© Lloyd

1. Galactic Dynamics Using 1/r Force Without Dark Matter - http://arxiv.org/ftp/arxiv/papers/1305/1305.6847.pdf
2. Abstract: Dark matter, a conjectured substance not directly observable but which has tremendous mass, was proposed to explain why galaxies hold together and rotate faster at their edges than predicted by Newton's Inverse Square (1/r 2) Law of Gravity.
3. Here we propose an alternative, an Inverse Law (1/r), which explains galactic morphology and rotation without dark matter.
4. By varying initial conditions, the Inverse Law can systematically and easily generate realistic galactic formations including spirals, cartwheels (extremely difficult under Newtonian gravity), bars, rings, and spokes.
5. This model can also produce filaments and void structures reminiscent of the large-scale structure of the universe.
6. Newtonian gravity cannot do all this without dark matter.
7. Occam's Razor suggests that at galactic scales, gravity should be 1/r and dark matter is unnecessary.
8. This simple model with its self-organizing emergent properties, combined with dynamical systems theory, has broader implications.
9. It may help us understand more complex systems. 3
10. Main Text: Introduction - It is well known that Newton's Law of Gravity, when applied to galactic dynamics, leads to a large amount of missing matter that is not directly observable.
11. Early in the 20th century, Oort (1) and Zwicky (2, 3) first discovered this phenomenon which Zwicky named, dark matter. []
12. They observed that stars at the edge of galaxies move faster than predicted by Newton's 1/r 2 Inverse Square Law of Gravity.
13. This discrepancy is called the "Rotation Curve Problem" and currently is solved by adding dark matter around a galaxy.
14. In this work, we propose an unorthodox but simple change of Newton's 1/r 2 to 1/r for the weak-field approximation of gravity at the galactic scale.
15. This approach solves several significant problems in galactic dynamics using the naïve (in the technical sense) direct simulation of the N-Body Problem in 2 dimensions.
16. Surprisingly, it produced a series of realistic galaxy figures with flat rotation curves (which give correct rotation velocities at the edge of the galaxies) without dark matter.
17. Both ring galaxies (Figure 1) and spiral galaxies (Figure 2) can be generated form this same model by simple changes to the initial conditions.
18. Newtonian gravity cannot do this without dark matter [].
19. In particular, Toomre (6) showed the Cartwheel Galaxy can only be generated by the collision of two galaxies in Newtonian models
20. Compare our simulated Cartwheel Galaxy in Frame 100, Fig. 1, to the real Cartwheel Galaxy ESO 350 - 40 in the lower right corner of Fig. 1; both have an inner and outer ring connected by slightly curved spokes.
21. Also compare our simulated spiral galaxy in Frame 30, Fig. 2, with the Barred Spiral Galaxy NGC1300 in the lower right corner of Fig. 2; both have two spiral arms, a bar, and a central core.
22. Frame 110 of Fig. 2 shows the spiral galaxy evolved a typical ring surrounding the bar as well.
23. Use of the 1/r force for gravitation is not new.
24. Fabris and Campos (7) and references therein pointed out numerous modifications to the theory of gravity recurrently in the literature.
25. They cited no less than six different physical theories which added a 1/r term to gravity.
26. These theories range from general relativity, string theory, quantum gravity, TeVeS theory, their own theory, and MOND theory (Modified Newtonian Dynamics []).
27. What is novel about our approach is the choice to retain only the 1/r term as the weak-field approximation of gravity to study dynamics at the galactic scale.
28. This allowed us to simulate galaxies without dark matter or additional assumptions and constraints like relaxation techniques or needing the system to be in special equilibrium states.
29. This greatly simplified the modeling, solves the Rotation Curve Problem, but, apparently, still captured some of the key global features of galactic dynamics.
30. In the tradition of celestial mechanics, we call this model the "Restricted N-Body Problem" (RNBP).
31. RNBP galaxies undergo significant evolution in morphology with nearly periodic coherent structures.
32. The periodic evolution of the morphology and the simplicity of the model suggest the use of dynamical systems theory to analyze the evolution of these coherent structures.
33. In addition to giving insights into galactic dynamics, the RNBP may also provide a useful model to study more complex evolutionary processes with emerging structures such as in molecular dynamics or biological systems. 4
34. Figure 1. Case A (Table S1.) Simulated evolution of the Cartwheel Galaxy at Frames 0, 60, 100, compare d with the Multispectral false color image of Cartwheel Galaxy ESO 350 - 40 (Credit: NASA/JPL - Caltech) in the lower right corner.
