Galaxy Spiral Arms Gravitational Stability

Gravitational Kinematics, Dynamics, Structure and Persistence

A hundred years before Isaac Newton lived, Johannes Kepler did amazing observational astronomy and mathematics.   He created some "Kepler Laws" regarding the Solar System where he assumed that essentially all the mass in the Solar System happens to be inside the Sun.   That was a good assumption for the Solar System, as Newton confirmed a hundred years later with his much better math (including Fluxions, which we now call Calculus).   At that time of Newton's death, no one yet knew what a galaxy was or even that we happened to be in one the Milky Way).   More recently, scientists have simply accepted Kepler's math regarding our Milky Way Spiral Galaxy, and they have been wrong in that assumption.   Newton's later Fluxions (calculus math) showed him that gravitational force is affected when mass is distributed over a large area, rather than where all the mass is localized as it is in the Sun in our Solar System.   People still use Kepler's equation regarding our Galaxy when they try to calculate distances and velocities of the Sun inside our Galaxy, and Newton totally proved that such calculations are quite wrong.   No one seems willing to use the more correct Newton's Fluxions in doing such calculations, as they are a lot more complex to do.   Still, it is sad that our scientific Community chooses to use Kepler's earlier and simpler math and logic, where many of the popular accepted calculations regarding our Galaxy are simply not accurate.

Our Solar System happens to be in one of the Spiral Arms of our Milky Way Galaxy, one we call the Orion Arm, and we happen to be near the very inner edge of that Arm.   If we look inward toward the Core of our Galaxy, there is really not much that is near us for at least 10,000 light years, and by the time we get to around 28,000 light years in that direction, there is an enormous population of many billions of stars in what we call the Core of our Galaxy.   If, instead, we look in the opposite direction, there are thousands of stars quite near us and even hundreds of millions of stars which are only around 6,000 light years away from us, within our Orion Arm.

We need to look in very different ways regarding the math for these two situations.   Yes, those many billions of stars in our Core are relatively close to each other and far from us where Kepler's simple math is decently accurate.   But in the opposite direction within the Orion Spiral Arm which we happen to be in, it is critically necessary to use Newton's far more precise math, essentially by individually calculating the Newtonian gravitational force of each separate (nearby) star and the Sun.   This happens to be some really involved math, where modern high-speed computers are needed to calculate the Summation of all those local gravitational forces, which is essentially a Summation which represents the Integral Calculus total force.

Simple Newtonian gravitation is sufficient to explain the entire existence and behavior of Galaxy Spiral Arms, including their long-term stability.   The well-known Keplerian centripetal force effect certainly dominates even though Isaac Newton downplayed its importance, but a tapering-shaped Spiral Arm also has complex internal mutual local gravitation, which has surprisingly strong local effects!

Some really obvious logical errors exist in the understanding of the structure of our Milky Way Galaxy.   For example, there is much evidence that the Sun and Earth are currently about 250 light years "above" the centerline of our Galaxy, and for more than 200 years, we have known that the Sun and Earth are currently traveling "upward at an angle" toward a spot in the sky we call the Apex of the Sun's Way.   We know that the Galaxy is very "thin", less than one- hundredth as thick as it is in diameter.   Prior to the math presented below, no one seemed to notice that at the rate we are moving "upward" we now can expect to entirely leave the Galaxy in only around 9 million years.   Which means that our understanding is extremely wrong when the Sun and Earth are certainly 500 times older than that.   This current reasoning establishes and mathematically proves that the Sun and Earth oscillate up and down in our Galaxy, likely around 150 complete cycles so far. This gravitational reasoning also establishes and mathematically proves that the Sun and Earth also oscillate radially in our (Orion) Arm, apparently doing around 100 complete radial cycles so far.   This radial cycling has resulted in the Solar System having passed through a very crowded region of our Arm's Centerline about every 26 million years, where we may have been intensely pounded by many massive asteroids on a regular basis.   One and one-fourth-cycle ago, about 65 million years ago, we had passed through that congested region of our Arm Centerline, and we wonder if debris that we encountered might have resulted in the final dinosaurs dying off.   While we were passing through (inward) at our greatest possible inter-Arm velocity, we also must have passed countless objects of various mass which were oscillating and then headed the other direction across our Arm.   This would result in "head-on collisions" between the Earth and those many other objects, all at extremely high impact velocity.   Other possible "mass extinctions" have apparently occurred on a schedule which agrees with this.   We also find it interesting that the most recent time we passed through that "junkyard", about 13 million years ago, the orientation of the Solar System was apparently severely tilted at around 63 degrees, such that the southern hemispheres of the Moon, of Mars, of Mercury and even of our Earth may have been leading us into intense bombarded by large objects which we encountered at very high head-on velocity.   I have not noticed that anyone had tried to explain why our Moon has a LOT of craters in its southern hemisphere and relatively few in the northern hemisphere.   Mercury looks similar. This reasoning also implies that most of the current craters on the Moon may have occurred just 13 million years ago, and not four billion years ago.

It is not necessary to speculate about "Gravity Waves" or "Dark Matter" or bizarre patterns of invisible structure within the Galaxy to fully explain all of the details of Galaxy Spiral Arms, in full compliance with all the Conservation Laws of science.

