#RotationCurves

[scepticaladventure]

16  A Machian Solution for the Physics of Spiral Galaxies  27Aug17

Preface:  This is a long blog written in slightly more formal style.  If its central suggestion has merit it could make Cold Dark Matter unnecessary.

Abstract : There are two main attempts to explain the apparently anomalous rates of rotation of stars in the discs of spiral galaxies. One approach assumes the existence of large amounts of exotic cold dark matter in the halo of the galaxies. The other modifies either the force of gravity or the acceleration of the stars at their very low rates of rotation. This paper presents a third hypothesis. It suggests that the rotational velocities observed in spiral galaxies are consistent with Kepler’s Law if careful attention is paid to the rotational reference frames in which the motion of such stars is described. It suggests that the rotational reference frames for stars in galactic orbits have a relationship to the vast amount of rotating matter in the local galaxy and not just to the reference frame provided by distant galaxies. The hypothesis lends itself to experimental and observational tests. If verified, the hypothesis could help resolve several major issues in our understanding of the physics of spiral galaxies.

Keywords:  Rotational reference frames, Mach’s Principle, galactic rotation curves, spiral galaxies, cold dark matter, Kepler’s Law, Modified Newtonian Dynamics, MOND, Mixed Rotational Reference Frames Effect.

Email summary:  This paper suggests that the physics of spiral galaxies becomes simpler under a hypothesis that the appropriate reference frame for rotations has a relationship to the vast amount of rotating matter in the local galaxy and not just to the reference frame provided by distant galaxies. The hypothesis lends itself to observational and computational verification and if verified would reduce the need for cold dark matter or modifications to Newtonian dynamics.

Introduction

For stars in the disc of a spiral galaxy, the acceleration due to gravity at orbital radius r is GM/r2   where M is the collective mass of the matter inside a sphere of radius r, and G is the gravitational constant.

If external observers use Kepler’s Law and a reference frame aligned to the distant universe, they calculate that for the stars to be in their observed orbits they should have a centrifugal acceleration equal to v2/r. Hence

                                          GM/r2  = v2/r                                        (1)

In the late 1960s and early 1970s, Vera Rubin, an astronomer at the Department of Terrestrial Magnetism at the Carnegie Institution of Washington, worked with a new sensitive spectrograph that could measure red shifts of stars in the discs of edge-on spiral galaxies to a high degree of accuracy. At a 1975 meeting of the American Astronomical Society, Rubin announced that the stars in the discs of spiral galaxies travel at roughly the same speed as each other, instead of decreasing in proportion to the inverse square root of r as predicted by Kepler’s Law. Rubin presented her results in an influential paper in 1980. (Rubin, Thonnard & Ford, 1980).

Although initially met with skepticism, the results have been confirmed by data gathered from tens of thousands of stars in millions of galaxies, including those able to be observed in detail by space based telescopes e.g. (M. Persic,1996).

Rubin suggested that the evidence implied that galaxy masses must grow approximately linearly with radius well beyond the galactic bulge. This developed into a hypothesis that there must be an enormous amount of hitherto undetected dark matter in the halo of spiral galaxies.

A large amount of theoretical and experimental work has been undertaken since 1980 to put parameters around this hypothesis and into inventing exotic types of matter that might fit such parameters. It was calculated that most of the mass in galaxies must consist of dark matter over and above normal dark matter such as dust, cold gases and brown dwarf stars, that the new dark matter has to be “cold” and it has to interact very weakly with normal matter.

An extensive range of experiments have been undertaken over the last 35 years in attempts to find direct evidence for any constituents of the conjectured cold dark matter. In spite of this not being very successful so far, the cold dark matter hypothesis has become a mainstream feature of the most generally accepted model of the universe.

The fact that equation (1) comes from the fundamentals of classical dynamics, yet is violated on a grand scale by direct observational evidence from countless galaxies, is a marvellous opportunity for physics to learn something new. As such, every part of equation (1) and every assumption and derivation on which it rests, ought to be given careful scrutiny.

Though dark matter is the most accepted explanation of the rotation curve problem, other proposals have been suggested. In 1983, Israeli physicist Mordehai Milgrom proposed that modifications to Newtonian dynamics might be able to explain fact that the orbiting speeds of stars in galaxies were observed to be much larger than expected. (For an extensive discussion of the data and its fit to MOND see (Milgrom, 2007) and more recently).

Milgrom noted that the discrepancy could be resolved if the gravitational force experienced by a star in the outer regions of a galaxy was proportional to a modified form of the centripetal acceleration (as opposed to the centripetal acceleration itself, as in Newton's Second Law), or alternatively if very weak gravity varies inversely with radius (as opposed to the inverse square of the radius in Newton's Law of Gravity). 

