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Monday, September 12, 2016

Flat Earth Follies: Planes would have to constantly pitch down to fly!

Flat Earth Claim

Planes would have to constantly pitch down to fly!

Short Version

In the Globe model "LEVEL" follows a sphere of uniform gravity potential. Feel free to disprove that is the case if you can but that is the model.

In an airplane, Elevator Trim simply controls the RATE of pitch of an aircraft - not the actual pitch of the plane.

A 747 flying at 35,000' would have a flight circumference of  about 2π*(3959+(35000/5280)) ≈ 24916 miles
Flying at 570 miles per hour, this plane would fly a fraction of the 360° around given by 360*570/24916
Since that's 1 hour we can divide by 60 to get the degrees per hour: 360*570/24916/60 ≈ 0.137°/minute

That is down in the noise since pitch rate is affected by temperature, density, power, and control settings on the plane you simply trim the plane for as close to ZERO vertical speed as you can and you've automatically compensated for all of them at once.  Of course, on a 747 computers do that.  In a small plane the pitch rate is microscopic.

Pitch Rate/minute = 360*groundspeedMPH/[2π*(3959+(altitudeFEET/5280))]/60

So yes, airplanes DO pitch down constantly at an EXTREMELY slow rate which presents absolutely no issue what-so-ever.  Once again, Flat Earthers failed to understand a phenomena and proceed to run around making absurd claims about it.

I address how the gyroscopically driven instruments work and other issues/questions below.


This usually starts out with a Flat Earther pointing out the 8" miles squared formula and how many bazillions of miles the plane has to drop out of the sky to fly the curve - imagining that the pilot has to constantly shove the controls further forward in a fever pitch to avoid flying off into space with their Attitude Indicator spinning like a fan.

That image is, of course, pure nonsense - even if you remove my slight hyperbole.  And given how loudly they shout that this PROVES the Earth is flat my hyperbole might even be understated.

First, please read my blog post on what 8" times miles squared means and how it was derived.  The important thing to note here is that while it is accurate enough for distances from about 1 mile to about 100 miles it is not an accurate formula for the curvature of the Earth on smaller or larger scales. It is and always was an old APPROXIMATION used by surveyors.  It was 'good enough' for what they used it for and I'm fine with using for distances between 1 mile and 100 miles.   However, it is clearly not a linear formula so you can't say well..  8" / 5280 = drop over 1 feet NOR can you go the other direction in a linear fashion (as Jeranism tries to do, his method would have the shape of the Earth being a giant wedge with constant slope).   It should be obvious (once I point it out) that 8"*d^2 is a parabola and NOT a circle.

[better approximation formula, use same units for both, R=Earth Radius:  √[d² + R²] - R ]

But what is TRUE is that the curvature drop over about 1 mile is about 8 inches.

But what is also TRUE is that the drop over the NEXT MILE is ALSO JUST 8 inches.

WHAT? How can that be?  8"*4 = 32 inches so it should be another 24 inches!?  RIGHT?

No.  WRONG.  Look at the geometry again...  (we're looking at the triangle on the right in this case)

The formula assumes your distance is ALWAYS ALONG THE SAME TANGENT TO THE SURFACE.

However, once you move YOUR HORIZON HAS ROTATED!!!!!!

Planes don't fly out 1 mile, jump down 8 inches, and fly some more - that's silly.  Planes constantly FLY THE CURVATURE because that is the ONLY line of equipotential gravity!

Over that 1 mile they would CONSTANTLY AND CONTINUOUSLY rotate at a rate that is too slight to measure and, in fact, the constant buffeting by variations in the atmosphere BURIES THAT SLIGHT ROTATION INTO NOISE.  Any variation from this equipotential would (and does) result in the plane climbing or descending, which would immediately show up as a change in the energy configuration of the plane.

Ok, that is the basics of why this argument is just wrong.  Now I'll add 1 level of detail on the major points.

What does up and down even MEAN?

I know Flat Earthers deny that Gravity exists but when we drop an item it falls to the ground - that IS GRAVITY.  It really doesn't matter WHY you think it happens or what you call it.  For our purposes here it doesn't matter -- all that matters is that you agree if you dangle something so that it can pivot freely that the little thing hanging down will point DOWN towards the ground.  Can we agree on that?  If I tie a rock to a string and hold the string the ROCK will be closer to the ground.  Right?

We can do this in a vacuum so we KNOW it has nothing to do with air pressure.

