## MOND (MOdified Newtonian Dynamics) - What is involved

### 6 Questions:

 Question 1 What is the exact definition of MOND. (When do you apply) Question 2 Is it possible to simulate a galaxy rotation curves using MOND Question 3 Is it possible to simulate each type of rotation curve using MOND Question 4 How do you compare MOND versus Newton's Law for galaxy simulation. Question 5 Is the cosmological constant a0 a true constant. Question 6 Is MOND a good theory to solve the darkmatter issue

### Purpose

The background of the four questions is to investigate if MOND is a possible solution to simulate galaxy rotation curves without dark matter.
The most obvious way to simulate galaxy rotation curves (GRV) is Newton's Law. Starting point of any simulation is to estimate all baryonic (Visible) mass based of a galaxy xyz on surface brightness measurements and a Mass to Light function. The problem is that you cannot simulate the measured rotation curve of galaxy xyz strictly based on Newton's Law. In order to do an accurate simulation with Newton's law non-baryonic mass (dark matter) has to be included.
MOND is a modification to Newton's Law.
• Newton's Law is based on the formula: a = G * m / r^2.
• MOND is based around the formula a^2 / a0 = G * m / r^2. MOND includes universal constant a0. MOND is valid when a is much smaller than a0.
The theory is that you can simulate any rotation curve starting from the baryonic mass distribution without dark matter using MOND.

### Literature

1. For the original paper "A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis" by M. Milgrom See: 1983ApJ...270..365M
3. From the Mond pages: A comparison between Newton's Law and MOND for NGC1560 CDM & MOND rotation curve fit See also: NGC1560
4. The original article in Physics World on which the above picture is based.
5. From the MOND pages: A comparison between Newton's Law and MOND for 12 galaxies NGC5533 etc
6. https://cds.cern.ch/record/645887/files/0310005.pdf "Squeezing MOND into a Cosmological" by Scenario Arthur Lue and Glenn D. Starkman. (30 sep. 2003) This document is dicussed in Q 5
7. http://arxiv.org/pdf/1404.7661v2.pdf "MOND theory" by Mordehai Milgrom (Submitted on 30 Apr 2014) Q 5

### Answer Q1: What is the definition of MOND. When do you apply MOND

In order to study what MOND is please study Reflection 4 MOND and Reflection 5 MOND
These same two sections also give an impression when or when not to apply.

### Answer Q2: Is it possible to simulate a Galaxy using MOND

It is possible to simulate stable galaxies using MOND in 2D. The same also can be done using Newton'Law.

In each program the galaxy is represented by many objects (or stars) which all are situated in the same 2D plane.. Each object has approximate the same mass. The actual mass is calculated in the program. Each object is a collection of many real stars.

### Answer Q3: Is it possible to simulate all types of Galaxy Rotation Curves

You can only simulate Galaxy Rotation Curves using MOND which are flat.
The "problem" with MOND is that when you want to simulate a curve the curve stays flat ad infinitum. That means the speed does not decrease.
That also means you cannot use MOND to simulate the planets of our Solar system. This is a huge disadvantage of MOND.

### Answer Q4: How do you compare MOND versus Newton's Law.

The above "Answer Q3" more or less answers this. With Newton's Law you can simulate each type of Galaxy Rotation Curve. The problem is the resulting mass distribution does not match the visible mass distribution. You need more mass. That is the dark matter issue.
With MOND a similar problem exists. Because with MOND you can only simulate flat Galaxy Rotation Curves you should not observe any mass behind (further away) the point where you measure the highest rotation speed!

### Answer Q5: Is the cosmological constant a0 a true constant.

The constant a0 is a true constant if when ever it is used the same value applies.
In my simulations the standard value is 8. The importance is any value can be used, because it has no influence on the simulations as such. The most important influence is: the total mass of the galaxy calculated.
Literature (6) in the introduction reads:
We find that the critical acceleration a0 must have a slight source-mass dependence (a0 =approx M^1/3) etc
When that is the case a0 is not any more a constant.