In Milgrom’s modified Newtonian Mechanics (MOND), violation of Newton's Laws occurs at the extremely small accelerations found in the outer reaches of galaxies and far below anything typically encountered in the Solar System or on Earth.

Both of the above attempts to resolve the issue have problems. The cold dark matter hypothesis struggles to explain the features of spiral galaxies without considerable fine tuning (see for example the Cuspy Halo Problem).

More significantly, nearly forty years of searching have gone by without finding conclusive direct evidence for cold dark matter in any of its predicted manifestations (e.g. massive compact halo objects, or weakly interacting massive particles). Success has been claimed several times, e.g. using interpretations of what is going on in the Bullet Cluster of colliding galaxies, but never conclusively.

The MOND approach and variations along similar lines have enjoyed greater success in explaining the observed features of spiral galaxies, see e.g. (S. S. McGaugh, 2013). However, most physicists are reluctant to accept modifications to the basic laws of physics and MOND is not widely accepted (which by itself is not proof that it is not correct). General Relativity would also need to be modified since it is asymptotically equivalent to classical dynamics. The MOND approach works well at a galactic level but has not yet been able to be used as the basis of a satisfactory cosmological model.

Mach’s Principle

Mach’s Principle was the name given by Einstein to part of the work of the German physicist and philosopher Ernst Mach. Mach was a forerunner of Einstein in realizing that basic physical properties in nature do not have an absolute nature, but can only be described relative to each other. Mach noted that whenever rotational effects were present there was also a rotation relative to a reference frame provided by the “fixed stars” and he postulated that this was not a coincidence.

There have been scores of papers written on and about Mach’s Principle, and there are many versions of it. One interpretation of Mach’s Principle is that distant matter is responsible for local inertia. However, this raises the question of how can stars at enormous distances contribute to inertia here and now?

Einstein tried to include Mach’s Principle into General Relativity. Initially he felt he had included it without requiring action at a distance by making matter responsible for the geometry of spacetime and by applying boundary conditions on the initial–value equations of his geodynamical models that gave rise to inertial reference frames consistent with Mach’s Principle. Opinions amongst astrophysicists on whether Einstein was able to include Mach’s Principle are divided (Barbour & and Pfister, (1995)).

Einstein himself eventually felt he had not succeeded in incorporating Mach’s Principle into General Relativity or any of its resultant cosmologies. And while Einstein was grateful for the inspiration he derived from Mach’s thinking, Mach himself never fully accepted Einstein’s General Theory of Relativity. 

A discussion of how rotational inertia is treated in General Relativity is outside the scope of this paper. However, as an aside, it appears to the author that while the Principle of Equivalence is straightforward for gravity and linear (translational) accelerations, it is not so straightforward for rotations.

A local observer can easily, always and everywhere tell when rotations are taking place and there is no gravitational field that easily produces normal rotational effects, such as the spheroidal surface of the water in Newton’s spinning bucket.

Furthermore, the basic equations of General Relativity survive unharmed if the spacetime metric tensor is split into a background metric tensor plus a local Riemannian tensor in a certain way (Quale,1973).

What are the “Fixed Stars”

The meaning of “fixed stars” in the early 20th century was generally taken to be the stars plainly visible in the night sky. It was not until 1923-29 that Edwin Hubble was able to demonstrate that many of these objects were in fact other galaxies and that our Milky Way is just one galaxy amongst countless others.

Mach, and even Einstein at first, did not know about the other galaxies. Hence Mach did not discuss whether he meant that rotations were always relative to the stars in our own galaxy or the full extent of matter in the Universe more generally. As far as Mach was aware, the stars in the Milky Way constituted the whole cosmos. And without recognising the existence of other galaxies (or the cosmic microwave background) there are no points of reference that make it clear that the Milky Way is itself rotating.

(Our solar system lies within the disc of the Milky Way at a distance of about 28,000 light years from its centre. Our sun is travelling at around 828,000 kilometers per hour. At this speed it takes about 240 million years to complete a full orbit.)

Newton’s laws of motion and gravity work superbly well at explaining the observations and model of the solar system put forward by Kepler. The stars of the Milky Way provide such a convenient reference frame for rotational effects that this is invariably taken for granted.

Rotations in physics generally typically occur in the range of microseconds to hundreds of years. However, on a galactic scale the period of rotation relative to the distant galaxies takes hundreds of millions of years, an increase of six orders of magnitude. On this sort of timescale the usefulness of the “fixed stars” for determining rotations becomes questionable, particularly as many of the stars will no longer be fixed at all, but will be moving in a variety of different directions and rates.

Hence there is an implicit assumption about the appropriate reference frame in Equation (1) that warrants a lot more careful consideration.

It is not unreasonable to imagine that when the scale of a dynamic system under study is a billion times bigger than the solar system, rotations are taking place as slowly as one every 300 million years or so, the masses involved are about 50 billion times bigger than the mass of the sun and the gravitational field strength is extremely weak, then some hitherto undetected effects might be taking place.

