Perturbations of Keplerian Orbits in Stationary Spherically Symmetric Spacetimes
aa r X i v : . [ g r- q c ] M a r Perturbations of Keplerian Orbits in Stationary Spherically Symmetric Spacetimes
Matteo Luca Ruggiero ∗ DISAT - Dipartimento di Scienza Applicata e Tecnologia,Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino, Italy andINFN, Sezione di Torino, Via Pietro Giuria 1, Torino, Italy (Dated: November 4, 2018)We study spherically symmetric perturbations determined by alternative theories of gravity to thegravitational field of a central mass in General Relativity. In particular, we focus on perturbationsin the form of power laws and calculate their effect on the secular variations of the orbital elementsof a Keplerian orbit. We show that, to lowest approximation order, only the argument of pericentreand mean anomaly undergo secular variations; furthermore, we calculate the variation of the orbitalperiod. We give analytic expressions for these variations which can be used to constrain the impactof alternative theories of gravity.
I. INTRODUCTION
In recent years, there has been much interest in studying theories of gravity alternative to General Relativity(GR) [1–4]: these theories have been motivated by the current observations, which seem to question the GR modelof gravitational interactions on large scales, such as the galactic and cosmological ones. Just to mention someexamples, one can consider f ( R ) theories of gravity [5–7], MOdified Newtonian Dynamics (MOND) [8], f ( T ) gravity[9], braneworld models [10], the relativistic MOdified Gravity theory (MOG) [11, 12], curvature invariant models[13, 14], Hoˇrava-Lifshitz gravity [15–17].In order to get a deeper insight into these theories, it is important to test their predictions in a suitable weak-fieldand slow-motion limit: to this end, for instance, central mass solutions which generalize the one obtained in GR (theSchwarzschild solution) are investigated and their predictions compared to the data available from astronomical andastrophysical observations, in the Solar System and beyond. For these purposes, a general theoretical frameworkthat can deal with classes of metric theories of gravity has been developed: the parametrized post-Newtonian (PPN)formalism. In the latter, the weak-field and slow-motion limit of these theories is studied in terms of suitable param-eters, in such a way that experiments may fix their values: as a matter of fact, the current best estimates of the PPNparameters are in agreement with GR [18]. As a consequence, it is reasonable to believe that the effects of theoriesalternative to GR are small, so that they can be dealt with as perturbations of the GR background.In this paper we want to suggest a simple approach that, without requiring the complex and comprehensive PPNframework, allows to evaluate the predictions of alternative theories of gravity on the Keplerian orbit of a test particle,which can be thought of as a simplified model of the dynamics of celestial bodies in planetary systems. In particular,we consider a general stationary and spherically symmetric (SSS) spacetime metric, that can be thought of a solutionof the field equations of a generic alternative theory of gravity describing the gravitational field around a point-likecentral mass, and we work out the perturbations of the Keplerian orbital elements. We focus on perturbations inform of power laws, which have been recently considered in the literature [19–28]: these perturbations are interestingbecause they reproduce some known solutions of the field equations of alternative theories of gravity; moreoverarbitrary spherically symmetric perturbations can be written in terms of power series, so that our results can be usedquite generally. In particular, by using the Gauss perturbation scheme, we obtain, to lowest approximation order,the expressions of the secular variations of the orbital elements and the orbital period in terms of hypergeometricfunctions. These expressions can be used to place bounds on the parameters of alternative theories of gravity.The paper is organized as follows: in Section II we write the perturbing acceleration in a generic SSS spacetimeand obtain its expression to lowest approximation order, while the Gauss perturbation equations are briefly reviewedin Section III. In Section IV we focus on perturbations in the form of power laws, while conclusions are in Section V. ∗ Electronic address: [email protected]
II. THE PERTURBING ACCELERATION