### Answer Q6: Is MOND a good theory to solve the darkmatter issue.

The whole dark matter issue is closely related to Galaxy Rotation Curves.
A galaxy rotation curve is curve which shows the relation of the (average) speed of a galaxy versus the distance towards the center of the galaxy.
In many cases GRC's show in the first part that the speed increases linear and than the speed stays the same. Such curves are called flat. The problem starts when you want to calculate such a GRC using vissible matter. The shape is different. In the first part we see that the speed increases linear, but in the second part the speed does not stay the same but slowly decreases. The next step is to calculate the amount of matter required to simulate a flat curve using Newton. This amount calculated is more than the amount visible. That means a certain amount of matter is missing using Newton's Law
MOND does not have this problem because all the GRC's are flat. The problem with MOND is that you cannot use MOND to simulate GRC's which are not flat.

There are two other issues involved: It is not realistic to simulate Galaxies based on the amount of matter that is visible. In general this are only star sized objects. In reality in many galaxy's there is a BH in the center, which falls in the category of invisible baryonic matter. A second problem is that all planet sized objects, all kuiper belt sized objects and all objects which belong through the Oort cloud are invisible. That means a galaxy harbours a lot of baryonic matter which is invisible but which should be taken into account before non-baryonic matter should be considered. See also: Reflection part 3

### Reflection part 1

Literature #3 makes for NGC1560 a comparison between MOND and Newton's Law.
The programs GAL_MOND.BAS, "VB Gal Mond" and "VB2010 Gal Mond" each can be used to simulate 2D galaxies with Newton's Law and MOND specific an example like NGC1560
What the results for NGC1560 show is that simulation of the GRC with MOND matches what is observed. That is easy in this case because the GRC always increases in speed at higher distances.
What the information also shows is the simulated curve using Newton's Law. That is the red Dark Matter line.
What the information also shows is the calculated GRC based on the amount of visible matter (stars and Gas).
The difference is an indication for the amount of missing matter
What is important is to know the shape of the observed GRC at much larger distances at present.

### Reflection part 2

Literature #5 makes for NGC5533 (and 15 other galaxies a comparison) between MOND and Newton's Law. The same comments as in Reflection part 1 apply. They are both from "The MOND pages".
What is important that each MOND simulation should also use the same universal constant a0. This is not clear.
What is also interesting that in many of the MOND simulations (NGC2903 and NGC4100) the speed of the GRC first increases and than decreases. IMO this is not possible to simulate using MOND.

### Reflection part 3

Is MOND a good tool to be used for galaxy simulations and to explain dark matter?
IMO not.
1. One reason is the universal constant a0, which allows you to scale up or down the rotation curve.
2. The second reason is that the speed of the rotation curves with MOND always go up.
3. The third reason is when does MOND applies?
General speaking the full law is a mixture of both. Something like:
a = (1-alpha) * (G*m/r2) + alpha * SQR(G*m*a0/r2)
with alpha being 0 at small distances and alpha being 1 at large values.
Someting like alpha(r) = r/(1+r). IMO a horrible solution.

### Reflection part 4 - MOND

For a comment about MOND, read the document "MOND N-body: why a = SQR(a0gN) isn't enough" by Chris Mihos, Case Western Reserve University http://www.astro.umd.edu/~ssm/mond/mondnbody.ps This is a two page document.
The document ends with the sentence: "Note that this does not mean MOND is wrong, just that this kind of calculation does not work". I do not 'understand' that sentence. If this kind of calculation does not work (i.e. replacing n stars by 1 star with the same mass) in order to simulate galaxy behaviour with MOND than what is the than the correct method to apply ?

