My opinion is that the intent was to arrive in the area earlier than they had, possibly with a Northern intercept as was suggested however that appears not to have happened. As you have pointed out on your page, the 157/337 line is only perpendicular to the Sun's azimuth at sunrise. FN had probably pre-established this line of position on his map and that was the intent but it did not work out that way probably due to the delay of bypassing the storm outside of Lae that cost them over an hour of time.
What I actually said is the exact opposite,
"The myth of the "sunrise observation" was created by people who know "a little bit" about celestial navigation. The 157°-337° sunline line of position (LOP) was derived by an observation of the sun when the sun's azimuth was 67° true since the LOP is at right angles to the azimuth to the celestial body. When the sun rose in the vicinity of Howland on July 2, 1937 its azimuth was 67° true. Those with "a little bit" of knowledge fastened on this fact to claim that Noonan took a sunrise observation and used only it in planning the approach to Howland. (They apparently believe that Noonan then opened the door and dropped his sextant into the sea.) But what these people didn't understand is that
the sun's azimuth stayed at 67° until 1847 Z, an hour after sunrise at Howland! This means that Noonan would have computed the same 157°-337° LOP from any sight taken during this one hour period. He would have taken several additional sights as he approached the LOP then more after the interception to ensure staying on it...
As the day goes on and the azimuth of the sun changes the LOP plotted through the destination will rotate as though on a pivot stuck on the destination. So a one degree change in the azimuth of the sun to
066° at 1847 Z (which lasted until 1923 Z including the time of the "must be on you" message) results in a 156°-336° LOP and would result in only a one nautical mile (NM) error in you position only if you were at least 60 NM out from the destination on the LOP. ( The sine of one degree is approximately 1/60th. so a one degree change results in a one mile change for each 60 miles flown. It is actually one part in 57.3 but navigators use the approximation of one part in 60.) A one mile error falls well within the margin of error of the sextant shot.
The azimuth of the s
un stays at 066° until 1923 Z; 065° until 1945 Z; 064° until 2003 Z and was 063°at 2012 Z when they reported being on the LOP. So even at 2012 Z it would only produce a 4 NM error at 60 NM from Howland on the LOP and only 2 NM if 30 NM from the island. You would expect that this period would have allowed enough time for them to find Howland."
The few degrees of change in the azimuth results in such a small movement of the LOP that these changes can be ignored in practical navigation.
How would they know that they had reached the location of Howland and so know that it was time to start searching?
They tracked their speed since the last known fix and when the ETA was zero that is when they were on the mic stating "we must be on you". I believe the intent was to fly straight in and use the DF to guide them in. When they realized that was not going to happen they circled until they came up with the search strategy.
So they determined when they should be at Howland solely by dead reckoning. The problem with this is the uncertainty in the DR position after flying the leg since the last fix. We have been using as an example that they had a fix at 1623 Z and flew about 400 miles after that so the DR uncertainty, including the 10 NM uncertainty in the original fix, is 50 miles, not nearly accurate enough to find the island which is why getting a sun observation was so critical.
Of course not! Sunrise at Howland was at 1745 Z which is the same time that Earhart radioed "200 miles out."
Precisely why I believe it was a pre-planned line of position and nothing more. Why travel that line to take further observations? It would depend on the time that you arrived if you wanted to be perpendicular to the solar azimuth at the time of your new fix and that is not what they did. Again, I think it is evident that they never established a new fix while in the vicinity of Howland. The only thing that the sunrise could tell them was that they were on track longitudinally when they were 200NM out at 10,000ft at 17:47GMT.
You take additional sights of the sun to ensure staying on the LOP that should pass within 7 NM of the destination. This would require flying a course of 157 or 337 true to stay on the LOP that they were depending upon to take them to Howland. And no, they could not take a sight at 1747 Z, 200 miles west of Howland because the sun had not yet risen there, it had just risen at Howland, and they would also have to wait approximately a half hour after they saw the sun for it to be high enough to take a measurement, see
https://sites.google.com/site/fredienoonan/discussions/the-myth-of-the-sunrise-lop
If they ended up North of Howland, it was cloudy there. They were probably at 1,000ft and would not be above the cloud layer to take further observations. If the sun were obscured by clouds or a higher cloud layer toward the East there may have been no opportunity to shoot the sun or anything else.
That is possible that they ended up north of Howland but I they were flying south east towards Gardner then they would have flown into the clear area around and south of Howland which would have allowd them the opportunity to shoot the sun and to shoot the moon, establish a fix, and go back to look for Howland, their planned destination where there was runway to land on.
gl