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9-25-2018

Yukawa Potential Orbital Energy: Its Relation to Orbital as well to the Mediating the Interaction in Celestial Bodies

Connor Martz Wilfrid Laurier University, [email protected]

Sheldon Van Middelkoop Western University

Ioannis Gkigkitzis East Carolina University, [email protected]

Ioannis Haranas Wilfrid Laurier University, [email protected]

Ilias S. Kotsireas Wilfrid Laurier University, [email protected]

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Recommended Citation C. Martz, S. Van Middelkoop, I. Gkigkitzis, I. Haranas, and I. Kotsireas, " Orbital Energy: Its Relation to Orbital Mean Motion as well to the Graviton Mediating the Interaction in Celestial Bodies," Advances in Mathematical Physics, vol. 2019, Article ID 6765827, 10 pages, https://doi.org/10.1155/ 2019/6765827

This Article is brought to you for free and open access by the Physics and Computer Science at Scholars Commons @ Laurier. It has been accepted for inclusion in Physics and Computer Science Faculty Publications by an authorized administrator of Scholars Commons @ Laurier. For more information, please contact [email protected]. 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 . 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 � 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 ’s 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 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 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 , 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 () 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 . 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 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- 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. Finally, if 0 � ,then(15)forthemean � � motion becomes (1 + ) ln ( )=1 (28) � �0 �/� � (E) =��−3� /�(1+�/�), (17) 0 Moreover, if in (28) we impose the condition � =a,weobtain decreasing for �>0and increasing for �<0.Moreover, that solving for the range of the potential � we fnd that �=�0√�. (29) � �=− , 1+�[3(E − E )/���(�) (�/� )] (18) 0 ln 0 4. Elliptical Orbits � where is the Lambert function of the indicated argument. Similarly, for elliptical orbits we have the following expres- Similarly, solving for the coupling constant � we obtain sion: �/� 3� (E − E ) dE ��� � −�/� �� 0 = [� (1 + )� ] (30) �=− . (19) � 2�2 � �� (1+�/�) � (�) ln (�/�0) d where � is the orbital radial vector. Using (8) we fnd that Inthecasewhere(E − E0)=�(�)we fnd that the range lambda � of the Yukawa potential is equal to 2 2 �� 3 �� 3� � +2�� =0. (31) � �� �� �=− , (20) 1+�[3/��ln (�/�0)] Te resulting simplifed equation is expressed in terms of the eccentric anomaly, i.e., �=�(1−�0 cos �) where �0 is the and the coupling contract � becomes of the secondary and � is the eccentric 3��/� anomaly; equation (31) becomes �=− . (21) (1+�/�) (�/� ) ln 0 � 2� � 2� (1 − �0 cos �) � d =− d =− d (32) � 3� � 3� � Te semimajor axis � is equal to the range of potential � and d d d according to (20) must satisfy the following equation: Next substituting (33) in (30) we fnd that

3(E − E0) dE 1+�[ ]=−1, (22) ��� (�) ln (�/�0) d� �(1−� �) from which we fnd that the gravitation satisfes the following ��� 0 cos =− [� (1 + ) (33) equation: 3�� (1 − �0 cos �) �

3� (E − E0) −�(1−� cos �)/� d� � (�) =− . (23) ⋅� 0 ] , 2� ln (�/�0) d� 4 Advances in Mathematical Physics from which we obtain the following equation for the change Similarly, for � we fnd that of orbital energy w.r.t. the mean motion �: �(1−�0 cos �)/� 3� (E − E0)(1−�0 cos �) � dE �= . (43) �� (�) (�+�(1+�0 cos �)) ln (�/�0) d� ��� �(1−� �) =− [� (1 + 0 cos ) 5. The Yukawa Potential Range and the � (34) 3�� (1 − �0 cos �) Particle Mediating the Interaction