Perhaps the acceleration term on the right hand side of equation (1) is the issue. Milgrom’s MOND approach explores this possibility in a certain way and with some success.

This paper will also focus attention on the right hand side of Equation (1), but will do so using an approach in accordance with Mach’s thinking, and without having to violate Newtonian physics.

Significantly, if one considers the motion of the so-called “fixed stars” arising from the orbit of a star around its galactic centre, it is clear that these stars and other celestial objects do not provide a simple and unambiguous frame of reference for detecting rotations. To observers on a star which is not spinning (apart from being in a galactic orbit), most of the stars in the “heavens” appear to be fixed and those which are moving are moving in arcs through the sky at variable rates which are all incredibly slow. Furthermore, the drift rates of any stars that do appear to be moving are doing so at much slower rates than the drift rates of the distant galaxies.

Observers on such a star are then faced with a dilemma – what exactly is the correct reference frame for understanding their own rotational motion around their galaxy?

Observations from a test star in galactic orbit

Consider a non-spinning star in a galactic orbit around the centre of a spiral galaxy. Define a regular Cartesian coordinate system with its origins on the star, the x direction pointed to the galactic centre, the y direction away from the direction of travel and the z direction orthogonal to both. The x and y axes are in the plane of the galactic rim and the star’s orbit.

The x axis can be thought of as the In/Out direction, the y axis as the Back/Front direction, and the z axis as the Up/Down direction. Call the direction of the orbit anti-clockwise. Hence observers on the star will see the distant galaxies rotating in a clockwise direction.

Now consider what observers based on such a star would see in the heavens all around.

The x direction contains the centre of the galaxy and most of the stars in the galaxy. The vast majority of these stars will appear to be fixed. Some of the nearer stars will show variable and very slow drifts in a clockwise direction. Some of the stars in the galactic disc on the other side of the bulge will be observed to be drifting anticlockwise. 

The stars in the front and back directions will be travelling in much the same fashion as the observers’ star. Hence observers will see these stars as being fixed. Likewise the nearby stars in the up and down directions.

There will be relatively few stars in the minus x direction (out through the rim). These will be orbiting more slowly than the test star and so will be observed to be slowly drifting in a clockwise direction. The further these stars away they are the faster they will be drifting.

If observers on the test star look beyond the vast number of stars in the local galaxy they will be able to see a vast number of distant galaxies. All of these galaxies will be wheeling in slow circular arcs about the z axis in a clockwise direction. This motion will be faster than any drifts of local stars, but will still be very slow e.g. of the order of 1 degree of arc per million years.

If the observers want to set up a non-rotating reference frame on their orbiting star they will be faced with a dilemma. Should they use the evidence from the vast preponderance of stars in their local galaxy? Or the evidence from the distant galaxies? Or something in between?

References to 20th century physics textbooks will not help because these invariably make simplistic assumptions about such matters.

In principle the observers could resort to experimental determinations. An incredibly sensitive Foucault pendulum or gyroscope could reveal whether or not a local reference frame was rotating. Or a comet orbiting their star with a long elliptical orbit in the x-y plane might reveal the answer through the precession of its perihelion. There might also be a slight distortion to light emerging from the centre of a galaxy and out through its rim. The observers could also project a satellite out through the rim and observe its path in very close detail.

The main difficulty with such experiments is that the effects would be very small because the angular velocities in question are so incredibly tiny. So for the moment we will resort to a hypothesis. There is a chapter at the end of this paper discussing a range of theoretical, computational and observational tests that could possibly test hypotheses about how the local inertial reference frame should be decided.

We will hypothesize that the reference frame that experimenters would discover is non-rotating (and hence free of effects such as Coriolis forces) would be one that has a significant correlation with the vast amount of rotating matter in the local galaxy and not just to the reference frame provided by distant galaxies.

Mixed Rotational Reference Frame effect (MiRRFe) hypothesis

In more detail, the MiRRFe hypothesis is:

The difference between the externally observed rates of orbit of stars in the discs of spiral galaxies and the rates predicted by Kepler’s Law is due to a dragging of the rotational reference frame for the stars in question. 

Localised observers will be able to establish that their local non-rotating reference frame is being dragged around and spun around at an extremely slow rate through an largely connected to the gross movements of matter in their local galaxy. 

Relative to the actual true localised inertial reference frame, the orbital speed of the stars is consistent with Kepler’s Law. 

The orbiting stars experience centrifugal forces related to a smaller rate of rotation than is apparent to external observers. 

There is no need to assume that spiral galaxies contain large amounts of cold dark matter. 

The Newtonian equation for gravity and Kepler’s Law do not need to be modified to account for galactic rotation curves. 

