system
Given the following post, respond as if you're commenting on Reddit:
human
Mercury isn't moving at a speed close to that of light. Why did Newtonian gravity fall short in predicting its orbit?

My understanding is that relativistic effects are negligible at speeds far, far below that of light (~50 km/s, give or take, in the case of Mercury's orbital speed). Does that rule of thumb apply on to special relativity?
gpt
The issue here is the Sun's mass much more so than velocity (relativistic effects also happen due to gravitation). Mercury is close enough to the Sun to cause an easily measurable difference in orbital precession relative to Newtonian theory. This too is actually a comparatively small change, if you were looking at your watch face as an analog of a complete orbit about the Sun the relativistic precession of perihelion would be less than 1/5 of a percent of the space swept out by one second hand, or equivalent to the angle a constantly moving second hand sweeps out in 2 milliseconds. Per century. But because orbits are large, this is a measurable amount on the scale of the Solar System (corresponding to a difference in location at perihelion equal to the diameter of the Earth).

It just happened to correspond to something that was possible to measure to that level of precision in the early 20th century. Today there are lots of things where we need to account for relativity. GPS satellites, of course, which rely on very precisely synchronized clocks in orbit, without accounting for relativity measurements would diverge by several km per day between re-calibration. Astrometry space telescopes (like Hipparcos and GAIA) need to account for the fact that the apparent positions of stars will be different depending on whether or not they are viewed along a sight-line that is parallel or perpendicular relative to the position of the Sun, due to the bending of light caused by the Sun's gravity. It's a small effect but for very precise measurements it makes a difference.