−�(1−� cos �)/� ⋅� 0 ]. At this point I would like to bring the readers’ attention to an issue that might have some of them concerned. Tis is the issue of closed bounded orbits under the light of Bertrand’s � Integrating and rearranging (34) with respect to ,weobtain theorem which proves that only a Newtonian as well as a that Hook type of potential results in closed bounded orbits. Tis �(1−�0 cos �)/� � 3(E − E0)(1−�0 cos �) � is an issue that the authors of this paper will extensively ln ( ) = . (35) deal with in a paper to come soon. As a preliminary result �0 ��� (�) (1 + (�/�) (1 − �0 cos �)) we say that all the applications of the Yukawa potential Solving (35) for the mean motion � we fnd that in our research are related to the solar system and within the boundaries of the solar system dimensions where the � (E) following condition, namely, �≫�,istrue[16].Te (36) � ((3(1−� cos �)��(1−�0 cos �)/� /�� (�)(1+(�/�)(1−� cos �)))(E−E )) parameter is the Compton wavelength of the exchange =�� 0 � 0 0 . 0 particle, which in the present case is a graviton. In other words, the range of the Yukawa potential is always larger than As a check of (36), let us note that if e = 0 we can easily recover � (15) for the circular orbits. In the case where �=�,(36) any distance away from the primary star the sun in our case simplifes in the following way: that the secondary body orbits. Our preliminary analysis of nearly circular orbits has shown that the ratio of a harmonic � (E) oscillator with frequency � oscillating around a point situated (37) � �̇ � ((3(1−� cos �)/�(2−� cos �))((E−E )/� (�))�−(1−�0 cos �)) at to that of the orbital frequency at is given the ratio =�� 0 0 0 � . ̇ 2 0 �/�≅1−(�/√2�) . In the case of the solar system with sun as the primary gravitating body producing the interaction, and At this point we fnd that value of the mean motion � again since �≫�, we obtain to a second order approximation that depends on the energy diference (E − E0).Ifagain(E > E0) ̇ �/�≅1and therefore the orbits are bounded and closed. As then (E − E0)>0implies that the mean motion reduces exponentially to the energy diference. On the other hand, a support of our fndings we refer to a paper by Mukherjee and Sounda [4]. In this paper the authors fnd closed and if (E < E0) and (E − E0)<0the motion increases bounded orbits for various values of the coupling constant � exponentially. Finally, if (E − E0)=�(�),(36)forthemean motion becomes of the Yukawa potential. �(1−� cos �)/� One of the primary motivations for today’s interest in (3(1−�0 cos �)� /�(1+�(1−�0 cos �)/�)) � (E) =�0� . (38) non-Newtonian gravity emanates from our interest to probe and understand long-range . Tis interest addresses the (E − E )=�(�) �=� Next, if simultaneously 0 and , we fnd following question: Why a certain class of theories predict that the mean motion of the orbiting body takes the following the existence of gravity - like interactions. At this point we form: must say that many various models in which gravity is unifed ((3(1−� cos �)/�(2−� cos �))�(1−�0 cos �) ) � (�) =�� 0 0 . (39) with other forces have been studied in the recent years. 0 Tese models are formulated along the consideration that the −�/� product �, Furthermore, if �=�and (E − E0) = (1/3)��(�)�� and �=�(1−� �) cos solving for the mean motion n we fnd that �2 �≡( )� ≅10−10 /�2, (1−�)(1−�0 cos �) � eV (44) (((1−�0 cos �)/(2−�0 cos �))� ) ℏ� � (�) =�0� . (40)

−�/� Finally, if (E − E0) = (1/3)��(�)�� then we fnd that determines the mass of the new feld [17]. Te Compton wavelength (or “range”) � associated with the feld charac- (�/�)(1−�(�+�0 ) cos �) ((1−�0 cos �)� /(1+(�/�)(1−�0 cos �))) � (�) =�0� . (41) terises the distance at which the corresponding feld acts and is given by the equation below: Going back to (36) which is the most general equation and � � ℏ solving for and , respectively, we obtain that �= . (45) �� � In Haranas et al. [13] and Mofat and Toth [16], the strength �(1−� �) (42) =− 0 cos . � of the potential near the sun in a point source scenario is 1+�[−3(E − E )(1−� �) /��� (�) (�/� )] 0 0 cos � ln 0 given by the following equation: Advances in Mathematical Physics 5

