling that we considered outside the scope of this simple experiment. We consider the increased accuracy of a computer model to be less significant compared to the assumptions involved in the “smaller is further” principle, and the approximation errors measuring core diameters.
Conclusion
Shapley’s experiment was straightforward to duplicate, especially since we substituted the measurement of core diameters from DSS images for Shapley’s more involved analysis of RR Lyrae variables. Although the experiment made large assumptions and numerous approximations, the result obtained (10.5 kpc) had very little error and was in fact more accurate than Shapley’s original calculation of 18 kpc. Because of this, we believe the experiment to be successful.
Appendix B - References
Belkora, L. 2003, “Minding the Heavens”
Digitized Sky Survey (DSS).
Space Telescope Science Institute (STSCI)
http://stdatu.stsci.edu/dss/index.html
Eisenhauer, F., Schoedel, R.
et al 2003 “A geometric determination of
the distance to the galactic center”,
Astrophysics Journal 597
Freedman, R., Kaufmann, W.
2001, “Universe Sixth Edition”
Harris, W. E. 1996, “Catalog of parameters for Milky Way globular
clusters”, AJ, 112, 1487
Harris, W. E. 2003, “Catalog of parameters for Milky Way globular
clusters: The Database” http://physwww.physics.mcmaster.ca/%7Eharris/mwgc.dat
Trimble, V. 1995, “The 1920 Shapley-Curtis Discussion:
Background, issues and outcome”
http://spiff.rit.edu/classes/phys230/lectures/ism_dust/ism_dust.html
Wolfram Research: Math World:
http://mathworld.wolfram.com/PolarCoordinates.html
Appendix C - Supplementary Reference Material
Encyclopedia of Astronomy
& Astrophysics 2000, “Shapley, Harlow (1885–1972)”
http://oasis.harvard.edu/html/hua03998.html
Lengyel, D., “The Distance to the galactic center”
http://junior.apk.net/~arstar50/galacticcenter.html
Trumpler, R. J. 1930, “Globular Star Clusters”, Astronomical
Society of the Pacific