Consider this top-view image of the middle part of the (Orion) Spiral Arm of our Milky Way Galaxy. Keplerian central force The stability of galaxy spiral arms has long troubled the Astrophysics community.   It has been assumed that only a Keplerian (centripetal) central force was acting, as suggested in the (partial) drawing at the right.   The orange dot represents the current location of the Sun, within the blue-outlined Orion Arm that we happen to be in.   We are currently fairly near the inner edge of that Arm.   For important local gravitational reasons, the tapering shape of the Arm turns out to be very important.   The rest of the Galaxy was left out of this drawing.   Traditionally we make the usual assumption that the Sun is revolving in the Milky Way Galaxy with a tangential velocity of around 250 km/sec, and that the Sun is currently around 28,000 light years from the center of the Galaxy (the black arrow represents that radius vector).   It is then easy to calculate that a centripetal (Keplerian) acceleration must exist of a = v2/r or 2.36 * 10-10 meters/second2. This acceleration must also equal G * Mgalaxy/r2 which tells us that the effective Keplerian mass of the Galaxy must be around 125 billion solar masses.   This is only true given the assumption that only a Keplerian central force is causing the Sun to have its revolution motion.

However, Isaac Newton made very clear that such a basic assumption is absolutely wrong. When Newton analyzed Kepler's Laws, he quickly realized that the Kepler Laws are all approximate. Newton derived the exact formulations of Kepler's Laws, but even then, only for within our Solar System. Kepler had (incorrectly) assumed that only the Sun's mass applied, and it acted at the exact center of the Sun, and the Gravitational attraction between the Sun and Earth would therefore be given by:

Force = G * MSun * mEarth / r2

Newton discovered that the sum of the mass of both of the objects must be used, that is:

Force = G * (MSun + mEarth) * mEarth / r2.

It was later determined that the Sun has a mass which is 330,000 times greater than the Earth, which makes these two equations virtually identical. However, the first is approximate and the second is far more precise. Newton also used the Calculus (Fluxions) which he invented, to discover another important fact. When a mass such as the Sun has its entire mass Integrated regarding gravitational effect (for any location outside the Sun itself) the gravitational effect IS as though the entire mass of the Sun is gravitationally acting as though it is all at a point at the exact center of the Sun. This is only true if the object is symmetric regarding density, and this is a basic problem given to beginning Physics students to solve as homework! In the case of the Solar System, this caused very minor differences, entirely because the Sun is so massive, that is, that virtually all of the mass in the Solar System resides in the Sun. The results of using Kepler's Laws, within the Solar System, therefore always gave wonderfully useful results.

But Newton therefore knew that Kepler's Laws only (approximately) apply for a point source mass such as the Sun. Our Milky Way Galaxy has distributed mass, where hundreds of billions of individual stars all contribute to the total mass of the Galaxy. On a gross scale, that mass is relatively symetrical, so for our gravitational effect on any other galaxy, yes, our entire Galaxy acts as though it is a point mass (as Kepler might have assumed). But for locations inside our Galaxy, both of these assumptions which are embedded in Kepler's Laws happen to be very wrong.

Having overlooked this obvious fact, modern Astrophysics has felt it necessary to create countless unsupported speculations to try to explain how and why our Galaxy can rotate as it does and still not quickly fly apart! So they have come up with Dark Matter, Hidden Matter, Gravity Waves, massive numbers of Neutrinos and many other wild assumptions and speculations. If they had recognized, as Newton did, that Kepler's Laws only apply for point-source masses like the Sun, and they certainly do not apply for distributed masses like a galaxy, they might have seen that standard Newtonian gravitation was the only explanation which is necessary.

Concept developed by November 1997

Specifically, the local irregularities regarding where stars are, is extremely significant. If our Galaxy was a uniform distribution of stars and other mass, the usage of Kepler's Laws would not be too terribly wrong. But our Sun and Earth happen to be near the (inner) edge of one of the Spiral Arms of our Galaxy. That is, radially inward from where we are, there are very few nearby stars (for several thousand Light Years distance). In contrast with that, radially outward from where we are, there are many millions of stars within that same several thousand Light Years distance. In other words, the uniformity of mass density upon which Kepler's Law must be based, is not remotely valid.

This concept was invented and Engineered by November 1997. This presentation was first placed on the Internet in August 1998.

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It would have been great if the tremendous simplification of using Kepler's Laws would apply for our Galaxy, but it certainly is not true. So, to do a proper analysis of our Galaxy, we need to set aside Kepler's Laws and instead use Newton's Gravitation, where we need to mathematically Integrate the gravitational effects on the Sun of ALL of the individual stars and debris in the Milky Way!

This is actually the same Gravitation which Newton had quantified, but it is an entirely different perspective. Think of any galaxy spiral arm as a very large and massive "open cluster" of stars. We all accept that open clusters have a structural integrity due to mutual gravitation, even while revolving with the galaxy as a whole. We all believe that the Pleiades Cluster stays together due to mutual gravitation between the component stars and gases.

We can make some decent estimates of characteristics of this very large, somewhat cone-shaped, open cluster that we all call our Orion Spiral Arm.

If we consider a galaxy to primarily have two (symmetric) main spiral arms, it seems reasonable to estimate that the total mass of each of those arms is around 1/10 of the mass of the entire galaxy (leaving 80% of the mass as being in the Core). In the case of the Milky Way Galaxy, that would be around 12.5 billion solar masses in each of two major Arms.