� � �= ��� ( ∞ −1), potential must be modifed. For an extended matter dis- � 2 � (46) (√����1 +�1) � tribution and in relation to the MOG feld, there are not presently any solutions. In this case, authors treat the problem � where �∞ ≅ 20�� and �1 = 25000√����. Similarly, the phenomenologically, seeking to fnd an efective mass distri- range � can be written as follows: bution �(�, �) that could be used in (46) and (47). What this function does simply determines an “efective mass” which √� � � �= ��� , determines and . Tis way, the gravitational infuence of � (47) � �2 matterlocatedatapointofdistance on the test particle located at � is calculated. Tus, the point source function � −1 and �2 = 6250√���� kpc . Terefore equating (42) and (45) can simply yield a mass proportional to the volume if the we obtain that the mass of the mediating of the interaction distribution is constant. Terefore, following Haranas et al. graviton is [13], we can write

(−�|(�−�)/(�−�)|) 3 ℏ � (�, �) = ∫ � (�) � d �, (52) ��� =− [1 � �� (1−�cos �) (48) where the coefcient � is to be determined by observation, 3(1−� �) +�(− 0 cos (E − E ))] , for example, comparison with Bullet Cluster Data. For a mass 0 �(�) = � �(�,�)∝ ���� (�) ln (�/�0) density function, 0; then we will have that 3 |� − �| . Furthermore, in the case of a nonconstant density �, and fnally using (46) the equation takes the form the coupling constant � and range of the potential � become (ibid. 2016) ℏ [ ��� =− 1 � (�, �) � �� (1−�cos �) �= ( ∞ −1), [ 2 (53) √ � �� (49) ( � (�, �) +�1) � 2 3�� (√���� +� ) (E − E0)(1−�0 cos �) +�(− 1 )] . and ���� (�∞ −��)�� (�) ln (�/�0) ] √� (�, �) �= . � (54) In the case of circular motion, i.e., e = 0, (49) simplifes to �2 In the case of an extended mass distribution, calculation of � ℏ � =− [1 and � will result in better estimates of the two corresponding �� �� [ parameters and therefore a better estimate of the efect. (50) With reference to Haranas et al. [13] in the case of an � 2 extended source we can write the gravitational potential in 3�� (√���� +�1) (E − E0) +�(− )] . the following way: � (� −� )� (�) (�/� ) ��� ∞ � � ln 0 ∞ ] �→ �2 � �(�,�)=2���∞ ∫ � �� Similarly, in the case of circular motion and if (E − E0)= 0 �(�) , (50) becomes � −√��2+�2−2��� cos ��/� (55) � � � � ⋅ ∫ sin � ( )�� �� 0 √��2 +�2 −2��� cos �� � 2 ℏ 3� (√� +� ) (51) which fnally integrates to =− [1+�(− � ��� 1 )] . �� � (� −� ) (�/� ) �(�→�,�) [ ��� ∞ � ln 0 ] �−�/� � Equations (48) to (51) give the mass of the graviton feld [ Φ( )�≥�] (56) =� � [ � � ] mediating the interaction, i.e., a massive graviton feld. ∞ [ 6��2 � (�/�) ] According to (48) we see that the mass of graviton continu- [1 − (1 + )�−�/� sinh ]�≤� [2�3�2 � �/� ] ously varies along elliptical orbit with a maximum value at the perihelion/periastron point (binary star scenario). Its mass where also depends on the ratio of (E−E0)/��(�) as well as on ratio 3 Φ (�) ≡ (� �− �) , of the initial to fnal mean orbital motion �/�(0) and fnally s �3 cosh sinh (57) onthemassoftheprimarybodyproducingtheYukawafeld, �=�/� Mofat and Toth [16]. and , and it has the following limits: �2 �4 Φ =1+ + +...�≪1, (58) 6.TheExtendedSourceCase s 10 280 � In the case of an extended mass distribution following 3� Φs ≅ �≫1. (59) Haranas et al. [13], we say that the equation of Yukawa 2�2 6 Advances in Mathematical Physics

