Astrophysicists have long accepted that small structures, such as nebulae, clusters, gas clouds, globulars, etc, each have a persistent structure, which they maintain due to mutual gravitation even while revolving around the Galaxy. This presentation simply asserts that the same is true of Spiral Arms, which have a surprising amount of mutual gravitation due to having billions of component stars and other materials. In the same way that a star cluster retains its local structure while revolving in the Galaxy, so do the Spiral Arms. Simply an application of Newton's gravitation!
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An amazing assortment of speculative ideas have been presented to try to explain the peculiar way our Galaxy rotates, including undocumented assumptions of "dark matter", "hidden matter", "gravity waves, causing density shock waves" and many others. Each of those speculations creates more problems than it solves, but oddly, each has become adopted by substantial numbers of Astrophysicists! Amazing!
This concept was invented and Engineered by November 1997. This presentation was first placed on the Internet in August 1998.
The various concepts that depend on additional invisible mass (even sometimes including astounding numbers of speculative neutrinos) never seem to address just where that mass would have to be. The mass distribution of invisible material would have to be very peculiar, so much so as to be in unstable configurations, such as a super-massive doughnut (torus) around the Core region. The suggestion that there is a massive but invisible halo around the Galaxy would not cause the observed effects (as it would be EXTERIOR to everything in the Galaxy, and by Newton, not having any effect on anything interior to it. The gravity-wave speculation seems to overlook that such a wave would have to come from some direction (if and when gravity waves are even ever discovered!) and that there is NO way that such a planar wavefront could selectively cause density shock waves to "light up" starburst patterns in a spiral arm pattern!
It turns out that all those speculative assumptions are probably unnecessary! Simple Newtonian gravitation seems to be able to completely explain all the mysteries!
It has long been known that our Galaxy does not seem to rotate as it should. Specifically, the Spiral Arms stay together, rotating as though being a single object. Since the Arms are actually made of billions of individual stars, prevailing logic seems to require that they individually rotate in accordance with Keplerian physics. That is, regions of inner stars should rotate much more rapidly, immediately tearing the Arm structure apart.
In recent decades, the astrophysics community felt it necessary to "invent" surprisingly wild and unsupported ideas to attempt to explain this unusual behavior of the galaxies' rotations. This essay is meant to show that dark matter and the other speculations are not necessary, and that standard Newtonian gravitation easily and completely explains all the observed behaviors.
A previously undescribed simple gravitational phenomenon exists. Essentially, the mutual perturbations among the component stars in a Spiral Arm can be shown to have far greater effects than previously noted. The inverse-square nature of gravitation causes the effect to be very strong at the relatively short distances within a Spiral Arm.
This is NOT any new "mechanism" like gravity waves or some electric or magnetic phenomenon. Rather, it is merely the effect of standard Newtonian gravitation acting for very long periods of time. One interesting consequence of this research is the realization that the Sun and all other stars slowly weave back and forth across the Spiral Arm!
Consider this simplified representation of a top-view of a flattened, cone-shaped Spiral Arm (straightened for simplicity here), with random stars within it. The Arm actually continues to the right, to join the Core of the Galaxy. The Arm actually contains billions of stars rather than the hundred shown in this diagram.
We have now marked a particular star on the central axis (in yellow). Consider for a moment how many of the stars are gravitationally attracting it to the left and to the right. There is certainly an unavoidable conclusion that a net gravitational attraction MUST exist on that star toward the right, toward the "front" of the Arm. This effect has not apparently previously been considered. It continually acts to accelerate each object, such that they all move faster than they otherwise could.
Now see that we have marked a star that is NOT on the central axis of the Arm. This is the situation that generally applies to most stars, including our Sun. Note that we still have the net gravitational attraction to the right. Note also that there is also a net gravitational attraction toward the centerline of the Arm (upward in our drawing). This is even if the Center of the Galaxy is downward in this illustration.
This Arm-transverse net gravitational force causes an Arm-transverse acceleration for the Sun and every other component of the Arm. An intra-Arm weaving motion must therefore occur for each and every object within the Arm. Only the Sun is shown moving in this drawing, but every object in the Arm would be involved in this complex dynamic movement. This consequence appears also not to have been previously described.
The logical and mathematical analysis below will establish that this situation causes stars within a specific Arm to interact with each other to create intra-Arm dynamics, in addition to the large traditional effect of drawing each component object toward the Center of the Galaxy and a general revolution around that Center.
The analysis of this field of intra-Arm dynamics seems to have several
These several predictions include a variety of phenomena that might be investigated. Some could provide research evidence either for or against confirmation.
Previous attempts at explaining the dynamics of the Galaxy have always assumed that only a Keplerian-Newtonian Central Force exists. However, this new premise suggests that within the Spiral Arms, a substantial previously undescribed conventional gravitational net force vector can be shown to also exist, primarily forward along the axis of the Arm. The tapering shape of a Spiral Arm always means that there are MORE massive objects ahead of any component object in an Arm than behind it. This situation, summed for all the intra-Arm gravitational attractions, ALWAYS results in a forward-directed force vector to pull and accelerate every star and molecule forward. This results in a meta-stable situation that establishes the stability and persistence of the Spiral Arm, including the circumstance where the Arm revolves in the observed non-Keplerian way.
Traditional analysis has always assumed that the Galaxy is isotropic, such that only a central gravitational force acts on each component part and so Keplerian calculations could be used. As a result, all sorts of exotic explanations have been put forward to try to explain the observed non-Keplerian rotation. A currently popular one is that the vast majority of the Galaxy (and all galaxies) are invisible dark matter. There has never been the slightest evidence for such material, and the only reason why it was suggested was to supply the great amount of additional mass that would be necessary to allow a traditional Keplerian description of the rotation. This particular explanation has become so popular that many spin-off ideas have already been based on it, such as a continuous disk of material in all spiral galaxies, with traveling shock waves traveling through it that initiate bursts of star formation where we see spiral arms. With no actual basis for the first idea, this premise has even less basis. There are dozens of logical reasons why such schemes are improbable, particularly regarding complete obscuration of all but the very nearest parts of our own Galaxy, and the fact that a few distant spiral galaxies appear to allow views of more distant galaxies between their Arms.