We suppose that the gravitational field of a point mass in a generic alternative theory of gravity is described bythe SSS spacetime metric ds = (1 + φ ( r )) dt − (1 + ψ ( r )) (cid:0) dr + r d Ω (cid:1) (1)where d Ω = dθ + sin θdϕ , and φ ( r ) , ψ ( r ) are functions depending on the mass M of the source and, possibly, onother parameters of the theory.We point out that, since we are considering spherically symmetric spacetime, gravitomagnetic effects [29] areintentionally neglected: in other words in calculating the test particle equation of motion (see equation (5) below), wedo not take into account the perturbing acceleration deriving from the off-diagonal element of the spacetime metric(see e.g. [30–32] for gravitomagnetic perturbations of the orbital elements)The metric (1) is written in isotropic polar coordinates, such that the spatial part of the metric is proportionalto the flat spacetime metric dr + r d Ω = dx + dy + dz . Consequently, it is possible to write the metric (1) inCartesian coordinates in the form ds = (1 + φ ( r )) dt − (1 + ψ ( r )) (cid:0) dx + dy + dz (cid:1) (2)where r = p x + y + z .Since the effects of the generic alternative theory of gravity are expected to be small, we suppose that they can beconsidered as perturbations of the known GR solution, i.e. the Schwarzschild spacetime. This amounts to saying that φ ( r ) and ψ ( r ) must approach their GR values: in other words we suppose that in a suitable limit, the metric takesthe form φ ( r ) = φ GR ( r ) + φ A ( r ) (3) ψ ( r ) = ψ GR ( r ) + ψ A ( r ) (4)where the GR values are given by φ GR ( r ) = − M/r + 2 M /r , ψ GR ( r ) = 2 M/r (see e.g. [33]) and the perturbations φ A ( r ) , ψ A ( r ) due to the alternative gravity model are such that φ A ( r ) ≪ φ GR ( r ), ψ A ( r ) ≪ ψ GR ( r ).In order to calculate the perturbations of the orbital elements, we must calculate the perturbing acceleration. Tothis end, first we assume that in the given theory, the matter is minimally and universally coupled, so that testparticles follow geodesics of the metric (2) (or, equivalently (1)). Then, we consider the (post-Newtonian) equationof motion of a test particle (see [34])¨ x i = − h ,i − h ik h ,k + h ,k ˙ x k ˙ x i + (cid:18) h ik,m − h km,i (cid:19) ˙ x k ˙ x m (5)where “dot” stands for derivative with respect to the coordinate time. Since in our notation it is h = φ ( r ) and h ij = ψ ( r ) δ ij , we can write the perturbing acceleration W in the form W = − (cid:8) Φ( r ) [1 + ψ A ( r )] + Ψ( r ) v (cid:9) ˆ x + [Φ( r ) + Ψ( r )] (ˆ x · v ) v (6)where we set x = ( x, y, z ), v = ( ˙ x, ˙ y, ˙ z ), ˆ x = x / | x | , andΦ( r ) . = dφ A ( r ) dr , Ψ( r ) . = dψ A ( r ) dr (7)We aim at investigating the lowest order effects on planetary motion of φ A ( r ) and ψ A ( r ): to this end, it is sufficientto apply the Gauss perturbation scheme to a Keplerian ellipse.Moreover, since we are interested in the lowest order effects, we may also neglect the non linear terms (i.e. nonlinear perturbations with respect to flat spacetime). To this end, we start by noticing that to Newtonian order, it is v ≃ φ GR ( r ) ≃ M/r . As a consequence, in (6) we may neglect the terms proportional to v and to (ˆ x · v ) v (which If not otherwise stated, we use units such that c = G = 1; bold face letters like x refer to spatial vectors while Latin indices refer tospatial components. is also proportional to the orbital eccentricity) and also the term Φ( r ) ψ A ( r ). In summary, in the weak-field andslow-motion limit the perturbing acceleration that we are going to consider is purely radial and is given by W = −
12 Φ( r )ˆ x . = W r ˆ x (8)We point out that even though we started from the study of a SSS spacetime with the aim of setting bounds onalternative theories of gravity, what follows can be applied to generic radial perturbations (in particular, to those inform of a power law) of the orbital elements of a Keplerian motion. III. PERTURBATION EQUATIONS
We start from the expression (8) of the acceleration and apply the Gauss perturbation scheme to obtain the secularvariations. Because of spherical symmetry, the motion of test particles is confined to a plane and, in this plane, theunperturbed Keplerian ellipse is r = a (cid:0) − e (cid:1) e cos f (9)where a is the semimajor axis, e the eccentricity, f the true anomaly describing the particle angular distance from thepericentre. For purely radial perturbations, the Gauss equations for the variations of the Keplerian orbital elementsread [35] dadt = 2 en √ − e W r sin f, (10) dedt = √ − e na W r sin f, (11) dIdt = 0 , (12) d Ω dt = 0 , (13) dωdt = − √ − e nae W r cos f, (14) d M dt = n − na W r (cid:16) ra (cid:17) − p − e dωdt , (15)where M is mean anomaly, I is the orbital inclination, Ω