And we also KNOW that it has nothing to do with buoyancy, because buoyancy is powered by gravity in exactly the same way our rock is, without the force of gravity there is no density gradient.  We know this because we can observe the behavior of density in the absence of this downward force we call Gravity.  Here is what density does when you remove the force of gravity, things no longer 'sort' by density:

You can also spin things really fast and find that the density gradient follows the resulting vector of combining the spin vector (centrifugal acceleration) with the gravity vector.


How much rotation?

Ok, so how much rotation are we talking about here?  Surely the pilot would notice, right?

The Earth is ~3959 miles radius and there are 360 degrees in a full circle so that means we have approximately 360/(2*3959*pi) = 0.01447° per mile.

Let's say we're in a 747 going 500 mph, so the plane needs to rotate 500 * 0.01447° per hour, or 1/3600 of that per second

(360/(2*3959*pi))*500/60/60 = 0.00201°

So that's a constant VERY SLOW PITCH rate of 0.00201° per second or just 0.12° per minute.

But does the pilot actually has to keep pitching forward?

No.  Again, you have to understand how aircraft work.

Planes control pitch using a control surface called the Elevator (usually on the trailing edge of the Horizontal Stabilizer)

And along the Elevator you'll find another smaller control surface called the Trim Tab.

Here is the most important point in this discussion:

The Elevator does not directly control the Pitch of the Airplane - it controls the RATE OF PITCH.

I think that Flat Earthers don't understand this concept.

To maintain level flight the pilot must find BOTH the elevator trim and power setting which maintains a constant altitude - they mostly use the Vertical Speed indicator to make fine adjustments to elevator trim to find the constant pitch rate that keeps Vertical Speed near zero and then makes POWER SETTING adjustments as needed to hold that Vertical Speed with a fairly constant airspeed.   If you want a higher airspeed you need to both increase power AND adjust the elevator trim so the pitch rate matches OR ELSE YOU WILL START CLIMBING.

This is a fact, I have personally flown small planes and they teach you about power control pretty much from Day #1.  You climb & descent mostly by changing the power setting (which changes when you make other configuration changes such as increasing flaps).

Now, how is a pilot supposed to tell that a TINY fraction of the elevator trim has to do with the curvature rate as opposed to all of the other forces acting on the airplane?  They couldn't possibly.

A curved gravity equipotential presents ABSOLUTELY ZERO issues for an airplane in flight.

So YES, the PLANE is (technically) constantly pitching forward as it flies the curvature of the Earth.  But it doesn't feel like pitch because DOWN is changing at the same time and it's an incredibly slight rate of pitch overwhelmed by other dynamic forces acting on the aircraft.

What about this FE Meme about the SR-71?

Wow - there are so many things absolutely wrong in this meme I almost don't know where to start.

Where did they get 2193.13 MPH from?  I get 668.9 mph for the speed of sound at 85,000' and SR-71 nominally goes Mach 3.2 (it can go slightly faster for short periods), that gives me 668.9*3.2 = ~2140.48 mph at 85,000'.   I guess that is "close enough", so I'll use their 2194 mph for this example.  But we're also at 85,000' so I'll add that into our radius of curvature (makes very little difference).

And where did they get '1/4 MILE OF CURVATURE EVERY HOUR' from?  Wow, that is just stupidly wrong.  At 2194 mph, you go 2194 miles in 1 hour and that gives 696.83 Miles OF CURVATURE DROP not 1/4 mile!  *I've assumed no wind so 2194 mph ~ ground distance and I've used CURVED GROUND DISTANCE for the curvature calculation: r/cos(d/r)-r

And even if it was a 1/4 mile 5280/4/60 = 22 FEET, not 23.  Maybe they rounded the .0?  They clearly cannot manage division.

But their worst error is assuming that 1/4 mile of curvature MEANS 22 feet per minute of vertical speed in the first place.  As I've already shown VERTICAL SPEED WOULD BE ZERO AT EVERY POINT.

The curvature 'drop' is BUILT INTO the flight dynamics - the plane isn't flying a linear path and having to drop down, it is flying along the curve -- you have to compute the integral.

The pitch RATE would be:

(360/(2*pi*(3959+(85000/5280))))*(2194)/60 = ~0.527° per minute (~0.00878°/sec)

Still only a very slight rotation that wouldn't be specifically noticeable and would very easily be compensated for in pitch rate - in this case, by the computer that is actually flying the airplane.  But again, the computer doesn't even have to understand the Earth curvature to do this, it just needs to keep VS near zero which is determined by pitch rate and power settings.


