Galaxy Spiral Arm Stability Reasoning

A different perspective regarding Galaxy Spiral Arms can make it far easier to understand their gravitational stability. It is universally accepted that enormous numbers of star clusters, nebulae and gas clouds have some amount of stability, in other words that they are traveling with a general common velocity, such that the component particles have minimal velocity relative to each other. No one questions that.

The premise here is to consider the stars and gases which are components of a Spiral Arm to be a similar situation. That the situation can be considered as a combination of two behaviors, of a mutually shared motion around the Core of the Galaxy, and of motions related to each other.

The Pleiades star cluster obviously has both of these behaviors, the Cluster has a Proper Motion that is shared by all components, but they clearly also locally interact gravitationally. A significant point is that their relative nearness to each other causes the mutual interactions to be especially prominent, because of the inverse square nature of gravitation. This fact is how and why that Cluster is able to maintain a cohesiveness, while many gravitational effects are simultaneously trying to disperse the component stars and gases.

Therefore, the current perspective is to consider a Galaxy Spiral Arm as a variation of a very large star cluster. The Arm would therefore have a mutual revolution about the Galaxy Core, with a period that is around 240 million years. But it also has internal interactions due to gravitation, again, because the components are relatively near each other.

It is generally accepted that the entire mass of our Galaxy is around 125 billion sun-masses. Roughly half of that seems likely to be in the Core region, with the other half spread out in the Arms. It seems reasonable to therefore assume that a major Arm structure would have a total mass of at least 10 billion solar-masses. From our position, it is difficult to determine the width of the Arm we are in, but some research seems to suggest that it might be around 12,000 light years wide in the region we are in, wider toward the front of the Arm and tapering off to nil width at the very tail end. We will simplify things by assuming a uniform thickness of 1000 light years. Later adjustments can be made regarding the irregularity of the thickness.

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First Approximation

The Sun is currently near the inner edge of the Arm that we are in. This means that the gravitational attractions on the Sun by the other Arm components are all currently creating a force outward, toward the centerline of the Arm. Given the estimated numbers above, we can say that the Arm centerline must be around 6,000 l.y. away from us, and roughly (Galaxy-)radially outward from our location. If we use the Newton-Kepler concept that distributed masses around such a point gravitationally act as though they all were at that point (which is technically not quite always true), then we can generate a first approximation regarding the total Arm gravitational force currently being applied on the Sun. We would then just use F = G * M * m / r2, the standard Newton gravitation equation. We can eliminate m from this equation to find the instantaneous gravitational acceleration. r is the 6,000 l.y. distance and M is the total mass of the materials in the Arm.

This gives a present acceleration of the Sun as 4.17 * 10-10 m/s2.

This seems like a rather small acceleration, however, it acts continuously. Given that tiny acceleration alone, with no other intra-Arm motion, the Sun would move the entire 6,000 light years to the Arm centerline in around 16.5 million years.

However, as the Sun moves toward the centerline, two important additional effects occur. As the Sun get closer to the mass of the Arm, the acceleration increases. If no other effect occurred, once the Sun had moved halfway in to the centerline, the inverse square would indicate that the acceleration at that point would be four times as great. However, there is a competing effect that also occurs. Once the Sun is no longer at the inner edge, it will be within a population of stars and other materials on all sides. Newton proved that, if the distribution of such materials is uniform, the net effect is that the object is drawn equally in all directions (by that surrounding uniform distribution) and therefore there is no net acceleration. This effect means that the "effective mass" of the Arm which is drawing the Sun toward the centerline is reduced by the mass of nearby stars and material that are within a vertical cylinder, whose outer edge is at the edge of the Arm and the center is at the Sun.

This second effect results in a complete elimination of net acceleration at the instant the Sun crosses the Arm centerline. The effect is more complex due to the extended shape of the Arm.

The end result is that the acceleration currently in effect on the Sun will increase substantially for several million years, as the Sun gets nearer to all those masses that are attracting it. But then the acceleration will later decrease, as the Sun passes masses which will then cause a reverse acceleration effect.

The exact shape of that acceleration curve is dependent on the mass distribution within the Arm. With the simplest assumption, of a uniform mass distribution, the net effect is to increase the effect of the acceleration (up to a maximum of around 4 times the current rate) which reduces the 16.5 my to around 13 million years for the Sun to accelerate from the edge to the centerline of the Arm.

That is not likely to be a good assumption. It seems almost certain that greater mass accumulations exist near the Arm centerline. In that case, the current acceleration would not be any different, but the acceleration graph would go much higher as the Sun more closely approached the great mass before passing so much local mass. The graphs depend on the assumption regarding the density distribution, but the time to the centerline can get down to around 10 million years.

There may be one approach to the data that might assist in making an educated guess regarding that distribution. Given the width, thickness, taper and length estimates given above, we know the volume of the entire Arm. We have also estimated the total Arm mass. These figures result in an average Arm density of one solar mass per 36 cubic light years. It is generally believed that the local density near the Sun is around one solar mass per 700 cubic light years. That might suggest that the local density near the Arm centerline might be as much as 20 times greater than currently near the Sun. If this is actually the case, the acceleration graph gets far higher, and the effects of this premise are even more pronounced.

This page was first placed on the Internet in February 2006.

It is a newer text of a web-page at Galaxy Spiral Arms Stability and Dynamics of August 16, 1998.

This page - - - - is at
This subject presentation was last updated on - -
Link to the earlier 1998 Galaxy Spiral Arm Stability Issues. Explaining Galaxy Arm Stability / Apparent Rotation Inconsistencies


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C Johnson,
Theoretical Physicist,
Physics Degree from University of Chicago