We can immediately apply some simplifications. We can assume that the Core of the Galaxy is far enough away and uniform enough that Kepler's Laws do apply, specifically because we are outside that mass of Core stars. In addition, the Spiral Arm opposite our Arm is on the other side of the Galaxy, where again we can treat the contents of that Arm as complying with Kepler's Laws. Therefore, we specifically need to focus on nearby stars, that is, the stars which share our Orion Spiral Arm with us. The fact that gravitation is an inverse second-power dependence, these nearby stars have far more gravitational effect on the Sun than the many similar stars which are farther away from us have.

Specifically, we immediately have a rather simple estimate which is available. Treat the 90% of the Galaxy's mass as Kepler's Laws would do. They are far enough away where the assumptions are generally minimal. But then realize that we have nearby neighbor stars, which are essentially all outward of where we are! With very few stars inward from us (that is, not in any Spiral Arm) we can create a simple estimate of the gravitational effect of the 10% of the Galaxy's mass which is near us and outward of us. (A full Calculus Numerical Integration is really needed for a more precise calculation.)

Given the known dimensions of the Galaxy, reasonable estimates of the dimensions of an Arm are 15,000 light years wide at the start, with a uniform taper to end 60,000 light years away. If the Spiral Arm is estimated to have a uniform thickness of 1000 light years, this results in credible mass density figures, as discussed below.

Intra-Arm accelerations added If the mass of the Spiral Arm is assumed to be uniformly distributed over this volume (as 450 million discrete equal masses), a numerical integration of the trillions of mutual gravitational attractions can be done, which results in two new consequences, as noted in this drawing. There is a rather strong gravitational "restoring force" acting on the Sun to transversely keep it within our Arm, and there is also a "forward boost" acceleration which aids the Sun in "keeping up" with the portions of the Arm that are forward and inward of it. The first exists because all the nearby masses are currently pulling us outward, since gravitation is an inverse-square force law. The second exists because the part of the Arm that is ahead of us is much fatter, meaning that there are far more massive objects in front of us than behind us.

It is important to note that the scale of these accelerations are comparable to the Keplerian central force. They exist because the assumption of "symmetric mass distribution of the Galaxy" on which the Keplerian formulas are based, is not true, because of the presence and massiveness of the Spiral Arm structure, and our proximity to the massive objects in the Arm we are in. That is, galaxy spiral arm dynamics is primarily a combination of this "intra-arm" self gravitation and the traditional Keplerian Galaxy Core central force effect. Both are simple applications of standard Newtonian gravitation.

As a clear indication of this, the discussion below shows that there is an "intra-Arm gravitational restoring force" currently acting on the Sun [and on everything else within the Arm] (toward the centerline of our Arm) which currently causes an Arm-lateral acceleration on the Sun of around 8.62 * 10-10 m/s2. This varies with the distance from the centerline of the Arm but is currently over three times the acceleration the Sun experiences due to the Keplerian central force in revolving around the Galaxy. The point here is that this intra-Arm effect is significant, on the same order of magnitude as the effects of the universally accepted Keplerian central force. This Arm-restoring acceleration is large enough to keep the Sun within the Arm, meaning the persistence of the Arm Structure. All other objects also experience restoring accelerations toward the center-line of the Arm, such that nearly all are kept within the region of the Arm.

This top-view graphic of the flattened Arm shows the net long-term effect on all the objects that are within an Arm, each constantly weaving back and forth across the width of the Arm, the Sun being represented by the yellow dot.

A spiral arm's stars in motion

Note also that there is a significant net forward acceleration imparted on the Sun (not shown in the graphic above), because of the greater amount of mass in front of the Sun than behind it. (The current forward boost due to material in front of us is actually around 5.43 * 10-10 m/s2 and the rearward drag due to material behind us is actually around -3.67 * 10-10 m/s2, so the net (forward) acceleration is the 1.76 * 10-10 value shown above). This forward effect enables the Sun to revolve around the Galaxy more quickly than suggested by Kepler alone. This effect is actually enhanced by the fact that the Arm centerline is angled in toward the Core, which causes this effect to not only be "forward" but also to have a component that adds to the Keplerian central force acceleration.

If the Arm center-line was inward at a 30 degree angle, then there would be an inward (radial) component of this 1.76 * 10-10 m/s2 of about 0.88 * 10-10 m/s2. If so, that would imply that the ACTUAL Kepler central acceleration would only need to be 1.48 * 10-10 m/s2. THAT would imply a far lower total mass of the Galaxy!

The subject of the Galaxy Spiral Arm Stability has long been the cause of massive research and countless speculations and theories. There seem to be a couple solidly established facts which cannot be ignored, but for some reason, nearly always are!

(1) Our Milky Way Galaxy, like millions of other galaxies, has "spiral arms", which tend to be relatively symmetric. We have good experimental evidence that suggests that our Galaxy has rotated around 60 times already, which suggests that the Spiral Arms are also a persistent characteristic. The fact that there are millions of other galaxies that have spiral arms seems to suggest that the arm structures are either stable or meta-stable in nature.

(1b) This being the case, consider a child's pinwheel. Because it is of stable structure, it rotates as a unit. Therefore the actual velocity of rotation of any point on it is proportional to the radius of the location of that point; points farther out have to have higher velocity to rotate with the whole structure.

