Which way, Amelia?

Copyright 2009 by Paul Niquette. All rights reserved.

Lockheed ElectraThe analysis set forth below applies information developed throughout this series of 10 puzzles.  The links indicated in {braces} correspond...

        {0} Explicit Assumption
        {1} The Clock Won't Wait
        {2} Here Comes the Sun
        {3} Landfall Navigating
        {4} Live Reckoning
        {5} Point of No Return 
        {6} Simplexity Aloft
        {7} Gathering Range
        {8} Shoot the Moon
        {9} Wages of Flight
      {10} Which way, Amelia?

Take-off at  Lae, New Guinea
0000 GCT -- Time Aloft 0 hr 
Amelia Earhart (AE) and Fred Noonan (FN) depart Lae, New Guinea for Howland Island, 2,556 miles to the east. 

The Lockheed Electra Model 10-E NR16020 is equipped with two Pratt & Whitney R-1340 Wasp S3H1 engines capable of developing a maximum of 650 hp at take-off, with a fuel consumption rate of 95.3 gph total. The plane is loaded with 1,100 gallons of fuel. 

FN's flight plan included a headwind estimate of 15 mph. The flight takes up a true heading of 083o on a direct course to Howland. 

Begin Cruise-Climb
0005 GCT -- Time Aloft 0.08 hr
Fuel consumed at take-off 8 gallons, 1,092 gallons remaining.  AE throttles back to a cruise-climb power setting that consumes 50 gph at an airspeed of 115 mph and a sustained rate-of-climb of 250 fpm.{9}

With the estimated headwind of 15 mph, the Electra would make good a groundspeed of 100 mph while climbing.

Level at 8,000
0037 GCT -- Time Aloft 0.62 hr
Flight reaches cruise altitude of 8,000 feet.  Distance from Lae 62 miles, with 2,494 miles to go on the 083o course direct to Howland.

Total fuel consumed in climb 35 gallons, 1,065 gallons remaining.  AE levels off and increases airspeed to FN's planned 157 mph consuming 50 gph.{9}

With the estimated headwind of 15 mph, the Electra would make good a groundspeed of 142 mph in level flight.

Abeam Nukumanu 
0708 GCT -- Time Aloft 7.13 hr
AE maintains airspeed at FN's planned 157 mph, consuming 50 gph.

Flight flies abeam north of the island of Nukumanu at 8,000 feet.  Distance from Lae 850 miles, with 1,706 miles to go on the 083o course direct to Howland.  Position report was logged late, at 0718 GCT.  Report included measured headwind of 26.5 mph. {1}

With the measured headwind of 26.5 mph, the Electra makes good a groundspeed of 130.5 mph in level flight.  Total fuel consumed at abeam Nukumanu 361 gallons, 739 gallons remaining.{9}
 

Overhead USS Ontario
1034 GCT -- Time Aloft 10.57 hr

Distance from Lae 1,300 miles, with 1,256 miles to go on the 083o course direct to Howland.

AE repeated attempts to contact USS Ontario 200 miles ahead were all in vain.

FN becomes concerned about whether RDF will be operative to support gathering range {7}, calls for increase in airspeed to shorten dead reckoning distance following sunrise and to reduce steering errors after Last Celestial Fix.{0}

AE increases airspeed to 168.5 mph, consuming 59.3 gph. Total fuel consumed at this location 528 gallons, 572 gallons remaining.{9}

Last Celestial Fix
1737 GCT -- Time Aloft 17.61 hr
Total fuel consumed at this location 946 gallons, 154 gallons remaining.

FN takes Last Celestial Fix as flight approaches the beginning of Nautical Twilight.{2}

Distance from Lae 2,300 miles, with 256 miles to go on the 083o course direct to Howland.  With the measured headwind of 26.5 mph, the Electra made good a groundspeed of 142 mph between USS Ontario and Last Celestial Fix.

AE reduces power for descent at 500 feet per minute, maintaining 168.5 mph and consuming 23 gph.{9}

Landfall Offset Procedure -- at the Last Celestial Fix, FN calls for heading change from 083o to 067o, which is an offset to the left side of direct course, thereby [a] compensating for maximum dead reckoning errors to Howland of plus or minus 16o and [b] selecting the shorter distance to 157-337 LOP. {3}

Level at 1,000
1751 GCT -- Time Aloft 17.84 hr
Total fuel consumed at this location following descent 951 gallons, 149 gallons remaining.{9}

With the headwind of 26.5 mph at 8,000 ft, the Electra was making good a worst-case groundspeed of 142 mph in level flight.  Headwind naturally slows with decreasing altitude to 15 mph at 1,000 ft following the descent.{0}

AE sets up level flight at reduced power to extend endurance aloft, using 30 gph for 120 mph airspeed.{9}

Alternative #1 Actual steering error turns out to be 16o to the right of heading, equal to the allowance by the Landfall Offset Procedure selected at Last Celestial Fix. The Electra is delivered directly to Howland Island.
2003 GCT -- Time Aloft 20.05 hr
Total fuel consumed on arrival overhead Howland Island, 1,024, landing with 76 gallons in reserve, 2:30 search time available.

