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Dark Matter or Stars that are hurled away

 29–4–24

In recent decades, the term Dark Matter has caused a stir. It is one of those terms that no one knows what meaning to give, but where your imagination immediately runs away: matter that you cannot observe, but of which there is perhaps a hundred times as much as normal matter.

Wow!

 It involves the observation of stars on the outside of a galaxy that are moving much faster than expected based on Newton's law of gravity. These stars orbit at a great distance from the center of the system and, because of their great speed, must be attracted much more strongly than Newton prescribes, otherwise they would fly away from the system. A good example of this is the Triangle Nebula galaxy in the Triangle constellation, which is located 3 million light years away. The nebula contains about 40 billion stars with a total mass of 50 billion solar masses. The diameter of the galaxy can be assumed to be 60 000 light years.

The scientific designation for this galaxy is Messier 33 or NGC 598.

 In this system, stars have been observed at a distance of 40 000 light years with speeds of 125 km/sec, while at that distance a speed of about 35 km/sec is expected if Newton's laws of gravity are applied (see Figure 1). The thought is that this must be caused by matter – Dark Matter – which is invisible.

On this basis it was concluded that the share of Dark Matter in galaxies exceeds the amount of observable matter many times over (see https://en.wikipedia.org/wiki/Triangulum_Galaxy#Etymology ). According to this source, the center of the galaxy contains no or only a 'small' black hole of less than 1500 solar masses.

The Triangle Nebula is suitable to test our new gravity theory - the Obstruction Theory.

It is important to keep in mind that in a galaxy all stars move in the same direction around the center of the galaxy. Furthermore, two individual stars will always attract each other. In the rotating system, a star near the center will be slowed in its motion by a more distant star if it moves in front of it, and will be accelerated in its motion if it moves behind it. According to Newton's theory, the two effects cancel each other out during an orbit.

This is not the case in Obstruction Theory. This is a consequence of the alternative interpretation given to gravity in the Obstruction Theory.

For our calculations we assume that in the center of the galaxy Messier 33 there is a mass of N = 1000 solar masses M, so NxM = 1000M. We imagine a star M₁ with one solar mass M moving in a circular orbit with a radius R of 2000 light years around the center. We look at the effect of the star M₁ on a second star M₂ with also a mass of one solar mass M at a distance ten times greater r = 20 000 light years.

The star M₁ has an orbital speed of v₁ m/sec and rotates one orbit in T=2πR/v₁ sec. According to the measured speeds in the figure, the speed at a distance of 2000 light years is about 50 km/sec. At that speed the orbital period T ≈ 75 million years.

We will now show what new physical insights the Obstruction Theory leads to.

This is because due to the speed of light, the passage of the farthest point is only observed at a later time than the passage at the same time of the closest point.

Fig. 1 The spiral nebula M33 and its associated rotation curves. The dashed curve shows the rotation at increasing distances according to Newton's laws of gravity, but the solid line shows the speed according to observations. Source https://nl.wikipedia.org/wiki/Donkere_materie Extended rotation curve of M33 by Stefania.deluca.

According to the obstruction theory, the force between the two stars will therefore last 4R/c sec longer as M₁ moves away than as M₁ approaches. The result is that the center of gravity shifts some distance from the system's pivot point, which coincides with Newton's center of mass, to a point in the direction of the trajectory the star follows during removal. That is an unexpected physical consequence of the Obstruction Theory interpretation.

With the given speed of 50 km/sec we can calculate how many solar masses are within that distance to the center, namely: 350 million. The volume of that area is 4πR³/3=33.5 billion cubic light years. This gives a density of approximately 0.01. So the stars are sparsely spread.
For comparison, the closest star to the Sun is more than four light-years away. This indicates a star density of 0.003.  

Ø           The formula for the speed shows that a higher speed at a certain radius of the orbit requires a quadratic higher mass with an unchanged distribution of the mass. For example, we see that at 40 000 light-years from the center of Messier 33, the speed of the stars is about 4 times higher than expected. This therefore requires - for the same mass distribution - a mass that is approximately 16 times larger. That is then attributed to Dark Matter. However, the Obstruction Theory gives a different explanation, as we will see.

The second star M₂ moves in a 'large' orbit of 20 000 light years around the center at a speed of about 105 km/sec. For the second, we can calculate the orbital period as 360 million years. It takes 5 times as long to complete an orbit as M₁.  

The force M₂ exerts on M₁ slows down M₁ as it moves away from M₂. M₂, on the other hand, is accelerated in its orbit. This is a consequence of the law of conservation of angular momentum. As the two stars approach each other, the opposite happens.

From M₂, M₁ is observed longer when it is moving away than when it is approaching. So, according to the Obstruction Theory, the force comes from the direction of the retreating star M₁ for a longer period of time than from the approaching star. On average, the force does not come from the pivot point of the system. We call the point where the resultant force originates from the dynamic center of gravity of the system.  

This dynamic center of gravity moves back and forth between the system's center of rotation and a maximum deviation. This maximum excursion occurs when M₁ is at the peripoint or at the maximum distance. It coincides with the pivot point when the two masses of the system are in their extreme lateral positions.

Since, according to this theory, the dynamic center of gravity generally does not coincide with the center of rotation, a force exerted on the system through the dynamic center of gravity can give the system more or less rotational speed.  

Ø           Think of a bicycle wheel that is also turned by the force of the chain on the sprocket whose teeth are also outside the pivot point. The larger the diameter of the gear, the easier - that is to say with less force - the wheel starts moving.

 The rotational speed therefore changes due to the moment of force exerted on the system. This changes the angular momentum of the system. What the system loses in angular momentum due to the force of M₂ on the dynamic center of gravity, the star M₂ must gain in angular momentum. This causes faster orbital motion of M₂.

