But back to the point that I was making, Noonan had clear enough skies to take observations of the sun and moon from at least 40 NM north of Itasca continuing well south of Itasca so would have been able to obtain such sights if they were preceding in the direction of Gardner. These observations, would have shown them that they were closer to Howland than to any other island so that the logical thing for them to do was to fly a search pattern to locate Howland.
Ok, lets run through a scenario. If they ended up North of Howland, say in the clouds just 60NM North of Howland at 19:28GMT ("We are circling"). Since they could not obtain a fix while circling they had to pick a direction to fly so that Fred could take a fix. Since as you say Fred knew that a degree change in the solar azimuth was insignificant, lets run with that idea. So for Fred to take a reading he would have wanted AE to fly the heading 157/337 so he could take an observation correct?
The reason for using the landfall (offset) approach is to know for certain what side of the destination you are on when you intercept the LOP so if he ended up north then it was on purpose and he knew to fly a course of 157° true.
I believe your theory would require that they would have flown South in order to obtain a fix. I also believe that would require that they knew it was clear to the South.
They didn't have to know that the sky was clear to the south, they would have had to fly the same course regardless.
But for the sake of argument lets say that this was the case. Say they flew South at 130MPH, they would have been in clear skies in only 5 minutes. How long did you say it requires to take a fix? Lets say 3 readings at 5 minutes per? So say at 19:50GMT they had a fix.
Three minutes for the observation itself plus two more minutes to average the readings and one minute to refer to the precomputed altitude graph, so six minutes for each object observed. You also have to allow two minutes between observations if shifting to a different object. There were only two objects available to observe, the sun and the moon, so only two observations, fourteen minutes total so 1947 Z, close enough, 1950 Z.
See example here.
By this time, the would have flown about 38NM to the South of where they started. Assuming that they were 60NM North originally, this would be roughly 22NM North of Howland. The would have perhaps made a longitudinal adjustment, and would have been on the line by 20:13GMT. In any case, they should have flown within a very close distance to Howland, well within the visual range as they continued South.
So why did this scenario not pan out?
The fix should not have been in error more than 10 NM in any direction. If they did find that the fix placed them 22 NM from Howland they would have measured the course from there to Howland, turned to a heading to make good that course, allowing for the wind that they had determined already, and flying 22 NM would cause the uncertainty to increase slightly to 12 NM, so they should have seen the island, if everything was working right. Why didn't they, possibly some damage to the octant causing all readings to be off by a fixed amount. It is unlikely that it was off by a great amount or the problem should have been found early in the flight so a smaller fixed error, say 1/4 to 1/2 degree which would produce errors of 15 to 30 NM in each LOP, which could put resulting LOPs off from Howland by up to 40 NM.
When they arrive at where they expected to find the island, and they didn't see it, they should have started a standard search pattern which, if continued long enough, has a high probability of finding the island.
Continuing with this scenario, based on a one-half degree constant error in the octant plus being at the extreme of the uncertainty of an observation, assume the plane is actually 40 NM east and 40 NM north of Howland and the visibility is 20 NM then a standard expanding square search pattern would bring them to Howland after flying 80 NM, if the first leg was flown to the west and then south or to the south and then west; 160 NM if the first leg was to the east then followed by a leg to the south and then a leg to the west or if the first leg was north, then west and then south; or, worst case, after flying 240 NM if the first leg was east, then north, then west, then finally to the south. See attached diagram. The first two legs are twice as long as the visibility, the second two legs are four times visibility, the third two legs are six times visibility, etc.
gl