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Fw: Re: [earhart] Navigation to Dakar, fourth attempt [1 Attachment]
From: Gary LaPook
Date: 2011 Feb 12, 17:31 -0800

﻿A further analysis of Noonan’s navigation makes it even more certain that they planned to land at
Saint Louis and not Dakar. In my first post I showed the areas of uncertainty around Noonan’s
fixes but I didn’t elaborate one what these represented. As a plane flies along its course, the
possibility of wandering further and further off course increases as the distance flown increases.
It is generally accepted that you are very unlikely to be off course more than 10% of the distance
flown and, even then, it is rare that you would be even that much off course and that you are
much more likely to be near where you think you are, your DR position, than near the edge of the
area of uncertainty.

What is important in the analysis of the Dakar story is just the deviation north of course. The
only information that Noonan had concerning his position north and south was the 1341 Z sun
line LOP. The initial band of uncertainty was 7 NM meaning that it was possible that he was
7 NM north (or south) of this LOP but he was more likely to be much closer to the LOP and
unlikely that he was even 7 NM away. As they flew on towards Africa the width of the band of
uncertainty increased at the rate of 13 NM per hour, 10% of the distance flown each hour. The
first running fix after this was at 1625 Z, 2:43 later. Cruising at 130 K (150 mph) the plane had
flown 354 NM in this period so the band of uncertainty grew by 10% of this distance, about 35
NM, which we add to the original uncertainty of 7 NM making the band extend 42 NM north of
the running fix at 1625 Z which is what I illustrated on my chart. Doing the same computations,
the band extended 50 NM at 1700 Z; 63 NM at 1800 Z; and 71 NM at 1836 Z, all of which I
have shown on my chart.

Amelia said:

“When we first sighted the African coast, thick haze prevailed and for some time no position
sight had been possible. My navigator indicated that we should turn south. Had we done so, a
half hour would have brought us to Dakar. But a "left turn" seemed to me in order and after
fifty miles of flying along the coast, we found ourselves at St. Louis, Senegal.” (Last Flight, page
119)

We know that she landed a 1927 Z and that they were cruising at 150 mph (130 K.) To cover the
fifty miles up the coast that Earhart claimed would have taken twenty minutes meaning that they
were at the coastline at least twenty minutes before landing, but probably about 25 minutes to
allow for flying the landing pattern, so they hit the coast at about 1902 Z, according to Earhart.
Doing the same computation as before, the band of uncertainty extended 77 NM north of
Noonan’s plotted position at 1902 Z.

Most of us have heard of standard deviation and this is the concept governing the uncertainty of
dead reckoning. We can consider that the band of uncertainty contains about 95% of the possible
actual positions of the aircraft so there is only a about a 5% chance that you would be outside the
band. In standard deviation terms, the edge of the band of uncertainty is 2 standard deviations
from the DR position. As you exceed this distance the probability that you are further away
decreases very quickly. In 1 case out of a 21 you will be beyond 2 S.D.s; in 1 case out of 370 will
you be more than 3 S.D.s ; in 1 case out of 15,787 will you be further out than 4 S.D.s; in 1 case
our of 1,744,278 will you be out 5 S.D.s; and in only 1 out case out of 506,800,000 will you be
our more than 6 S.D.s.

See: http://en.wikipedia.org/wiki/Standard_deviation

Going the other way, 68% of the time you will be within half of the uncertainty band, at 1 S.D.,
of the DR position which means that only about 32% of the time will you be in the outer one-half
of the error band. The uncertainty at 1902 Z of 77 NM was 2 S.D.s so one S.D was 38.5 NM.

I have attached another image showing where Earhart claims that she hit the shoreline. This is
about 210 SM (180 NM) north of where Noonan’s chart work showed them to be. The error of
180 NM divided by the S.D. of 38.5 NM means that the spot Earhart claimed was 4.7 standard
deviations from Noonan’s position. The probability of being this far off course is less than one
chance in a million!

But what if they had unusual winds or the plane flew at a different speed than planned or Earhart
didn’t stay on the correct heading? Well all those things were taken into account when figuring
the band of uncertainty so none of those things change the conclusion that there was less than one
chance in a million that they hit the coast like Earhart said they did based on Noonan’s chart
work.

If the math isn’t convincing then lets look at it with just common sense. Noonan had just missed
his landfall on the continent of Africa by more than two hundred miles and now Earhart was
going to have him navigate her to a one mile long island in the middle of the Pacific. As my kid
would say, “I don’t think so!”

gl

zσ Percentage within CI Percentage outside CI Fraction outside CI
0.674σ 50% 50% 1 / 2
68.2689492% 31.7310508% 1 / 3.1514872
1.645σ 90% 10% 1 / 10
1.960σ 95% 5% 1 / 20
95.4499736% 4.5500264% 1 / 21.977895
2.576σ 99% 1% 1 / 100
99.7300204% 0.2699796% 1 / 370.398
3.2906σ 99.9% 0.1% 1 / 1000
99.993666% 0.006334% 1 / 15,787
99.9999426697% 0.0000573303% 1 / 1,744,278
99.9999998027% 0.0000001973% 1 / 506,800,000
99.9999999997440% 0.0000000002560% 1 / 390,700,000,000

--- On Sat, 2/12/11, Gary LaPook <garylapook@pacbell.net> wrote:

From: Gary LaPook <garylapook@pacbell.net>
Subject: [earhart] Navigation to Dakar, fourth attempt [1 Attachment]
To: earhart@yahoogroups.com
Date: Saturday, February 12, 2011, 12:21 AM

[Attachment(s) from Gary LaPook included below]

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