We use the corrections to the Newton-Einstein secular precessions of the longitudes of perihelia 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 an uniform cosmological constant Lambda 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 neatly negative, even by further re-scaling 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.
Introduced for the first time by Einstein (1917) to allow static homogeneous solutions to the Einstein's equations in the presence of matter, the cosmo-logical constant Λ, which turned out to be unnecessary after the discovery of the cosmic expansion by Hubble (1929), 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 (Riess et al., 1998;Perlmutter et al., 1999). For the relation between the cosmological constant and the dark energy see (Peebles and Ratra, 2003). For a general overview of the cosmological constant see (Carroll, 2001) and references therein. Theoretical problems concerning the cosmological constant are reviewed in (Weinberg, 1989).
Since, at present, there are no other independent signs of the existence of Λ 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 precession1 of the perihelia ω of the Solar System’s inner planets (Islam, 1983;Cardona and Tejeiro, 1998;Wright, 1998;Kerr et al., 2003;Kraniotis and Whitehouse, 2003;Dumin, 2005;Iorio, 2006;Jetzer and Sereno, 2006;Kagramanova et al., 2006;Sereno andJetzer, 2006, 2007;Adkins et al., 2007;Adkins and McDonnell, 2007).
Starting from the radial acceleration (Rindler, 2001)
where c is the speed of light, imparted by an uniform cosmological constant Λ in the framework of the spherically symmetric Schwarzschild vacuum solution with a cosmological constant, i.e. the Schwarzschild-de Sitter (Stuchlk and Hledk, 1999) or Kottler (1918) space-time, Kerr et al. (2003) by using the Gauss equations for the variation of the Keplerian orbital elements (Roy, 2005) worked out the secular, i.e. averaged over one orbital revolution, precession of the pericenter of a test-body induced by the cosmological constant
(2)
In it n = GM/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 eq. ( 2) based on the use of the Lagrange perturbative scheme (Roy, 2005). The Lagrange equation for the pericentre is
where i is the inclination angle to the equator of the central mass and V pert is the perturbing potential V pert averaged over one orbital revolution. For the Schwarzschild-de Sitter spacetime the cosmologically-induced additional potential is (Kerr et al., 2003)
By evaluating eq. ( 4) onto the unperturbed Keplerian ellipse defined by
where E is the eccentric anomaly, and integrating over one orbital period
(1 -e cos E)
By inserting eq. ( 8) into eq. (3) one obtains just eq. ( 2). Jetzer and Sereno (2006), Sereno and Jetzer (2007), Adkins et al. (2007) and Adkins and McDonnell (2007) obtained, in different frameworks, the same result of eq. (2). Note that ω Λ ∝ a3 (1 -e2 ), where, for a uniform Λ, the proportionality factor is common to all the bodies orbiting a given central mass. Moreover, eq. ( 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 (Estabrook, 1971), so that eq. ( 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 2 ̟ (Pitjeva, 2005a). 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 (Pitjeva, 2005b) 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 unmodelled force existing in nature.
Since the cosmological accelerated expansion yields Λ ≈ 10 -56 cm -2 , Kerr et al. (2003) concluded that the precession of eq. ( 2) is too small to be measured in the Solar System. Iorio (2006), Jetzer and Sereno (2006), Kagramanova et al. (2006), Sereno and Jetzer (2006), Sereno and Jetzer (2007) Adkins et al. (2007) and Adkins and McDonnell (2007) used eq. ( 2) and the extra-precessions of the inner planets of the Solar System estimated by Pitjeva (2005a) to put constraints on Λ. In particular, Jetzer and Sereno (2006), after working out the effect of Λ on the pericentre of a general twobody 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 (2006) discussed the possibilities offered by futur
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