35. Particles are color-coded by distance fro m galaxy center in Frame 0 by the color bar.
36. Little mixing occurs as shown by the colors.
37. Rings form by density waves travel[]ing through the spokes which look like invariant manifolds of resonant periodic orbits in dynamical systems theory.
38. See the additio nal figures in the Supporting Information attached, and the animation http://martinlo.com/Home/CartwheelMovie.html on-line. 5
39. Figure 2. Case B (Table S1). Simulated evolution of the barred spiral galaxy with ring at Frames 0, 30, 110.
40. Compare Frame 30 with the Barred Spiral Galaxy NGC1300 [].
41. Particles are color coded by distance from galaxy center in Frame 0 by the color bar.
42. Little mixing occurs as shown by the colors.
43. A bar, ring, and two spiral arms develop with the bar spinning around the ring.
44. See the additional figures in the Supporting Information attached, and the animation http://martinlo.com/Home/SpiralGalaxyMovie.html on-line.
45. N-body Model of a Galaxy with 1/r Force The equations of motion for the Restricted N-Body Problem (RNBP) are given by Eqs. (S1) and (S2) in the Supporting Information Appendix.
46. Objects in the Supporting Information are number as S1, S2, etc.
47. Our particles are assumed to be abstract point masses each of which should be thought of as a large cluster of stars and not as single stars.
48. The 1/r force law is a central force with potential equal to log(r). 6
49. We restrict our attention to the 2-dimensional case to keep the model simple for analysis.
50. Ring galaxies and spiral galaxies are typically very flat, so a 2D planar model is reasonable.
51. Extension to 3 dimensions is straightforward by adding more components to the position, velocity, and force vectors.
52. To gain insight for interpreting the simulations, we examine the Two Body Problem first.
53. This is a central force, so energy, momentum, and angular momentum are conserved.
54. Hence, a Two Body orbit moves in a fixed orbital plane normal to the angular momentum vector [].
55. Bertrand's Theorem [] implies that the 1/r force has no periodic orbits besides circular orbits.
56. Interestingly, all circular orbits in this potential have constant velocity, v c = Γ M, (1) where Γ is a new gravitational constant different from Newton's gravitational constant G and M is the mass of the central body.
57. This is seen by equating centripetal acceleration with gravitational acceleration v c 2 / r = Γ M / r and simplify.
58. Another interesting property of the Two Body orbits of the 1/r force is that they are all bounded in an annulus.
59. They cannot collapse to 0 or escape to infinity!
60. This is seen from the Two Body Energy, E = v 2 / 2 + Γ M log(r), which with algebra yields r < exp(E/ΓM) < ∞.
61. The 1/r force is much stronger than the 1/r 2 force.
62. Simulations Of Galaxies - With 1/r Force We use the symplectic Euler integrator with difference equations Eq. (S5) (Lambers and Lof (13), Lambers (14)).
63. The units are non-dimensional and normalized.
64. For simplicity, all particles have mass = 1.
65. The initial conditions for the position and velocity of the i th particle are given in Eq. (2) m r r i 0: x i 0 = ρi cos (θ i), y i 0 = ε ρi si n(θ i), r v i 0: x i 0 = − V ρi sin(θ i), y i 0 = V ρi sin(θ i).
66. (2) The ρi's are uniformly random numbers from 0 to 0.25; the θ i's are uniformly random angles from 0 to 2π.
67. V is the velocity scale factor around ~50.
68. The shape factor ε is set to 1 for a circular distribution (for ring galaxies) and ε < 1 for an elliptical distribution (for spiral galaxies).
69. Since the uniform distributions are in polar coordinates, the effect in Cartesian coordinates is a distribution with particle density increasing towards the center of the galaxy at (0, 0).
70. This simulates the greater mass at the center of galaxies.
71. This turns out to be important for the evolution of the galaxies (see Case C below).
72. Each particle is given a velocity transverse to its position vector from the center of the galaxy, (0, 0), to create a spin and avoid total collapse to the center.
73. The particles are color coded by their initial distance from the gala ctic center so they & & 7 can be tracked by their colors as the galaxy evolves.
74. The gravitational constant Γ = 0.05 and the time step Δ t = 10 − 4.
75. Recall that Γ is a new gravitational constant different from Newton's G.
76. The shape of the initial conditions determines the morphology of the galaxy and its development.
77. The shape factor ε set to 1 gives a circular distribution for Case A which resulted in cartwheel galaxies.
78. Set ting ε < 1 gives an elliptical distribution for Case B which resulted in spiral galaxies.
79. Hence, in the RNBP model, the morphology of galaxies are easily controlled by the density and shape of the initial distribution of particles.
