Motion of a Test Particle According to the Scalar Ether Theory of Gravitation and Application to Its Celestial Mechanics

Z. Naturforsch. 2019; aop

Mayeul Arminjon* and Rainer Wolfgang Winkler Motion of a Test Particle According to the Scalar Ether Theory of Gravitation and Application to its Celestial Mechanics https://doi.org/10.1515/zna-2018-0470 1 Introduction and Summary Received October 17, 2018; accepted December 9, 2018 There are two empirically indistinguishable interpreta- Abstract: The standard interpretations of special relativ- tions of special relativity: the Lorentz–Poincaré (L-P) inter- ity (Einstein–Minkowski) and general relativity (GR) lead pretation, which sees the relativistic effects as following to a drastically changed notion of time: the eternalism from the “true” Lorentz contraction of all objects in their or block universe theory. This has strong consequences motion through the ether, and the standard (Einstein– for our thinking about time and for the development of Minkowski) interpretation [1, 2]. The latter is currently pre- new fundamental theories. It is therefore important to ferred, despite the fact that the L-P interpretation is com- check this thoroughly. The Lorentz–Poincaré interpreta- patible with the classical notion of time and has definite tion, which sees the relativistic effects as following from advantages regarding the physical understandability, in a “true” Lorentz contraction of all objects in their motion particular regarding the causal explanation of the rela- through the ether, uses a conservative concept of time tivistic effects such as time dilation [1, 3]. (The L-P interpre- and is in the absence of gravitation indistinguishable from tation of special relativity is also known under the name the standard interpretation; but there exists currently no “Lorentz ether theory,” which we consider inappropriate accepted gravitation theory for it. The scalar ether theory because it hides (i) the decisive contributions of Poincaré of gravitation is a candidate for such a theory; it is pre- and (ii) the full compatibility of this interpretation with the sented and discussed. The equations of motion for a test physics of special relativity.) An important argument for particle are derived; the case of a uniformly moving mas- the standard (Einstein–Minkowski) relativity is that gen- sive body is discussed and then specialized to the case eral relativity (GR) is an extension of it and is not compat- of spherical symmetry. Formulas for the acceleration of ible with the L-P interpretation [4]. In standard relativity, test particles are given in the preferred frame of the ether the concept of time is completely different from what we and in the rest frame of the massive body that moves with experience: there is no observer-independent flow of time, velocity V with respect to the ether. When the body rests in and there is no simple concept of present. Instead, the = −2 the ether (V 0), the acceleration is up to order c iden- idea of eternalism (the block universe theory) appears to ̸= tical to GR. The acceleration of a test particle for V 0 is be the notion of time that best corresponds with standard given; this makes it possible to fit observations in celestial relativity. These are significant changes to our understand- mechanics to ephemerides with V as a free parameter. The ing of the world, which we should challenge if we wish to current status of such fits (although to ephemerides and make sure that they are really correct. We will therefore not to observations) is presented and discussed. examine in this article a theory of gravity that is consistent with the L-P interpretation. That theory interprets gravity Keywords: Alternative Theories of Gravitation; Celestial as Archimedes’ thrust exerted by a perfect fluid or “ether” Mechanics; Lorentzian Metric; Preferred Reference Frame; on the matter particles – those being viewed as extended Test Particle. objects, more precisely as organized flows in that same fluid. Archimedes’ thrust is due to the (macroscopic) pressure gradient in the postulated fluid. This interpretation of gravitation and its natural coupling with the L-P interpretation of special relativity have been discussed in detail in [5]. In short, the L-P interpretation of special relativ- *Corresponding author: Mayeul Arminjon, University of Grenoble ity sees the metrical effects (space contraction and time Alpes, CNRS, Grenoble INP, 3SR Lab., F-38000 Grenoble, France, E-mail: [email protected] dilation) as absolute effects of the uniform motion. When Rainer Wolfgang Winkler: Schäufelweg 4, D-81825 Munich, combined with the interpretation of gravity as being due Germany, E-mail: [email protected] to the pressure gradient of the ether, this leads to attribute

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM 2 | M. Arminjon and R. W. Winkler: Motion in the Scalar Ether Theory of Gravitation similar metrical effects to the gravitation. One thus builds The experimental check of such an alternative theory of a relativistic theory of gravitation with a preferred refer- gravitation involves of course many points, a good number ence frame, based on a unique scalar field [6], hence the of which have been already checked, see, among others, name “scalar ether theory” or SET.¹ The preferred refer- [6, 15, 16]; for a summary, see [17], Section 1. (As noted ence frame ℰ of the theory is the frame that moves with there, this scalar theory differs from all known scalar the- the macroscopic (or averaged) velocity field of the fluid ories.) In particular, the celestial mechanics has also been ether considered by that theory. In other words, the aver- checked for this theory [15, 18], but this was for an ear- age velocity field of the ether with respect to ℰ is zero by lier version of the theory (“v1,” see [16] and references definition of that frame [5]. The global frame ℰ is a natu- therein), which had to be modified to the current version ral inertial frame. Hence, the existence of global inertial (“v2,” see [6]). The aim of the present research is to begin frames is unproblematic for this theory. (Global inertial the check of the celestial mechanics of the “new” version, frames occur of course in Newton’s theory but also, to a v2. That beginning consists essentially in assuming for certain extent, in the celestial mechanics based on GR, simplicity that the mass centres of the N bodies move as through the use there of the harmonic gauge condition [9], test particles in the gravitational field of the other bodies. which can be interpreted as defining a generalization of In this framework, the main task of the present work was the Newtonian global inertial frames [10].) Special relativ- to derive a tractable equation of motion for a test particle ity is the limiting situation in which the pressure gradi- in SET. ent is negligible, thus making the ether (macroscopically) In Section 2, we present the main equations of the the- homogeneous. In summary, that theory has a physical ory. We note there that a general expression of the acceler- interpretation for gravity and extends the L-P interpreta- ation of a test particle obtained for v1 holds true for v2, and tion of relativity to the heterogeneous ether implied by we show the spatial covariance of the equations. Section 3 gravity. derives a simple and exact expression of the acceleration The other motivation for that theory is to avoid some in the most general case. In Section 4 we prove that, in problems that are suffered by GR, despite its experimental the case where the gravitational field is produced by a success. uniformly moving massive body, the source of that field (i) In SET, the gravitational collapse does not lead to a can be defined in the uniformly moving frame as a time- singularity [11]; neither does the past state of very independent scalar field. We obtain then the explicit exact high density implied by the cosmic expansion [12]. solution for the gravitational field (50). That case is fur- (ii) In SET, the spacetime manifold is given, and there is ther specialized in Section 5 by assuming that the mas- no need for any gauge condition. sive body has spherical symmetry. We provide there the (iii) Because that theory has a preferred reference frame, expression of the acceleration both in the preferred frame its coupling with quantum theory is easier than for and in the moving frame. Section 6 discusses the applica- GR; e.g. the energy operator of the Dirac equation tion to the effective calculation of an ephemeris.² We show does not have any nonuniqueness problem [13]. that the approximation done in the present work to calcu- (iv) In SET, the cosmic expansion is necessarily accel- late the post-Newtonian (PN) correction, that each planet erated, without the need to introduce any dark moves as a test particle in the field of the spherical Sun, energy [12]. plus the fact that comparison is made with an ephemeris (v) In order to formulate a consistent electrodynamics based on GR, necessarily lead to the result found that the in the presence of gravitation for that theory, one velocity of the barycentre is unrealistically small. There- had to postulate an interaction energy, and that fore, the next steps should be (i) to derive fully consistent energy turns out to be a possible candidate for dark equations at the PN level that take into account the self matter [14]. fields (as was previously done for v1 [18]) and, most importantly, (ii) to adjust the theory on “direct” data, as little affected as possible by a reduction using GR.