(2) There is strong empirical evidence that our Sun is currently moving (locally) toward a point in the direction of mu Herculis, which is around 26° degrees UPWARD of the Plane of our very thin Galaxy. The upward (Z-direction) velocity we have would cause us to entirely leave the Galaxy in only around 9 million years, barely a moment in the lifetime of most stars or a Galaxy.

Virtually all of the currently popular theories seem to ignore both of these established facts! The concept presented here shows how those facts, and all others, are easily explainable, without violating any established principle of Physics, by simple Newtonian gravitation.

Globular Star Cluster
Open Star Cluster
Spiral Arm
Spiral Arm
(NCG 1566)
Centaurus Orion Pleiades M100 NGC 1566
Which of these Galaxy components do NOT have gravitational self-cohesion?

They ALL do!

The Milky Way has countless star clusters, gas clouds, and many nebulae.

They certainly all revolve with everything else around the Galaxy center. But everyone accepts that clusters, nebulae and gas clouds maintain their structures due to mutual gravitational attraction, due to their relative nearness to each other and the inverse square nature of gravitation. They all certainly revolve around with the Galaxy, but their individual structures persist.

The same situation must be true for a Spiral Arm structure.

The billions of stars and countless other masses within a Spiral Arm certainly have mutual gravitational attraction. There is a "relative nearness" of the component parts, as there are in those smaller structures. The claim here is that a Spiral Arm should also have a gravitational integrity. In fact, the total amount of mass involved is far greater than in any nebulae or cluster, and so a meta-stable shape-persistence must occur.

The complexity of a galaxy is such that BOTH the principles of Keplerian revolution AND this gravitational Arm self-cohesiveness must apply. The great size and mass of a spiral arm structure requires additional consideration, but the situation is essentially that of a normal star cluster or nebula. This combination then explains all the observed evidence.

Current Thinking

It seems that virtually every researcher regarding the Galaxy starts off with the assumption that Keplerian calculations are the absolute starting point, and then they find various ways, such as there being an extremely massive but invisible halo of dark matter surrounding the Galaxy, to explain the observed data. Since this is the case, essentially all Papers start off by assuming either of two pre-conditions, that stars which are farther from the Galaxy Core revolve more slowly (per Keplerian thinking) OR that stars might revolve at the same velocity at any radii.

It is hard to see why this would be believed, except for an absolute reliance on Kepler's approach! We know that spiral galaxies are numbered in the millions, so the configuration is not a fluke. We believe that ours has probably already rotated at least 60 revolutions, which implies that there is some sort of stability or meta-stability. Doesn't this imply something similar to the rotation of a "rigid toy pinwheel"? The critical point is that the local rotational velocity of any point on such a pinwheel is directly proportional to radius? Larger radius should therefore imply proportionately larger revolutionary velocity, at least in the Spiral Arms, in our Galaxy.

This reasoning is based on the far larger relative size of the Sun's velocity of revolution (roughly 200 kps) than the local recognized velocity of around 1/10 that. Those individual small velocities are just too small to account for the velocities required to keep up with the revolution of the Galaxy or its Spiral Arm.

Available data regarding differential velocity rates at different radii inside of other galaxies tend to have rather large error factors. We are primarily interested here in that the error factors presented are often comparable in size to the variations of the data itself.

Unfortunately, the existing error factors in determining rotational speeds in galaxies, or in our own Galaxy, are great enough as to not even provide us whether the galaxy rotation velocity curve increases, decreases, or stays constant with increasing distance from the center. This situation seems to be true for all the data that appears to yet be available.

The data regarding the rotation of our own Galaxy also has many unavoidable large error factors, which again can be interpreted as supporting any of the general concepts. The individual motions of each of the stars are often large enough to mask the effects of localized differential rotation. Around 80 years ago, Eddington, Oort, Lindblad and others developed generalized descriptions of apparently systematic patterns of star movements, which are essentially still the data we rely on today.

Solid Evidence that Current Theories are Incorrect!

By about 1783, it was already known that the Sun appears to be moving toward a spot that Herschel called the Apex of the Sun's Way, in the Constellation of Hercules. This motion is implied by the examination of proper motions of thousands of other nearby stars, so the Sun is definitely moving toward that direction, in a "local" sense.

However, Stromberg, Kapteyn, Eddington, Oort, Lindblad, and many others noted what Eddington referred to as "Star Drifts A and B". Large-scale patterns of movements in two opposite directions were noticed in the proper motions of stars, which was realized as being generally tangential to the Galaxy. The analysis was that, given that the Galaxy is revolving with the Sun as part of it, then the Galaxy must be rotating at different velocities at different distances from the Core. Lindblad and others devised formulas to associate Galactic radius with these Drift velocities.

More recent and more accurate research has confirmed these differential velocity effects, which even then implied that we are revolving in the Galaxy toward a point that we might refer to as a Tangent, which is along the Galaxy Plane but 90° from the direction of the Galaxy Core. A star chart has been provided here that shows the exact locations of the Galaxy Plane (yellow line) along with small green dots to indicate the locations of the Galaxy Tangent (which we are certainly moving toward at over 200 km/sec) and the Galaxy Core (the center of our revolution motion). On the same star map is another green dot which shows the location of the Apex of the Sun's Way.