The currently popular star-burst, shock-wave premise includes another obvious difficulty. If it were really true that the appearance of Spiral Arms was somewhat fictitious in this way, there seems to be an implication that no differential rotation of the Galaxy would exist. But, by a hundred years ago, Kapteyn, Eddington, Schwarzschild, Stromberg, Lindblad, and Oort showed the pattern of Star Drifts, which were caused by differential speeds of stars as a result of the rotation of the Galaxy. The direction of these Star Drifts are very close to 90° from the direction of the Galaxy Core, meaning that they are directed essentially tangential in the Galaxy. This evidence was developed to establish that our Galaxy was rotating, and it is definitely valid. This fact seems to be overlooked in the shock-wave premises, as their assumption would require exactly the opposite of what the evidence shows.
By the way, those Star Drifts might give us somewhat misleading data if this premise is true, because of the important additional accelerations that would be due to intra-Arm dynamics.
In this essay, we will consider a traditional view of the Galaxy, where the visible Arms are actually Arms and that there are basically empty spaces between them. This is far more logical than in speculating about invisible massive material that there is no evidence of. Therefore, it will be shown that, for objects within a specific Spiral Arm, a central-force-only assumption is an incorrect assumption in that virtually all of the nearby stars are also within that same Arm. Being nearby, the inverse-square law tells us that these nearby stars have particularly great gravitational effect. Thus, the following mathematical analysis shows that the intra-Arm forces are surprisingly strong and that there is therefore a resultant substantial gravitational attraction along the axis of the Arm, both forward and rearward. We will show that the tapering shape of the Arm is absolutely necessary in that it always means that there are more objects drawing each object forward than rearward, resulting in a significant force vector forward, generally parallel to the Arm axis. This Net Forward along-Arm Force vector will be shown to have a magnitude as great as one-fourth that of the Central Force of the Galaxy. This results in the Sun and all other component stars and molecules of the Spiral Arms being continually accelerated forward. This results in a faster rate of revolution than would otherwise be possible, which also enables and maintains the integrity of the Arm structure, while apparently violating Keplerian physics.
A number of significant consequences result from this understanding. A very straightforward explanation for the formation and persistence of Spiral Arm structures will be established. An explanation for the specific tapering shape of Spiral Arms will be shown, including aberrant examples such as M51, and including a self-perpetuating characteristic of a tapering shape of a Spiral Arm. An explanation for the observed increase in tangential velocity of rotation with increasing radius, which is the opposite of what would be expected from Keplerian calculations, will be shown. An additional result of this premise is that the total mass of the Galaxy might be substantially less than previously thought.
It has long been known that the Milky Way Galaxy does not rotate in accordance with the Keplerian model of Physics. If it did, the inner parts would rotate far more rapidly than parts farther out, and its Spiral Arms would quickly wrap themselves around the central Core and it would soon cease to be a Spiral Galaxy. This clearly hasn't happened, and empirical evidence implies that the bulk of the Galaxy's Arms revolve relatively as a unit. The existence of a large number of other spiral galaxies implies that this is a reasonably common, stable and persistent structure. The proposition presented here is that the resolution of this mystery lies in the tapering shape of the Spiral Arms themselves.
There has been a basic flaw in the logic applied in trying to explain the rotational dynamics of the Galaxy. It has always been assumed that the Central force of gravitation, toward the exact center of the Galaxy is the ONLY force that causes and affects the revolution of the Sun and all the other stars in the Galaxy. This assumption was a seemingly logical carry-over from the situation of our Solar System. However, that assumption is wrong in the case of the Galaxy, even though it is admirably correct for the situation of the Solar System.
In the Solar System, virtually all the mass is concentrated at the center, in the Sun. The planets cause perturbations in the paths of each other, but those perturbations are rather minor. The gravitational Central Force between the Earth and the Sun is what actually causes the generally elliptical path of the Earth around the Sun. Kepler and Newton recognized this hundreds of years ago.
In the Galaxy, the mass is distributed quite broadly. Much of it is concentrated in the many billions of individual stars distributed throughout the Galaxy. These stars are NOT all concentrated in the Central Core of the Galaxy. More specifically, most of the stars that are relatively near any star in an Arm are either forward of, or rearward of, the star itself, within that same Arm. This is NOT the isotropic situation that had been assumed.
Hundreds of years ago, Newton proved mathematically that a UNIFORM sphere of matter, or spherical shell of matter, or even a uniform circular ring of matter, will act gravitationally (for any object outside of it) as though all the mass of the sphere, shell or ring was concentrated at the point at its center. This proof made it possible to develop and advance Physics to the point today of being able to reasonably well describe and explain the motions of the objects in Earth orbits, Solar orbits, and many other empirical gravitational phenomena. The central gravitational force that exists between the Earth and Sun, for example, is overwhelmingly strong in comparison with the forces due to the other planets. This fact enabled Johannes Kepler to recognize and define his orbital rules long ago.
That central force is certainly very important in the dynamics of the Galaxy. But it is NOT the only force that affects the paths of the Sun and stars in revolving around the Galaxy's center. The environment in and near a Spiral Arm does not involve an isotropic distribution of mass. The tapering shape of the Spiral Arm that the Sun is in, necessarily causes an additional force vector to exist. There are more nearby stars that are pulling the Sun forward than there are nearby stars that are pulling it rearward.
The actual magnitude of this force vector (on each star or molecule), calculated below, is apparently less than that of the Central force vector, but it is very significant. It is also relatively tangential in the Galaxy, always directed forward and slightly inward of the star in its path of revolution in the Galaxy. This Net Forward Force Vector therefore continuously imparts a forward acceleration on each star and molecule (in the Spiral Arms), thereby causing each of these constituent parts to act more cohesively and to travel faster in its revolution than would be the case if just a central (Keplerian) force vector existed. Even though this Arm-axial force vector is apparently smaller in magnitude than the Central force vector, its orientation of permanently acting to increase the magnitude of the velocity vector allows it to have considerable cumulative long-term effect.
