NASA’s astonishing evidence that c is not constant: The pioneer anomaly E. D. Greaves Universidad Simón Bolívar, Laboratorio de Física Nuclear, Apartado 89000, Caracas 1080 A, Venezuela. E-mail: [email protected] For over 20 years NASA has struggled to find an explanation to the Pioneer anomaly. Now it becomes clear the solution to the riddle is that they have uncovered evidence that c, the speed of light, is not quite a universal constant. Using J. C. Cure’s hypothesis that the index of refraction is a function of the gravitational energy density of space and straightforward Newtonian mechanics, NASA’s measurements provide compelling evidence that the speed of light depends on the inverse of the square root of the gravitational energy density of space. The magnitude of the Pioneer anomalous acceleration leads to the value of the primordial energy density of space due to faraway stars and galaxies: 1.0838. x 1015 Joule/m3. A value which almost miraculously coincides with the same quantity: 1.09429 x 1015 Joule/m3 derived by J. C. Cure from a completely different phenomenon: the bending of starlight during solar eclipses. PACS numbers: 95.55.Pe, 06.20.Jr, 04.80.Cc, 95.10.-a Introduction Anderson and collaborators at the Jet Propulsion Laboratory (JPL) have reported [1] an apparent, weak, long range anomalous acceleration of the Pioneer 10 and 11 with supporting data from Galileo, and Ulysses spacecraft. [2, 3] Careful analysis of the Doppler signals from both spacecraft have shown the presence of an unmodeled acceleration towards the sun. By 1998 it was concluded from the analysis, that the unmodeled acceleration towards the Sun was (8.09 +/- 0.20) x 10-10 m/s2 for Pioneer 10 and of (8.56 +/- 0.15) x 10-10 m/s2 for Pioneer 11. In a search for an explanation, the motions of two other spacecraft were analyzed: Galileo in its Earth-Jupiter mission phase and Ulysses in a Jupiter-perihelion cruise out of the plane of the ecliptic. It was concluded that Ulysses was subjected to an unmodeled acceleration towards the Sun of (12 +/- 3) x 10-10 m/s2. To investigate this, an independent analysis was performed of the raw data using the Aerospace Corporation’s Compact High Accuracy Satellite Motion Program (CHASMP), which was developed independently of JPL. The CHASMP analysis of Pioneer 10 data also showed an unmodeled acceleration in a direction along the radial toward the Sun. The value is (8.65 +/- 0.03) x 10-10 m/s2, agreeing with JPL’s result. Aerospace’s analysis of Galileo Doppler data resulted in a determination for an unmodeled acceleration in a direction along the radial toward the Sun of, (8 +/- 3) x 10-10 m/s2, a value similar to that from Pioneer 10. All attempts at explanation of the unmodeled acceleration as the result of hardware or software problems at the spacecraft or at the tracking stations have failed. A very detailed description of the Pioneer anomaly, of the measurements and of the analysis was given by the JPL team [4]. Two conferences have been carried out on the subject, in 2004 [5] and in 2005 [6] and although several explanations have been advanced, no clear consensus exists of the cause of the weak [7] anomalous acceleration experienced by the various spacecraft. With no plausible explanation so far, the possibility that the origin of the anomalous signal 1 is new physics has arisen.[8] Very recently evidence of the puzzling phenomenon was found in the motion of other spacecraft. [9] The Pioneer anomalous acceleration a is derived from the Doppler drift Δf of the base frequency f detected: Δf = f a In this paper the anomalous drift is shown to be due to o o ( c ) a change of the index of refraction of vacuum, a function of the gravitational energy density of space predicted by the Curé hypothesis [10]. It affects c the speed of light in space far from the influence of the sun. 1.