1 Note that extensions of GR having a preferred reference frame have been proposed, in particular the “Einstein-aether” theory. That theory adds a vector field (the 4-velocity of the preferred frame) into the Hilbert-Einstein action [7, 8] and has thus a “dynamical” preferred 2 In order to test this theory in celestial mechanics, it is preferable frame. The preferred reference frame of SET is “prior-geometrical” to calculate ephemerides. In any case, one cannot use the parameter- instead of dynamical. SET is not an extension of GR; instead, it ized PN formalism, as in that theory the test particles generally do not proposes a fully different view of gravity. have a geodesic motion [6, 19].

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM M. Arminjon and R. W. Winkler: Motion in the Scalar Ether Theory of Gravitation | 3

2 Main Equations of the Theory where √ β = γ (A) Metrics. The theory that is studied (SET) is a scalar the- : 00. (3) ory with a preferred reference frame and two Lorentzian This means that, in a gravitational field, the mea- spacetime metrics: a flat one (Minkowski’s metric) γ0 and suring rods are contracted and the periods of clocks are a curved or “physical” one γ, the latter being related to dilated, in the same ratio β (usually β ≤ 1). The reason for γ0 through a scalar field [6]. The preferred reference fluid³ this assumption in the framework of the hypothesis of a ℰ assumed by the theory is an inertial frame for the flat µ perfectly fluid “ether” is explained in detail in [5]. Thus, metric γ0; i.e. there are spacetime coordinates x that are 0 0 in Cartesian coordinates that are adapted to the reference both adapted to ℰ⁴ and Cartesian for γ – that is, γµν = fluid ℰ, we can write the line elements of the flat metric ηµν, where ηµν (µ, ν = 0, ... , 3) are the components of the and the “physical” metric, respectively, as standard matrix η = (ηµν):= diag(1, −1, −1, −1). The 0 0 2 0 µ ν 0 2 i i time T := x /c is well defined up to a constant shift⁵ and (ds ) := γµνdx dx = (dx ) − dx dx , (4) is a preferred time: the inertial time in the inertial frame 2 µ ν 2 0 2 −2 i i ds := γµνdx dx = β (dx ) − β dx dx . (5) ℰ. (Here, c is the velocity of light.) In such coordinates, we have also, by assumption: (B) Equation of Motion of a Test Particle. In SET, motion is defined by an extension to curved spacetime γ i = 0 (i = 1, 2, 3). (1) 0 of the special-relativistic form of Newton’s second law: force = time − derivative of momentum, the latter involv- The latter equation (“synchronization condition”) ing either the velocity-dependent relativistic mass or the implies that the spatial metric g associated in the refer- energy of the photon (divided by c2) [19]. That extension ence fluid ℰ with the spacetime metric γ [20, 24, 25] is just involves an acceleration vector g of gravitation, defined as the spatial part of the spacetime metric γ, i.e. gij = −γij follows: (i, j = 1, 2, 3) in such coordinates [24]. The spatial metric 0 associated in the reference fluid ℰ with the flat metric γ 2 grad g β g := −c , (6) is a Euclidean (i.e. flat and Riemannian) spatial metric g0 β that is time independent, i.e. ∂g0 /∂x0 = 0 in any coordi- ij where nates that are adapted to ℰ. The metric g is assumed to have a simple relation with g0: i ij (grad g β) := g β,j , (7)

− g = β 2g0 (︁ ij)︁ (︀ )︀−1 (︀ )︀ (2) g := gij being the inverse matrix of matrix gij . The precise writing of the curved-spacetime Newton second law has been discussed in detail in [19] and has been 3 A reference fluid is a 3-dimensional congruence of reference world summarized, for example, in [17]: Section 2, Point (iii). lines, each of which defines the trajectory of a point bound to that Here, we will need only the “coordinate acceleration,” reference fluid [20]. This notion does not imply the presence of a real which has been deduced from it in [26], (18) where fluid. However, in the case of the preferred reference fluid ℰ, we do i (︂ )︂ imagine (at a heuristic level) that it represents the averaged motion du 1 ∂β j i i j k = + 2β j u u − Γjk u u of some kind of fluid, the “micro-ether,” see Section 3.3 in [5]. dT β ∂T , 4 Spacetime coordinates xµ adapted to a given reference fluid ℱ are 2 ones for which the reference world lines have constant spatial coor- 1 ij ∂gjk k c i i − g u − f (∇0f ) , (8) dinates x (i = 1, 2, 3) [21, 22]. For a change from one set of coordi- 2 ∂T 2 nates adapted to ℱ to another one (i.e. for an “internal transformation i i 2 of coordinates” [20]), the change in the spatial coordinates has to with u := dx /dT, f := β , ∇0 := gradg0 , and where the i be independent of the time coordinate, but the change in the time Γjk’s are the Christoffel symbols associated with the spa- coordinate is arbitrary [20]. Hence, we can speak of adapted spatial tial metric g. In [26], which belonged to a first version of coordinates xi as well. By a reference frame, we mean a reference SET (“v1”), the assumed relation between the two spatial fluid endowed with a given time coordinate map. For more details on these notions and their development, see [23] and references metrics differed from (2) above. However, the derivation therein. of (8) above depended on that relation only through the 5 If x′µ are other coordinates that verify those two conditions, we following reexpression of the space vector (6) ((2) in [26]): ∂x′ρ ∂x′σ ∂x′i have µ ν ηρσ = ηµν and 0 = 0 (i = 1, 2, 3). It follows easily that ∂x ∂x ∂x 2 ∂x′0/∂xi = 0 and ∂x′0/∂x0 = ±1. Therefore, asking ∂x′0/∂x0 > 0, c g = − ∇0 f . (9) the time coordinate is well defined up to a constant shift. 2

