Stable Galaxy - What is involved

7 Questions:

Question 1 What is the definition of a Galaxy Rotation Curve
Question 2 What is the definition of a stable Galaxy
Question 3 Is it possible to simulate a flat galaxy rotation curve using Newton's Law
Question 4 What are the consequences relative to 2D versus 3D
Question 5 Is it possible to simulate a galaxy using General Relativity?
Question 6 In order to simulate a flat galaxy rotation curve is dark matter required?
Question 7 Is it possible to simulate a flat galaxy rotation curve using MOND

Contents

Answer question 1 - What is the definition of a Galaxy Rotation Curve.

The Galaxy Rotation Curve shows the average speed of the stars revolving around the center of a Galaxy as a function of the distance from the Center 
          x
       x     y  y  y   y   y  y
              z   
     x          z
                   z        
    x                 z
                          z  
   x                          z
   
  C----------------------------
Figure 1
Figure 1 shows two aspects of a Galaxy Rotation Curve.
  • The shape of the letters "x" show the GRC of the Bulge of the Galaxy. What it shows that the speed almost linear increases.
  • The shape of the letters "y" show the observed GRC. The shape is almost horizontal and stops suddenly. This is called a flat GRC.
  • The shape of the letters "z" show the calculated GRC based on the masses of the visible stars. The shape indicates that the speed of the stars decreases.
It is possible to calculate the mass responsible for the flat GRC. This mass is called m_flat. The masses of the visible stars.is called m_vis. m_flat is much larger as m_vis. The difference is called the missing matter problem . The solution is called dark matter

Answer question 2 - What is the definition of a stable Galaxy

This question has to do with physical implications of what stability means.
There are five different point of views.
  1. First of all is our solar system stable?
    The anwer is: Yes and No
    • The answer is "Yes" when you consider a relative short time frame. Stable means that the configuration of the planets does not change and its future can be predicted as a function of observations in the past.
    • The answer is "No" when you consider a much longer time frame. No implies that the solar system becomes unstable. Unstable means that the configuration of the planets changes and its future becomes more or less unpredictable. This changes are mainly because the masses of the objects involved change (infalling material) or the number of objects change. (collisions and mergers)
    In short: One important point is the time frame considerd
  2. Is a globular star system stable.?
    A globular star system is a collection of more or less the same stars collected in a sphere, swirling around each other.
    Such a collection is considerd stable, with the exception that some times stars can be ejected from this collection.
    For more details about a related issue see: Virial Theorem
  3. Is a binary star system stable?
    Binary star system are very stable. Specific when they are situated more or less isolated in space.
    The problem comes when a third object approaches such a binary system. Than dependent about the mass of the object the results can be dramatically. For more details about binary stars see: Hubble's Law - 7 Questions
  4. When you consider our galaxy the answer is more or less indentical. Yes and No.
    • The answer is "Yes" when you consider a relative short time frame. Within such a time frame the Galaxy Rotation Curve will not change.
    • The answer is "No" when you consider a long time frame. No implies that the Galaxy Rotation Curve becomes unpredictable. This is for example the case when Our Galaxy the Milky Way start to collide with the Andromeda Galaxy.
  5. Is he Universe stable?
    The universe is by definition stable.
  6. A different question is: What is the future of the Universe?
    No body knows for sure.
    For more details about this issue see: Friedmann's Equation which discusses the size issue.

Answer question 3 - Is it possible to simulate a flat galaxy rotation curve using Newton's Law

This question is related to the mathematical implications of galaxy evolution.
A galaxy rotation curve is simulated by dividing a galaxy in rings and stars (in each ring). 

         y
     y        y
         x     
       x   x 
   y  x  B  x  y
       x   x
         x 
     y        y
         y
Figure 2
Figure 2 shows a symetrical configuration of a galaxy.
  • The number of rings is 2.
  • The numbers of stars in each ring is 8
  • There is a Black Hole in the center.
  • All the stars in the same ring have the same mass.
  1. The Quick Basic program "Gal_2D" (For more details see below) demonstrates that stable galaxies can be simulated with almost any shape of rotation curve when the stars are configured in rings and when the # of stars in each ring is identical.
  2. For a good simulation the number of rings should be large and the number of stars (objects) in each ring should be large. Good values are 10 rings and 50 stars.

