Hindawi Advances in Mathematical Physics Volume 2019, Article ID 6765827, 10 pages https://doi.org/10.1155/2019/6765827 Research Article Yukawa Potential Orbital Energy: Its Relation to Orbital Mean Motion as well to the Graviton Mediating the Interaction in Celestial Bodies Connor Martz,1 Sheldon Van Middelkoop,2 Ioannis Gkigkitzis,3 Ioannis Haranas ,4 and Ilias Kotsireas4 1 University of Waterloo, Department of Physics and Astronomy, Waterloo, ON, N2L-3G1, Canada 2University of Western Ontario, Department of Physics and Astronomy, London, ON N6A-3K7, Canada 3NOVA, Department of Mathematics, 8333 Little River Turnpike, Annandale, VA 22003, USA 4Wilfrid Laurier University, Department of Physics and Computer Science, Waterloo, ON, N2L-3C5, Canada Correspondence should be addressed to Ioannis Haranas; [email protected] Received 25 September 2018; Revised 25 November 2018; Accepted 4 December 2018; Published 1 January 2019 Academic Editor: Eugen Radu Copyright © 2019 Connor Martz et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Research on gravitational theories involves several contemporary modifed models that predict the existence of a non-Newtonian Yukawa-type correction to the classical gravitational potential. In this paper we consider a Yukawa potential and we calculate the time rate of change of the orbital energy as a function of the orbital mean motion for circular and elliptical orbits. In both cases we fnd that there is a logarithmic dependence of the orbital energy on the mean motion. Using that, we derive an expression for the mean motion as a function of the Yukawa orbital energy, as well as specifc Yukawa potential parameters. Furthermore, various special cases are examined. Lastly, expressions for the Yukawa range � and coupling constant � are also derived. Finally, an expression for the mass of the graviton ��� mediating the interaction is calculated using the expression its Compton wavelength (i.e., the potential range �). Numerical estimates for the mass of the graviton mediating the interaction are fnally obtained at various eccentricity values and in particular at the perihelion and aphelion points of Mercury’s orbit around the sun. 1. Introduction through the interaction of a Yukawa potential. Terefore, the experimental and observational search for such deviation Any scientist will agree that Einstein’s general relativity theory might result in new type of physics [3]. (GR) is one of the most mathematically elegant theories In a recent paper by Haranas et al. (2018), the authors invented in the human history. Even though the theory examinedustparticleorbitsundertheinfuenceofPoynting- explains many physical phenomena, it is unable to shed light Robertson efect in which Newtonian gravity has been on the problem of the observed accelerating universe. To do modifed by a Yukawa term. Similarly, in Mukherjee and that, GR introduces a cosmological constant lambda Λ as well Sounda [4], the authors investigate the orbits resulting from as the so-called dark energy. Various gravitational theories various coupling constants � (alpha) of a Yukawa correc- exist today which try to explain the observed acceleration of tion to the Newtonian potential. Quite ofen in celestial the universe [1]. Tese gravitational theories are nonsymmet- mechanics, a Yukawa-type potential is proposed to modify ric, scalar-tensor, quantum gravitational, or �(�) theories of the Newtonian gravity [5–9] (Iorio 2007), and its efect on gravity, etc. As a common denominator in the weak limit, all various gravitational, astrophysical, and orbital scenarios is theories result in a Yukawa type of gravitational potential. In examined. In this contribution, we examine the time rate of a paper by Chan [2], the authors put forward the idea that the Yukawa orbital energy of circular and elliptical orbits, observational evidence for the existence of cold dark matter and from that we derive relations for the orbital energy as a particles in the cores of dwarf galaxies could be explained function of mean motion � of the orbiting body. To relate � 2 Advances in Mathematical Physics � ⟨�⟩ and to the orbital parameters of the secondary, expressions E = ⟨�⟩ + ⟨�⟩ = . (3) are derived that relate them to various orbital parameters. 2 � In particular, considering the expression for ,weobtain Anticipating small perturbations, we can add the average a Lambert function that relates the mass of the graviton value of the perturbing term to the average of the Newtonian along the orbit of the secondary to the Yukawa parameters, potential to that of the perturbative term making also use that eccentric anomaly, orbital energy, and eccentricity. Tis is the semimajor axis � is the average value of the radial distance done using the already derived expression for lambda and � along the orbit; we can further write that as substituting it into the corresponding equation for the range � � ��� of graviton and then solving for its mass ��.Itiswell E ≅− (1 + ��−�/�) known that if gravitation