Virgo Cluster of Galaxies and Dark Matter

Week 17 Exercise:  Virgo Cluster of Galaxies and Dark Matter

Astronomy 25 Section 60490

May 7, 2024

Purpose(s) of Lab: 

To use the Radial Velocities of Galaxies in the Virgo Cluster to find evidence for Dark Matter

Apparatus used in the Exercise:

You will need to use several tools to calculate the radial velocity of Galaxies in the Virgo Cluster from tabulated data.  You will ‘bin’ the data into ‘Virgo Centric Angular Distances’ and then plot the data on graphing paper.  You will need to use the materials provided along with:

1. scissors,
2.  a calculator,
3. a thumb pin or similar pins to keep the ‘orientation of the paper correctly,
4.  a small piece of cardboard or other material to push the thumb tack into,  
5. and graphing paper (or use of a plotting program, like Excel) – I’ll bring some to class

Background:

The radial velocities of galaxies in comparison to the Earth can be measured using the Doppler Effect on emission lines of stars within the Galaxies, most likely the Hydrogen Balmer absorption lines of stars in the Galaxy.  The average radial velocity of the entire Virgo Cluster can be measured by averaging these velocities.

While the orbits of the individual Galaxies in the cluster is very complicated, if we assume that they are all ‘bound’ to the cluster by the combined Gravity of all the Galaxies and any other mass within the cluster then we can estimate that there is a relationship similar to

Velocity² ~ Mass of the Entire Cluster/(Radius of Cluster)²    or

                                       V²  = Constant x M_Virgo/R_Virgo                                                              Eq 1

If the Galaxies were all in circular orbits then the Constant = 1.  If the Galaxies are in highly elliptical orbits, almost unbound by Gravity,  then the Constant ~ 2.  Either way, we can estimate the Mass of the Virgo cluster by manipulating the equation

                                    M_Virgo  =   V² x R_Virgo/Constant                                        Eq. 2

For simplicity, let’s assume Constant =1 and we won’t be off by much.

So if we pick a few Galaxies in the Cluster near the outer edge of the Cluster and measure their Radius from the center of the Cluster, then we can estimate M_Virgo.  This will be one of your tasks.

But there are complications since we are not measuring the ‘total’ velocity when we use the Doppler Effect but just the radial velocity.  So unless we accidently pick a Galaxy moving either only radial away or towards us, we will underestimate V².

So, we have to do this calculation for many of the Galaxies in the cluster and assume the Galaxy with the HIGHEST V² is actually moving only radial away or towards us (no transverse motion).  So we need to calculate M_Virgo for all the galaxies and use the maximum value of M_Virgo .

Lastly, Nature threw us a curve.  If most of the mass was all near the core of the Cluster, where most of the Galaxies reside, then M_Virgo would peak at some radius and remain constant afterwards.  This is what happens in the Solar System.  But does that happen here?

To test this assumption, we need to tabulate M_Virgo as a function of Radius from the Core of the Cluster.  If M_Virgo  is not a constant with ‘Virgo Centric Radius’ then something else must be creating the Gravity.  Can you prove that it exists?

Task 1:  You are given a map of the Virgo cluster with the location and name of each Galaxy in the Cluster. 

I will give you the location of the Center (you could do that by averaging all the RAs and Dec of the objects!) of the Virgo Cluster.  You then use a thumbtack on a piece of cardboard to place the map on the board with the pin at the Cluster center.

I also supply an image of circular rings drawn on it depicting radial ‘Rings’.  You’ll need one sheet but print two in case of oops!  You will use this sheet and the scissors to create ‘circles’ of varying ‘Virgo Centric Radii’ to mask the inner Galaxies.

You’ll need to tabulate your results as you measure the ‘locations’ of each Galaxy and then look up its radial velocity.  Task 1 only asks you to locate the Galaxies outside of the  largest ‘Ring’ radius.  Those galaxies are in Ring 7.  We assume their radius is always = 7.