is the ascending node, ω is the argument of pericentre, n = p M/a is the Keplerian mean motion. In a Keplerian orbit the mean anomaly is a linear function of time definedby M = n ( t − t ), where t is the time of a passage through pericentre. The orbital period P b is related to the meanmotion by n = 2 π/P b .In order to obtain the secular effects we must evaluate the Gauss equations onto the unperturbed Keplerian ellipse(9), and then we must average them over one orbital period of the test particle. From (10) and (11), and takinginto account (9), we see that the secular variations of the semimajor axis and eccentricity are null, because whenaveraging them the arguments of the integrals are odd functions. Hence, we obtain that in SSS spacetimes, tolowest approximation order (i.e. when the perturbations are purely radial) semimajor axis, eccentricity, beside theinclination and the node (which are only affected by normal i.e. out-of the-plane perturbations), are not affectedby secular variations. We point out that, once that the secular variations of the argument of the pericentre < ˙ ω > and mean anomaly < ˙ M > have been determined, it is possible to obtain the corresponding variation of the meanlongitude < ˙ λ > . = < ˙ ω > + < ˙ M > [35].Starting from the equation describing the variation of the mean anomaly, it is possible to evaluate the perturbationof the orbital period. In fact (see [36]), on using df = (cid:16) ar (cid:17) p − e d M (16)from (15) it is possible to write dfdt = n (cid:16) ar (cid:17) p − e (cid:20) − n a W r (cid:16) ra (cid:17) + 1 − e n ae W r cos f (cid:21) (17)We aim at performing a first order calculation in the perturbation, which is supposed to be small (see e.g. [34, 37, 38]).As a consequence, first we may write dtdf ≃ (cid:16) ra (cid:17) n √ − e (cid:20) n a W r (cid:16) ra (cid:17) − − e n ae W r cos f (cid:21) (18)and then, by integrating over one revolution along the unperturbed orbit (9) we obtain P ≃ P b + P A (19)where P b . = Z π (cid:16) ra (cid:17) dfn √ − e = 2 πn (20)is the unperturbed period, and P A . = Z π (cid:16) ra (cid:17) (cid:20) n a W r (cid:16) ra (cid:17) − − e n ae W r cos f (cid:21) dfn √ − e (21)is the variation of the orbital period due to the perturbation.The above equations (14), (15) and (21) allow to calculate the variations of the argument of pericentre, meananomaly and orbital period. This can be accomplished at least by numerical methods for arbitrary perturbations,however, as we are going to show in next Section, when the perturbation is in the form of a power law, it is possibleto obtain analytic expressions. IV. POWER LAWS PERTURBATIONS
After having discussed in the previous Section the expressions for the secular perturbations of the orbital elementsfor a generic purely radial perturbation, here we focus on the particular case of perturbations in the form of a powerlaw. As we are going to show, in this case we can obtain analytic expressions for the secular variations. We pointout that an arbitrary spherical symmetric perturbation that is expressed by an analytic function can be expanded inpower series up to the required approximation level: hence, by knowing the contribution of each term of the series,it is possible to evaluate the whole effect of the perturbation, within the required accuracy. Eventually, even thoughour approach is motivated by the study of the effects of alternative theories of gravity, the results that we are goingto obtain are quite general, and apply to radial perturbations of Keplerian orbits by means of arbitrary power laws.In particular, we consider two kinds of perturbation that we write as follows. Given a constant α , which is aparameter deriving from the gravity model alternative to GR, the perturbations that we focus on are in the form:(i) φ A ( r ) = αr N where the integer N is such that N ≤ −
1, or differently speaking φ A ( r ) = αr | N | , | N | ≥
1; (ii) φ A ( r ) = αr N where the integer N is such that N ≥
1. The perturbations of the first kind are asymptoticallyflat, while the second ones are not: in the latter case, we assume that there exists a range 0 < r < ¯ r for which φ A ( r ) ≪ φ GR ( r ), and our results are valid within this range. We notice that, for a logarithmic perturbation in theform φ A ( r ) = β log( r/r ), it is W r = − βr , which can be dealt with as in the case (i) above.In what follows we perform first-order calculations in the perturbation to obtain the secular variations; the lattercan be used to place constraints on α by means of a comparison with the available data. A. Perturbation of the argument of pericentre
