Amelia Earhart's point of no return

Boy, that term "Point Of No Return" makes shivers run down my spine and has been used in many movies to heighten tension. "We can't turn back, we must go forward, no matter what!"

In reality, it's not that dramatic.


The "point of no return" is just another example of the more general "radius of action" calculation. This was known in 1937 since it was published by Noonan's friend Weems in Air Navigation, 1938, and even earlier in Navigation and Nautical Astronomy, Dutton, 1934.

The formula is quite simple and there are various ways to write it depending on how you want the result. The inputs are the endurance (based on the fuel on board divided by fuel flow) the head (or tail) wind component for the approximate first half of the flight and the true airspeed of the plane. The true airspeed and the wind component are combined to determine the ground speed out (on course towards the destination) and the ground speed for the return leg.

So:

Time (to PNR) = (Endurance X GS return)/(GS return + GS out)

Since the wind component causes an equal but opposite effect on the ground speed for the GS out and the GS return the divisor is simply 2 X TAS so the formula can be rewritten:

Time PNR = (Endurance X GS return)/(2 X TAS)


We know for sure that the plane had an endurance of at least 20:13. Using the TAS of 130 knots (150 mph) and the wind component of 23 knots as determined in flight, and reported by Noonan, we can substitute into the formula:

GS out = 107 K
GS return = 153 K
2 X TAS = 2 X 130 = 260 K

T = (20:13 X 153)/260

T = 11:54
If we multiply this time by the GS out we find the distance to the PNR.

Dist PNR = 107 K X 11:54 = 1273 NM

Alternatively, if you just want the distance to the PNR you can use the formula:

Dist PNR = (Endurance X GS out X GS return)/(2 X TAS)

D = (20:13 X 107 X 153)/ 260

D =  1273 NM

To confirm that his is a correct result we can divide the distance by the GS return:

1273 NM / 153 K = 8:19

8:19 + 11:54 = 20:13, the endurance.

So we can be sure that had they turned around prior to 11:54 Z  they could have made it back to Lae. If, in fact, the endurance was only 20:13 then we also know that if they turned around any time after 11:54 Z that they could not make it back to Lae, they would have been past the "point of no return."

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Using a 15 knot wind component from the July 1st forecast you get:

11:16 and 1297 NM

Using the 25 knot wind component from the July 2nd forecast you get:

12:03 and 1265 NM

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If Noonan thought he had a 24 hour endurance using the 23 knot wind component then he would have calculated the PNR as 14:07 at 1511 NM. Using a 15 knot wind component then he would have gotten 13:23 at 1539 NM and with a 25 knot wind he would have gotten 14:18 and 1502 NM, just past Nauru, so the decision had to be made more than 100 NM west of the Gilberts at about the time that Itasca first heard from the plane at 1415 Z.


See excerpts of navigation manuals relating to this topic here.

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We have now looked at one simple example of the “point of no return” so this would be a good point to do some
more computations. There is also a theory that Earhart made it back to the island of New Britain and a point of no
return calculation may help in an analysis of this theory.

The PNR is a simple case of the “radius of action” calculation. These calculations determine how
far away you can fly and still make it back within the endurance of the aircraft. If you go beyond
the PNR or the “radius of action” then you can’t make it back to the departure airport, that’s why
it is called the “point of no return.”

Navy pilots flying off of aircraft carriers have to do a more complicated radius of action
calculation because if they just make it back to the point where they took off from, there won’t be
an airport there, the carrier has moved on. It should be obvious that if the carrier is steaming in
the opposite direction from the plane's outbound course that the plane will have to turn around
sooner to go back and chase after the carrier.

The way this “radius of action from a moving base” calculation is done is by drawing a vector
diagram including the normal wind vector and then adding a vector to represent the speed and
course of the carrier. Then the radius of action (PNR) calculation is done with the combined
effect of these two vectors. Conceptually, the calculation is done based on the wind that would
have been measured by the moving carrier.

