Why Does 'a' Being Smaller Than 'a0' Eliminate μ in MOND?

[mesa]

I'm trying to figure out the formula adjustment to Newtonian mechanics using MOND but I'm getting stuck. Wikipedia states:

"Assuming that, at this large distance r, a is smaller than a0 so: μ × (a/a0) = a/a0"

Why does 'a' being smaller than 'a0' get rid of μ?

 

[Jonathan Scott]

I'm trying to figure out the formula adjustment to Newtonian mechanics using MOND but I'm getting stuck. Wikipedia states:

"Assuming that, at this large distance r, a is smaller than a0 so: μ × (a/a0) = a/a0"

Why does 'a' being smaller than 'a0' get rid of μ?

It doesn't actually say what you've quoted; you've inserted the multiplication sign.

In MOND, μ(x) is actually defined to be a function of x such that μ(x) = 1 when x is much larger than 1 and μ(x) = x when x is around 1 or smaller.

 

[mesa]

I am not very familiar with functions like this. That makes more sense

So what is μ?

 

[Jonathan Scott]

I am not very familiar with functions like this. That makes more sense

So what is μ?

The MOND scheme only specifies how the interpolation function behaves for large and small x, without giving a specific form.

For illustration purposes, you could for example use μ(x) = x/(1+x).

The whole idea of a function which effectively cuts off the effect at a certain acceleration is very odd anyway. MOND is typically used to describe how stars move when in a very weak gravitational field at the edge of a galaxy, but if you consider the atoms within the star, they are all subject to much greater accelerations due to the star itself. This seems to require some sort of magic whereby the motion of the star as a whole is not the same as the average motion of its component atoms.

 

[twofish-quant]

So what is μ?

The tooth fairy. Seriously...

The idea behind MOND is to insert a fudge factor into the gravity equations and see if you can get the observed behavior of galaxy rotation curves. Now if it turned out that there was some pattern in that fudge factor, you could then start thinking about what that fudge factor could be.

But it hasn't worked out. It turns out that every galaxy seems to have a different fudge factor, which makes dark matter a more convincing explanation for what is causing the rotation curves.

There's something one of my advisors called the "tooth fairy rule." Which is that in any theoretical astrophysics paper, you are allowed one wave of the tooth fairy's magic wand. If you assume one crazy thing and if everything works, you win. Dark matter is another tooth fairy, but you just wave it once and lots of problems disappear.

The problem with MOND is that right now you need to wave the magic wand several times.

 

[mesa]

The MOND scheme only specifies how the interpolation function behaves for large and small x, without giving a specific form.

For illustration purposes, you could for example use μ(x) = x/(1+x).

The whole idea of a function which effectively cuts off the effect at a certain acceleration is very odd anyway. MOND is typically used to describe how stars move when in a very weak gravitational field at the edge of a galaxy, but if you consider the atoms within the star, they are all subject to much greater accelerations due to the star itself. This seems to require some sort of magic whereby the motion of the star as a whole is not the same as the average motion of its component atoms.

Okay, I'm going to look at this again.

 

[mesa]

The tooth fairy. Seriously...

The idea behind MOND is to insert a fudge factor into the gravity equations and see if you can get the observed behavior of galaxy rotation curves. Now if it turned out that there was some pattern in that fudge factor, you could then start thinking about what that fudge factor could be.

But it hasn't worked out. It turns out that every galaxy seems to have a different fudge factor, which makes dark matter a more convincing explanation for what is causing the rotation curves.

There's something one of my advisors called the "tooth fairy rule." Which is that in any theoretical astrophysics paper, you are allowed one wave of the tooth fairy's magic wand. If you assume one crazy thing and if everything works, you win. Dark matter is another tooth fairy, but you just wave it once and lots of problems disappear.

The problem with MOND is that right now you need to wave the magic wand several times.

That much I understood, it would be interesting to see (on average) how much that 'fudge factor' is. If it does somewhat represent the actual observations for the speed of stars (which to some degree it does) then it is at least a place to start to help understand how gravity works on the galactic scale.

The dark matter theory seems a little shaky too though with it's "halo" and in essence is just another 'fudge factor' is it not to preserve Newtonian Mechanics?

