Are the planetary orbital effects of the solar dark matter wake detectable?

 Recently, there has been some discussion in the literature about the effects of the anisotropy in the spatial density of dark matter in the Solar neighbourhood arising from the motion of the Sun through the Galactic halo. In particular, questions have been asked about the orbital motions of the Solar system’s planets and whether these motions can be effectively constrained by the radiotechnical observations collected by Cassini. I show that the semilatus rectum p, the eccentricity e, the inclination I, the longitude of the ascending node Ω, the longitude of perihelion ϖ, and the mean anomaly at epoch η of a test particle of a restricted two-body system affected by the gravity of a dark matter wake undergo secular rates of change. In the case of Saturn, they are completely negligible, being at the order of ≃0.1 millimetres per century and ≃0.05–2 nanoarcseconds per century: the current (formal) accuracy level in constraining any anomalous orbital precessions is of the order of ≃0.002–2 milliarcseconds per century for Saturn. I also numerically simulate the Earth–Saturn range signature Δρ(t) arising from the dark matter wake over the same time span (2004–2017) as covered by the Cassini data record. I find that it is as low as ${\simeq}0.1{-}0.2\, \mathrm{m}$, while the existing range residuals, computed by astronomers without modelling any dark matter wake effect, are of the order of ${\simeq}$ 30 m. The local dark matter density ϱDM would need to be larger than the currently accepted value of $\varrho _\mathrm{DM}=0.018\, \mathrm{M}_\odot \, \mathrm{pc}^{-3}$ by a factor of 2.5 × 106 in order to induce a geocentric Kronian range signature large enough to make it discernible in the present-day residuals.