In a paper by Branco et al. [18], by examining the ��� Equation (62) admits the following limiting cases (ibid. 2014): component of the metric the authors identify a Newtonian 2 potential plus a Yukawa of the form (ibid. 2014) (�� ) �(�,� )≅1+ � ≅1.0 �� ≪1 (63) � 10 if � �� ��� 1 −�/� 3� � � (�) =− (1 + ( −4�)�(�,�)� ), (60) � 3 � � (�, ��) ≅ if ��� ≫1. (64) 2(���)

For a sun like central body (ibid. 2014) the more accurate where �=1/�is the characteristic length, � = 1/3 − 4� is the NASA model has been used for the density of the sun strength of the Yukawa addition to the feld, and �(�, ��) is that obeys the following conditions �(����)=0and the form factor defned as follows [18]: d�(����)/d�=0.Tus,thedensityfunctionisoftheform � 4� � 2 �(�,��)= ∫ �� (�) sin (��) d�, (61) � � ��� 0 � (�) =�0 [1 − 5.74 ( ) + 11.9 ( ) �� �� (65) � 3 � 4 which integrates to − 10.5 ( ) +3.34( ) ]. �� ��

��� cosh (���)−sinh (���) Integrating (65) the authors obtained the following func- � (�, ��) =3( ) . (62) tion for the form factor �(�, ��) (http://spacemath.gsfc (�� )3 � .nasa.gov/(2014)):

4.6 × 104� + 2.1 × 103�3 +�(2.7×104 + 131�2) � −7 [ cosh ] �(�,��)=� ( ) , (66) − (7.3 × 104 +3.6×103�2 − 14.6�4) � [ sinh ]

with limiting cases given by �=���,and�=1/�with the In other words, we fnd that the rate dE/d� has the units of limiting cases angular momentum. Similarly, in the elliptic case, (33) results −2 2 in (71) below: � (�, ��) ≅1+6×10 (���) ≈1 if ��� ≪1, (67) dE ��ℎ �� =− (1 � � �(�,� )≅7.3 �� ≫1. d� 3√1−�2 (1 − � �) � 3 if � (68) 0 0 cos (���) �(1−�0 cos �) −�(1−� cos �)/� �ℎ (71) Inthecaseofanextendedsourceweonlyconsiderasunlike + )� 0 =− � 2 star. In this case according to Haranas et al. [13] and Mofat 3√1−�0 15 and Toth [16] � ≅ 4.937 × 10 mwillimply��� =��/� ≪ 1 −2 2 and therefore �(�, ��) ≅ 1+6×10 (���) ≈1. Terefore, in ⋅���� (�, �) , the case of a sta,r our analysis above does not need to involve where we defne as “Yukawa function”thetermappearing the �(�, ��) factor function. in (71) and (72) that is originally derived from the term −1 −�/� �� (1 + �/�)� . Assuming circular orbits and a semimajor 7 7. Discussion and Numerical Results axis equal to the semimajor axis of Mercury � =5.79× 10 km � × −8 To obtain numerical results, we frst start with circular orbits. and Yukawa coupling constants =3.039 10 Mofat and 2 3 −10 15 Using (13) and the relation �� = � � to eliminate � from Toth [16] and � =3.57× 10 and range � =4.937× 10 mas the equation, we obtain that in [13] in the case where �>�we obtain that E dE 1 � −�/� d −10 =− ��√��� (1 + )� . (69) = −1.19 × 10 ℎ. (72) d� 3 � d� �=� Furthermore, using the fact that ℎ=√���(1 − �2)= Similarly, if we obtain that √��� or angular momentum per unit mass and for circular dE −11 = −8.755 × 10 ℎ. (73) orbits, i.e., e = 0, we write (69) in the following way: d� dE 1 � −�/� 1 Finally, if �=�and using � as it is given in Mofat and Toth =− ��ℎ (1 + )� =− �ℎ���� (�, �) . (70) d� 3 � 3 [16], we obtain the following relation: Advances in Mathematical Physics 7