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM 4 | M. Arminjon and R. W. Winkler: Motion in the Scalar Ether Theory of Gravitation

Now, with the relation between the two spatial met- Similarly, γ0i (i = 1, 2, 3) are manifestly the compo- rics assumed in the second version of SET (“v2”) [5, 6], (2) nents of a spatial vector; hence, (1), if it is valid in some ij ij above, we have for the inverse matrices (g ) and (g0 ): coordinate system, remains valid after any spatial change.

ij 2 0 ij Moreover, rewriting (4) and (5) after a change (14) is easy: g = β g , (10) i i 0 i j replace dx dx with gijdx dx . The coordinates obtained after a change (14) are still adapted to the preferred refer- so that the gravity acceleration (6) is i ence frame ℰ, but of course the new spatial coordinates x′ ij 2 0 ij 0 i 2 g β,j 2 β g β,j 2 0 ij are generally not Cartesian for g . As to the acceleration g : = −c = −c = −c βg β,j β β (8), it is not a spatial vector [26], but (8) becomes (mani- i j k 2 2 festly) space covariant if one puts the term Γjk u u on the c 0 ij c i = − g f,j := − (∇0 f ) . (11) l.h.s., thus expressing the “absolute” derivative of the 3- 2 2 vector u w.r.t. T (and based on the metric g of the time Therefore, (9) remains valid for v2, and hence, the considered [19]), instead of its total derivative w.r.t. T – so same is true for (8). We emphasize that the equation of that also (8) holds true after any purely spatial coordinate motion (8) is valid for a massive test particle as well as for change (14). Thus, as one expects from a preferred-frame a photon [26]. theory, all equations written above are spatially covari- Equation for the Scalar Field. (C) This is the flat- ant, except for (4) and (5), which are easily rewritten in a ψ = − β spacetime wave equation for the scalar field : Log spatially covariant form. [6]: In addition, the “synchronization condition” (1), as 4πG well as the definition of the gravity acceleration (6), is sta- ψ := ψ − ∆g0 ψ = σ. (12) ,0,0 c2 ble also by a change of the time coordinate having the special form: Here, ∆g0 is the usual Laplace operator, defined with 0 the Euclidean space metric g (thus, ∆g0 ψ = ψ,i,i in Carte- 0 0 x 0 = φ x0 sian coordinates for g , i.e. such that gij = δij). And σ is ′ ( ). (16) the energy component of the total energy-momentum ten- sor T of matter and nongravitational fields in the reference However, the relation (2) between the flat and the frame ℰ (i.e. the reference fluid ℰ endowed with the pre- curved spatial metrics, as well as the field (12), is valid ferred time coordinate x0 = cT, with T the inertial time in only if the time coordinate is x0 = cT – and in any spatial ℰ): coordinates that are adapted to ℰ. Indeed, β is not invariant under a change (16) of the time coordinate, in contrast (︁ 00)︁ σ := T . (13) 0 ℰ with g and g , hence (2) is not covariant under such a change. Hence neither is the rewriting (11) of the vector g, T c2σ (We take in mass units; i.e. it is that is truly a nor the equation of motion in the form (8). The scalar field volume energy density.) of the theory has therefore to be defined more precisely (in Covariance. (D) A first point is that all equations writ- spatial coordinates adapted to ℰ) to be [6, 16]: ten above, except for (4), (5), and (8), are manifestly covariant under any purely spatial coordinate change: √ ˜ = (︀ )︀ β : γ00 x0=cT . (17) i i x′0 = x0, x′ = ψ (x1, x2, x3). (14) In this article, we take x0 = cT as the time coordinate; For instance, the fact that (6) is covariant under any hence, we may forget the distinction between β and β˜. change (14) results immediately from the facts that β (3) is obviously invariant under such a change and that then (6) and (7) manifestly define a spatial vector⁶; i.e. after such a i change, the components g become 3 Exact Equation of Motion of a

i i ∂x′ j Test Particle g′ = g . (15) ∂xj The equation of motion (8) does not fully take into account g 6 A “geometric” definition also exists for this: a spatial vector is a the explicit form (2) of the spatial metric ; e.g. it is valid vector in the tangent space, at some point, to the space manifold also for the first version of SET, for which it was first associated with the reference fluid ℰ (see [23], Section 4.2). derived. With (2), we have in Cartesian coordinates for the

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM M. Arminjon and R. W. Winkler: Motion in the Scalar Ether Theory of Gravitation | 5

0 −2 Euclidean metric g : gij = ϕδij with ϕ = β ; hence, the 4 Case of a Uniformly Moving Christoffel symbols of g are given by Massive Body i 1 il(︀ )︀ Γjk : = g glj,k + glk,j − gjk,l 2 Consider a massive body, say B, which is in a translation, −1 at a uniform and constant velocity V, with respect to the ϕ il(︀ )︀ = δ ϕ,k δlj + ϕ,j δlk − ϕ,l δjk ℰ 2 preferred reference frame . This means that, at any time T, the spatial velocity vector u of any point bound with B −1 ϕ (︁ i i )︁ V = ϕ,k δj + ϕ,j δk − ϕ,i δjk is the same spatial vector .⁷ Hence, the spatial positions 2 of that point at time T = 0 and at time T fulfil −1(︁ i i )︁ = β k δj + β j δk − β i δjk . (18) β , , , x(T) − x(T = 0) = V˜T. (26)

