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The Origins of Dark Matter, Dark Energy, and their Causal Linkage Charles D. Cole, III1 1Dinosaur, CO, United States Email: [email protected] Cover: This paper offers an alternative explanation for Dark Matter, Dark Energy, and their unexpected causal linkage by creating an alternative version of Laplace's and Le Sage’s Shadow theory. A geometric proof is derived that explains the origins of Dark Energy and Dark Matter, predicting both in amounts that agree with observed data. This leads to a surprising double conclusion about the speed of gravity and a different interpretation of r in Newton’s Law for orbiting masses. This research affects multiple topics of which these five stand out: 1. Currently Dark Matter is considered a different type of matter than visible or ordinary. 2. There’s no accepted explanation of Dark Energy. 3. There’s no current proof of a linkage between Dark Energy and Dark Matter. 4. Mechanistic gravity is currently considered to travel infinitely fast. 5. The r in Newton’s Law seems inviolate. Atomic decay, black hole and the big bang theories are also affected in that we can compare more exactly, the inward pull of all matter versus the outward pull resulting from Dark Energy’s tangential acceleration vector. Sincerely, Charles Cole The Origins of Dark Matter, Dark Energy, and their Causal Linkage Abstract: Newton’s Law of universal gravitation provides that any two bodies in the universe attract each other with a force that is proportional to the product of their masses and inversely proportional to the square of the distance between them. However, in observed galactic rotation, gravitational forces are found to be larger than can be explained by visible or ordinary matter. To address this apparent discrepancy, termed cosmology’s “missing mass” problem, Zwicky accommodates the 5.6-­‐fold greater force observed than predicted, by postulating that a further invisible mass termed “Dark Matter” makes up the difference. 𝑚! 𝑚!
𝐹! = 𝐺 ! 𝑟
(1) The following alternative approach works by instead manipulating the r in the denominator. 𝑚! 𝑚!
𝐹! = 𝐺 ! ! 𝑟
(2) Looking at this concept through the lens of a revised shadow theory without Fatio’s “ultramundane corpuscles”, we see how the relatively slow speed of gravity generates dark forces that dwarf the original non-­‐orbiting visible matter’s force in a specific ratio based on the velocity of gravity and the speed of the orbiting bodies. Introduction There is a theoretical phenomenon that seems to pop up every century or so before quickly becoming discredited. In 1690, Nicolas Fatio de Duillier (1664-­‐1753) was the first to record the concept and almost certainly shared it with his friend Sir Isaac Newton (1642-­‐1727). It was eventually given the name “Shadow Theory” by the French mathematician Georges-­‐Louis Le Sage (1676-­‐1759) and examined with respect to orbits by French mathematician and astrophysicist Pierre-­‐Simon Laplace (1749–1827). He was interested in proving the stability of the orbits, so he had to address the Shadow Theory, which predicts unstable orbits. By studying historical records of lunar eclipses Laplace looked at the lengthening of the moon’s cycle and concluded it was off by one arc second per century. He then used that number to calculate that the speed of the force of gravity must be seven million times the speed of light for the moon to stay in its one arc second per century-­‐off orbit, if the Shadow Theory and eclipse records were right. Modern relativistic physics has the speed of the gravity traveling at 𝓬, but “mechanistic” physics in this case states gravity must be traveling infinitely fast or else all orbits would be unstable. At this point, we should take a quick look at this Shadow effect. Figure 1. The Shadow Theory effect. Two stars 𝐴 and 𝐵 orbit each other. 𝐵! ! is directly across from 𝐴! ! but because of the slowness of gravity, is pulled toward where 𝐴 was at time 𝑡! , when the gravity was emitted. Current theory says mechanistic gravity travels infinitely fast; therefore frictionless orbits are stable since the gravity from 𝐴𝑡! hits 𝐵 at 𝑡! . Figure 2. 