For perturbations in the form φ A ( r ) = αr | N | , | N | ≥
1, we start from eq. (14), and we average it over one orbitalperiod by taking into account the expression of the unperturbed orbit (9) and the relation [39] dt = (1 − e ) / n (1 + e cos f ) df (22)On using (A3), we then obtain < ˙ ω > = − πα | N | ( | N | − (cid:0) − e (cid:1) −| N | n a | N | F (cid:18) − | N | , − | N | , , e (cid:19) (23)where F is the hypergeometric function (see Appendix A). In particular, this result is in agreement with the corre-spondent expression found in [19], where the effects of a central force were considered.On the other hand, for perturbations in the form φ A ( r ) = αr N with N ≥
1, we start again from (14) but, in orderto average it over one orbital, it is useful to introduce the expression of the unperturbed orbit r = a (1 − e cos E ) (24)in terms of the eccentric anomaly E , which is is related to the true anomaly f by the following relations:cos f = cos E − e − e cos E , sin f = √ − e sin E − e cos E . (25)On using the relation [39] dt = 1 n (1 − e cos E ) dE (26)and the integrals (A3) and (A4), we obtain < ˙ ω > = − παN √ − e n a − N (cid:20) ( N − F (cid:18) − N , − N , , e (cid:19) + 2 F (cid:18) − N , − N , , e (cid:19)(cid:21) (27)or, equivalently < ˙ ω > = − παN ( N + 1) √ − e n a − N F (cid:18) − N , − N , , e (cid:19) (28)In particular, eq. (28) is in agreement with the correspondent expression found in [19].These general expressions can be used to reproduce known results. For instance, on setting N = 1 in eqs (28) weobtain the variation of pericentre due to a constant perturbation acceleration < ˙ ω > = − παn a p − e (29)This result is in agreement with Sanders [40] (also reported by [19]), who, working in MOND framework, consideredthe case of the constant acceleration to put constraints on the effects of anomalous acceleration by analyzing Mercury’sadvance of perihelion.The perturbation in the form φ A ( r ) = αr coincides with the effect of a cosmological constant Λ [41],[42] in GR,while in the case of f ( R ) gravity, it describes the vacuum solution in Palatini formalism [43, 44]. In this case, bysetting N = 2 in eq. (28) we get < ˙ ω > = − παn p − e (30)On setting α = − Λ and using the relation n a = M which holds for the unperturbed orbit, eq. (30) becomes < ˙ ω > = π Λ a M p − e (31)in agreement with [41, 44–46].On setting | N | = 4 in (23), we obtain the perturbation determined by a vacuum solution of Hoˇrava-Lifshitz gravity[47, 48]; by approximating the hypergeometric function, the variation of the argument of pericentre becomes < ˙ ω > ≃ − παn a (cid:0) e (cid:1) (1 − e ) (32)in agreement with [47].If we consider a logarithmic perturbation, in the form φ A ( r ) = β log( r/r ), from eq. (23), by approximation up to e we obtain < ˙ ω > = − πβ (cid:0) − e (cid:1) n a F (cid:18) , , , e (cid:19) ≃ − πβ (cid:0) − e (cid:1) n a (cid:18) − e (cid:19) (33)in agreement with [19].Eventually, a perturbation in the form φ A ( r ) = αr is obtained both in the case of Reissner-Nordstr¨om spacetimes[49] and in recent works pertaining to the constraints for f ( T ) gravity deriving from the Solar System observations[50]. On setting | N | = 2 in (23), we do obtain < ˙ ω > = − παn a − e ) (34)in agreement with [49, 50]. B. Perturbation of the mean anomaly
We may proceed as in the previous section to calculate the secular variation of the mean anomaly. However, wepoint out it is hard to use the mean anomaly in observational tests, because of the unavoidable uncertainty arisingfrom the Keplerian mean motion n .For a perturbation in the form φ A ( r ) = αr | N | , | N | ≥
1, we start from eq. (15), and focus on the part that isproportional to W r ; after averaging over the unperturbed ellipse (9) making use of eqs. (22) and (A4), we get < ˙ M > = − πα | N | (cid:0) − e (cid:1) / −| N | n a | N | F (cid:18) − | N | , − | N | , , e (cid:19) − p − e < ˙ ω > (35)where < ˙ ω > is given by (23).Similarly, for perturbations in the form φ A ( r ) = αr N , with N ≥
1, averaging the part of (15) which depens on W r ,making use of (26) and (A4), we obtain < ˙ M > = 2 παNn a − N F (cid:18) − N , − − N , , e (cid:19) − p − e < ˙ ω > (36)where < ˙ ω > is given by (28).On setting N = 2, from eq. (36), by approximating the hypergeometric function, the variation of the mean anomalyis < ˙ M > = 4 παn F (cid:18) − , − , , e (cid:19) + 3 παn (cid:0) − e (cid:1) ≃ παn (cid:18)
73 + e (cid:19) (37)in agreement with [44]. C. Perturbation of the orbital period
Starting from eq. (21), we may calculate the variation of the orbital period. In particular, for a perturbation inthe form φ A ( r ) = αr | N | , | N | ≥