We get also from (2): i 3 i Here, x(T):= (x (T)) ∈ R , the x ’s being any Carte- ℰ ij −β T i sian spatial coordinates adapted to , and accordingly 1 , i g gjk,T = δk . (19) V˜ = V ∈ 3 ℰ 2 β : ( ) R .⁸ The energy density relative to , (13), follows that translation and verifies, hence We thus obtain from (8): σ T x + VT = σ T = x i ( , ) ( 0, ). (27) du 1(︁ j)︁ i = β T + 2β j u u T β , , d This relation remains true in a more general situation, (︁ )︁ in which the body has, in addition to its uniform trans- + 1 β ui uk + β uj ui − β uj uj β ,k ,j ,i lation, a stationary rotation with an axis that follows the translation at V, and around which the energy distribu- 2 β T i c i + , u − f(∇ f ) . (20) tion σ is rotationally symmetric. Consider now the global β 0 2 reference fluid, say ℰV, which follows (only) the uniform translation, at velocity V, of body B. That is, at any time In Cartesian coordinates for the Euclidean metric g0, T, the spatial velocity vector u of any point bound with ℰV the last term is is the same spatial vector V – but now the initial position c2 x T = x ∈ 3 − ∇ i = − 2 3 ( 0) in (26) can be any vector 0 R . The reference f( 0 f) c β β,i . (21) 0 2 fluid ℰV is also an inertial frame for the flat metric γ , as is ℰ. The reference fluid ℰV, endowed with its own iner- Hence, we can rewrite (20) as tial time T′, will henceforth be called “the moving frame.” i du 1[︁(︁ j)︁ i j j]︁ = 2β T + 4β j u u − β i u u dT β , , , 7 The velocity of a given point bound with B is in general a time- 2 3 ℰ − c β β,i . (22) dependent spatial vector in the reference frame , having components ui(T) = dxi /dT in any spatial coordinates xi adapted to ℰ. The We still reexpress this in terms of the field ψ := −Log β uniformity, at any given time T, of the 3-vector V, refers to the connec- g0 V that enters the field (12). We have tion associated with the Euclidean metric ; it means that is trans- ported parallel along any spatial curve xi = xi(ξ). It thus means that Vi V i ξ = β,µ the components stay unchanged (d /d 0) along any curve = Log β = −ψ µ (23) β ( ),µ , in any Cartesian spatial coordinates adapted to ℰ (Note 4); in other words, Vi does not depend on the spatial position. The constancy and of V means that this spatially uniform vector field actually does not depend on T, dV i /dT = 0. Thus, Vi is a true constant in any Cartesian 3 4 β,i −4ψ spatial coordinates adapted to ℰ. β β i = β = −e ψ i (24) , β , 8 Thus, x and V˜ are “coordinate 3-vectors.” They are not spatial vectors [see after (15)]. This is because to define the spatial position as a so that (22) rewrites as spatial vector one needs to choose an origin point. When the coordinate system is changed, x and V˜ change to x′ := (x′i) and V˜′ := (V′i), i i du i j i but the spatial vector V that has the V ’s as components in the first = −2ψ T u − 4ψ j u u i dT , , coordinate system and the V′ ’s in the second one is the same vector. Hereafter, for the simplicity of notation, V will denote both the spatial j j 2 −4ψ + ψ,i u u + c e ψ,i (25) vector and the “coordinate vector.”

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM 6 | M. Arminjon and R. W. Winkler: Motion in the Scalar Ether Theory of Gravitation

We can define the field σ, as it is “seen” in the moving Let us evaluate σ′(T′, x′)−σ′(T′ = 0, x′). To apply (28), frame, as follows: we define

−1 00 (T, x): = F (T′, x′) and σ′(X′):= σ(X(X′)) (σ(X):= (T )ℰ (X)). (28) −1 µ (T , x ): = F (T′ = 0, x′). (33) Here, X′ := (x′ ) are spacetime coordinates adapted to 0 0 ℰ x 0 = cT the moving reference fluid V and with ′ ′, whereas T x T = µ To obtain ( 0, 0) we first apply (32) with ′ 0 and X := (x ) are spacetime coordinates adapted to the pre- get ferred frame ℰ and with x0 = cT. Of course, σ′ is in general not the T00 component in the new coordinates, i.e. Vx′ T0 = γ , x0 = γx′, y0 = y′, z0 = z′. (34) σ′(X′) ̸= T′00(X′) in general, as c2

0 0 Then, as from (33) we have (T′, x′) = F(T, x), we enter ∂x′ ∂x′ µν T′00(X′) = T (X) ̸= T00(X) = σ(X), (29) ∂xµ ∂xν precisely (31) into the latter equation to get

γ2V(x − VT) and as the latter is by the definition (28) equal to σ′(X′). T = , x = γ2(x − VT), 0 c2 0 Proposition. If the relation is true, then the field σ (27) ′ y0 = y, z0 = z. (35) defined by (28) does not depend on the inertial time T′ in ℰV: Now we want to use (27). We rewrite it as

1 2 3 σ′ = σ′(x′ , x′ , x′ ) = σ′(x′). (30) σ(T1, x + VT1) = σ(T = 0, x) = σ(T2, x + VT2) (36)

Proof. Note first that the energy density (13) involved in (for any given x and any two times T1 and T2), from which the relation (27) is invariant under any purely spatial coor- we see that dinate change (14). Hence, if (27) is true for one set of spa- i σ(T , x ) = σ(T , x ) if x − x = V(T − T ). (37) tial coordinates x , which are adapted to ℰ and are Carte- 1 1 2 2 2 1 2 1 sian for the Euclidean metric g0, then it remains true if (Note from (35) that y and z are constant in our cur- we change the spatial coordinates for ones with the same i rent manipulations, corresponding with the fact that V = property. Also, if (30) is true with one set of coordinates x′ (V, 0, 0).) We apply (37) with (T1, x1):= (T0, x0) defined (whether they are Cartesian for some Euclidean metric or in (33)2 and computed in (35). So (37) allows us to write not), it holds true after a purely spatial coordinate change x i (14) applied to the ′ ’s. So we can choose the spatial coor- σ(T0, x0) = σ(T0, x0, y0, z0) = σ(T0, x0, y, z) i dinates x (adapted to ℰ and Cartesian for g0) and the spa- i tial coordinates x′ (adapted to ℰV) as we wish. Starting = σ(T2, x, y, z), (38) µ from coordinates (x ) = (cT, x, y, z), which are adapted to ℰ and with x, y, z being Cartesian for g0, we obtain coor- provided that dinates adapted to the moving frame by doing a Lorentz V(T − T ) = x − x . (39) transformation, which can be made special through the 2 0 0 choice of the axes: From this and (35), we compute the corresponding (︂ Vx )︂ time T2 as F : T′ = γ T − , c2 2 2 x − x0 γ V(x − VT) x − γ (x − VT) T2 = T0 + = + , x = γ x − VT y = y z = z V c2 V ′ ( ), ′ , ′ (31) (40)