A better understanding of the interaction between matter, rotational inertia and rotational reference frames will help to explain the evolution and structures of spiral galaxies.

Matter, gravitational mass, gravity, inertial mass and rotational inertia all emerged in an inter-related way during the evolution of the universe.

It should come as no surprise that the hypothesised frame dragging effect is not readily noticeable – the angular velocity involved in this relative motion is of the order of 10-8 radians/year and the inertial effects at a local scale are almost immeasurable. It is only on the scale of the galaxy itself that they become very noticeable.

The hypothesis does not necessarily imply that rotating galactic matter is directly responsible for dragging the local inertial reference frame around with it. Although this would be the most obvious explanation if the MiRRFe hypothesis turns out to have validity, it would be premature to rule out more exotic explanations. For example, it might be that spiral galaxies form because of a “whirlpool in inertial spacetime” and not the other way around.

The MiRRFe hypothesis is not at variance with the Principle of General Covariance. In essence this principle states that physical outcomes should not change if the reference frame in which they are studied is moved or rotated. Physics is immune to being rotated - but it is not immune to becoming rotating!

(Note - the English language as used by normal people and even most physicists is a bit confusing when it comes to revolutions, once off rotations, changes from static to rotating states, rotational accelerations, spins about an internal axis, orbits around an external axis and so on.)

MiRRFe applied to spiral galaxies

Spiral galaxies can be thought of as large concentrations of matter in their bulge, surrounded by spiralling arms in a broad flat disc extending well beyond the luminous region, all enclosed in a spherical halo and quite often featuring a bar across much of the disc. For purposes of analysis, the disc can be thought of as concentric thin rings of matter each rotating at lower and lower angular velocities as r increases. These rings contain luminous stars, luminous gases and normal dark matter but as r increases they consist mainly of hydrogen, helium and dust. The spherical halo surrounding the budge and disc is relatively free of stars.

Redshift data provides evidence of tangential velocities. External observers and observers travelling with the stars in question can agree on these velocities, and also on the radius r to the galactic centre. So both sets of observers will conclude that the stars at radius r have angular velocity w(r).

For stars in the rim of a galaxy, the acceleration at orbital radius r due to gravity is GM/r2 where M is the collective mass of the matter inside a sphere of radius r. Most of this mass is inside the core of the galaxy.

If the external observers use Kepler’s Law and a reference frame aligned to the distant universe, they will calculate that for the stars to be in their observed orbits they should have an angular velocity equal to √(GM/r)/r. However this value is much lower than the angular velocity observed in practice.

This is the conundrum of galactic rotation curves. GM/r2 is supposed to equal w2r but the latter is observed to be anything up to an order of magnitude too large.

The cold dark matter hypothesis responds by boosting M with dark matter. The MOND hypothesis responds by replacing G/r2 with something stronger, or by weakening the effect on stellar matter of the acceleration term w2r.

The MiRRFe hypothesis suggested here is that the issue is resolved by a better understanding of the rotational reference frames that apply for the dynamics of spiral galaxies.

Frame Dragging

The concept of inertial frames of reference being dragged around by rotating masses is not new. Einstein predicted a relativistic frame dragging effect for the interior of a massive thin shell and local neighbourhood of a massive rotating object. In 1913 he wrote a letter to Mach in which he said:  

“For it necessarily turns out that inertia originates in a kind of interaction between bodies, quite in the sense of your considerations on Newton’s pail experiment... The following additional points emerge: (1) if one accelerates a heavy shell of matter S, then a mass enclosed by that shell experiences an accelerative force. (2) If one rotates the shell relative to the fixed stars about an axis going through its center, a Coriolis force arises in the interior of the shell; that is, the plane of a Foucault pendulum is dragged around (with a practically immeasurably small angular velocity).”  (C. W. Misner, 1970)

In 1918 Austrian mathematicians Lense and Thirring showed that there is a relativistic frame spinning effect in the external vicinity of a massive rotating sphere of matter. The Lense Thirring frame dragging effect can be visualized as the motion of a small planetary gear on the shoulder of a large rotating gear wheel. It is a small relativistic effect for intense gravitational fields.

Hence the idea that the enormous amounts of matter in a spiral galaxy might have a substantive influence over the establishment of rotational reference frames in the disc of the galaxy is perhaps not unreasonable.

The MiRRFe hypothesized here is a small effect on a very grand scale. It takes place in very low gravitational fields and very low rates of rotation in the outer reaches of galaxies.

The local rotational reference frames are hypothesised to be substantially fixed in orientation within the local galaxy but with a slow rotation in the direction of the apparent motion of distant galaxies.

MiRRFe applied to galactic rotation curves

The diagram below is for the Milky Way and is fairly typical of the galactic rotation curve issue.