 

[Drakkith]

That much I understood, it would be interesting to see (on average) how much that 'fudge factor' is. If it does somewhat represent the actual observations for the speed of stars (which to some degree it does) then it is at least a place to start to help understand how gravity works on the galactic scale.

Except that as has been pointed out the value is always different and there doesn't appear to be any pattern at all.

The dark matter theory seems a little shaky too though with it's "halo" and in essence is just another 'fudge factor' is it not to preserve Newtonian Mechanics?

What does Newtonian mechanics have to do with anything? And as twofish stated, one wave of the wand is ok, but many many waves is obviously not working. Dark matter explains the most amount of observations and is the one that makes the least amount of wand waving, so currently there's not much of a reason to think it's mistaken entirely.

 

[Jonathan Scott]

The tooth fairy. Seriously...

The idea behind MOND is to insert a fudge factor into the gravity equations and see if you can get the observed behavior of galaxy rotation curves. Now if it turned out that there was some pattern in that fudge factor, you could then start thinking about what that fudge factor could be.

But it hasn't worked out. It turns out that every galaxy seems to have a different fudge factor, which makes dark matter a more convincing explanation for what is causing the rotation curves.

Where did you get that from?

The really weird thing about MOND is that it actually works for a huge range of different galaxies using the same a0 value, and correctly predicted the results for Low Surface Brightness (LSB) galaxies before any measurements had been made on them.

However, it doesn't work at larger scales (such as galaxy clusters and interacting galaxies) nor at smaller scales (globular clusters within galaxies) without further tweaking.

 

[Dotini]

http://www.scilogs.eu/en/blog/the-dark-matter-crisis/2011-03-21/question-c.ii-mond-works-far-too-well

Pavel Kroupa is highly enthused over MOND.

Respectfully submitted,
Steve

 

[mesa]

Except that as has been pointed out the value is always different and there doesn't appear to be any pattern at all.

Dark matter explains the most amount of observations and is the one that makes the least amount of wand waving, so currently there's not much of a reason to think it's mistaken entirely.

I thought MOND was supposed to be a better predictor than the dark matter theory for star velocities in galaxies without screwing up mechanics for smaller systems?

What does Newtonian mechanics have to do with anything?

I thought that was the point of introducing the dark matter into the system, so the laws of gravitation still work without re-inventing the wheel like in quantum mechanics.

 

[Jonathan Scott]

As far as I know, MOND is extremely successful on the scale of galaxies. The problem is mainly that it is simply an arbitrary rule which has been found to work experimentally, while attempts to find an underlying theory behind it have not been very convincing; even though progress has been made, for example with TeVeS, some of the concepts still seem to violate fundamental physical principles.

Calculations on the scale of galaxies are typically done using mainly Newtonian gravity theory, with the occasional check to ensure that General Relativity effects are small enough to ignore in specific cases.

 

[Drakkith]

I thought MOND was supposed to be a better predictor than the dark matter theory for star velocities in galaxies without screwing up mechanics for smaller systems?

I thought that was the point of introducing the dark matter into the system, so the laws of gravitation still work without re-inventing the wheel like in quantum mechanics.

I apologize, I think I misunderstood what MOND was and was thinking of something else. I'll remove myself from this thread now!

 

[mesa]

I apologize, I think I misunderstood what MOND was and was thinking of something else. I'll remove myself from this thread now!

No worries. Let's continue the discussion,

Anyone know how to run the calculations for MONDS?

In DMT (dark matter theory) it looks like they are just using a certain mass of dark matter outside the galaxies to account for the discrepancies in velocities of the stars and to preserve the laws of gravitation. Does that sound right?

 

[mesa]

...but if you consider the atoms within the star, they are all subject to much greater accelerations due to the star itself. This seems to require some sort of magic whereby the motion of the star as a whole is not the same as the average motion of its component atoms.

I'm not sure I am getting this, how do the atoms in the star affect the overall velocity of the star? Or are you saying this just a way of looking at the effect of gravity on the scale of the very large vs small and that it seems silly to have different rules for both systems?

