I don't think she consciously headed for either group. She was flying on the LOP hoping to find Howland - first to the north, then to the south.
Flying to the Gilberts was not an option. You can't chart a course to anywhere - regardless of the wind - if you don't know where you are. The LOP was all they had. There were islands on or near the LOP. Stay on the LOP and hope to find one. Leave the LOP and you're dead for sure.
Just to be pedantic about terminology (and with apologies to Gary LaPook)... but this may help people understand some of the navigation issues... there was no
the LOP; there was
a LOP that they thought they were on.
Heading is the direction in which the airplane is pointing. A pilot can fly a heading simply by turning the airplane until it's pointed in the desired direction.
Course is the airplane's track over the ground, which differs from heading to the degree that the wind is blowing. If the airplane is pointing due north, but the wind is coming from the east, then the airplane will follow a track to the northwest. You cannot fly a
course unless you have information that allows you to adjust your
heading to account for the effect of the wind. That information might be knowledge about actual wind speed and direction, or frequently updated fixes that tell you where you actually are. Neither of these were available to Fred Noonan on the morning in question, although he could estimate to some degree.
A
Line of Position is a line that extends in a specific direction and that passes through a specific point on Earth. Right now, where you're standing, point your left hand north and your right hand south. That defines the 000 (N) / 180 (S) LOP that passes through where you're standing. I'm standing on an 000/180 line of position, too; almost certainly not the same one you're standing on. Point your left hand east and your right hand west; that's the 090/270 line of position on which you're standing. Point your left hand northwest (337) and southeast (157), and that's the 157/337 line on which you happen to be standing now. There are, of course infinitely many 157/337 lines around the world.
So why the significance of 157/337?
Imagine looking at Earth from space, and seeing the line that separates the illuminated part (day) from the dark part (night). That line crosses the surface of the earth at different angles at different times of year. On the morning in question, that line of shadow traversed the earth at the angle 157 /337 (on or about March 21, the line would be straight north/south; on or about Dec 21, it would be tipped the other way, from northeast to southwest).
Given the rotation of the earth, that line, dawn, right now, is hundreds of miles west of where it was an hour ago. But if you have an accurate timepiece, and some navigation tables, you can calculate exactly where the line is at the moment you observe dawn.
When Fred Noonan saw the dawn, what that told him is that he was somewhere on a line that extends in the direction 337 (northwest) to 157 (southwest) stretching from the arctic to the antarctic; and his navigation tables and calculations told him exactly which 337/157 line he was on at that instant. but not, of course, where on that line he was. He could, using other information and previous fixes, narrow that down to some degree: obviously he wasn't in Siberia or Antarctica.
In addition to the 157/337 line that Fred knew he was on at the instant of dawn, there's another important 157/337 line: the one that passes through Howland Island. Unlike the one he plotted at dawn, Fred could have drawn that second line on his chart well in advance of the flight: simply drop a ruler onto the chart at Howland; align it 157/337, and pencil in the line.
So now you're Fred. It's dawn, you've just completed your calculations and drawn a 157/337 line on the chart -- you know with great certainty you're on that line. Your job is now to get yourself to the
other 157/337 line -- the one that you drew previously through Howland. Once you're there, you'll know with good certainty that Howland is either northwest of you (bearing 337) or southeast of you (bearing 157). You could even deliberately offset your course, so you'd be pretty sure which side of Howland you were on.
What next? You measure the distance between the two lines of position
along the course that you're flying and find that they're (say, for example) 150 miles apart. And you know that you're flying 150 miles per hour, so that in exactly an hour, you'll be on the 157/337 line that passes through Howland.
Umm... except that you don't really know your course or groundspeed; all you know is your heading and airspeed. So, if you guess that the wind is coming out of the east at 20 miles per hour, but it's actually blowing 30, then, in an hour, you'll be 10 miles further west than you thought you were going to be. And you'll be on a 157/337 line that passes through a point 10 miles west of Howland, not on the 157/337 line that passes through Howland, and 10 miles is enough that you could fly up and down that line all day and never see Howland.
So we have two problems on our hands. First, we can't really put ourselves onto the 157/337 line that passes through Howland; all we can do is approximate it given the LOP we knew were on at dawn and our best estimate of course and speed. And secondly, once we're on what we think is the right 157/337 line, we can't really fly up and down that line... once again, we can't choose a course, only a heading, and the degree to which we can fly a course is completely dependent upon our estimation of wind speed and direction.
So when AE radioed, "we're on the 157/337 line" -- that's not really very interesting; everyone's on one 157/337 line or another. Undoubtedly she meant, "we're on what we hope is the 157/337 line that passes through Howland," but there was some estimation involved in getting there, and there could easily have been 10 or 15 miles of error, more than enough to miss Howland completely.