# Putting Yukawa-Like Modified Gravity (MOG) on the Test in the

Authors: Lorenzo Iorio
Journal: Scholarly Research Exchange
 (2008)

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## Abstract

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 anomalous retrograde secular precessions of the planetary longitudes of the
periheliaϖ.  Their 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 3σ level in the Solar
System. If and when other teams of astronomers will independently estimate
their own corrections to the usual precessions of the perihelia, it will be possible
to repeat such a test.


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## Body

## 1. Introduction

The  modified
gravity (MOG) theory put forth in [1] was used successfully to describe
various observational phenomena on astrophysical and cosmological scales
without resorting to dark matter (see  [2] 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., [3] and references therein). An approximate solution of the
MOG field equations [4] 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 massM is [2, 4] (1)AMOG=−GNMr2{1+α[1−(1+μr)exp(−μr)]}, where GN is the Newtonian gravitational constant and
[2, 4] (2)α=M(M+C1′)2(G∞GN−1),G∞≈20GN,C1′≈25000M⊙1/2,μ=C2′M,C2′≈6250M⊙1/2kpc−1. Such values have been obtained
in  [4] 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.In this paper, we will put (1) on the test in the
Solar System in order to check if it is compatible with the latest
determinations of the corrections 〈Δϖ˙〉 to the usual Newtonian/Einsteinian planetary
perihelion precessions [5, 6]. Such quantities are solved for parameters
phenomenologically estimated in a least-square sense by fitting about one
century of observations of various kinds concerning planetary motions with
their values theoretically predicted by simultaneously integrating the
equations of motions of the major bodies of the Solar System written in terms
of a complete suite of dynamical force models. They encompass all the known
Newtonian effects along with those predicted by general relativity, except for
the Lense-Thirring effect [7–9] which was left unmodelled (see below). Thus, 〈Δϖ˙〉 account for any unmodelled/mismodelled
dynamical features of motion and are good, in principle, to put on the test
MOG, provided that they are used in a suitable way.Note that Moffat and Toth explicitly write in [2]  that (1), with (2), 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 (nonzero) 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 [10].  Now, we outline the procedure that we will follow.Generally speaking, let long-range modified model of
gravity  (LRMOG) be a given exotic model of modified
gravity parameterized in terms of, say,K,
in 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,
for example, the secular precession of the perihelion of a planet. For all the
exotic models considered, it turns out that(3)𝒫(LRMOG)=Kg(a,e),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). (In our case, it will be K=−αμ2,
as we will see in Section
2).  Now, let us take the ratio of 𝒫(LRMOG) for two different systems A
and B, for example, two Solar System's
planets: 𝒫A(LRMOG)/𝒫B(LRMOG)=gA/gB. The model's parameter K has now been canceled, but we still have a
prediction that retains a peculiar signature of that model, that is, gA/gB.
Of course, such a prediction is valid if we assume K is  not zero, which is just the case both
theoretically (if  K was zero, no modifications of
gravity at all would occur) and observationally because K is usually determined by other independent
long-range 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 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 [5, 6]), we can construct 𝒪A/𝒪B and compare it with the prediction for it by LRMOG,
that is, with gA/gB.
(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.) Note that δ𝒪/𝒪>1 only means that 𝒪 is compatible with zero, being possible a
nonzero value smaller than δ𝒪. Thus, it is perfectly meaningful to construct 𝒪A/𝒪B.
Its uncertainty will be conservatively evaluated as |1/𝒪B|δ𝒪A+|𝒪A/𝒪B2|δ𝒪B.
As a result, 𝒪A/𝒪B will be compatible with zero. Now, the
question is “is it the same for gA/gB as well?" If yes, that is, if (4)𝒪A𝒪B=𝒫A(LRMOG)𝒫B(LRMOG) within the errors, or,
equivalently, if (5)|𝒪A𝒪B−𝒫A(LRMOG)𝒫B(LRMOG)|=0 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 challenged.