Answer question 4 - What are the consequences relative to 2D versus 3D

2D simulations only use x and y directions. the z direction is zero
3D simulations use x,y and z directions. Actual simulations are much more complex.
The problem is not so much in case of a galaxy to calculate the galaxy rotation curve. This is a static problem.
The first serious problem is to perform the actual simulation. This is a dynamic issue. 
  
   +z1  +z2  +z3  +z4


  . 1 .. 2 .. 3 .. 4 .  


   -z1  -z2  -z3  -z4  

     ----------->
Figure 3
Figure 3 shows 12 objects or stars in the same ring i.e. distance from the center.
The 12 stars rotate towards the right. They all have the same speed in the xy direction.
  • the 4 stars marked 1,2,3 and 4 are rotating in the equator of the galaxy.
  • the 4 stars marked +z1 +z2 +z3 and +z4 are rotating above the equator
  • the 4 stars marked -z1 -z2 -z3 and -z4 are rotating below the equator.
  • in the case of a 2D simultion the three stars +z1, 1 and -z1 are combined in one object etc
There are three possible scenario's
  1. All the stars have the same speed vz = 0 in the z direction.
    When this is the case the stars +z1 and -z1 will both move towards the equator while star 1 will stay in the plane of the equator. When they reach the equator "all" will collide i.e. the simulation is not stable at a short time frame.
    This does not mean that the actual galaxy is not stable.
  2. Consider the following situation: the initial speed vz of star +z1 is slightly positif, of star 1 vz = 0 and the speed of star -z1 is slightly negatif.
    If you do that correct "in theory" you can get a stable configuration, but in practice the stars will also collide.
  3. The third possiblity is to combine the stars +z1, 1 and -z1 in one object at position +z1. The next three stars at position 2. The next three at position -z3 and the 4 stars at the left at position 4.
    When you do that than all the stars in one ring are in a wave form. This is may be a better solution, but also not perfect.
The second serious problem with any 3D simulation of a galaxy of roughly 1000 objects is that each object is not a star but more a Black Hole because of its hugh mass. When any object in such a simulation is ejected it has a hugh influence on the positions on many other objects i.e. specific on the behaviour of the whole galaxy.
When instead of 1000 objects you use 10000 objects (that means all BH's become a factor 10 smaller) and you position these smaller objects more or less randomly in the space occupied by the original BH (you do the same with the speed) than the individual behaviour of 1 BH has less influence on the galaxy in total. Any ejection or collision becomes less critical.

Answer question 5 - Is it possible to simulate a galaxy using General Relativity.

IMO the answer is: No.
What I have read is an article discussing a galaxy in 2D. That means the z coordinate is 0.
What people use is the "parametrized post-Newtonian" or PPN formalism.
Accordingly to the book "Gravitation" at page 1049 it is a good approximation whenever, as in the solar system, the sources of the field gravitate weakly and move slowly.

Answer question 6 - In order to simulate a flat galaxy rotation curve is dark matter required?

Using the "Gal_2D" program it is possible to simulate any galaxy rotation curve.
When you consider figure 1 it is both possible to simulate the shape of the GRC indentified with the letters X and Y as with the letters X and Z.
What program ""Gal_2D" does it calculates how much mass is responsible for each. For the "Y" curve the mass is larger as for the "Z" curve. In principle this Missing Matter can be either baryonic, nonbaryonic or any combination of each.

Answer question 7 - Is it possible to simulate a flat galaxy rotation curve using MOND

With MOND you can only simulate flat galaxy rotation curves.
When a galaxy rotation curve is not flat you cannot use MOND.
With Newton's Law you can simulate any type of galaxy rotation curve.

For all the issues related to MOND see: MOND - What is involved

Stable Galaxy in 2D

The proof that stable galaxies in 2D are easy to simulate is the program: "GAL_2D".
Program "Gal_2D" is written in Quick Basic. The source file is called: Gal_2D.bas.
Select: Program "Gal 2D.bas: Galaxy simulation in 2D"

For a copy of gal_2d.exe select:gal_2d.zip. This zip file also contains the source file gal_2d.bas

The galaxy is represented by many objects (or stars). Each object has approximate the same mass. The actual mass is calculated in the program. Each object is a collection of many real stars.

In this simulation the central bulge of the galaxy is not represented by one object but by more. The speed of the objects that form the central bulge increases linear with distance.

The same program "Gal_2D" is also written in Visual Basic 5.0
For all information select: Visual Basic program "VB Gal 2D" - Description and operation

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Created: 8 November 1998
Updated: 17 Aug 2015
Updated: 19 Februari 2016

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