## Virial Theorem

### Question 1

Can the Virial Theorem be used to calculate the total mass of a Globular Cluster and the mass of the individual stars?

### Question 2

Can the Virial Theorem be used to calculate the total mass of a Galaxy and the mass of the individual stars (objects)?

### Question 3

Can the Virial Theorem be used to calculate the the total mass of a Galaxy Cluster like the Virgo Cluster and the mass of the individual galaxies?

### Question 4

What is the relation between the Virial Theorem and the dark matter issue? Is it solved ?

### Background

The Virial Theorem is based on two concepts:
The Total Kinetic Energy KE and the Total Potential Energy PE
And the Virial Theorem states that: KE = -0.5 * PE

For more detail go to: http://math.ucr.edu/home/baez/virial.html by John Baez
In fact this documents states: "it (The Virial Theorem) is really the reason we think that dark matter exists." which is the real reason behind the first three questions.

In order to answer the three questions we need the theorem in a different form like:
Mtot = 2 * Rtot * v^2 / G.
and: M = Mtot / N
For an explanation See: http://astrosun2.astro.cornell.edu/academics/courses/astro201/vt.htm
With:
• Rtot (or Reff) = the Radius of system studied
• v = the average speed of the objects studied
• N = the number of Objects of the system studied
• M tot = the total mass of the system studied
• M is the mass of the individual objects.

### Simulation Programs of the Virial Theorem

In order to demonstrate the virail Theorem two simulation programs are supplied:
one written in Quick Basic called virialth.bas
and a program in Visual Basic called VB VirialTh
1. For a copy of the source of the QB program select: VIRIALTH.BAS.
2. For a listing of the QB program select: virialth.HTM.
3. To execute the QB and VB programs select: virialth.ZIP.
4. For an explanation of the QB program See: progrm21.htm
5. For an explanation of the VB program See: VB Virialth x.htm
6. For Implementation details Select: Implementation

### Answer question 1

In order to calculate the total mass of a Star Cluster you must know the average speed of all the stars and the total size of the Star Cluster
The total size is not so difficult to observe. Reff is half that size.
The speed of each star is relativ "easy" to observe in the radial direction after a time dt and with doppler shift in the axial direction.
If that is done for all the stars (moving randomly) you can calculate an average speed v.
The real Question to answer is if the Virial Theorem a good candidate to estimate the total mass of the whole Cluster. Based on the results of the simulation I have my doubts because the answer can be wrong with a factor of five.
Assuming that the total mass is correct the average mass of the individual stars can easily be calculated. If this also a good estimate for each of the stars depents if all the stars are identical.

### Answer question 2

There are broadly speaking two types of Galaxies: Elliptical Galaxies and Spiral Galaxies.
Elliptical Galaxies are more or less the same as Star Cluster but much larger and much more stars are involved. This makes the calculation specific of the average speed of all the individual stars much less reliable. As such becomes the calculation of the total mass a rather big guess (or in dutch a wet finger approach).
In order to calculate the average mass of each stars you must count all the stars and that is not so easy
And finally to claim that that value is a good estimate for each individual stars is an overstatement, because there are stars in all sizes and masses.

Spiral Galaxiesdon't consists of stars which move randomly, which is one precondition to use the Virial Theorem, but to a large extend in circles. This immediate makes the Virial Theorem not a reliable candidate.

### Answer question 3

To a large extend a Galaxy Cluster can be compared with an Elliptical Galaxy however the problems are more severe.
First of all it is very difficult to decide who belongs to the cluster and to calculate the total size of the cluster. The problem is that number is probably not a very good number to be used in the formula to calculate mtot. Most probably that value is too large. But what is the correct number?
To calculate the average speed is may be not so much of a problem and the same with the total mass.
I expect that the biggest problem is too claim that all galaxies in the cluster are identical in mass and that each individual mass is equal to the total mass divided by n. This is IMO sheer guesswork.
Most probably the smallest Galaxies have the largests speeds and this has to be taken into account. Unfortunate The Virial Theorem does not do that.

### Answer question 4

IMO Because of the problems in answering question 3 can the Virial Theorem be used to solve the dark matter problem. Nor can the Virial Theorem be used to calculate the amount of dark matter in Galaxies which belong to a Cluster. Nor can the Virial Theorem be used to prove that there is at least some form of dark matter involved in Galaxies

### Reflection

The Local Cluster contains two large Galaxies: The Andromeda Galaxy (M31) and The Milky Way.
• First the movement of those two predominant Galaxies is "slow" and towards each other i.e. not random.
• Secondly each galaxy is surounded by a cloud of smaller galaxies which move relatif fast"
As such there are two reasons why the Virial Theory can not be used to calculate the total mass of the Local Cluster and the mass of the individual galaxies
If this is true, for which Cluster can the Virial Theory than be used reliable? (I do not want to pessimistic)

For an review of the article: The night sky is a field of stars in Scientic American of March 2013 read this: The night sky is a field of stars This document discusses the same issues (ie. the evolution of clusters) as in this document.

### Feedback

None

Created: 6 December 2005 Modified: 15 March 2013

Back to my home page: Contents of This Document