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Concepts in Special Relativity (1905) and of General relativity (first developed between 1907 and 1915) substantially predated spaceflight; they were well known and had been tested well before objects were put in space.
I'd like to ask when relativistic corrections were first required in spaceflight mission calculations, and if possible which missions absolutely required relativistic corrections in order to not fail.
I think that the timing of GPS signals was the first real necessity to apply General Relativity to spaceflight, or else precision would be much lower.
According to (Ashby, 1997) and other sources I found the first GPS satellite launched in 1977 was used to prove that General Relativity will have a noticeable effect on the clocks. It turned out that the clock onboard it was off by 38 microsec/day or a resulting positional error of 11km/day. This would make the GPS the first space application that required relativistic corrections.
As far as I know, there has not been a space mission that would have been impossible without a theory of relativistic physics.
It is true that the relativistic effects are clearly visible in GPS clocks. However, if the theory didn't exist, they'd just classify it under "weird observation" and trim the clocks to match ground station clocks. The weird observation could very well lead to the development of a new theory, but even if it didn't, GPS could be made function just as well.
It is conceivable that there could be a mission that would have needed relativistic physics for e.g. orbit planning in advance. Normally small inaccuracies would just be corrected mid-course. In very long distances, fast speeds or high gravity, the course corrections needed would be so large that fuel would run out if theory didn't give accurate predictions. Only at that point I would call it "absolutely required". But so far all space missions have been quite firmly in the Newtonian range of speeds.
Some records so far:
Juno: Speed of $209\,000\text{ km/h} \approx 0.0002 \cdot c$. According to "A possible new test of general relativity with Juno", the relativistic effects in orbit are less than 900 meters. Closest approach to Jupiter's "cloud tops" is 3500 km, so the difference doesn't sound large enough to cause a mission failure.
Parker Solar Probe: Speed of $343\,180\text{ km/h} \approx 0.0003 \cdot c$. It's also close to the most massive body in our solar system, and executes multiple Venus flybys to change orbit over several years, so any errors can accumulate for a long time. I didn't find a calculation on how much of an effect it would have.
According to this answer, relativistic effects due to gravity are, however, quite critical to accurately determine the orbit. So it is quite likely that the science results obtained by this mission would be inaccurate if relativity wasn't accounted for, even though we might not detect it.
JPL's DESCANSO website links to online books describing spacecraft navigation. Page 4-19 of Volume 2 states "The point-mass Newtonian acceleration plus the point-mass relativistic perturbative acceleration ... is given by Eq. (54) of Moyer (1971)." So JPL was incorporating relativistic effects in its navigation calculations at least as early as then.
I don't know which mission was the first to require such corrections.
As others have pointed out, a civilization with zero knowledge of relativity could have carried out all space missions so far, and even constructed the GPS network, if they had simply added in various ad hoc corrections based on experience, without any understanding of the underlying physics. However, it's easy to come up with examples where if you failed to do these ad hoc corrections on the fly, there would be big effects.
As a silly example, suppose that the Mariner 10 mission in 1975 had been planned based on orbital elements of Mercury that had been measured in 1875. These data would have been a century out of date. Now suppose that they understood the dominant reasons for the precession of Mercury's perihelion, which have to do with Newtonian effects from other planets' gravity, but they didn't understand enough GR to know about the anomalous precession. Let $e=0.2$ be the eccentricity of the orbit, $a$ the semimajor axis, and $\delta\theta=2\times10^{-4}$ rad the anomalous precession of its perihelion. Then the error in predicting Mercury's position would be on the order of $ea\delta\theta\sim10^6$ m, which is certainly enough to make the mission fail. Of course this is a silly example, because there's no reason they wouldn't have updated their data on Mercury's orbit for a whole century.
