Breaking Kepler's Law

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1 Breaking Kepler's Law 0% 15% 16.7% 25% 50% 75% The Kepler Curve Modified for a Finite Speed of Gravity 100% Charles D. Cole, III PO Box 160 Dinosaur, CO USA Charles.Cole@skysthelimit.com orcid.org/

2 orbit velocity Kepler s third law states a relationship between a mass s orbital velocity and its orbiting radius. v = $ & radius Figure 1. Kepler Curve The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 1

Planets line up quite nicely on a Kepler curve. Figure 2.

Galaxies, in contrast, have little central mass and the orbiting stars are of similar masses to each other. Figure 5.

5 Kepler assumes an immovable central mass orbited by a negligible mass. Figure 4. Immovable Central Mass Like the Kepler curve assumptions, solar systems exhibit a large mass (star) circumnavigated by much smaller orbiting masses (planets, moons, satellites). Galaxies, in contrast, have little central mass and the orbiting stars are of similar masses to each other. Figure 5. No Central Mass The following math modifies the Kepler curve to assume orbiting masses of similar size as is observed in galaxies and galactic clusters. A different curve arises if one derives a Kepler type law wherein one sets the centrifugal force equal to the radial pull of gravity when taking into consideration a finite speed of gravity. The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 4

6 The orbit geometry diagram assumes two similar masses A and B orbiting each other. Forces are calculated assuming the gravitational force on B at t + comes from A at t $, an earlier position of A on the orbit path. The gravity then arrives at an angle θ. At any finite speed of gravity, this reduces the distance gravity travels, both increasing its magnitude and introducing a tangent vector. Figure 6. Orbit geometry Like Newton s law, Kepler s law assumes that the force of gravity travels infinitely fast. Following are the steps taken to modify Kepler s law for a finite Vfg. Equations are derived from the orbit geometry diagram and plotted using the Desmos graphing program. The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 5

7 Define r, the distance that the force of gravity must travel. At any finite speed, the r between orbiting masses must decrease. In our Desmos program, r is called r 0. r { } Figure 7. r to r θ The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 6

8 Therefore, the gross gravity increases based on r in Newton s law: Gross radial pull from gravity: g $ = cos (θ) g 0 Gross tangential pull (dark energy) from gravity: gross gravity g + = sin (θ) g 0 gross radial pull x = 90 x = 90 gross tangential pull Figure 8. Gross gravity distribution θ The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 7

9 Corrections distributing the gross tangential dark energy vector among the gross radial and tangential vectors are shown on the orbit geometry diagram. Since the masses are in an orbit, not all of the gross tangential force ends up tangent to the orbit. Additional centrifugal force is added by a fraction of the gross tangential force. These correction vectors adjust the final radial and tangent vectors by breaking up the gross tangent vector into these radial and tangent correction vector sub-components. These are the Newtonian equations for relativistic corrections. If one misses these, the dark energy will be grossly overstated as Laplaces s calculations were. gross radial correction gross tangential correction gross radial correction: θ c 0 = g + cos (90 2θ) gross tangential correction: c $ = g + sin (90 2θ) Figure 9. Tangential acceleration vector distribution The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 8

10 Corrected radial pull: i $ = g $ c 0 Corrected tangential pull: d $ = g + c $ After corrections, they need to be added together again to find the net gravity s magnitude and direction. Combining these gives a net gravity g3 of: net gravity force θ corrected tangential pull corrected radial pull Figure 10. Net gravity calculations The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 9

11 Of which the radial vector f, the net inward pull is Kepler s law was derived by setting the formula for centrifugal force equal to Newton s law of gravity to hold the planets in balance, Instead of Newton s law, which also assumes an infinitely fast Vfg, set the centrifugal force (v^2/r) equal to the net inward pull f or, v + r = f (θ) Because the r in the centrifugal force equation is ½ the r in Newton s, this simplifies to Velocity v(θ) in units of [Vfg = 1] Net inward pull f(θ) in units of gravity where Vorb = 0 modified Kepler curve vs θ θ net inward pull Figure 11. The modified Kepler curve measured against θ The black dotted line is the Kepler curve, modified for a finite speed of gravity (measured against θ starting at -90 degrees). The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 10