In practice it is easier to generate this curve for nearby galaxies such as Andromeda or M33 than for our own Milky Way due to large clouds of dust, the masking effect of the galactic bulge and the need to correct for all the relatively large motions of the Sun and Earth for telescopes based in Earth orbit.

One feature that is more apparent here than on most other rotation curves is the wavy feature of the velocity curve as r increases. The rotation curves for several thousand spiral galaxies can be found on-line.

[Data for several million galaxies are available from huge online databases e.g. Hypercat (Lyon/Meudon Extragalactic Database http://www-obs.univ-lyon1.fr/hypercat/) classifies references to spatially resolved kinematics (radio/optical/1-dimension/2-dim/velocity dispersion, and more) for 2724 of its over 1 million galaxies. NED (NASA/IPAC Extragalactic Database /index.html) contains velocities for 144,000 galaxies.]

Fig 1: Observed velocity curve (in this case for Milky Way) and rate expected if Kepler’s Law is applied in a frame at rest to the distant universe. (The diagram is taken from a Wikipedia article on galactic rotation curves.) In similar diagrams for other galaxies the turning point on the observed velocity curve is closer to the crossover with the calculated Keplerian line. The flat part of the velocity curve can be higher or lower for different galaxies and sometimes has either a gentle slope up or down). The solid line shows the tangential velocities of stars in the rim. The dashed line shows the rotation curve derived from Kepler’s Law under the assumption that the appropriate reference frame for such an analysis is the universe at large.

The MiRRFe hypothesis is that the observed angular velocities arise from a combination of rotational speeds consistent with Kepler’s Law plus an effect arising from a very slow rotation of the inertial frame of reference at each value of r. 

Accordingly there is a need to explain why the hypothesised effect becomes bigger as r increases through the luminous part of the galactic rim and out as far as observations permit.

MiRRFe might arise in various conceptual ways:

One avenue to explore is that the enormous mass of a galaxy (~1041 kg) has, over billions of years of evolution, managed to create a kind of whirlpool in spacetime that affects or determines the non-rotating reference frames and hence rotational inertia for stars in its disc. 

Another, more exotic possibility is that a whirlpool in spacetime is somehow responsible for the formation of the galaxy.

A third possibility is that when the big bang created matter out of energy, it gave rise to gravity, gravitational mass, and inertia (both translational and rotational) in an interrelated way that has persisted ever since.

The section below will give an equation for the frame dragging effect across the full rotation curve along with a suggested physical interpretation based on Mach’s Principle.

Components of frame dragging effect

Consider a test star orbiting at radius r around a galactic centre. Denote the angular velocity with respect to the universal reference frame as observed by external observers by we(r). However, observers on the star think that they have an angular velocity of wk(r) in accord with Kepler’s Law. Denote the difference by wg(r).

With an appropriate sign convention    we(r)  =  wk(r) + wg(r)                      (2)  

i.e. the observed angular velocity relative to the distant galaxies is the sum of the angular velocity according to Kepler’s Law, plus angular velocity wg(r) that can be interpreted as the rate at which local reference frames are being rotated at radius r, as viewed by external inertial observers.

(Conversely, observers on the star in question are entitled to think that wg(r) is the rate the rate at which the distant galaxies, their own galactic core and everything else is rotating around them.)  

From equations (1) and (2)         wg(r)  =  ve/r - √(GM) / (r√r)                        (3)

From observations we know that ve is approximately constant. Choose to express this as ve = k√(GM) where k is a value derived from observation for each galaxy under consideration.

Detailed analysis of the large number of galaxies for which data has been gathered might enable k to be replaced by a general expression that applies to all spiral galaxies. (Note that Milgrom’s MOND theory derives ve4 = (a0GM) where a0 is a new constant.)

Hence wg(r)  has the form  wg(r)  =  k√(GM) / r – √(GM) / (r√r)                    (4)

This is not profound. We have simply subtracted the Keplerian angular velocity from the externally observed angular velocity. The equation holds for r well outside of the core and as far out as observations have been made so far. M is often treated as being constant, since most of the mass of a galaxy lies in its inner regions, but strictly speaking it increases slightly with r until there is no more matter to be found.

If the equation holds for every test star at radius r then it holds for the whole ring of stars at radius r.

What follows is an attempt to explain the hypothesised frame dragging effect by reference to Mach’s Principle. It is interesting that expression (4) as written has both a positive and a negative component.

A Machian explanation for local frame dragging

It may be helpful to think of a gyroscope in circular orbit at radius r with the axis of the gyroscope pointed towards the galactic centre at time t = 0 and completing a complete orbit (as seen by external observers) in time 

T = 2πr/ve.

The back of the gyroscope points towards some distant galaxies in the minus x direction. If there is no MiRRFe then the back of the gyroscope will remain pointed at the same set of distant galaxies as the gyroscope completes its galactic orbit.

