# Solar System Tests of Some Models of Modified Gravity Proposed to Explain Galactic Rotation Curves without Dark Matter

Authors: Lorenzo Iorio; Matteo Luca Ruggiero
Journal: Scholarly Research Exchange
 (2008)

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## Abstract

We consider the recently estimated correctionsΔω¯˙ to the standard Newtonian/Einsteinian secular precessions of the longitudes of perihelia ω¯ of several planets of the Solar System in order to evaluate whether they are compatible with
the predicted anomalous precessions due to models of long-range modified gravity put forth to account for certain features of the rotation curves of galaxies without resorting to dark matter. In particular, we consider a logarithmic-type correction and a
f(R) inspired power-law modification of the Newtonian gravitational potential. The results obtained by taking the ratio of the predicted apsidal rates for different pairs of planets show that the modifications of the Newtonian potentials examined in
this paper are not compatible with the secular extra-precessions of the perihelia
of the Solar System's planets estimated by E. V. Pitjeva as solve-for parameters
processing almost one century of data with the latest EPM ephemerides.


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## Body

## 1. Introduction

Dark matter and dark energy are, possibly, the most
severe theoretical issues that modern astrophysics and cosmology have to face,
since the available observations seem  to question the model of gravitational
interaction on a scale larger than the Solar System. In fact, the data coming
from the galactic rotation curves of spiral galaxies [1] cannot be explained on
the basis of Newtonian gravity or general relativity (GR) if one does not introduce
invisible dark matter compensating for the observed inconsistency of the
Newtonian dynamics of stars. Likewise, since 1933, when  Zwicky [2] studied the
velocity dispersion in the Coma cluster, there is the common agreement on the
fact that the dynamics of cluster of galaxies is
poorly understood on the basis of Newtonian gravity or GR [3, 4] if the dark
matter is not taken into account. In order to reconcile theoretical models with
observations, the existence of a peculiar form of matter is postulated, the
so-called dark matter, which is supposed to be a cold and pressureless
medium, whose distribution is that of a spherical halo around the galaxies. On
the other hand, a lot of observations, such as the light curves of the type Ia
supernovae and the cosmic microwave background (CMB) experiments [5–8],
firmly state that our Universe is now undergoing a phase of accelerated
expansion. Actually, the present acceleration of the Universe cannot be
explained within GR, unless the existence of a cosmic fluid having exotic
equation of state is postulated, the so-called dark energy.The main problem dark matter and dark energy bring
about is understanding their nature, since they are introduced as ad hoc
gravity sources in a well-defined model of gravity, that is, GR (or Newtonian
gravity). Of course, another possibility exists; GR (and its approximation,
Newtonian gravity) is unfit to deal with gravitational interaction at galactic,
intergalactic, and cosmological scales. The latter viewpoint led to the
introduction of various modifications of gravity [9, 10].However, we must remember that, even though there are
problems with galaxies and clusters
dynamics and cosmological observations, GR is in excellent agreement with the
Solar System experiments (see [11, 12]). Hence, every theory that aims at
explaining the large-scale dynamics and the accelerated expansion of the
Universe should reproduce GR at the Solar System scale, that is, in a suitable
weak field limit. So, the viability of these modified theories of gravity at
Solar System scale should be studied with great care.In this paper, we wish to quantitatively deal with
such a problem by means of the secular (i.e., averaged over one orbital
revolution) precessions of the longitudes of the periheliaϖ˙ of some planets of the Solar System (see also
[13, 14]) in the following way. 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 the
secular precession of the perihelion of a planet. For all the exotic models
considered, it turns out that(1)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). 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 is now disappeared, 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 that K is not zero, which is just the case both
theoretically (If K was zero, no modifications 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, as already done, for example, in
[13, 14].If we have observational determinations𝒪 for A
and B of the effect considered above
such that they are affected also by LRMOG (if they are differential quantities
constructed by contrasting observations to predictions obtained by analytical
force models of canonical effects, 𝒪 are, in principle, affected also by the
mismodelling in them) (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 [15–17]), we can construct 𝒪A/𝒪B and compare it with the prediction for it by LRMOG,
that is, with gA/gB.
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(2)𝒪A𝒪B=𝒫A(LRMOG)𝒫B(LRMOG)within the errors, or,
equivalently, if(3)|𝒪A𝒪B−𝒫A(LRMOG)𝒫B(LRMOG)|=0within 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.The paper is organized as follows. In Section2, we apply this approach to a test particle in motion around a central
mass M whose Newtonian gravitational potential
exhibits a logarithmic-type correction. Then, in Section 3, we consider
the power-law modification of the gravitational potential inspired by f(R) extended theories of gravity. Comments and
conclusions are outlined in Section 4.