Note: The flight would have already arrived at Howland before the Last Transmission by Amelia Earhart at 2013 GCT.  Inasmuch as the landing did not take place, Alternative #1 must be rejected.

Alternative #2 Actual steering error turns out to be 0o, which is 16o to the left of direct course to Howland as established by the Landfall Offset Procedure at Last Celestial Fix. The Electra intercepts the 157-337 LOP with 71 miles to go after the right turn to Howland Island.
2033 GCT -- Time Aloft 20.55 hr
Inasmuch as the flight reaches the landfall turning point on 157-337 LOP 71 miles north of Howland, only 2 minutes of flying time are required to reach the visibility limit of 47 miles from Howland at 1,000 ft AGL.

Note: Radio failures mean that no RDF gathering range is established.{7}

Total fuel consumed on arrival overhead Howland, 1,039, landing with 61 gallons in reserve, 2:02 search time available. The flight would have reached 157-337 LOP and turned south 17 minutes before the Last Transmission by Amelia Earhart, after which only  minutes are required to reach visibility limit. 

Note: Since the arrival did not take place, Alternative #2 must be rejected.

Alternative #3 Actual steering error turns out to be 16o to the left of heading, which is beyond the allowance in the Landfall Offset Procedure selected at Last Celestial Fix. The Electra is delivered to the 157-337 LOP with 142 miles to go after the right turn to Howland.
2015 GCT -- Time Aloft 21.25 hr
With the flight reaching the landfall turning point on 157-337 LOP 142 miles north of Howland, a total of 48 minutes of flying time would be required to reach the visibility limit of 47 miles. 

Note: Radio failures mean that no RDF gathering range is established.{7}

If the flight had continued south, the total fuel consumed on arrival overhead Howland, would be 1,060, landing with 40 gallons in reserve, 1:20 search time available.

Note: Since safe arrival was quite possible, Alternative #3 must be considered.

Alternative #3 Last Transmission by Amelia Earhart
2013 GCT -- Time Aloft 20.22 hr
The flight would have already flown 11 minutes on a heading of 157o toward the south at the time of the Last Transmission by Amelia Earhart, thus the Electra was at a position at most 120 miles north of Howland on the 157-337 LOP.
 
Between 2013 GCT and 2113 GCT -- unable to see Howland, AE decides that FN's offset to the north and thus that turn to the south at the 157-337 LOP was the wrong landfall procedure to use.  AE reverses course to a heading of 337o. {0

That assumption complies with all the known conditions, including the profile of radio receptions from the Electra transcribed onboard Itasca.{6}

Ironic Duplication of Conditions:   The analysis in {7} refers to an incident that occurred nearly a month earlier, on June 8, 1937.  AE and FN crossed the Atlantic Ocean eastbound, with Dakar as the destination.  FN had to use dead reckoning for that leg, which covered 1,400 miles from Natal to Dakar. He chose a heading north of Dakar, applying a landfall navigation procedure with an offset of 4o.  Upon arrival at the African coastline, the Electra was supposed to turn right and fly south to the airfield at Dakar.  AE did not believe that procedure to be correct.  Exercising her prerogative as pilot in command, she turned left and flew along the coastline searching for the destination at Dakar. 
Fortunately, there was an airport in French West Africa 120 miles north of Dakar.
As shown in this analysis, conditions approaching Dakar were nearly the same while approaching Howland a month later:
[1] requirement for dead reckoning (but hoping for RDF);
[2] application of landfall navigating procedure (with the
157-337
      LOP
in place of an actual coastline);
[3] FN's prescribed offset of the Electra's course
toward the north by
    
16o {0}, which allows for imprecision in the Last Celestial Fix as
     the 'point of departure' (unlike the beginning of the Atlantic
     crossing, which departed from an aerodrome);
[4] AE's rejection of the turn toward the south
in favor of taking a
     
heading to the north (or AE's reversal of course after not seeing
      destination from cockpit while facing south); and
[5] conducting final search away from the destination on Howland
      Island.  Unfortunately, there was not an alternate airfield -- only
      the Pacific Ocean -- along the
157-337 LOP toward the north.

The puzzles in the Amelianna Collection do not make use of information from flights prior to July 2, 1937. 

Fuel Exhaustion
2248 GCT -- Time Aloft 22.81 hr
Most likely location of the ditching is between 234 and 334 miles north of Howland Island on a heading of 337o from Howland Island.
End-of-Flight Summary (as depicted in the sketch below)...

At 2002 GCT, AE turns southeast  (heading 157) and flies for 11 minutes before making the last transmission at 2013 GCT.  Perhaps as much as an hour later, without receiving radio contact from Itasca and not seeing Howland, AE concludes that FN's offset-north landfall navigating was wrong.  AE turns northwest (heading 337), away from Howland and Itasca.

In Simplexity Aloft AE's 2013 GCT transmission is described as "An ambiguous -- indeed, an impossible -- flight maneuver and altogether useless for guiding subsequent searches." 

Based on the logic in the analysis set forth above, your immediate answer to the question, "Which way, Amelia?" 
must be "South."  Some time later, your answer must be "North."
Some authors (and I) have speculated that the last line in Amelia Earhart's last radio transmission was transcribed in error -- that she really said...

WE ARE RUNNING ON LINE NORTH NOT SOUTH.


 
 


 
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