# Solar System Motions and the Cosmological Constant: A New Approach

Authors: Lorenzo Iorio
Journal: Advances in Astronomy
 (2008)

---

## Abstract

We use the corrections to the Newton-Einstein secular
precessions of the longitudes of periheliaϖ
˙ of some planets (Mercury, Earth, Mars,
Jupiter, Saturn) of the Solar System, phenomenologically estimated
as solve-for parameters by the Russian astronomer E. V. Pitjeva in a
global fit of almost one century of data with the EPM2004 ephemerides,
in order to put on the test the expression for the perihelion precession
induced by a uniform cosmological constant Λ in the framework of the
Schwarzschild-de Sitter (or Kottler) space-time. We compare such an
extra rate to the estimated corrections to the planetary perihelion precessions
by taking their ratio for different pairs of planets instead of using
one perihelion at a time for each planet separately, as done so far in
literature. The answer is  negative, even by further rescaling by
a factor 10 (and even 100 for Saturn) the errors in the estimated extra
precessions of the perihelia released by Pitjeva. Our conclusions hold
also for any other metric perturbation having the same dependence on
the spatial coordinates, as those induced by other general relativistic
cosmological scenarios and by many modified models of gravity. Currently
ongoing and planned interplanetary spacecraft-based missions
should improve our knowledge of the planets' orbits allowing for more
stringent constraints.


---

## Body

## 1. Introduction

Introduced for the first time by Einstein [1] to allow static homogeneous solutions to Einstein's equations
in the presence of matter,  the cosmological constant Λ, which turned out to be unnecessary after the discovery of the cosmic expansion by Hubble [2], has been recently brought back mainly
as the simplest way to accommodate, in the framework of general relativity, the
vacuum energy needed to explain the observed acceleration of the universe
[3, 4]. For the relation between
the cosmological constant and the dark energy see [5]. For a general overview of
the cosmological constant see [6] and references therein. Theoretical problems
concerning the cosmological constant are reviewed in [7].Since, at present, there are no other independent
signs of the existence ofΛ apart from the cosmological acceleration
itself, attempts were made in the more or less recent past to find evidence of
it in phenomena occurring on local, astronomical scales with particular
emphasis on the precession of the perihelia ω of the Solar System's inner planets [8–20] (for other local effects induced by Λ  like
gyroscope precession, mean motion change, geodetic precession, gravitational red-shift,
deflection of light, gravitational time-delay, and doppler tracking
of spacecraft on escape trajectories, see, e.g., [15, 18]).


