Lae, sighting Nauru, to Howland [extensive formatting and images have not transferred into this: please see attachments]
The details below are to illustrate the principle of how a 5600ft high light could have thrown them way off course, and whilst intending to be reasonably representative are not definitive.
For simplicity I leave to others the accurate calculation and conventions of true/magnetic, sm/nm, altitude/speed/fuel, metric/imperial, dead-reckoning/celestial/lunar/solar, nautical/Google map subtleties, air/ground speed, etc.
Below are simple rounded figures, and a large chunk of what MyFred might have considered! I use MyFred in attempting to alleviate future Forum-searches using extracts out of context.
They illustrate how with the wrong height for the Nauru light the closer one flies to Nauru the less the subsequent navigational error. Directly over-flying the Nauru light only provides head- and not cross-wind figures.
For indication of winds and weather reports see:
http://tighar.org/Projects/Earhart/Archives/Research/ResearchPapers/Worldflight/2ndattemptweather.htmlThe calculations below do not address whether the before Nauru, and after Nauru, winds are from different, or one, weather system. Here they are regarded as steady for each leg. See footnote: Winds
Nomenclature:
a) Figures for a heading change to counteract a wind I’ve called a course correction.
b) Figures to counteract flying parallel with their intended course I’ve called a course offset.
Course M (sighting Nauru, ignoring Myrtlebank)
Direct flight to Howland on bearing circa 77degrees, with Nauru way-point course correction.
Aim is to arrive 70m north of Howland (range 2560miles) and pass 110m south of Nauru.
110miles is near the maximum distance the 560ft high Nauru light could be seen from the plane at 10,000ft.
Lae-Nauru: Flight altitude 11,000ft.
Range 1500miles at 150mph (air speed) plus 20minutes for the initial climb and subsequent rain/squall avoidance, then mostly. Headwind 10mph, 1500miles/140mph(ground speed)=10.7hrs; 10.7+0.3hours=11hrs.
S to N crosswind component 5mph. Course corrected to S by 11hrsx5mph=55miles over the 11hrs.
Nauru light bearing taken when tangential/opposite it:
Simply:
Say light is circa 500ft asl, thus 11,000ft-500ft=10,500ft=circa 2miles below.
They are on course, so, a fix on the light reads 89degrees, at 2mile high, thus they’d be in their correct position 110miles to its south.
See attachment: 20150106 Nauru lights 110miles.jpg
Believing light is at 5600ft:
Calculation becomes the same angle reading of 89degrees BUT calculated at only 1 mile below, gives range of only 55miles.
They are too far north indicating the southerly crosswind has increased pushing them 55 miles further north.
Two factors ensue:
1: MyFred revises his original 5mph southerly crosswind course correction to 10mph; to bring them back south an additional 55miles (11hrsx10mph=110 miles total) over the Nauru-Howland leg.
2: Believing they are offset 55miles north of intended position he offsets heading for the Nauru-Howland leg to bring them back south by 55miles. [Otherwise they’d continue and be flying parallel with their intended course & arrive 70+55=130miles north of Howland.]
Their heading is now a potential 110+55=165miles to south.
But they are not 55miles off position and after Nauru the southerly crosswind disappears as the wind becomes (veers?) northerly. Thus leaving them with a 165miles of unwanted course correction.
Nauru-Howland leg.:
Range 1050miles, at 150mph air speed.
Headwind component increases to 20mph, =130miles ground speed, flight duration 1050miles/130mph(ground speed)=8hours.
The crosswind is now North to south at 10mph.
Without course correction the crosswind will push them 10mphx8hrs=80miles to south.
However at Nauru they are (as planned) 110miles south, and are on correct course to arrive 70miles north of Howland.
Result:
Intending to be 70miles north of Howland, but now: less 110miles erroneous correction against non-existent southerly crosswind; less 55 miles Nauru-Howland course offset to south; less 80miles of unchecked northerly crosswind = 70-110-55-80=175miles south of Howland (circa 135miles S of Baker).
The GoogleEarth image below, modified in Photoshop, is only illustrative & it is not to scale:
See GoogleEarth attachment: 20150106 Howland nav diagram-02.jpg
What might the weather have been like 100+miles south (& west) of Howland?
A side observation:
Its said* MyFred was an adherent of Sir F Chichester’s method of navigating to an isolated island by flying to a position 200miles N (or S) of the destination. Thus MyFred might well have targeted 200miles north of Howland On a 338degree bearing from Howland this intersects W177deg53’, thus forming a mathematical triangle with 200mile hypotenuse, a 185mile N-S side & a 90mile E-W side. With the same navigational errors and conditions in Course M above they could be 175 miles south but longitude W177deg53‘ thus 185-175=10miles north of Howland but Longitude W177deg53’ passes 90 miles west of Howland’s actual position.
Thus if they then flew southerly along this 338-138 LOP (Line Of Position) they could be nearly 90miles west of Howland and pass more than 80 miles west of Baker. Too far to have spotted Howland especially if there were at a mere 1000ft and there was scattered cloud casting shadows on the sea.
