Description of Excel programs: Circ11.xls, Circ12.xls, Circ13.xls, Circ14.xls and Circ15.xls

The purpose of the Excel programs is to calculate the densite profile of a disc in order to simulate flat galaxy rotation curves as a function of mass (density) distribution. Each galaxy consist of a bulge, a disc and an halo of dark matter. This halo is an option
There is also an Excel program which only calculates a rotation curve: grotc.xls and grotcexp.xls. For more details See: grotc.xls and grotcexp.xls
The examples of Circ11.xls and Circ15.xls are used to demonstrate rotation curves without dark matter.
The examples of Circ12.xls, Circ13.xls and Circ14.xls are used to demonstrate flat rotation curves with dark matter with two profiles: NFW and Hernquist

For a copy of the programs in zip format select circ11.xls, circ12.xls or circ.zip. The last one contains all the five programs.

All the programs are identical. The difference lies in the parameters of the tests performed. The main physical assumption is that visible matter and darkmatter behave similar. In this simulation this is Newton's Law for both.
The main problem or chalenge is not to include to much darkmatter, at least such that the darkmatter density should not be higher than the visible matter, because otherwise it will become visible.

The radius of each rotation curve is 100 units.
The shape of the standard "target" rotation curve (v sel = 1) is such that over the first 3 units (rbulge = 3) the speed of the rotation curve increases lineair. There after the speed is constant.
Each program consists of 5 or 6 tests: Test1, Test2, Test3, Test4, Test5 and Test6
Each test has 9 parameters: "NFW", "r bulge", "r disc", "r disp", "v sel", "rs" , "rhos", "a" and "Mh"

• The variable "NFW" (or "Hernquist") is used to select the NFW profile. When NFW is 1 the NFW profile is selected. When NFW is 0 the Hernquist profile is selected. When this is the case the description changes into "Hernquist" and the value into 1. When Hernquist is 0 the reverse happens.
• The variable "r bulge" shows the length of the bulge in units. The maximum value is 100.
  
• The variable "r disc" shows the length of the disc in units. The maximum value is 100.
• The variable "v sel" identifies the "target" rotation curve. Standard value is 1. The maximum value for "v sel" is 5.
  Page "V SEL" shows the rotation curves for "v sel" values 2 to 5. Additionnal user defined rotation curves can be added.
  rs and rhos are two parameters of the NFW dark mattter profile. See http://lanl.arXiv.org/pdf/astro-ph/0301144 page 17 for more details.

  Mh and a are two parameters of the Hernquist dark mattter profile. See http://lanl.arXiv.org/pdf/astro-ph/0506015 page 2 for more details.

In order to start the simulation select START button.

Each simulation consists of 4 phases or passes. When no dark matter is involved (rhos = 0 when NFW = 1 or Mh = 0 when hernquist = 1) the simulation proceeds as follows:

1. In pass 1, First the density and the mass of the bulge is calculated as a function of r (within "r bulge") in order to match the rotation curve. The result is stored in column E under "rho bulge" and in column F under "m bulge".
   Second the force F1 of the bulge and a test mass as a function of r is calculated.
   The disc is divided in a set of rings
   Third the force F2 of test mass at distance r for each of the rings at distance rcnt is calculated assuming that density of the disc is 1. The result is stored in a matrix of 100 by 100 as a function of r and rcnt. The solution of this matrix is the density curve of the disc
   Fourth a target Force F3 function is calculated. This target Force is equal to a rotating mass at distance xx and is equal to m1 * v *v / xx with v equal to the rotation curve. Finally a delta Force is calcuted as a function of r. This force is equal to F3 - F1. The result is stored in vsave.
2. In pass 2 The matrix is solved with vsave. The result is stored in column D under "rho calc" and in vsave.
3. In pass 3 a high order lineair equation is calculated to represent the density curve of the disc. The result is stored in column J under "rho disc".
4. In pass 4 first again the force F of the bulge and a test mass as a function of r is calculated. Secondly the force F of each of the rings of the disc mass and a test mass is calculated as a function of r, but now using the density curve. Finally the rotation curve of the whole galaxy is calculated. The result is stored in column I under "v disc".