𝑽𝑭𝒈 = ∞. These gravitational forces point exactly toward each other if 𝑉𝐹𝑔 = ∞. If 𝑉𝐹𝑔 < ∞, a tangential acceleration vector results. In the original Shadow Theory as studied by Fatio, Le Sage, and Laplace, this was the only vector considered. The reduction of 𝒓 would be so small at high gravitational velocities as to be irrelevant and besides, the tangential acceleration vector was clearly impossible for orbits to be stable, so why look any further? Figure 3. The Shadow Theory effect. However, if gravity takes any amount of time to get from 𝐴 to 𝐵, then there must be a difference between locations 𝐴𝑡! and 𝐴𝑡! , meaning that there arises a tangential acceleration vector pushing the stars out of orbit and a larger 𝐹! resulting from a reduced 𝒓 or 𝒓’. The Shadow effect refers to the idea that each star chases the other’s shadow, or where it was rather that where it is. To further digress, another way of looking at the speed of gravity and the speed of light is: Figure 4. Gravity and Light Aberration. *The sun only wobbles a bit in its “orbit” around the earth, so the orbit is very small, the axis being inside the sun’s diameter and fairly close to the sun’s actual center of gravity -­‐ not what people think when they think about the sun moving across the sky. They equate amount of travel across the sky to a certain speed of light but the effect is exaggerated by the spinning of the earth making the sun’s location appear to change much more that it really does. Although not technically accurate, this exaggeration makes the idea of light and gravity taking time to travel more clear in many people’s minds. This delayed effect is sometimes referred to as aberration. In Figure 4, does gravity pull us towards 𝐴𝑡! or 𝐴𝑡! or somewhere in between? In the earth to sun’s case, the time it takes for light to reach the earth is a little over eight minutes. This means that the sun you see is not in the same location as where it really is. You see it where it was eight minutes previously because that’s how long it took for the sun’s light to arrive here -­‐ but how about the gravity? Are you pulled to where the sun is or where it was eight minutes before, or somewhere in between? Any time delay at all would cause instability of all frictionless orbits. Physicists and mathematicians have concluded during the intervening centuries that this Shadow effect must prove that in a mechanistic sense gravity travels infinitely fast. This was most recently studied in 1998 and it was reconfirmed that the Shadow effect is completely and irreparably discredited, best relegated to the dustbin of historical curiosities. Coincidentally, and perhaps ironically, it was also in that same year astronomers discovered Dark Energy. It followed the 1930’s claim by Dutch astronomer Jan Oort (1900-­‐1992) that there was too much gravity in the universe, confirmed by Vera Rubin (1928-­‐present) at 5.6 times the expected amount. Fritz Zwicky (1898-­‐1974) speculated that to achieve this amount of gravity, there must be “Dark Matter”. Like Dread Pirate Roberts’ Iocane powder, Dark Matter is an odorless and tasteless invisible substance scattered about the universe in just the right amounts, adding the necessary gravity to hold it together. For neither Dark Matter nor Dark Energy is there external corroboration or triangulating evidence to affirm their existence beyond a shadow (ahem) of doubt. Their effect reveals that visible or ordinary matter makes up only 5% of the universe, Dark Matter 27%, and Dark Energy 68%, with physicists in debate as to where Dark Matter and Dark Energy come from, and indeed what they are. A Geometric Proof Our stars 𝐴 and 𝐵 are in their usual orbit but 𝐵 is feeling 5.6 times as much pull towards 𝐴 as 𝐵 should. Zwicky approached this problem by considering Newton’s Law: 𝑚! 𝑚!
𝐹! = 𝐺 ! 𝑟
(3) On the one hand, theorizing Dark Matter provides the most straightforward approach since, intuitively, more mass yields more gravity, so if one or a mix of masses are increased 5.6 times, there will be an increased gravitational pull of 5.6 times Fgexpected satisfying both the gravitational evidence and Newton’s Law.
