User:James Antoniadis/sandbox

Lagrange orbital Theory for Galaxy Formation == HYPOTHESIS FOR GALAXY FORMATION CONFORMING TO OBSERVED ORBITAL VELOCITIES WITHOUT INVOKING COLD DARK MATTER OR NEW PHYSICAL LAWS.

A hypothesis by James Antoniadis

Summary of this Hypothesis

Attempts to explain observed galactic structures and orbital velocities using essentially Newtonian level physics and only that matter which is directly observable. Proposes that the motion of matter in the outer aspects of a galactic disc does not orbit the center of the galaxy directly from a gravitational perspective. This outer matter orbits more proximal parts of the disc with a period of orbit that is the same as the time it takes those more proximal parts of the disc to orbit the galactic center. This creates the illusion that the outer disc is also orbiting the galactic center. A mechanism of galaxy formation that results in this orbital structure is then considered. Concludes that if the proposed orbital structure was what occurred in nature then there would be no need to invoke as yet undiscovered Dark Matter or new physical laws such as Modified Newtonian Dynamics MOND to explain the way matter moves around a galaxy.

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Lagrange points

Introduction.

This paper aims to present a mechanism for galaxy formation which could explain the fact that in most galaxies the orbital speeds of the distal parts of the galaxies do not match the speeds predicted by the distribution of visible matter within those galaxies. Unlike a solar system the outer bodies in a spiral galaxy have a similar orbital period to the inner bodies. In effect the velocity of rotation increases linearly with the distance from the galactic core. The hypothesis is derived from a series of thought experiments. This hypothesis does not require the existence of Cold Dark Matter CDM,(Cold, because it emits no radiation signature that we can detect) or any other strange physics such as variations of the laws of gravity ( see MOND) to describe what is seen in nature. This is not to say that CDM or other physical laws do not exist, only that they may not be necessary to explain the formation of spiral galaxies. Currently the computational models for exploring gravitational interactions between many objects simultaneously lack the ability to simulate galactic rotation adequately. In this absence, MOND and Dark Matter have been invoked to explain the strange rotation velocities of spiral galaxies.

== Hypothesis for Spiral Galaxy Formation without Dark Mater ==

Thought experiment.

Imagine a satellite in an orbit around the Earth with an orbital period of 365.25 days. If this satellite were to be opposite the sun from the Earth it would appear to be in an orbit around the sun further out from the Earth but with an identical period of revolution around the sun. The satellite would be further from the sun than the Earth and yet would appear to be orbiting the sun at a higher speed than the Earth. If in fact the satellite were orbiting the sun further than the Earth the orbital speed would be expected to be slower and the orbital period would exceed the year that the Earth takes for that journey. See figure 1.

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Figure 1

Now imagine a three star system with a large star in the centre and two smaller stars in identical orbits but at opposite sides of the central star. Let us call the central star M1 for and the two identical orbiting stars M2A and M2B as they have approximately the same mass and orbit. Now imagine each of the M2 stars having a companion body M3 being M3A and M3B. If these companion bodies were in an orbit around their respective M2’s with the same orbital period that the M2’s require orbiting M1 i.e. “Y” then the M3’s would appear to be orbiting M1 instead of their respective M2's. If the M3’s were at the furthest positions in their orbits relative to M1 they would keep this relative position throughout their orbits. To an observer looking at the system from above it would appear that the M3’s were orbiting M1 directly with the same orbital period as the M2’s and yet much further out from this body. In fact each body M3 is really orbiting a point somewhere between M1 and M2 as its orbit is affected not only by its nearer M2 but also less so by the objects M1 and the opposite bodies M2 and M3.

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Figure 2. This shows how even though the M3 bodies are really orbiting their respective M2s the paths they take give the appearance that they are orbiting M1

Each M3 reduces the gravitational pull on their respective M2 by counteracting some of the pull of M1. This means that each M2 can orbit M1 more slowly than would be expected by their distance from M1 alone. Thus in this five body system the M2s would be orbiting slower and the M3s faster than the expected orbital period that their distance from M1 would suggest. Both sets of bodies would take the same time to go around M1.

The five-body system can be extended to involve a symmetrical seven or nine-body system with the addition of smaller bodies M4 A& B and M5 A & B. In these systems the inner bodies would appear to have slower than expected orbits and the outer objects faster than expected orbital speeds. All would have the same orbital periods around the galactic center. In effect momentum is transferred from the inner bodies to the outer bodies in a sling shot type effect similar to the technique used to speed the outward journeys of planetary probes via encounters with our planetary gas giants. The result of which is to speed up the probes and imperceptibly slow down the gas giants.

In these hypothetical five, seven and nine body systems the inner bodies are effectively dragging the outer bodies around the central body M1 faster than they would be expected to go move on their own and at the same time preventing them from flinging away from the system itself.

Now substitute the bodies M1 through to M5 A & B for masses comprised of interstellar gas, dust, stars and star clusters and an image more closely resembling a galaxy with arms at opposite sides becomes apparent.

