Perturbations of Keplerian Orbits in Stationary Spherically Symmetric

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II. THE PERTURBING ACCELERATION

We suppose that the gravitational ﬁeld of a point mass in a generic alternative theory of gravity is described by the SSS spacetime metric1

ds2 = (1+ φ(r)) dt2 (1 + ψ(r)) dr2 + r2dΩ2 (1) − where dΩ2 = dθ2 + sin2 θdϕ2, and φ(r), ψ(r) are functions depending on the mass M of the source and, possibly, on other parameters of the theory. We point out that, since we are considering spherically symmetric spacetime, gravitomagnetic effects [29] are intentionally neglected: in other words in calculating the test particle equation of motion (see equation (5) below), we do 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 proportional to the flat spacetime metric dr2 + r2dΩ2 = dx2 + dy2 + dz2. Consequently, it is possible to write the metric (1) in Cartesian coordinates in the form

ds2 =(1+ φ(r)) dt2 (1 + ψ(r)) dx2 + dy2 + dz2 (2) − where r = x2 + y2 + z2. Since the eﬀects of the generic alternative theory of gravity are expected to be small, we suppose that they can be consideredp 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 takes the form

φ(r) = φGR(r)+ φA(r) (3)

ψ(r) = ψGR(r)+ ψA(r) (4) where the GR values are given by φ (r)= 2M/r +2M 2/r2, ψ (r)=2M/r (see e.g. [33]) and the perturbations GR − GR φA(r), ψA(r) due to the alternative gravity model are such that φA(r) φGR(r), ψA(r) ψGR(r). In order to calculate the perturbations of the orbital elements, we must≪ calculate the≪ perturbing acceleration. To this end, ﬁrst we assume that in the given theory, the matter is minimally and universally coupled, so that test particles follow geodesics of the metric (2) (or, equivalently (1)). Then, we consider the (post-Newtonian) equation of motion of a test particle (see [34])

i 1 1 k i 1 k m x¨ = h00 h h00 + h00 x˙ x˙ + h h x˙ x˙ (5) −2 ,i − 2 ik ,k ,k ik,m − 2 km,i where “dot” stands for derivative with respect to the coordinate time. Since in our notation it is h00 = φ(r) and hij = ψ(r)δij , we can write the perturbing acceleration W in the form 1 W = Φ(r)[1+ ψ (r)] + Ψ(r)v2 xˆ + [Φ(r)+Ψ(r)] (xˆ v) v (6) −2 A · where we set x = (x,y,z), v = (x, ˙ y,˙ z˙), xˆ = x/ x , and | |

. dφA(r) . dψA(r) Φ(r) = , Ψ(r) = (7) dr dr

We aim at investigating the lowest order effects on planetary motion of φA(r) and ψA(r): to this end, it is sufficient to 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. non linear perturbations with respect to flat spacetime). To this end, we start by noticing that to Newtonian order, it is v2 φ (r) M/r. As a consequence, in (6) we may neglect the terms proportional to v2 and to (xˆ v) v (which ≃ GR ≃ ·

1 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 to spatial components. 3 is also proportional to the orbital eccentricity) and also the term Φ(r)ψA(r). In summary, in the weak-ﬁeld and slow-motion limit the perturbing acceleration that we are going to consider is purely radial and is given by 1 . W = Φ(r)xˆ = W xˆ (8) −2 r We point out that even though we started from the study of a SSS spacetime with the aim of setting bounds on alternative theories of gravity, what follows can be applied to generic radial perturbations (in particular, to those in form 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 secular variations. Because of spherical symmetry, the motion of test particles is confined to a plane and, in this plane, the unperturbed Keplerian ellipse is a 1 e2 r = − (9) 1+ e cos f where a is the semimajor axis, e the eccentricity, f the true anomaly describing the particle angular distance from the pericentre. For purely radial perturbations, the Gauss equations for the variations of the Keplerian orbital elements read [35] da 2e = Wr sin f, (10) dt n√1 e2 − de √1 e2 = − W sin f, (11) dt na r dI = 0, (12) dt dΩ = 0, (13) dt dω √1 e2 = − W cos f, (14) dt − nae r d 2 r dω M = n W 1 e2 , (15) dt − na r a − − dt p where is mean anomaly, I is the orbital inclination, Ω is the ascending node, ω is the argument of pericentre, n = M/aM 3 is the Keplerian mean motion. In a Keplerian orbit the mean anomaly is a linear function of time defined by = n (t t0), where t0 is the time of a passage through pericentre. The orbital period P is related to the mean Mp − b motion by n =2π/Pb. 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 taking into account (9), we see that the secular variations of the semimajor axis and eccentricity are null, because when averaging them the arguments of the integrals are odd functions. Hence, we obtain that in SSS spacetimes, to lowest approximation order (i.e. when the perturbations are purely radial) semimajor axis, eccentricity, beside the inclination and the node (which are only affected by normal i.e. out-of the-plane perturbations), are not affected by secular variations. We point out that, once that the secular variations of the argument of the pericentre < ω˙ > and mean anomaly < ˙ > have been determined, it is possible to obtain the corresponding variation of the mean . longitude < λ>˙ =< ω˙ >M+ < ˙ > [35]. Starting from the equation describingM the variation of the mean anomaly, it is possible to evaluate the perturbation of the orbital period. In fact (see [36]), on using a 2 df = 1 e2d (16) r − M p from (15) it is possible to write