Consider the Sun's situation for a moment. It is in one of the Spiral Arms. From observation of other galaxies, it is clear that Spiral Arms generally taper as they get farther from the central core of a galaxy. We cannot actually see the structure of the Arm we are in very well, but it is reasonable to believe that would also be true in our Galaxy. Therefore, we can safely conclude that there are more nearby stars forward of the Sun than there are behind it. Because gravitation is an inverse square law force, all these nearby stars (both forward of and rearward of the Sun) have relatively much greater effect than more distant stars.
The Sun actually has hundreds of billions of individual gravitational interactions (attractions) with all of the other objects in the entire Galaxy. We will show that the larger objects (primarily stars) that are within our same Arm, have tremendously greater resultant force because of their relative closeness. Each individual gravitational force vector acting on the Sun, from other objects of the Arm, has a component force parallel to the axis of the Arm. Since there are more stars forward of the Sun in the Arm, the summation of these many Arm-axial component forces must necessarily be directed forward parallel to that axis. The net result is a gravitationally induced force vector forward parallel to the centerline of the Spiral Arm. This situation must be true for each and every star and molecule in a tapering Arm.
The relative strength of this net forward force vector is primarily dependent on one parameter: the rate of taper of the Spiral Arm. For an (hypothetical) Arm that had no taper, there would be no net forward force vector, and only the traditional Keplerian central force vector would be in effect. (The current premise suggests that such an Arm would quickly evolve into a tapered shape.) For the situation in an Arm that had a rapidly tapering shape, each and every component star of the Arm would be subject to a more extensive forward force vector, and each would then be accelerated forward more greatly.
This situation enables each star that is part of a Spiral Arm to revolve around the Central Core at a faster rate than would be possible if it were not part of the Spiral Arm. This explains the discrepancies from Keplerian Physics that have long been known in the rates of revolution of different parts of our Galaxy. It also explains the existence and the durability of the Spiral Arm structure. The fact that each star and molecule in the Arm is revolving faster than would be possible apart from the Arm structure, makes this a meta-stable solution. Up to a certain point, very stable Arm structure would persist and even grow in integrity. Beyond that point, stability would be catastrophically lost and that part of the Arm (and everything behind it) would break away to be hurled radially outward. The M51 galaxy may be an example of this occurring. It is currently asymmetric, with a substantial part of one Arm missing and a large nearby whorl.
For visualization purposes, there is an analogy available. Consider an ice skating rink, with a vertical pole sticking up out of the center of it. A 10-meter long string is looped around it, and you hold on to the other end of the string. This will cause you to skate in 20-meter diameter circles around the pole. When you skate as fast as you can, someone times your lap (orbit) time.
Now, consider adding more skaters, all holding hands, with one keeping his free hand on the central pole, and the others circling outward, until the last one holds YOUR hand. (Skaters sometimes do this sort of thing! The arrangement is called cracking the whip.) (You still have hold of your string, too, that will maintain the radius of your path). As the inner skaters pick up speed, there is a net effect of an additional forward acceleration on your hand. If you are out along this string of people, you will soon find yourself being pulled along, skating faster than you could have just due to your own efforts.
When skaters do this, as speeds build up, the outermost person invariably eventually loses his grip on the neighbor's hand, primarily due to the increasingly greater force necessary between those hands. That force is notable as being the reason why you were able to go much faster than otherwise possible, and also as being the reason why the meta-stable situation eventually catastrophically breaks down.
Similarly, this solution regarding Spiral Arms in the Galaxy is a meta-stable solution. As with the skaters, the along-Arm Force Vector is both the reason for being able to go faster than otherwise possible and the reason that the meta-stable situation can break down if that Force exceeds a certain limit. Below that limit, it is a stable arrangement, but above it, the apparent stability can be instantly and irrevocably disrupted by breaking the hand/gravitational link with the person/objects ahead.
In each case, this means that there is a limitation to the effectiveness of this configuration, before the meta-stability breaks down. In the case of the skaters, that point of breakdown is dependent on the strength of the grip between EACH two people. A similar situation must equally be true regarding the along-Arm Force effect in the Galaxy Arm. In each case, the forces involved are greater with greater radius, so the breakdown of the meta-stability will tend to occur for the outermost parts.
Before leaving this analogy, consider also that as the speed kept increasing, the originally straight (radial) line of skaters wound up curving back, as the outer skaters would have great difficulty keeping up with the identical rate of revolution. At the maximum speed situation, the outermost skaters would be being pulled almost straight forward (but ALWAYS slightly inward as well). The point being considered here is that the ANGLE of the line of hand-to-hand acceleration is related to the FORCE being transferred by that hand-to-hand coupling. The outermost part of this grouping would appear substantially tangential to the circle path, if it is near the limit of the meta-stability. This closely resembles the backwards curvature of Spiral Arms in galaxies.
Similar mathematical and geometric arguments regarding the dynamics of Spiral Arm structure suggest that the same should be true. The angle of a Spiral Arm segment from being tangential should be related to the Net Forward along-Axial Forces of the component stars, and therefore to the rate of taper of that Arm at that point. Since that Net Force is along the center axis of the Arm, it has both tangential and radial (central) components as related to the Galaxy as a whole. The tangential component increases the velocity of revolution of the object in the Galaxy. The central component maintains the meta-stability of a circular or elliptic orbit, by adding to the main Keplerian central force. The relative strengths of the two components should be an indication of the dynamic stability of that portion of the Arm. This premise may suggest that this relationship changes as the Arm ages, as velocities become successively greater than Keplerian velocity of revolution.
This situation should be confirmable by measuring the geometries of images of Arms of actual galaxies. Spiral Arm structures in many thousands of photographs generally seem to show this Arm curvature from reasonably radial at the inward end to moderately tangential at the outer end. (Some investigators feel that the Arms represent logarithmic spirals, where the angle between the Arm centerline [tangent] and the direction of the Center of the Galaxy is constant. This doesn't materially affect the central premise of this argument except regarding the limiting case of the outermost end of an Arm. The difference would be between an Arm structure that was everywhere near the limit of stability or of one that had areas of greater stability.) Accurate analysis of existing photographic evidence should establish the true relationship.