## 2. The Perihelion Precession Due to a1/r Force

Logarithmic corrections to the Newtonian gravitational
potential have been recently used to phenomenologically tackle the problem of
dark matter in galaxies [18–23]. For example, Fabris and Campos [22]
used(4)Vln=−αGMln(rr0),where α has the dimension of L−1,
to satisfactorily fit the rotation curves of 10 spiral galaxies
getting(5)α≈−0.1kpc−1.The extra potential of (4)
yields an additional 1/r radial force (for another example of a 1/r extra force and its connection with galaxy
rotation curves see [24]):(6)Aln=αGMrr^.Various theoretical
justifications have been found for a logarithmic-like extra potential. For
example, according to Kirillov [21], it would arise from large-scale
discrepancies of the topology of the actual Universe from the Fridman space;
Sobouti [23] obtained it in the framework of the f(R) modifications of general relativity getting
preliminarily flat rotation curves, the Tully-Fisher relation (admittedly with
some reservations), and a version of MOND, while Fabris and Campos [22] pointed
out that string-like objects [25, 26] would yield a logarithmic-type potential
with α related to the string tension.Alternative tests of such proposed correction to the
Newtonian potential, independent of the dark matter issues themselves, would
be, of course, highly desirable and could be, in principle, conducted in the
Solar System. This is, indeed, considered one of the tasks to be implemented in
further investigations by Sobouti [23]; Fabris and Campos [22] argue that, in
view of their extreme smallness due to (5), no detectable effects induced by
(6) would be possible at the level of Solar System.Pitjeva has recently processed almost one century
of data of different types for the major bodies of the Solar System in the
effort of continuously improving the EPM2004/EPM2006 planetary ephemerides [15, 17]. Among other things, she also simultaneously estimated corrections Δϖ˙ to the secular rates of the longitudes of
perihelia ϖ of the inner [16] and of some of the outer
[17, 27] planets of the Solar System as fit-for parameters of a global solution
in which she contrasted, in a least-square way, the observations to their
predicted values computed with a complete set of dynamical force models
including all the known Newtonian and Einsteinian features of motion. As a
consequence, any unmodelled exotic force present in nature is, in principle,
entirely accounted for by the obtained apsidal extra-precessions Δϖ˙ (of course, since modelling is not perfect, in
principle, Δϖ˙ include also the mismodelled parts of the
standard Newtonian/Einsteinian effects, but it turns out that including them in the following computation does not
alter our conclusions). See Table 1 for the inner planets and Table
2 for the gaseous giant ones.  In regard to
them, we must note that modern data sets cover at least one full orbital
revolution only for Jupiter, Saturn, and, barely, Uranus; this is why no
secular extra precessions of perihelia for Neptune and Pluto are today
available.Table 1
Inner planets. First
row: estimated perihelion extra precessions, from [16, Table 3]. The quoted
errors are not the formal ones but are realistic. The units are arcseconds per
century (cy−1′′). Second row: semimajor axes, in Astronomical
Units (AU). Their formal errors are in [15, Table IV] in meter.
Third row: eccentricities. Fourth row: orbital periods in years.
MercuryEarthMars〈Δϖ˙〉(cy−1′′)−0.0036±0.0050−0.0002±0.00040.0001±0.0005a (AU)0.3871.0001.523e0.20560.01670.0934P (yr)0.241.001.88Table 2
Outer planets. First row: estimated perihelion extra precessions [17, 27]. The quoted uncertainties are the formal, statistical errors rescaled by a
factor 10 in order to get the realistic ones. The units are arcseconds per
century (cy−1′′). Second row: semimajor axes in Astronomical Units (AU). Their formal errors are in [15, Table IV] in
meter. Third row: eccentricities. Fourth row:
orbital periods in years.
JupiterSaturnUranus〈Δϖ˙〉 (cy−1′′)0.0062±0.036−0.92±2.90.57±13.0a (AU)5.2039.53719.191e0.04830.05410.0471P (yr)11.8629.4584.07In order to make a direct comparison with them, we
will now analytically work out the secular effects induced by the extra
acceleration of (6) on the pericentre of a test particle. To this aim, we will
treat (6) as a small perturbation of the Newtonian monopole. The Gauss equation
for the variation of ϖ under the action of an entirely radial
perturbing acceleration Ar is(7)dϖdt=−1−e2naeArcosf,in which a is the semimajor axis, e is the eccentricity, n=GM/a3 is the (unperturbed) Keplerian mean motion
related to the orbital period P by n=2π/P,
and f is the true anomaly. After being evaluated
onto the unperturbed Keplerian ellipse, (6) must be inserted into (7); then,
the average over one orbital period P must be performed. It is useful to use the
eccentric anomaly E by means of the relations(8)r=a(1−ecosE),dt=(1−ecosE)ndE,cosf=cosE−e1−ecosE,sinf=sinE1−e21−ecosE.On using(9)12π∫02π(cosE−e1−ecosE)dE=−1+1−e2e,it is possible to
obtain(10)〈ϖ˙〉ln=−αGM(1−e2)a(−1+1−e2e2).Note that (10) is an exact
result with respect to e.
Equation (10) agrees with the precession obtainable divided by P,
the adimensional perihelion shift per orbit Δθp(log) worked out by Adkins and McDonnell [28] within
a different perturbative framework; note that for Adkins and McDonnell [28], the
constant α has the dimensions of ML2T−2 (indeed, V(r) in  [28, (47)] is an additional potential
energy), so that that the substitution α/m→−GMα must be performed in  [28, (50)] to retrieve
(10).It may be interesting to note that for the potential
of (4), the rates for the semimajor axis and the eccentricity turn out to be
zero; it is not so for the mean anomaly ℳ,
but no observational determinations exist for its extra rate.