 

[twofish-quant]

The dark matter theory seems a little shaky too though with it's "halo" and in essence is just another 'fudge factor' is it not to preserve Newtonian Mechanics?

In essence, yes. It's just that there is a *lot* less fudging that you have to do to fit the observations. You wave the magic wand once, and not only can you fit galaxy curves, but observations of CMB, and various cosmological quantities make sense.

But things can change.

 

[twofish-quant]

http://www.scilogs.eu/en/blog/the-dark-matter-crisis/2011-03-21/question-c.ii-mond-works-far-too-well

Pavel Kroupa is highly enthused over MOND.

True, and looking just at the data that he is looking at, I probably would be too.

The trouble is that the main reason people think that there is dark matter has to do with large scale cosmology which Kroupa doesn't talk about. Basically in order to get the right lumpiness factor and deuterium abundances, you have to assume dark matter.

Modified gravity theories don't quite work in that context. One reason for this is that things go in the wrong direction. With dark matter, the denser things are, the weirder things get, whereas with modified gravity, you end up with things getting weirder the less dense things get. This matters for things like deuterium abundances.

Look at point 9) that Kroupa makes. In order to get the CMB distributions with MOND he has to assume a 11eV sterile neutrino. That's fine, but 1) sterile neutrinos are dark matter and 2) that's another wave of the magic wand, and it's not a small wave. Once you put in a new particle, then you have to recalculate all of the big bang nucleosynthesis numbers.

What Kroupa is saying is that MOND + a hypothetical particle makes everything work. Trouble is that you can get everything to fit by dropping MOND and just assuming a hypothetical particle, and you save one wave of the magic wand.

 

[twofish-quant]

In DMT (dark matter theory) it looks like they are just using a certain mass of dark matter outside the galaxies to account for the discrepancies in velocities of the stars and to preserve the laws of gravitation. Does that sound right?

Not quite. The thing about dark matter is that you wave that magic wand once and lots of seemingly unrelated things make sense. For example, there seems to be more deuterium than you would expect if the universe were all "normal matter" and you can calculate the "lumpiness factor" of the universe.

Galaxy rotation curves are only one "weird thing", and frankly, if galaxy rotation curves were the only "weird thing" that we see, then MOND would make more sense to me than dark matter.

Also the possibility exists that both are correct (i.e. that there is dark matter and gravity doesn't behave the way we think it does).

 

[twofish-quant]

Where did you get that from?

I was misremembering something that you seem to have remembered correctly...

The really weird thing about MOND is that it actually works for a huge range of different galaxies using the same a0 value, and correctly predicted the results for Low Surface Brightness (LSB) galaxies before any measurements had been made on them.

When people saw that this was like *wow* there might be something here. It's this particular observation that gave MOND quite a bit of credibility for a time.

However, it doesn't work at larger scales (such as galaxy clusters and interacting galaxies) nor at smaller scales (globular clusters within galaxies) without further tweaking.

Yup. The trouble is that the more "tweaking" you have to do to get things to work, the less strong the theory is. Both dark matter and modified gravity require tweaking to get the right fit with observations, but at this point dark matter seems to require a lot less tweaking than modified gravity, but this is one of those things that could change quickly.

One other thing about arguments toward elegance is that different people can weight things differently. If someone looks at the data that modified gravity requires less tweaking than dark matter, it can be hard to argue otherwise because these are somewhat subjective.

 

[twofish-quant]

The other problem is that Krupa seems to misunderstand the applicability of LCDM. The idea behind LCDM is that the big bang produced large scale clumps and that these clumps influences where galaxies form. How galaxies actually form is outside of the theory, so LCDM really says nothing about things at small scales.

No one has been able to reproduce the cosmological observations with only modified gravity (lots of people have tried). Once you assume that some dark matter is necessary, then it becomes easier to assume (unless you have some reason otherwise) that it's all dark matter.

 

[Jonathan Scott]

I'm not sure I am getting this, how do the atoms in the star affect the overall velocity of the star? Or are you saying this just a way of looking at the effect of gravity on the scale of the very large vs small and that it seems silly to have different rules for both systems?