Our Galaxy has long held a mystery. Keplerian Physics predicts that the outer portions of the Spiral Arms should be moving at proportionally slower velocities, but empirical evidence has shown just the reverse, that they move at faster velocities than stars in the Arms nearer the Central Core. All Physicists accept these empirical findings, because if that were NOT true, then the Spiral Arms would rapidly wind themselves around the core. That would mean that Spiral Galaxy structure was extremely unstable, which does not appear to be the case. Relying exclusively on the central force model has then required a LOT of Physics trickery to explain all the observed conditions. For example, "dark matter" is proposed to be distributed, in enormous quantities, throughout the Galaxy to provide enormous amounts of additional mass necessary to try to create a Keplerian explanation for the apparent paradox of revolution rates. The necessary distribution of this dark matter would be very peculiar indeed! Among other things, an extremely massive dark matter toroid ("doughnut") would have to encircle the Core of the Galaxy. Many Physicists today have become convinced from this situation that far more dark matter exists than matter that we can sense! They seem to overlook the very peculiar mass distribution necessary for that mass! Additionally, the dynamic stability of such peculiar mass distributions would be doubtful. All this exotic speculation has been postulated to explain Galaxy rotation characteristics that seemed otherwise difficult to comprehend in Keplerian terms.
Newton's proof of an extended object acting gravitationally as though all of its mass was concentrated at its center had two caveats. One was that the solution was true ONLY regarding effects on objects beyond the dimensions of the object being mathematically integrated. The other is that ONLY radial variations of density were allowed while still ensuring the correctness of his calculations.
In the case of the Sun, the Earth, all the planets and all of the individual stars, that uniformity generally DOES exist, and so the premise is rather accurately true. This has allowed much advancement in Physics and Astronomy over the past 400 years, by allowing us to develop equations based on effectively point masses gravitationally attracting each other. But it represents an incorrect assumption in the case of the Galaxy, because of anisotropic distribution of the mass within it.
First, we will consider the standard approach to the calculations regarding the Galaxy. Only a very few facts need to be known. In Newton's universal law of gravitation, the constant G is accurately known to be 6.672 * 10-11 (metric system). The mass of the Sun is accurately known to be 1.9891 * 1030 kg.
Only three other variables exist, the Sun's velocity of revolution, the distance the Sun is from the center of the Galaxy, and the total mass of the Galaxy. In order to determine the third, only the two other variables need be known, the distance the Sun is from the center of the Galaxy and the velocity of the Sun's movement in its orbit around that center. It might first seem that these two values would be easily determined, but that is not actually the case.
From the location of the Earth, we cannot actually see the center of the Galaxy, and the methods that have been used to determine our distance to the center of the Galaxy are all rather indirect and possibly susceptible to error. Nearly one hundred years ago, Shapley determined the distances to a number of Globular Clusters, which generally appear to be above and below the Core of the Galaxy. He assumed that these Globulars were in a spherical Halo that is centered on that Core. He methodically used several methods during the 1910s to determine the distances to the Globulars. As a result, he determined that the "center of gravity" of the Globular Cluster system was 52,000 LY from us, implying that that was our distance to the center of the Galaxy. Later work by Stebbins found evidence of an absorbing layer between us and the Globulars, which reduced our calculated distance to the center of the Galaxy to 33,000 LY. Most current estimates of this distance range from 24,000 light years (LY) to 33,000 LY. (Zielek p.391) A popular modern estimate is 28,000 LY. It is pretty well established that the Core of our Galaxy is in the direction of the point in the sky described by R.A. 17h45.6m Dec. -28°56'. So, even though we can't actually see it, that is the direction that the center of the Galaxy must be in.
The velocity of the Sun in its orbit within the Galaxy is even more uncertainly known. A major difficulty is that everything in our vicinity is certainly traveling at roughly the same velocity as we are (revolving around the Galaxy center), so they cannot be used as fixed reference points. Actually, there is virtually nothing that can be used as a fixed reference point for attempts to determine this value. All of the methods used are extremely indirect and subject to the possibility of error.
We can consider two DIFFERENT velocities for the Sun, one in relation to nearby stars and the other as part of the whole rotation of the Galaxy. They are certainly in different directions (around 41° apart in our sky) and at rather different velocities.
Regarding the first, some of the best of such methods appear to be the statistical analysis of Proper-motions and Radial-velocities of other stars in the sky. Patterns were noted early in the Twentieth Century, and by 1911, Wilson had shown that the Sun is apparently moving (relative to other nearby stars) at about 20 km/sec. toward a spot in the constellation of Hercules (approximately at R.A.18h02m Dec. +29.2°) that is referred to as the Apex of the Sun's way. Proper-motion analysis (beginning with Boss) and Radial-velocity analysis (beginning with Campbell and Moore) give slightly different location for this point.
With this result, it is possible to determine the Parallactic Motion of each star, which is how the star (if actually stationary) would seem to move solely due to the motion of the Sun. This Parallactic Motion is then subtracted from the measured values of Proper-motion and Radial-velocity for each star, leaving a Peculiar Motion, which indicates the actual motion of that star. In 1904, Kapteyn reported that these remaining Peculiar Motions showed a decided tendency toward two groups or "streams" that are apparently moving in opposite directions. Eddington later called these Star Drifts (Drift A and Drift B), which became more commonly used. Schwarzschild and Stromberg developed the evidence further. Lindblad showed that these Drifts were caused by differential speeds of stars as a result of the rotation of the Galaxy (the second of our two Sun velocities). By 1927, Oort showed a method of analyzing these results to determine the rotation of the Galaxy, including the actual revolution velocity of the Sun in that process. Modern analysis of this data suggests that our part of the Galaxy is rotating in the direction R.A. 21h12.0m Dec. +48°19' at around 250 km/sec, but some good arguments for values around 220-225 km/sec have also been offered. (Zielek p.391). This direction is exactly 90° away from the direction of the center of the Galaxy. If this evidence is reliable it implies that our part of this Arm of the Galaxy is moving tangentially, and therefore revolving in an approximately circular path. There is a substantial difference in the direction of the Apex of the Sun's Way (our apparent 'local' movement) and this implied Galaxy-tangential overall movement (upward by 26° and inward by 37°). Had the Sun NOT had any intra-Arm motion, no such significant difference would exist.