## 2.1. Comparison with Data

According to
the general outline of Section1, we may now consider a pair of planets A
and B, take the ratio of their estimated extra rates of perihelia, and compare
it to the prediction of (10) in order to see if they are equal within the
errors:(11)ΨAB=|Δϖ˙AΔϖ˙B−aB(1−eA2)aA(1−eB2)(eBeA)2(−1+1−eA2−1+1−eB2)|.If the modification of the gravitational
potential (4), not modelled by Pitjeva in estimating Δϖ˙,
accounts for what is unmodelled in the perihelia precessions, that is, just for Δϖ˙,
then (11)
must be compatible with zero, within the errors. It must be
noted that our approach is able to directly test the hypothesis that the
proposed 1/r exotic force is not zero, irrespective of the
magnitude of α because our prediction for the ratio of the
extra rates of perihelia, that is,(12)𝒫A𝒫B=aB(1−eA2)aA(1−eB2)(eBeA)2(−1+1−eA2−1+1−eB2),is just independent of α itself and is a specific function of the
planets' orbital parameters a and e.
Of course, the ratio of the perihelion rates cannot be used, by definition, to
test the zero hypothesis which, instead, can be checked by considering the
apsidal precessions separately; indeed, in this case the prediction for the
ratio of the perihelion precessions would be, by definition, 0/0.
Note that, being Δϖ˙ observational quantities, their ratio Π=Δϖ˙A/Δϖ˙B is a well-defined quantity, with an associated
uncertainty δΠ which will be evaluated below.From (11) and Table 3, it is possible to
obtain(13)ΨMarMer=0.5±0.2,ΨMerJup=4.2±4.1,ΨEarJup=2.3±0.2,ΨMarJup=1.8±0.2,ΨMerSat=4.91±0.02,ΨEarSat=3.090±0.001,ΨMarSat=2.4984±0.0008,ΨJupSat=1.36±0.06,ΨMerUra=6.9±0.1,ΨEarUra=4.383±0.009,ΨMarUra=3.543±0.005,ΨJupUra=1.9±0.3.The exact eccentricity-dependent
factor(14)F(eA,eB)=1−eA21−eB2(eBeA)2(−1+1−eA2−1+1−eB2)of (11) is always close to
unity, so that its impact on the results of (13) is negligible. The errors in ΨAB due to δa and δΔϖ˙ have been conservatively computed
as(15)δΨAB≤|Δϖ˙AΔϖ˙B|(δΔϖ˙A|Δϖ˙A|+δΔϖ˙B|Δϖ˙B|)+12(aBaA)3/2(δaAaA+δaBaB)F(eA;eB).We did not optimistically summed
the biased terms in a root-sum-square fashion because of the existing
correlations among the estimated extra precessions, although they are low with
a maximum of about 20% between Mercury and the Earth (L. Iorio thanks
E. V. Pitjeva for such an information). It turns out that by rescaling the
formal errors in the semimajor axes released by Pitjeva [15] by a factor
10, 100, or more, the results of (13) do not
change because the major source of uncertainty is given by far by the errors in
the perihelion precessions. In regard to them, let us note that the tightest constraints come from the pairs involving Saturn and Uranus (but not for the
pairs A = Uranus/Saturn, B = Saturn/Uranus yielding values of ΨAB compatible with zero) for which Pitjeva [15]
used only optical data, contrary to Jupiter (this is why a rescaling of 10 of
the formal error in its perihelion extra precession should be adequate).
Now, even by  further rescaling by a
factor 10 the uncertainties, that is, rescaling by a factor 100 their formal, statistical errors, the results do not change (apart from A = Jupiter
B = Uranus). In the case of the pairs  A = Earth, B = Uranus and A = Mars, B = Uranus even
if the real error in the correction to the perihelion precession of Uranus was
1000 times larger than the estimated formal one, the answer of (13) would still
remain negative. For A = Mars and B = Saturn, a rescaling of even a factor 10 000
of the formal error in the  perihelion extra rate of Saturn would be possible
without changing the situation. In the case A = Earth, B = Saturn, a rescaling of
1000 for the Saturn's perihelion extra precession would not alter the result.Table 3
A B denotes
the pair of planets used:Π=Δϖ˙A/Δϖ˙B and 𝒜=(aB/aA)1/2;
F(eA,eB) is given by (14). The perihelion extra rates
for the inner planets have been retrieved from [16]; their errors are not the
formal statistical ones. The perihelion extra rates for the outer planets come
from [27]; their formal errors have been rescaled by a factor 10. The
uncertainties in the semimajor axes have been retrieved from [15, Table IV].
They are the formal ones, but, as can be noted, their impact is negligible.
While Π is always compatible with zero, this is
definitely not the case for 𝒜.
The eccentricity function F is always close to unity.
A BΠ𝒜F(eA,eB)Mars Mercury−0.03±0.20.504±𝒪(10−12)1.008Mercury
Jupiter−0.6±4.13.666±𝒪(10−9)0.989Earth
Jupiter−0.03±0.252.281±𝒪(10−10)1.000Mars
Jupiter0.02±0.171.847±𝒪(10−10)0.998Mercury
Saturn0.004±0.0174.963±𝒪(10−9)0.989Earth
Saturn0.0002±0.00113.088±𝒪(10−9)1.000Mars
Saturn−0.0001±0.00092.501±𝒪(10−9)0.998Jupiter
Saturn−0.006±0.0601.353±𝒪(10−9)1.000Mercury
Uranus−0.006±0.1527.041±𝒪(10−8)0.989Earth
Uranus−0.0003±0.00874.380±𝒪(10−8)1.000Mars
Uranus0.0002±0.00483.459±𝒪(10−8)0.983Jupiter
Uranus0.01±0.311.920±𝒪(10−8)0.999As a conclusion, the logarithmic correction to the
Newtonian potential of (4) is ruled out by the present-day observational
determinations of the Solar System's planetary motions.


