Adam Hibberd
Juno the wife and sister of Jupiter in Greek mythology, is also a spacecraft currently orbiting the planet Jupiter. The spacecraft suffered a major setback in 2016 when the main engine became unusable necessitating key alternative arrangements to be made to the Juno mission plan.
A paper by Avi Loeb, Adam Crowl and I will shortly come out - do look out for it - on the possibility of diverting the Jupiter probe from its current highly elliptical Jupiter orbit in order to execute a flyby of 3I/ATLAS. Juno's current orbit has a perijove altitude of 73,400 km and apojove at 5.8 million km. Just as a reminder, the perijove altitude is the closest approach of Juno to Jupiter and apojove is the farthest point.
This highly eccentric orbit has various advantages should Juno wish to escape Jupiter's pull, in that the extra velocity need at perijove to perform this feat would only be a few 100 m/s, it seems that Juno is teetering on the brink of escape, and only needs a tiny nudge to do so.
This assertion however is not quite correct in the case of 3I/ATLAS due to a problem with Juno's orbital plane (to be precise its Longitude of Ascending Node, LOAN). It turns out not to be at a good angle for 3I/ATLAS missions.
Let's examine this object as a target then. 3I/ATLAS, the third interstellar object to be discovered in our Solar System, will actually get quite close and personal to Jupiter on 16 March 2026, as can be seen on the plot below. Could Juno intercept this and provide invaluable data for scientists on Earth?

Let us assume that there IS sufficient DeltaV for Juno to undertake such an intercept, but how much DeltaV would be needed, to wit, what is the lowest possible DeltaV needed from the engines?
Below is the colour contour plot which provides precisely this information, and assuming that the path to 3I/ATLAS is a direct one without any intervening impulses before the target is intercepted.

Bear in mind, in a final Thelma and Louise moment, the spacecraft is planned to plunge into Jupiter's atmosphere towards the end of September this year. This is a shame because funding for the Juno mission beyond this point is simply not available (even though significant science data could still be returned from extending the mission). There's plenty of life left yet in the old bird.
It seems from the above contour plot that a well timed DeltaV around mid-August, or alternatively early September of 2025 would enable an intercept with 3I/ATLAS around the time of its closest approach to Jupiter, with the single DeltaV for reaching 3I/ATLAS being executed at a magnitude of 3.3 km/s. But is there an alternative lower DeltaV solution?
The answer to this question is a resounding yes, that is by conducting what is known as a Jupiter Oberth Manoeuvre (JOM). For a JOM, first of all an adjustment is necessary to line up the orbital plane of Juno to enable it to reach 3I/ATLAS at the appropriate moment, and secondly another impulsive DeltaV is applied with a low perijove to slingshot the spacecraft to the target. A table of this is provided below.

Observe that the overall DeltaV amounts to 2.1574 + 0.5181 = 2.6755 km/s, which is a few hundred metres per second lower than the direct method.
Given the state of Juno's main engine, would this intercept be pracitical? I am merely a humble astrodynamicist and the key-players for this NASA mission are in a far better position to make the call on this.
Please look out for the preprint on arXiv.