The numbers show almost a lineair increase with distance. What that means is that mass far away (in its outer limits) is as much important as nearby.

Test1, Test2 and Test3

Test4, Test5, Test6 and Test7 of Circ16.xls

• The primary purpose of the Test5, Test6 and Test6 is to calculate the rotation curve and based on the observed amount of visible matter. More strictly: luminosity
  The parameter VSEL for those three tests is respectivily: 2, 3 and 4
  The final velocity values for those three tests are respectivily: 200, 150 and 100 km/sec.
  The "disktype" used is 3.
• The secondary purpose of Test5, Test6 and Test7 is to calculate how much mass is required if you make the galaxy rotation curve more flat.

The mass value to support the flat galaxy rotation curve (v = 250 km/sec) of Test4 is 111 * 10^40 Kg.
The mass values for the Test5, Test6 and Test7 are resp 77, 44 and 22 * 10^40 Kg.
That means respectivily that 34, 64 and 89 *10^40 Kg is missing.

Test 4 (top yellow line), 5, 6 and 7 (bottom yellow line)

For a typical example which demonstrates the difference of a galaxy rotation curve expected from visible matter and which is observed see this: Galactic Rotation Curves
You can compare that with the test4 versus test5. That means roughly 30% is missing.
You can postulate that this is caused by darkmatter.
Much simpler is the solution that it is ordinary matter caused by planet sized invisible objects

Parameters of Excel program Circ16.xls - No dark matter

Parameters of Excel program Circ161.xls - No dark matter

Parameter	Test1	Test2	Test3	Test4	Test5	Test6	Test7
1 NFW	1	1	1	1	1	1	1
2 rbulge	10000	10000	10000	10000	10000	10000	10000
3 disk	25000	50000	100000	200000	200000	200000	200000
5 Vsel	1	1	1	1	2	3	4
10 tot m bulge	8,86	8,86	8,86	8,86	8,86	8,86	8,86
11 tot m disk	6,62	20,51	49,14	107,25	69,96	40,49	20,88
12 rho bulge	0,250E-10	0,250E-10	0,250E-10	0,250E-10	0,250E-10	0,250E-10	0,250E-10
13 rho disk	0,676E-10	0,861E-10	0,927E-10	0,945E-10	0,825E-10	0,717E-10	0,620E-10

Circ161

Parameters of Excel program Circ162.xls - No dark matter

Parameter	Test1	Test2	Test3	Test4	Test5	Test6	Test7
1 NFW	1	1	1	1	1	1	1
2 rbulge	5000	5000	5000	5000	5000	5000	5000
3 disk	25000	50000	100000	100000	100000	100000	100000
5 Vsel	1	1	1	1	2	3	4
10 tot m bulge	4,43	4,43	4,43	4,43	4,43	4,43	4,43
11 tot m disk	10,18	24,52	53,37	53,40	41,30	30,77	22,03
12 rho bulge	0,999E-10	0,999E-10	0,999E-10	0,999E-10	0,999E-10	0,999E-10	0,999E-10
13 rho disk	1,641E-10	1,795E-10	1,839E-10	1,838E-10	1,709E-10	1,588E-10	1,474E-10

Circ162

Page "V SEL"

Introduction

Page "V SEL" is divided into two parts:

• The left part is used to calculate galaxy rotation curves.
  The left part consists of 5 columns marked:
  "VSEL = 2", "VSEL = 3", "VSEL = 4", "VSEL = 5" and VSEL = 6
  Each of those colums shows a user defined target rotation curves for r going from 0 to R disk in Lightyears in increments of 1000 Lightyears.
  To perform the calculations select INITIAL.

  The right part is used to calculate the NFW profile or the Hernquist profile.
  The right part consists of 4 columns marked:

  "rho bulge/disk", "mass bulge/disk", "rho dark matter", and "mass dark matter"

  To perform the calculations select: NFW.

