# Could stronger gravity replace dark matter?

First up, I am not a physicist (just a meek mathematician and engineer). I was watching a television show last night on Discovery Channel, where they were summarizing and simplifying the current train of thought in cosmology. Therein, the expounded how dark matter is needed to hold spiraling galaxies from spinning apart.

Then I struck me that perhaps a revised equation of gravity (Newton's inverse square law) might be able to also explain the observed "adhesion" within the galaxies. Here is the idea:

- The force of gravity, F.g, which is holding galaxies together is stronger than Newton's equation (F.g=G*M1*M2/r^2) but only over large distances, when r is of the magnitude of galaxies. So, 1/r^2 term would be replaced by some polynomial that doesn't deminish at the same rate as 1/r^2, such as [C/r + 1/r^2] where C is a constant.
- The centrifugal force, F.cf, which is trying to fling the galactic stars out of their orbits, remains the same, as per classical mechanics, F.cf = d(M1*V)/dt
- The centripetal force, F.cp, which matches gravitation force would remain the same, as per classical mechanics, F.cp = M1*V^2/r.

Nabeshin
Looks like you've stumbled upon MOND (MOdified Newtonian Dynamics).

http://en.wikipedia.org/wiki/Modified_Newtonian_dynamics

In general, dark matter is a much more accepted explanation than throwing away Newtonian dynamics. We have more advanced theories which reduce to ordinary Newtonian mechanics in the respective limits. Therefore it would be a theoretical upheaval indeed if MOND were correct.

Thanks.

1) Could you summarize why dark matter theory is so well accepted?

2) Do I understand right that, in theory, dark matter predominantly resides on the periphery of a galaxy's planar boundary?

Janus
Staff Emeritus
Gold Member
Thanks.

1) Could you summarize why dark matter theory is so well accepted?
It matches the observations that best. One of the reasons that MOND has failed to get a footing is that you can't get it to fit observation.
2) Do I understand right that, in theory, dark matter predominantly resides on the periphery of a galaxy's planar boundary?

No, it forms a roughly spherical cloud in which the visible galaxy is imbeded.

It matches the observations that best. One of the reasons that MOND has failed to get a footing is that you can't get it to fit observation.

No, it forms a roughly spherical cloud in which the visible galaxy is imbeded.

re bold: This is a semi-tangent... that's very much like a obsolete model of the atom, but I forget the name of the modeler! I remember my physics teacher decades ago saying, "like raisins in pudding". It just occurred to me that it's not a bad way to describe CDM's relation to our galaxies, although obviously its just as flawed.

Nabeshin
re bold: This is a semi-tangent... that's very much like a obsolete model of the atom, but I forget the name of the modeler! I remember my physics teacher decades ago saying, "like raisins in pudding". It just occurred to me that it's not a bad way to describe CDM's relation to our galaxies, although obviously its just as flawed.

Thompson's Plum Pudding model?

http://en.wikipedia.org/wiki/Plum_pudding_model

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D H
Staff Emeritus
It matches the observations that best. One of the reasons that MOND has failed to get a footing is that you can't get it to fit observation.
That's not necessarily true, is it? From what I've read MOND, or a relativistic extension, can be made to fit observation.

You just need to add dark matter.

That's the heart of the problem. Various people have created some rather complex models, chock full of ad hoc tuning parameters, and they still need to invoke dark matter to make the model match a set of observations that are outside of the suite of observations used to set those tuning parameters.

[strike]Compare that to general relativity, which has one tuning parameter, and that one tuning parameter is exactly the same as the one tuning parameter used in Newton's law of gravitation.[/strike]

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Compare that to general relativity, which has only two tuning parameters, one of which is G, the same tuning parameter used in Newton's law of gravitation. The other is the cosmological constant Λ.

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The reason why gravity obeys the inverse square law is that the total gravity is the same at any distance d from the object, but the gravity becomes spread out along the surface of the sphere of radius d from the object, and surface area is 4pid^2. However, the universe is not completely euclidean, and over large distances and where gravity is strong, the curvature of space makes the surface area of a sphere slightly less than 4pid^2, and therefore gravity is observed to be slighly stronger. However, on an even larger scale outside galaxies the surface area of a sphere could be greater than 4pid^2 if the universe is saddle-shaped.

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Well we don't know if the universe is curved. Also, you're kind of forgetting that Einstein came up with a better model for gravity, one that's been tested many times already. I haven't done any simulations or calculations, but it looks like you'd have to bend the universe a considerable amount to be able to even see a small difference in gravitational potential. That being said, most observational experiments point to the universe being flat to a very small error margin (though this could mean that it's curved, but nowhere near enough).

MOND works in some situations but utterly fails at explaining the bigger picture. Of course there's lots of work to be done, they say, but there are a few examples (look up some information on the Bullet Cluster) that make it extremely difficult for MOND theorists to explain, if not impossible.

Chronos
Gold Member
I fail to see how gravity 'knows' how far away it is from its source under MOND. A point source force that falls off with the square of the distance seems a lot more sensible. Applying the logic of MOND to the electromagnetic force would have stars increasing in luminosity with distance.

Jonathan Scott
Gold Member
I fail to see how gravity 'knows' how far away it is from its source under MOND. A point source force that falls off with the square of the distance seems a lot more sensible. Applying the logic of MOND to the electromagnetic force would have stars increasing in luminosity with distance.

Something very similar to MOND arises trivially if you assume that the universe is finite and that the solid angle deficit of the boundary around a given part of it is proportional to the enclosed mass m, so the distant asymptotic limit of the shape of space-time around a given central mass is slightly "conical" rather than flat. The only parameter is the "effective mass of the universe" M as used in the angle ratio and the result matches MOND when that's of the order of 10^54 kg. This gives an extra acceleration as follows:

$$\frac{c^2}{r} \, \sqrt{\frac{2m}{M}}$$

However, that something does not "turn off" above a minimum acceleration, which is a feature of MOND that I find totally implausible.

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