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@PearsonArtPhoto's answer links to the page Apollo 8, 21 – 28 December 1968 by Hamish Lindsay at honeysucklecreek.net from where I have quoted below.

Question: What exactly turned on the light indicating Apollo 8 was starting to fall towards the Moon? Was this a calculation based on elapsed time and inertial guidance, or did the Apollo computer use some doppler shift information from signals received from Earth?

Was there a separate subroutine in the Apollo 8's computer program to generate this trigger? Considering memory space was so precious it seems surprising for there to be a separate subroutine dedicated to processing data and triggering a signal just to turn on a light, as cool as that sounds.

EQUIGRAVISPHERE

Borman, Lovell and Anders were the first humans to leave the Earth’s gravity. They also never felt any physical change when the spacecraft slowed down to 3,578 kilometres per hour relative to Earth and crossed over into the Moon’s gravity field at 55:38:40 GET (0629:40 AEST). They were 326,415 kilometres from Earth and 62,598 kilometres from the Moon.

Capcom Mattingly, Welcome to the Moon’s sphere.

Borman, The Moon’s fair?

Mattingly, The Moon’s sphere – you’re in the influence.

Borman, That’s better than being under the influence.

Jack Garman, in the Guidance Support Room in Mission Control, remembered that moment,

“One of my anecdotal stories as they were coasting towards the Moon, which was a long and fairly boring period, a number of us decided to take a bet on exactly when a light would light up on our consoles. This light was connected to a telemetry reading from the on-board computers, that signalled when they determined that they were falling towards the Moon instead of rising away from the Earth, that is, when they had escaped from the Earth.

When that light came on there was silence – it was a kind of dawning – we were witnessing the first time human beings were falling away from the Earth.”

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    $\begingroup$ It was checking the projected trajectory around the moon, when it turned from brownish to blue they knew they where there. :-) $\endgroup$ – Diego Sánchez Feb 18 at 14:53
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The Apollo Guidance Computer used a state vector either centered at the Earth or the Moon. The switchover point is the the lunar sphere of influence, defined in the AGC as 64,373,760 meters (https://www.ibiblio.org/apollo/NARA-SW/R-577-sec5-rev4-5.6-end.pdf PDF page 127).

When in the idle program P00 the AGC will periodically check if it needs to update the stored state vector. When that happens it also checks if it needs to switch the center of its coordinate system. In that case it calculates a lunar centered state vector from its current Earth centered one together with a stored lunar ephemeris. Or vice versa. And to keep track of the currently used coordinate system origin the AGC has a flag (a single bit). The symbol name in the AGC code for this is MOONFLAG: https://www.ibiblio.org/apollo/listings/Colossus237/ERASABLE_ASSIGNMENTS.agc.html#4D4F4F4E464C4147

The flagwords are all downlinked via telemetry so what the flight controllers will have done in this case is to dedicate a light on their console for the state of this flag in the AGC. When it switched over to lunar sphere of influence (wouldn't have happened at exactly the radius given above, because the AGC only updates periodically) then mission control will have known immediately.

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    $\begingroup$ This is great, thank you. I can see now that this was a very important light, and a very important bit! $\endgroup$ – uhoh Feb 17 at 22:46
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    $\begingroup$ Great explanation! What kind of sensor was used to determine when it needed to switch? The phrase that the crew "never felt any physical change" makes me wonder if measuring the direction of gravity in free fall is actually a non-trivial problem, or if it was only imperceptible to humans. $\endgroup$ – Christoph Burschka Feb 18 at 15:51
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    $\begingroup$ @ChristophBurschka: "The direction of gravity in free fall" sounds like an oxymoron. In free fall you're not experiencing any gravity -- if you want to measure your acceleration relative to some particular coordinate system (which is caused by gravity), that is the coordinate system's problem, not yours. You can't measure what a coordinate system is doing; they're abstract constructs. $\endgroup$ – Henning Makholm Feb 18 at 16:04
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    $\begingroup$ @ChristophBurschka No sensor, just orbital mechanics calculations. When it is controlling a burn then the AGC uses accelerometer data to figure out where it is and where it is going, but at all other times it is just doing math to calculate its trajectory. $\endgroup$ – indy91 Feb 18 at 17:05
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    $\begingroup$ @Uwe: If you know at what day you're supposed to be there, you can calculate the expected distance at that date. For a single-trip, days-long mission, that is sufficient - no point in burdening the computer with a calculation when you can just store a constant. $\endgroup$ – Piskvor Feb 26 at 15:13

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