So your scenario of about 45 minutes flying to the northwest, then 45 minutes flying back to the southeast, at 2013Z back where they started from, doesn't fit the primary source data. Did you just goof on the times? Care to revise your scenario?
Yeah, I just goofed on the times. Gotta slow down. Let's try again.
Hypothetical: After sunrise Fred took a sun shot and established an LOP. Additional sun shots (speed lines) gave him their ground speed. He then gave AE an ETA for when they would reach the LOP that passes through Howland. He probably wrote it on a piece of paper - maybe just "ETA 19:10."
0742 on ITASCA (1912Z): "We must be on you but cannot see you ..."
Fred would never say something like that. Amelia doesn't understand that the ETA is for arrival at the LOP. It will only be the ETA for arrival at Howland if they were dead on course after a whole night of no celestial sightings due to overcast conditions. Fred knows that's not likely.
Fred knows they'll need to search up and down the LOP to find Howland. There are alternative islands to the southeast of Howland but not to the northwest so the sensible thing to do is search northwest hoping that you've hit the LOP southeast of the island - but you can't afford to search northwest very far because if you're going to search to the southeast you'll have to back-track to your starting point.
Fred tells her to head northwest. After about 15 minutes she still hasn't heard any reply from Itasca so she gives up on asking them to take a bearing on her and decides to try to use her loop antenna
0758 on ITASCA (1928Z): "We are listening but cannot hear you. Go ahead on 7500 with a long count either now or on the scheduled time on half hour." (For AE the "scheduled time on half hour" is in two minutes. She has forgotten, or never realized, that ITASCA has no voice capability on 7500 Kcs. They can't send a "long count." They can only send code.)
0800 on ITASCA (1930Z): "We received your signals but unable to get a minimum ..."
How far do you dare go before you turn around? Fifteen minutes? Half an hour? Forty-five minutes? We'll never know and it doesn't really matter. The fact that Itasca heard a strong signal during the entire period from 1912Z to 2025Z suggests that the plane was within the highest probability of reception range during that time.
0843 on ITASCA (2013Z): "We are on the line 157-337 ..."
0855 0n ITASCA (2025Z): "Running on line north and south"
Also, what is your evidence that AE and FN initially chose to fly northwest (vice southeast) along the LOP?
Logic. All of the alternative islands are to the southeast. Use your remaining fuel heading in the direction that could bring you to an island. Also, she said, "Running on line north and south" not "south and north."
Ric