When dark matter is included in the simulation in the case of NFW = 1 then the parameters rs and rhos are non zero. Dark matter is assumed outside the radius "r bulge".In the case of Hernquist = 1 then the parameters Mh and a are non zero.
In pass 1 the density of the dark matter is calculated using the following formula:

1. In the case of NFW = 1:
   
   rhoNFW = rhos/(r/rs*(1+r/rs)^2)
   In the case of hernquist = 1:
   

   rho = (Mh/a) * a/(r+a)^3)

In pass 1 the dark matter mass is calculated. The result is stored in column E under "rho dark m" and in column F under "m dark m"

Test1 of Circ11.xls

Test1 shows a flat rotation curve where the length of the bulge is maximum. In this case there is no disc.
The variabele v sel = 1, r bulge = 100.

• Column F shows the mass of the bulge which increases lineair to 10000 units
  The most important result is in column E which shows the density of the bulge resposible for the flat rotation curve. For the first 10 units the density is constant, decreases sharply at r = 11 and then decreases slowly.
  
• This is an interesting result. The implication is that the bulge becomes "invisible" at higher distances from the centre. The easiest explanation for this "invisible" bulge is a halo of very small stars or dwarf stars.
  
• Test1 can also be seen as a simultion of a mixture between a bulge (until r = 10) and dark matter (starting from r = 11) . In that case Test1 shows the maximum amount of dark matter in order to simulate a flat rotation curve.

Test2 of Circ11.xls

Test2 shows a flat rotation curve where the length of the bulge is zero. In this case the density of the disc is at maximum in order to simulate the flat rotation curve. It is important to remember that the total height of the disc is 1 unit. That means 0.5 units in both positive and negative directions.
The total mass of the disc is approx. 6500. THe density of the disc is almost equal as the density of the bulge.
The variabele "r bulge" = 0.
Column J shows the density of the disc. The first value is almost 10 units. This is the maximum density possible. The density at a distance of 4 units is close to 5. This is an important value for the next two tests.

Test3, Test4 and Test6 of Circ11.xls

Those three test are almost identical

• Test3 shows the simulation for the smallest bulge. The variable "r bulge" = 3. In Test4 this is 6 and in Test6 10.
  
• Column E shows the density of the bulge. For the three tests the values are 2.6, 0.6 and 0.38
• Column J shows the density of the disc. For Test3 the density at a distance of 4 units (r = 4) is 2.8. For Test4 this value is approx 1.4 and for Test6 0.7
• For each of those tests the sum of the total mass of the bulge and the disc is the same and approx. equal to 6500. This is also the value for Test2

Test4 and Test5 of Circ11.xls

Test5 is almost identical as Test4.

• The parameter "r bulge" = 6. The main difference is that the target rotation curve is not flat but has a small bent and bulge. The selection of the shape is done via the V SEL page.
• the sum of the total mass of the bulge and the disc is smaller of Test5 compared to Test4: 4500 versus 6500.

Parameters of Excel program Circ11.xls - No darkmatter

Test1 to Test5 of Circ12.xls

1. In Test1 the values of rs and rhos are both zero. At r = 4 Rho dark matter = 0 and "rho disc" = 2.8. This is the target curve.
2. In Test2 rs = 6 and rhos = 0.9. At r = 4 Rho dark matter = 0.49 and "rho disc" = 0.6. The density is almost the same. THis raises the question if the name dark matter is correct.
3. In Test3 rs = 10 and rhos = 0.28. At r = 4 Rho dark matter = 0.36 and "rho disc = 0.9
4. In Test4 rs = 28 and rhos = 0.035. At r = 4 Rho dark matter = 0.19 and "rho disc" = 1.42
5. In Test5 rs = 70 and rhos = 0.0057. At r = 4 Rho dark matter = 0.09 and "rho disc" = 1.96. The dark matter density is much smaller.
6. The mass of the disc in all the five Tests is almost the same and half that of the target curve of Test1
7. Total mass of the dark matter is much larger as that of the visible matter. For Test4 and Test5 much more than a factor 10.