- Energy density of space. By energy density of space we mean the classical energy density (Energy per unit volume) associated with the potential energy of all forms of force: electric, magnetic, gravitational or any other force in nature. In particular, to be associated to gravitational energy of nearby massive bodies such as the Sun and the Earth which we can readily calculate, and to the gravitational energy density produced by the gravitational field of the stars and far away galaxies, not so easily estimated. The energy density of space associated with the presence of static electric E and magnetic B fields are given by: 1 2 1 2 ρ = ε o E + B (1) 2 2μo Where ε 0 and μ0 are the electric permittivity and the magnetic permeability of space respectively. The equivalent energy density associated with a gravitational field g (m/s2) is given by 1 g 2 ρ = (2) 2 4πG with G the Universal constant of gravitation. Hence any volume of space is immersed in the universal primordial field of energy ρ * which includes the immediate gravitational field due to the presence of our own galaxy superimposed on the energy fields of all far-away galaxies. Thus the energy density in the surface of the Earth and in the proximity of the Sun is given by: * (3) ρ = ρ + ρS + ρ E where the energy density due to the Sun ρS produced by the gravitational effect of the 2 mass of the Sun M S is obtained from (2) with g = GM S / r GM 2 (4) ρ = s S 8πr 4 Here r is the distance from the centre of the Sun to the point in question. And ρ E is the energy density due to the gravitational effect produced by the mass of the Earth and is calculated in analogous manner. The acceleration of gravity g S due to the Sun at the radius -2 of the Earth’s orbit is g S = 0.00593 m s . Hence the Sun’s energy density at the Earth orbit 4 3 is ρS = 2.097 x 10 Joules/m . With the Earth’s acceleration of gravity the energy density +10 3 due to the Earth at the surface is ρ E = 5.726 10 Joules/m and the universal primordial 2 energy density ρ * is estimated [10] at 1.09429 x 1015 Joules/m3. This is a value arrived at by an analysis of the deflection of light by the Sun’s energy field considered as a refraction phenomenon as reviewed below. [11] J.C. Curé [10, p. 276] explains the energy density of space in the following illuminating words: “Every celestial body is surrounded by an invisible envelope of gravitostatic energy caused by the matter of the body and given by Eq. (104). (Our Eq. 4) To proceed with a colorful description, let us assign a yellow color to the sun’s gravitostatic energy. Let us picture the background cosmic energy with a bluish color. Now we can see, in our imagination, that the sun is surrounded with a green atmosphere of energy. The green color fades away into a bluish color as we recede from the sun.” 2.- Effect of energy density of space Now let us consider the hypothesis that the speed of light is a function of the energy density of space ρ which in the neighbourhood of the sun is determined by a constant background value due to the distant galaxies plus a smaller value due to the gravitational presence of the Sun’s mass as seen by (3) above. We assume the speed is inversely proportional to the square root of the energy density by the use of the Curé hypothesis [10, p 173] given by relation (5): k c'= (5) ρ * +ρ S + ρ E This implies that the speed of light decreases near the Sun and increases far away from the sun. We may then assign an index of refraction n to space such that n = 1 in vacuum space near the Earth, as we usually do, and assign an index n' < 1 far away from the Sun, in deep space, where the speed of light c' is greater and is given by: c c'= (6) n' so that the index of refraction there is n'= c / c' . Using (5) we may write expressions for c and c' and obtain the index of refraction, n' , far away from the Sun in terms of ρS1AU the energy density of the Sun at the distance of the Earth: one Astronomical Unit ( r = 1 AU), ρ E the energy density of the Earth at the surface, ρSfar , the energy density of the Sun, relatively far away but in the vicinity of the Sun and ρ * the interstellar primordial energy density in the vicinity of the Sun [12] as: ρ *+ρSfar + ρ Efar n'= (7) ρ *+ρS1AU + ρ E Strictly speaking, relation (7) should contain in the numerator and denominator the gravitational energy density due to all the other planets. However, the contribution is negligible due to the 1/ r 4 factor in the energy density, unless n' is being calculated near a planet. At this point it is fitting to address the order of magnitude of the quantities being discussed. With n = 1 at the Earth at 1 AU from the Sun, the index of refraction n' further away from the Sun is dependent on the relative magnitudes of the energy density values that enter into 3 Eq. (7), i.e. the relative value of the Sun’s energy density, the Earth’s energy density and the primordial energy density ρ * of space due to the stars and far-away galaxies.