Why does the precession of the earth affect other planets? In the Wikipedia article on numerical integration of planetary motion it states:


  ...the flattening of the Earth causes precession, which causes the
  axial tilt to change, which affects the long-term movements of all
  planets.


Why would precession affect the motion of the other planets?
Why would precession affect the motion of the other planets?


First things first: That's an unreferenced portion of a wikipedia article.





That said, a perfectly spherical body acts exactly like a point mass in Newtonian mechanics. A non-spherical body does not. The Earth's equatorial bulge has a significant effect on satellites in low Earth orbit. Sun synchronous orbits would be impossible without that bulge. For example, a sun synchronous orbit about Venus with its very slow rotation rate and hence very small equatorial bulge is not possible (reference).

The Earth's equatorial bulge adds a quadrupole (inverse quartic, or $1/r^4$) term to the inverse square law relation for point masses, and this inverse quartic term depends not only on distance but also on latitude. The added term is
$$3 J_2 \frac {GM_\oplus a^2}{r^4}\left(\frac 3 2 \cos^2 \lambda - 1\right)$$
where $J_2$ is the Earth's second dynamic form and $\lambda$ is the geocentric latitude.

Since this term drops off as $1/r^4$, it becomes very small beyond the Moon's orbit. But it still exists beyond the Moon's orbit. However, I've yet to see a paper that does account for (or says we need to account for) Earth's $J_2$ effect on the other planets.