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Astronomical Distances The Act of Measurement I: Astronomical Distances B. F. Riley The act of measurement causes astronomical distances to adopt discrete values. When measured, the distance to the object corresponds through an inverse 5/2 power law – the Quantum/Classical connection – to a sub-Planckian mass scale on a level or sub-level of one or both of two geometric sequences, of common ratio 1/π and 1/e, that descend from the Planck mass and may derive from the geometry of a higher-dimensional spacetime. The distances themselves lie on the levels and sub-levels of two sequences, of common ratio π and e, that ascend from the Planck length. Analyses have been performed of stellar distances, the semi-major axes of the planets and planetary satellites of the Solar System and the distances measured to quasars, galaxies and gamma-ray bursts. 1 Introduction Using Planck units the Quantum/Classical connection, characterised by the equation (1) maps astronomical distances R – in previous papers only the radii of astronomical bodies [1, 2] – onto sub-Planckian mass scales m on the mass levels and sub-levels1 of two geometric sequences that descend from the Planck mass: Sequence 1 of common ratio 1/π and Sequence 3 of common ratio 1/e.2 The sequences may derive from the geometry of a higher-dimensional spacetime [3]. First, we show that several distances associated with the Alpha Centauri system correspond through (1) to the mass scales of principal levels3 in Sequences 1 and 3. We then show that the mass scales corresponding through (1) to the distances from both Alpha Centauri and the Sun to the other stars lie on the levels and sub-levels of Sequences 1 and 3. Then we show that the distances to the stars are also directly related to Planck scale: the distances lie on the levels and sub-levels of two sequences, of common ratio π and e, that ascend from the Planck length. We go on to perform an analysis of the mean orbital radii (semi-major axes) of the planets and planetary satellites of the Solar System and then analyse the distances to quasars, galaxies and gamma-ray bursts. 2 The Alpha Centauri System The binary star Alpha Centauri AB lies in the middle of the Main Sequence. The mass scales corresponding through (1) to the two stellar radii (A: 1.2234(53) ; B: 0.8632(37) [4]) are symmetrically arranged about Level 35 in Sequence 1, close to Level 40 in Sequence 3, as shown in Figure 1. Nearly-coincident super-levels (levels whose level-numbers are multiples of 5) are rare, and are important locations for physics: the quarks of each generation are arranged symmetrically about mass super-levels that are adjacent to nearly-coincident super-levels [5]. The sub-Planckian mass scale (48.8 MeV) characteristic of Alpha Centauri AB, which is equal to the geometric mean of the mass scales corresponding through (1) to the radii of Alpha Centauri A and B, is plotted on the levels of Sequences 1 and 3 in Figure 2 together with the mass scales corresponding through (1) to the various orbital distances associated with the Alpha Centauri system. Alpha Centauri A and B orbit each other. Proxima Centauri orbits Alpha Centauri AB. The planets Proxima Centauri b and c orbit Proxima Centauri. The mass scales corresponding through (1) to the semi-major axis, or mean orbital radius, of each orbit are included in Figure 2. Each distance corresponds through (1) to a mass scale that is coincident with a principal level in either Sequence 1 or Sequence 3 suggesting that the classical world emerges from a quantum world. 1 Half-levels, quarter-levels, eighth-levels etc 2 Sequence 2 is of common ratio 2/π. 3 Levels of integer level-number 40.5 Alpha Centauri A n3 40 Alpha Centauri B 39.5 34.5 35 35.5 n1 Figure 1: Sub-Planckian mass scales corresponding through (1) to the radii of Alpha Centauri A and B. Shown on the levels and sub-levels of Sequence 1 (common ratio 1/π) and Sequence 3 (common ratio 1/e). 47 46 E 45 44 D n3 43 C 42 41 B 40 A 39 34 35 36 37 38 39 40 41 n1 Figure 2: Sub-Planckian mass scales corresponding through (1) to: A The geometric mean radius of Alpha Centauri A and Alpha Centauri B (1.028 ) B The orbital radius of Proxima Centauri b (0.0485(41) AU [6]) C The orbital radius of Proxima Centauri c (1.489(49) AU [7]) D The semi-major axis of the binary Alpha Centauri AB’s orbit (23.520(36) AU [8]) E The semi-major axis of Proxima Centauri’s orbit around Alpha Centauri AB (8.7 +0.7/-0.4 kAU [9]) Shown on the levels of Sequences 1 and 3. The mass scales are confined to a straight line since the level-numbers in the two sequences are in constant ratio. 