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arXiv:1302.7129v1 [physics.class-ph] 28 Feb 2013

LETTER TO THE EDITOR
Stellar parallax in the Neo-Tychonian planetary system
Luka Popov
University of Zagreb, Department of Physics, Bijeniˇcka cesta 32, Zagreb, Croatia E-mail: lpopov@phy.hr
Abstract. The recent paper published in European Journal of Physics [1] aimed to demonstrate the kinematical and dynamical equivalence of heliocentric and geocentric systems. The work is performed in the Neo-Tychonian system, with key assumption that 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 the very 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 the very motion of the celestial bodies as seen from the static Earth. After the paper was published, the question was raised if that same potential can explain the motion of the distant stars that are not affected by the Sun’s gravity (unlike Mars), and if it can be used to reproduce the observation of the stellar parallax. The answer is found to be positive.
PACS numbers: 45.50.Pk, 96.15.De, 45.20.D-
Submitted to: Eur. J. Phys.

Letter to the Editor
Distant star θ

z Distant star at t=0

2
Distant star at t=0.5 years

y θ

Earth at t=0.5 years

Sun

Earth at t=0

x Earth

Figure 1. Illustrations of the stellar parallax in the heliocentric (left pannel) versus geocentric (right pannel) frames of reference.

1. Introduction
The well-known effect of stellar parallax can be explained in two ways. The first and most common one is in the heliocentric system, in which the Sun and the observed stars are approximately considered to be at rest. While the Earth moves around the Sun, its position relative to the stars changes, and that results with the effect of motion of the near 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 of the stars is in fact the real motion in the pseudo-potential that is, according to Mach’s principle [3], generated by the very fact of the simultaneous accelerated motion of all the bodies in the Universe, including the distant stars.
The comparison between two approaches is given in the Figure 1, with the appropriate 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 stellar parallax in the Neo-Tychonian system, we will calculate the trajectory of the star Proxima Centauri in the pseudo-potential given by Eq (4.4) in [1, 4],

Ups(r)

=

GmMS rS2 E

ˆrSE

·

r.

(2.1)

Here G stands for Newton’s constant, MS stands for the mass of the Sun and rSE(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’s

pseudo-potential is therefore given by (gravitational interaction between the star and

the Sun is, of course, neglected):

L

=

1 2

mr˙ 2

−

GmMS rS2 E

ˆrSE

·r,

(2.2)

Letter to the Editor
y AU
1.5

3
z

x AU

1.5

1.5

y

1.5
x Earth
Figure 2. Left pannel displays the result of the numerical solutions for equations of motion derived from the Lagrangian (2.2) over the period of 1 year. It represents the trajectory of the star in the x-y plane, as seen from the Earth. Right pannel illustrates the 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 motions are mass-independent, as expected.
The Euler-Lagrange equations for this Lagrangian are solved numerically in the Cartesian coordinate system, using Wolfram Mathematica package. The numerical solutions over the period of 1 year are presented in the Fig 2.
Stellar parallax can now be geometrically calculated:

arctan θ = rx(t = 0.5 y) , 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). The

result is

θ = 3.705 × 10−6 rad = 0.76′′ ,

(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 pseudopotential previously derived from Mach’s principle [1]. The obtained results are in accord with the observed data. The kinematical and dynamical equivalence of NeoTychonian and Copernican systems has once again been demonstrated.

Acknowledgments
Author kindly thanks Dr Karlo Lelas for bringing up this issue. This work is supported by the Croatian Government under contract number 119-0982930-1016.

Letter to the Editor

4

References

[1] Popov L 2013 Newtonian-Machian analysis of Neo-tychonian model of planetary motions Eur. J. Phys. 34 383 (Preprint arXiv:1301.6045v2)
[2] Ostlie D A and Carrol B W 2007 An Introduction to Modern Stellar Astrophysics 2nd ed (San Francisco: Addison Wesley) pp 57–59
[3] Barbour J 2010 The definition of Mach’s principle arXiv:1007.3368 [gr-qc] [4] Popov L 2013 Corrigendum to “Newtonian-Machian analysis of Neo-tychonian model of planetary
motions” (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 1086