MOND has a different gravitational acceleration rule for cases where the gravitational acceleration is of the order of a0 or weaker. If this rule were just treated as additional to Newtonian gravity, it appears that corrections due to MOND would already have been necessary to match solar system experiments (although it's not completely conclusive, because MOND accelerations don't add up in the same way as Newtonian gravity). For this reason, MOND assumes an interpolation function which means that the acceleration of an object in a very weak field obeys the MOND rule but in a stronger field it obeys Newtonian rules (or GR where that level of accuracy is necessary).

A star on the edge of a galaxy is treated by MOND as being very weakly accelerated as a whole by the galaxy, so the MOND rule applies. However, if you consider the component atoms of the star, they are all within the gravitational field of the star itself, so the overall gravitational acceleration on those atoms would be expected to be much greater than a0, which means they would obey Newtonian gravitation and be "immune" to MOND. It is difficult to see how the atoms of a star can accelerate in one way but the star as a whole accelerate in a different way.

Similarly, a system of masses such as a binary star or a star and planets at the edge of the galaxy is also treated by MOND as a single object in the low-acceleration regime, even though the components are clearly subject to higher accelerations from each other.

Note however that the MOND force is quite tricky to work with anyway, in particular because it is not linear in the source mass.

 

[mesa]

No one has been able to reproduce the cosmological observations with only modified gravity (lots of people have tried). Once you assume that some dark matter is necessary, then it becomes easier to assume (unless you have some reason otherwise) that it's all dark matter.

How is the formula set up for the dark matter halo? Can we work out an example? Perhaps predict the velocity of a star using basic Newtonian Mechanics vs DMT vs MOND

Galaxy rotation curves are only one "weird thing", and frankly, if galaxy rotation curves were the only "weird thing" that we see, then MOND would make more sense to me than dark matter.

So there is a great deal more to the predictions of these systems than star velocity alone. Is star velocity the predominating area or are there other aspects of equal or greater importance? I would like to put some chalk to a board on star velocities unless you feel there is a better place to start, can we work out an example?

A star on the edge of a galaxy is treated by MOND as being very weakly accelerated as a whole by the galaxy, so the MOND rule applies. However, if you consider the component atoms of the star, they are all within the gravitational field of the star itself, so the overall gravitational acceleration on those atoms would be expected to be much greater than a0, which means they would obey Newtonian gravitation and be "immune" to MOND. It is difficult to see how the atoms of a star can accelerate in one way but the star as a whole accelerate in a different way.

Okay, I understand what you were saying now.

 

[mesa]

I was looking at the MOND equation, it looks like the adjustment is 'hidden' at smaller scales allowing Newtonian mechanics to work on our scale as the function brings it's value to 1 while adjusting to increased values for 'a' as the effects of gravity would become weaker as distance 'r' is increased.

I can not figure out how the function μ(a/a0) actually works except that a0 becomes more significant with respect to an increase in the value for 'r' as it reduces 'a' to a lesser value than a0 = 1^-9m/s^2, a very tiny value.

So the equation has terms in it I am unfamiliar with:
∇ - ?
ρ - this is a function for the spread of mass in a galaxy is it not? If so how does it work?
I don't see the symbol to the right for gravitational potential as written in the function

Any thoughts?

 

[Jonathan Scott]

I was looking at the MOND equation, it looks like the adjustment is 'hidden' at smaller scales allowing Newtonian mechanics to work on our scale as the function brings it's value to 1 while adjusting to increased values for 'a' as the effects of gravity would become weaker as distance 'r' is increased.

I can not figure out how the function μ(a/a0) actually works except that a0 becomes more significant with respect to an increase in the value for 'r' as it reduces 'a' to a lesser value than a0 = 1^-9m/s^2, a very tiny value.

So the equation has terms in it I am unfamiliar with:
∇ - ?
ρ - this is a function for the spread of mass in a galaxy is it not? If so how does it work?
I don't see the symbol to the right for gravitational potential as written in the function

Any thoughts?

The symbol ∇ or "nabla" is used as the mathematical operator called "Del" which is the vector differential operator, used as a short notation for the differential operators grad, div and curl (depending on whether it is applied to a scalar, or to a vector via dot product, or to a vector via cross product). If you don't know about those, it's probably beyond the scope of this forum to explain. Technically, it is equivalent to a sort of vector with the following partial derivative operator components:
$$
\left ( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right )
$$
For example, if you apply the gradient operator to the gravitational potential, you get the vector field describing the gravitational acceleration.