On the other hand, there is another way:
instead of increasing the mass product 𝑚! ⋇ 𝑚! to get ( 5.6 ∗ 𝐹𝑔expected ) as Zwicky did, (4) it is also possible to decrease the 𝑟 in Newton’s Law to get the same ( 5.6 ∗ 𝐹𝑔expected ) (5) Figure 5. r and r’. As you see from Newton’s Law, decreasing 𝒓 increases 𝐹! Or 𝑭𝒈 ’ = 5.6𝑭𝒈 𝑤ℎ𝑒𝑛 𝒓’ = 𝑟/√5.6 (6) So by reducing 𝑟 to 𝑟’ 𝒓’ = 𝑟/√5.6 𝑜𝑟 0.423 ∗ 𝒓 (7) As will be shown, by using 𝒓’ in Newton’s Law, we can see where both the effects ascribed to Dark Energy and Dark Matter come from, without need of such hypotheses.
But how can the distance between 𝐴 and 𝐵 be 𝒓’ when astronomers assure us that they are at a real distance of 𝒓? Figure 6. Swing 𝒓’ to meet 𝑨’s orbit. There are two points on 𝐴’𝑠 orbit when it is at a distance of the exactly 𝒓’ from 𝐵. If you swing the 𝒓’ from 𝐵𝑡! , you see the two intersections with 𝐴’𝑠 orbit. The only one that is logical is the 𝐴𝑡1 point ahead of 𝐵𝑡! . This is the only spot that is at the right distance to emit the “correct” gravity of 𝐹! ′ that is also on 𝐴’𝑠 orbit path. 5. 6 𝐹𝑔𝑒𝑥𝑝𝑒𝑐𝑡𝑒𝑑 = 𝐹𝑔’, 𝑤ℎ𝑒𝑛 𝒓 = 𝒓’ (8) This spot 𝑨𝒕𝟏 , is the only place that 𝑨 could have been when it emitted the correct amount of gravity. (This statement is revised in the comment section.) Figure 7. Breakdown of gravity’s force between 𝐴𝒕𝟏 and 𝑩𝒕𝟐 . If we look at this gravitational force vector, it is pointed at an unexpected 65° off of where it should be if indeed gravity does travel infinitely fast. The angle is expected to be a 𝟎° and 𝑉𝐹𝑔 is expected to equal ∞ resulting in stable orbits. This means that 𝐴𝑡! was at an angle of 65° pulling 𝐵𝑡! in and forward at a 65° angle instead of the expected 0° angle. This shows the effect of the revised Shadow Theory. The visible matter force grows in addition to a Dark Energy component forming as the angle 𝜃 increases. At 𝜃 = 65°, the net gravitation can be broken down into 𝑠𝑖𝑛 (65) ∗
𝒓’ 𝑎𝑛𝑑 𝑐𝑜𝑠(65) ∗ 𝒓’. The inward energy as a percent of the total is: cos θ
= 32% 𝑜𝑓 𝑎𝑙𝑙 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 sin θ + cosθ
(9) The tangential energy being added is sin θ
= 68% 𝑜𝑓 𝑎𝑙𝑙 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 sin θ + cosθ
(10) Thus, this rethinking of Shadow Theory obviates the need to postulate a distinct source of either Dark Matter or Dark Energy. The tangential acceleration vector can be relabeled Dark Energy. The inward pulling vector can now be relabeled “All Matter”. There really is no Dark Matter other than the effect caused by the Shadow Theory prediction resulting from non-­‐infinitely fast 𝑉!! . The empirical predictions Dark Matter and Dark Energy were invented to satisfy are, in this conceptualization, the natural consequence of the calculated non-­‐infinitely fast 𝑉!! . The predicted result matches astronomers’ empirical results by using the corrected Shadow Theory which takes into the account the slowness of gravity and expands the gravitational pull of orbiting objects by significantly decreasing their 𝒓 to 𝒓’ and correcting the angle θ at which they are pulled by gravity. Figure 8. Relabeled axes of Figure 7. This surprising result, that of gravity coming in at a 65° angle leading to unstable orbits, allows a better calculation of the speed of gravity than Laplace was able to derive over 200 years ago. We