So it isn't the 696.83 miles of curvature drop you need to worry about, let's look at that over 1/1000th of a second:

The plane moves forward ~3.218 feet
The plane pitches forward 0.00000878°
The plane DOES NOT CLIMB 0.00000024669 of a foot (the 'drop' over that distance)

After 1 second the plane HAS NOT CLIMBED 0.24667 feet but it is entirely INACCURATE to say that it has DROPPED 0.24667 feet.   If you take 0.00000024669 * 1000 you'll note that you get 0.00024669  instead of 0.24667 feet -- you are a OFF BY ONE THOUSAND TIMES HERE by assuming this is a linear function.  Every microscopic bit of pitch rotation in between ALREADY ALTERED OUR COURSE.


It is the sum total of these CONTINUOUS microscopic adjustments that result in the curved flight path.

That is what these guys don't understand.  With the atrocious math they cited in this Meme is that any surprise?

What about the Attitude Indicator?

Gyroscopically driven Attitude Indicators are TIED TO GRAVITY.  They constantly adjust themselves to find the vertical axis using gravity to force gyroscopic precession into the vertical.   They adjust very slowly and evenly and constantly but they do adjust and if you hold the plane in a non-level attitude they will even give you the incorrect attitude as a result.

Here is a very detailed video showing how the Artificial Horizon works.  At 4:00 you can see the 'pendulous vanes' that open and close little air vents which cause the gyroscopic precession until the vanes close the vents by hanging straight down under the gyroscope.

The curvature rate is well below the rate at which the gyroscopes precess to remain in the correct orientation.

And here is an actual test showing the AI aligning itself to Gravity.

The Horizon Rises to Eye-Level

No, it just does NOT do any such thing.  Aiming a camera to put the horizon in the middle or cropping an image to put the horizon in the middle does not demonstrate any such thing - in fact they are obvious lies by Flat Earthers.

Here is a non-professional instrument (Theodolite app on phone) clearly showing the horizon dip:

And here is an actual aircraft cockpit showing the horizon dip in LEVEL FIGHT.

Professional pilot Wolfie6020 also just posted a video on this same topic (and covers a few points I didn't and has some great examples):


I think that about wraps it up for this Flat Earth Folly.

Planes are not 'dropping' 8 inches every mile (per se), they are flying along the constant curvature of the gravity equipotential, while constantly adjusting pitch ever-so-slightly by means of the elevator trim setting which controls the pitch RATE of the airplane.  ANY deviation from that rotation results in the plane climbing or descending which immediately shows up in the Vertical Speed indication and power settings would be adjusted accordingly.  This constantly rotates their 'tangent' so there is no 8" to drop at the end of each mile, it's a constant, smooth, and VERY SLIGHT curve that presents no problems for pilots, and would be virtually undetectable in the face of other forces acting on the airplane, even at 500 mph.  The plane is simply tweaked for near zero vertical speed and that's all that is required.

The Attitude Indicator is very clearly compensating for all kinds of precessional forces acting up on it and to remain accurate over the longer term must be tied to the gravity potential as well (so it remains vertically aligned over time, reacting only to sharper movements of the airplane over the short term).


  1. Very good article. Well written, logical and nice use of math calculations.

  2. Great article, good common sense and the science to back it.

    But looking up pendulous vanes, they seem to relate to compensation for procession of the gyro.

    Schuler tuning (also called schuler loop) is the feedback that compensates for earth's curvature. Here are some articles on it.

    Section 3.3

    Section 3.1

    Section 4.3 ( pages are backwards )

    And when someone says they know how a gyro works in a plane, show them this image of how complex the unit can be.

    1. That is correct for modern electronic INS but not for the much simpler mechanical/gryo-based Attitude Indicator discussed here. It is just indexed to gravity via the pendulous vanes (which dangle freely underneath it and open up little air vents if it's not upright). There are many reasons the gyro can be misaligned, curvature being one of the smaller ones - if you turn too hard the gryo will precess as well and the pendulous vanes correct for that also. Their job is to slowly align the AI gyro back to a vertical position regardless of the cause.

      The SR71 itself apparently used celestial navigation, at least in part, but I don't know enough about how that worked to comment on it. I'm sure it has INS as well and that would have incorporated Schuler tuning as you mentioned. Thanks for the links!

  3. explains the basic functions according to the Operations Manual of the Blackbird. The system could combine an automated starfinder, an inertial system analog to the gyroscope and simply computer based "dead reckoning" if both other systems failed... e.g when the sight to the stars was impaired
    Basically it's kind of a facial recognition software for Stars and if it can identify 2 or more of them the navigation computer can correct any deviations delivered by the inertial part of the system (and even will use that to find out HOW much that deviates so it can cancel out variations better when only inertial system data is used!)

  4. only,..., how is mechanical gyro keeping the horizon in the plane again,..., ever since the right brothers?


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