[ggm]

The craft has aerodynamics and speed. It might be figuratively true "unrecoverable" but if it takes e.g. 2 weeks to complete a return, their oxygen and food and batteries ran out. Alternatively if it enters too fast they return ... in pieces.

I think you're being a pedant, if your point is a grazing entry causing rebound skip ultimately returns to some orbital path downward.

[420official]

You seem to intentionally be ignoring the original quote that any error may have caused them to be flung into space. This is patently false unless the one math error is pumping in hundreds of pounds more propellant and burning far longer than the scheduled burns. NASA would need to make a significant series of mistakes beyond orbital math for the "flung out into space" statement to be true.

They certainly could've gotten the return wrong but with a perigee of 119 miles they arent even in a stable orbit and likely could deorbit themselves using only rcs thrusters at apogee, or by just waiting a few orbits.

[dgfl]

This is underselling the risks. On top of the many trajectories which push them into unrecoverable situations, leaving them stranded in orbit, there can be trajectories where the moon gives a gravity assist strong enough to fling the spacecraft into escape velocity, fulfilling the OP.

In fact, the trajectory they chose for this mission exploited the opposite effect to yield a free return without propellant expense.

In the modern day, the chance of a math error being the root cause behind this failure mode are vanishingly small, but minor burn execution mistakes that do not require hundreds of extra pounds of propellant are definitely plausible. They were extremely common in the early days of spaceflight and plagued most of the very first moon exploration attempts. Again, with modern RCS this is unlikely. But reentry is still incredibly tight and dangerous. Apollo famously had a +-1° safe entry corridor, and Orion is way heavier and coming in even faster. If their perigee was off they could’ve easily burned up or doubled their mission time, which they may not have been able to survive.

[420official]

The amount of things that would have to go wrong for the craft to get an accidental gravity boost and be ejected would be significant.

I feel like the original claim paints the whole thing as on a knife edge and barely achieved by virtue of not making a single mistake. In today's age with so many moon landing deniers and worse I feel like we should be specific about where the actual dangers challenges and unknowns there were here. In reality, the orbital mechanics are one of the simplest parts of the entire problem, at least when we're talking about a moon flyby

[dgfl]

Yes, this is a fair point. I agree that orbital mechanics is trivially easy compared to everything else. The chances of a math mistake in particular are null, these trajectories have all been calculated years in advance.

[WalterBright]

The lumpiness of the moon's gravity is not well mapped out.

[accrual]

It is now better mapped after the GRAIL mission: https://en.wikipedia.org/wiki/Gravity_Recovery_and_Interior_...

[WalterBright]

The moon's gravity turns out to be "lumpy" because its density is not constant. This was detected by the Apollo missions and caused them to make errors in orbit calculations. This source of error could have influenced the flyby.