− that is (here γ = (1 − (V2/c2)) 1/2 is the Lorentz factor). The (︂ 2 2 )︂ (︂ 2 2 )︂ inverse transformation is γ V γ − 1 2 γ V T = x − + T γ − = T. (41) 2 c2 V c2 (︂ )︂ − Vx′ F 1 : T = γ T′ + , x = γ(x′ + VT′), c2 This and (38) mean that

y = y′, z = z′. (32) σ(T0, x0) = σ(T, x). (42)

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM M. Arminjon and R. W. Winkler: Motion in the Scalar Ether Theory of Gravitation | 7

But by the definitions (28) and (33), we have precisely The exact solution (50) for the gravitational potential ψ is thus just like the Newtonian potential, but remind σ T x = σ x σ T x = σ T x ( 0, 0) ′(0, ′) and ( , ) ′( ′, ′). (43) that it has to be transformed to the preferred frame ℰ by a Lorentz transformation. In order to use (50) in the equation We thus obtain that, for any value T′ of the time of the of motion (25), we have to express the derivatives w.r.t., moving frame and for any position x′ in this frame, µ the “fixed” coordinates x in terms of derivatives w.r.t., the µ σ′(0, x′) = σ′(T′, x′). (44) “moving” coordinates x′ . This will be done in the next section in the case of spherical symmetry. In practice, the This proves the Proposition. velocity vector V is an unknown; more precisely, it is a “solved-for parameter” in the optimization software [18], Remark: If instead of the Lorentz transformation (31), the see Section 6. As a consequence, we cannot use a spe- coordinate transformation is the Galileo transformation: cial Lorentz transformation; instead, we use the general version of the Lorentz transformation (e.g. Weinberg [28], (T′, x′) = F(T, x):= (T, x − VT), (45) Section 2.1): then the same Proposition is true also, its proof being then (︂ )︂ V.x much simpler. On the other hand, if the coordinate trans- T′ = g(T, x):= γ T − , (51) c2 formation F is fully general (and in particular nonlinear), then one cannot deduce anything like this. What one can γ − 1 write then is x′ = f(T, x):= x + V.x V − γTV. (52) V2 ( ) − σ′(T′, x′): = σ(T, x) with (T, x):= F 1(T′, x′) From this, we find easily using the fact that ∂ψ′/∂T′ = = σ(0, x − VT) 0 [(50)]:

= σ′(T′0, x′0) with (T′0, x′0):= F(0, x − VT), (∇xψ)(T, x) = (∇x′ψ′)(f(T, x)) (46) and this cannot be transformed further to obtain (30). (︀ )︀ V + (γ − 1) V.(∇x ψ′)(f(T, x)) (53) In the same way as in (28), we define ′ V2

ψ X = ψ X X ′( ′): ( ( ′)). (47) V −4 = ∇x ψ′ + (V.∇x ψ′) + O(c ) (54) ′ ′ 2c2 As the flat wave operator is (in particular) Lorentz −n invariant, it follows from (12), (28), and (47) that we have (all equations without an O(c ) remainder are exact), and µ when the coordinates x′ (adapted to ℰV and such that also x′0 = cT′) are moreover Cartesian for the Minkowski met- γ0 ∂ψ ∂ψ ∂f j ric : = ′ (T, x) j (f(T, x)) (T, x) ∂T ∂x′ ∂T ∂2ψ ∂2ψ πG ψ = ′ − ′ = κσ κ = 4 ′ 2 i i ′, : 2 . (48) ∂ψ ∂ x 0 ∂x′ ∂x′ c = − j ′ = − ∇ ( ′ ) γV j γV. x′ψ′. (55) ∂x′ The relevant solution for an (assumed) isolated massive body B is the pure retarded potential (i.e. without the addition of a solution of the homogeneous wave equation), 5 Case of a Spherical Uniformly because it corresponds to the situation without an external Moving Massive Body field [24, 27]. Thus,

(︂ )︂ 3 σ κ ∫︁ |x′ − y′| d y′ Let us assume that the source ′ in (50) is spherically ψ′(T′, x′) = σ′ T′ − , y′ . (49) 4π c |x′ − y′| symmetric, i.e. B σ x = σ r r = |x − x | However, due to the time independence of σ′ (30), the ′( ′) ′( ′), ′ : ′ ′b , (56) retardation has no effect, and we get a stationary field: where x′b is the (fixed) position, in the moving frame ℰV, G ∫︁ σ y ψ = ψ x = ′( ′) 3y of the centre of spherical symmetry (the mass centre of ′ ′( ′) 2 d ′. (50) c |x′ − y′| |x | = g 0 x x 1/2 = x i x i g 0 B body B) and ′ : ( ′ ( ′, ′)) ′ ′ , with ′ the

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM 8 | M. Arminjon and R. W. Winkler: Motion in the Scalar Ether Theory of Gravitation

Euclidean spatial metric associated with the Minkowski in which from (53): 0 metric γ in the moving frame ℰV.⁹ Then we get c2 V.h′ h := (∇xψ)(T, xa(T)) = h′(T) + (γ − 1) V GM GM V2 ψ′(r′) = , (57) c2 r′ V.h′ − = h′ + V + O(c 4), (65) and 2c2

GM x′ − x′b and from (55): (∇x′ψ′)(x′) = − , (58) c2 r′3 hV := −γh′.V, (66) with ∫︁ with, moreover (to be calculated at x = xa(T)): M := σ′(r′)d3x′. (59) β = β T x = e−ψ(T,x) B ( , ) , (67)

(Of course, (57) and (58) are valid only outside the mas- where, according to (57), sive body B, i.e. for r′ > R, where R is the radius of B, i.e. GM σ′(r′) = 0 for r′ > R. Note that the assumption that the ψ(T, x) = ψ′(T′, x′) = . (68) c2r′ energy density be spherically symmetric is indeed more correct in the frame that moves with the body, for there is We want to compute the (first) PN approximation, −2 then no Lorentz contraction to account for.) We note xa(T) which includes the O(c ) corrections to the Newtonian −2 and xb(T), the positions in the preferred frame of a test par- acceleration. The latter is order zero w.r.t. c and comes GM 2 4 4 ticle (which could be the mass centre of a planet) and of the from the c2 c β h = GMβ h term in the acceleration (64). centre of body B (which could be the Sun), and Indeed, we compute from (67) and (68):