## 3. The Power-Law Corrections in thef(R) Extended Theories of Gravity

In the recent
years, there has been a lot of interest in the so-calledf(R) theories of gravity [29]. In them, the
gravitational Lagrangian depends on an arbitrary analytic function f of the Ricci scalar curvature R (see [30] and references therein). These
theories are also referred to as “extended theories of gravity,” since they
naturally generalize, on a geometric ground, GR, in the sense that when f(R)=R the action reduces to the usual
Einstein-Hilbert one, and Einstein's theory is obtained. It has been showed [30]
that these theories provide an alternative approach to solve, without the need
of dark energy, some puzzles connected to the current cosmological observations
and, furthermore, they can explain the dynamics of the rotation curves of the
galaxies, without requiring dark matter. Indeed, for instance Capozziello et
al. [31], starting from f(R)=f0Rk,
obtained a power-law correction to the Newtonian gravitational potential of the
form(16)Vβ=−GMr(rrc)β,(where β is related to the exponent k of the Ricci scalar R), and they applied (16) to a sample of 15 low
surface brightness (LSB) galaxies with combined HI and Hα measurements of the rotation curve extending
in the putative dark matter dominated region. They obtained a very good
agreement between the theoretical rotation curves and the data using only
stellar disk and interstellar gas when the slope k of the gravity Lagrangian is set to the value k=3.5 (giving β=0.817) obtained by fitting the SNeIa Hubble diagram
with the assumed power-law f(R) model and no dark matter.Here, we wish to put on the test (16) in the Solar
System with the approach previously examined.First, let us note that an extra-radial
acceleration(17)Aβ=(β−1)GMrcβrβ−2r^can be obtained from (16); let
us now work out the secular precession of the pericentre of a test particle
induced by (17) in the case β−2<0.
By proceeding as in Section 2 we get (see also [28])(18)〈ϖ˙〉β=(β−1)GM2πrcβaβ−3/2G(e;β),with(19)G(e;β)=−1−e2e∫02πcosE−e(1−ecosE)2−βdE.Since we are interested in
taking the ratios of the perihelia, there is no need to exactly compute (19);
the eccentricities of the Solar System planets are small and similar for all of
them, so that we will reasonably assume that G(eA;β)/G(eB;β)≈1.
Note that β=0 would yield a vanishing apsidal precession
because G(e;0)=0 (the case β=1 is not relevant because it would yield a
constant additional potential and no extra force). In fact, the case β=0 is compatible with all the estimated extra
rates of Tables 1 and  2; β=0,
within the errors, is also the outcome of different tests performed in the
Solar System by Zakharov et al. [32].


## 3.1. Comparison with Data

According to Capozziello et al. [31], a value of β=0.817 in (16) gives very good agreement between the
theoretical rotation curves and the data, without need of dark matter. Again,
if we consider a pair of planets A and B and take the ratio of their estimated
