We apply our recent results on orbital dynamics around a mass-varying central body to the phenomenon of accretion of Dark Matter-assumed not self-annihilating-on the Sun and the major bodies of the solar system due to its motion throughout the Milky Way halo. We inspect its consequences on the orbits of the planets and their satellites over timescales of the order of the age of the solar system. It turns out that a solar Dark Matter accretion rate of approximate to 10(-12) yr(-1), inferred from the upper limit Delta M/M - 0.02 - 0.05 on the Sun's Dark Matter content, assumed somehow accumulated during last 4.5 Gyr, would have displaced the planets faraway by about 10(-2) - 10(1) au 4.5 Gyr ago. Another consequence is that the semimajor axis of the Earth's orbit, approximately equal to the Astronomical Unit, would undergo a secular increase of 0.02 - 0.05 m yr(-1), in agreement with the latest observational determinations of the Astronomical Unit secular increase of 0.07 +/- 0.02 m yr(-1) and 0.05 m yr(-1). By assuming that the Sun will continue to accrete Dark Matter in the next billions year at the same rate as putatively done in the past, the orbits of its planets will shrink by about 10(-1) - 10(1) au (approximate to 0.2 - 0.5 au for the Earth), with consequences for their fate, especially of the inner planets. On the other hand, lunar and planetary ephemerides set upper bounds on the secular variation of the Sun's gravitational parameter GM which are one one order of magnitude smaller than approximate to 10(-12) yr(-1). Dark Matter accretion on planets has, instead, less relevant consequences for their satellites. Indeed, 4.5 Gyr ago their orbits would have been just 10(-2) - 10(1) km wider than now. Dark Matter accretion is not able to explain the observed accelerations of the orbits of some of the Galilean satellites of Jupiter, the secular decrease of the semimajor axis of the Earth's artificial satellite LAGEOS and the secular increase of the Moon's orbit eccentricity.