## 2. The Predicted Perihelion Precessions and the Confrontation with the Estimated Nonstandard Rates

In the case of
the Sun, (2) yield (6)α⊙≈3×10−8,μ≈3×10−5AU−1, so that (7)α⊙μ2=3×10−17AU−2. Since in the Solar System μr≈10−5−10−4, we can safely assume exp(−μr)≈1−μr, so that (8)AMOG≈−GNMr2(1+αμ2r2). As a result, a radial, uniform
perturbing acceleration (9)A=−GNMαμ2≈10−19ms−2 is induced.The secular, that is, averaged over one orbital
revolution, effect of a small radial and unform perturbing acceleration on the
longitude of the perihelion of a planetϖ has been perturbatively worked out, for example, by Sanders  in [11]; it amounts to (10)〈dϖdt〉=Aa(1−e2)GNM=−αμ2GNMa(1−e2). 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 to us by (9). 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 αμ2r2≪1 in the Solar System so that the perturbative
approach can be applied to (8); 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=−αμ2 and g(a,e)=GNMa(1−e2),
one can construct (11)Π≡〈Δϖ˙A〉〈Δϖ˙B〉 with the estimated corrections 〈Δϖ˙〉 to the standard Newtonian/Einsteinian
perihelion precessions of planets A and B, listed in Table 1, and compare
them to the theoretical prediction (12)𝒜≡aA(1−eA2)aB(1−eB2), obtained from (10), for that
pair of planets A and B. The results are
in Table 2. |Π−𝒜| is different from zero at more than 3σ level for A = Venus, B = Mercury, A = Earth, B
= Mercury and A = Mars, B = Mercury. It is important to note that the errors
have been conservatively evaluated as (13)δΠ≤|Π|(δ〈Δϖ˙A〉|〈Δϖ˙A〉|+δ〈Δϖ˙B〉|〈Δϖ˙B〉|) because of the existing
correlations among the estimated corrections to the precessions of perihelia.
(The maximum correlation, 26%,
occurs for  the Earth and Mercury (Pitjeva, personal communication to the
author, November 2005)).Table 1
Inner planets. First
row: estimated corrections to the standard
precessions of the longitudes of the perihelia in10−4′′cy−1 (′′cy−1→ arcseconds per century), from [6, Table 3] (apart from Venus). The quoted errors, in 10−4′′cy−1, are not the formal ones but are realistic.
The formal errors are quoted in square brackets (Pitjeva, personal
communication to the author, November 2005). The units are 10−4′′cy−1. Second row: semimajor axes, in astronomical
units (AU). Their formal errors are in [5, Table 4], in m. Third row:
eccentricities. Fourth row: orbital periods in years. The result for Venus
has been recently obtained by including the Magellan radiometric data (Pitjeva, personal communication to the author, June 2008).
MercuryVenusEarthMars〈Δϖ˙〉(10−4′′cy−1)−36±50[42]−4±5[1]−2±4[1]1±5[1]a (AU)0.3870.7231.0001.523e0.20560.00670.01670.0934P (yr)0.240.611.001.88Table 2
First two columns: pair of planets. Third column:Π for that pair of planets. The errors come from
the realistic uncertainties in 〈Δϖ˙〉.
Fourth column: 𝒜 for that pair of planets. Fifth column: σ level of discrepancy between Π and 𝒜 for that pair of planets.
ABΠ𝒜σVenusMercury0.1±0.31.44EarthMercury0.05±0.181.648MarsMercury−0.03±0.182.0211If we
repeat our analysis by subtracting from〈Δϖ˙〉 the main canonical unmodelled effect, that is,
the general relativistic Lense-Thirring precessions [7–9] induced by the Sun's
angular momentum [12] shown in Table 3, that is, if we use (14)Π⋆≡〈Δϖ˙A〉⋆〈Δϖ˙B〉⋆=〈Δϖ˙A〉−ϖ˙A(LT)〈Δϖ˙B〉−ϖ˙B(LT), 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.Table 3
General relativistic Lense-Thirring
precessions of the longitudes of perihelia of the inner planets of the Solar
System in10−4′′cy−1.
MercuryVenusEarthMarsϖ˙(LT)(10−4′′cy−1)−20−3−1−0.3Table 4
First two columns: pair of planets. Third column:Π⋆ for that pair of planets including the
unmodelled general relativistic Lense-Thirring effect. The errors come from the
realistic uncertainties in 〈Δϖ˙〉.
Fourth column: 𝒜 for that pair of planets. Fifth column: σ level of discrepancy between Π and 𝒜 for that pair of planets.
ABΠ⋆𝒜σVenusMercury0.06±0.511.42.7EarthMercury0.06±0.441.643.5MarsMercury−0.08±0.562.023.7The 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 (1) can also be recast in the
commonly used Yukawa form [4] (15)AY=−GYMr2[1+αY(1+rλ)exp(−rλ)], where (16)GY=GN1+αY,αY=−(G∞−GN)M(G∞−GN)M+GN(M+C1′)2,λ=1μ.  In the case of the Sun, (17)αY⊙=−3.04×10−8,GY=1.00000003040GN,λ=33000AU.A Yukawa-type acceleration of the form of (15) has
been tested  [13] in the Solar System without any a priori assumption on
the size of αY;
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 [13] found (18)λ=aB−aAln(aB/aAΠ)=0.182±0.183AU, which contradicts (17).
(The strength parameter α used in [13] can be identified with αY here.) Using the data for Venus in the equation
for αY [13] (19)αY=2λ2〈Δϖ˙〉GYMaexp(aλ)yields(20)αY=(−1±4)×10−11, which is three orders of
magnitude smaller than the result of (17).
(According to (17), using GN in (19) instead of GY,
as done in [13], does not produce appreciable modifications of the
results.)If we useΠ⋆ for the Earth and Mercury in (18) and 〈Δϖ˙〉⋆ for Venus in (19), the results do not change
appreciably; indeed, we have (21)λ=0.2±0.4AU,αY=(−0.3±2.7)×10−11.


## 3. Conclusions

In the
framework of the attempts of explaining certain astrophysical and cosmological
features without invoking dark matter, MOG [1, 4] is a long-range modified
model of gravity, based on a vector field and three scalar fields representing
running constants, which assumes a Yukawa-like form. Recent developments of
this theory allowed their proponents to fix [2, 4] the values of the
constants entering it. We have shown that, on Solar System scales, MOG yields an anomalous uniform acceleration inducing retrograde planetary
perihelion precessions. 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 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, that is,
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 [2, 4]. In
assessing the results presented here, it must be considered that, at present,
no other researchers have estimated the nonstandard 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 standard perihelion precessions as well.



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