Parameters of Excel program Circ12.xls - NFW model

Test1 to Test5 of Circ13.xls

1. In Test1 the values of rs and rhos are both zero. At r = 4 Rho dark matter = 0 and "rho disc" = 2.8. This is the target curve.
2. In Test2 a = 10 and Mh = 3800. At r = 4 Rho dark matter = 0.55 and "rho disc" = 0.52 The density of darkmatter is higher. This raises the question if the name dark matter is correct.
3. In Test3 a = 20 and Mh = 5500 At r = 4 Rho dark matter = 0.31 and "rho disc = 1.1
4. In Test4 a = 40 and Mh = 11000 At r = 4 Rho dark matter = 0.2 and "rho disc" = 1.4
5. In Test5 a = 80 and Mh = 20000 At r = 4 Rho dark matter = 0.1 and "rho disc" = 1.8 The dark matter density is much smaller.
6. The mass of the disc in all the five Tests is rougly the same and half that of the target curve of Test1
7. Total mass of the dark matter in the case of the Hernquist profile is much larger as that of the visible matter but not so much as in the case of the NFW profile.

Parameters of Excel program Circ13.xls - Hernquist model

Test1 to Test5 of Circ14.xls

Parameters of Excel program Circ14.xls - Hernquist model - Mh var

Test1 to Test6 of Circ15.xls

1. For Test1 "V SEL" = 1. For Test2 "V SEL" = 2 etc. For Test6 "V SEL" = 6.
2. The total mass disc value is resp: 6118, 5079, 4140, 3302, 2565 and 1995. What this means is that, if you want to straigten the rotation curve (Starting from "V SEL" = 6) not so much visible matter is required. At least much less mass compared to dark metter in the case of the Hernquist profile and the NFW profile (which is the worst).

Parameters of Excel program Circ15.xls - No Dark Matter - Vsel

Additional Tests

In the program itself, in the subroutine "main", you can change the variabele vrc(V rotation curve) from 10 to 100 and Select START. The result is that all speed variables increase with a factor 10 and all density (rho) and all mass variables increase with a factor 100. The "net" result is no change. Undo this change.

In the program itself, in the subroutine "main", you can also change the variabele G from 1 to 10 and Select START. The result is that all density (rho) and all mass variables decrease with a factor 10. The "net" result is no change. Undo this change.

Page V SEL

Page "V SEL" shows:

1. 4 columns "VSEL = 2", "VSEL = 3", "VSEL = 4" and "VSEL = 5".
   Each of those colums shows a user defined target rotation curves for r going from 0 to 100.
   
2. A button INITIAL. This button is used to calculate the 4 target rotation curves. In order to calculate select INITIAL.
3. 2 columns "rho bulge" and "mass b".
   The first shows the Rho density of the bulge and the second column shows mass of the bulge. Each for r going from 0 to 200.
   
4. 2 columns "rho dm" and "mass d m".
   The first shows the Rho dark matter values of a dark matter profile as a function of rs and rhos for r going from 0 to 200 (IF NFW = 1). The second shows the total dark matter mass.
   
5. A button NFW. In order to calculate a new profile select NFW.
6. Input parameters for the NFW profile are "NFW" = 1, "rs" and "rhos".
7. Input parameters for the Hernquist profile are "NFW" = 0, "a" and "Mh". When "NFW" = 0 the description changes into "Hernquist" = 1.

Discussion

1. One of the simulations discussed above "Test2 of Circ11.xls" is a galaxy with only a flat disc. This simulation is not realistic. On the other hand it gives information how much mass there should be in the bulge if you squeeze the bulge to the same height of the disc i.e. 0.5 units.
2. The document "General Relativity Resolves Galactic Rotation Without Exotic Darkmatter by F.I. Cooperstock and S. Tieu http://lanl.arXiv.org/pdf/astro-ph/0507619 does exactly the same. If you compare the density functies of page 9 and 12 of that document with the density functions Test2 of Circ11.xls and Test3 of Circ15.xls you will see that they match. This will start you thinking: Maybe there is no dark matter required at all to simulate flat galaxy rotation curves.
3. One of the simulations discussed above "Test1 of Circ11.xls" is a galaxy which consists only of a bulge and has darkmatter. This simulation is very realistic because it reflects an elliptical galaxy which supposedly does not contain darkmatter. For more information See: http://universe.gsfc.nasa.gov/press/2003/030409d.html
4. All the other galaxies are mixtures of those two extremes.
   The three main questions to answer are:
   • what is the size of the visible bulge, including planet size objects.
   • what is the minimal density of the visible bulge, including planet size objects.
   • what is the maximum density of dark matter.