1, on evaluating eq. (21) over the unperturbed ellipse (9) making use of the integrals(A3), (A4), we obtain P A = πα | N | (cid:0) − e (cid:1) / −| N | n a | N | (cid:20) F (cid:18) − | N | , − | N | , , e (cid:19) −
12 ( | N | − F (cid:18) − | N | , − | N | , , e (cid:19)(cid:21) (38)As for perturbations in the form φ A ( r ) = αr N with N ≥
1, on evaluating eq. (21) over the unperturbed ellipse (24)making use of the relation (16) and the integrals (A3), (A4), we obtain P A = − παNn a − N " F (cid:18) − N , − − N , , e (cid:19) + (cid:0) − e (cid:1) N + 1) F (cid:18) − N , − N , , e (cid:19) (39)On setting | N | = 4 , from eq. (38), by approximating the hypergeometric functions, we obtain P A ≃ πα (cid:0) − e (cid:1) − / n a (cid:18) e (cid:19) (40)in agreement with [48].Furthermore, on setting N = 2 in (39), by approximation of the hypergeometric functions, the perturbation of theorbital period turns out to be P A ≃ − παn (cid:0) e (cid:1) (41)in agreement with [42]. V. CONCLUSIONS
We considered a general stationary and spherically symmetric spacetime, and worked out the perturbations deter-mined by a generic alternative theory of gravity to the GR solution describing the gravitational field around a centralmass. In order to evaluate the effects of such perturbations, we showed that, in the weak-field and slow-motion limit,to lowest approximation order, these effects can be described by a purely radial acceleration. Then, we considered theKeplerian orbit of a test particle, which can be thought of as a model of the dynamics of celestial bodies in planetarysystems, and we evaluated the impact of such perturbations on the orbital elements.In particular, we obtained analytic expressions, in terms of hypergeometric functions, for the secular variationsof the advance of pericentre, mean anomaly and orbital period determined by perturbations in form of power laws;the other orbital elements do not undergo secular variations to lowest approximation order. The expressions for thevariation of the argument of pericentre are in agreement with the results recently obtained, pertaining to the orbitalprecession due to central force perturbations.Our results are quite general, since even though were motivated by the study of the effect of alternative theoriesof gravity, they can be applied to arbitrary radial power laws perturbations of the orbital elements of a Keplerianmotion.Spherically symmetric perturbations in the form of a power law were obtained both in GR, for instance when theeffect of a cosmological constant is considered, or in Reissner-Nordstr¨om spacetimes, and in alternative theories ofgravity; we have shown that our results are in agreement with those already available in the literature. For an arbitraryperturbation it is possible to obtain the secular variations by means of numerical approaches, or by expanding it inpower series and applying our results to each term of the series.The possibility of testing the predictions of alternative theories of gravity on planetary motion is important toverify their reliability on scales different from those typical of galactic dynamics or cosmology, where they are usuallytested. The simple approach that we considered here allows to place bounds on the theory parameters, by evaluatingtheir impact on the Keplerian dynamics and allowing a comparison with the available data. In this context, it is veryimportant the development in the study of Solar System ephemerides, due to the works of Russian [51, 52] and French[53] research teams.
Appendix A: Solutions of Integrals by means of Hypergeometric Functions
In order to evaluate the secular variations of the Keplerian elements and of the orbital period determined by theperturbations in the form of power laws that we have considered, it is necessary to solve the following integrals I N = Z π cos u [1 + e cos u ] N du (A1) L N = Z π [1 − e cos u ] N du (A2)They can be solved by subsituting cos u = z , and then using the binomial theorem for (1 + z ) N . The solutions aregiven in terms of hypergeometric functions: I N = πN eF (cid:18) − N , − N , , e (cid:19) (A3) L N = 2 πF (cid:18) − N , − N , , e (cid:19) (A4)where F ( a, b, c, x ) . = F ( a, b, c, d ) defined by (see e.g. [54]) F ( a, b, c, x ) = ∞ X n =0 ( a ) n ( b ) n ( c ) n x n n ! (A5)where the Pochhammer symbol ( a ) n is defined by( a ) n . = ( a + n − a − a ) . = 1 (A7) [1] S. Tsujikawa, in Lectures on Cosmology Accelerated Expansion of the Universe by Georg Wolschin , Lecture Notes in Physics , 99 (2010)[2] V. H. Satheeshkumar, P. K. Suresh,
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