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As of November 2020, Pioneer 10 is the 2nd most distant spacecraft from the Sun and Voyager 2 is the 3rd.
However Voyager 2's heliocentric velocity is significantly higher than Pioneer 10 and it seems destined to overtake it sooner rather than later.
When will that date come?
There is useful current data at Heavens Above. The relevant bit is
Voyager 2 is currently (Nov 2020) 125.4 AU from the Sun travelling at 3.231 AU/year, and Pioneer 10 at 127.2 AU and 2.512 AU/yr.
So we expect them to change places in the list in $$\frac{127.2-125.4}{3.231 - 2.512} = 2.50\quad\mathrm{ years}$$ so roughly May 2023.
Currently, PIONEER 10 is at a distance of 134.353 AU from the Sun and VOYAGER 2 is at a distance of 134.356 AU from the Sun, overtaking it at a relative speed of approximately (3.224-2.509)/365 = 0.002 AU per day. So it overtook Pioneer in distance from the Sun about 1.5 days ago, somewhere in the first half of September 10, 2023.
EDIT: At https://heavens-above.com/SolarEscape.aspx at 20:43 UTC distance Pioneer 10 134.353 AU > 134.354 AU, and at 20:45 UTC the distance of Voyager 2 134.356 AU > 134.357 AU, so this difference was almost perfectly 0.003 AU, so 0.003 / 0.0019575 = 1.53252 days = 1 day, 12 hours and about 47 minutes earlier, which gives with high precision September 10, 2023, 08:00 UTC (at more precise calculations around 08:03 UTC with an accuracy of approximately 5 minutes).
@Sheldon points out
I think it's been 20 years since the last contact with Pioneer10, but I assume these numbers are nonetheless very accurate
The blurb in the Horizons output for Pioneer 10 says:
The last fully successful acquisition of signal was March 3, 2002. Pioneer 10 signal at the Earth (<= 185 dBm) is now at DSN threshold limit of reception.
With that important caveat in mind, we can still look at the trajectories in Horizons which will be based on all observations of the spacecraft and an excellent model of the solar system's gravity. Barring some unknown venting events or new Pioneer anomalies, these should be pretty accurate.
The crossover for distance to the Sun (which moves around a bit relative to the solar system barycenter due to mostly Jupiter, Saturn, Neptune and Uranus, in order of decreasing effect) was roughly at about 2023-Jul-18 14:00 UTC at about 20.0343 billion km, and for the distance to the solar system barycenter it came a day earlier at about 2023-Jul-17 15:00 at about 20.0328 billion km.
script: https://pastebin.com/YCxGc6S5
