Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: Ken Hughes on 29/05/2011 14:32:03
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Specifically, if the gravitational effects within a hypothetical thin sphere of matter are zero at all internal locations, then what is the justification for the argument that the CDM halo speeds up the rotation of stars?
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The distribution of dark matter is believed to be, more or less, a solid sphere−−not a hollow one. A solid sphere can be divided, mathematically, into many concentric hollow spheres. Let's suppose that Newton's law of gravitation is accurate at the scale of galaxies. If so, gravity inside a hollow spherical shell (due to the shell) is zero, but gravity outside the shell (due to the shell) is the same as the total mass of the shell concentrated at the center. The mass, M, of a solid sphere of uniform density is proportional to r³.
For a body whose maximum distance from the center is r, the orbital period, P, (of a circular orbit) is determined by the formula P²= 4π²a³/MG , where a is the semimajor axis, M is the mass contained inside the sphere of radius r, and G is the universal gravitational constant. (I specify a circular orbit because M will vary for an elliptical orbit inside a solid sphere.)
If all the mass were concentrated at the center, M would be constant, and the orbital period, P, at radius r would be proportional to ∛r². (That is Kepler's 3rd law (http://en.wikipedia.org/wiki/Kepler's_laws_of_planetary_motion#Third_law).) But if M is proportional to r³, the orbital speed will be proportional to r^2. Instead, the orbital period at r (near the outer arms of a galaxy) is roughly proportional to r. Therefore, M must be roughly proportional to r², instead of r³, and the galaxy's density, including both visible matter and dark matter, must be roughly proportional to ∛r².
Math ain't my strong point. I'm sure somebody will be able to find an error in this analysis.
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Hi,
Thanks for that.
The thing that surprises me is that you say the halo is effectively solid! Well not solid exactly but with constant density from the galactic center outwards? I always thought that a halo was exactly that, like a shell, albeit thicker than our fictitious empty shell and with the stars rotating within a void at the centre.
To me, this sounds like the Aether again with some magical undetectable substance, yet to be identified with unheard of properties that allow all the stars to move through it, invisible, non reflective, and so on.
I must say, I remain very sceptical about the very notion of CDM. We cannot justify it with the Neptune analogy, since all we did with Neptune was to forecast another planet, a type of known entity, whereas with CDM, we've invented something new, anything, with any properties required to make it work, however unlikely. I question whether the ultimate fate of CDM might be similar to the now defunct Aether.
What do others think?
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Ken,
I sympathize with your skepticism. In case you missed it, I did preface my analysis with, "Let's suppose that Newton's law of gravitation is accurate...." You are right; dark matter was introduced as a fudge factor, meant to rescue Newton's law of gravitation. After all, the law of gravitation is among an astronomer's favorite security blankets−−along with conservation of energy, momentum, etc. Heavens forbid that any of our laws be wrong!
I personally doubt that dark matter is as plentiful as they say, but I also doubt that we can see all the matter that is present. As for stars magically moving thru the dark matter, you seem to be neglecting how thin the dark matter is; even the upper estimates make it a better vacuum than anything we can produce on Earth. Stars may be dense, but look at how small they are compared to the distance between them. With a handful of exceptions, all the visible stars are but one pixel in our best photographs.
As for objects moving thru the æther, I have my own æther model (http://www.thenakedscientists.com/forum/index.php?topic=37454.0), in which fundamental particles of matter consist of orbiting pairs of photons, held in orbit by the Higgs force. Particles move thru the æther as easily as photons because particles ARE photons. Masses don't drag the æther because waves don't drag their own medium.
EDIT: To change external link to internal. I thought I was on another site, where the rules are more relaxed.
EDIT to add: The term "halo" has deep roots in religious lore. It is probably synonymous with the term "aura". Many people claim to see auras, which they describe in a variety of ways. Many of the religious paintings (http://en.wikipedia.org/wiki/Halo_(religious_iconography)) that depict auras or halos probably were drawn by artists who had never seen one, but only heard second and third hand accounts of what they look like.