## 2. The Perihelion Precession Induced by a Uniform Cosmological Constant and the Confrontation with the Data

Starting from
the radial acceleration [21] (1)
A
Λ
=
1
3
Λ
c
2
r
, where c is the speed of light, imparted by a uniform
cosmological constant Λ in the framework of the spherically symmetric
Schwarzschild vacuum solution with a cosmological constant, that is the
Schwarzschild-de Sitter [22] or Kottler [23] space-time, Kerr et al. [16] by using the
Gauss equations for the variation of the Keplerian orbital elements [24] worked out the secular,
that is, averaged over one orbital  revolution, precession of the pericenter of a
test-body induced by the cosmological constant Λ finding (2)
〈
ω
˙
〉
Λ
=
1
2
(
Λ
c
2
n
)
1
−
e
2
, where n
=
G
M
/
a
3 is the Keplerian mean motion of the planet
moving around a central body of mass M, G is the Newtonian constant of gravitation, and a and e are the semimajor axis and the eccentricity,
respectively, of the test-body's orbit.Here, we wish to offer an alternative derivation of
(2) based on the use of the Lagrange perturbative scheme [24]. The Lagrange equation for
the pericenter is (3)
d
ω
d
t
=
1
n
a
2
1
−
e
2
tan
i
∂
〈
𝒱
p
ert
〉
∂
i
−
1
−
e
2
n
a
2
e
∂
〈
𝒱
p
ert
〉
∂
e
, where i is the inclination angle to the equator of the
central mass and 〈
𝒱
p
ert
〉 is the perturbing potential 𝒱
p
ert averaged over one orbital revolution. For the
Schwarzschild-de Sitter space-time, the cosmologically induced additional
potential is [16] (4)
𝒱
Λ
=
−
1
16
Λ
c
2
r
2
. By evaluating (4) onto the
unperturbed Keplerian ellipse defined by (5)
r
=
a
(
1
−
e
cos
⁡
E
)
, where E is the eccentric anomaly, and integrating over
one orbital period P
b
=
2
π
/
n
(6)
d
t
=
(
1
−
e
cos
⁡
E
n
)
d
E
,
⁢ by means of the result(7)
∫
0
2
π
(
1
−
e
cos
⁡
E
)
3
d
E
=
π
(
2
+
3
e
2
)
, the following yields(8)
〈
𝒱
Λ
〉
=
−
1
12
Λ
c
2
a
2
(
2
+
3
e
2
)
. By inserting (8) into (3), one
obtains just (2).Jetzer and Sereno [14], Sereno and Jetzer [19], Adkins et al. [8] and Adkins and McDonnell [9] obtained, in different frameworks, the
same result of (2). Note that 〈
ω
˙
〉
Λ
∝
a
3
(
1
−
e
2
), where, for a uniform Λ,
the proportionality factor is common to all the bodies orbiting a given central
mass. Moreover, (2) was obtained by using the standard radial isotropic
coordinate which is commonly used in the Solar System planetary data reduction
process to produce the ephemerides [25], so that (2) can meaningfully be used for comparisons
with the latest observational determinations of the non-Newtonian/Einsteinian
secular precessions of the longitude of the perihelia ϖ (the longitude of
perihelion ϖ is defined, for orbits nearly equatorial like
the Solar System's ones, as the sum of the argument of perihelion ω and of the longitude of the ascending node Ω;
the latter one, sensitive to the out-of-plane disturbing forces, is not
affected by the entirely radial Λ-induced extra
acceleration) [26]. Indeed, they were estimated by contrasting, in a
least square sense, almost one century of data of different kinds with the
suite of dynamical force models of the EPM2004 ephemerides [27] which included all the
standard Newtonian and Einsteinian dynamics, apart from just any exotic effects
as the ones by Λ on both the geodesic equations of motion and
of the electromagnetic waves. Thus, such extra precessions of perihelia,
estimated independently of our goal, account in principle for any unmodeled
force existing in nature.Since the cosmological accelerated expansion yieldsΛ
≈
10
−
56
cm
−
2, Kerr et al. [16]
concluded that the precession of (2) is too small to be measured in the Solar
System. Iorio [12], Jetzer and Sereno [14], Kagramanova et al. [15], Sereno and Jetzer [18], Sereno and Jetzer [19], Adkins et al. [8] Adkins and McDonnell [9] used (2) and the extra precessions of the inner
planets of the Solar System estimated by Pitjeva [26] to put constraints on Λ.
In particular, Jetzer and Sereno [14], after working out the effect of Λ on the pericenter of a general two-body system
with arbitrary masses in the standard post-Newtonian gauge, used various binary
pulsar systems and planets of the Solar System one at a time separately; Sereno and Jetzer [18] discussed the possibilities offered by
future interplanetary ranging, especially for Pluto; Sereno and Jetzer [19] and Adkins et al. [8] obtained (2) as a particular case in the framework of cosmological models with a nonnull acceleration. Adkins and McDonnell [9] worked out the
perihelion precessions under the action of arbitrary central forces obtaining
(2) as a particular case. In all such cases—and also in previous analyses
performed when only data for Mercury existed such as [10, 13, 17, 20] (in
fact, also the non-Newtonian perihelion precession of the highly eccentric (
e
=
0.826
) orbit of the asteroid Icarus was
investigated [28–31], but it was never used for putting constraints on Λ. We will not use it in the present analysis)—the
general scheme followed was to assume the planets, or the pulsar systems
considered, separately one at a time and to derive constraints on Λ for each of them by considering it as a free
parameter.


## 3. Taking the Ratio of the Perihelia

Here we will
follow, instead, a different approach which is able to tell us something much
more definite about (1) and (2). We will construct the ratios of the estimated
extra precessions of perihelion for different pairs of planets A and B and will
compare them to the corresponding ratios of the precessions of (2) for the same
planets. Thus, it is possible to construct the following
quantity: (9)
λ
AB
=
|
ϖ
˙
A
ϖ
˙
B
−
a
A
3
(
1
−
e
A
2
)
a
B
3
(
1
−
e
B
2
)
|
. If (2) is correct, independently
of the value of Λ provided that, of course, it is nonzero and
small enough to assure that the perturbation approach followed to derive (2) is
appropriate, then (9) must be compatible with zero within the errors in ϖ
˙ and a [27]. It must be noted that such an approach holds, in
general, also for any other extra acceleration term, whatever origin it may
have, of the form (10)
A
=
𝔎
r
,
𝔎
≠
0, so that our conclusions will not
be restricted to the Schwarzschild-de Sitter space-time only. Indeed, there are
other general relativistic cosmological models [32] and many long-range models
of modified gravity [33] which are able to induce a perturbing acceleration
like that of (10) [34].


