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Cosinusoidal Poten0al, Gravita0onal Lensing, and Galac0c Magne0c Fields

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Cosinusoidal Poten0al, Gravita0onal Lensing, and Galac0c Magne0c Fields Cosinusoidal Potenal, Gravitaonal Lensing, and Galacc Magnec Fields David Bartle & John P. Cumalat University of Colorado Long Beach, CA January 8, 2013 www‐hep.colorado.edu/Cosinusoidal/ Original Cosinusoidal Manuscript August, 1996 D. F. Bartle, Department of Physics, University of Colorado, Boulder, Colorado 80309‐0390 ABSTRACT A replacement for Newtonian gravity is proposed: mφ(r) = −(GmM/r) cos(2πr/λo). (This replacement is movated by the recent observaon that only a very few central point potenals have an associated uniqueness theorem.) The spacing of external shells around the ellipcal galaxy NGC 3923 gives a tentave value of 1800 light‐years for the universal constant λo. This value of λo also is accommodated by the observed distribuon in the diameters of lenses associated with bright barred disk galaxies and by the spacing of gravitaonally lensed images in the Einstein Cross (2237+0305). The potenal is consistent with the flat rotaon curves and the Tully‐Fisher law for disk galaxies. It also explains several features of our Galaxy. A previously catalogued asymmetry in the azimuthal velocity distribuon of stars near the sun is interpreted as evidence for the hypothesis and against a smoothly‐varying spherical halo of galacc dark maer. The observed broad distribuon in radial space velocies of nearby stars is only understood if the sun is near an inner turning point. This point is confirmed directly. Circular features near the Galacc center are consistent with the potenal as is a central bar. The bar is related dynamically to the spiral arms and, surprisingly, to the dwarf spheroidals. The dispersion in the radial velocity within each of the nearby spheroidal dwarf galaxies is seen as a consequence of the Galacc potenal, rather than internal dark maer. Catalogued radial velocies of the globular clusters test both the proposed and the Newtonian potenals. Original Abstract Manuscript cont. The cosinusoidal potenal is wrien in a form appropriate for general relavity. This is done by adding a term −ηµνΛ to the usual term gµνΛ which expresses the cosmological constant. Λ is 2 idenfied with λo: . Λ = −(1/2)(2π/λo) . A consequence of the relavisc formulaon is that the bending of light by gravitaonal lenses, even those with no apparent lens, can probably be explained without dark maer. Alternavely, the phase and rapid oscillaon of the cosinusoidal potenal reduces to nearly zero its contribuon to the me delay between images. The remaining, geometric, me delay is predicted to be 25 days between the two semi‐circular rings in the radio lens MG 1634+1346. This predicon is yet to be tested. The cosmological implicaons are invesgated. Surprisingly, the self‐interacon of a large −1 homogeneous sphere is explosive, a feature which assures that the age of the universe ≃ H o −1 (not (2/3)H o ). The proposed value of λo together with the present density of baryons point to an inflaonary period at a red shi z ≃ 100,000. This is a lookback me which is consistent with that expected from observed periodicies in present galacc densies. The potenal also provides a natural explanaon for the stability of the recently discovered chain galaxies. Finally, this proposal requires that gravitaonal radiaon have a dispersive group velocity, −2 −2 2 1/2 vg = c(1 + ν λo c ) > c. The consequences of this tachyonic behavior are discussed. The appendix addresses the complementary issue of whether the photon has a small mass, −2 −2 2 1/2 mγ = h/λo, and thus a velocity, vγ = c(1 − ν λo c ) . Evidence for such a massive photon is found in the magnec fields within the Galaxy, M31, and the Coma cluster. Full arcle can be found on website: www‐hep.colorado.edu/Cosinusoidal/ Evoluon since AAS 220 (Anchorage, AK) June 2012 This poster addresses two topics not discussed in the website; namely Magnec Fields in galaxies and gravitaonal lensing by galaxies. Massive Photon Hypothesis Exemplified by the MW Double Helix • Assume a universal constant k0, for both gravity and electromagnesm, whose value is roughly 2 