Sky Chart Note that the 20 km/sec locally identified motion of the Sun toward a spot in the constellation of Hercules (approximately at R.A.18h02m Dec. +29.2°) is generally toward the Tangent direction (at R.A. 21h12.0m Dec. +48°19') but is also aimed around 26° above the Galaxy Plane, as well as about 37° inward from that Tangent direction. These are significant facts.

Given that our local motion is believed to be around 20 km/sec toward the Apex, this means that the Sun has a Z-axis (vertical) component velocity of around 8.7 km/sec upward relative to the Galaxy Plane (toward the North Galactic Pole). It also has a radially-inward (toward the Core) component velocity component of around 12.0 km/sec. The bulk of this local motion is the third component, along the direction of the revolution motion of the Sun around the Galaxy, with that component being around 16.0 km/sec.

This is in general agreement with currently accepted figures:
(found in Wiedenhoff) "the galactic circular velocity components, which give [for the Sun] U = -9 km/sec, V = +12 km/sec, and W = +7 km/sec." where "Space motions comprise a three-dimensional determination of stellar motion. They may be divided into a set of components related to directions in the Galaxy: U, directed away from the galactic centre; V, in the direction of galactic rotation; and W, toward the north galactic pole."

We have long known that the Galaxy where we are is roughly 1,000 light years thick. We also see that a great-circle for us is actually around 1.6° above the center of the distribution of the components of the Milky Way Galaxy. Therefore, it has long been universally accepted that we are currently ABOVE the Plane of the Galaxy, at a Z-distance which seems to be around 250 light years. This means that there is only around 250 light years of Galaxy currently above us (and 750 light years of Galaxy currently below us). We are moving upward, as noted above. It is easy to calculate that, at the rate we are currently moving upward, we would entirely escape the Galaxy in under 9 million years! Another similar calculation regarding the fact that we are near the inner edge of the Arm that we are in, and we are moving inward, shows that we would escape the inner edge of the Arm in roughly 13 million years. These are very brief intervals of time, considering that a single revolution of the Galaxy is believed to take around 200 million years.

None of the current popular theories regarding the Galaxy seem to account for these well-established facts. It seems that such theories would not have much of a Galaxy to work with, after even 50 million years if a majority of all the stars had exited toward the Galactic Poles!

This current presentation insists that each Spiral Arm has gravitational self-cohesion. This results in the component stars and other objects being decelerated when moving away from the Galactic Plane, by the billions of solar-masses "behind" it collectively providing that deceleration. In fact, the result certainly is that all such stars and other objects must be oscillating in the Arm (vertically) along the Z-axis. In the case of the Sun, the current "upward" velocity would therefore be slowing, eventually stopping and reversing, at or before the upper limit of the Galaxy. Then it will proceed "downward" through the thickness of the Galaxy to eventually approach the bottom limit of the Galaxy. After some "period" of this oscillation (probably around 12 million years), we will again be at about the Z-position and Z-velocity that is currently true.

Incidentally, this concept helps explain the previously unexplained "waviness" seen in many spiral galaxies, as being short-term, local gravitational, effects, which will self-correct within a few million years.

The exact same effect must certainly be happening for every component of a Spiral Arm, along a radial direction. The current radially-inward velocity of the Sun must "soon" stop and reverse, such that the Sun remains part of this Arm. Otherwise, there could be no recognizable structure in a spiral galaxy! The Sun must therefore weave back and forth across the width of our Arm. In our case, it appears that we must oscillate across nearly the entire width of the Arm that we are in.

A popular common idea is to attribute elliptical galactic orbits to many stars, while others have relatively circular orbits in their paths around the Galaxy. (That approach has some real problems when the Z-axis velocities are considered, due to the extreme small dimension of the thickness of the Galaxy!) This new approach shows the same net effect, of arm-transverse velocities, but attributes them to a different cause, that of a constant weaving back and forth across the Arm. It results in rather different calculations, though, because the elliptical orbit hypothesis requires a period of that of the rotation of the Galaxy (~200 million years) while the intra-Arm weaving can have an independent period, which appears to be ~50 million years.

If this new reasoning is valid, and it seems to be, then the situation is actually a little different than it first appears. Yes, we may be currently moving upward, but it seems logical that a significant number of stars in our vicinity are actually currently moving downward. This would result in an apparent value for our upward velocity that is larger than is actually true. With certain reasonable assumptions, we might conclude that the ACTUAL upward velocity of the Sun might be around half of what we observe. This might be a better value to use in gravitational acceleration calculations regarding the nearly harmonic motion due to the Z- restraining force of gravitation.

These matters are discussed a little more below, and in the web-page calculations of this (referenced below), it appears that the likely period of such a radial, cross-Arm oscillation is around 50 million years.

Arm-Longitudinal Effects

We will use 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 the 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 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 the following discussion and the fact that we are currently near the inner edge of the Arm that we are in. The reasoning below suggests that it is quite possible that the mass density may easily be twenty times as great along the Arm centerline region.

