In article , "Jonathan
Thornburg [remove -animal to reply]"
writes:

(c) Considering this as a "post-Newtonian" power series in v/c,
in turns out that other GR effects cancel out the v/c term.
And still other GR effects cancel out the (v/c)^2 term.
And still other GR effects cancel out the (v/c)^3 term.
And still other GR effects cancel out the (v/c)^4 term.

So, the actual leading order correction to the Newtonian
equations of motion is actually at (v/c)^5, describing the
lowest-order (quadrupole) gravitational-radiation emission.

This is of course a HUGE advantage to people doing N-body simulations
and the like: just use Newtonian physics and any errors due to using the
wrong theory will be much less than other sources of error (finite
resolution (in time and space), simplified astrophysics etc).