It is conjectured that this would not happen in practice. The MiRRFe hypothesis is that a gyroscope would rotate clockwise by a full circle during the orbit, less a counter rotation that depends on the radius r. And since the gyroscope is in free fall around the galactic core, its behaviour is a good indication of what is happening to the local reference frame.

In more detail the conjecture is that there are two main effects that contribute to the behaviour of the gyroscope. For ease of reference, the two effects will be called the Core Effect and the Distant Galaxies Effect. There may also be some second order effects.

Core Effect: Mach’s Principle associates rotation with movement relative to the “fixed stars” and non-rotation with lack of movement relative to the “fixed stars”. Observers travelling with the gyroscope and facing the galactic centre will see all of the matter in the bulge as being more or less fixed. In fact they would also perceive the stars above and below them, and in front and behind, to be fixed as well.

The Core Effect is a conjecture that a gyroscope pointed at the galactic centre will try to remain pointed at the galactic centre as it completes its orbit. External observers would see the gyroscope rotated by 2π radians every orbit and in the same direction as the orbit itself (which we have called anti-clockwise). The core effect tries to make a local rotational reference frame roll around a complete circle every orbit. The rate at which this happens is

 2π/T = ve / r = k√(GM) / r.

Distant Galaxies Effect:  With the core effect in place a gyroscope at radius r has to travel at the Keplerian rate, relative to this local inertial reference frame, in order to stay in a stable orbit. While it is completing its very slow galactic orbit, observers traveling with it would perceive the distant galaxies to be drifting in a clockwise direction at a relative angular velocity of

wk(r) = √(GM)/(r√r).

The traditional interpretation of such an observation is that the observers would deduce that they themselves must be rotating at this very low rate.

However, our observers are not so sure. They are getting a conflicting Machian message from the galactic core plus most of the rest of the stars in their local galaxy that is telling them that they are non-rotating when they remain oriented to the galactic centre.  

The stars from their own galaxy are telling them to stay oriented towards the galactic centre if they want to be non-rotating but the distant galaxies are telling them to roll around once every orbit so that they remain pointing at some fixed distant galaxy.

This gives rise to a kind of tussle between the two effects. The core effect is trying to keep the gyroscope oriented towards its initial position relative to the core while the distant galaxies effect is trying to keep the gyroscope oriented towards the gyroscope’s initial orientation relative to the distant galaxies.

Net Effect:  A physical interpretation for the MiRRFe hypothesis is that the reference frame with the least evidence of local rotational effects (i.e. the best local inertial reference frame) will be determined the combined influences of the Core Effect and the Distant Galaxies Effect.

External observers will see this to be a rotation of the frame at the rate given by the core effect less the rate given by the distant galaxies effect i.e.                             wg(r)  =  k√(GM) / r - √(GM) / (r√r)                                   (5) which is the same as equation (4).  Which is what we are trying to explain.  

An orbiting star is moving at wk(r) over and above that, making up the balance of the rate of angular rotation as seen by an external observer.

Second order effects:  Since we have argued that local non-rotating reference frames are influenced by large amounts of matter within the spiral galaxy, we need to acknowledge that there might be some second order effects as well.

At time t = 0 consider a line of stars either side of the gyroscope lying in the x and -x directions. The gyroscope is in the middle. The gyroscope and the stars are all lined up like runners in a race around a circular track. As observed and agreed by all, all the stars have the same speed. However, stars nearer the centre have the advantage of a smaller orbit and so by the time the gyroscope completes a lap they will be further advanced. Similarly the stars on the outside will have fallen behind. For every light year that inner stars are closer to the centre they will be 2π light years ahead, and for every light year that outer stars are further from the centre they will be 2π light years behind. Hence the starting line will now not be straight but will be at an angle  = arctan (2π). This will have taken place in the orbital period of the gyroscope = T = 2πr/ve .

So the gyroscope will note that the starting line of stars around it is being rotated at a rate equal to  arctan(2π) ve / 2πr = 0.225 k√(GM) / r              (6)

This is a rotation is the clockwise direction. It is similar to an effect that can be seen in the spinning teacups ride often featured in a traditional fairground roundabout. It may or may not have an influence on the behaviour of the gyroscope. But since the number of stars in the local neighbourhood is much smaller than the number of stars in the core, it is reasonable to assume that any effect like this is of the second order. Another small effect might arise in conjunction with proximity to the galactic bar.

The Galactic Core

Equation (1) only holds for objects outside of the galactic core. Even then M should be recognized as increasing slightly as r increases due to the extra mass of stars in the disc and halo as r increases.

Putting to one side the cold dark matter hypothesis, most of the mass in a galaxy is contained in the central bulge. It is of course not concentrated as a point source right in the middle, but is distributed as a kind of flattened globular cluster.