This suggests that our mass and dimensions, of 1/10 the Galaxy mass in each of two major, relatively symmetric arms, are credible estimates. When a numerical integration is done regarding a UNIFORM distribution of the mass in the Arm, we get the following results:

Consider a random distribution of masses in a tapered wedge region as indicated here, as representing a Spiral Arm. Consider the yellow star indicated. We note that geometrically, there is three times the area to the right of it as to the left. Forward of it (to the right) there are 9,300,000,000 solar masses of material. Behind it (to the left) there are 3,100,000,000 solar masses of material. The summation of gravitational attractions certainly is a net force to the right (forward). For the current situation of the Sun, with the above estimates in numerical integration, we get a forward acceleration of around 1.76 * 10-10 m/s2. As noted above, the Keplerian central force from the entire mass of the 125 billion Suns of the Galaxy causes a radially inward acceleration of around 2.36 * 10-10 m/s2. This "forward boost" is therefore very significant.

This must certainly be true for every star and object in the pattern. All the component objects would therefore receive a (forward) incremental acceleration. In a Spiral Arm, this is a forward- and somewhat inward- directed (due to the angle of the Arm's centerline to the Galaxy) acceleration, for all components. All objects in the Arm therefore revolve FASTER than they would if the (tapered) Spiral Arm structure was not present. Numerical integrations for an object at the very end of the tail of the Arm describe above would receive a forward boost of 8.0 * 10-10 m/s2. This acceleration is around four times the magnitude of the Keplerian central force acceleration for that object, a strong evidence that this indicates why such trailing materials are able to keep up with the Arm.

Angle of Taper is Important

If the taper is broad, the differential masses of stars are physically closer, more masses are at nearer distances for gravitational effect and so the effect is stronger. If the taper is narrow and long, the differential masses are physically farther away, and so the effect is weaker. The angle of taper of the Arm is very important regarding the meta-stability of the Arm structure.

Effects-b Even though this yellow star is not along the centerline of the pattern, there is certainly still the effect of a net gravitational attraction toward the right (forward) side.

An analysis of current data suggests that the Sun is currently getting this "forward boost" which results in about 1/4 of the effect of the Keplerian Central Force. This therefore causes the Sun to revolve at a rate around 20% FASTER than it would if the Spiral Arm was not present. This differential acceleration enables the different parts of a Spiral Arm to "keep up" with the revolution rate of inner portions of it.

One effect of this is that the Keplerian calculated total mass of the Galaxy is therefore much less than previously thought. If only Keplerian effects were present, the period of revolution of the Sun would therefore be about 5/4 what we believe it to actually be. By Kepler, the Galaxy mass would therefore be 16/25 as great, less than 2/3 of what we now believe it to be.

Arm-Transverse Effects

Effects-c Stars and material that are not along the centerline of the Arm must also be accelerated toward the centerline of the pattern. Therefore all stars and materials must constantly weave back and forth across the width of the Spiral Arm. Using the Arm specifications given above, numerical Integration gives a current "Arm restoring acceleration" for the Sun to be about 8.62 * 10-10 m/s2. Again, this is several times stronger than the Keplerian acceleration which also acts on the Sun, keeping the Sun securely part of this Arm.

The acceleration that the Sun is subjected to decreases as the Sun passes some of the attracting mass and approaches the Arm centerline, where it becomes zero and the Sun is moving transversely across the Arm at the highest velocity. If we have a net average of 4 * 10-10 m/s2, using standard Newtonian equations, we find that the Sun should be traveling at around 170 km/sec as it crosses the Arm centerline. Those standard Newtonian calculations indicate that the Sun should "fall" to the Arm centerline in around 13 million years. This would result in a full cycle of oscillation across the Arm and back in around 52 million years. The Sun must therefore weave back and forth across the width of our Arm around four times for each revolution around the Galaxy.

After the Sun crosses the centerline, there is obviously no further acceleration, and with the bulk of the Arm mass then behind it, from then on the effects are in deceleration. The differential mass density within the Arm affects the value of this acceleration, so until better mass density distributions are known, this value must be considered to have a high error factor.

The motion then resembles a periodic sinusoidal motion across the Arm, but it is actually more complicated, having variable acceleration along its motion. But the gravitational restorative force must certainly exist, and it is of significant amplitude, probably around 8.62 * 10-10 m/s2.

For comparison, if we accept the distance of the Sun from the center of the Galaxy to be 28,000 light years, and the total mass of the Galaxy to be 125 billion Suns, by Kepler we get a central acceleration of 2.36 * 10-10 m/s2. These are not to be compared literally, due to the many possible inaccuracies in the data within the Arm, but it is clear that the effects are on the same order of magnitude. The Intra-Arm gravitational effects are strong enough to enable Arm Stability and Persistence. In fact, they are strong enough to enable Arm-genesis.

Possible Implications

Implications-a Due to collisions and the effects of near passes, it seems logical that the local density of material would become greater near the centerline of the Arm, as compared to near the inner edge of it where we currently are. Such a differential may even indicate the age that an Arm has had to evolve. As noted above, the local mass density near the centerline may be twenty times what it is around us now. That may be an indication that the Spiral Arms of the Milky Way Galaxy have persisted for a long time. We calculated above that the Sun is likely to pass through that congested area at extremely high transverse velocity, around 170 km/sec.