Alternate methods have been presented. They tend to be less desirable. For example, a statistical analysis of Radial-velocities of external galaxies gives a result that might have substantial error.
These brief histories have been presented to suggest the difficulty in accurately determining the two values needed (radial distance and velocity), and the potential for error in the currently accepted values for each. The premise of this paper is not changed by whatever corrections might later be made in improving the accuracy of these values.
In any case, the 20 km/sec (local) motion toward the Apex is around 41° away from the tangential direction (26° upward and 37° inward) that all of the Arm stars must be collectively moving. This implies that the Sun is currently advancing forward along the Arm at around 16 km/sec, moving away from the Arm Centerline at around 12 km/sec and moving upward at nearly 9 km/sec.
For the following discussion, we will use 28,000 LY and 250 km/sec for these two values. It would be quite simple to replace either or both with some other values to re-calculate the equations' results.
Newton's universal law of gravitation is:
Using the accepted values for G, m1 (mass of the Sun), and r (distance of the Sun from the center of the Galaxy - 28,000 light years), and choosing 125 billion Suns as the mass of the Galaxy, we get:
This equation assumes that all of the component materials that comprise the Galaxy are symmetrically distributed around the center point such that they effectively act collectively as though their masses are all located at that point. This has always been taken to be a reasonable assumption but there really is no way to confirm that. This gives a central force of:
This central force acts on the Sun according to:
which solves to give a central acceleration of:
If we consider a one-second interval, this acceleration will cause a velocity of 2.359 * 10-10 meter/second toward the center of the galaxy. During this second, the Sun moves at 250,000 meters/second forward in the orbit. For a circular orbit, these two velocity components are at right angles to each other. This deflection is less than one part in one quadrillion, but that is enough to cause the Sun to follow a circular orbit within the Galaxy. This amount of deflection or curvature of the orbit corresponds to a circular orbit with a radius of 28,000 light years. (We had initially chosen a Galaxy mass of 125 billion Suns in order to demonstrate this [circular] orbit with the known radius and velocity).
There are very few variable in these calculations. The constant G and the mass of the Sun are very well known. This only leaves three remaining variables, (1) the Sun's distance from the center of the Galaxy, (2) the total mass of the Galaxy, and (3) the velocity of the Sun in moving in that orbit. If any two of these are known the third can be determined (assuming the validity of Kepler's formulae). This obviousness and simplicity are basically why the traditional results have been so universally unchallenged.
Note: The actual entire (generally assumed) mass of the Galaxy would be a little more than this 125 billion Suns mass. The reasons are complex, but they relate primarily to parts of the Galaxy that are radially farther outward than the Sun. Therefore, the effect is relatively small, increasing the probable total mass of the Galaxy to around 135 billion Solar masses.
The assumption of uniform, isotropic mass distribution in the Galaxy is substantially incorrect. There are significant systematic variations in the mass density of different parts of the Galaxy. In the local vicinity of the Sun, for example, most of the stars are in the same Spiral Arm we are in. There are virtually no nearby stars beyond the edges of the Arm. This necessarily results in a net force vector roughly parallel to the centerline of that Arm. Since gravitational attraction falls off as the square of the inverse of the distance, these few hundred thousand relative nearby stars (say 1000 LY ahead of and behind the Sun in the Arm) have the net gravitational effect on the Sun of eight hundred times as many stars uniformly distributed throughout the Galaxy, which would act as though they were 28,000 LY away from us.
The proposed modification of understanding will suggest that our Galaxy is less massive than previously thought. Previous thought has always assumed that the velocity of revolution of the Sun is entirely due to the Keplerian Central force. Since we are demonstrating that the Sun's velocity of revolution is actually the sum of two components, the Keplerian Central force and a Net Forward along-Arm force, this implies that the Keplerian velocity alone would be less. The above calculations then indicate a less massive Galaxy. This then has significant implications on our Galaxy's gravitational effect on other galaxies, but which means that Keplerian and Newtonian logic and calculations are accurate for those situations. For gravitational effects on the Sun and the other constituent parts of the Spiral Arms of the Galaxy, it changes everything!Calculations of an Intra-Arm Gravitational Effect
An important consideration is that a Spiral Arm tapers. Since this is the case, for any star or molecule within the Arm, the width (and mass) of the Arm ahead of it are greater than at an equivalent distance behind it. That means that there are more stars ahead of it gravitationally pulling it forward than there are behind it pulling it rearward. The net effect of adding these opposing forces ALWAYS results in a Net Forward Force along the center axis of the Arm.
It has been reasonably well estimated that the cross-sectional area of the (full, complex) Arm we are in is generally elliptical, approximately 7,500 LY wide (where we are) and 1,000 LY thick. We have considered the Spiral Arm as having been straightened out for this discussion, and of a total length of 60,000 light years.
We are therefore assuming a flattened cone-shaped Arm with a taper angle of around 15 degrees (7-degree-angle on each side of a central line). We will assume that the Arm has a constant taper. For simplicity, we are also going to assume that the thickness of all the Arm is all 1000 light years.
Because of our location inside a Spiral Arm (called the Orion Arm), it is not really possible to determine the actual tapering angle of the Arm we are in. For the discussion above, we chose 15 degrees (with a constant taper) since that is appropriate to an Arm that extends about 30,000 LY outward beyond us, and from photographs of other Spiral Galaxies, that seemed like a reasonable length. The magnitude of the along-Arm Net Force Vector is very dependent on the taper angle of the Arm. The logical argument presented here applies for any Arm taper angle.
Our discussion has been about a constant taper cone-shaped Arm, the most conservative possibility. An exponential Arm taper would greatly increase the along-Arm Net Force Vector. In some scenarios the magnitude is more than five times that calculated for a constant-taper Arm.
In addition, our discussion has assumed a constant mass density distribution throughout the length and breadth of the Arm. It seems highly likely that the local mass density will be greater nearer the Core (where it may be as high as 1.0 solar masses per cubic light year, around 700 times as great as in our vicinity) and lower farther out in the Arm. If these situations are true, the effect becomes far greater yet.