For inverse square law forces such as gravity and electricity, the Shell Theorem shows that the net effect anywhere inside a sphere is zero. Hence only the matter inside an imaginary sphere of radius r has a net gravitational pull on a test particle. A simple symmetry argument also shows that the net effect at the middle of such a sphere has to be zero.

Calculations of Keplerian orbits are problematic as the stars interact as an n-body system where n is very large. But since the net gravitational effects inside the core are small, the orbital motions are also small. Hence the redshifts related to motions of stars within to core are expected to be small and experimental observations agree with this.

Discussion of Galaxy Rotation Curves with MiRRFe The MiRRFe hypothesis is that the stars are actually orbiting at their correct Keplerian rates relative to their local reference frames and that the local reference frames are turning at a rate equal to the sum of a core effect and a distant galaxies effect. The end result is the tangential velocities of the orbiting stars are as we have observed in practice since Vera Rubin’s work in 1975.

Returning to Figure 1. Note that there is generally a point at which the observed tangential velocity equals the Keplerian velocity, i.e. there is a point where the observed velocity curve crosses the calculated Keplerian velocity curve.

In our suggested explanation for the hypothesised MiRRFe, this occurs when the distant galaxies effect exactly matches and counterbalances the core effect so that wg(r) is zero.

In other words, the Keplerian orbit is so fast that its period equals the observed period. (Fast might be a misleading term as the period may still be in the order of several hundred million years).

From equation (5) we have that this occurs when k√r = 1.

For values of r a bit smaller than this crossover radius, the calculated Keplerian rotation is actually bigger than the observed rate of rotation. Figure 1 suggests that this may in fact be the case for the Sun within the Milky Way. 

Not all calculations of Keplerian curves in galaxies show a radial zone with a negative discrepancy, but many do. It is not clear to this author how adding extra mass in the form of cold dark matter would help to resolve the rotation curve issue for these stars.

Under the MiRRFe hypothesis the explanation is simply that the distant galaxies effect has become bigger than the core effect and so the stars are still at their correct Keplerian rates relative to their correct local rotational reference frames.

For r bigger than the cross over radius the observed tangential velocities tend to be constant, so our hypothesized frame dragging effect has to show an increase. This is demonstrated in Equation (5) due to the extra √r term in the denominator of the second term.

In Machian terms the core effect weakens but the distant galaxy effect weakens faster, in accord with the slowing down in the observed drift rate of the distant galaxies. Thus stars have to “hurry up” to stay in stable orbits and this gives rise to the flat rotation curves reported by Vera Rubin in 1975.

Finally a comment about what happens as r becomes very large. As the local galaxy recedes into the distance it eventually starts to look like any other galaxy. One would expect any frame dragging effects to disappear. This is suggested in expression (4) and (7) by the fact that wg(r) tends to zero as r becomes large.

Potential Tests, Explanations and Predictions

A hypothesis can become a useful theory if it meets three criteria:

1. it provides a plausible explanation for a range of observed effects

2. it makes predictions which turn out to be true, and

3. it is not contradicted by the results of reliable relevant experiments.

(Accordance with existing paradigms and dogma is convenient, but not obligatory.)

The MiRRFe hypothesis would be fundamental to the dynamical structure of spiral galaxies and hence to a lot of other astrophysics and cosmology. Because it is a large scale and rather basic effect, it lends itself to a wide range of tests under each of the above three categories.

As well as providing a novel but not outrageous interpretation of the flat rotation curve issue, the MiRRFe hypothesis may be helpful to a range of other issues in the dynamics of spiral galaxies.

Virial Theorem:  The virial theorem in dynamics implies that the total kinetic energy of an n-body system should be half of its total energy. Applied to galaxies this suggests that their total kinetic energy should be half of the total gravitational binding energy. However, observationally, the total kinetic energy appears to be much greater.

However, if the apparent rotational velocities of the stars are adjusted by a rotational frame dragging effect back to their Keplerian levels, the total kinetic energy of the galaxies would be reduced to levels more consistent with the virial theorem prediction.

Winding Problem:  The winding problem is that since matter nearer to the centre of a spiral galaxy rotates faster than the matter at the edge of the galaxy, the arms would become indistinguishable from the rest of the galaxy after only a few orbits. However, spiral arms in spiral galaxies are clearly quite persistent.

MiRRFe may provide a partial remedy to this issue. It suggests that, viewed in the correct reference frame, the stars in the outer parts of the arms are actually rotating at much lower angular velocities that are consistent with Kepler’s Law. 

