The goal of this project is to produce visuzalizations, 3 Dimensional if possible) of the distribution of galaxies in nearby space. There are two parts of this project, the first is the study of just the Local Group of galaxies, and the second is a somewhat larger volume --- still local, however, to enable us to see how the Local Group fits into our view of the very nearby Universe. This volume studied should be a sphere or cube of diameter 400 million light years (about 13 Megaparsecs) which means we need to collect all the galaxies within about 7 Megaparsecs of the Milky Way. Ideally we would code the software/data used for this project so as to be able to extend the plots arbitrarily as additional data sets are created.

Input dataset: lowz.dat a file containing all the low redshift galaxies (and perhaps other things!) in the CfA Redshift catalog.

You should:

Software for simple mapping and visualization has been produced by several different organizations and companies; one well known commercial variant is Mathematica (Wolfram & Co., there is a student license available through Harvard ITS), and a public software package called ``Partiview'' has been written by Stuart Levy of the National Center for Supercomputing Applications (NCSA) and is distributed by the American Museum of Natural History & the Hayden Planetarium as part of the Digital Universe Project. Partiview and its documentation are available at:

and is available in versions for Windows, Mac OS-X and Linux.

Some examples of local maps include:

A 2-D plot (very crude) looking down on the LG, missing lots!)

And quasi 3-D plots of the Local Group can be found at:

2-D sky plots of the local supercluster: http://astrosun.tn.cornell.edu/courses/astro201/lsc.htm http://www.cfa.harvard.edu/~huchra/zcat/local.gif

that last was produced by the map program in your directory,

Many of these are just for the Local Group. There are a couple of books and print atlases, Fairall's "Large Scale Strcture" (perhaps too complicated) and Tully & Fisher's "Nearby Galaxy Atlas" that are worth looking at.

Larger scale maps (beyond just the local supercluster) can be found at:

Some of these have plots that might be of interest for figuring out what you might want to produce.

Here are additional links to data for the Local Group:

Tables of local group members:

Images of individual galaxies can be found either just by Googleing them and using Google's image search (works pretty well for named galaxies) or by going to the NASA/IPAC extragalactic database:

By definition, the Local Group is getting larger all the time --- not necessarily in size but definitely in numbers of galaxies. In the last few years, several new members have been discovered and should be added to the plots. If you can find them, we'll add them to the list of members for the course.

After you have identified all the known members of the Local Group, determine the mass and the integrated luminosity of the group --- and thus its Mass-to-Light ratio. Remember that the Locl Group is actually only those galaxies at the center of the larger, "Nearby Universe" map you have made above. Most LG galaxies are inside a radius of 1 Megaparsec from the Milky Way. See the tables of Local Group members given above. The mass of a group or cluster can be calculated two ways, one by applying the Virial estimator and one by applying the Projected Mass estimator. I've written a more complete description of these two estimators and of the Virial Theorem from which the first is derived which can be viewed in the following power-point file: Mass Estimators. Both give you general formulae to use for either of these calculations, if you can determine the group center, the separations of the galaxies from the center and the velocities of the galaxies with respect to the center.

The ** Virial Mass Estimator **, after corrections for projection effects
both in the projected radius (to go from 2-D to 3-D, and
in the radial velocity (to go from 1-D to 3-D), is:

where R_{ij} is the separation between the ith and jth particle (the limitation
i < j is meant to eliminate double counting) and v_{i} is the velocity
if the ith galaxy w.r.t. the average cluster velocity, V_{C}. I.e.
v_{i} = v_{k} - V_{C}, where v_{k} is the velocity
of the kth galaxy in the sample.

The average cluster velocity V_{C} can be defined in several ways, the most
common is to just average all the velocities of known cluster members, the second
most common is to weight the velocities in the average by the luminosities of the galaxies.
Here N is the number of galaxies in the sample (in the cluster).

where L_{k} is the * Luminosity * of the kth galaxy, and is usually expressed in
terms of Solar Luminosities.

Note that the luminosity of a galaxy (or anything else for that matter) is usually calculated from its absolute magnitude and will depend on the bandpass or color it has been measured in, e.g. blue or visial or whatever. The absolute magnitude, M, is calculated from the apparent magnitude, m, (but for this Local Group Exercise it is already given) and the distance, D, to the object.

For this exercise, again, use only objects in the Local Group table with measured distances.

Now magnitudes are logarithmic quantities so do not "add" --- luminosities add. Absolute magnitudes can be converted to luminosities in solar units using the known absolute magnitude of the Sun in the appropriate bandpass. For blue, visual or infrared (B, V or K) magnitudes, the absolute magnitude of the Sun is:

B_{Sun} = +5.48

V_{Sun} = +4.83

K_{Sun} = +3.28

so, for V band (visual) magnitiudes, the Luminosity of any galaxy is given by

As an example, the absolute magnitude of the Milky Way is given in the table as
M_{V} = -20.6. Thus its luminosity in Solar units is 1.5 x 10^{10} L_{Sun}.

If you calculate both the weighted and unweighted mean "velocity" of the Local Group, how different are those numbers?

The mass you calculate will depend on the units used for the constants and the parameters.
You need to be consistent. If G is given in the cgs system, i.e.
G = 6.67259x10^{-8} cm^{3} gram^{-1} s^{-2}, then
v and R should be expressed in those same units, cm/s and cm, and the output mass
will then be in grams. Ditto for the MKS or SI units based on meters/kilograms/seconds.
To convert to Solar masses, remember that the mass of the Sun
= 1.989x10^{33} grams = 1.989x10^{30} kg. It has also been pointed out
by a group of students that you can work in "natural" units where
G = 4.3x10^{-9} km^{2} Mpc M_{Sun}^{-1} s^{-2}
and directly use separations in Megaparsecs and velocities in km/s rather than
converting all units to either cgs or mks! Then your estimated mass will be in Solar masses.

The **Projected Mass Estimator ** of Heisler, Bahcall & Tremaine is:

where R_{i,c} is the projected separation between the galaxy and the
cluster center, and v_{i} is as above. The projection factor, f_{p},
depends on the average orbital eccentricity of the system,

How different are the PM and Virial estimates of the Local Group's mass?

Finally, a key number in extragalactic astronomy and cosmology is the Mass-to-Light ratio (M/L) for a given object or system of objects. What, in Solar Units again, is the M/L of the Local Group?

Copyright John P. Huchra <huchra@cfa.harvard.edu> 2009