Table 1: Table of the rate dE/d� as a function of eccentric anomaly for an elliptical orbit in a Yukawa potential. 1.2 E Eccentric Anomaly d 2 −1 ∘ [kgm s ] �[ ] d� 1.1 13 40 -4.738×10 h YukawaFunction

13 − 60 -4.450×10 h 1.0 13 80 -4.141×10 h 13 100 -3.856×10 h Mercury 0.9 13 120 -3.622×10 h 13 100500 150 200 250 300 350 140 -3.451×10 h 13 Eccentric − Anomaly E (deg) 160 -3.348×10 h 13 180 -3.314×10 h Figure 2: Plot of the Yukawa function for Mercury as a 13 function of eccentric anomaly during a full revolution. 200 -3.348×10 h 13 220 -3.451×10 h 13 240 -3.622×10 h 12 13 -3.856×10 h 260 10 13 280 -4.141×10 h 13 8 300 4.450×10 h 13 320 -4.738×10 h 6 13 340 -4.946×10 h m(per)/m(apo) 13 4 360 -5.023×10 h 2

0.0 0.2 0.4 0.6 0.8 2.5 Eccentricity [e]

Figure 3: of the mediating the 2.0 particle as it is felt by Mercury at perihelion and aphelion is plotted 15 as a function of eccentric anomaly for �=5.79× 10 mand�= 3.04 × 10−8,3.57×10−11

Function . 1.5 −

0.205, olive green e = 0.6, and green e = 0.9, and for eccentric Yukawa 1.0 anomaly values during one full revolution. Similarly, in Figure 2 we plot the Yukawa function for planet Mercury as a 0.5 function of eccentric anomaly and during one full revolution. 100500 150 200 250 300 350 Next, in Table 2 we tabulate the parameters ΔE = E− E0 and Eccentric − Anomaly E (deg) � as related to the graviton mass at perihelion as is “felt” from an orbiting around a sun like star body. For this numerical Figure 1: Plot of the “Yukawa function” for elliptical orbits of various −8 eccentricities and eccentric anomaly during one full blue e =0.01,red calculation for planet Mercury, � =3.04× 10 , � =4.937× 15 e = 0.205, olive green e = 0.6, green e =0.9. 10 m, and e = 0.205. Similarly, in Table 3 we tabulate the same parameters as in Table 2 for Mercury using � =3.57× −10 15 10 and � =4.937× 10 m. In Table 4 we tabulate the ratio of the mediating of the interaction graviton mass at perihelion dE −8 and aphelion points as a function of various eccentricities = −1.013 × 10 ℎ. (74) � � /� d� using 0 = 138.326. In Figure 3 the mass ratio � �� of graviton particle mediating the Yukawa interaction as is felt from Mercury at perihelion and aphelion position is plotted, Similarly, in Table 1 looking at elliptical orbits, we consider 15 Mercury’sactual orbit with semimajor axis a =57,909,050km, as a function of eccentric anomaly for � =5.79× 10 m −8 −11 and eccentricity e = 0.205 and thus we have obtained the and � = 3.04 × 10 ,3.57×10 , respectively. Here �� values shown in Table 1 for the rate dE/d� as a function of and ��� indicate the mass of the massive graviton mediating eccentric anomaly. the interaction at the perihelion and aphelion points on the In Figure 1 the Yukawa function is plotted for elliptical orbit of the secondary. In Table 5, we tabulate the ratio of the orbits of various eccentricities, i.e., blue e = 0.01, red e = graviton mass ratios as a function of orbital eccentricities and 8 Advances in Mathematical Physics

Table 2: Tabulation of parameters ΔE = E − E0 and �0 relating to the mediating graviton mass at perihelion for a sun like star as is “felt” by −8 15 the secondary i.e. Mercury for � =3.04×10 and � =4.937×10 m, e= 0.205.