A very conservative (longitudinal) exponential mass density variation would be from 0.0555 solar mass per cubic light year at the inner end of the Arm, to the known 0.0014 value for where the Sun is, to 0.00004 at the very outer end. This would account for about 12 billion solar masses per Arm, which would seem reasonable if the total Galaxy mass was 125 billion solar masses. This variation in the local mass density increases the net along-Arm Forward Force Vector by an additional factor of about twelve.
Very little solid data exists on these matters. That is why the most conservative possibility was chosen for these calculations. As just indicated, with very reasonable estimates for longitudinal mass density distributions and exponential Arm tapering, the effect can be more than fifty times as great as calculated above. Reasonable assumptions regarding taper and density variations can therefore result in an along-Arm Force even greater in magnitude than the Keplerian Central Force!
There is yet another variable that may also cause this effect to be greater still. The value used here for the local mass density (0.0014 solar masses per cubic light year) has been considered to be uniform for any specific transverse cross-section of the Arm. In reality, this is unlikely. It seems quite likely that the local density near the centerline of the Arm is somewhat higher than near the edge of an Arm. The Sun is near the inner edge of our Arm. That could imply a relatively low density in our area because of our nearness to the edge. Our local vicinity is the only area where investigators have established a firm mass density value. If the mass density does have this sort of an Arm-radial dependence, then there are actually even more masses pulling the Sun inward, toward the Arm centerline, within the Arm.
We have used the following estimates here: The total mass of the Milky Way Galaxy is 125 billion Suns. The bulk of that mass is in the Core area, but a pair of major spiral arms can realistically each be expected to have a total Arm mass of 1/10 of this, around 12.5 billion Suns. If an Arm were straightened it would extend for 60,000 light years in length, with a taper from a maximum of 15,000 ly width to zero at the tail. The thickness is here assumed to be a uniform 1000 light years. These assumptions have a volume of 450 billion cubic light years. With the assumed mass of 12.5 billion Suns, that represents an average mass density of one Solar mass in every 36 cubic light years. In our immediate vicinity, it is believed that the mass density is around 1/20 of that, around one solar mass in every 700 cubic light years. This has a reasonable explanation, regarding this discussion and the fact that we are currently near the inner edge of the Arm that we are in. This reasoning suggests that it is quite possible that the mass density may easily be twenty times as great along the Arm centerline region.
These various effects suggest that the true effect of these intra-Arm forces and accelerations may be far greater than the conservative estimates indicated here. Detailed analysis of photographs of other spiral galaxies might give an approximate value for this variation.The Effect of these Net Force Vectors
Considering the effect of the Tangential component of the Net Forward Force Vector:
Computer simulations have been done (beginning in 1997) where each of the 450 million cubic light years inside the Arm have been considered to contain one star with mass of 27 suns, to account for the entire 12.5 billion solar masses inside the specified volume. Those simulations have resulted in a net forward Intra-Arm acceleration of 1.76 * 10-10 m/s2
This acceleration is nearly comparable in magnitude with the Keplerian central force acceleration. This force ALWAYS acts to increase the magnitude of the velocity vector.
One result of this is that the velocity of revolution of the Sun around the Galaxy is greater than it would be if only the Keplerian central force were acting.
Such a result would suggest that our Galaxy may only have around 2/3 the total mass previously thought. If our Galaxy has such a small mass, then all of the efforts at trying to find "hidden mass" or "missing mass" within our Galaxy will be unnecessary, including the speculations that there is a massive black hole in its center. Additional scientific pressure has been exerted toward finding such missing mass in order to try to explain the existence and persistence of Spiral Arms. These researches as well would become unnecessary since this premise demonstrates the mechanism behind the existence of Spiral Arms.
Another consequence of this is to allow for a much more realistic stellar population in the Central Core area of our Galaxy. Present theories are all very speculative on this matter, but they all include many billions of individual stars in a remarkably small volume of space (Zielek p.378, one star per cubic LY). If present theories were true, the average separation between individual stars would be less than one light year, around 700 times the star density in the vicinity of the Sun. This would certainly represent an extremely unstable situation, with collisions occurring rather regularly and all of their paths continually being perturbed and disrupted.Please note that these calculations have been based on the most conservative situation possible, that of a cone-shaped Arm that has a uniform mass density everywhere in it. When these calculations are repeated for a realistic exponentially tapering Arm with the reasonable longitudinal mass density variation described above, we sometimes found a permanent forward acceleration of at least 60 times as great.
On a related point, we know that many Globular Clusters of stars are present relatively near the Central Core but above or below the plane of the Galaxy. It is gravitationally not possible that these Globulars forever remain above or below that plane and they must therefore pass through the plane of the Galaxy twice per orbit. Simple calculations show that their orbital periods must be on the scale of tens of millions of years. That means the Globulars, which are known to be extremely old (because of their Population II stars), have passed through the plane of the Galaxy many hundreds of times already. In those passages, if the Galaxy stellar population is as great as others have postulated, the individual stars of the Globular would be each gravitationally affected by close approaches to Galaxy stars, which would disrupt the organized structure of the Globular. This is not the observed case. The current premise, with a far lower stellar density necessary in the Core, is far more compatible with observed conditions. (I have written a separate essay on these problems regarding Globular Clusters).
In addition, calculations regarding the distances of the Magellanic Clouds which orbit the Galaxy may be wrong and they may be much closer than generally accepted. In similar ways, the Great Andromeda Galaxy and everything else in the Universe may be much closer to us than has been thought. If THIS is true and the Universe only has one-half the scale as has been generally thought, its volume is 1/8. Researches into whether the Universe is expanding or contracting would only need to account for 1/8 as much mass (actually even less than that!) to account for a closed Universe. All such theories and many others could be impacted.
With a Universe that is much smaller, objects such as quasars would necessarily be much closer, and therefore not as spectacular as energy sources. It may be possible to explain such sources in less exotic ways than is currently the case. All of the reasoning that first encouraged the speculation of black-hole existence may also become irrelevant.Computer Simulations
Computer simulations are extremely beneficial in establishing the validity of these ideas. Refinements are appropriate in several major areas.