### 3.1.  The Inner Planets

Let us start
with the inner planets whose relevant orbital parameters are listed in Table1. The answer they
give is negative. Indeed, for the pairs A
=
Mars
,
B
=
Mercury and A
=
Earth
,
B
=
Mercury we have (11)
λ
MarMer
=
7.8
±
0.2,
λ
E
arMer
=
4.1
±
0.2
; a negative result at 40
−
σ and 20
−
σ levels, respectively.
The other four pairs of inner planets yield, instead, results compatible with
zero. The uncertainties in (11) have been conservatively worked out by
propagating the errors (the eccentricities are
negligible) in ϖ
˙’ and a in (9) and linearly summing the resulting
biased terms (12)
δ
λ
AB
≤
|
ϖ
˙
A
ϖ
˙
B
|
(
δ
ϖ
˙
A
|
ϖ
˙
A
|
+
δ
ϖ
˙
B
|
ϖ
˙
B
|
)
+
3
2
(
a
A
a
B
)
3
/
2
(
δ
a
A
a
A
+
δ
a
B
a
B
)
; the dominant sources of
uncertainty are by far the perihelion rates. It is important to stress that
even if δ
ϖ
˙ were 10 times larger than the errors released
by Pitjeva [26] and reproduced in Table 1—which are
not the mere formal, statistical ones—we would still be forced to rule out
(2) at some σ level.Table 1
Estimated semimajor axesa,
in AU  (
1  AU
=
1.49597870691
×
10
11
m) [27], and phenomenologically estimated corrections to the
Newtonian-Einsteinian perihelion rates, in arcseconds per century (
cy
′
′
−
1
), of Mercury, the Earth, and Mars [26]. Also the associated errors are quoted: they are in m
for a [27] and in cy
′
′
−
1 for ϖ
˙ [26]. For the semimajor axes, they are the formal,
statistical ones, while for the perihelia, they are realistic in the sense that
they were obtained from comparison of many different solutions with different
sets of parameters and observations [35].
However, the results presented in the text do not change if δ
ϖ
˙ are rescaled by a factor 10.
Planet
a (AU)
δ
a (m)
ϖ
˙  (
cy
′
′
−
1
)
δ
ϖ
˙  (
cy
′
′
−
1
)
Mercury
0.38709893
0.105
−
0.0036
0.0050
Earth
1.00000011
0.146
−
0.0002
0.0004
Mars
1.52366231
0.657
0.0001
0.0005


### 3.2. The Outer Planets

It is
interesting to use also some of the outer planets for which it was possible to
estimate the corrections to the perihelion precessions [35], quoted in Table 2. The giant
planets whose extra precessions of the perihelion are at our disposal are
Jupiter, Saturn, and Uranus because the temporal extension of the data set used
covered at least one full orbital revolution just for such planets: indeed, the
orbital periods of Neptune and Pluto amount to about 164 and 248 years,
respectively. For the external regions of the Solar System only optical
observations were used, apart from Jupiter [27]; they are, undoubtedly, of poorer accuracy with
respect to those used for the inner planets which also benefit of radar-ranging
measurements, but we will show that they are accurate enough for our purposes.
Let us stress that in Table 2 we rescaled by a factor 10 the otherwise formal,
statistical errors in the estimated extra rates of perihelia. The pair
A
=
Jupiter
,
B
=
Saturn yields (13)
λ
Jup
Sat
=
0.41
±
0.06
; it is incompatible with zero at
about 7
−
σ level. If we further rescale by 10 the error
of Table 2 in the extra precession of Saturn, for which Pitjeva did not use
radiometric data, that is, by 100 its formal, statistical error, we
get (14)
λ
Jup
Sat
'
=
0.4
±
0.2, negative at 2
−
σ level. In regard to Jupiter, since Pitjeva [27] used also
some radiometric data for it, we believe that a rescaling of 10 of the formal
error in its estimated perihelion extra rate is adequate.Table 2
Estimated semimajor axesa,
in AU (
1  AU
=
1.49597870691
×
10
11
m) [27], and phenomenologically estimated corrections to the
Newtonian-Einsteinian perihelion rates [35], in arcseconds per century (
cy
′
′
−
1
), of Jupiter, Saturn, and Uranus. Also the associated errors are
quoted: they are in m for a [27] and in cy
′
′
−
1 for ϖ
˙ [35]. For the semimajor axes, they are the formal,
statistical ones, while for the perihelia, they are the formal ones rescaled by
a factor 10 in order to yield realistic estimates for them.
Planet
a (AU)
δ
a (m)
ϖ
˙  (
cy
′
′
−
1
)
δ
ϖ
˙  (
cy
′
′
−
1
)
Jupiter
5.20336301
639
0.0062
0.036
Saturn
9.53707032
4 222
−
0.92
2.9
Uranus
19.19126393
38 484
0.57
13For the pairA
=
Jupiter
,
B
=
Mercury we
have (15)
λ
Jup
Mer
=
51
±
12 by using the figure of Table 1
for the uncertainty in the extra precession of Mercury; a negative result at
more than 4
−
σ level. If, in a very pessimistic approach, we
rescale by 10 the error of Table 1 in Mercury's extra rate—although it is not
the formal one—we get (16)
λ
Jup
Mer
'
=
51
±
34, which is incompatible with zero
at 1.5
−
σ level. The errors in the semimajor axes, even
if rescaled by 10 or more, do not affect at all our results. The other pairs of
planets yield results compatible with zero.


