20251206_1535_01kbs9a6baer0vrynnmwynjtyp.mp4

A toy-model look at 1/r² vs 1/r⁷․⁵ non-gravitational acceleration

If you’ve been following the drama around interstellar comet 3I/ATLAS, you’ve probably seen some variation of this claim:

“3I/ATLAS’s non-gravitational acceleration has ‘fine-tuned’ its path so that in March 2026 it will skim exactly at Jupiter’s Hill radius — the sweet spot where an alien mothership could drop probes into orbit with minimal fuel.”

Harvard astrophysicist Avi Loeb has popularized this idea in several Medium essays, pointing out that early JPL Horizons solutions for 3I/ATLAS’s trajectory had its closest approach to Jupiter at virtually the same distance as Jupiter’s Hill radius – the radius of the region where Jupiter’s gravity dominates over the Sun’s.

Later JPL solutions, using a different model for the comet’s non-gravitational acceleration, moved this closest-approach distance outward by about 0.085 million km. Loeb argues that if you adjust the radial dependence of the acceleration to be much steeper – something like an r⁻⁷․⁵ profile instead of r⁻² – the match to the Hill radius might be restored.

Cue headlines about “alien engines” and “motherships orbiting Jupiter.”

I’m not interested here in dunking on the possibility of alien technology in general. I am interested in the much narrower technical question:

How big a dynamical effect is that 0.085-million-km shift, really?

And how radical a change in the non-gravitational acceleration law would you actually need to produce it?

To get a feel for that, I built a deliberately simple 1-D toy model of 3I/ATLAS’s non-gravitational acceleration near perihelion. It’s not a full JPL-grade orbit fit — think of it as a calculator for intuition.

The short version of what I found:

• In the toy model, changing the profile from 1/r² to 1/r⁷․⁵ (a much steeper law) moves the effective time at which the acceleration acts forward by about 9 days.
• The perijove shift in question (53.587 → 53.502 million km) corresponds, at Jupiter’s relative speed, to an arrival-time shift of order tens of minutes, not days.
• In the same toy model, that kind of tiny timing shift arises from tweaking the exponent from n = 2.00 to about n ≈ 2.02 — a 1% change.

In other words: if you really want to adjust perijove by 0.085 million km, you don’t need to jump from 1/r² to 1/r⁷․⁵. A tiny adjustment in the radial profile – well within the ordinary uncertainty of cometary outgassing models – is already more than enough.

Let me walk through the reasoning.