80. The number of particles for the simulations ranged from 2500 ≤ N ≤ 4000.
81. The simulations were programmed in Matlab and F 77.
82. A simulation with 4000 particles and 2000 time steps requires ~10 minutes on a MacBookPro laptop.
83. Due to the symplectic nature of the integrator, the energy variations (Eq. S3) of the simulations are always under 1%.
84. We examined four cases: (A) The Cartwheel Galaxy, (B) A Barred Spiral Galaxy, (C) A Uniformly Distributed Galaxy without a massive central core, and (D) Collision of Two Galaxies (A and B).
85. Table S1 lists the parameters for each case.
86. For the discussion of these cases to follow, please view the animations, additional figures and text in the Supporting Information for more details.
87. Case A. The Cartwheel Galaxy: The Cartwheel Galaxy (ESO 350 - 40) is a one of the rare ring galaxies that has a ring within a ring connected by slightly curved spokes.
88. Zwicky (15) discovered it and called it "one of the most complicated structures awaiting its explanation on the basis of stellar dynamics".
89. Under Newtonian gravity, Toomre (6) showed the only known method to simulate the Cartwheel Galaxy is by colliding two galaxies.
90. However, with the 1/r force, the Cartwheel Galaxy evolves naturally from a circular distribution of randomly placed particles described by Eq. S5 and Table S1.
91. Fig. 1 provides 3 Frames from the Movie S1 to show the evolution of the Cartwheel Galaxy simulation.
92. Frame 0 shows the initial circular distribution with a radius of 0.25 and a higher concentration of particles at the origin.
93. The units are selected to keep the simulation within the square [-1, 1] × [-1, 1].
94. The particles have a counterclockwise rotation.
95. Initially, the entire galaxy contracts towards the center.
96. In Frame 60, one can see the radius of the galaxy shrank from 0.25 to 0.2.
97. The dark yellow and red center is much denser in Frame 60 where the greenish ring has now formed which grew out of the center from a density wave.
98. In Frame 100 the ring has expanded to the edge of the galaxy and the color has now changed to blue.
99. This shows that as the density wave pulses outwards, particles are not being carried along very far.
100. In Frame 100, one can also see a new yellow ring formed closer to the center.
101. In fact, a series of three rings can be see n at one time in the animation.
102. Between the rings, spokes have formed, giving the Cartwheel Galaxy its distinct wheel shape.
103. These spokes are highly reminiscent of invariant manifolds of unstable resonant orbits in the Restricted Three Body Problem.
104. In fact, Athanassoula, Romero-Gomez, & Masdemont (16, 17) have shown, for a static galactic potential, that rings, bars, and spiral arms are indeed formed from invariant manifolds of periodic orbits.
105. For our dynamic potential, these manifolds appear to be Lagrangian coherent structures. 8
106. Case B. The Barred Spiral Galaxy: Spiral galaxies are the quintessential galaxies we typically think of.
107. For Case B, ε = 0.25, Frame 0 of Fig. 2 shows the initial conditions of the simulation.
108. The galaxy first contracts from a radius of 0.25 to 0.2 as shown in Frame 30 of Fig. 2 where it has developed two spiral arms and a central bar.
109. Traces of spoke-like structures can be seen connecting the arms to the central bar.
110. From the animation, a density wave pulses from the center trying to form a ring.
111. But due to the elliptical distribution, only two fragments of the ring are formed which became the arms.
112. In Frame 110, an elliptical ring has formed to contain the bar.
113. The bar actually spins inside this eye-shaped structure.
114. The arms break up par tially in Frame 110, but reform again in later Frames.
115. In both Cases A and B, the galaxy is constantly evolving and going through a series of nearly periodic structures.
116. Recent work showed strong evidence that disk galaxies have undergone strong dynamical evolutions throughout the last 8 – 11 Gyr [].
117. The RNBP shows both ring and spiral galaxies undergo substantial evolutions which are nearly periodic.
118. All of the intermediate shapes and structures produced by RNBP appear to be consistent with observed galaxies.
119. Case C. Uniform Galaxy: For Case C, the initial particles are uniformly distributed in Cartesian coordinates as shown in Figure 3, Frame 1, and Table S1.
120. This example shows the significance of a massive central core to the morphology and evolution of the galaxy which we now describe.
