The purpose of this question is to see if MOND is possible solution to simulate galaxy rotation curves without dark matter.
The most obvious way to simulate galaxy rotation curves is Newton's Law. Starting point of any real simulation is to estimate baryonic mass based of a galaxy xyz on surface brightness measurements and a Mass to Light function. The problem is that you cannot simulate the corresponding 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.
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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
- For some more information about MOND see The Mond Pages
- A comparison between Newton's Law and MOND for NGC1560 NGC1560
- The original article in Physics World on which the above picture is based.
- A comparison between Newton's Law and MOND for 12 galaxies NGC5533 etc
The proof that stable galaxies in 2D are possible to simulate galaxy rotation curves is the program: GAL_MOND.BAS.
Select: Galaxy simultion in 2D with MOND
For a copy of gal_mond.exe select: gal_2d.zip. This zip file also contains the file gal_2d.exe.
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.
For a copy of EXCEL program mond.xls select: mond.xls.zip. For comments about this program see "Reflection part 5"
Reflection part 1
The above mentioned url (item 3) makes for NGC1560 a comparison between MOND and Newton's Law. Starting point between such a comparison should always be the same baryonic mass and the galaxy simulation should be stable.
The program GAL_MOND.BAS tests both Newton's Law and MOND.
In my opinion the sketch is wrong. You cannot have a simulation with the following rule vMOND(r) = 2 * vNewton(r)
Reflection part 2
The above mentioned url (item 5) makes for NGC5533 (and 15 other galaxies a comparison) between MOND and Newton's Law. Starting point between such a comparison should always be that the Galaxy should have the same baryonic mass and the galaxy simulation should be stable.
Each MOND simulation should also use the same universal constant a0
In many of the MOND simulations the speed of the rotation curve first increases and than decreases. IMO this is not possible using MOND.
Reflection part 3
Is MOND a good tool to be used for galaxy simulations and to explain dark matter?
IMO not.
- One reason is the universal constant a0, which allows you to scale up or down the rotation curve.
- The second reason is that the speed of the rotation curves with MOND always go up.
- 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)
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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
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
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 methode 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 = SQR(M * G * a0)
That means at a certain distance from the BH we get a flat galaxy rotation curve.
Now replace the the 1 BH with mass M, with n stars of mass M/n
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 = SQR(n * M * G * a0)
That means with Newton no difference, but with MOND the reverse.
This IMO places MOND in a dark corner.
- At close distance the behaviour resembles Newton at a large distance the speed becomes flat.
- There is an inbetween range where the discrepancy with Newton slowly increases. In this range a little bit of both Newton and MOND apply.
- The conctant speed with MOND can be adjusted by the parameter a0.
- A galaxy simulation with MOND should include all objects or a0 should be a function of n.
- When two BH collide the speed of a test particle in orbit should decrease.
- And last but not least this can not be tested by an experiment.
Reflection part 5
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 r2 and r1, r5 the distance between r5 and r1
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 by MOND and the mass M5 by 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
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