know the distance gravity must really travel is 𝒓’. It is possible to calculate the time it takes star 𝐴 to trace 130° along its arc if we plug in velocity figures from the fringe stars of the Milky Way. Unfortunately galaxy rotation charts indicate a wide range of data between 150 and 250 km/sec as the speed at which these stars travel. Using the average of 200km/sec we can figure out how long it took for a star to trace a path of 130° in our Milky Way. The time it takes for star 𝐴 to trace its orbit from 𝐴𝑡! to 𝐴𝑡! (or 130°) is the same time it takes for gravity to travel from 𝐴𝑡! to 𝐵𝑡! You have distance over time with which to calculate 𝑉!! . 𝑉!! ≅ 166,000 mph (11) Or 𝑉!! ≅ 𝑐
4000
(12) Since the astronomical data going into these equations is not that precise, the speed of the force of gravity as calculated herein must be taken with a grain of salt. The one thing that we can absolutely see from the geometric proof is that 𝑉𝐹𝑔 << 𝑐 (13) Three Positions of Relevance for Mass: 1. where it is 2. where you see it 3. where you feel it gravitationally Going back to our earth/sun example, the earth is being pulled not towards the actual position of the sun, not to where it was eight minutes ago, and not to a position in between (it was a trick question). Gravity is so slow, that the earth is pulled in the direction of where the sun was — over seven months ago. Of course the sun barely wobbles at all so the direction of pull is virtually the same. The steps taken above are on the Figure 9: Figure 9. Proof Results. Conclusion: This revised Shadow Theory addresses the empirical observation that the gravitational force is 5.6x as great as predicted. Rather than solving this by postulating increased mass via Dark Matter, it explores what would take place if the denominator of Newton’s Law were instead reduced, making a smaller r through a re-­‐
envisioning of Shadow Theory by which a finite speed of gravity causes the gravitational pull of a body at the “opposite” point in the orbit to be exerted from a prior location of that body. The resulting prediction obviates the need to invent the concepts of Dark Matter or Dark Energy to satisfy astronomers’ observations. A total inward pulling force which is 5.6 times larger than the predicted visible mass inward pull indicates that the speed of gravity travels surprisingly slowly at approximately c/4000 or around 166,000 miles per hour which then arrives at the orbiting partner at around 65 degrees instead of the expected zero degree angle. Dark Energy represents the tangential acceleration vector component of the total gravitational force between orbiting masses. The inward pulling vector represents what people refer to as Dark Matter plus visible matter although it is due to a smaller r’ and not to any real Matter that is ‘dark’. Because of the speed, or rather the slowness of gravity, Dark Matter and Dark Energy appear in a specific ratio based on the angle at which gravity arrives. The most important conclusion we can draw from this is that Newton’s “Universal” Law of Gravity is only a specific case of gravity, applicable to non-­‐-­‐-­‐orbiting objects like we see in our everyday life on earth. The proper universal or general law of gravity should be thus: Universal or General Law of Gravity 𝐹! = 𝐺 𝑚! 𝑚!
𝑟! !