π/400pc. • ˆ Assume vector potenal A = A0k0r /cosh(kr) φ • Assume equiparon between total energy stored in the magnec field B2/8π and the total energy stored in the vector 2 2 potenal , k 0 A /8π €A k [2/cosh(kr) - kr (sinh(kr)/cosh2 (kr)] zˆ • B = ∇ × A = 0 0 A k = 0 0 [2 - kr (tanh(kr)] zˆ cosh(kr) • Using Equiparon: B2dV = k 2 A2dV € ∫ ∫ 0 • Yields an equaon which is solved numerically and gives € k=3.416k0 which is approximately 1/19pc € Sets limit on transverse dimension of Axial B‐field • At the galacc center and with our assumpons there should be a uniform B field only over 19 pc. • Using the distance of sun to GC, 8 kpc, we find a Galacc longitude range of π 8000pc × =140pc /deg 180 19 pc /140pc /deg = 0.136 deg Referring to the figure by Morris, et al. this esmate is about twice as large as the width of the € double helix. Vector Potenal and Axial Magnec Field that were integrated for Equiparon Vector Potential Magnetic Field A B k0 r k0 r Az = r sech[kr] k=3.4k0 A magnec torsional wave near the Galacc Centre traced by a ‘double helix’ nebula Mark Morris, Keven Uchida, & Tuan Do – Nature 440, 2006, p.308 Figure 1 | The double helix nebula (DHN), observed at the infrared wavelength of 24 mm with the MIPS camera on the Spitzer Space Telescope. The spaal resoluon is 6 arcsec. At the 8 kpc distance of the Galacc Centre, 1 arcmin corresponds to 19 pc 2.5 pc. The full region observed extends well to the lower right of the region shown, and consists of a long strip centred on the bright infrared source AFGL537625, upon which we will report separately. Add Two Force‐Free Fields Note J is parallel to B, different for currents in wires. J × B = 0 Arbitrary amplitudes A0 and A1 Ax = Aosin(kz); Ay = Aocos(kz) + A1sin(kx); Az = A1cos(kx) B = A ksin(kz); B = A kcos(kz) + A ksin(kx); B = A kcos(kx) € x o y o 1 z 1 2 2 2 2 2 2 2 2 Jx = Ao(k0 +k )sin(kz); Ay = Ao(k0 +k )cos(kz) + A1(k0 +k )sin(kx); Az = A1(k0 +k )cos(kx) Use Equiparon Assumpon Again Using M31 Polarizaon Data! 800pc Slice through torus in M31 z 0 k B field out of slide k r 0 B field into of slide Amplitudes A0 and A1 not the same Still satisfies equipartition requirement z 0 k k0 r Polarizaon of B‐vectors in M31 Standard Interpretaon Fletcher, et al A&A 414, pg. 53(2004) Resolution is insufficient to observe variations on a scale of 400pc. Deflecon Angle α versus Image Displacement Angle θ Deflection angle α versus image displacement angle θ for light from a quasar Q not quite on axis of the intervening galaxy. Image occurs whenever the line intersects the solid curve. Since magnification is inversely proportional to angle between the curve and the line, only detectable images are at points of tangency, as illustrated for positive θ. Figure 16 in the original manuscript, 1996. Bending of Axial Photons • Assume deflecng galaxy is in‐line with distant extended source • Use impulse approximaon to calculate deflecng angle α Δp Force [b] α = = ( rel )(Collision time) p p 2GMeff mγ l0 α = − 2 Sin[k0b]× 2 b mγ c c bλ € Where l = r 2 − b 2 ≈ 0 is stationary when 0 max 2 -Sin [k0 b] is a maximum or b=(N+3/4)λ0. € € Bending of Axial Photons l 0 < γ α b λ r = b + 0 max 4 € € Gravitaonal Lensing Rings (from Gravitaonal Lensing) have radii of 4¾ λ0 & 5¾ λ0 Channels (from Mag. Field ) A = A sech[kr] zˆ 0 have a half‐width 1/k, with k = 1.32k0 € Galaxy ESO 325‐G004 Russell J. Smith et al (2005) Fig. 3.—Observations and models of lensing in ESO 325-G004. Panel a shows the 1101 s F475W image, with color map optimized to show the arcs. In panel b, the 18,882 s F814W image is shown after subtracting a smooth boxy-elliptical profile model. In panel c, we show the color-subtracted image, with results from a lensing model for a singular isothermal ellipse (see text). Galaxy ESO 325‐G004 Russell J. Smith et al (2005) Fig. 3.—Observations and models of lensing in ESO 325-G004. In panel b, the 18,882 s F814W image is shown after subtracting a smooth boxy-elliptical profile model. In panel c, we show the color-subtracted image,with results from a lensing model for a singular isothermal ellipse (see text). Acknowledgements We thank Russell Smith, Mark Morris, and Andrew Fletcher for permission to use figures from their published arcles. .
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