 Ø           Because all inner stars in the system are slowed down by the outer stars and all outer stars gain speed as a result, the principle described works for the entire galaxy. This is how Dark Matter can be explained. We can further imagine that local density differences in the number of stars in the galaxy's disk contributed to the development of the galaxy's spirals.

 We will use a simple calculation to show how much speed M₂ will gain from its force action with M₁ when M ₁ has completed one revolution around the center compared to M₂. 
We must first determine by integration where the dynamic center of gravity is located on average. To do this, we look at the locations of P and Q, see Figure 2, and the resulting distances SP and SQ. We use this to determine the distance difference dℓ= |SQ–SP| .

 

Fig. 2 Seen from M₂, M₁ is moving away over a longer period of time than it is approaching.

As a result, the moment of force on the rotating system is greater when M₁ moves away than when the star approaches. This also applies to the heavy central mass, which although it moves over much smaller distances, has a much larger mass to the same extent.

With the measured values ​​v₁= 50 km/sec and v₂= 105 km/sec for R = 2000 light-years and r = 20 000 light-years respectively, we find: Δv= 0.89x10^–9 m/sec after one orbit.

That is a very small number. Yet this leads to substantial speed changes over time if we consider that in the galaxy the number of stars within a distance of 5 000 light years can easily exceed 30 billion solar masses. The previous train of thought applies to each of these stars. Then the speed of M₂ could increase by a factor of 30x10⁹. This gives M₂ a joint speed increase of Δv= 27 m/sec after an assumed average of one orbit in a period of 84 million years.

Ø           Because the orbital period of M₂ is five times longer than that of M₁, one orbit, viewed from M₂, will take 5/4x75= 94 million years.

 If we assume that the galaxy has been around for 10 billion years, then the stars have completed an average of more than 100 orbits. The speed increase would then have increased to at least 2.7 km/sec.

However, the speed v₂ of M₂ is difficult to calculate because the distribution of the stars seen from M₂ is not spherical but that of a disc with a spherical center. However, for v₂ we can fill in the values ​obtaine by measurements (see fig.1).

We see several variables in the formula for Dv. The first is the radius R of the orbit of M₁. An increase by a factor of 2 leads to a decrease by the same factor of Dv . But this increase also leads to an increase in the number of stars N that M₁ is attracted to. If we maintain a spherical star distribution at a constant density, N increases with the third power so that in total Dv will increase quadratic. At a large distance R, however, this no longer holds because the mass distribution is then no longer spherical.

 A second variable is the distance r of M₂ . As the distance increases, Dv decreases quadratic so that the velocity v₂ becomes almost constant.
 The number N of stars involved that attract M₂ is practically constant at these large distances.

This explains why the velocity of the outer stars in the graph (fig.1) shows a value  that does not depend strongly on the distance.

From this we see that if we choose R and r favourable – at a relatively short distance from the galactic centre – and then double R and halve r the increase in velocity can be increased by a factor 16. The value of 2.7 km/sec then increases to 43 km/sec.

 This example shows that the velocity increase of the outer stars – provided sufficient data are available – can be easily calculated via the Obstruction Theory, with the velocity increase being of the order of magnitude observed in systems where Dark Matter is assumed.

On the other hand, the star M₂ is slowed down by stars that are even further away. At a certain distance from the mass center, the effect of the increase in velocity due to the transfer of angular momentum from the inner stars to the outer stars will be in equilibrium with the loss of angular momentum by these stars to stars that are even further away. We also realize that the outermost stars no longer lose any angular momentum to the not– existent, even more  distant stars. They will then gain some speed after each lap.

 There is a danger that they will be thrown out of the galaxy, but this may be prevented by a second mechanism that follows from the Obstruction Theory, namely that the stars that are in a row amplify each other's time delay. The deflection of the light by the first star causes the light at the next star to show an amplified deflection compared to the situation where the first star is absent. In this way, masses form a lens system similar to an optical lens system. According to the Obstruction Theory, the resulting increased time delay results in a direct increase in the gravitational acceleration.

However, this is generally an exceptional situation because the moments when they are aligned are very short, although the alignment must be understood as meaning that the mechanism is at work as long as there is some overlap – from the observer's point of view – of the Einstein rings of the two masses.

 However, it is more favourable for a galaxy with a large black hole at the center. The Einstein ring of the black hole has a solid angle of ω=2πR₀/r sr where R₀ is the radius of the black hole. This is a relatively large solid angle within which, seen from a large distance, a number of intermediate stars can be located. Each star increases the time delay at the location of the observer, so that the amplification of the gravitational acceleration exerted by the black hole becomes the product of the amplifications provided by each of the stars involved.

 The amplification by a single star can, under favourable conditions, amount to a factor of 4/3 (see "Time and Cosmos" p.122). In general it will not be more than a few percent. However, when the number of stars directly between the observer and the black hole is large, the gravitational acceleration can become many times as strong. When this amplification occurs as seen from M₂, it can result in the star M₂ not flying out of the system, but still being held in an orbit within the system by the greater gravitational acceleration.

The extent to which both mechanisms can balance each other needs to be further investigated.

In the previous section we have shown, through freehand calculations, that the Obstruction Theory offers great promising possibilities to explain the mystery of Dark Matter. On the website www.einsteingenootschap.nl   you can also find the explanation of the mysterious increase in velocity of the interstellar comet Oumuamua by applying the Obstruction Theory.

Given these results, further research into the potential of the (physical) Obstruction Theory is urgently needed so that our understanding of physics can develop further. Perhaps this can even explain the strongly deviating elliptical orbit of Mercury and the Titius–Bode rule for our planetary system.

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