It is the observation of orbital velocities in galaxies that has stumped theoreticians and caused them to postulate the existence of, so far unobserved, cold dark matter. The dark matter is needed if one believes that the outer aspects of galactic arms are orbiting the galactic centres at a speed in excess of the inner aspects of the galactic arms in order to keep up with them. As in the five, seven and nine body systems described above it may be possible that the outer parts of the galactic arms are not directly orbiting the galactic centre but are in orbit around centres of gravity comprising a much larger effect from the galactic arms themselves and a lesser effect from the galactic centre. These orbits are synchronised with the orbital period of the proximal aspects of the arms and so the stars on the distal parts of the arms erroneously appear to be orbiting the galactic centres.

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Figure 3. The stretching of gas clouds by tidal force causes the elongated bodies to appear to merge into single long arms

Figure 3 shows a system composed of nine diffuse bodies M1 MA (2-5), MB (2-5) Each body M (n), except M1, orbits a body M (n-1). Each body is composed of a cloud of material with its center being the center of mass and not necessarily embodied in a discreet solid object within it. The margins of each body may merge with the margins of the adjacent bodies thus giving the appearance of a continuous long arm.

The Evolution of a Spiral Galaxy.

Imagine a huge gas cloud like the ones that form planetary systems but billions of times more larger. The cloud begins to collapse under its own gravity and as gas falls inwards to a developing core its velocity imparts an angular momentum on the cloud and core. For now let us ignore the presence of stars or bright objects within the cloud and consider only that areas of higher and lower density form within the cloud. As the cloud coalesces you would imagine that the majority of the matter would be attracted by gravity to the center of mass of the cloud. This would most likely be spinning rapidly. This spinning central amorphous mass would also transfer some of its angular momentum to the surrounding gas cloud. The surrounding gas cloud would eventually attain sufficient angular momentum to overcome the gravitational pull of the centre, and instead go into orbit around it. As in a solar system this matter spinning around the central mass would concentrate itself around the equatorial plane of the spinning mass thus forming a spinning disc. Any material spinning at a different orbital plane would have to pass through the dense disc material and would be slowed by it until after enough passes it would be captured by the disc. As in planetary systems the masses closer to the central mass would orbit at higher speed with shorter orbital periods than the peripheral parts. Furthermore the fast moving inner regions would interact with slower gases in slightly wider orbits resulting in a transfer of angular momentum from the inner gases to the outer gases.

The inner gases having lost some of their momentum would no longer be able to stay in orbit and would fall into the inner core. The core could conceivably accumulate enough mass to begin to collapse further resulting in a very dense region which could eventually become a very massive star cluster or even a huge black hole (this point is not essential to the hypothesis of galaxy formation that I am presenting).

At present the gas cloud still has the properties of a solar system with the inner orbital velocities exceeding those on the periphery. Now consider that at some point in the disc of gas away from the center and in the slower moving periphery a secondary region of increased density develops. If the central condensation is M1 then this peripheral condensation is M2A. This is similar to how we think planets are formed in planetary systems but in this case the clumped matter does not yet have to be any denser than a gas cloud.

The effect of M2A on the gas cloud is that it could cause an intensification of the gravitational field pulling the region opposite it from M1 in the direction of M1. In this way the region of gas cloud opposite it could become more massive much as the moon creates a bulge in the oceans of the earth opposite it and not just adjacent to it. This region then forms a second mass focus M2B with the same orbital period “Y” and hence in the same orbit as M2A. See figure 4.

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Figure 4

The tidal effect of M1 on M2A and M2B will cause them to bulge inwardly in the direction of M1 so that they become egg shaped. The existence of the opposite M2 body on each of the M2’s is that the gravitational force pulling each of them toward M1 will be greater, thus the M2s will move faster in their orbits than gas in other parts of the same orbital path. As a result the M2s will rapidly overtake and incorporate the remaining gas along their orbital paths thus clearing the paths and becoming more massive. See figure 5.

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Figure 5, Elongated M2s in largely cleared gas cloud.

The tidal bulge of each M2 could extend all of the way into the outer reaches of the M1 gas cloud. This could occur if, each M2 is suitably massive and the outer reaches of M1 are composed of gas orbiting around its centre. (Rather than if it were sitting on its surface at sub orbital velocity like an atmosphere).

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Figure 6. Structure of M1 body.

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Figure 7. Line drawing in which the higher the number of arrow heads on each line indicates a higher orbital speed.

The inner aspects of the M2’s are moving more slowly than the adjacent outer layers of M1 (see figure 7). If the M2’s inner aspects actually dipped into the outer layers of M1 the effect would be like a finger dipped into a fast moving stream.

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Figure 8. Thicker lines indicate gas being ejected from M1 to join the far side of the M2's.

Figure 8 shows how M1 would also bulge out towards the M2’s so that the outer orbiting layers of M1 would change direction towards the M2’s as they bulge out from their otherwise circular orbits. As higher energy systems tend to transfer energy to adjacent lower energy systems the result would be that some of the mass and angular momentum of M1 would be transferred to the M2’s. With enough velocity this gas could even be flung beyond the distal edge of the M2s thus extending the M2s furthest reaches away from M1. This new distant part of M2 could even be considered to become the bodies M3A and M3B. See figure 9.

