We consider a non-rotating , massive test particle acted upon by a “ pressure ” -type , non-geodesic acceleration arising from a certain general class of gravitational theories with nonminimal coupling between the matter and the metric . The resulting orbital perturbations for a two-body system are investigated both analytically and numerically . Remarkably , a secular increase of the two-body relative distance occurs . In principle , it may yield a physical mechanism for the steady recession of the Earth from the Sun recently proposed to explain the Faint Young Sun Paradox in the Archean eon . At present , the theorists have not yet derived explicit expressions for some of the key parameters of the model , such as the integrated “ charge ” \xi , depending on the matter distribution of the system , and the 4-vector K ^ { \mu } = \ { K ^ { 0 } , \boldsymbol { K } \ } connected with the nonminimal function F . Thus , we phenomenologically treat them as free parameters , and preliminarily infer some indications on their admissible values according to the most recent Solar System ’ s planetary ephemerides . From the latest determinations of the corrections \Delta \dot { \varpi } to the standard perihelion precessions , estimated by the astronomers who produced the EPM2011 ephemerides without modeling the theory considered here , we preliminarily obtain | \xi K| \lesssim 0.1 kg s ^ { -1 } for the Sun and Mars . From guesses on what could be the current bounds on the secular rates of change of the planetary semimajor axes , we get | \xi K ^ { 0 } | \lesssim 1249 kg s ^ { -1 } for Mars . More effective constraints could be posed by reprocessing the same planetary data sets with dedicated dynamical models including the effects studied here , and explicitly estimating the associated parameters . The Earth and the COBE and GP-B satellites yield | \xi K| \lesssim 2 \times 10 ^ { -4 } kg s ^ { -1 } and | \xi K { { } ^ { 0 } } | \lesssim 2 \times 10 ^ { -10 } kg s ^ { -1 } , respectively .