A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding
From: Frank Reed
Date: 2019 Aug 3, 20:10 -0700
I had hoped someone else might reply, but so far no takers. Anyone else? Please feel free!
You say that you're clear on the idea of the cone of position as an extension of the normal circle of position. In that case, I suspect that what you're having trouble picturing is the intersection of that cone of position with the Earth when a sight has been taking using the "Moon's horizon" (another way of thinking about a lunar).
If I measure some angle between the Moon's limb and a star in deep space, that places me on a "cone of position". Everywhere on that infinite cone, the (corrrected) angle between the star and the center of the Moon would have exactly the same vndialue. It would also have that same value where the cone intersects the Moon's surface in a circle of position. The apex of the cone is at the Moon's center. Picture one of these giant cones extending out from the Moon all the way to the Earth. The distance to the Moon compared to the diameter of the Earth is about 30-to-1. So for any but extreme cases, the portion of the cone that intersects the Earth (assuming it does at all) will be nearly a flat plane. When I shoot a lunar from the Earth's surface at known UT, it places me on a nearly planar surface that extends out into space. Shoot a second lunar, and you have two planes. Those two planes will intersect along a line extending from the center of the Moon. Your position is where that line intersects the Earth's surface.
And this is also how the Apollo sextant was originally intended to work for position determination (which did not happen in practice except in Jim Lovell's experiments on Apollo 8, excluding some occasional system tests on later flights). In deep space, you could shoot two lunars. Those would place you on a ray extending out from the center of the Moon. Then, theoretically, you could shoot one "earther" (like a lunar but using the earth), and the ray from the Moon would intersect the cone of position from the earther sight in a single point. A bit less theoretically, you would always know that you're on some family of similar trajectories at any time. For example, you had a known position in high orbit above the Earth at some point and then fired your rocket, and you want to assess your change in apogee. Under certain assumptions, you would known that you're somewhere along an arc of possible positions at a given time. Shooting a single lunar would run a cone through those possible positions and determine which one is correct directly.
So why didn't they do all of that on Apollo with such a fine sextant built right into the hull? First, it wasn't really a sextant. It was closer to a theodolite in many ways, and it could read out in two dimensions (almost always the computer read the results of observations, by the way). It served well in that role determining the spacecraft's orientation relative to the stars. It was an "astro-compass". This was a critical role because the orientation of the inertial navigation system drifted, and there was really no alternative. Automated star trackers would become available in just a few years, but they weren't ready for Apollo. The position-finding role turned out to be much less critical since Mission Control could track the spacecraft based on timing delays of signals received at different ground stations (reverse GPS) and then "feed" the resulting orbit vector back up to the spacecraft automatically.
There's another reason the sextant was less useful than expected. The whole sextant system had an interesting technical flaw which was apparent on the outbound leg of every mission except Apollo 8. The sextant was near the "pointy end" of the Apollo Command Module. But that's also where the Lunar Module was docked after it was pulled out of the SIV-B stage. The LM got in the way! Even when it wasn't directly blocking the sextant's view, it frequently beamed "stray light" into the sextant making sights much more difficult. This was a fairly serious design flaw, and if there had been more time in the schedule (maybe if JFK had suggested "before this millennium is out..." instead of "before this decade is out...") and if the sextant was still considered valuable, it might well have been re-positioned in a Block III or IV or V version of the CM.
Does that help? You have looked at the 3-dim diagrams of this in the NavList archive, right? If not, I can probably find the relevant posts for you.