4 GM (︁ −4)︁ rab(T):= xa(T) − xb(T). (60) β = 1 − 4 + O c . (69) c2 r′

The positions in the moving frame are related to xa(T) The 1 on the r.h.s. of (69) gives the term GMh in the and xb(T) by the Lorentz transformation (51) and (52): acceleration (64); this contains the Newtonian acceleration GMh′, see (65). x T = f T x T ′a( ): ( , a( )), The acceleration (64) can be reexpressed in the moving frame ℰV, by using the Lorentz transformations of the f(T, xb(T)) = x′b = Constant. (61) velocity:

Setting [︂(︂ −1 )︂ ]︂ 1 u.V(1 − γ ) − u′ = − 1 V + γ 1u (70) 1 − u.V/c2 V2 r′ab(T):= x′a(T) − x′b , R′ab(T):= |r′ab(T)|, (62) and the acceleration: we have from (58): u u u u d ( d .V)V(γ − 1) ( d .V)u d ′ = dT − dT + dT GM r′ab(T) 2 3 3 . ∇ x = h h = − dT′ 2 u.V 2 3 u.V 2 2 u.V ( x′ψ′)( ′a(T)) ′, ′(T): . (63) γ (1 − 2 ) V γ (1 − 2 ) c γ (1 − 2 ) c2 R′3 (T) c c c ab (71) c−4 We can then use (53) and (55) to rewrite the accelera- Neglecting all terms of order or higher and set- r = r r = R tion (25) more explicitly as ting ′ : ′ab and ′ : ′ab, we obtain after a somewhat tedious calculation: du GM [︁ 2 2 4 ]︁ (︂ )︂ = −2hV u − 4(h.u)u + u h + c β h , (64) du′ GM A − dT c2 = − r′ + + O(c 4), (72) dT′ r′3 c2

with 9 Since the Lorentz transformation (51) and (52) transforms the Carte- sian coordinates xµ for γ0 to Cartesian coordinates x′µ for γ0, it follows A = −(r′.V + 4r′.u′)(V + u′) that the new spatial coordinates x′i are Cartesian for the Euclidean 0 0 2 0 2 i i 0 2 [︁ ]︁ GM spatial metric g′ . Simply: (ds ) = (dx ) − dx dx = (dx′ ) − + u 2 + u V + V2 r − r i i 0 2 0 i j ′ 4( ′. ) 2 ′ 4 ′. (73) dx′ dx′ = (dx′ ) − g′ij dx′ dx′ . r′

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM M. Arminjon and R. W. Winkler: Motion in the Scalar Ether Theory of Gravitation | 9

Thus, that part of the acceleration, which is indepen- to minimize the least-squares residual [29]. The solved-for dent of V, is parameters of the optimization are the initial conditions (position and velocity) for the N bodies, their masses, and (︂ u )︂ d ′ (for SET only) the “absolute” velocity of their barycentre. T d ′ V=0 That software had been built for v1, and it had been indeed {︂ [︂(︂ )︂ ]︂}︂ used for v1 [18], after having been tested (i) with the New- GM 1 2 GM = − r′ + u′ − 4 r′ − 4(r′.u′)u′ tonian equations of motion (for spherical attracting bod- r′3 c2 r′ ies) [29] and (ii) with the Newtonian equations of motion (︁ −4)︁ modified to include, for the planets, the PN correction + O c . (74) that comes from the Sun considered as a spherical body [30]. Here, besides using the same ODE integration scheme This is exactly the PN acceleration of a test particle and the same optimization algorithm as in [18, 29, 30], we in an SSS (static spherically symmetric) field as found in also use the same approximation as in the latter work [30] GR from the spatially isotropic SSS solution of GR. See e.g. for the equations of motion. That is, in view of (64), we Weinberg [28], (9.5.3) and (9.5.14), or [6], (89). Equivalently, compute the acceleration, in the reference frame ℰ, of the (74) is exactly the SSS case of the PN acceleration of a test planet with number a (a = 1, ... , N − 1), as particle as found in GR when one starts from the so-called “standard PN metric.” (The latter is spatially isotropic, N−1 ua ∑︁ GM xa − x GM see e.g. Weinberg [28], (9.1.60).) In fact, (74) can be easily d = − d ( d) + N 3 2 V = 0 dT |xa − x | c checked directly by doing in the acceleration in the d=1 d preferred frame, (64), as we have then d̸=a [︁ ]︁ 3 × − h u − h u u + u2h + c2 β4h u = u′, hV = 0, h = h′ = −r′/r′ , (75) 2 V a 4( . a) a a . (78) and as we can use (69). The other part of the acceleration The last body, with number N, is the Sun. Its accelera- is tion is computed as (︂ )︂ (︂ )︂ du′ du′ − N−1 T T duN ∑︁ GMd (xN − xd) d ′ d ′ V=0 = − . (79) T 3 d = |xN − xd| GM d 1 = {[r′.(V + 4u′)]V c2 r′3 Thus, in this model, we take into account the New- (︁ 2)︁ }︁ tonian attractions of the planets on the Sun; hence, the + (r′.V)u′ − 4u′.V + 2V r′ velocity uN of the Sun w.r.t. ℰ is not exactly a constant. To (︁ − )︁ compute the contribution of the Sun to the acceleration of + O c 4 . (76) body a (a < N), i.e. the term with the square bracket on (︀ − )︀ the r.h.s. of (78), we substitute uN for V into the expres- Up to O c 4 , we can absorb the last term into sions (65) and (66). By doing so, essentially, we neglect the the Newtonian acceleration through a redefinition of the effect of the very small (and variable) acceleration of the active mass M to Sun on the gravitational field that it produces. This is in [︁ ]︁ addition to neglecting the departure from spherical sym- M′ := M 1 + 2(V2/c2) . (77) metry of the gravitational fields of the Sun and the planets, to neglecting the PN corrections on the motion of the Sun 6 Implementation in a Software for that are due to the planets, and perhaps most importantly [6, 18], to considering that the mass centres of the N bod- Ephemeris Calculation ies move as test particles in the gravitational field of the other bodies, i.e. essentially to neglecting the effect of the (A) Principle, Equations of Motion. The equation of motion self fields, which depend on the structure (density profile, (64) has been implemented in a software for ephemeris self-rotation, etc.) [18]. calculation with parameter optimization. The main pro- (B) Different Frames. The data are referred to the helio- gram loops on the numerical integration of the equations centric reference frame, say ℋ, whereas the equations of of motion for the major bodies of the solar system in order motion (78) and (79) are written in the preferred reference