12 To convert the θ angle to distance, set the appropriate multiple of the arc of travel for At $ to At + equal to the cord of gravity s travel from At $ to Bt +. Distance is the multiple n of Vfg) z + z + = θ π {0 < θ < 90} z = ($Y0 234(5)) Z [ {0< θ < 90} , Enter orbit velocity as a multiple n of Vfg s, or distance as a multiple n of V \] n = 1 z n is orbit speed in Vfg s, or distance in [r ] s. Here we have entered an orbit speed of 1 Vfg. The intersection of the curves gives the number of degrees in θ, in this example, θ Figure 12. θ to distance or θ to velocity conversion For convenience, a conversion chart is shown next, relating the θ angle to a linear distance measured in units of r when Vorb = Vfg The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 11

$θ θ to distance conversion chart Distance in units of r at V w&x = V \] or velocity in units of V \] Figure 13.$

13 θ θ to distance conversion chart Distance in units of r at V w&x = V \] or velocity in units of V \] Figure 13. θ to distance or velocity conversion chart The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 12

14 The classic Kepler curve is shown here as connected purple dots and the modified Kepler curve in connected black dots. This modified Kepler curve exhibits the shape of typical galactic rotation curves. Compare with typical rotation curve observations shown in Figure 13. The modified Kepler curve eventually plateaus at a velocity of 70.7% of Vfg, with the velocity gap above the Kepler curve proper increasing the farther out one goes. Although there is a regression proof indicating that there must exist a speed at around Vfg = 350 kps where all the dark effects are predicted, including the 5.6 extra gravity found in galaxies, there is a fun quick method to estimate the (same) speed of gravity in your head so that the y axis can be labeled for velocities in kps. classic Kepler curve Kepler curve modified for a finite speed of gravity. Figure 14. Classic Kepler curve The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 13

15 Figure 15. Milky Way Galactic Rotation Curves, Y. Sofu, Figure 16. Solid line is smoothed Milky Way Galactic Rotation Curve, Y. Sofu, The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 14

16 To label the y axis with actual speeds in kps, Vfg must be estimated. 1. The largest known orbiting structures are galaxy clusters. The largest of these have a radius of 5 mpc. Unit Terminology kps kilometers per second (velocity) kpc kiloparsecs (distance) mpc megaparsecs (distance) 2. A Hubble constant of 70kps/mpc indicates the separation speed at a distance of 5mpc is 5mpc }0~ = 350 kps orbital velocity in kps or km sec $ traditional Kepler Vfg = modified Kepler Vfg = 350 kps Somewhere around this speed, the galaxies are separating at a rate faster than the gravity connecting them. distance in kiloparsecs or kpc Figure 18. Kepler curve when Vfg = 350 KPS Much like the stars blink out never to be seen again once they are outside the Hubble sphere, gravity blinks out at distances when/where the Hubble constant separation speed exceeds the speed of gravity. This appears to occur at 5mpc as evidenced in galactic clusters. The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 15

17 This chart s predicted rotational velocities plateau at 250kps with a dip to about 185kps at a distance of about 2.7kpc, similar to what we see in many galactic rotational charts. By assigning a value of 350kps to Vfg, and using 8 kpc as the average estimate for the distance of the sun to the galactic center, the sun s location can be identified on the chart. 220kps is used for an estimate of the speed of the sun on the Milky Way rotation curve. Note that the sun (orange spot) lies somewhat below the modified Kepler curve. The highest connected black dots represent the Kepler curve modified for a finite speed of gravity, in orbits of equal masses. The purple solid line shows the unmodified Kepler curve. The sun is overlaid on this graph at a distance of 8kpc and 220kps x = 8kpc y = 220kps orbital velocity in kps or km sec 1 Sun distance in kiloparsecs or kpc Figure 19. Sun s location on modified Kepler s curve when Vfg = 350 KPS The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 16