We can use the “radius of action from a moving base” computation to look at the case of the
plane departing from Lae and returning to New Britain. We do this by using a “fictitious aircraft
carrier.” The east end of New Britain is 344 NM east of Lae on the course line to Howland. If a
fictitious carrier departed  Lae at the same time as Earhart, steaming towards Howland, it would
have arrived at the east end of New Britain at the end of 20:13 (the proven endurance of the
plane) by steaming at 17 knots. Fortunately, the required vector diagram is as simple as it could
be since the plane and the ship were heading directly into the 23 knot headwind measured by
Noonan. So the fictitious carrier would have measured a direct headwind of 40 knots. We use
this 40 knot value instead of the true wind of 23 knots to do the calculation for the PNR for a
return to New Britain.

Doing the calculation:

TAS = 130 K   (2 x TAS = 260 K)

Speed of relative movement out   = 90 knots.

(The plane is moving away from the fictitious carrier at only 90 knots because the carrier is
chasing after the plane.)

Speed of relative movement return = 170 knots (130 K + 40 K)

PNR time = (20:13 x 170 K)/260

PNR time = 13:13

Multiplied by the speed of relative movement out of 90 K places the plane 1190 NM from the
fictitious carrier. But since the real ground speed was 107 K it would be 1414 NM from Lae.
This is 141 NM further and 1:19 later than in our first computation of  PNR for a return to Lae.
To check our math we can subtract this 13:13 from the endurance of 20:13 giving us 7:00 hours
to return to New Britain. Seven hours multiplied by the actual return ground speed of 153 knots
means the plane will travel 1071 NM back towards Lae. Since it would be starting 1414 NM
from Lae it will end up 344 NM east of Lae at the eastern end of New Britain, just as we expected.

Doing the same computation using an endurance of 24 hours we use a slightly slower speed for
the fictitious carrier since it now has 24 hours to travel the 344 NM resulting in a fictitious speed
of 14.3 K and a relative wind of 37.3 K. The PNR for New Britain then occurs at 15:26 Z, 1653
NM from Lae. This is 1:19 later and 142 NM further from Lae than the similar calculation for the
return to Lae. So even using a 24 hour endurance and a planned return to New Britain, the
decision to turn around would have had to have been made prior to passing the Gilberts. Since
we know the plane went past this PNR and proceeded for at least 4:47 further, to the vicinity of
Howland, it would not have been possible for the plane to make it back to New Britain even with
a 24 hour endurance.

O.K., cutting to the chase. We know that prior to passing the PNR they had the ability to return safely and try again another day. We know that they did not turn around prior to the PNR because they continued on to the vicinity of Howland which is well past the PNR. We know that on the planned flight from Hawaii to Howland they considered the possibility of turning around and, after Noonan had computed a PNR for that leg, had taken on additional fuel to allow for a return to Hawaii against the existing wind which makes it logical that they would have done the same if they had encountered a problem on the last flight. The entire "around the world flight" was planned around the need for celestial navigation on the leg to Howland and just two days before takeoff Earhart had sent a radiogram from Lae saying "FN MUST HAVE STAR SIGHTS."




Putting all this together we can conclude that they were satisfied with the navigation (FN was getting star sights) until at least passing the PNR which rules out the idea that they were just dead reckoning.

The second thing we can determine from these calculations is that they also could not return to New Britain from the vicinity of Howland thus making that theory very unlikely.

I have attached two charts depicting the PNRs I discussed in the two prior posts.


A and B are for a return to Lae and C and D are for return to New Britain.

 PNR "A" is the first case, 20:13 fuel on board, time at PNR 1154 Z, 1273 NM from Lae and 949 NM short of Howland.

PNR "B" is 24 hours of fuel on board, time at PNR 1407 Z, 1511 NM from Lae and 711 NM short of Howland.


PNR "C" is 20:13 hours of fuel on board, time at PNR 1313 Z, 1414 NM from Lae and 809 NM short of Howland.


PNR "D" is 24 hours of fuel on board, time at PNR 1526 Z, 1653 NM from Lae and 569 NM short of Howland and only 50 NM short of the Gilberts.








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Gary LaPook,
Apr 18, 2012, 9:55 PM
ą
PNR 1.jpg
(84k)
Gary LaPook,
Apr 18, 2012, 9:46 PM
ą
PNR 2.jpg
(51k)
Gary LaPook,
Apr 18, 2012, 9:47 PM
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