Relationship between luminosity and rotational velocity:  The Tully–Fisher relation is derived from observations and shows that for spiral galaxies the rotational velocity is well related to its total luminosity. A consistent way to predict the rotational velocity of a spiral galaxy is to measure its bolometric luminosity and then read its rotation rate from its location on the Tully–Fisher diagram. Conversely, knowing the rotational velocity of a spiral galaxy gives its luminosity. Thus the magnitude of the galaxy rotation is related to the galaxy's visible mass. However, there is not yet any straightforward explanation as to exactly how and why the observed scaling relationship exists. MiRRFe may be able to contribute to such an explanation.

The cuspy halo problem:  The cuspy halo problem is that cold dark matter (CDM) simulation models predict halos have a core which too dense, or have an inner profile that is too steep, compared to calculations of the dark matter densities required by the CDM hypothesis as applied to low mass galaxies.

Nearly all simulations form dark matter halos which have "cuspy" dark matter distributions, with density increasing steeply at small radii, while the rotation curves of most observed dwarf galaxies suggest that they would need to have a flat central dark matter density profile. MiRRFe does not require CDM and hence does not suffer from this problem.

Cosmological models:  A criticism of the MOND approach is that it has not proved to be helpful in constructing full cosmological models of the universe. MiRRFe however, does not modify classical dynamics and MiRRFe itself disappears outside of galaxies.

Having said that, the implications of the MiRRFe hypothesis would extend well beyond the physic of spiral galaxies and would speak to major issues such as what causes inertia to exist in the first place.  This would provide a Machian influence or constraint upon cosmological models more generally.

Computational Tests:  Powerful computers and simulation models have been used for several decades to try to model the evolution of galaxies in an attempt to replicate their observed features. A relatively easy test for MiRRFe is to use it in such models in place of cold dark matter and examine whether reasonable outcomes can be achieved.

MiRRFe predicts substantive effects, e.g. that the angular momentum of spiral galaxies is less than it seems, so the outcomes ought to be noticeable. If not MIRRFe exactly as proposed here, then perhaps variations of the basic idea.

Note however that the MiRRFe would only emerge as the galaxy itself emerges.

In addition to the mainstream family of spiral galaxy types, there are several hundred non-standard spiral galaxies that have been studied in detail with modern high-resolution instruments. MiRRFe may or may not prove useful in helping to understand how their peculiarities came about.

Dynamics of the Core: If the MiRRFe hypothesis is valid it has some predictions for stars inside the core. A star in the core which is not in an orbit with respect to the distant galaxies would still be adjudged to be in an orbit by local observers due to the rotation of the galaxy. Similarly a cloud of stars in the core would feel itself to be rotating on account of MiRRFe and would either rotate in sympathy with the aggregate effect from the rest of the galaxy, or it would feel itself pulled apart slightly. If the later case applies then this might be help to stop the cloud from collapsing.

Test Experiments

In principle, observers could resort to experimental determinations

Sensitive Gyroscopes: An incredibly sensitive Foucault pendulum or gyroscope could reveal how fast a local reference frame is rotating in response to galactic orbital motion. Comets:  A comet orbiting a star (e.g. the Sun) with a long elliptical orbit in the plane of the galactic disc would demonstrate a MiRRFe in the precession of its perihelion. Similarly a satellite launched at very high speed towards the rim or centre of the Milky Way would tend to be swept around slightly by the frame dragging effect. Supernovae:  Assume the pattern of matter ejected from a supernova has a distinctive shape. If MiRRFe exists there is likely to be some distortion in the pattern of the ejected matter due to frame dragging effects. The local reference frame will be undergoing bigger rotations at larger r and smaller rotations at smaller r. This will affect the trajectory of matter flung towards or away from the galactic core. Light bending effects:  The way in which electromagnetic radiation from distant quasars is bent when it passes just above and across each side of the disc of an intervening galaxy might shed some light on the answer. A relativistic elaboration of MiRRFe probably requires that light travelling near to the disc in the same direction as the galaxy’s rotation will arrive sooner or be bent more than light passing through or above the disc against the direction of rotation. Negative Tests - Cold Dark Matter:  MiRRFe has been suggested as an alternative to the hypothesis that cold dark matter makes up 27% of the universe and over half of the mass of typical spiral galaxies, including the Milky Way. It follows that if CDM is ever detected in an unambiguous physical experiment or observation, then there would not be a need for MiRRFe or other alternative ideas to CDM. Conversely, the longer that time goes on without CDM being conclusively detected then the greater the need for a plausible alternative.

Concluding Comment

The apparent violation of classical physics in the orbits of stars in spiral galaxies has given rise to hypotheses about the existence cold dark matter or else a need to modify Newtonian physics. This paper suggests a third hypothesis based on Mach’s Principle. It suggests that the stellar rotation velocities are actually in accord with Kepler’s Law if the reference frames against which these are measured are themselves rotating at rates of up to several degrees per million years. The hypothesis may or may not be correct but it lends itself to observational and computational verification and may well be worth some detailed consideration.

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