18 The evolution from the Kepler curve proper to the Kepler curve modified for a finite speed of gravity, shows the dependence of the curve shape on the percentage of central mass. The highest line represents a pure twobody orbit of equal masses with no central mass. Modified Kepler Curve % of Central Mass 0% As the percentage of central mass of the orbiting system increases, the modified curve drops to the regular Kepler curve which represents that 100% of the mass of an orbiting system is a central immovable mass. Milky Way Predicted Galaxy Average (ESO) 15% 16.7% 25% The ESO estimates the average percentage of central mass in galaxies to be 16.7 %, represented by the third line down in green. 50% The sun s location lies on a curve which implies that the central mass of the Milky Way is 15%. This is the second curve down in orange. Alternatively, a velocity of kps instead of the given 220 kps would put the sun on ESO s galactic average curve. Kepler Curve 75% 100% Thus, it is evident that Kepler s law better predicts the velocities of masses in orbit when it is modified for a finite speed of gravity. Figure 20. Dependence of the Kepler Curve on the percentage of central mass to total orbiting system mass The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 17

Modified Kepler for a two body orbit and no central mass Milky Way appears to lie on the 15% CM curve Galactic Average = 1/6 Kepler Curve % of Central Mass 0% 15% 25% 50% 75%!

19 Modified Kepler for a two body orbit and no central mass Milky Way appears to lie on the 15% CM curve Galactic Average = 1/6 Kepler Curve % of Central Mass 0% 15% 25% 50% 75%! 100% Sun distance to galaxy s axis.!"#$! The Shape of Kepler s curve is dependent on the percentage of central mass to total system mass. Figure 21. Kepler curve dependence on central mass percentage The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 18

20 Conclusion and Ramifications Dark Matter The sole reason that Dark Matter was theorized was to explain the systemic differences between the observed galaxy (or galaxy cluster) rotation curves and the rotation curve predicted by Kepler s law, which seems only to accurately forecast rotation speeds of planets and moons. By recognizing that there is a fundamental difference in the distribution of mass in galactic orbits and solar system orbits, a more general curve is derived. This curve, or really a field of curves, shows the dependence of the rotation curve on the [central mass to total system mass] percentage. When a Vfg of 350 kps is used in an analysis of a 2-body orbit of equal masses (no central mass), curves that match the observed galactic rotation curves result. The sun appears to be on the line representing a Milky Way central mass of 15%. This is very close to the ESO s claim that the average central mass of galaxies is 1/6 or 16.7%. Given the implied lack of precision in a fraction, the Milky Way could well be on this curve. Newton s Law Newton derived his law of gravity from Kepler s third law. This brought into Newton s law the Kepler assumption that Vfg = infinity which worked perfectly for measurements of the era. Using a Vfg considerably less than c, a different radial gravity formula is developed for orbiting masses which, when set equal to centrifugal force, predicts all rotationally supported orbiting system rotation curves, from moons to planets to stars to galaxy clusters, when the percentage of central mass is known. In the interest of simplicity, there is an assumption in Kepler s law that one of the masses of the orbiting system is unaffected by the other. This paper presents math correcting this, although the elegant simplicity of V = $ is sacrificed. & The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 19

21 General Relativity In General Relativity, the speed of gravity is usually considered c. In any mathematically driven world, this fast yet finite speed of gravity would generate a small tangential acceleration force. Einstein made this leftover vector disappear by saying that, in this one case only, the speed of gravity was infinitely fast and placed it outside the general relativity paradigm that created it, putting it instead in a mechanistic physics category which then makes it disappear by changing the speed of the gravity which created it, from c back to infinity. This unwanted leftover tangential acceleration vector will derive from any finite Vfg and predicts unstable orbits (although friction matching this vector makes for stable orbits). This vector would then create a net expansion in orbiting systems and explain the outward forcing dark energy, at least in direction. To match the magnitude of dark matter and dark energy, more aberration is required than when one uses c as Vfg. Using the same math with any finite speed of gravity creates identical curves where only the y-axis velocity labels are different. By using Vfg = 350 kps, a velocity hundreds of times slower that c, the ratio of dark matter to dark energy matches observed. This does in no way change GR s conclusions concerning time, height and width, because unlike gravity, they are measured by light which does indeed travel at c. In fact, by using Vfg = 350 kps, General Relativity s gravity predictions are improved since the masses of galaxies are increased to the point where no extra mass is required, completely obviating the need for dark matter. The same logic would imply that magnetism or any other force would also dilate at high relative speeds. The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 20