This might suggest that the Sun and Earth passed through a region of far more debris approximately 13 million, 39 million, 65 million, 91 million, 117 million, etc, years ago. Paleobiologists seem to have found some evidence for repetitive mass extinctions of species at around those pseudo-periodic dates. The Sun would pass through that cluttered region at extremely high (Arm-transverse) speed, while other debris would simultaneously be passing through at equally high speed going the opposite direction. It therefore seems that a possible astronomical explanation of such mass extinction periodicity might exist.

If such an object would hit the Earth head-on, the relative speed could be over ten times as fast as any Solar System meteorite could impact the Earth. This would therefore involve over 100 times the kinetic energy to cause destruction, such as the speculated K-T boundary event meteorite.

Actual Physical Evidence

There may be actual physical evidence for this. The orientation of the plane of the Solar System is such that the southern hemisphere would have been in front as the Sun and planets passed through that cluttered Arm centerline region, roughly 13 million years ago. (The Northern hemisphere of each Solar System object would lead on the return path across that region (such as 13 million years in the future).

It has long been noted that the Moon has many more large impact craters in its southern hemisphere than in the northern. The same has been found true of the planet Mercury. Possibly even for Mars and some satellites. This new approach may provide a logical explanation for that long-known asymmetry, that the (leading) south side of the Moon, Earth, Mercury, Mars, etc, would have been susceptible to far faster collision impacts than would be true of the trailing side. We note the same effect every night in that evening meteors tend to be less brilliant than early morning meteors, because of being on the trailing/leading sides of the Earth in its orbital motion.

This also implies that the many meteorite craters we see on the Moon's surface are NOT four and a half billion years old, but instead only about 13 million years old!

There is another possibility as well as the brute force impact.

Such an impacting object would have come from well outside our Solar System, quite possibly carrying simple molecules or even amino acids that are not present in the Earth's environment.

This might provide a biological source for mass extinctions, and/or rapid bursts of new biological species.

The unexplained bursts of biological diversity might therefore have an astronomical explanation.

Moon The cratering of the Moon might also therefore occur primarily in cycles of around 26 million years.

Many of the craters visible on the Moon might be from a barrage around 13 million years ago rather than at the dawn of the Solar System.

Stars and material near the very tail of a Spiral Arm would have virtually all of the locally attracting mass forward of it. We already calculated this boost acceleration as 8.0 * 10-10 m/s2. This would have the effect of enhancing the forward- and inward-acceleration imparted on materials in that region. This effect creates a meta-stable "adhesion" of such materials to the main body of the Spiral Arm. This provides an explanation of how Spiral Arms can be stable for extended periods of time, and without the Arms spiraling themselves around the Core as is often suggested. The summed accelerations caused by the relative nearness of billions of solar masses ahead of it are significant enough to have large effects.

Yes, a star near the end of the tail of a Spiral Arm is affected by the Keplerian Central Force of maybe 100 billion solar masses, but it is also attracted by maybe 12 billion solar masses within its own Arm, which are generally closer to it and therefore more gravitationally effective.

Our discussion above has assumed a constant mass density distribution throughout the length and breadth of the Arm. We have already mentioned that it seems likely that the local mass density near the Arm centerline may be 20 times our local mass density. It seems highly likely that due to "forward migration" the local mass density will be greater nearer the Core (forward of us) (where it is thought to be as high as 1.0 solar masses per cubic light year, around 700 times as great as in our vicinity (Zielek p. 352)) and lower farther out in the Arm. If these situations are true, the Arm-longitudinal effect described here becomes far greater yet.

Around 1998, I did extensive computer simulations regarding this concept. NO assumptions were made, and only simple Newtonian gravitation was calculated, between each and all of the component stars. The available computer had limited speed, so my simulations used 1000 equal mass mega-stars, with a total mass approximating that of an entire Spiral Arm. In the simulations, those 1000 objects were given initial random positions within a two-dimensional tapering triangle shaped area, (indicated above) approximating the shape of a Spiral Arm.

A single iteration therefore involved approximately a million gravitational attractions of mega-star to mega-star, which essentially devolved to determining the actual distance between the two, to quad precision. Some simulations ran for weeks! The pattern was seen to gradually change along the length of the Arm, with the objects tending to migrate together toward the wider (forward) end of the pattern. There were also transverse bunchings near the centerline, mostly due to close passes of the objects to each other, where some transverse velocity was sometimes altered into longitudinal velocity. I interpret these things as indicating a self-sustaining structure where the taper of the Spiral Arm actively maintained itself. All the individual components continually weaved / danced back and forth across the width of the pattern.

M51 Additional simulations were done where an additional mega-star was given an initial location outside of the triangular region. The exact position and initial velocity were critical, as sometimes it would be drawn into the Arm pattern and sometimes it would drift away. It also occasionally happened that one of the included mega-stars became severely perturbed by a close pass of another mega-star and then exited the pattern area. Further study in this area indicated that there is a meta-stability rather than a true stability, and that there are some conditions where objects can break away.

The whorl of M51 that appears to have broken away from a now shortened Arm is potentially an example of this.

NGC 981 There are also z-axis effects and oscillations for all components in the Spiral Arm. Since the thickness of the Galaxy is rather small, these effects seem to be somewhat irrelevant, and their shorter time periods of oscillation seem to be unimportant. In the case of the Sun, the period of this Z-axis oscillation may be around 12 million years. The current Solar velocity toward the North Galactic Pole seems to be such that it will stop around 400,000 years from now, only a few light years above where we are now. This effect appears to provide an explanation for the "waviness" of the Galaxy's Plane, where no other explanation seems to have been presented.

