When the Voyagers passed the Pioneers

At this point, the United States has launched 5 spacecraft (and a bunch of junk associated with them¹) onto a path out of the Solar System. These spacecraft are going in different directions and at different speeds and since their launches, there has been some jockeying for position - with more to come.  What isn't clear is when each of these changes in position happened. The spacecraft in question, and the date and order of their launch, are listed below

Pioneer 10 - 3/3/1972
Pioneer 11 - 4/6/1973
Voyager 2 - 8/22/1977
Voyager 1 - 9/5/1977
New Horizons - 1/19/2006

But they have not all been travelling the same speed. By speed they're currently ordered:

Voyager 1
Voyager 2
New Horizons
Pioneer 10
Pioneer 11

And, because they will be travelling for so very very long (possibly "forever") this will eventually be their final order in distance from the Sun (assuming New Horizons doesn't slow down too much - see below). But that hasn't sorted itself out yet and so as of 2017, this is their current order.

Voyager 1
Pioneer 10
Voyager 2
Pioneer 11
New Horizons

This means there has been 4 times that one of the spacecraft have passed one of the other, and there are 3 more position changes yet to come. The future position changes can only be estimated, and might not ever be known with any certainty, but they are

Voyager 2 past Pioneer 10 - ~2019
New Horizons past Pioneer 11 - ???
New Horizons past Pioneer 10 - ???

I have not seen an official estimate for when New Horizons will pass the Pioneers, but we can figure out an estimate.

And the position changes that have already occurred were

Voyager 1 past Voyager 2 - December 19, 1977
Voyager 1 past Pioneer 11 - ???
Voyager 2 past Pioneer 11 - ???
Voyager 1 past Pioneer 10 - February 17, 1998

Unfortunately, when Pioneer 11 was passed was not noted at the time, or if it was, was not recorded anywhere that I can find. But we can come up with an estimate of that too.

Let's do Voyager 1 passing Pioneer 11 first. We can easily put it in a window. Voyager 1 passed Saturn in 1980 about a year after Pioneer 11 did, so it would have to be after that. And in 1987, it was 2.9 billion miles from Earth, a distance Pioneer 11 wouldn't get to until 1990 when it passed beyond Neptune's orbit. So, Voyager 1 must have passed Pioneer 11 between 1980 and 1987.  But that's a pretty big window. We can do better.

We can use the data from a now defunct, but archived, website that gave updated position and velocity data for all four satellites. The oldest page is from 2002. If we use that data, and assume constant speed [(85.037-3.634x)=(61.948-2.474x)], we have an overtake date of 7/19/1982. The problem is we can't assume constant speed because they're slowing down.

So I plugged in the speeds and dates for a bunch of the older pages and fit a linear curve to them and then used that curve to calculate their speed every day and then that to backwards calculate their position. After all that, I got an overtake date of 8/31/1983.

So, I'm willing to say that the actual date was in the 1982-1983 range, but I can't narrow it down anymore than that.

Similarly for Voyager 2 and Pioneer 11, we can make a window between August of 1981 (when Voyager 2 visits Saturn ~24 months after Pioneer 11) and August of 1989 (When Voyager 2 passes Neptune, 6 months ahead of Pioneer 11). But during that time, Voyager 2 passes both Uranus and Neptune and both those encounters change its speed and direction. This makes the speed in 2002 and after somewhat meaningless. About the best I can do is use the numbers above to say that it gained 30 months over 7 years. If we extrapolate that, it means that Voyager 2 overtook Pioneer 11 around March of 1988. But it was probably earlier because Voyager 2 was travelling away from the Sun faster between Saturn and Uranus than it was between Uranus and Neptune.

Figuring out when New Horizons will pass Pioneer 10 and 11 involves extrapolating the speed and location into the future. I have to do it so far out, that tiny errors propagate until they're very large, so this is low confidence. Further complicating things is that New Horizons is slowing down about 10 times faster than Pioneer 11 is right now, and it's very hard for me to say where that will settle out. In fact if it continues to slow down at its current rate, it might not catch Pioneer 10 at all. 3 years ago is was going 0.67 AU/per year faster than Pioneer 10, but now it's only going 0.47 AU faster. It might not even catch Pioneer 11. But at current speeds (as of December 2017), it won't catch Pioneer 11 until 2113 and won't catch Pioneer 10 until 2187, and neither of those account for the rate at which the spacecraft are slowing down.

1. In addition to the five satellites, four of their third stages (all but Pioneer 11's), and a pair of yo-yo de-spin weights from New Horizons are also leaving the Solar System. But these items are untrackable.
2. Lee Siegel (August 30, 1987). "Voyager Spacecraft on Eternal Mission" , New Braunfels Herald-Zeitung