* Ref: M S Lovell in Sound of Wings pg357
Course N (nearer Nauru)
Aim is to arrive 70m north of Howland (range 2560miles) and pass 40m south of Nauru, with intended course alteration after Nauru.
Lae-Nauru: Lae-Nauru bearing 75degrees.
Flight altitude 11,000ft.
Range 1500miles at 150mph (air speed) plus 20minutes for the initial climb and subsequent rain/squall avoidance, then mostly. Headwind 10mph, 1500miles/140mph(ground speed)=10.7hrs; 10.7+0.3hours=11hrs.
S to N crosswind component 5mph. Course corrected to S by 11hrsx5mph=55miles over the 11hrs.
Nauru light bearing taken when tangential/opposite it:
Nauru light fix of 87.1degrees at 2mile high puts them 40miles to south.
See attachment: 20150106 Nauru lights 40miles.jpg
Believing light is at 5600ft: the reading is still 87.1degrees but calculated at only 1 mile below, puts them only 20miles to south of Nauru.
They’ve apparently been pushed 20miles to north so MyFred perceives southerly crosswind has increased by 20/11= approx 2mph.
Two factors ensue:
1) MyFred increases his original 5mph southerly-crosswind correction from 5 to 5+2=7mph, spread over the next 8hours for the Nauru-Howland leg, so as to correct them to the south by 7mph; viz 11hrsx7mph=circa 75miles to the south.
2) Believing they are 20miles north of their intended position he offsets heading for the Nauru-Howland leg to bring them back south by 20miles.
But for the next leg the crosswind has changed (veered?) from southerly to northerly and increased, and they are not 20miles off position.
Nauru-Howland leg:
The now north to south crosswind component increases to 10mph, of which 0mph is corrected against.
This northerly crosswind is now set to push them south by 11hrsx10mph=110miles.
However at Nauru they are 40miles south, and are on correct course to arrive 70miles north of Howland.
Result:
70miles north of Howland, less 75miles non-existent southerly wind correction; less 20 miles Nauru-Howland course offset, less effect of northerly wind pushing them 110miles south: 70 -75 -20 -110 =135miles S of Howland (circa 95miles south of Baker).
BUT they may also not be on the correct 338-158 LOP
In the Nauru examples above they intend to arrive 70miles north of Howland, & need to obtain a sunrise and longitudinal reading to calculate the LOP.
In 1935 Howland was shown as approx 6 miles west of its present-day position. MyFred may or may not have had the correct position.
70miles north of 1935-Howland on bearing 338 is longitude circa 177-03’W
But if they were actually well south and on longitude 177-03degreesW they could be:
Course M (Myrtlebank): 175 miles south and 30 miles west of Howland
Course N (Nauru): 135 miles south and 30 miles west of Howland
Plus the Date Line: (see
http://www.datelinetheory.com/p/effects-of-date-line.html)
If MyFred used the wrong date in his celestial calculations after midnight on 2nd when they were still west of (ie before) the Date Line, the date onboard becomes the 3rd. Continuing to fly east and crossing the Date Line (IDL) it’d restore the date ‘back’ to the 2nd
Using the wrong date after midnight and before crossing the Date Line, erroneously indicates to them that they are 60nm(69sm) being 1 degree longitude east of their actual position. The difference is due to the 4minutes daily change between sidereal day (23hrs 56minutes) and our 24hour day [http://en.wikipedia.org/wiki/Sidereal_time] Thus they’d be about 70m further west in both the examples above!
Course M (Myrtlebank): 175miles south of Howland and 30+70=100 miles west
Course N (Nauru): 135miles south of Howland and 30+70=100 miles west
… umm, a miss is as good as a mile! [and a mistress is even better . . . so the saying goes!]
Yet they made Niku!! So wherever they were might MyFred have decided, after aborting Howland, realise they must be too far south of Howland and thus their LOP was too far west, by at least 30miles, and thus, that they needed to fly east 30+miles, to obtain a fix on longitude 174deg30’W, then south . . . Perhaps 200 to 250miles, circa 2hrs flying and . . . Niku!?
Winds:
a) If the winds are from different weather systems then arguably one could use the average of the increase in wind speed for each leg.
Namely: Lae-Nauru 5mph, increasing to 10mph, being 7.5mph average.
Nauru-Howland 10mph increasing to 20mph, being 15mph average.
b) If it is one veered wind then the effect is mathematically more involved. Perhaps averaging from 5 to 20+mph over the whole route and steadily veering from southerly to northerly which continuously changes the cross- and head-wind component; and additionally the wind’s effect due to the plane bearing:
Roughly: 1st leg: wind somewhat SE, aircraft bearing somewhat NE.
2nd leg: wind somewhat NE, aircraft course-changed to somewhat SE.
. . . a navigators’ paradise!
Noon Sight
Arguably a Noon Sight might have been the best to locate Howland but they‘d have been out of fuel before noon! Should they have left Lae later in the day? (20150108 RFW edit-save 35)