The foregoing discussion makes it clear that it is extremely important to determine the long-term effect on the velocity of the Sun of this Along-Axis Force Vector. If the Spiral Arms were believed to exist for most or all of the existence of the Galaxy, the foregoing discussion suggests that a Along-Axis induced velocity greater than the total current velocity of the Sun would be indicated, and that is obviously impossible. There are clearly long-term meta-stable limitations on such effects and only long-term computer simulations are likely to uncover the breakdown of the meta-stability for different circumstances. In this same area, stars farther out radially than the Sun will be exposed to even greater accelerations and limitations regarding these matters. In those limiting situations, Spiral Arm angle would necessarily become almost entirely dependent on the relative magnitudes of the radial and tangential components of the along-Arm Force Vector. The tangential component keeps the object revolving with the Arm, and the radial component keeps it from being flung free if the meta-stability would break down. Computer simulations will be necessary to establish the predicted Arm angles.
For stars that are at the very outer end of an Arm, virtually all of the stars in the Arm are forward of it, with virtually no gravitational attraction rearward. This logically agrees with the fact that such outermost stars NEED the greatest Net Forward along-Arm Force vector magnitudes in order to maintain meta-stability. Anything that used to be farther out has been previously lost to the Galaxy.
An analogy exists for this, and NASA may have already done the analysis for it. A normal (unpowered) Earth satellite remains in orbit strictly due to Keplerian Central force. Consider an Earth satellite that contained a constant-burning rocket engine that continually accelerated it forward. Such a rocket engine could not be directed tangentially because that would cause the orbit to change radius and become elliptical. Rather, it would have to be directed forward AND INWARD in order to maintain the current orbit radius with a higher orbital velocity. With a proper angle, this continually-powered satellite would have a (meta-stable) circular orbit but a higher-than-Keplerian orbital velocity. This situation would not be an intentional goal for NASA because it would continually consume fuel, but they may have done analysis for situations where they wished to change orbital parameters. As the satellite orbital velocity increased, the necessary angle of thrust would probably have to be adjusted to maintain a circular orbit.
The third of the simulations mentioned above involves the motion of stars within a Spiral Arm, and predicts oscillating motions due to Arm-transverse components of the generally along-Arm Net Force Vectors. Each component object of an Arm that was not on the centerline of the Arm should experience an acceleration within the Arm toward that centerline. That acceleration should be calculable from the foregoing analysis and the star-centerline distance. (Our earlier summation of the gravitational effects within the Arm had concentrated on the along-Arm summation. A similar Arm-transverse summation [probably currently around 8.62 * 10-10 m/s2] determines this force and acceleration.) Individual star motions within the Arm, including resonant period intervals (probably around 52 million years), of the Sun and everything else constantly weaving back and forth across the width of the Arm seem certain to be true.
Statistical analysis of the empirical motion of stars within 1000 LY of the Sun should establish a pattern of relative internal Arm movements based on location in the Arm. Such a pattern could then be matched with these calculated / predicted values. As an example, the Sun is believed to be near the inner edge of the Orion Arm. Therefore, if this premise is correct, the Sun should be currently experiencing the intra-Arm acceleration indicated above, toward the centerline of the Arm (actually outward in the Galaxy). (In an exponentially tapering Arm, the force and acceleration is virtually the same.)
This effect should be large enough to empirically measure for the various stars in different parts of the Arm's cross-section. The acceleration will continue to increase the transverse speed of the Sun until it actually crosses the Arm's center-line, at which time the transverse velocity will begin to slow down.
At the time when the Sun would cross the centerline of the Arm, with this configuration, it should have a high transverse velocity within the Arm of about 170 km/sec. This corresponds with a period of oscillation for the Sun within our Arm of around 50 million years. These values are based on an assumption of a constant mass distribution throughout the cross-section of the Arm. This is almost certainly not the case. With a reasonable assumption regarding the transverse mass distribution in our Arm, this force and acceleration becomes somewhat higher as the Sun falls nearer many more of the massive objects. Many of the necessary measurements (overall shape of the Arm, central mass density of the Arm, mass density transverse distribution in the Arm, actual Arm width, etc.) have not been accurately determined, so these figures are based or apparently reasonable estimates.
There could be even other consequences of this oscillation. If it is true that the mass density near the center-line of the Arm is significantly greater than where we are now, this might regularly put the Sun in an environment where a lot more large objects pass through the Solar System. With the Sun traveling through that congested area at around 170 km/sec relative speed, collisions seem certain to occur. Even worse is the fact that there are other stars crossing the center-line going the opposite direction, and the relative velocity to them would be ferocious!
The fact that this would be a repeating situation might have some relationship with the intriguing cyclical extinctions that seem to have occurred in the Earth's ancient history. Since the Sun is near the inner edge of the arm now, this would suggest that we crossed the centerline around 13 million years ago and also 39 million, 65 million, etc.
This would imply the possibility that the Sun and planets are REGULARLY subjected to massive meteor storms every 26 million years or so, but that the source of these larger bombarding objects is actually external to the Solar System. If there is validity in this premise, several phenomena may need to be looked at in a new light. Recurrent mass extinctions could clearly result from massive bombardments of very large objects that we encountered in that crowded portion of the Arm. These extinctions would not have to be precisely regular, as would be necessary with a "Nemesis-type" hypothesis, but would represent several million year long periods of heavy meteoric activity. In addition, such EXTERNAL additions to our planet might be associated with bringing new or different organic molecules into our atmosphere, enabling the sorts of spurts of evolutionary development that seem to occur. Finally, our general assumption that all large craters on the Moon and Earth are from four billion years ago may need to be reviewed, because new large impact craters might be produced with this 26 million year cyclicity.
There may be empirical evidence for this intra-Arm oscillation. As mentioned earlier, the rotation of the Galaxy in our vicinity appears to be at about 250 km/sec in the direction R.A. 21h12.0m Dec. +48°19'. However, the local motion toward the Apex of the Sun's Way is at about 20 km/sec in the direction R.A. 18h02m Dec. +29.2°. In other words, the Sun locally appears to be moving in a direction 40.84° inward and upward of the direction as well as significantly forward along the Arm. Since the Sun is believed to be near the inner edge of our Arm, it figures to be moving in an Arm-transverse direction and that much of the inward directed component might just represent the radially inward component of its velocity with the Arm. This might allow developing an estimate of the Arm-angle in our neighborhood of the Galaxy.