Operation page V SEL - The left part

1. In order to calculate the Rotation Curves 3 parameters are important:
   
   R bulge, R disk and Disk type
   • R bulge defines the distance of the bulge for All curves. The Standard value is 6000 Lightyears or 6 Units.
     The calculated curve from 0 to R bulge is lineair going from 0 to 250 km/sec
   • R disk defines the maximum distance of the disk. The standard value is 100000 Lightyears or 100 Units.
   • Disk type defines the shape of the velocity curve. The are 4 velocity curves.
     For more detail see: program2 - Disktype 1-4

   There is one more very important issue and that is that each of the curves requires two parameters:

   • A value at R bulge and a value at R disk. These are the two lines in Yellow.
   • The value at R bulge is standard 250 km/sec
   • The value at R disk should be lower. In most cases for the 5 selection curves the values are: 200, 150, 100, 50 and 0 km/sec

   A button INITIAL. This button is used to calculate the 5 target rotation curves. In order to calculate select INITIAL.

Operation page V SEL - The Right part

In order to calculate a darkmatter profile 10 parameters are important:

NFW/Hernquist, R bulge, R disk, rs, rhos, a, Mh, rho disk start, rho disk end and Filter

• NFW/Hernquist defines the profile.
  When NFW = 1 or Hernquist = 0 the NFW profile is calculated.
  When Hernquist = 1 or NFW = 0 the NFW profile is calculated.
• R bulge defines the distance of the bulge for All curves. The Standard value is 6000 Lightyears or 6 Units.
  The calculated curve from 0 to R bulge is lineair going from 0 to 250 km/sec
• R disk defines the maximum distance of the disk. The standard value is 100000 Lightyears or 100 Units.
• rs and rhos are two parameters for the NFW profile.
• a and MH are two parameters for the Hernquist profile. MH means Mass Halo
• rho disk start, rho disk end and Filter are three parameters to define the density profile of the disk.
  Smallest value parameter Filter is 0.01

A button NFW. In order to calculate a new profile select: NFW.

Example 1 - Circ12.xls Test3

Program Circ12.xls is used to demonstrate the NFW profile.
Page "V SEL" demonstrates Test3. The parameters rs and rhos are respectivily 5000 and 1.7E-10
In order to demonstrate Test3 the parameter Filter should be 3.
Important: Before doing the calculation set the parameter rho disk start equal to zero.

Example 1 - Circ13.xls Test5

Program Circ12.xls is used to demonstrate the Hernquist profile.
Page "V SEL" demonstrates Test5. The parameters a and Mh are respectivily 3000 and 0.1
In order to demonstrate Test5 the parameter Filter should be 8.
Important: Before doing the calculation set the parameter rho disk start equal to zero.

Reflection part 1

1. One of the simulations discussed above "Test2 of Circ11.xls" is a galaxy with only a flat disk. 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 disk 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 disk. This is the parameter "r disk". 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 disk" 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 disk. 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 disk, 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 disk.

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 disk" 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 disk" is 1.95 (2.8). The values in brackets are for Test1
   At r = 20 "rho dm" is 0.014 (0.0) and "rho disk" is 0.25 (0.68).
   At r = 35 "rho dm" is 0.005 (0.0) and "rho disk" is 0.09 (0.36).
   At r = 50 "rho dm" is 0.0027 (0.0)) and "rho disk" is 0.056 (0.23)
   That means only for larger values of r "rho disk" 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 disk" 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.

Reflection part 2

• The two most important tests are Test 1 of Circ11.xls and Test 2 of Circ11.xls.
  The first one simulates a flat rotation curve when there is only a bulge. You could also say when there is only a spherical halo.
  The second one simulates a flat rotation curve when there is only a flat disk. This is is the most realistic. The issue of course is here the height of the disk.
• When you do a simulation you must always include all the mass of the disk. It is "wrong" to stop a simulation at a certain radius r if you know that more disk mass is available outside r. The reason is that the disk mass outside r effects the rotation curve of the disk inside r.
  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.

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)

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