The force of gravity 𝐹! , is equal to a constant 𝐺 , multiplied by the product of the two masses 𝑚! 𝑚! , whose mutual gravity is being measured, all divided by the distant r’ that each gravitationally perceives the other to be, squared. In Newton’s case, he is correct if: 1. Gravity travels infinitely fast or 2. The masses rotational velocity is zero and they are not rotating about each other or in acceleration. This more comprehensive Law of Gravity both predicts and explains the hitherto unexplained empirical evidence of Dark Matter and Dark Energy provided to us by our hard working astronomers. The difference between Newton’s Law and this General Law of Gravity can range from zero in the case of non-­‐-­‐-­‐orbiting bodies, to substantial amounts in the case of masses orbiting near the speed of light. Since all galaxies and the entire universe are in an orbit of one kind or another, this more comprehensive law applies to free-­‐-­‐-­‐floating bodies in space. Bodies “orbiting” at a zero rotational velocity or not undergoing acceleration will obey Newton’s specific law of gravity and generate no Dark Energy or the extra gravity that is currently defined as Dark Matter. Although our calculated speed of gravity about c/5000 seems very slow and the effects because of it seem anti-­‐-­‐-­‐intuitive, it is easy to forget that even this slow speed of gravity is very, very fast and far beyond what we see, feel, or experience here on earth or even in our solar system. Much like the relativistic corrections that Einstein makes due to very high relative velocities, this General Law of Gravity also makes a correction noticeable only at high rotational velocities or accelerations. COMPARING DARK ENERGY'S OUTWARD PULL WITH MATTER'S INWARD PULL At this point, gravity's inward pull is compared with dark energy's outward pull at different orbital velocities. It is easy to forget that what we have examined thus far is one specific case and speed of orbit. In our geometric proof and subsequent conclusions, we only considered two masses orbiting at a speed of 200 KPS or the universal or galactic averages derived from the same data. Different speeds of orbits yield different gravitational vectors changing the amounts and percentages of dark energy and ordinary matters inward pull. In my first geometric comparison of forces, Figure 10. Force Proportions At Different Speeds Of Rotation. I could see that dark energy becomes the dominant force at high speeds and matter becomes the dominate force at low speeds but there was no scale to see what is going on between speeds as to the magnitude of the forces or vectors. In Figure 11, I plotted what 𝐵𝑡2 would feel as its orbit speed increases and it gravitationally feels the “closer” 𝐴𝑡1. The first chart shows how the forces, vectors and speeds would interact when the speed of gravity, 𝑉𝐹𝑔, is 70kps. This chart finally shows us what shadow theory predicts for all orbit speeds or accelerations. We can see that the “dark” forces go to zero at zero orbit velocity where Newton's specific law of gravity applies. We can now also see how, at high orbital velocities, there would be an accelerated expansion of the Universe. This net outward force occurs at speeds a little faster than the speed of Gravity or 𝑉𝐹𝑔 and increases exponentially as the orbit speed rises. Figure 11. Force Vectors To Scale At Various Orbital Velocities. I then started worrying that the star speed that I used to calculate the 𝑉𝐹𝑔 was not necessarily the right star -­‐ that is to say that the star was already locked in an orbit speed that could be at other distances that what I looked at. So I redid the force chart to be based on the star's orbit velocity compared with 𝑉𝐹𝑔. !
Figure 12. Forces On Bt2 Based On The Angle And Distance of Perceived At1. Interestingly the 𝑉𝑓𝑔 angle never changes. You can plug in any 𝑉𝐹𝑔 and all the forces and angles recalculate in multiples of 𝑉𝐹𝑔. I subsequently added the forces that 𝐵𝑡2 would feel if it started “feeling” it's own shadow when it reached or exceeded the 𝑉𝐹𝑔 speed. This chart shows a sudden inward jerk as the speed of orbit creates an apparent third mass also attracting 𝐵𝑡2. The only case that a third mass would not eventually see its own shadow would be if 𝑉𝐹𝑔 was indeed c as the masses could never accelerate to 𝑉𝐹𝑔. C as a limit is itself some multiple of 𝑉𝐹𝑔. Figure 13. Net Of Forces In A Three Out Of Two Mass System. These charts in particular and shadow theory in general predicts the accelerated expansion of the universe and graphically illustrates how Newton's law of gravity is only a specific case of the more general law of gravity using shadow theory and 𝑟′. This step function increase in inward pull might be the reason that spiral galaxies have arms that suddenly change direction at the ends of their bars. Older spiral galaxies have arms that detach from the center. This might be where the inward pull is exceeded by the outward pull. The next task is to go back to square one and recalculate 𝑉𝐹𝑔 based on what we have learned from shadow theory and galactic rotation charts. !