The reason that you cannot divide 1 star by n stars (or vice versa) can easy be shown with 1 BH and one test mass. The Black Hole has a mass m. The speed of the test mass is v

```With Newton we get:
a = M * G / (r * r) = v * v / r
or
M * G / r = v * v  or  v = SQR( M * G / r)

With MOND we get
a = SQR( M * G *a0/ (r * r)) = v * v / r
or
SQR( M * G *a0 ) = v * v  or v = (M * G * a0)^1/4
```
That means at a certain distance from the BH we get a flat galaxy rotation curve, because v = constant.
Now replace the the 1 BH with mass M, with n stars of mass M/n (refining)
```With Newton we get:
a = n * (M/n) * G / (r * r) = v * v / r
or
M * G / r = v * v  or  v = SQR( M * G / r)

With MOND we get
a = n * SQR( (M/n) * G *a0/ (r * r)) = v * v / r
or
SQR( n * n * (M/n) * G *a0 ) = v * v  or v = (n * M * G * a0)^1/4
```
That means with Newton there is no difference, but with MOND this is not the case: the speed increases when a large mass is broken down.
With Newton it does not matter if a star rotates at far distance around one object with a certain mass or 10 smaller objects in a cluster with the same mass. With MOND it does. This IMO places MOND in a difficult situation considering star merges and explosions.
1. At close distance the behaviour resembles Newton at a large distance with MOND the speed always becomes flat.
2. There is an inbetween range where the discrepancy with Newton slowly increases.
3. The conctant speed with MOND can be adjusted by the parameter a0, which should apply for all galaxies.
4. A galaxy simulation with MOND should include all objects or a0 should be a function of n.
5. When two BH collide the speed of a test particle in orbit should decrease. This is the reverse as refining.
6. And last but not least this can not be tested by an experiment.
This whole issue also has a different consequence. When you want to simulate a fixed Galaxy Rotatation curve with Newton and you consider two cases one with 100 objects and a second one with one 1000 objects this refining with Newton has no consequences for the total mass considered. With MOND (because you want to keep the total speed for each ring constant) the total mass will decrease.

### Reflection part 5 - MOND

In the above mentioned document the behaviour (acceleration a) of a test star with a mass M5 around a cluster of 4 stars with each a mass M is described by the following formulas:
(The 4 stars are identified by M1, M2, M3 and M4, r1 is the distance between M5 and M1 etc)
```           M2

M1        M3                                           M5

M4

a(M5) = G*M1/r1*r1 + G*M2/r2*r2 +G*M3/r3*r3 + G*M4/r4*r4
With  Newton

a(M5) =SQR(G*M1*a0)/r1 + SQR(G*M2*a0)/r2 +SQR(G*M3*a0)/r3 + SQR(G*M4*a0)/r4
With  MOND

```
The question is what is the acceleration of M1 with Newton and MOND ? (r2 is the distance between M2 and M1, r3 the distance between M3 and M1 etc)
```
a(M1) = G*M2/r2*r2 +G*M3/r3*r3 + G*M4/r4*r4 + SQR(G*M5*a0)/r5
With Newton and Mond

```
The above equation is a mixture between Newton and Mond. The 3 masses close to M1 are described with Newton and the mass M5 with MOND.
Is that correct. ? Suppose the four masses M1, M2, M3 and M5 are at a larger distance from each other. What is than the correct formula to describe the physical situation ?

To study a much simpler situation I have written a program MOND.xls in EXCEL.
In that program first the four masses are combined into one and the acceleration and the speed of a test mass is shown using Newton. This is done in the first 3 colums as a function of distance r.
In the nect two columns the acceleration and speed of a test mass is given using MOND. At close distance those values are equal to Newton. MOND is used when the acceleration of the test star becomes smaller than a0. The program shows that the speed becomes constant. (See also Reflection part 4)
The question to answer is: Is this according to the physical reality.. Does a test mass behave that way i.e. how far away you are the speed around this single mass stays constant and the test mass moves in a circle!

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Created: 13 March 2007
Updated: 13 August 2015
Updated: 19 Februari 2016