Cosinusoidal Poten al, Gravita onal Lensing, and Galac c Magne c 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 mo vated by the recent observa on that only a very few central point poten als have an associated uniqueness theorem.) The spacing of external shells around the ellip cal galaxy NGC 3923 gives a tenta ve value of 1800 light‐years for the universal constant λo. This value of λo also is accommodated by the observed distribu on in the diameters of lenses associated with bright barred disk galaxies and by the spacing of gravita onally lensed images in the Einstein Cross (2237+0305). The poten al is consistent with the flat rota on 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 distribu on of stars near the sun is interpreted as evidence for the hypothesis and against a smoothly‐varying spherical halo of galac c dark ma er. The observed broad distribu on in radial space veloci es of nearby stars is only understood if the sun is near an inner turning point. This point is confirmed directly. Circular features near the Galac c center are consistent with the poten al 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 Galac c poten al, rather than internal dark ma er. Catalogued radial veloci es of the globular clusters test both the proposed and the Newtonian poten als. Original Abstract Manuscript cont.
The cosinusoidal poten al is wri en in a form appropriate for general rela vity. This is done by adding a term −ηµνΛ to the usual term gµνΛ which expresses the cosmological constant. Λ is 2 iden fied with λo: . Λ = −(1/2)(2π/λo) . A consequence of the rela vis c formula on is that the bending of light by gravita onal lenses, even those with no apparent lens, can probably be explained without dark ma er. Alterna vely, the phase and rapid oscilla on of the cosinusoidal poten al reduces to nearly zero its contribu on 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 predic on is yet to be tested. The cosmological implica ons are inves gated. Surprisingly, the self‐interac on 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 infla onary period at a red shi z ≃ 100,000. This is a lookback me which is consistent with that expected from observed periodici es in present galac c densi es. The poten al also provides a natural explana on for the stability of the recently discovered chain galaxies. Finally, this proposal requires that gravita onal radia on 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 magne c fields within the Galaxy, M31, and the Coma cluster.
Full ar cle can be found on website: www‐hep.colorado.edu/Cosinusoidal/ Evolu on since AAS 220 (Anchorage, AK) June 2012 This poster addresses two topics not discussed in the website; namely Magne c Fields in galaxies and gravita onal lensing by galaxies. Massive Photon Hypothesis Exemplified by the MW Double Helix
• Assume a universal constant k0, for both gravity and electromagne sm, whose value is roughly 2 π/400pc. • ˆ Assume vector poten al A = A0k0r /cosh(kr) φ • Assume equipar on between total energy stored in the magne c field B2/8π and the total energy stored in the vector 2 2 poten al , 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 Equipar on: B2dV = k 2 A2dV € ∫ ∫ 0 • Yields an equa on 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 galac c center and with our assump ons there should be a uniform B field only over 19 pc. • Using the distance of sun to GC, 8 kpc, we find a Galac c longitude range of π 8000pc × =140pc /deg 180 19 pc /140pc /deg = 0.136 deg Referring to the figure by Morris, et al. this es mate is about twice as large as the width of the € double helix. Vector Poten al and Axial Magne c Field that were integrated for Equipar on
Vector Potential Magnetic Field
A B
k0 r k0 r
Az = r sech[kr] k=3.4k0 A magne c torsional wave near the Galac c 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 spa al resolu on is 6 arcsec. At the 8 kpc distance of the Galac c 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 J × B = 0 Note J is parallel to B, different for currents in wires. 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 Equipar on Assump on Again Using M31 Polariza on 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 Polariza on of B‐vectors in M31 Standard Interpreta on Fletcher, et al A&A 414, pg. 53(2004)
Resolution is insufficient to observe variations on a scale of 400pc. Deflec on 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 deflec ng galaxy is in‐line with distant extended source • Use impulse approxima on to calculate deflec ng 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 €
€ Gravita onal Lensing
Rings (from Gravita onal Lensing)
have radii of 4¾ λ0 & 5¾ λ0 Channels (from Mag. Field ) ˆ A = A0 sech[kr] z 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 ar cles.