Does one need to consider relativistic effects when simulating the (or any) Solar system?

Question

There was a question over at Physics asking how to improve a home-brew numerical simulation of the Solar system.

Even though in that case the defects were likely numerical, I started to wonder whether a "Newtonian" approach to the system's dynamic is simply inadequate: Gravitation does not act in an "instantaneous action at a distance"¹. Instead, its influence on spacetime and hence massive bodies progresses with light speed like everything else. The gravitational pull of a fast-moving body points to some location that body has long left, much like the apparent noise source is trailing a passing jet.

Rømer's determination of the speed of light is a famous example for the significance of the light speed at interplanetary distances. Not only the light, also the gravitation vector of remote, fast-moving objects is ever so slightly off.

Do these small differences accumulate to "observable" (say, with 19th century technology) deviations of a Newtonian model from the relativistic reality in human timescales (at most thousands of years)? Do modern simulations account for them?²

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_{¹ Indeed, the theory of relativity has so thoroughly done away with the concept of "action at a distance" that "spooky" is the first thing that comes to mind when we hear the term ;-).}

_{² For some bodies coming close to the Sun general relativistic effects become relevant (and interact with Jupiter's influence) but that's not what I mean here.}

Answer 1

Does one need to consider relativistic effects when simulating the (or any) Solar system?

It depends.

The answer is a resounding all-caps NO

• When trying to model the orbit of an artificial Earth satellite in low Earth orbit. The uncertainties in accelerations due to atmospheric drag overwhelm relativistic effects by multiple orders of magnitude.
• When trying to model and correct the orbit of an artificial satellite in geosynchronous orbit. The frequency of orbital corrections make the small relativistic effects (which grow over time) a bit irrelevant. The uncertainties in accelerations due to not quite perfect orbital corrections make those relativistic effects even more irrelevant.
• When trying to send a vehicle from the Earth to the surface of the Moon and back to Earth in less than two weeks. The time span is too short for relativistic effects to have much of an affect.
• When trying to precisely model the behavior of the outer solar system over multiple decades. Mid-19th century astronomers and mathematicians discovered Neptune by noticing discrepancies in the orbit of Uranus, which had only been discovered 65 years earlier. They did this without using relativity theory, which had not yet been developed.

The answer is a resounding all-caps YES

• When trying to precisely model the behavior of the solar system over centuries or longer. Mid- to late- 19th century astronomers and mathematicians discovered a very small discrepancy in Mercury's argument of latitude. Astronomy in the 19th century (and well into the 20th) used true-of-date coordinates. This alone meant a ~5000 arcseconds per century apparent precession in Mercury's orbit (and that of all other planets) due to the precession of the Earth's rotation axis. Accounting for the influence of other planets on Mercury's orbit added another ~500 arcseconds per century of precession to Mercury's orbit. Note well: Those 19th century astronomers and mathematicians did all of this work by hand.
  Despite their hard work, a tiny 43 arcsecond discrepancy could not be explained by the Earth's axial precession or by perturbations from other planets. The hunt for another planet, tentatively named Vulcan, interior to Mercury's orbit was on! Vulcan was never found. The 43 arcsecond per century discrepancy remained an unsolved enigma until Einstein developed the theory of general relativity, which explained that discrepancy to well within measurement uncertainty.
• When trying to precisely model the behavior of atomic clocks in space. While the relativistic effects on the orbit of an artificial satellite are small, the effects on how precise clocks tick is significant. If a clock on a global navigation satellite (e.g., a GPS satellite) was used on Earth it would be a lousy timekeeper. Atomic clocks in space are subject to measurable relativistic effects. The atomic clocks on global navigation satellites are intentionally made to tick at a different rate than atomic clocks on the surface of the Earth.
• When trying to fly a fast moving probe closer to the Sun than Mercury and that uses multiple gravity assists with Venus to make the probe get even closer to the Sun at perihelion passages. There is no way that the Applied Physics Laboratory in Columbia MD (not to be confused with the Jet Propulsion Laboratory in Pasadena CA) is not using relativity in their modeling of the Parker Space Probe.

Answer 2

Does one need to consider relativistic effects when simulating the (or any) Solar system?

Yes!

In addition to Besides retarded gravitation, anything else to worry about when calculating MU69's orbit from scratch? for which @PM2Ring has linked to @DavidHammen's excellent answer in a comment, there are also several answers to How to calculate the planets and moons beyond Newtons's gravitational force? which explain how to include a very popular approximation for GR effects in n-body simulations on the scale and speed of the solar system, by just adding an extra acceleration term to the simulation:

$$\mathbf{a_{GR}} = GM \frac{1}{c^2 |r|^3}\left(4 GM \frac{\mathbf{r}}{|r|} - (\mathbf{v} \cdot \mathbf{v}) \mathbf{r} + 4 (\mathbf{r} \cdot \mathbf{v}) \mathbf{v} \right),$$

See all those answers for several references and an example of how to use.

For example, NASA JPL certainly does use GR corrections for Horizons. From a very recent The JPL Planetary and Lunar Ephemerides DE440 and DE441 they use much more than what I've listed above:

3.1. Point-mass Acceleration

The point-mass interaction between planetary bodies is governed by the parameterized post-Newtonian (PPN) formulation (Will & Nordtvedt 1972; Moyer 2003)

(very long Equation 27. not copied here)

where the summations are over all bodies, and β and γ are the Eddington–Robertson–Schiff parameters representing the measure of nonlinearity in the superposition law for gravity and the amount of space curvature produced by a unit rest mass, respectively, and are constrained to unity as predicted by the general theory of relativity (GTR).