The Aeronautical Calculations Found in al-Bayoumi’s Birmingham Residence

The Italian version of this article is available here.

Note: In the most recent video of the series with Adam Fitzgerald, I gave an incorrect explanation of the equation found in Omar al-Bayoumi's notes. What follows is the correct explanation. I apologize to the viewers of the video.

Source: New York Times

Among the notes found in the house where Omar al-Bayoumi was staying in Birmingham after 9/11, there was a handwritten page with what appears to be aeronautical calculations to estimate the distance between an aircraft and a ground object when the object appears on the horizon. The reported formula is:

d=√(h(h+8000))

In al-Bayoumi's original text, the closing parenthesis is missing, but this is clearly a mistake. According to what was verified by the FBI in one of the documents declassified following Executive Order 14040 by former President Joe Biden, the formula may be the application of the Pythagorean Theorem to the right triangle (shown in the diagram below, which is highly out of scale for clarity) formed by:

• the aircraft (P)
• a ground object when it appears on the horizon (G)
• the center of the Earth (C)

The sides of the triangle are:

• Earth's radius (r) plus aircraft altitude (h)
• distance between the aircraft and an object on the horizon (d)
• Earth's radius (r)

It should be noted that the triangle is a right triangle, since the second side is a tangent segment to Earth's surface and is therefore perpendicular to the radius.

Source: FBI.

Starting from the equation where the area of the square built on the hypotenuse equals the sum of the areas of the squares built on the two legs, we obtain:

r²+d²=(r+h)²

r²+d²=(r+h)(r+h)

r²+d²=r²+h²+2rh

d²=h²+2rh

d=√(h²+2rh)

d=√(h(h+2r))

By substituting the value 4000 for Earth's radius (approximately 3,963 miles), we arrive at the equation written by al-Bayoumi. At this point in the calculations, al-Bayoumi replaces d² with the values 5000 and 2500, which correspond to values of d of about 70 and 50 miles respectively, and writes the following equations.

2500=h²+8000h

h²+8000h=5000

In al-Bayoumi's notes follow numbers and sums that are hard to decipher. What is noticeable is the appearance of the numbers 52 and 76, which are close to the d values in the above calculations, and also this sum appears:

225+20=245

20 is the distance in miles between the Pentagon and Dulles Airport from which American Airlines Flight 77 departed; 245 is the distance traveled from the departure airport at the moment the terrorists took control of the aircraft. Al-Bayoumi's calculations thus result in the distance between the aircraft and the target at the time of hijacking. It is therefore likely that the hijackers were instructed to begin their assault when flying over a predetermined visible point on the ground, which the FBI identified as possibly:

• Kanawha River, 245 miles from Dulles Airport
• Elk Fork Lake, 225 miles from Dulles Airport
• Ohio River, 231 miles from Dulles Airport
• Interstate 77, 230 miles from Dulles Airport

Returning to the equation for calculating altitude, it is therefore likely that the terrorists used the same approach: they may have used this formula to calculate altitude after estimating horizontal distance from their target based on what they were flying over when the Pentagon appeared on the horizon. Starting from altitude and horizontal speed, they could then calculate the vertical speed to enter into the flight instrumentation in order to strike the Pentagon.

It should be noted that the formula used is very rudimentary, for example because it does not take into account atmospheric refraction, i.e., the deviation of light due to air density variation. In any case, the terrorists on American Airlines 77 did not care about absolute precision, only about hitting a point somewhere on a huge building like the Pentagon.