The symbol ρ in the MOND article is simply the local density of mass (in mass per unit volume).

 

[mesa]

If you don't know about those, it's probably beyond the scope of this forum to explain.

Lets give it a shot. Where would you like to start?

 

[123 mark]

Sorry if I'm in the wrong section to ask this question.

I'm trying to find out when astronomers discovered that the solar system oscillates through the galactic plane. I just can't imagine the Mayans having the ability to determine that it actually occurs on a 26,000 (or whatever) year period.

Thanks for the consideration

123 mark

 

[twofish-quant]

Lets give it a shot. Where would you like to start?

Let me just sketch out the problem.

You have a function X. You have a set of rules to convert that function into another function Y.

The problem is here is that learning what those rules are is a one semester course in calculus. Look at 18.02 on MIT OCW.

 

[mesa]

Let me just sketch out the problem.

You have a function X. You have a set of rules to convert that function into another function Y.

The problem is here is that learning what those rules are is a one semester course in calculus. Look at 18.02 on MIT OCW.

So let's start with the basics and go from there, how do they calculate for ρ? Do they just take the average for the mass of the entire galaxy or is it based on what is inside the area swept by a star?

With DMT I was told by an astrophysicist that the dark matter is put into the model and essentially is an adjustment to the mass to have the stars match Newtons gravitation formula. Is that right? Is Gm/r^2 modified in any way?

 

[Jonathan Scott]

So let's start with the basics and go from there, how do they calculate for ρ? Do they just take the average for the mass of the entire galaxy or is it based on what is inside the area swept by a star?

The local density of matter in the form of stars or gas is estimated from the luminosity of that part of the galaxy in various parts of the spectrum.

The Newtonian acceleration is then calculated in the usual way by integration (summing the effect of all the mass). With spherical symmetry, Newtonian gravity would simplify to being equivalent to having all the mass inside a given orbit concentrated at the center, but for galaxies the shape is more complicated. The MOND acceleration can then be calculated in terms of the Newtonian acceleration.

With DMT I was told by an astrophysicist that the dark matter is put into the model and essentially is an adjustment to the mass to have the stars match Newtons gravitation formula. Is that right? Is Gm/r^2 modified in any way?

Yes, Dark Matter simply adds additional invisible source mass obeying the standard Newtonian gravitational formula (as an approximation to GR).

A very weird feature of the MOND rule is that for a wide range of galaxies it correctly predicts the velocity distribution based only on the distribution of visible matter. If dark matter is the real explanation, this suggests that the distribution of the dark matter in galaxies must somehow be strongly linked with the distribution of the visible matter in such a way as to reproduce the MOND result, but so far there is no theoretical explanation for this.

 

[mesa]

Sorry it took a few days to get back to you, had finals last couple days.

The Newtonian acceleration is then calculated in the usual way by integration (summing the effect of all the mass). With spherical symmetry, Newtonian gravity would simplify to being equivalent to having all the mass inside a given orbit concentrated at the center, but for galaxies the shape is more complicated. The MOND acceleration can then be calculated in terms of the Newtonian acceleration.

I'm a little surprised that would work, how is the integretion setup? Is it a function of the gravity of each sun and it's affect on the next by putting together an artificail layout based on average distances apart or is it simply the sum of all the masses thrown into the center for the swept area of the galaxy by a particular star?

I was told by an astrophysicist that it has only been recently that papers were published changing the model from a spherical density to a more disc like shape, I found this surprising as well.

A very weird feature of the MOND rule is that for a wide range of galaxies it correctly predicts the velocity distribution based only on the distribution of visible matter. If dark matter is the real explanation, this suggests that the distribution of the dark matter in galaxies must somehow be strongly linked with the distribution of the visible matter in such a way as to reproduce the MOND result, but so far there is no theoretical explanation for this.

That's very interesting, so MOND at least is able to show a possible correlation between matter and dark matter (that is if DMT is correct).