It's not the distance between the camera and Aldrin's feet, it's the distance between the astronauts' feet: between Armstrong's feet and Aldrin's feet. If you're going to do an estimate based on apparent shadow length (starting from one astronaut's feet) compared to where another astronaut appears to be standing (with his feet), you are estimating the distance between their feet.
OK, let's start with your 3.12m between camera and visor, and assume it's correct. Allowing for, say, 150mm increase per astronaut for forward leaning and, say, 100mm for the camera projection (only as far forward as the imaging plane), that puts the astronauts' feet at something around 3.5m apart, give or take.
... so why don't you just go on ahead now and reconcile that with the shadow length, or is the 'uncertainty' over the shadow length because of the many, incalculable unknowns simply a non-starter?!
The latter is the case. The
reflection in the visor is way too distorted to draw meaningful conclusions about objects' relative positions. To illustrate, when one looks at the
reflection in the visor, it seems
unambiguous that when one takes the line connecting the two module pads visible in the reflection, the SWC experiment (the rightmost vertical object) is on the
same side of that line as Armstrong, and on the
opposite side than Aldrin and the sun. From looking at the reflection, it seems inconceivable for the SWC experiment to be on the
same side of that line as Aldrin, and on the
opposite side than Armstrong.
And yet, that is actually the case. What seems clear and unambiguous may not be true, due to the extreme spherical distortion of the reflection, which is greater than one might suspect.
I agree that the spherical distortion of the visors poses major optical problems, and I thank you for illustrating how extreme this can be. In fact,
this photo , that you posted, makes the point equally well. When I first looked at the reflection in Aldrin's visor in this shot I thought it looked extremely odd, in particular the directions that the shadows follow across the visor compared to the directions across the ground. The more you study and analyse them, though, the more they make sense. I'm actually very pleased that you posted this photo, for reasons I'll come on to in a moment.
Having said all that, whilst relative positions of objects can be very difficult to establish just from studying a reflected image in the visor, what's not so difficult is when two objects, or reflections, coincide, or come very close to coinciding, as the distortion then becomes irrelevant. Let's take another look at
this photo, for example (I note that this image is presented as a mirror image of the orginal, for some reason, possibly to re-create the image of Armstrong as if looking from Aldrin. I will continue to comment based on the original imagery). We can clearly see that Armstrong's right foot is planted very close to, if not right on, the intersection of the shadow of the LM strut and the broader shadow of another part of the LM that runs roughly perpendicular, forming a 'T'. We can actually clearly identify that position on
this photo to within a few cm each way; the broader shadow being roughly the length of Armstrong's boot.
Fortuitously, I believe we can also identify reasonably accurately from the same photo the position where Aldrin was actually standing. We can see from
the famous Aldrin photo that Aldrin is standing about a metre or so back from the Lunar Surface Sensing Probe (LSSP), and that he's standing in an obvious indentation, with his right foot fairly centrally planted within it and his left on the slope, close to a small rock. If we go back to
this photo we can identify the indentation immediately to the left of the SWC as we look at the photo, level with a distance about 1/4 the distance up the SWC pole. The cross-hair reticule immediately to the left of the SWC marks an edge of the indentation. I believe the small rock/shadow is actually identifiable midway between the said reticule and a position about 1/4 distance up the SWC pole. Luckily, this photo was taken with the camera aiming very close to perpendicular to the line between Armstrong and Aldrin in the famous Aldrin photo, so we can take relative measurements directly from the photo without worrying unduly about inaccuracy due to perspective.
Now, taking the landing pads as 940mm diameter, as identified from
this article, we have a quantified reference point. Going back to
this photo, we can measure the diameter of the foremost landing pad as 21mm, which we know sits right next to the line between the two astronauts in the famous Aldrin photo. The distance between the mid-point of the 'T' shadow where Armstrong was standing, as identified above, and the position within the indentation where Aldrin's right foot was planted, also identified above, as measured from the photo is about 115mm. 115mm / 21mm x 940mm = 5.2m.
Now, in planning on reminding ourselves what the shadow analysis would derive, I realize I've made a grave error previously. In focusing on the image reflected in Aldrin's visor I mistakenly took Armstrong to be standing
up-sun of Aldrin, whereas he was actually standing
down-sun. This would, of course, given a downward slope in the lunar surface away from Aldrin towards Armstrong, give the effect of
lengthening Aldrin's shadow and not shortening it, as I've previously assumed (I'm surprised nobody's picked this up, and pointed it out; or am I?!
). So, let's recalculate the ball-park distance as derived from a shadow analysis:
An astronaut standing 1900mm tall overall on a level lunar surface with a 15-degree sun angle would cast a shadow 7.1m long. Let's reduce Aldrin's overall height by, say, 300mm to allow for both the depth of the indentation in which he's standing (say 250mm) and a reduction due to forward leaning (say 50mm). That reduces the shadow length to around 6m. And let's assume an upward slope of, say, 3 degrees (we can see from
this photo that the lunar surface appears to slope upwards from right to left, given that Aldrin's forward-facing body is clearly very close to the true vertical, and we can measure that slope by reference to the three cross-hair reticules towards the upper left of the photo (the vertical lines of which are parallel to the near-true vertical plane in which Aldrin is standing, so the horizontal lines of which must be very close to the true horizontal) as around 4mm in 73mm, i.e. 3 degrees). So, allowing for a reduced equivalent sun angle of 12 degrees instead of the actual 15 degrees gives us a compensating increase in the shadow length to 7.5m.
It would seem, therefore, that the actual distance between the two astronauts on the famous Aldrin photo lies somewhere within the range 5.2-7.5m. Whilst that's a large range, it's wholly inconsistent and entirely irreconcilable with the 3.5m calculated from the FOV above. Having identified this it seems more and more obvious, just by casually studying the famous Aldrin photo, that the position where Armstrong was
really standing to take the photo, as determined by the position of the LM landing pad in the bottom of the photo, and the apparent distance away of Aldrin, that he
most certainly was not standing where he is shown to be in the reflection. i.e. at the intersection of the 'T' shadow of the LM landing strut. A casual glance at
this photo clearly confirms that to be the case, especially when you bear in mind that the image centre of Armstrong's camera was aiming
well below Aldrin's right knee, which would have taken in far more of the lunar surface between Armstrong and the LM LSSP than the roughly two boot-lengths that we see in the famous photo.
In closing this particular post, there is one related anomaly that might well serve to contradict everything I've written above, but which, if so, raises similar questions:
Looking again at
this photo, the relative lengths of the SWC pole extending
below the fabric to the lunar surface and the shadow cast by that section of pole are 5mm and 6mm. Assuming a flat lunar surface that would require a sun angle of 40 degrees. In order for what we observe to be consistent with the actual sun angle of 15 degrees the ground would need to slope
upwards in the direction of the shadow, i.e. in the opposite direction to the general slope of the lunar surface(!), at an angle of around 25 degrees (40-15), or steeper than 1:2! That is most definitely not the case, so we can only conclude that the 'sun angle' was far nearer to 40 degrees than 15 degrees. What could possibly account for that?!
