Putting Yukawa-like Modified Gravity (MOG) on the test in the Solar System

We deal with a Yukawa-like long-range modified model of gravity (MOG) which recently allowed to successfully accommodate many astrophysical and cosmological features without resorting to dark matter. On Solar System scales MOG predicts retrograde secular precessions of the planetary longitudes of the perihelia \varpi whose existence has been put on the test here by taking the ratios of the observationally estimated Pitjeva's corrections to the standard Newtonian/Einsteinian perihelion precessions for different pairs of planets. It turns out that MOG, in the present form which turned out to be phenomenologically successful on astrophysical scales, is ruled out at more than 3sigma level in the Solar System. If and when other teams of astronomers will independently estimate their own extra-precessions of the perihelia it will be possible to repeat such a test.


Introduction
The modified gravity (MOG) theory put forth in (Moffat 2006) was used successfully to describe various observational phenomena on astrophysical and cosmological scales without resorting to dark matter (see Moffat and Toth (2008) and references therein). It is a fully covariant theory of gravity which is based on the existence of a massive vector field coupled universally to matter. The theory yields a Yukawa-like modification of gravity with three constants which, in the most general case, are running; they are present in the theory's action as scalar fields which represent the gravitational constant, the vector field coupling constant and the vector field mass. Actually, the issue of the running of the parameters of modified models of gravity is an old one, known in somewhat similar contexts since the early 1990s (see, e.g., (Bertolami et al. 1993) and references therein). An approximate solution of the MOG field equations (Moffat and Toth 2007) allows to compute their values as functions of the source's mass.
The resulting Yukawa-type modification of the inverse-square Newton's law in the gravitational field of a central mass M is (Moffat andToth 2007, 2008) where G N is the Newtonian gravitational constant and (Moffat andToth 2007, 2008) Such values have been obtained by (Moffat and Toth 2007) as a result of the fit of the velocity rotation curves of some galaxies in the framework of the searches for an explanation of the flat rotation curves of galaxies without resorting to dark matter.
(3), is not in contradiction with the present-day knowledge of Solar System dynamics. We will show that it is not so also for any other (non-zero) values of α and µ, with the only quite general condition that µr ≪ 1 in Solar System, as it must be for any long-range modified model of gravity. It is interesting to point out that Yukawa-like modifications of Newton's law might also be tested in the context of stellar dynamics (Bertolami and Páramos 2005).
Here we outline the procedure that we will follow.
Generally speaking, let LRMOG (Long-Range Modified Model of Gravity) be a given exotic model of modified gravity parameterized in terms of, say, K, in a such a way that K = 0 would imply no modifications of gravity at all. Let P(LRMOG) be the prediction of a certain effect induced by such a model like, e.g., the secular precession of the perihelion of a planet. For all the exotic models considered it turns out that 1 where g is a function of the system's orbital parameters a (semimajor axis) and e (eccentricity); such g is a peculiar consequence of the model LRMOG (and of all other models of its class with the same spatial variability). Now, let us take the ratio of P(LRMOG) for two different systems A and B, e.g. two Solar System's planets: The model's parameter K has now been canceled, but we still have a prediction that retains a peculiar signature of that model, i.e. g A /g B . Of course, such a prediction is valid if we assume K is not zero, which is just the case both theoretically (LRMOG is such that should K be zero, no modifications of gravity at all occurred) and observationally because K is usually determined by other independent longrange astrophysical/cosmological observations. Otherwise, one would have the meaningless prediction 0/0. The case K = 0 (or K ≤ K) can be, instead, usually tested by taking one perihelion precession at a time. If we have observational determinations O for A and B of the effect considered above such that they are affected also 2 by LRMOG (it is just the case for the purely phenomenologically estimated corrections to the standard Newton-Einstein perihelion precessions, since LRMOG has not been included in the dynamical force models of the ephemerides adjusted to the planetary data in the least-square parameters' estimation process by Pitjeva (Pitjeva 2005a,b)), we can construct O A /O B and compare it with the prediction for it by LRMOG, i.e. with g A /g B . Note that δO/O > 1 only means that O is compatible with zero, being possible a nonzero value smaller than δO. Thus, it is 1 In our case it will be K = −αµ 2 , as we will see in Section 2.
2 If they are differential quantities constructed by contrasting observations to predictions obtained by analytical force models of canonical Newtonian/Einsteinian effects, O are, in principle, affected also by the mismodelling in them.
perfectly meaningful to construct O A /O B . Its uncertainty will be conservatively evaluated As a result, O A /O B will be compatible with zero. Now, the question is: Is it the same for g A /g B as well? If yes, i.e. if within the errors, or, equivalently, if within the errors, LRMOG survives (and the use of the single perihelion precessions can be used to put upper bounds on K). Otherwise, LRMOG is ruled out.

