Top of 2.5° Descent

Eddie sez:

We long ago figured out that 3 times your altitude (in thousands) gives a nice 3° descent from en route altitude. When the 60-to-1 gurus came out, they were convinced this was an offshoot of the flight levels to lose technique. In my Hawaii squadron our airplanes couldn't handle a 3° descent at some altitudes without losing pressurization or resorting to speed brakes. So we came up with 4 times your altitude (in thousands) gives a nice 2.5° descent. Are either of these due to the 60-to-1 relationship? No, but they do work. Here's why.

We'll define the rule, provide an example of its application, and then we'll provide a mathematical proof of the rule.

Last revision:

20151213

Top of 2.5° Descent

Definition

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Start descent at four times your altitude to lose in thousands of feet to achieve a 2.5 degree gradient.



Example

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Figure: 2.5° Top of Descent, from Eddie's notes.

Let's say you are cruising at FL410 and are planning a descent into an airport that is just over 1,000 feet MSL elevation. So you have 40 thousand feet to lose. If you want a gentle 2.5 descent, when should you start your descent?

Top of Descent = (Thousands of Feet) 4 = (40) 4 = 160 nm

Proof

If you multiply your altitude in thousands of feet by four, you will arrive at the ideal distance to start your descent and end up with about a 2.5 degree descent gradient. The 60-to-1 mavens will have you believe this is an offshoot of the earlier flight levels to lose technique:

Gradient = Flight Levels Nautical Miles

If you do the math, that would mean:

Top of Descent = Flight Levels 2.5

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Figure: Top Descent trigonometry, from Eddie's notes.

That works, but that isn't the "school solution." We can verify the technique using trigonometry. We will use K to represent the altitude in thousands of feet and D for the distance in nautical miles. Of course that means the formulas will require two conversion factors:

Top of Descent (D) = ( K tan θ ) ( 1 nm 6076 feet ) ( 1000 feet 1 K )

For a 2.5° angle, tan(θ) = 0.0437 and the math works out to:

Top of Descent (D) = ( K ) ( 3.77 )

That is an error of less than 6% from our time-tested rule of thumb.

Bottom Line

So what about the claims this rule of thumb is based on 60-to-1? My conclusion: It has more to do with trigonometry.

Rule of Thumb 60-to-1? Trigonometry π
Top of Descent (3°)    

60-to-1 — Rule of thumb is based on the mathematical relationship of a 360° circle and/or 6076' to 1 nm.

Trigonometry — Rule of thumb is based on the relationship to a right angle and the derived trigonometric functions.

π — Rule of thumb is based on the relationship of a 360° circle, the number π, and/or 6076' to 1 nm.


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