2 3 The Distances of the Nearby Stars from Alpha Centauri On the basis of Figure 2 a first thought was to calculate the mass scales corresponding through (1) to the distances between Alpha Centauri and the nearby (to Earth) stars.4 This has been done for six such stars, of various types: the results are shown in Figure 3. Despite the inconstancy of interstellar distances each mass scale calculated lies on a sub-level in either Sequence 1 or Sequence 3. This result immediately tells us that the act of measurement causes the distance to a star to adopt a discrete value that is codified through (1). 48 F E D 47.5 n3 C B A 47 41 41.5 42 n1 Figure 3: Sub-Planckian mass scales corresponding through (1) to the distance from Alpha Centauri of: the nearest brown dwarfs5 to Earth; the next nearest red dwarf to Earth after Proxima Centauri; the nearest solitary white dwarf to Earth; and the nearest stars of F-type, G-type and K-type to Earth. Distances stated are from Alpha Centauri B. A Luhman 16: a binary system of brown dwarfs; distance 1.097 pc B Sun: G-type; distance 1.350 pc C Barnard’s Star: red dwarf; distance 1.984 pc D Epsilon Eridani: K-type; distance 3.871 pc E Procyon A: F-type; distance 3.997 pc F van Maanen’s Star: white dwarf; distance 5.226 pc Shown on the sub-levels of Sequences 1 and 3. 4 All stellar distances in this paper are taken from WolframAlpha [10]. 5 Brown dwarfs are not true stars. 3 4 The Distances of Nearby FGK Stars from Earth The mass scales corresponding through (1) to the distances from Earth of nearby FGK stars lie on ‘higher-order’ sub-levels in Sequences 1 and 3, as shown in Figure 4. The Sun (G-type), Alpha Centauri A (G-type), Alpha Centauri B (K-type), Sirius A (A-type) and dwarf stars will be considered later. 47.625 E D C B n3 A 47.5 41.5 41.625 n1 Figure 4: Sub-Planckian mass scales corresponding through (1) to the distance from Earth of all FGK stars other than the Sun and Alpha Centauri A and B within 4 pc. A Epsilon Eridani: K-type; distance 3.212 pc B 61 Cygni: a binary system of K-type stars; distance 3.496 pc C Procyon A: F-type; distance 3.51 pc D Epsilon Indi: K-type; distance 3.639 pc E Tau Ceti: G-type; distance 3.646 pc Shown on the sub-levels of Sequences 1 and 3. 5 The Distance of the Sun from Earth (Earth’s Orbital Radius) One might imagine that the mass scale corresponding through (1) to the semi-major axis of Earth’s orbit, which is equal to the mean distance of the Sun from Earth, would lie on a principal level or low- order6 sub-level in either Sequence 1 or Sequence 3. That it does not do so suggests that this distance corresponds through (1) to a particular sub-Planckian mass scale or sub-atomic length scale, much as the radius of the Sun, which corresponds through (1) to a mass scale closely adjacent to but not on (35, 40) in Sequences 1 and 3, corresponds to the mass of a specific nuclide (53Cr) [1, 2]. 6 Low-order sub-levels are half-levels and quarter-levels. 4 7 11 Earth’s semi-major axis = 1.496 x 10 m corresponds through (1) to a mass scale -18 5.76 GeV and a length scale, of 34.3 x 10 m, which may be the scale of a key parameter in the quantum theory, perhaps the effective quark radius. A quantum/classical equation connects with the mass of the up quark: (2) -10 where 0.529 x 10 m is the Bohr radius and 2.16 MeV is the central value of the Particle Data Group’s evaluation of the up quark mass [12]. From (2) with precisely equal to 2.16 MeV, 11 1.495 x 10 m. Equation (2) shows a connection between a classical parameter ( ) measured in units of Bohr radius and a quantum parameter ( ) measured in Planck units. In [2] a connection was shown between stellar mass measured in units of inverse-Bohr radius and corresponding atomic radius measured in Planck units. 6 The Distances from Earth of Alpha Centauri and Sirius The mass scales corresponding through (1) to the distances of the component stars of Alpha Centauri (the third brightest star in the night sky after Sirius and Canopus) and Sirius lie on high-order sub- levels in Sequences 1 and 3, as shown in Figure 5, but the distances themselves occupy lower-order sub-levels, Sirius actually occupying a principal level, in geometric sequences of distance from Earth that ascend from the Planck length with common ratio π (Sequence 1c) and common ratio e (Sequence 3c), as shown in Figure 6.
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