The predicted perihelion precessions and the confrontation with the measured non-standard rates
In the case of the Sun, eq. (2) and eq. (3) yield so that Since in the Solar System µr ≈ 10 −5 − 10 −4 , we can safely assume exp(−µr) ≈ 1 − µr, so that As a result, a radial, uniform perturbing acceleration is induced.
The secular, i.e. averaged over one orbital revolution, effect of a small radial and unform perturbing acceleration on the longitude of the perihelion of a planet ̟ has been worked out by, e.g., Sanders (2006); it amounts to Clearly, using only one perihelion rate at a time would yield no useful information on MOG due to the extreme smallness of the perturbing acceleration, as told us by eq. (10). Thus, let us take the ratios of the perihelion precessions. It must be noted that the following analysis is, in fact, truly independent of the values of α and µ, provided only that αµ 2 r 2 ≪ 1 in the Solar System so as that the perturbative approach can be applied to eq. (9); the condition µr ≪ 1 is the cornerstone of any long-range modified models of gravity, and should α ≈ 1 the planetary orbits would have been distorted in a so huge manner that it would have been detected since long time. Applying the scheme outlined in Section 1 to our case in which K = −αµ 2 and g(a, e) = G N Ma(1 − e 2 ), one can construct with the estimated corrections ∆̟ to the standard Newtonian/Einsteinian perihelion precessions of planets A and B, listed in Table 1, and compare them to the theoretical obtained from eq. (11), for that pair of planets A and B. The results are in   because of the existing correlations 3 among the estimated corrections to the precessions of perihelia.
If we repeat our analysis by subtracting from ∆̟ the main canonical unmodelled effect, i.e. the general relativistic Lense-Thirring precessions (Lense and Thirring 1918) induced by the Sun's angular momentum (Iorio 2007a) shown in Table 3, i.e. if we use the situation does not substantially change, apart from the sigma level at which |Π ⋆ − A| is not compatible with zero, as shown in Table 4.   The availability of the corrections to the usual rates of perihelia of several planets allows us to put on the test MOG also in another way as well. The acceleration law of eq.
(1) can also be recast in the commonly used Yukawa form (Moffat and Toth 2007) where In the case of the Sun A Yukawa-type acceleration of the form of eq. (16) has been tested by Iorio (2007b) in the Solar System without any a-priori assumption on the size of 4 α Y ; concerning λ, it was only assumed that λ ae. By using the corrections to the standard rates of the perihelia of A = Earth and B = Mercury quoted in Table 1 Iorio (2007b) found which contradicts eq. (20). Using the data for Venus in the equation for α Y (Iorio 2007b) which is three orders of magnitude smaller than the result of eq. (20).
If we use Π ⋆ for the Earth and Mercury in eq. (21) and ∆̟ ⋆ for Venus in eq. (22) the results does not change appreciably; indeed, we have

Conclusions
In the framework of the attempts of explaining certain astrophysical and cosmological We put on the test the possibility that such exotic precessions exist by comparing the ratio of them A for different pairs of planets to the ratio Π of the corrections to the usual Newtonian/Einsteinian precessions estimated by E.V. Pitjeva which account for any unmodelled/mismodeleld dynamical effects. It turns out that Π = A at more than 3σ level even by including in Π the main unmodelled canonical effect, i.e. the general relativistic Lense-Thirring precessions. Conversely, using the estimated corrections to the planetary perihelion rates to phenomenologically determine the strength parameter of the putative MOG Yukawa force and its range yields values which are neatly incompatible with those of MOG (Moffat andToth 2007, 2008). In assessing the results presented here it must be considered that, at present, no other people have estimated the non-standard part of the planetary perihelion motions; it would certainly be useful to repeat the present analysis if and when other teams of astronomers will estimate their own set of corrections to the -12standard perihelion precessions as well.