Comments: In our solar system, forces act on the earth in addition to the sun’s gravity. The solar wind, which is not inconsiderable, also travels slowly in comparison with c and therefore works against the Dark Forces being generated by the orbiting earth or any other planet. The wobble of the sun due to other orbiting planets dwarf the effect generated by gravitational Dark Energy making it easily mistaken for noise. As convenient as it is, our solar system is not the ideal laboratory for gravity. A far better lab is when the masses are far enough apart to be treated as points, as stars in a galaxy can be. In our revised Shadow theory using 𝒓’, we use the same data from the (orbiting too fast) stars 50 million light years from the galactic center of the Milky Way that remain part of cosmology’s “Missing Mass” problem in order to derive the speed of gravitational force. One of the things that made this puzzle hard to solve was that 𝒓 seems to be inviolate, a constant and not subject to manipulation. By using 𝒓 = diameter of orbit, gravity was assumed to be instant at all distances. However, using 𝒓’ allows for a velocity of gravity that is less than infinitely fast. Secondly, after using 𝒓’, gravity between orbiting bodies is then predicted to come in from a 65 degree angle, far off from the expected zero degrees. After making these two unobvious assumptions plowing ahead with the discredited Shadow Theory, the proportions of visible matter, Dark Matter and Dark Energy in the universe can be calculated. Comment Postscript and Additional Calculations After “finishing” this paper, I realized that another strange effect can exist. Because 𝑉𝐹𝑔 is slower than the speed of the stars, they outrun their own gravity — that is, if they are traveling in a straight line. In a circular orbit, they will eventually have to run back into and feel their own gravity that was emitted at an earlier time, putting at least one additional shadow or a third apparent mass in a two mass system. An analogy would be if a hypersonic fighter jet flies in a circle. It, at some point, must hear and experience the sonic boom it created at an earlier time. If 𝐵 feels its own shadow from a previous time, the location of 𝐵’𝑠 shadow would change the location 𝐴𝑡! must be at in order to have the net effect still pull at a 65° angle. The geometric proof above only showed the net effect of gravity on 𝐵𝑡! and its direction of pull. With additional points of attraction for 𝐵𝑡! , we see that the 𝐴𝑡! point would have to be the net of the gravity of star 𝐴 and any shadow that 𝐵 may feel even if it was emitted by itself at an earlier time. The distance gravity travels in a specific time (𝑡! -­‐ 𝑡! ) is the distance that gravity travels from point 𝐴𝑡! to 𝐵𝑡! . This represents the time that both stars 𝐴 and 𝐵 take to travel about 130 degrees of an orbit, putting 𝐴 at point 𝐴𝑡! on the diagram. Going back two time periods to near 𝑡! , the gravity 𝐵 emitted earlier would hit Bt2 at around a 52° angle. This means that At1 was really at an angle of 70° changing its location from the earlier spot on the geometric proof. That spot marked 𝐴𝑡! in the proof can actually be relabeled “the net effect of gravity on 𝐵𝑡! .” Figure 14. Three Masses in a Two Mass System. In the diagram above, new calculated positions of locations 𝐴𝑡! and Bat an earlier time represent the net effect of gravity on 𝐵𝑡! . 𝐴𝑡! becomes closer to 𝐵𝑡! which then balances the farther and weaker force coming from Bt2’s shadow. This new arrangement of 𝐵𝑡! being attracted to not only 𝐴 but also its own shadow lowers the calculated speed of gravity to about 120,000 mph or c/5500. Depending on where they emanate from, additional shadows would change the 𝑉𝐹𝑔. 23. This is one of the weirdest things I have seen in (non-­‐quantum) physics. That is, a mass in a fast enough orbit becoming attracted to itself at first seems too strange to be true, but upon reflection there seems to be no reason for it not to. Additionally, since our calculated 𝑉𝐹𝑔 is around 70km per second, any orbiting masses with a higher velocity would suddenly start reacting to their own shadow. It