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.

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.

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.

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.

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