I may have actually witnessed such an aura, myself, a couple of times. What I "saw" was more of an awareness than than a visual sensation. On the campus of Arizona State, I saw a cute chick about 100 meters away, and it was like she filled a sphere ten meters across with her presence. Not far from there, on another occasion, a Bible thumper had been shouting his message, and the whole area where he had been was permeated by an impenetrable repulsive force field. I couldn't see it, but I perceived it as a cold, dark cloud. The area was normally crowded, but for several hours after the Bible thumper left, it was like a giant hole in the crowd.
The term "halo" is also applied to certain optical phenomena (http://en.wikipedia.org/wiki/Halo_(optical_phenomenon)) associated with the sun, moon and clouds. Some of them are bright rings, dark in the middle; others are difuse glows.
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Hi Practality,
Thanks for your response. I want to state that I believe in Newton's laws being applicable anywhere and everywhere, since they are fundamental laws (or at least, rules) and so are not negotiable.
Whatever we observe, however contradictory it may seem, our first responsibility is to try and explain these observations using known science, and that means Newton.
I think we have neglected this scientific approach with the Galactic rotation anomaly and perhaps hurriedly "invented" a convenient excuse to explain the problem away.
Our first question should have been;-
"How can we explain these (flat curve) rotation speeds whilst at the same time still obeying Newton's rule of gravitation".
THIS is the question we must address BEFORE we invent any new entity or give up on Newton.
If I continue, I will get driven into non mainstream arguments, which is not surprising, since the answer to this question is unknown and will, of necessity require an expansion of our understanding.
Regarding the halo question, I merely referred to the current scientific misuse of the word.
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Ken, the notion of dark matter and its distribution is to explain that very question you ask. It does not give up on Newton at all but is to try to remain consistant with it. The distribution of Kinetic energies and potential energies withing any closed system (which galaxies and clusters of galaxies are, at least to a reasonable approximation) follows that given by the Virial Theorem. To fit in with observations, and Newton's laws, a distribution of dark matter as a three dimensional approximate sphere of roughly even density seems to be the simplest "fit". I don't think there is any easier explanation that does not throw up contradictions with physics as we know it, although there are an probably a lot (an infinite number) that could be invented that could reproduce the observed effects with changes to the laws that we currently know (or think we do).
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Hi Graham,
"THIS is the question we must address BEFORE we invent any new entity OR give up on Newton."
I used "OR" as the operative word. I agree, CDM does not give up on Newton. It's just that the alternative postulates do (MOND, etc) and I wanted to dismiss them for the time being since they are as yet, un proven.
Regarding CDM, we have indeed invented a new entity to try and match observations with the known laws of physics.
Occam's Razor guides us to avoid doing this until we have explored alternatives which already match existing science and do not require new entities. This is what I am saying we have failed to do.
My question should have read;-
"How can we explain these (flat curve) rotation speeds whilst at the same time still obeying Newton's rule of gravitation and without inventing any new entities".
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"How can we explain these (flat curve) rotation speeds whilst at the same time still obeying Newton's rule of gravitation and without inventing any new entities".
There aren't a lot of options left after those two. The point of the rotation curve data is that we don't see enough matter in galaxies to keep objects in orbit around the galactic center at the speeds they are seen to be moving. So our options are:
1. There is more matter than what we see. This violates your no new entities requirement. Now if you are specifically uncomfortable with non-baryonic matter, MACHOs may be able to solve the galactic rotation curve problem. There are other forms of the "dark matter problem" that don't seem to be solvable with purely baryonic dark matter. However, I'll concede that there are other theoretical assumptions (e.g. Big Bang nucleosynthesis) that are being made there that you might take separate issue with.
2. We see all the matter that is there, but the force of gravity does not drop off as quickly as we expect, i.e. the Newton is wrong option.
Moving into new territory
3. We see all the matter that is there, and the force of gravity is as Newton predicts, but we are just underestimating how massive the stars are. If this were only an issue with our models of stars, this might be a good option. The problem is the density of the alleged missing mass does not track with the density of stars. Which means that stars that look identical to us can't necessarily have the same or even similar masses. Furthermore, for some reason the more massive than average stars are biased towards being farther from galactic center.