## 3.1.  The Inner Planets

Let us start
with the inner planets whose relevant orbital parameters are listed in Table1. The answer they
give is negative. Indeed, for the pairs A
=
Mars
,
B
=
Mercury and A
=
Earth
,
B
=
Mercury we have (11)
λ
MarMer
=
7.8
±
0.2,
λ
E
arMer
=
4.1
±
0.2
; a negative result at 40
−
σ and 20
−
σ levels, respectively.
The other four pairs of inner planets yield, instead, results compatible with
zero. The uncertainties in (11) have been conservatively worked out by
propagating the errors (the eccentricities are
negligible) in ϖ
˙’ and a in (9) and linearly summing the resulting
biased terms (12)
δ
λ
AB
≤
|
ϖ
˙
A
ϖ
˙
B
|
(
δ
ϖ
˙
A
|
ϖ
˙
A
|
+
δ
ϖ
˙
B
|
ϖ
˙
B
|
)
+
3
2
(
a
A
a
B
)
3
/
2
(
δ
a
A
a
A
+
δ
a
B
a
B
)
; the dominant sources of
uncertainty are by far the perihelion rates. It is important to stress that
even if δ
ϖ
˙ were 10 times larger than the errors released
by Pitjeva [26] and reproduced in Table 1—which are
not the mere formal, statistical ones—we would still be forced to rule out
(2) at some σ level.Table 1
Estimated semimajor axesa,
in AU  (
1  AU
=
1.49597870691
×
10
11
m) [27], and phenomenologically estimated corrections to the
Newtonian-Einsteinian perihelion rates, in arcseconds per century (
cy
′
′
−
1
), of Mercury, the Earth, and Mars [26]. Also the associated errors are quoted: they are in m
for a [27] and in cy
′
′
−
1 for ϖ
˙ [26]. For the semimajor axes, they are the formal,
statistical ones, while for the perihelia, they are realistic in the sense that
they were obtained from comparison of many different solutions with different
sets of parameters and observations [35].
However, the results presented in the text do not change if δ
ϖ
˙ are rescaled by a factor 10.
Planet
a (AU)
δ
a (m)
ϖ
˙  (
cy
′
′
−
1
)
δ
ϖ
˙  (
cy
′
′
−
1
)
Mercury
0.38709893
0.105
−
0.0036
0.0050
Earth
1.00000011
0.146
−
0.0002
0.0004
Mars
1.52366231
0.657
0.0001
0.0005