This photo of NGC 891 shows a slight waviness in its plane.

Note that since all spiral arms have centerlines that angle inward toward the front, the resulting acceleration not only acts to instantaneously increase the orbital speed but also acts to add an incremental Central Force to the existing Keplerian Central Force. The results of these two effects are different, but they both act to inspire self-formation of spiral arms and also persistence of existing arms.


This general effect is somewhat similar to ice skaters holding hands and circling in a "crack-the-whip" action. The image here is meant to show an aerial view of a skating rink with seven people standing still in a line. The lines between the seven are meant to suggest their arms as they are holding hands with each other. The small lumps are to represent their hands gripping each other.


They have now started to skate SLOWLY around the one (larger) person who represents a pivot point. Notice that the inner people are hardly moving while the people farther out are skating fairly quickly, in order to keep the line straight. They each must skate at a speed that is exactly proportional to the radius of the circle they are following.

There is NO significant tension in the hand-to-hand grips, and they are essentially only needed to keep them all aligned in a row.


Here they have picked up rotary speed. The inner skaters still have no trouble skating fast enough to keep up, but the outer skaters are now unable to skate fast enough to keep up. They are only able to because of the additional forward (and inward) thrust they receive from the hand-to-hand connection with the next person in.

The outermost skaters are now in a meta-stable situation, where they are actually revolving faster than they could otherwise skate! The are receiving an additional force / acceleration from the hand-to-hand pulling of the person just inside them. That hand-to-hand force/acceleration is both forward and inward to create that effect. Notice that the angle of that effect changes with the radius /speed involved. In this particular example, the outermost person is hardly any farther out than the next person in, and is pulled nearly directly forward by the hand-to-hand tension.

That effect is meta-stable, as when any hand-to-hand grip slips, one or more outer skaters shoot off radially outward! Therefore, in any large such skater formations, except at very low rotational speed, the outermost skaters tend to be somewhat behind a desired straight line, which makes the somewhat logarithmic curve pattern which is essentially the same as the observed curvatures in all galaxy spiral arms.

At MODERATE rotation speed, most of the inner people can generally maintain the original straight line formation, where only the outermost skaters cannot keep up and are pulled forward. This is extremely similar to the pattern of the Barred-Spiral galaxies, which have rarely been decently explained otherwise.

The currently popular density-wave concepts do not adequately explain several things. There seems no provision for keeping most of the component stars from quickly escaping along the Z-axis. The universal logarithmic curvature/convexity seen in the shape of spiral arms would not be a consequence of any (planar) density wave concept. In addition, no one seems to have considered that an incoming planar gravity wave would have had to have come from some initial direction, and there would therefore NOT be any "spiral pattern" of a shock or density wave at all, but instead a rather planar wave which would pass through the region of a galaxy, allegedly causing starbursts. There is no conceivable way that would cause a pattern of stars "lighting up" that were ALWAYS in a spiral pattern! And a generally symmetric one at that! So the common claims of a fairly uniform actual (but invisible) distribution of a disk mass, with the alleged density waves causing localized ignition, cannot be causing spiral arm patterns. In addition, many existing theories do not seem to properly consider Keplerian considerations as they describe them.

There are claims that enormous amounts of invisible mass must exist, as in a giant massive halo around the galaxy, but such claims seem to never have contemplates where that mass would have to be! In some such theories, a super-massive halo is supposed to exist OUTSIDE the Galaxy, but Newton showed us that such external mass would have no gravitational effects interior to it. In some such theories, a very massive torus of invisible material (dark matter, hidden matter, neutrinos) would have to exist around the rim of the Core, and additional peculiar distributions of that mass would have to exist, in order to cause the non-Keplerian fast revolving of the Spiral Arms. The alleged distribution is often simply referred to as an invisible halo! Such a structure would not be stable or even meta-stable.

The alleged invisible matter could not be uniformly distributed, but it would have to be arranged in a very unstable pattern. Such an unstable arrangement would have to exist in all of the many observed spiral galaxies.

There are people who make a popular claim that 90% or 99% of all the mass of the entire Universe is actually neutrinos. It is a rather silly idea, I think! Consider the known source of neutrinos. A neutrino is emitted when a neutron breaks apart into a proton and electron, and needed back when they re-combine to form a neutron again. That seems to me to indicate a pretty solid limitation that there are not likely to be more neutrinos in existence than there are neutrons. Yet those very speculative theories claim that there are billions or trillions of times as many neutrinos than neutrons or protons. If there actually were such an astounding number of them, then someone should explain where they all originally came from! The claims seem to force a conclusion where all those neutrinos were created at the start of the Universe, with no actual function or purpose, just somehow existing to account for a lot of mass that people think is needed! Sounds pretty funny to me!

Calculations and an extended presentation are presented in a web-site at Galaxy Spiral Arm Stability Issues. Explaining Galaxy Arm Stability / Apparent Rotation Inconsistencies


Michael Zeilek Astronomy: The Evolving Universe

First Developed, Nov 1997,
First Published on the Web: Aug 16, 1998
Updated occasionally since.

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Carl Johnson, Theoretical Physicist,
Physics Degree from Univ of Chicago