The bulge consists of three sections.

• The first section or central part is around the centre of the galaxy. In this section the density is supposed to be constant. The speed of the rotation curve increases almost lineair.
• The second section is a sphere around the central part. In this layer the density decreases until the bulge becomes invisible.
• The third section is the area outside the visible part of the bulge. The third section reaches until the outside border of the visible disc. This is the parameter "r disc". In this area are the globular clusters. This area contains dark matter. The question is how much.
• You can also define a fourth section, which is not part of the bulge. The fourth section or sphere starts outside the parameter "r disc" and contains dark matter. Again the question is how much.

1. The consequence of the first section of the bulge is that the speed of the rotation curve increases lineair.
   
2. The consequence of the second section of the bulge is that this increase levels off and finally stops.
   The most important question to ask is how large is (i.e. what is the radius of) this section. The longer this section the less mass is required for the disc. You can see the significance by comparing Test3 (r bulge=3) with Test4 (r bulge = 6)
   
3. You can argue that section 3 with the globular clusters does not belong to the bulge. That is true. But this is not so important. The true question to ask is how many small stars are there in the sphere outside the bulge and the disc, which are not individual visible. This number can be relative small, but its significance can be large. The maximum situation is demonstrated in Test1 when there is no disc.

There are three problems with the Dark Matter simulations of Circ12.xls:

1. When rs is small the rhoNFW dark matter values for small values of r are only slightly smaller as the "rho disc" values. That means the dark matter halo is visible, which it should not.
   This implies that the supposed dark matter halo becomes an extension of the bulge.
2. When rs is large in Test5 at r = 4 "rho dm" is 0.09 (0.0) and "rho disc" is 1.95 (2.8). The values in brackets are for Test1
   At r = 20 "rho dm" is 0.014 (0.0) and "rho disc" is 0.25 (0.68).
   At r = 35 "rho dm" is 0.005 (0.0) and "rho disc" is 0.09 (0.36).
   At r = 50 "rho dm" is 0.0027 (0.0)) and "rho disc" is 0.056 (0.23)
   That means only for larger values of r "rho disc" decreases.
3. The most important question to answer is which density is still visible en which density value is invisible.
   In the above "rho dm" at r=4 is identical as "rho disc" at r=35. The first density should be invisible and the second density should be visible. This is difficult to understand.
   The same problem is also mentioned in Test4a above.

One more general remark.

• When you do a simulation you must always include all the mass of the disc. It is "wrong" to stop a simulation at a certain radius r if you know that more disc mass is available outside r. The reason is that the disc mass outside r effects the rotation curve of the disc inside r.
  To solve this problem the user defined rotation curves for "v sel" = 2, 3 or 4 are so defined as if there is no mass outside r = 80.
  For dark matter assuming that dark matter is uniform spherical distributed does not have this problem because all mass within a certain radius can be considerd as situated at the centre and all mass outside has no influence.
• You can also do that on purpose.

Technical Information

• For an overview article about Darkmatter See: http://www.astr.ua.edu/keel/galaxies/darkmatter.html for more details.
• For a different article about Rotation Curves and Darkmatter See: http://astron.berkeley.edu/~mwhite/darkmatter/rotcurve.html.
  When you read this article you should answer the question:
  Is a Mass to Light (M/L) ratio of 3 the same as a "Total Mass" to "Visible Mass" ratio of 3 ? (And if not what is the relation between those two ratios.)
  For more detailed information see the following article Lecture 5: Introduction to stellar dynamics. I. This article gives more information about the Hernquist profile:Hernquist(1990)

Back to my home page Contents of This Document