is propagated by a massive feld, 2� (4) the velocity of the gravitational waves (gravitons) will depend upon their frequency, and the efective Newtonian potential Now rewriting (1) only the Yukawa orbital energy we have is −1 −�/��� will have a Yukawa form, i.e., �(�) ∝ � � ,where��� = ��� ℏ/� � E =− (��−�/�). �� is the graviton Compton wavelength. Today’s research 2� (5) for the mass of the graviton includes both theoretical and observational work. For example, in Stavridis and Will [10] Next diferentiating (5) with respect to time we obtain that the authors try to bound the graviton mass using gravitational the total time rate of the energy of the Yukawa correction is efects and their efect in the spin precessions of massive hole E ��� � �� binaries. Similarly, in Mureika and Mann [11], the authors d = [� (1 + )�−�/�] . 2 (6) use an entropic gravity approach to estimate a bound for d� 2� � �� the mass of the graviton. Finally, in Zacharov et al. [12], the authors consider Yukawa gravity interactions of S2 star 3. Circular Orbits orbits near the galactic plane to improve expectations for graviton mass bounds. At this point we must say that in To examine the case of circular orbits, namely, we let the � today’s gravity research various methods have been employed eccentricity 0 =0andr = a,(6)becomes in the determination of the graviton mass. Our motivation dE ��� � −�/� �� for paper emanates from the fact that this work can serve as = [� (1 + )� ] . (7) � 2�2 � �� another possible observational test in setting solar system as d � well as binary system bounds on graviton mass ��,where Next, following Haranas et al. [9], we consider the unper- the bound depends on the mass of the source, which in this turbed relative orbit of the secondary body, in this case case is a sun like type of star. a Keplerian ellipse. If � is the semimajor axis, �0 is its eccentricity and � is its mean motion. On the unperturbed 2. The Yukawa Potential Keplerian ellipse, this law is expressed as [15] 2 3 Let us consider a two-body problem, where a secondary body �� = � � . (8) of mass � orbits under the infuence of a primary body of mass �. With the potential being central, the two-body Diferentiating (8) w.r.t. time, the rate of change of the mean problem can be reduced to a central-force problem, and the motion can then be expressed as motion of the secondary body can be examined. Te efects d� 3� d� of gravity on the secondary in the presence of a Yukawa =− , (9) correction can be described by the modifed potential energy d� 2� d� per unit mass [13]. from which we obtain that �� � (�) =− (1 + ��−�/�), (1) d� 2� d� � =− . (10) � 3� � In (1), � denotes the distance between the centers of the d d two bodies, � is the Newtonian gravitational constant, �= Terefore, substituting (10) in (7) and simplifying, we obtain ��/���,where� and � are the coupling constants of the dE ��� � −�/� new force to the bodies relative to the gravitational one, and =− [� (1+ ) � ] . (11) � is the range of this interaction (ibid. 2016). Next, let us now d� 3�� � write down an expression of the orbital energy by making � use of the virial theorem which states that for a bounded Next defning the potential energy at semimajor to be unperturbed system the following relation holds [14]: 1 1 ��� � (�) =− ( ) , ⟨�⟩ 3 � 3 � (12) ⟨�⟩ =− (2) 2 the Newtonian part of the potential, (11) can written as � where is the time average on the system’s kinetic energy and follows: � the time average of the system’s potential energy. Terefore, dE �� (�) � −�/� conservation of energy implies that the total energy is equal =− [� (1 + )� ]. (13) to (ibid. 2002) d� 3� � Advances in Mathematical Physics 3 Integrating, we obtain the orbital energy dependence on the Consequently, the energy diference satisfes the following mean motion � to be equation: ��� (�) � −�/� � 2� � E = E − (1+ ) � ( ) . (E − E )=− � � ( ). 0 3 � ln � (14) 0 ( ) ln (24) 0 3� �0 On the other hand, to obtain the dependence of the mean Using (24) we fnd that the mean motion � for �=�must motion on the Yukawa parameters and orbital energy E,we independently satisfy the equation use (14) and solving for � we obtain −3�(E−E )/2��(�) � (E) =�� 0 . �/� 0 (25) −3� (E−E0)/��N(�)(1+�/�) � (E) =�0� . (15) In particular if (E − E0)=�(�),(25)satisfestheequation In the case where the range of the potential equals the −3�/2� semimajor axis � = a, (15) becomes � (E) =�0� . (26) −3�(E−E0)/2���(�) −�/� � (E) =�0� . (16) Furthermore, using (19) if (E − E0)=−��(�)� /3 we fnd that Atthispointwecanseethatvalueofthemeanmotion� (E − E ) (E > E ) ln (�/�0) depends on the energy diference 0 .If 0 then �= �. (27) (E − E0)>0and therefore the exponent remains negative (1 − ln (�/�0)) and the mean motion reduces exponentially with the energy � diference. On the other hand, if (E < E0) and (E − E0)< Using, (19) it is impossible to fnd an expression for since 0, the exponent is positive and therefore the mean motion the resulting equation takes the form (E − E )=� (�) increases.