df a 2 2 r 1 e2 = n 1 e2 1 W + − W cos f (17) dt r − − n2a r a n2ae r p 4

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 dt r 2 1 2 r 1 e2 1+ Wr − Wr cos f (18) df ≃ a n√1 e2 n2a a − n2ae − and then, by integrating over one revolution along the unperturbed orbit (9) we obtain

P P + P (19) ≃ b A where 2 . π r 2 df 2π Pb = 2 = (20) 0 a n√1 e n Z − is the unperturbed period, and

2 . π r 2 2 r 1 e2 df P = W W cos f (21) A 2 r −2 r 2 0 a n a a − n ae n√1 e Z − 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, mean anomaly 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 possible to obtain analytic expressions.

IV. POWER LAWS PERTURBATIONS

After having discussed in the previous Section the expressions for the secular perturbations of the orbital elements for a generic purely radial perturbation, here we focus on the particular case of perturbations in the form of a power law. As we are going to show, in this case we can obtain analytic expressions for the secular variations. We point out that an arbitrary spherical symmetric perturbation that is expressed by an analytic function can be expanded in power 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 though our approach is motivated by the study of the effects of alternative theories of gravity, the results that we are going to 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 a parameter deriving from the gravity model alternative to GR, the perturbations that we focus on are in the form: N α (i) φA(r) = αr where the integer N is such that N 1, or differently speaking φA(r) = r|N| , N 1; (ii) N ≤ − | | ≥ φA(r) = αr where the integer N is such that N 1. The perturbations of the first kind are asymptotically flat, while the second ones are not: in the latter case,≥ we assume that there exists a range 0

A. Perturbation of the argument of pericentre

α For perturbations in the form φA(r) = r|N| , N 1, we start from eq. (14), and we average it over one orbital period by taking into account the expression of the| |≥ unperturbed orbit (9) and the relation [39]

(1 e2)3/2 dt = − df (22) n(1 + e cos f)2 On using (A3), we then obtain

1−|N| 1 πα N ( N 1) 1 e2 N 3 N < ω>˙ = | | | |− − F 1 | |, | |, 2,e2 (23) −2 n2a2+|N| − 2 2 − 2 5 where F is the hypergeometric function (see Appendix A). In particular, this result is in agreement with the correspondent expression found in [19], where the eﬀects of a central force were considered. N On the other hand, for perturbations in the form φA(r)= αr with N 1, we start again from (14) but, in order to 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 E e √1 e2 sin E cos f = − , sin f = − . (25) 1 e cos E 1 e cos E − − On using the relation [39] 1 dt = (1 e cos E) dE (26) n − and the integrals (A3) and (A4), we obtain

1 παN√1 e2 N 3 N N 1 N < ω>˙ = − (N 1) F 1 , , 2,e2 +2F 1 , , 1,e2 (27) −2 n2a2−N − − 2 2 − 2 − 2 2 − 2 or, equivalently

1 παN (N + 1) √1 e2 N 1 N < ω˙ >= − F 1 , , 2,e2 (28) −2 n2a2−N − 2 2 − 2 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) we obtain the variation of pericentre due to a constant perturbation acceleration πα < ω>˙ = 1 e2 (29) −n2a − p This result is in agreement with Sanders [40] (also reported by [19]), who, working in MOND framework, considered the case of the constant acceleration to put constraints on the eﬀects of anomalous acceleration by analyzing Mercury’s advance of perihelion. 2 The perturbation in the form φA(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, by setting N =2 in eq. (28) we get 3πα < ω>˙ = 1 e2 (30) − n2 − p On setting α = 1 Λ and using the relation n2a3 = M which holds for the unperturbed orbit, eq. (30) becomes − 3 πΛa3 < ω˙ >= 1 e2 (31) M − p 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