The local calculations might be misleading! If this premise is true, then roughly half the local stars would be moving toward the edge of the Arm and the other half would be moving back toward the Arm Centerline. In this case, the Sun's current Arm-transverse speed might actually be half what it appears to be, around 2 km/sec.
If we can determine that there IS a net remaining transverse velocity within the Arm, we may be able to determine whether the Sun is still moving "outward" toward the innermost edge of the Orion Arm or whether it has already passed that point of greatest excursion and is moving back toward the Arm centerline. The Intra-Arm transverse Force will continually affect the Sun's transverse velocity with the necessary result of bringing the Sun back toward the Arm's centerline in a permanent transverse oscillation.
Similarly, a z-axis effect will keep the Sun from ever being lost above or below the Galaxy. The computer simulations for this effect suggest that the Sun will continue to travel upward (+Z) for maybe 400,000 years, where the Sun will then reverse and travel downward for slightly over 3 million years before crossing the Z-axis Galaxy Plane, with the entire period taking a little over 12 million years. This last effect may be worth investigating, since it represents a good explanation for the observed warpiness of the Galaxy, which may just represent temporal appearances of z-axis oscillations.Arm Integrity
An additional result of this premise is a native cohesiveness of the Spiral Arm. As described above, as long as the Arm tapers as it extends outward, there is significant force active on each constituent star to pull it along and also toward the axis of the Arm. The relative scale of these Arm-radial and Arm-axial forces are dependent on the rate at which the Arm tapers, but the discussion above suggests a substantial 8.62 * 10-10 m/s2 acceleration acting on the Sun toward the centerline of the Arm. There seems little doubt that the Arm-radial force would enhance Arm edge definition and the Arm-axial force would define longitudinal characteristics.
This therefore explains the lasting integrity of the Arm structure, and also suggests a much less massive galaxy. It may remove the need for dark matter, exotic particles, materials, or objects to account for a lot of unseen distributed mass in the Galaxy. The question of Missing Mass as to explaining the rotation of the Galaxy is no longer necessary or appropriate.
Any stray material that is just outside the Spiral Arm could have a tendency to be drawn into the Arm, depending on the initial position and velocity.Origination of Spiral Arms and Galaxy Evolution
Two of the computer simulations mentioned above may shed light on the initial formation of Spiral Arms. If the second, regarding Barred-Spirals, shows validity, then the following possibility might be explored.
This premise might suggest some new thoughts about the classification and evolution of galaxies. There are certainly many Spiral and many Elliptical galaxies. The current premise might suggest that there is an evolutionary cycle that could exist. Spirals may eventually have their Arms close in on themselves and effectively become rotating and possibly oscillating Ellipticals. The current premise also seems to suggest the possibility that instabilities in Ellipticals could cause them to evolve into Barred Spirals and eventually, normal Spirals, as such computer simulations might show. Much more study is needed in this area, but these preliminary results might suggest a cyclic existence.Spiral Arm Angle
The premise under consideration suggests a strong relationship between different Arm angles and different Arm taper rates. A straightforward check of the validity of this premise would be to see if there is such a correlation in actual galaxies' photographs.
This premise suggests that the whole spiral galaxy rotates pretty much as a whole, (with the possible exception of the Core) but that the outermost ends of the Arms should probably slightly lose ground on the inner portions. This would suggest that the Arms don't rapidly wrap themselves around the Core, but that they very slowly stretch out in length. This would fit in with the concept of Spiral Arms beginning with minimal length and gradually evolving into long Arms. Such long Arms could only develop after billions of years of gradually building up additional tangential velocity for the constituent materials. Eventually, the Arms would become so long as to either be lost to the galaxy or to blend in with the other coils of that and other Arms. Confirmation of this would be if empirical evidence showed slight discrepancies near the outer edges in revolutional velocity from that expected of a rotating integral disk structure. Some empirical evidence already seems to exist on this.
A true logarithmic spiral does not have a logical outer limit. Spiral galaxies seem to have a fairly distinct outer edge. This may support the meta-stable aspect of this solution. Up to a certain point, the Net Forward Arm-Axial Force is strong enough to maintain the meta-stability. If a star, group of stars or molecule is at the outer border of a Spiral Arm, it is in an environment where the stability is questionable.
Such a situation could also occur farther in, where a large number of stars break away from the galaxy in one breakdown of the meta-stability, an appearance might occur like that in M51, where there is a massive whorl just past the end of a shortened Arm. If empirical evidence indicates that that whorl is moving radially from the center of M51, this might suggest support for this premise.Conclusion
The current premise seems to comply with standard physical laws and principles much better than the exotic explanations of heretofore unconfirmed missing mass, gravity waves, black holes and other hypotheses. It does NOT require any new or exotic explanation, but rather just points out a logical flaw in the previous logic attempting to explain the Galaxy's rotation and the durability of the Spiral Arms. A consequence of it is a less massive Galaxy, possibly implying a significantly smaller Universe.
This logical approach may also hold clues to several unexplained mysteries within the Solar System. Using variations on the initial circumstances, several near synchronizations might find explanation. Jupiter's four largest moons have periods of revolution that are near a simple 1:2:4:8 relationship. Saturn's rings have gaps at radii that would have periods of revolution of simple fractions of those of the massive moons. The asteroid belt has gaps at radii that would have periods of revolution of simple fractions of that of Jupiter. Saturn's rings have certain persistent texture, such as "spokes". Bode's Law was an early observation of an apparent inter-relationship between the periods of revolution of the planets. Each of these phenomena appear to have characteristics in common with some synchronizations existant in the Spiral Arms that maintain their meta-stability. Further study might be beneficial in those areas.
This essay has several references to Astronomy: The Evolving Universe by Michael Zeilek
First Developed, Nov 1997,
First Published on the Web: Aug 16, 1998
C Johnson, Theoretical Physicist, Physics Degree from Univ of Chicago