Coupling constant Eccentric Anomaly ΔE = E − E0 ∘ � � �[ ] [J] 0 24 0.0 -7.708×10 −8 24 3.04×10 45 -7.685×10 24 13.83408952 90 -7.631×10 24 180 -7.555×10 24 270 -7.593×10 24 360 -7.708×10

Table 3: Tabulation of parameters relating to the ratio of the mediating graviton mass at perihelion for a sun like star as is “felt” from the −10 15 orbiting secondary i.e. Mercury for � =3.57×10 and � =4.937× 10 m and for various values of the eccentric anomaly. ΔE = E − E Coupling constant Eccentric Anomaly 0 � � �[∘] [�] 0 22 0.0 -9.052×10 −10 22 3.57×10 45 -9.025×10 22 138.326 90 -8.961×10 22 180 -8.872×10 22 270 -8.916×10 22 360 -9.052×10

Table 4: Ratio ��/��� of the mediating the interaction graviton Moreover, since the graviton mass must be a positive number, as a function of various eccentricities as is “felt” by the orbiting (48) implies that the following equation must be satisfed: secondary body �0 = 138.326 between perihelion and aphelion 1 positions. [1 (1 − � �) � � /� 0 cos Eccentricity 0 � �� (77) 0.0 1.000000 3(1−�0 cos �) +�(− (E − E0))] = −1. 0.001 1.001660 ���� (�) ln (�/�0) 0.1 1.182141 Tis can be true if the following conditions are met for all the 0.2 1.400340 corresponding parameters separately. First from (77) solving 0.3 1.671480 for the coupling constant �, we fnd that (83) is satisfed if 0.4 2.018540 (2−�0 cos �) 3(E − E0)(1−�0 cos �) � 0.5 2.482622 �= , (78) 0.6 3.142260 ��� (�) (2 − �0 cos �) ln (�/�0) 0.7 4.170730 where � is the exponential function. For circular orbits, (78) 0.8 6.049680 takes the form 0.9 10.90870 3� (E − E0) �= . (79) 2�� (�) ln (�/�0) for �0 = 13.83408952. Using (48) we fnd that at perihelion Similarly, for the mean motion � from Eq. (77) we fnd that the mass of the graviton is given by satisfed if: 2−�0 cos � �� 3(E−E0)(1−�0 cos �)� /����(�)(2−�0 cos �) ln(�/�0) �=�0� , (80) ℏ 3(E − E )(1−� ) (75) =− [1 + � (− 0 0 )] . which for circular orbits becomes 3(E−E )�/2�� (�) ln(�/� ) �� (1 − �0) ���� (�) ln (�/�0) 0 � 0 �=�0� . (81) Similarly, at aphelion we fnd that Next, in relation to the fnal orbital energy, (77) is satisfed if

��� E = E0

−(2−�0 cos �) ℏ 3(E − E0)(1+�0) (76) ���� (�) (2 − �0 cos �) � � (82) =− [1 + � (− )] . + ln ( ), �� (1 + �0) ���� (�) ln (�/�0) 3(1−�0 cos �) �0 Advances in Mathematical Physics 9

Table 5: Ratio of the mediating the interaction particle as a function of various eccentricities as is “felt” by the orbiting secondary body for �0 = 13.83408952.

Eccentricity ��/��� ��(�) [g]��(��) [g] �0 −62 −62 0.0 1.00000 1.00000×10 1.000×10 −62 −63 0.001 1.00155 1.00155×10 9.984×10 −62 −63 0.1 1.16851 1.16851×10 8.558×10 −62 −63 0.2 1.36960 1.36959×10 7. 3 0 1 ×10 −62 −63 0.3 1.61576 1.61574×10 6.190×10 −62 −63 0.4 1.92698 1.92695×10 5.190×10 −62 −63 0.5 2.33765 2.33761×10 4.277×10 −62 −63 0.6 2.91303 2.91297×10 3.433×10 −62 −63 0.7 3.79632 3.79623×10 2.634×10 −62 −63 0.8 5.38523 5.38508×10 1.856×10 −62 −63 0.9 9.46986 9.46964×10 1.056×10

which for circular orbits takes the form and

2��� (�) � 1+�0 1 + � (138.326 (1 − �0)) E = E0 + ln ( ). (83) �� =( )[ ]��� , (89) 3� �0 1−�0 1 + � (13.326 (1 + �0))