1 2 6πα 1+ 4 e < ω˙ > 3 (32) ≃−n2a6 (1 e2) − in agreement with [47]. If we consider a logarithmic perturbation, in the form φA(r)= β log(r/r0), from eq. (23), by approximation up to e4 we obtain 1 πβ 1 e2 3 1 πβ 1 e2 1 < ω˙ >= − F 1, , 2,e2 − 1 e2 (33) −2 n2a2 2 ≃−2 n2a2 − 4 6 in agreement with [19]. α Eventually, a perturbation in the form φA(r) = r2 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 | | πα 1 < ω˙ >= (34) −n2a4 (1 e2) − 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, we point out it is hard to use the mean anomaly in observational tests, because of the unavoidable uncertainty arising from the Keplerian mean motion n. α For a perturbation in the form φ (r) = N , N 1, we start from eq. (15), and focus on the part that is A r| | | | ≥ proportional to Wr; after averaging over the unperturbed ellipse (9) making use of eqs. (22) and (A4), we get

3/2−|N| 2πα N 1 e2 1 N N ˙ | | − 2 2 < >= 2 2+|N| F | |, | |, 1,e 1 e < ω˙ > (35) M − n a 2 − 2 − 2 − − p where < ω>˙ is given by (23). N Similarly, for perturbations in the form φA(r)= αr , with N 1, averaging the part of (15) which depens on Wr, making use of (26) and (A4), we obtain ≥ 2παN N 1 N < ˙ >= F , , 1,e2 1 e2 < ω˙ > (36) M n2a2−N − 2 −2 − 2 − − p where < ω>˙ is given by (28).

On setting N = 2, from eq. (36), by approximating the hypergeometric function, the variation of the mean anomaly is 4πα 3 3πα 3πα 7 < ˙ >= F 1, , 1,e2 + 1 e2 + e2 (37) M n2 − −2 n2 − ≃ n2 3 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 in α the form φA(r)= r|N| , N 1, on evaluating eq. (21) over the unperturbed ellipse (9) making use of the integrals (A3), (A4), we obtain | |≥

3/2−|N| πα N 1 e2 N 3 N 1 N 3 N P = | | − 2F 1 | |, | |, 1,e2 ( N 1) F 1 | |, | |, 2,e2 (38) A n3a2+|N| − 2 2 − 2 − 2 | |− − 2 2 − 2 N As for perturbations in the form φA(r)= αr 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

παN N 1 N 1 e2 N 1 N P = 2F , , 1,e2 + − (N + 1) F 1 , , 2,e2 (39) A −n3a2−N − 2 −2 − 2 2 − 2 2 − 2 " # On setting N = 4 , from eq. (38), by approximating the hypergeometric functions, we obtain | | −5/2 πα 1 e2 5 P − 2+ e2 (40) A ≃ n3a6 2 7 in agreement with [48]. Furthermore, on setting N = 2 in (39), by approximation of the hypergeometric functions, the perturbation of the orbital period turns out to be πα P 7+3e2 (41) A ≃− n3 in agreement with [42].

V. CONCLUSIONS

We considered a general stationary and spherically symmetric spacetime, and worked out the perturbations determined by a generic alternative theory of gravity to the GR solution describing the gravitational field around a central mass. 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 the Keplerian orbit of a test particle, which can be thought of as a model of the dynamics of celestial bodies in planetary systems, 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 variations of 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 the variation of the argument of pericentre are in agreement with the results recently obtained, pertaining to the orbital precession due to central force perturbations. Our results are quite general, since even though were motivated by the study of the effect of alternative theories of gravity, they can be applied to arbitrary radial power laws perturbations of the orbital elements of a Keplerian motion. Spherically symmetric perturbations in the form of a power law were obtained both in GR, for instance when the effect of a cosmological constant is considered, or in Reissner-Nordström spacetimes, and in alternative theories of gravity; we have shown that our results are in agreement with those already available in the literature. For an arbitrary perturbation it is possible to obtain the secular variations by means of numerical approaches, or by expanding it in power 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 to verify their reliability on scales different from those typical of galactic dynamics or cosmology, where they are usually tested. The simple approach that we considered here allows to place bounds on the theory parameters, by evaluating their impact on the Keplerian dynamics and allowing a comparison with the available data. In this context, it is very important 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 the perturbations in the form of power laws that we have considered, it is necessary to solve the following integrals

2π N IN = cos u [1 + e cos u] du (A1) Z0

2π L = [1 e cos u]N du (A2) N − Z0 They can be solved by subsituting cos u = z, and then using the binomial theorem for (1 + z)N . The solutions are given in terms of hypergeometric functions: 1 N N I = πNeF , 1 , 2,e2 (A3) N 2 − 2 − 2

N 1 N L =2πF , , 1,e2 (A4) N − 2 2 − 2 8

. where F (a,b,c,x) =2 F1(a,b,c,d) deﬁned by (see e.g. [54])

∞ n (a)n(b)n x 2F1(a,b,c,x)= (A5) (c) n! n=0 n X where the Pochhammer symbol (a)n is deﬁned by . (a + n 1)! (a) = − (A6) n (a 1)! . − (a)0 = 1 (A7)

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