## 3.2. The Outer Planets

It is
interesting to use also some of the outer planets for which it was possible to
estimate the corrections to the perihelion precessions [35], quoted in Table 2. The giant
planets whose extra precessions of the perihelion are at our disposal are
Jupiter, Saturn, and Uranus because the temporal extension of the data set used
covered at least one full orbital revolution just for such planets: indeed, the
orbital periods of Neptune and Pluto amount to about 164 and 248 years,
respectively. For the external regions of the Solar System only optical
observations were used, apart from Jupiter [27]; they are, undoubtedly, of poorer accuracy with
respect to those used for the inner planets which also benefit of radar-ranging
measurements, but we will show that they are accurate enough for our purposes.
Let us stress that in Table 2 we rescaled by a factor 10 the otherwise formal,
statistical errors in the estimated extra rates of perihelia. The pair
A
=
Jupiter
,
B
=
Saturn yields (13)
λ
Jup
Sat
=
0.41
±
0.06
; it is incompatible with zero at
about 7
−
σ level. If we further rescale by 10 the error
of Table 2 in the extra precession of Saturn, for which Pitjeva did not use
radiometric data, that is, by 100 its formal, statistical error, we
get (14)
λ
Jup
Sat
'
=
0.4
±
0.2, negative at 2
−
σ level. In regard to Jupiter, since Pitjeva [27] used also
some radiometric data for it, we believe that a rescaling of 10 of the formal
error in its estimated perihelion extra rate is adequate.Table 2
Estimated semimajor axesa,
in AU (
1  AU
=
1.49597870691
×
10
11
m) [27], and phenomenologically estimated corrections to the
Newtonian-Einsteinian perihelion rates [35], in arcseconds per century (
cy
′
′
−
1
), of Jupiter, Saturn, and Uranus. Also the associated errors are
quoted: they are in m for a [27] and in cy
′
′
−
1 for ϖ
˙ [35]. For the semimajor axes, they are the formal,
statistical ones, while for the perihelia, they are the formal ones rescaled by
a factor 10 in order to yield realistic estimates for them.
Planet
a (AU)
δ
a (m)
ϖ
˙  (
cy
′
′
−
1
)
δ
ϖ
˙  (
cy
′
′
−
1
)
Jupiter
5.20336301
639
0.0062
0.036
Saturn
9.53707032
4 222
−
0.92
2.9
Uranus
19.19126393
38 484
0.57
13For the pairA
=
Jupiter
,
B
=
Mercury we
have (15)
λ
Jup
Mer
=
51
±
12 by using the figure of Table 1
for the uncertainty in the extra precession of Mercury; a negative result at
more than 4
−
σ level. If, in a very pessimistic approach, we
rescale by 10 the error of Table 1 in Mercury's extra rate—although it is not
the formal one—we get (16)
λ
Jup
Mer
'
=
51
±
34, which is incompatible with zero
at 1.5
−
σ level. The errors in the semimajor axes, even
if rescaled by 10 or more, do not affect at all our results. The other pairs of
planets yield results compatible with zero.


## 4. Conclusions

In this paper,
we used the latest observation-based determinations of the
non-Newtonian/Einsteinian precessions of the longitudes of periheliaϖ
˙ of some planets of the Solar System obtained
by E. V. Pitjeva (Institute of Applied Astronomy, Russian Academy of Sciences)
[26, 35] with the EPM ephemerides
[27] to constrain the
dynamical effects induced in our planetary arena by a nonzero uniform
cosmological constant in the framework of the Schwarzschild-de Sitter (or
Kottler) space-time. Such corrections to the standard perihelion precessions
were determined without modeling at all the effects of Λ on both the geodesic equations of motion of
planets and electromagnetic waves carrying information on them, so that they
fully account, in principle, for such effects. The ratios of different pairs of
planetary perihelia were used; by conservatively treating the errors in the
estimated extra precessions, it turns out that the expression of the Λ-induced perihelion precession is ruled out at
many σ levels. It is important to note
that our phenomenological approach is quite general, because it holds also for
any small extra acceleration of the form A
=
𝔎
r
,
𝔎
≠
0: such a functional form is, in fact, induced
not only by general relativistic cosmological models [32] but also by modified models
of gravity (see, e.g., [33, 34]).The present analysis relies only upon the extra
precessions estimated so far by Pitjeva; it would be useful if also other teams
of astronomers would estimate independently their own corrections to the
standard planetary perihelion rates by exploiting the huge records of modern
observations currently available. In regard to future perspectives, our knowledge
of the motion of the inner planets of the Solar System should improve in the
near future thanks to the ongoing Messenger (http://messenger.jhuapl.edu/)
and Venus Express (VEX) [36] missions to Mercury and Venus, respectively, and to
the planned BepiColombo (http://messenger.jhuapl.edu/) [37] mission to
Mercury; also planetary laser ranging (PLR) [38], for example, to a lander on Mars [39] or to the Mercury laser
altimeter [40], would
greatly increase the accuracy in planetary orbit determination. In regard to
the outer regions of the Solar System, the processing
of the radiometric data from the ongoing Saturnian mission of Cassini and from
Jupiter's flyby of New Horizons available at
http://pluto.jhuapl.edu/ (it took a gravity assist
by Jupiter in February 2007 and should reach the orbit of Saturn in
mid-2008), recently occurred, might improve our
knowledge of the motion of the outer planets as well in a not too far future.



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*Source: 268647-2008-04-08.xml*