Moreover, solving for the value of ��(�) satisfying (77) we where � is the Lambert function of the indicated argument. obtain In Table 4 we calculate and plot the variation of the mass (2−� ) of the mediating particle for the fxed values of the coupling 3(E − E )(1−� )� 0 0 0 constant alpha and range lambda of the potential as a function �� (�) = , (84) �� (2 − �0 cos �) ln (�/�0) of eccentricity e. With reference to Ohanian and Rufni [19] looking at which for circular orbits becomes (91) and referring to Yukawa gravitational potential according

3� (E − E0) to relativistic quantum theory, we say that the mass of the �� (�) = . (85) graviton is inversely proportional to the range �.Relyingon 2� ln (�/�0) 24 −62 the observational limit of �>10 cm we fnd that �� <10 −62 Next using (75) and (76) we can obtain a relation for the ratio g. Next, assuming the graviton mass to be �� =10 � of the graviton mediating the interaction at perihelion and we tabulate the corresponding mass values of graviton at aphelion correspondingly, to be perihelion and aphelion as “felt” by the orbiting Mercury.

1+�0 1+�(�0 (1 − �0)) �� =( )[ ]���, (86) 8. Conclusions 1−�0 1+�(�0 (1 + �0)) Inthispaperwehavederivedtherateofchangeofthe where the �0 is given by energy of an orbiting body with respect to mean motion in the infuence of Yukawa potential of coupling constant 3(E − E0) � � �0 =− . (87) and range .Wehavefoundthattherateofchange ���� (�) ln (�/�0) of the orbital energy of circular and elliptical orbits w.r.t. themeanmotionisalogarithmicfunctionofthemean Next calculating we calculate the orbital energy diference � −8 anomaly . Furthermore, using this expression we have ΔE = E−E0,forplanetMercuryusing� =3.04× 10 and � = 15 derived expressions for the mean anomaly as a function of the 4.937 × 10 m and for various values of the eccentric anomaly. Yukawa parameters (�, �) and the orbital energy diference We tabulate the results in Tables 2 and 3. which simplifes considerably in the case where that range Taking the numerical values of �0 into account as in of the Yukawa potential is equal to the semimajor axis, i.e., (87) we can write the mass of the graviton at perihelion and � = a. Furthermore we have found that in the case where aphelion to be the energy diference ΔE = E − E0 is negative the orbital � ΔE = E − E � mean motion increases exponentially. If 0 � is positive, then mean motion decreases exponentially. In ΔE = E − E =�(�) 1+� 1 + � (13.84208952 (1 − � )) (88) the special case where 0 � ,themean =( 0 )[ 0 ]� , motion also decreases exponentially for �>0andincreases 1−� 1 + � (13.84208952 (1 + � )) �� 0 0 for �<0. Similarly, for circular orbits we have derived 10 Advances in Mathematical Physics expressions for the Yukawa parameters � and �.Wehave [3] E. G. Adelberger, B. R. Heckel, and A. E. Nelson, “Tests of the found that � is given in terms of a Lambert function of Gravitational Inverse-Square Law,” Annual Review of Nuclear argument �(3(Δ�)/���(�) ln(�/�0)). Similarly, for circular and Particle Science,vol.53,pp.77–121,2003. �/� orbits we fnd that �=−3� Δ�/(1 + �/�)�(�)ln(�/�0). [4] R. Mukherjee and S. Sounda, “Single particle closed orbits in Analogous relations have been derived for elliptical orbits. Yukawa potential,” Indian Journal of Physics,vol.92,no.2,pp. Next expressions for the mass of graviton have been derived 197–203, 2018. in terms of the of the orbiting secondary for [5] L. Iorio, “Constraints on a Yukawa gravitational potential from circular and elliptical orbits, from which special conditions laser data of LAGEOS ,” Physics Letters A,vol.298,no. have been extracted for a positive graviton mass. We have 5-6, pp. 315–318, 2002. found that there is no graviton mass diference efect for [6] J. R. Brownstein and J. W. Mofat, “Te Bullet Cluster IE0657- circular orbits, as to what the mass of the graviton between 558 Evidence shows Modifed Gravity in the absence of Dark Matter,”vol. 23, page 3427, 2006. perihelion and aphelion is. Terefore, the strength of the gravitational interaction along the orbit propagates with the [7] J. R. Brownstein and J. W. Mofat, “Gravitational solution to the Pioneer 10/11 anomaly,” Classical and ,vol.23, help of a constant mass graviton. As the eccentricity increases, no. 10, pp. 3427–3436, 2006. the mass of the massive graviton appears to increase, being larger at perihelion where at aphelion it is always less by at [8] I. Haranas and O. Ragos, “Yukawa-type efects in dynamics,” Astrophysics and Space Science,vol.331,no.1,pp.115– least two orders of magnitude from its assumed mass, i.e., 119, 2010. � =10−62� � . Our motivation for this paper emanates from [9]I.Haranas,O.Ragos,andV.Mioc,“Yukawa-typepotential the hope that this work can add an extra possibility in the efects in the anomalistic period of celestial bodies,” Astrophysics estimation of the graviton mass. Tis will be the addition of and Space Science, vol. 332, no. 1, pp. 107–113, 2010. one more observational test in setting solar system as well � [10] A. Stavridis and C. Will, “Bounding the mass of the graviton as binary system bounds for the graviton mass �� in terms with gravitational waves: Efect of spin precessions in massive of orbital energy mean motion relations. Tis is in the case black hole binaries,” Physical Review D: Particles and Fields,vol. where the bound depends on the mass of the source, which 80, no. 4, 2009. in this case is a sun like type of star. Tus near the massive [11] J. R. Mureika and R. B. Mann, “Does entropic gravity bound the body we would expect the graviton mass to have its higher masses of the and graviton?” Modern Physics Letters A, valuewhereitslowestvaluewillbeattheaphelionpoint, vol.26,no.3,pp.171–181,2011. respectively. In other words, the strength of the interaction [12] A. Zakharov, P. Jovanovi´c, D. Borka, and V. B. Jovanovi´c, of gravity will reduce and this is something we already know. “Constraining the range of Yukawa gravity interaction from Te only diference is that now we can have an explanation for S2 star orbits III: improvement expectations for graviton mass the strength of gravitational interaction in terms of a massive bounds,” Journal of Cosmology and Astroparticle Physics,vol. graviton mass change as the new theories predict. 2018,no.04,2018. [13] I. Haranas, I. Kotsireas, G. Gomez, M. J. Fullana, and I. Gkigkitzis, “Yukawa efects on the mean motion of an orbiting Appendix body,” Astrophysics and Space Science. An International Journal of Astronomy, Astrophysics and Space Science,vol.10,2016. Te Lambert � function is defned as the inverse function � [14] H. Goldstein, C. P.Poole Jr., and J. L. Safo, Classical Mechanics, of the �| �→ �� mapping and thus solving the equation � Addison Wesley, 2002. �� =�. Tis solution is given in the form of the Lambert � �=�(�) ��� =� [15] C. D. Murray and S. F. Dermott, Solar System Dynamics, function, which according to the relation Cambridge University Press, Cambridge, UK, 1999. �(�)��(�) =� satisfes the following equation: [20]. Te [16] J. W. Mofat and V. T. Toth, “Fundamental parameter-free function is also known as Omega or Product Log function. solutions in modifed gravity,” Classical and Quantum Gravity, vol. 26, no. 8, 2009. Data Availability [17] E. Fischbach and C. L. Talmadge, Te Search for Non-Newtonian Gravity,Springer,Berlin,Germany,1999. Te data used to support the fndings of this study are [18] N. C. Branco, J. Paramos, and R. 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