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM 10 | M. Arminjon and R. W. Winkler: Motion in the Scalar Ether Theory of Gravitation frame ℰ. To transform the initial conditions for the plan- where the index N indicates the Newtonian acceleration, ets’ positions and velocities from ℋ to ℰ, we first trans- given e.g. by (79) for the Sun, and where the index PNc form them from ℋ to the barycentric frame ℬ, by using the means the PN correction to the acceleration of a planet. following coordinate transformation: The latter correction is given for this model by the term with the square brackets in (78) minus the Newtonian x′ = xh − xB h , (80) acceleration due to the Sun. (See around (69).) The first sum in the rightmost side of (84) is like the sum of the x where h is the set of heliocentric spatial coordinates, and Newtonian interaction forces in a system of point parti- (︃N− )︃ N cles (divided by Mtot) and is hence zero by the actio-reactio ∑︁1 ∑︁ xB h = Maxa h / Ma (81) principle. (This is easily checked directly.) We are thus left a=1 a=1 with is the heliocentric position of the barycentre (account- N−1 (︂ )︂ ing for xN h = 0). The two reference frames ℋ and ℬ are dV ∑︁ Ma dua = ̸= 0. (85) cT dT Mtot dT endowed with the same time coordinate ′. Accordingly, a=1 PNc the corresponding heliocentric and barycentric velocities exchange by In the present work (although not in the older work dx′a on v1 [18]), we have taken this into account. Updating V is u′a := = ua h − uB h (a = 1, ... , N), dT′ done by adding it to the unknowns in the ODE solver, with (85) as the corresponding ODE and with V(T0) as the ini- dxa h ua h := . (82) tial data. (It is now V0 = V(T0) that belongs to the solved- dT′ for parameters of the optimization program.) However, we Then, assuming that the velocity of the barycentre find that the time variation of V is fully negligible, at least w.r.t. the reference frame ℰ is the vector V, we transform over the time period investigated (one century). the initial conditions from the uniformly moving frame (D)Value of V. As was shown above, the equation of ℰV to ℰ by using the Lorentz transformation (51) and (52), motion of a test particle in the gravitational field of a or rather its obvious inverse. After having integrated the uniformly moving spherical object in SET, (64), coincides equations of motion (78) and (79), we do the reverse trans- for V = 0 with (74). Now some ephemerides are based formations. While going from ℰ to ℰV or conversely, we on (74) plus the Newtonian attractions due to the plan- have to correct the positions and velocities from “simul- ets, as with (78) and (79) of SET (and possibly includ- taneity gaps” [the fact that e.g. the values T′a of the time in ing also oblateness corrections for the Sun’s gravitational ℰV that correspond to the events (T, xa), which are simul- field): e.g. VSOP82 [31, 32], VSOP2000 [33], VSOP2013 [34]. taneous in ℰ, depend on a through (51)]; we do that by (Also, the Warsaw ephemeris WAW [35] is based on the using first-order Taylor expansions. same approximation, although it uses the Painlevé SSS (C) Variation of V. The velocity V of the barycentre solution of GR [36] instead of the spatially isotropic SSS w.r.t. the reference frame ℰ: solution of GR.) As mentioned at Point A, in our equa-

(︃ N )︃ N tion of motion of a planet that includes the Newtonian dxB d ∑︁ Ma ∑︁ Ma h V := := xa = ua , attractions of the other planets, (78), more precisely in dT dT Mtot Mtot a=1 a=1 and hV that enter this equation and that are given by the expressions (65) and (66), the velocity V is replaced by uN, N ∑︁ the velocity of the Sun – because the Sun does play the Mtot := Ma , (83) role of the supposedly unique massive body in (64). As a=1 the VSOP2000 ephemeris is based on (74) plus the Newto- is not exactly a constant in this model, as from (78) and nian attractions of the planets and as (64) coincides with (79) we have (74) for V = 0, it follows that the equations of motion for

N VSOP2000 and for the present calculation are equivalent V ∑︁ Ma ua d = d when uN = 0. We ran a parameter optimization for the dT Mtot dT a=1 Sun and the eight major planets, aimed at best fit the data of the DE403 ephemeris of the JPL [37] over one century. N (︂ )︂ N−1 (︂ )︂ ∑︁ Ma dua ∑︁ Ma dua Among the solved-for parameters is the velocity vector V. = + , (84) Mtot dT Mtot dT a=1 N a=1 PNc Now, the difference between VSOP2000 and DE403 is very

