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.
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.
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.
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.
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
(Centaurus) |
Nebula
(Orion) |
Open Star Cluster
(Pleiades) |
Spiral Arm
(M100) |
Spiral Arm
(NCG 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 Polaris.net 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.
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.
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
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
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.
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.
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.
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
public/galaxyzz.html
References
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