Earth and Moon orbital anomalies

Transcription

1 Eath and Moon obital anomalies Si non è veo, è ben tovato Ll. Bel axiv: v2 [g-qc] 18 Feb 2014 Febuay 19, 2014 Abstact A time-dependent gavitational constant o mass would coectly descibe the suspected inceasing of both: the Astonomical unit and the eccenticity of the Luna obit aound the Eath 1.- The model The gavitational model below, although it was initially motivated by a desie to show the potentiality of time dependent solutions of Einstein s equations, is essentially a pue Newtonian-like one, and diffes fom the classical model by the vey simple substitution: µ (1 pt)µ, µ = Gm (1) whee 1 p is a constant whose tentative value I assume to be s 1, andwheeptissupposed tobesmall enoughso that(pt) 2 isnegligibleduing the whole duation of the pocesses to be consideed below. Using obvious assumptions and notation, the equations of motion of a point body of unit mass moving in the gavitational field ceated by a point souce of mass m may thus be deived fom the time dependent Hamiltonian: H = 1 2 (ṙ2 + 2 ϕ 2 ) (1 pt) µ, µ = Gm (2) that leads to the following equations: wtpbedil@lg.ehu.es 1 p in this pape coesponds to the poduct 3p of [1]. Simila substitutions have been consideed befoe: they ae eminded in the intoduction of [4]. 1

2 ϕ 2 = µ 2(1 pt) (3) as well as: J = 2ṙ ϕ+ ϕ = 0 with J = 2 ϕ (4) Ḣ = pµ (5) 2.- Inceasing of the astonomical unit and the Moon to Eath distance The following example assumes that both the Eath and the Sun can be dealt with as two point bodies and that the Eath deviates fom a cicle of adius = a, the astonomical unit, at time t by a small constant amount δ afte a shot inteval of time δt. Moe pecisely, we assume that at the pesent epoch we have: = a = 0 (6) Using the second assumption above and the definition of J we get: that leads to the following esult: = J2 (1+pt) and so: ṙ = pj2 µ µ = pa (7) ȧ = pa = 0.07my 1 (8) Mutating mutandis the Sun by the Eath, the Eath by the Moon and the astonomical unit a by the mean distance b of the Moon fom the Eath I get: ḃ = pb = my 1 (9) 3.-Inceasing of the Eath aound the Sun and the Moon aound the Eath obits eccenticities Using the classical fomula satisfied by the eccenticity e : e = 1+ 2HL2 µ 2 (10) 2

3 whee the kinetic moment J is constant and the enegy H given by (2) is time dependent, we obtain: ė = J2 J2 p eµ 2Ḣ o using (5) ė = eµ a (11) With m being the mass of the Sun, a the astonomical unit and e the pesent eccenticity of the obit of the Eath, the esult is: ė = y 1 (12) Mutating mutandis the coesponding esult giving the inceasing of the eccenticity of the Luna obit is: Conclusion ė = y 1 (13) Fom the fou esults hee mentioned, ȧ, e, ḃ, and ė, only the fist and the fouth have been well documented, and supisingly both can be deived using the same paamete p. At this moment this can be consideed as a coincidence o as an eventual new paadigm 2. Notewothy is the fact that because the Hamiltonian is time dependent this model may have something to say about the anomalies of flybys. And last but not least it is also wothy to say that afte so many theoetical physics models dedicated to explain the Pionee s anomaly, now unnecessay [3], this one pedicts a negligible contibution to this effect. Appendix Let us conside the following spheically symmetic space-time model whose line-element is: ds 2 = A 2 dt 2 +A 2 d s 2, d s 2 = M 2 d 2 +N 2 2 dω 2 (14) whee to stat with we assume that: A 1 = 2 See also efeences[4], [5], [6] ( ) m 1/2 +(1+2pt) 1/2 1, G = c = 1 (15) +m 3

4 1 M = 1+p2 2, N = 1 m2 (16) 2 If p = 0 then (14) is the Schwazschild model, being the adial Fock coodinate (x,y,z:hamonic): A 2 = m +m, M2 = 1, N 2 = 1 m2 (17) 2 If m = 0 then (14) is Milne s flat space-time model: A 2 = 1+2pt, M 2 1 = 1+p 2 2, N2 = 1 (18) But t is not the global pope time that the model allows to use. It is the time that in both cases leads to a space model geomety: d s 2 = d2 1+p dω 2 (19) that is time independent and has constant cuvatue, thus fulfilling Helmholtz s fee motion postulate. The line-element (14) can be consideed in geneal as an appoximate vacuum solution of Einstein s equations whee the quality of the appoximation depends on the elevant domains of and t and the values of m and p. On the othe hand, fomal linea developments with espect to both p and m, of (15) and (16) yield: A 2 = 1 2m ( m ) pt, M = N = 1 (20) so that the cental foce pe unit mass is: f = dlna = m d 2(1 pt) (21) At this fomal appoximation one has: R α β 1 2 δα β = 0 (22) and the non zeo stict components of the Riemann tenso ae: R = m (1+pt), R1.313 = R 1.212sin 2 θ (23) R = m (1 3pt), R4.343 = R4.343 sin2 θ (24) R = 2m (1 3pt), 3 R3.232 = 2 m (1+pt) (25) 4

5 Acknowledgements I wish to thank L. Acedo whose apt questions and comments helped me to wite a bette manuscipt. Refeences [1] Ll. Bel, axiv:/ v1 [g-qc] [2] J. D. Andeson and M. M. Nieto, in Relativity in Fundamental Astonomy, Poceedings IAU Symposium No. 261, 2009 S. A. Klione, P. K. Seidelman & M. H. Soffel, eds. [3] Slava G. Tuyshef et al. axiv: v1 [4] C. Duval, G. Gibbons and P. Hováthy, Phys. Rev. D, 43, 12, (1991) pp [5] L. Acedo Phys. Essays 26, 4 (2013) pp [6] J. Bootello, Joun. Moden Physics,4, (2013) pp