extra rates of perihelia, by taking β=0.817,
we may define(20)ΞAB=|Δϖ˙AΔϖ˙B−(aBaA)0.683|.Of course, such kind of test could
not be applied to the β=0 case since, in this case, we would have the
meaningless prediction 𝒫A/𝒫B=0/0. If the modification of the gravitational potential of (16) exists and
is accounted for by the estimated corrections Δϖ˙ to the standard Newton-Einstein perihelion
precessions, then the quantities ΞAB must be compatible with zero, within the
errors. Instead, what we obtain from Tables 1, 2, 4, and (18) is:(21)ΞEarMer=0.5±0.2,ΞMarMer=0.4±0.2,ΞMerJup=6.5±4.2,ΞEarJup=3.1±0.2,ΞMarJup=2.3±0.2,ΞMerSat=8.92±0.02,ΞEarSat=4.666±0.001,ΞMarSat=3.4997±0.0008,ΞJupSat=1.52±0.06,ΞMerUra=14.4±0.1,ΞEarUra=7.523±0.008,ΞMarUra=5.642±0.005,ΞJupUra=2.4±0.3.It is  remarkable to note that
even if the formal error in the Saturn's apsidal extra precession was rescaled
by a factor 10 000 instead of 10, as done in this paper, the pairs A = Earth
B = Saturn and A = Mars B = Saturn would still rule out (18). A rescaling by 1000 of
the Uranus' perihelion extra precession would still be fatal, as shown by the
pairs A = Earth B = Uranus and A = Mars B = Uranus.Table 4
A B denotes the pair of planets used,Π=Δϖ˙A/Δϖ˙B,
and ℬ=(aB/aA)0.683 (correspondingtoβ=0.817).
The perihelion extra rates for the inner planets have been retrieved from [16];
their errors are not the formal statistical ones. The perihelion extra rates
for the outer planets come from [27]; their formal errors have been rescaled by
a factor 10. The uncertainties in the semimajor axes have been retrieved from
[15, Table IV]; they are the formal ones, but, as can be noted, their impact
is negligible. While Π is always compatible with zero, this is
definitely not the case for ℬ.
A
BΠℬEarth Mercury0.05±0.190.523±𝒪(10−13)Mars
Mercury−0.03±0.20.392±𝒪(10−13)Mercury
Jupiter−0.6±4.15.898±𝒪(10−9)Earth
Jupiter−0.03±0.253.084±𝒪(10−9)Mars
Jupiter0.02±0.172.313±𝒪(10−10)Mercury
Saturn0.004±0.0178.921±𝒪(10−8)Earth
Saturn0.0002±0.00114.665±𝒪(10−9)Mars
Saturn−0.0001±0.00093.499±𝒪(10−9)Jupiter
Saturn−0.006±0.0601.512±𝒪(10−9)Mercury
Uranus−0.006±0.15214.383±𝒪(10−8)Earth
Uranus−0.0003±0.00877.522±𝒪(10−8)Mars
Uranus0.0002±0.00485.642±𝒪(10−8)Jupiter
Uranus0.01±0.312.438±𝒪(10−8)Thus, also the power-law correction to the Newtonian
potential of (16) with β=0.817 is ruled out. Criticisms to Rk models of modified gravity were raised on
different grounds by  Nojiri and Odintsov [33] who proposed more realistic models in [34]. Moreover,
it is well established now that DM shows also particle-like properties. In this
respect, the proposal of Rk gravity as DM (thanks to a change of the
Newton potential) is not considered as a realistic one now. A more realistic DM
candidate from f(R) gravity was suggested by Nojiri and Odintsov
[35]. They show that not only a correction to the Newton potential appears, but
also composite graviton degree of freedom shows particle-like behavior, as
requested by DM data.