22 Dark Energy From the same geometry that allowed the calculation of the radial gravity at any relative velocity, the tangential acceleration vector or dark energy can be plotted. dotted orange line is the dark energy curve against θ in units of V w&x = 0] Figure 22. Close-up of the dark energy rotation curve at various θ s showing slowing acceleration before accelerating acceleration. This graph shows dark energy accelerated acceleration as a function of θ. e $ ] The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 21

23 dotted red is the dark energy curve against distance in units of = 0] Figure 23. Close-up of dark energy rotation curve at various distances, showing the different phases of tangential acceleration from a slowing acceleration to an accelerated acceleration. The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 22

24 dotted red is the dark energy curve against distance in units of = 0] Figure 24. Net tangential force or dark energy overlaid on a field of Kepler "type" curves. The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 23

25 Bibliography Yu. Baryshev. Expanding Space: The Root of Conceptual Problems of the Cosmological Physics. Astronomical Institute of the St.-Petersburg State University, St.-Petersburg, Russia. 1 Oct Wikipedia. Nicolas Fatio de Duillier. (accessed 2015). Richard Feynman. The Relation of Mathematics and Physics - Part 1- The Law of Gravitation (full Version Richard Feynman. YouTube, (accessed 2015). Hafele, Joseph C. Causal Version of Newtonian Theory by Time Retardation of the Gravitational Field Explains the Flyby Anomalies. April Web. Wikipedia. Pierre-Simon Laplace. (accessed 2015). Wikipedia. Georges-Louis LeSage. (accessed 2015). McGaugh, Stacy S., Federico Lelli, and James M. Schombert. The Radial Acceleration Relation in Rotationally Supported Galaxies. Department of Astronomy, Case Western Reserve University, 21 Sept arxiv: v1, Web. Newton, Isaac. The Preliminary Manuscripts for Isaac Newton's 1687 Principia. 1st ed. England. Cambridge UP; Wikipedia. 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; Van Flandern, Thomas. The Speed of Gravity What the Experiments Say. Physics Letters A 250:1-11 (1998) Zwicky, F. (1933) Die Rotverscheibung von extragalaktischen Nebeln. Helvetica Physica Acta 6: pp 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: The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 24

26 Table 1. Meaning of Commonly Used Symbols Symbol A Meaning One of two orbiting masses At 1 Location of A at t 1 At 2 Location of A at t 2 AU Astronomical units B The other orbiting mass Bt 1 Location of B at t 1 Bt 2 Location of B at t 2 c Speed of light c 0 c 1 Central mass d 2 g 0 i 1 kpc EOS e 1 F g G g 0 g 1 g 2 g 3 Gross radial pull Gross tangential pull Non-orbiting part of galaxy Connected tangential pull Gross gravity Corrected radial pull Kiloparsecs European Southern Observatory Dark energy as a function of θ Force of gravity Gravitational constant Gross gravity Gross radial pull Gross tangential pull Net gravity Km/s Km/s = KPS = KM sec 7$ mpc Megaparsecs m 1 m 2 r r r o r v V = 1 r V fg V orb One of two masses in Newton s Law One of two masses in Newton s Law Distance between two masses in Newton s Law of Gravity Radius Distance gravity travels when taking into consideration a finite speed for gravity bases a two-body analysis. Velocity Classic Kepler curve Velocity of force of gravity Velocity of orbit The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 25

27 V sun Velocity of sun around the Milky Way θ Angle at which gravity from At $ arrives at Bt + z Process converting θ to distance or velocity z 2 f (θ) At + is in the 12:00 position Net inward pull as a function of θ Listed above are symbols used throughout this paper and their meanings. The Kepler curve modified for a finite speed of gravity. Copyright 2017 [Charles D. Cole]. All Rights Reserved. 26

Fatal Flaw of Relativity

Fatal Flaw of Relativity Charles D. Cole, III Fatal Flaw of Relativity, Copyright 2016 [Charles D. Cole, III]. All Rights Reserved. 2/5/17 1:56 PM 1. I know this is a talk on gravity, which always traces