RASC Calgary Centre - Galaxy Centre
Galaxy Centre
by Jason Nishyama
Page last updated November 5, 2018
Science from your armchair
In the article on quantitative observing, I described a project where an amateur could
measure the period and orbital radii of Jupiter's moons. This required some simple observations to
gather the data. If standing around in the cold for hours at a time isn't your thing, there is
another way you can do some quantitative astronomy work without leaving the comfort of your own
living room.
In this modern age the instrumentation at professional observatories is capable of providing
considerably more data than can be reduced by any particular researcher. Much of this data is
on-line and available to anyone with internet access. This means much can be done by the arm chair
astronomer from the warmth of whatever room in their house their computer is located. In this
example project we'll look at determining the distance and direction to the centre of our
Galaxy.
Surprisingly the distance our Sun is from the Centre of the Galaxy is still not known to great
accuracy. Many methods have been used to determine the distance to the Galactic centre, with this
"observing" project you'll be using a method similar to that used by Harlow Shapley and others to
work out the distance to the centre of the Galaxy.
The method is actually quite straight forward given the assumption that the Galactic globular
clusters orbit the dynamic centre of the Galaxy. This means if you can determine the dynamic
centre of the system of globular clusters, you have determined the dynamic centre of the Galaxy.
To do this you need to know the direction and distance to each cluster.
The method you choose to determine the distance to each cluster depends on how much work you want
to do. The easy method is to simply pull the Sun-cluster distance off the Catalog of Parameters
for Milky Way Globular Clusters located at http://physwww.physics.mcmaster.ca/~harris/mwgc.dat.
This catalogue also provides the RA and Dec for each cluster as well, you'll need that too. Now
the astute amongst you will note that what you are about to work out is available in the catalogue
but no cheating because where would the fun be in that...
If you want to do a little more work, a first order approximation as to distance can be determined
by making an assumption that the clusters are all roughly the same size (not exactly the case, but
will work for a rough approximation). With this assumption, the apparent diameter of the cluster
is dependent on its distance. To work out the diameter you can use the images available from the
The STScI Digitized Sky Survey available at
http://archive.stsci.edu/cgi-bin/dss_form?target=ngc2808&resolver=SIMBAD. For this you want to
use the red plates from either the POSSI or POSSII surveys. Globular clusters tend to be made up
of older Population II stars so they are easier to see in the red plates. You want to record the
plate scale that you selected and probably want to save the image as a GIF and not in FITS format.
From the plate scale (angular hight and width of the plate divided by the number of pixels high
and wide the image is) you can work out the diameter of each cluster. Again, the astute observer
will note that further down in the Catalog of Parameters for Milky Way Globular Clusters the
half-light diameter is given so you can do this either way, depending on how much work you would
like to do.
To work out the distance, pick one cluster to use as your benchmark. This will be your reference
cluster, dref. The distance to this cluster is defined as 1. When selecting this reference cluster
it is important to select one for which the distance is well known such as NGC 5139. All other
cluster distances will be in terms of the distance to this cluster and can be determined by
dividing the reference cluster diameter by the test cluster diameter or r=dref/dcluster. The
actual distance to each cluster can then be found by multiplying the r value determined for each
individual cluster by the distance to the reference cluster.
So at this point you should have a list of Galactic globular clusters and their distances from
Earth either from the catalogue or by working them out. You will need to convert these distances,
using the clusters' right ascensions and declinations into a model of the clusters in Cartesian
space. If one defines the positive X axis pointing to the vernal equinox, the positive Z axis to
the North Celestial Pole and the positive Y axis 90 degrees counter-clock wise from the positive X
axis and uses the Sun as the origin, the following formula will do the conversion:
Rx = R cos RA
Ry = R sin RA
Rz = R sin Dec
This will give you an (x,y,z) coordinate for each cluster. It is important to note two things at
this point. First you will have to convert right ascension to degrees (1 hour of RA=15 degrees) to
use the formulae above. Second if you are doing this in a spreadsheet such as Excel or
OpenOffice.org Calc also note that the sine and cosine functions of these programs require the
angle to be in radians and NOT degrees. So multiply the degrees by pi/180 to do this
conversion.
Now that you have a lovely model of the positions of the Galactic globular clusters you can work
out the dynamic centre of these clusters using Formula 1 below on each axis individually:
Where R is the distance to the dynamic centre on the axis you are calculating. The sigma indicates
"sum of", so the formula has the total sum of each cluster's mass times it's axis coordinate,
divided by the total sum of each individual cluster's mass. Since the assumption that you used to
determine the distance was that each cluster is approximately the same diameter, you can make the
assumption that they are also approximately the same mass. This is a safe assumption since the
mass of the Galaxy far exceeds the cluster masses. This allows us to reduce Formula 1 to Formula
2:
Where we are now adding up the total of each coordinate and dividing by the number of clusters.
This basically means that by taking the average of each of the x, y, and z coordinates, we can
then find the coordinates in our system for the Galactic centre. Looking at the resultant x, y and
z coordinates as a vector allows us to determine the distance and direction to the Galactic centre
by using the Pythagorean theorem for the distance and by converting the x, y, and z coordinates
back to equatorial coordinates for the direction from Earth.
At this point you have now determined the distance and direction to the Galactic centre and you
didn't even have to get cold to do it.
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