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM M. Arminjon and R. W. Winkler: Motion in the Scalar Ether Theory of Gravitation | 11 small [33].¹⁰ Therefore, one expects that the optimization fitting the corresponding equations of v1 to the DE403 program tries to make uN as close as possible to 0, because ephemeris [18]. then the equations of motion of SET are equivalent to those used in VSOP2000, which give nearly the same result 7 Conclusion as DE403. This means that V, now defined as the velocity of the barycentre, should be found by the optimiza- By studying in detail the equations of motion of a test par- tion program to be as close as possible to the velocity of ticle in the investigated theory (SETv2), we have been able the barycentre with respect to the Sun. The latter velocity to put them in a tractable form. This allowed us to imple- varies in time but is approximately 40 km/h. And indeed, ment a first version of celestial-mechanical equations of we find the modulus V (or equivalently V0) to be of the motion for that theory in a software for ephemeris cal- order of 40 km/h. We believe that the foregoing discussion culation with parameter optimization. Those simplified shows that this result is fully expected. equations of motion coincide with equations used in the (E) Comments. Nevertheless, the latter finding does celestial mechanics of GR, when the absolute velocity of not imply that SET has an accurate celestial mechanics the Sun is zero. Therefore, they can lead to an equiva- only if V, the absolute velocity of the barycentre of the lent celestial mechanics, but with an unrealistically small solar system, is nearly equal to zero – the latter being velocity V for the barycentre of the solar system. To be of course difficult to accept given what we know about able to really check what the theory says about V, one will galactic motion. It does not imply that mainly for two need to develop a more realistic PN approximation, tak- reasons: ing into account the self fields. Above all, one will need (i) As we know, the ephemerides are not direct obser- to make comparison with “direct” observations instead of vations but are a fitting of the true observations, which are ephemerides and preferably with the observations being diverse in nature [39], by the equations of motion based “reduced” (corrected) by using the investigated theory on the standard PN approximation of GR (essentially (74) instead of GR. Especially the latter will be a hard special- plus the Newtonian attractions of the planets – see Point ized work. D above). Moreover, even these observations themselves, or at least many among them, are analysed and corrected precisely by using ephemerides or more generally by using References GR. For instance, the reduction of ranging data does use ephemerides [38]. This means that the observational data [1] S. J. Prokhovnik, The Logic of Special Relativity, Cambridge are in fact influenced by the theory (or more precisely by University Press, Cambridge 1967. [2] S. J. Prokhovnik, Z. Naturforsch. 48a, 925 (1993). the approximate equations by which it is replaced in prac- [3] G. Builder, Austral. J. Phys. 11, 279 (1958). tice) that is used to “reduce” them – in the present case, [4] P. Acuña, Stud. Hist. Philos. Sci. B: Stud. Hist. Phil. Mod. Phys. GR or more precisely its standard PN approximation. 46, 283 (2014). (ii) The equations for extended bodies got for v1 had [5] M. Arminjon, Found. Phys. 34, 1703 (2004). strong structure effects (including an effect of the self- [6] M. Arminjon, Braz. J. Phys. 36, 177 (2006). rotation), and it is likely that something similar will apply [7] T. Jacobson and D. Mattingly, Phys. Rev. D 64, 024028 (2001). [8] T. Jacobson, in: From Quantum to Emergent Gravity: Theory to v2. Thus, the correct equations of motion of the plan- and Phenomenology, Proc. of Science, Vol. 43 (2008). ets are quite different from those for test particles orbiting [9] M. Soffel, S. A. Klioner, G. Petit, P. Wolf, S. M. Kopeikin, etal., the Sun. Due to this difference, the correct 1PN equations Astron. J. 126, 2687 (2003). of motion for extended bodies do not coincide with (74) [10] V. A. Fock, The Theory of Space, Time and Gravitation, 2nd [plus the Newtonian attractions due to the planets, as in English edition, Pergamon, Oxford 1964. [11] M. Arminjon, Rev. Roumaine Sci. Tech.– Méc. Appl. 42, 27 (78) and (79)] for V = 0. As a matter of fact, it had been (1997). found higher velocities (of the order of a few km/s) while [12] M. Arminjon, Phys. Essays 14, 10 (2001). [13] M. Arminjon, Int. J. Geom. Meth. Mod. Phys. 10, 1350027 (2013). 10 In the accurate ephemerides such as DE403 and followers, [14] M. Arminjon, Open Phys. 16, 488 (2018). VSOP2000 and followers, WAW, EPM [38], etc., the numerical preci- [15] M. Arminjon, Int. J. Mod. Phys. A17, 4203 (2002). sion is greater than in the present calculation; more bodies are taken [16] M. Arminjon, Theor. Math. Phys. 140, 1011 (2004). into account: Pluto and many asteroids, etc. Hence, the present calcu- [17] M. Arminjon, Open Phys. 14, 395 (2016). lation cannot aim at a similar accuracy; e.g. the longitude differences [18] M. Arminjon, in: Recent Research Developments in with DE403 are here at the 10 mas level over one century, except for Astronomy & Astrophysics (Ed. S. G. Pandalai), Research Sign Mercury (0.2 arcsec). Post, Trivandrum 2003, Vol. 1, p. 859.

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM 12 | M. Arminjon and R. W. Winkler: Motion in the Scalar Ether Theory of Gravitation

[19] M. Arminjon, Arch. Mech. 48, 551 (1996). [30] M. Arminjon, Astron. Astrophys. 383, 729 (2002). [20] C. Cattaneo, Nuovo Cim. 10, 318 (1958). [31] J.-F. Lestrade and P. Bretagnon, Astron. Astrophys. 105, 42 [21] C. Cattaneo, in: Fluides et Champ Gravitationnel en Relativité (1982). Générale (Eds. A. Lichnerowicz, M.-A. Tonnelat), Editions du [32] P. Bretagnon, Astron. Astrophys. 114, 278 (1982). CNRS, Paris 1969, p. 227. [33] X. Moisson and P. Bretagnon, Cel. Mech. Dyn. Astron. 80, 205 [22] J. Norton, Stud. Hist. Phil. Sci. 16, 203 (1985). (2001). [23] M. Arminjon, J. Geom. Symmetry Phys. 46, 1 (2017). [34] J.-L. Simon, G. Francou, A. Fienga, and H. Manche, Astron. [24] L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Astrophys. 557, A49 (2013). 3rd English edition, Pergamon, Oxford 1971. [35] G. Sitarski, Acta Astron. 52, 471 (2002). [25] C. Møller, The Theory of Relativity, Clarendon Press, Oxford [36] G. Sitarski, Acta Astron. 33, 295 (1983). 1952. [37] E. M. Standish, X. X. Newhall, J. G. Williams, and W. M. Folkner, [26] M. Arminjon, Rev. Roumaine Sci. Tech. – Méc. Appl. 43, 135 JPL planetary and lunar ephemerides, DE403/LE403, Jet Prop. (1998). Lab. Interoflce Memo. 314.10–127 (1995). [27] J. D. Jackson, Classical Electrodynamics, 3rd edition, Wiley, [38] E. V. Pitjeva, Solar Syst. Res. 39, 176 (2005). Hoboken 1998. [39] W. M. Folkner, J. G. Williams, D. H. Boggs, R. S. Park, [28] S. Weinberg, Gravitation and Cosmology, John Wiley & Sons, and P. Kuchynka, IPN Progress Report 42-196 (2014). New York 1972. https://naif.jpl.nasa.gov/pub/naif/generic_kernels/spk/ [29] M. Arminjon, Meccanica 39, 17 (2004). planets/de430_and_de431.pdf.

Brought to you by | provisional account Unauthenticated Download Date | 1/14/19 6:16 PM