is probably not coincidental that many galactic rotation curves begin to noticeably depart from the expected Keplerian velocities when the masses exceed about 70 km per second. Most galactic rotation curves focus on the farther stars in the galaxies where they display the greatest divergence from Keplerian predictions. I think more interesting is the point at where they begin to diverge from Kepler’s predicted velocities. Although every galaxy has its own unique rotation curve, there are also similarities that they share. I have seen on several curves the divergence start when the speeds of the stars exceed about 70 km per second. This would be expected from our theory because at that point the stars are traveling faster than their own gravity and would, quite suddenly, start seeing their own shadows. Since galactic rotation curves rarely focus on the divergence, more data would help to verify this. This gravitational effect is caused by the slow speed of gravity between all rotating and orbiting masses. All spinning mass systems are affected from large to small, from orbits on a universal scale to atoms and their constituents, and everything in between. Although dark force vectors look large on paper, they are directly related to the visible matter vector, so it is well worth remembering that the visible matter and its dark gravitational forces, at any significant distance — let alone at a distance of 100 million light years between orbiting masses, is going to be breathtakingly small — so if you don’t feel 5.6 times as heavy as you think you should after reading this paper, that’s OK. *Two-­‐significant digit accuracy is all that can be claimed for the results herein since the percentages of missing gravitational matter and the speed of the outer stars that is used throughout this is of two-­‐
digit accuracy. 24. Bibliography Wikipedia. Nicolas Fatio de Duillier. https://en.wikipedia.org/wiki/Nicolas_Fatio_de_Duillier (accessed 2015). Richard Feynman. The Relation of Mathematics and Physics -­‐ Part 1-­‐ The Law of Gravitation (full VersionRichard Feynman. YouTube, https://www.youtube.com/watch?v=1SrHzSGn-­‐I8 (accessed 2015). Wikipedia. Pierre-­‐Simon Laplace. https://en.wikipedia.org/wiki/Pierre-­‐
Simon_Laplace (accessed 2015). Wikipedia. Georges-­‐Louis LeSage. https://en.wikipedia.org/wiki/Georges-­‐
Louis_Le_Sage (accessed 2015). Newton, Isaac. The Preliminary Manuscripts for Isaac Newton's 1687 Principia. 1st ed. England. Cambridge UP; 1684-­‐1685 Wikipedia. Jan Oort. https://en.wikipedia.org/wiki/Jan_Oort (accessed 2015). Rubin, V. C. Bright Galaxies, Dark Matters . 1st ed. Woodbury, NY. Amer. Inst. Phys. Press; 1997 Sanders, R. H. The Dark Matter Problem: A Historical Perspective. 1st ed. Cambridge. Cambridge University Press; 2010. Zwicky, F. (1933) “Die Rotverscheibung von extragalaktischen Nebeln”. Helvetica Physica Acta 6: pp. 110-­‐127. Zwicky, F. (1937a) “Nebulae as Gravitational Lenses”. Physical Review 51: 290. Zwicky, F. (1937b) “On the Masses of Nebulae and of Clusters of Nebulae”. Astrophysical Journal 86: 217-­‐246. 25. Background Material Supporting Information Symbol Meaning c Speed of light F g Force of gravity Fg expected The expected force of gravity without dark matter m 1 One of two masses in Newton’s Law m 2 One of two masses in Newton’s Law r Distance between two masses r’ Reduced distance between two orbiting masses that gives 5.6 times the gravity that would be expected from visible matter only. The reduced r or r’ is r * √of 1/5.6 VF g VF g’ A B At 1 At 2 Bt 1 Bt 2 Velocity of force of gravity Velocity of force One of two orbiting masses The other orbiting mass Location of A at t1 Location of A at t2 Location of B at t1 Location of B at t2 𝜶 Number of degrees that A or B travel in their orbit between t1 and t2 Gravitational constant Angle at which gravity arrives at 𝐵! 𝑮 𝜽 Listed above are symbols used throughout this paper and their meanings. PONE-­‐D-­‐15-­‐18507 Table 1. Meaning of Commonly Used Symbols PONE-­‐D-­‐15-­‐18507 26. Figure 11. Velocity Of Gravity Calculations. B sees its shadow and B doesn’t see its shadow. 27.