Consider NASA's Gravity Recovery and Interior Laboratory (GRAIL) mission. Not only would each probe's positions need to be precisely known, but the effects of lunar gravity at their locations. This pairing of position and gravitational measurement could qualify for Relativity being necessary for the success of the mission.
https://www.nasa.gov/pdf/582116main_GRAIL_launch_press_kit.pdf
Zuber and company will have to correct for pesky factors such as atmospheric drag, gravitational pull from other planets and general relativity, just to name a few.
https://science.nasa.gov/science-news/science-at-nasa/2008/22may_grail
I'm not sure that any spaceflight missions to date have really needed SR/GR.
The Apollo missions finished in ~1972. GPS didn't kick in until ~1978. The Hubble Space Telescope's Pointing Control System (PCS) uses guide stars for alignment. Interplanetary probes presumably don't rely on Earth GPS(!). The slingshot effect tends to be presented as a Newtonian exercise in momentum exchange, partly for simplicity, and partly because GR has some issues with velocity-dependent gravitomagnetic dragging effects, http://maths.dur.ac.uk/~dma0rcj/Psling/sling.pdf :
"... by its usage the slingshot effect is a modern triumph of Newtonian Mechanics."
The fastest ever space probe seems to be the Parker Solar Probe, listed as having reached 692,000 km/h ( ~192 km/s? ) . By comparison, the the speed of light is more like 300,000 km/s So we're talking about less than a thousandth of the speed of light. At v<0.001c, differences between SR and Newtonian calculations are likely to be small compared to other errors (chemical rockets!). NASA don't expect their initial probe trajectories to be perfect, and the craft have manoeuvring thrusters to make changes and trajectory corrections during their multi-year missions.
GPS and GR
GPS calculations are designed to incorporate SR/GR corrections, but analogous effects also show up under Newtonian-based theory. The two "important" GPS corrections that GR folk like to talk about are the gravitational time dilation effect down on the Earth's surface (which makes GPS' orbiting atomic clocks tick faster than us) and the SR transverse redshift (which slows them down again). The clocks are typically adjusted to compensate for these effects before launch.
The "Newtonian" gravitational shift effect that changes the energy of light as it crosses a gravitational gradient was described by John Michell back in 1783 (paper). Einstein then seems to have rediscovered the effect and in 1911 pointed out that the unavoidable consequence was gravitational time dilation ("On the Influence of Gravitation on the Propagation of Light"), but for simplicity Einstein's 1911 paper derives the effect within Newtonian gravity. So while gravitational time dilation isn't usually "historically" considered to be Newtonian theory, the Newtonian version predates GR by a few years, and could and should have been a Nineteenth-Century prediction, and presumably would have been if not for human fallibility.
Earth gravity is almost certainly going to be probably too weak to tell the two predictions apart, because the Earth's gravitational terminal velocity is only about 11 km/s (as opposed to light's three hundred thousand km/s).
Newtonian theory also gives a counterpart to the second "relativistic" GPS correction, the SR transverse redshift effect. The Newtonian version of this is an "aberration redshift" caused by the forward deflection of rays, which causes the light entering a "transverse-aimed" detector to receive a ray that has a slight recession redshift component. Oliver Lodge "The Ether of Space" p134-136:
"... a spurious or apparent Doppler effect due to common aberration. " http://www.gutenberg.org/files/40911/40911-h/40911-h.htm
The Newtonian "transverse" effect appears identical to the SR effect other than it's a Lorentz-squared redshift rather than a single Lorentz redshift.
As a result there's no obvious reason to believe that GPS wouldn't work just as well outside the SR/GR context, and Ronald Hatch, one of the independent GPS technical experts who served on the GPS board until he died this year (2019), spent some years arguing this point.
In general, in practical situations, we use the Newtonian relationships for astronomy rather than the SR/GR set, because the Newtonian set are simpler, and the differences are so small at normal relative velocities that there's no appreciable difference between them: http://spiff.rit.edu/classes/phys301/lectures/doppler/doppler.html
"... because the velocities of planets, binary stars and clusters of stars are hardly ever more than a few hundred kilometres per second, the classical red-shift equation is accurate enough for most astronomical problems". Foundations of Astronomy, Third edition, Michael A. Seeds (1992)
If we ever manage to get spacecraft travelling at "full percents" of the speed of light, then maybe the difference might start to be important.