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Figure 9. Transfer of matter from M1 to outer M2 resulting in formation of M3

With time the orbits of the M3s may become skewed so that they are no longer in the same line as the bar formed by M2A, M1 and M2B. They might then appear to be dragged around by the M2s like a rhythmic gymnast trailing a ribbon. The gravitational force between M2 and M3 is the force pulling M3 along rather than M3 orbiting M1 directly. See Figure 10.

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Figure 10. The appearance of spiralling arms caused by the trailing of the M3’s.

In this way the formation of a system resembling a spiral galaxy with the orbital periods as seen in nature could develop without cold dark matter or any new physics.

The presence of galaxies with four spiral arms instead of two could also be explained. If similar forces and a sufficient amount of material could allow the formation of two further bodies in the position of the poles of a compass. See Figure 11.

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Figure 11. The beginnings of a spiral galaxy with four arms.

After this point the further development would be identical to the development of the two-armed systems. I would suspect that the four-armed galaxies would require a higher total mass than the two-armed types. Beyond a certain mass it may be that the tidal forces between massive arms would be enough to disrupt them so that no arms survive. This would fit the observation that the smaller and more plentiful galaxies have arms and the largest galaxies tend to lack distinct arms.

Some illustrative Mathematics (Newtonian Physics)

A simple bit of mathematics may be useful to illustrate the point that even if the bulk of the mass of a galaxy is in its centre it is still possible for an outer body to be more affected by the arm it is in than by the galactic center.

Imagine a galaxy with a core and a massive bar bell type arm composed of an inner massive part and an outer less massive part. Imagine that the outer arm orbits a centre of mass between the inner arm and the core and has the same orbital period as the inner arm orbits the core. To an outside observer it would appear that the outer object would be travelling at a paradoxically high speed. Some brief calculations illustrate this point. If orbital velocity is V

V²= 2G(M+m)/R² Eqn(1)

Let, m, be a smaller part of a galaxy orbiting a larger part, M, where R is the orbital radius.

The orbital circumference is 2πR

Thus orbital period P = 2πR/V Eqn (2)

Combining Eqn (1) & Eqn (2) we derive

P = √(πR⁴/2G(M+m)) Eqn (3)

And

R⁴= P²G (M+m)/2π Eqn (4)

If the orbital period P is fixed at 1 time unit for the orbiting bodies in a galaxy then

R⁴= G/2π(M+m) Eqn (5)

R = k⁴√(M+m) Eqn (6)

Where k = ⁴√G/2π

For the sake of the experiment let the centre of the galaxy (core) have a mass of 2 units and an orbiting inner spiral arm (inner arm) with a mass of 1 unit. For a given orbital period, P=1 time units, the radius of rotation will be

R = k⁴√(2+1) = 1.32k Calculation (1)

Another object with mass 0.5, such as an outer part of the same spiral arm, (outer arm) further from the (core) but in a line with it and the (inner arm) will orbit at a distance of

R = k⁴√(3+0.5) = 1.37k Calculation (2)

from the centre of mass of the (core) and the (inner arm). As these have masses of 2 and 1 units and they are 1.32k units from each other then their centre of mass is at distance

⅓х1.32k = 0.44k Calculation (3)

from the (core) towards the (inner arm). Thus the (outer arm) is 0.44k + 1.37k = 1.81k Calculation (4) from the galactic core and yet (as has been designated in this experiment) it will have the same orbital period as the inner part of the spiral arm. If an observer were to assume that the outer arm orbited the centre of the galaxy with radius 1.81k then their calculations would lead them to believe that the mass of the system M+m was

1.814 = 10.73 mass units’ Calculation (5)

rather than the 3.5 mass units with which we started. This could lead them to believe that about 70% of the matter were invisible or dark. By this example it can be seen that despite very considerable distances within a galactic arm it is possible for an outer part of the arm to be attracted more strongly by an inner part of the same arm than by the much more massive but distant galactic center. Furthermore, if less of the mass were in the galactic core then the figures become even more favourable for this hypothetical galactic structure.

Discussion/Conclusion.

This is a hypothesis for galaxy formation that could explain the orbital velocities of the outer aspects of galactic arms particularly in spiral galaxies. Some Physicists have postulated the presence of Dark Matter in order to explain these orbits; others have proposed variations in the laws of gravity to explain the same observations. This hypothesis invokes no such unproven physics but relies on a theory of galaxy formation that has at its basis various assumptions about how matter clumps to form galaxies. One might argue that the series of steps suggested here might be unlikely and yet this would not be argument enough to dismiss this hypothesis out of hand. This hypothesis gives an example of a model of how galaxies form using physical laws and particles that are already understood and known to exist, as such it may be less speculative than the theories which require as yet unexplained Dark Matter or baseless changes to well established physical laws such as MOND.

References.

P. Salucci, F. Walter, A. Borriello The Distribution of Dark Matter in Galaxies: a Constant-Density Halo around DDO 47 http://arxiv.org/abs/astro-ph/0206304