The Table of Galaxies gives their names and the Radial velocities of each Galaxy.  I’ll also give you the Radial Velocity of the ‘Center’ of Virgo Cluster, V_Virgo.  This velocity is actually a combination of the expansion of the Universe due to the Big Bang and the local velocity of the Milky Way that is ‘falling’ into the Gravity of the Virgo Cluster (albeit slowly compared the Big Bang motion).

To compute V^2, look up the radial velocity, subtract from it the ‘V_Virgo’ (nevermind if the result is negative or positive), and then square it using a calculator.  To make the result easier to deal with, divide it by 10,000.  The result is V²/10,000for that galaxy.

Lastly, compare all the galaxies you find in the ‘outer ring’ values of V² and find the MAXIMUM value, V²_MAX.  This Galaxy will be closest to a circular orbit where Eqs 1 and 2 will be most accurate.  Write down the maximum value you find as V²_MAX on the graph paper you submit in Task 2.

Task 2:  Now that you figured out how to do this, keep doing it!  

But you’ll be doing the same calculation in smaller and smaller radius ‘Rings’.

Start by cutting the mask down to the next circle.  Place the mask back on the map and ‘closer to the center’ Galaxies are now visible.  Being careful not to recount a Galaxy further out (you can overwrite the names of the galaxies you already have measured in pencil on the map), compute V² using the same process as before:

1. Read off the Radial Velocity of the Galaxy
2. Subtract the V_Virgo  velocity
3. Square the result
4. Divide by 10,000 to make the result manageable
5. New Step!  Multiply the V² by the Radius of the Ring (e.g., 7, 6, 5, 4…)
6. Tabulate the result in your results as V²R

Do this also for the Galaxies that you did in the outer mask for Task 1

Once you have completed about half the Galaxies in the list (or going inward to the 3 or 4^th ring – I will tell you in class beforehand how far to proceed) you can complete this portion of the Exercise. 

Adding all the Galaxies in the cluster will get you 2 more pts of extra credit!

Task 3:  Plot your results as Plot 1

Using a sheet of graph paper place a mark on a graph of V²R versus R.  The horizontal axis should use R and the vertical axis V²R.

When you start placing each Galaxy’s numbers on this graph you should start to see a pattern emerge.  Since most galaxies radial velocity will be less than its ‘total velocity’ (radial velocity combined with transverse velocity), at any given radius galaxies will be distributed across a vertical range of V²R.  But the maximum V²R in the range at ANY radius will give us an estimate of the MASS interior to that Radius.

Task 4:  Compare to the Total Number of Galaxies in each ‘Ring. 

For each Ring you used, count up the total number of Galaxies in each Ring.  Plot the results of the ‘Ring number’ versus the total number of galaxies in each ring.  The horizontal axis should us R and the vertical axis the total number of galaxies in that ‘Ring’.

If you assume the galaxies are all about the same Mass, and THERE’S NO OTHER MASS IN THE CLUSTER,  than Plot 2 should be telling us about the total amount of VISIBLE  matter in each ring.

Now compare the results of PLOT 1 with PLOT 2.  What do you see?  What does this tell you about the distribution of mass in the cluster?  Is there ‘missing mass’ not seen in the galaxies?

Write any observations you see on Plot 2.  Consider the statements below…

If V²R is ‘flat’, i.e., no change with R, then the Mass of the Cluster is concentrated at the Center

If V²R increases with larger R, then there is more unseen mass in the Cluster, i.e., Dark Matter exists

If V²R does something else….

What You Need to Turn in For This Exercise:

1. A table of the values you computed.  I give you a Word doc and a PDF to be filled in with the numbers you calculate.
2. On that table of values note the result of Task 1 (the maximum value of V² x R that you computed.
3. A graph – Plot 1 – of the values in the table – V² x R plotted against R (the Ring number)
4. A second graph – Plot 2 – of the number of galaxies in each Ring plotted against R (the Ring number) with your observational comments on the results.   You can use any tool to graph the data including good old graph paper, Excel, or free hand, if you have to!