## 4. Comments and Conclusions

In this paper, we have studied the secular precession
of the pericentre of a test particle in motion around a central massM whose Newtonian gravitational potential
exhibits a correction which has a logarithmic and power-law behavior. In order
to put on the test the hypothesis that such extra forces are not zero, we
devised a suitable test by taking into account the ratios (and not the extra
rates of the perihelia of each planet at a time separately, or a linear
combination of them, since their uncertainties would fatally prevent obtaining
any useful constraints) of the corrections to the secular precessions of the
longitudes of perihelia estimated by E. V. Pitjeva for several pairs of planets
in the Solar System. The results obtained, resumed by (13) and (21), show that
modifications of the Newtonian potentials like those examined in this paper are
not compatible with the currently available apsidal extra precessions of the
Solar System planets. Moreover, the hypothesis that the examined exotic force
terms are zero, which cannot be tested by definition with our approach, is
compatible with each perihelion extra rate separately, in agreement with our
results. It must be noted that to give the hypothesized modifications of the
Newtonian law, the benefit of the doubt, and given that the formal errors in a and Δϖ˙ of the outer planets are probably
underestimates of the true uncertainties, we multiplied them by a factor ten or
even more, as suggested to one of us (L. Iorio) by
some leading experts in the ephemerides generation field like E. M. Standish,
but the answers we obtained were still negative. Another thing that should be
pointed out is that, in principle, in assessing ΨAB and ΞAB one should have used Δϖ˙∗=Δϖ˙−δϖ˙canonical,
where δϖ˙canonical represents the mismodelled part of the
modelled standard Newton/Einstein precessions. However, neglecting them did not
affect our conclusions, as can be easily noted by looking at the Π and 𝒜/ℬ columns in Tables 3 and 4.
Indeed, the residual precessions due to the imperfect knowledge of, for
example, the solar quadrupole mass moment [16] δJ2/J2≈10% are of the order of ≤10−3cy′′−1 [36], and even smaller are the mismodelled
precessions due to other potential sources of errors like the asteroid ring or
the Kuiper Belt objects [37].However, caution is in order because, at present, no
other teams of astronomers have estimated their own corrections to the
Newtonian/Einsteinian planetary perihelion rates, as it would be highly
desirable. If and when we have, say, two independent determinations of the
anomalous perihelion rate of a given planetx=xbest±δx and y=ybest±δy,
we will see if they are compatible with each other and take their difference |xbest−ybest| as representative of the real uncertainty
affecting the apsidal extra precession of that planet. Moreover, it would be
interesting to see if for different sets of estimated corrections to the
perihelion rates the figures for Π in Tables 3 and 4 change by
an extent sufficient to alter the conclusions of (13) and (21).



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*Source: 968393-2008-12-24.xml*