121. An interesting observation in Frame 100 (Fig. 3), the particles clumped to form voids and filaments reminiscent of the large scale structure of the universe.
122. Thereafter, this galaxy was never able to form a consistent massive core.
123. At Frame 343, it produced a loosely organized galaxy with shadows of multiple spiral arms for a short duration.
124. By Frame 2401, it evolved into an elliptical galaxy.
125. More than 10,000 Frames later, it remained an elliptical galaxy, the shape of the oldest known galaxies.
126. Case C suggests that a massive central core is critical for the formation of ring and spiral galaxies.
127. Case D. Colliding Galaxies: In Case D, we collided the Cartwheel Galaxy (4000 particles) with the Barred Spiral Galaxy (1500 particles).
128. The results are shown in Figure 4 below.
129. Frame 2 shows the two galaxies approaching one another.
130. Frame 101 shows the two galaxies after multiple collisions but still separate.
131. Frame 525 shows the two galaxies merged into an elliptical galaxy.
132. The same software used to simulate the Cartwheel Galaxy and the Barred Spiral Galaxy was used to simulate the colliding galaxies.
133. This shows how easy this kind of numerical experiments can be performed with the 1/r force because of its numerical stability. 9
134. Figure 3. Case C (Table S1). Evolution of Uniform Distribution Initial Condition.
135. Frame 1 shows the uniform random distribution initial condition.
136. Here the rotation is slowed down: V =25.
137. Frame 100 shows filament and void clumping reminiscent of the large scale structure of the universe.
138. Frame 343 shows multiple spiral arms and a central core forming.
139. Frame 2401 shows a final steady state elliptical galaxy.
140. See http://martinlo.com/Home/UniformGalaxyMovie.html on-line. 10
141. Figure 4. Case D (Table S1). Two Colliding Galaxies. Frame 2 shows the two galaxies approaching.
142. Frame 101 is after several collisions.
143. Frame 525 shows the two galaxies have combined and formed a steady state elliptical galaxy after multiple collision encounters.
144. See the animation http://martinlo.com/Home/GalaxyCollisionMovie.html on-line.
145. GALAXY WITH 1/r GRAVITY FORCE HAS CONSTANT ROTATAION CURVE - A key problem in galactic dynamics is the Rotation Curves of galaxies which describes the velocity of circular orbits around the galaxy center as a function of orbital radius.
146. Keplerian circular orbits with radius r have velocity v k = G M / r, M is the mass of the central body.
147. Thus, away from the center of the galaxy, circular velocity should become smaller.
148. However, Rubin & Ford (4) observed the opposite: away from the center of the galaxy, they found the circular velocity to be nearly constant which produced a flat rotation curve.
149. Recall the circular velocity for the Two Body Problem for the 1/r force is a constant, v c = Γ M.
150. Surprisingly, for any N particles in the RNBP, the average circular velocity is still a constant, v c = Γ M, where now M = total mass of the galaxy.
151. The rotation curve for N bodies at a distance r from the center of the N-b[odi]es is given by 11 v c (r) = r d 〈 Φ 〉 (r) dr, 〈 Φ 〉 (r) = 1 2 π Φ (r r) r d θ r r = r Ñ ∫, Φ (r r) = − Γ m i log r r − r r i i = 1 N ∑ = Φ i (i = 1 N ∑ r r), (7) where 〈 Φ 〉 (r) is the azimuthal average of the log(r) gravitational potential of the galaxy at distance r from the center of the galaxy []).
152. We showed in the Supporting Information Appendix that the rotation curve can be reduced to two definite integrals which by symmetry and cancellations reduces to the constant rotation curve, v c (r) = Γ M.
153. The significance of our calculations is that they are valid for ANY DISTRIBUTION of N particles in ANY CONFIGURATION during any time step of the simulation under the 1/r force.
154. Thus the rotation curve for the 1/r force and the RNBP model is always a constant Γ M and dark matter is not needed.
155. This solves the Rotation Curve Problem.
156. Of course, the actual rotation c[ur]ves of galaxies are not flat towards the center of the galaxy.
157. Since near the center of the galaxy, the assumption of the far-field approximation of gravitation fails, so one should not expect the 1/r force to be working in this regime.
158. The transition from 1/r to 1/r 2 force is an unsolved problem in fundamental physics beyond the scope of this paper.
159. As a corollary, since we are able to measure the rotation curves of real galaxies, this suggests a potentially new approach to estimate the total mass M of a galaxy.
160. Of course, since real rotation curves are not constant near the center of galaxies where most of the mass of the galaxy is concentrated, more careful thought is needed for estimating the total mass M of galaxies this way.