4. We see all the matter there is, our estimate of the mass is correct, Newtonian gravity holds, but our measurements of the rotation curves are faulty. The problem here is that we think we understand the Doppler effect pretty well, so why isn't it a reliable way of measuring rotation speed. Furthermore, it runs into the problem that the inaccuracy of our measurements for some reason increases as we look further from galactic center.
5. We see all the matter there is, our estimate of the mass is correct, Newtonian gravity holds and we've measured the rotation curves correctly, but the matter is not in fact in stable (or at least not in near circular) Keplerian orbits. This requires us to be living at a miraculous point in time where the galaxies we observe have everything moving just so for them to look like the stars are orbiting more or less circularly, whereas they have at best wildly eccentric orbits around galactic center and at worst will be flying off into intergalactic space.
6. We see all the matter there is, we estimate its mass correctly, we've measured the rotation curves correctly, and it moves in Keplerian orbits under the influence of Newtonian gravity, but Newton's second law is in fact faulty (or inertial mass is not always equal to gravitational mass, take your pick) so it requires less force to provide stars with the necessary acceleration. Unfortunately, this requires the ratio of force to acceleration to depend on distance from galactic center, i.e. either Newton's second law varies from place to place or the relationship between gravitational and inertial mass varies from place to place.
Maybe you can think of an alternative that I didn't. For my own part I listed these options in order of leas objectionable (in my opinion) to most, so the two options you are wanting to get away from strike me as the best of the lot.
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I want to state that I believe in Newton's laws being applicable anywhere and everywhere, since they are fundamental laws (or at least, rules) and so are not negotiable.
In the last three centuries, we have accumulated a lot of new evidence that might have change Newton's mind if it were known to him. We must not canonize any scientific "law" to the extent that it is immunized from future discoveries to the contrary.
To avoid unnecessarily complicating the theory of gravity, we should consider all the alternative explanations and ultimately choose the most elegant one, even if it turns out to be a combination unseen mass and missing parts of the formula. Occam's razor should be seen as a suggestion, not a law.
The thing that surprises me is that you say the halo is effectively solid! Well not solid exactly but with constant density from the galactic center outwards?
The orbital speed near the rim of a galaxy is nearly constant, meaning that the orbital period out there is directly proportional to the radius. If the density were constant, the orbital period would be proportional to r². To get a period proportional to r, you need to have density proportional to ∛r². So the halo gets thinner as you get farther out.
If the density were constant at all distances, all the way to infinity, you would run into the falacy of Newton's spheres. The gravity inside a thin spherical shell of uniform density and finite radius is zero, but you can't extrapolate that to include infinite space. No matter how large the radius of the finite sphere, the infinite mass outside the sphere must always cancel the gravity from inside the sphere. I don't have a rigorous proof up my sleeve, since I'm not a mathematician, but it is intuitively obvious; in a homogeneous, isotropic and infinite universe, every point is equivalent. Therefore, every point must be at the same gravitational potential, and the acceleration of gravity is the gradient of that potential. The gradient of a constant is the zero vector.
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To avoid unnecessarily complicating the theory of gravity, we should consider all the alternative explanations and ultimately choose the most elegant one, even if it turns out to be a combination unseen mass and missing parts of the formula. Occam's razor should be seen as a suggestion, not a law.
I'm not sure we should be considering all the alternative explanations. Perhaps we should start with the most elegant and work our way out from there? I think that's the general reason why modifying Newton's laws/general relativity wasn't the first recourse. Of course, if simpler models are exhausted, there's no reason why Newton's laws can't change if that explains things best.
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Exactly, JP. I now understand this is what you are saying to, Ken: sorry, I misinterpreted what you meant.
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Ken, I split off your latest post to New Theories since it was proposing a non-mainstream new theory of time dilation to explain the anomalous galactic rotation curve.
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OK guys, go to New Theories for my response.
What is the title?
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Same title as this thread. You're welcome to change it by clicking "edit post" if you'd like: http://www.thenakedscientists.com/forum/index.php?topic=39624.0;topicseen
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Thanks,
Ok guys, the title now is:-
"Explanation of the galactic rotation anomaly without CDM"