I work out the Newtonian and general relativistic effects due to an isotropic mass loss M˙/M of a body on the orbital motion of a test particle around it; the present analysis is also valid for a variation G˙/G of the Newtonian constant of gravitation. Concerning the Newtonian case, I use the Gauss equations for the variation of the elements and obtain negative secular rates for the osculating semimajor axis a, the eccentricity e, and the mean anomaly , while the argument of pericenter ω does not experience secular precession; the longitude of the ascending node Ω and the inclination i remain unchanged as well. The true orbit, instead, expands, as shown by a numerical integration of the equations of motion with MATHEMATICA; in fact, this is in agreement with the seemingly counter-intuitive decreasing of a and e because they refer to the osculating Keplerian ellipses which approximate the trajectory at each instant. A comparison with the results obtained with different approaches by other researchers is made. General relativity induces positive secular rates of the semimajor axis and the eccentricity completely negligible in the present and future evolution of the solar system.