aa r X i v : . [ phy s i c s . c l a ss - ph ] F e b LETTER TO THE EDITOR
Stellar parallax in the Neo-Tychonian planetarysystem
Luka Popov
University of Zagreb, Department of Physics, Bijeniˇcka cesta 32, Zagreb, CroatiaE-mail: [email protected]
Abstract.
The recent paper published in European Journal of Physics [1] aimed todemonstrate the kinematical and dynamical equivalence of heliocentric and geocentricsystems. The work is performed in the Neo-Tychonian system, with key assumptionthat orbits of distant masses around the Earth are synchronized with the Sun’s orbit.Motion of Sun and Mars have been analysed, and the conclusion was reached that thevery fact of the accelerated motion of the Universe as a whole produces the so-called“pseudo-potential” that not only explains the origin of the pseudo-forces, but also thevery motion of the celestial bodies as seen from the static Earth. After the paper waspublished, the question was raised if that same potential can explain the motion of thedistant stars that are not affected by the Sun’s gravity (unlike Mars), and if it can beused to reproduce the observation of the stellar parallax. The answer is found to bepositive.PACS numbers: 45.50.Pk, 96.15.De, 45.20.D-
Submitted to:
Eur. J. Phys. etter to the Editor Distant star θ Earth at t=0.5 years Earth at t=0Sun θ z Distant star at t=0.5 yearsDistant star at t=0 Earth y x Figure 1.
Illustrations of the stellar parallax in the heliocentric (left pannel) versusgeocentric (right pannel) frames of reference.
1. Introduction
The well-known effect of stellar parallax can be explained in two ways. The first andmost common one is in the heliocentric system, in which the Sun and the observed starsare approximately considered to be at rest. While the Earth moves around the Sun, itsposition relative to the stars changes, and that results with the effect of motion of thenear stars [2]. The parallax is observed using the more distant stars in the background.Second way to explain stellar parallax is by saying that the apparent movement ofthe stars is in fact the real motion in the pseudo-potential that is, according to Mach’sprinciple [3], generated by the very fact of the simultaneous accelerated motion of allthe bodies in the Universe, including the distant stars.The comparison between two approaches is given in the Figure 1, with theappropriate choice of coordinate axes that will be used in the calculation which follows.
2. Motion of Proxima Centauri in the Earth’s pseudo-potential
Now in order to demonstrate how one can arrive to the correct prediction of the stellarparallax in the Neo-Tychonian system, we will calculate the trajectory of the starProxima Centauri in the pseudo-potential given by Eq (4.4) in [1, 4], U ps ( r ) = GmM S r SE ˆ r SE · r . (2.1)Here G stands for Newton’s constant, M S stands for the mass of the Sun and r SE ( t )describes the motion of the Sun in the Earth’s pseudo-potential and was calculated in[1]. The Lagrangian that describes the motion of the Proxima Centauri in the Earth’spseudo-potential is therefore given by (gravitational interaction between the star andthe Sun is, of course, neglected): L = 12 m ˙ r − GmM S r SE ˆ r SE · r , (2.2) etter to the Editor - @ AU D - @ AU D xyzEarth Figure 2.
Left pannel displays the result of the numerical solutions for equations ofmotion derived from the Lagrangian (2.2) over the period of 1 year. It represents thetrajectory of the star in the x - y plane, as seen from the Earth. Right pannel illustratesthe stellar parallax effect, in consistence with the numerical results. where m is the mass of the star, and r ( t ) describes its motion. The equations of motionsare mass-independent, as expected.The Euler-Lagrange equations for this Lagrangian are solved numerically in theCartesian coordinate system, using Wolfram Mathematica package. The numericalsolutions over the period of 1 year are presented in the Fig 2.Stellar parallax can now be geometrically calculated:arctan θ = r x ( t = 0 . D , (2.3)where D = 4 .
24 ly is the well-knows distance of Proxima Centauri from the Earth [5].Using the numerical results obtained above, one can evaluate the expression (2.3). Theresult is θ = 3 . × − rad = 0 . ′′ , (2.4)which is perfectly consistent with the astronomical data [6].
3. Conclusion
We have analysed the motion of the star Proxima Centauri in the Earth’s pseudo-potential previously derived from Mach’s principle [1]. The obtained results are inaccord with the observed data. The kinematical and dynamical equivalence of Neo-Tychonian and Copernican systems has once again been demonstrated.
Acknowledgments
Author kindly thanks Dr Karlo Lelas for bringing up this issue. This work is supportedby the Croatian Government under contract number 119-0982930-1016. etter to the Editor References [1] Popov L 2013 Newtonian-Machian analysis of Neo-tychonian model of planetary motions
Eur. J.Phys.
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Preprint arXiv:1301.6045v2)[2] Ostlie D A and Carrol B W 2007
An Introduction to Modern Stellar Astrophysics (in press) [5] Wikipedia 28 Feb 2013 Proxima Centauri http://en.wikipedia.org/wiki/Proxima_Centauri [6] Benedict G F et al 1999
Astron. J.118