161. THE log(r) POTENTIAL SATISFIES GAUSS' LAW IN 2D BUT NOT IN 3D The log(r) potential is harmonic in 2D but not in 3D.
162. This means that it satisfies Gauss' Law in 2D, but not in 3D.
163. This has some significant consequences.
164. Gauss' Law states that the Newtonian gravitational flux, F Newton (S), through any closed surface, S, is proportional to the total mass, M, enclosed by S.
165. Assume S is a sphere of radius r enclosing the point mass M at the center of S.
166. Eq. (8) gives the flux for the general force laws, 1/r n, n=1, 2, 3,...
167. F (S) = Γ M r n S ∫ r 2 d Ω = 4 π r 2 − n Γ M (8) where Ω is the solid angle of the unit sphere and r 2 d Ω = d S is the infinitesimal area element of the sphere S of radius r.
168. For Newtonian gravity, n=2 and F Newton (S) = 4 π G M is a constant.
169. But, for the log(r) potential, n=1 and F log(r) (S) = 4 π r G M depends on r.
170. In fact F log(r) (S) depends on the size and shape of the surface S, and F log(r) (S) ≠ 0 even when S does not contain the mass M ! This suggests potentially a way to estimate the distribution of matter within a galaxy by measuring the flux F log(r) (S) and using inverse methods.
171. Thus, here on Earth at the far edge of the Milkyway, we 12 are able measure a non-zero F log(r) (S) which may yield some information about the mass distribution within the Milkyway.
172. Or by measuring F log(r) (S) in a small volume in the direction of another galaxy like M31, we may be able to extract information about its mass and distribution.
173. CONCLUSIONS AND FUTURE WORK - In this paper we proposed the Restricted N-Body Problem (RNBP) with 1/r force for the far field approximation of gravity at the galactic scale.
174. This enabled a naive direct N-Body simulation of galaxies in 2D with a flat rotation curve without dark matter.
175. Simulations of 2D RNBP galaxies produced realistic looking galaxies that do not require assumptions of equilibrium states or relaxation techniques.
176. The RNBP model is able to produce both the Cartwheel Galaxy and Barred Spiral Galaxies by simple variations in the initial conditions.
177. We also showed the RNBP does not satisfy Gauss' Law in 3D.
178. This suggests potentially new methods to measure the total mass and distribution of matter in galaxies.
179. The RNBP has many directions for future work.
180. The most important is to verify and understand the fundamental physics of 1/r gravity and how gravity scales from 1/r 2 at the scale of the solar system to 1/r at the galactic scale and beyond.
181. The simplicity of RNBP allows for a dynamical systems analysis of the evolution and structures of galaxies.
182. For example, the density waves creating the rings, spokes, spiral arms and bars appear, from the simulations, to be coherent structures controlled by invariant manifolds and symmetry properties of the initial conditions.
183. This approach may provide a systematic theory for the formation, evolution, and classification of galactic structures.
184. The RNBP from N= 2, 3, 4 to N >> 1 provide a new family of classical mechanics problems which have somehow escaped the attention of researchers (see Saari (24, 25)).
185. Every theorem in Newtonian celestial mechanics could be reinterpreted under the 1/r force.
186. Finally, the RNBP is perhaps the simplest of N-Body models with evolutionary and emerging properties.
187. Moreover, it is numerically robust and highly stable.
188. Simple mathematical models like the Three Body Problem or Henon's Map have played major roles in the development of science.
189. Given the success of dynamical systems theory in the analysis of galaxies with static potentials by Athanassoula, Romero-Gomez, & Masdemont (17), similar approaches to study the RNBP will help us unravel the complex dynamics and controls behind the morphogenesis of the RNBP.
190. Seeing the galactic structures emerge and evolve in the RNBP animations, one can imagine chem[ic]al or biological systems undergoing similar dynamics and evolutions.
191. The insights gained from studies of the much simpler RNBP will provide a foundation and stepping stones for approaching other N-Body systems with more subtle forces and more complex dynamics.
192. In addition to contributions to the scientific problems, the RNBP may also contribute to computational algorithms, simulations, and visualizations involving N-Bodies because of its simplicity, robustness and stability. 13

'17-03-08, 22:48   GenePreston

Re: Galactic Dynamics Using 1/r Force Without Dark Matter   If anyone reading this is modeling galaxy and star movements, could you test equation 5 with k=0.04 in http://egpreston.com/GravityMod.pdf ... thanks.