How far away is the horizon in a high altitude spyplane?

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Member 6L6 and I were amusing ourselves listening to the SR71 "Los Angeles Speed Check" audiotape, when I realized that a teeny fragment of it sounded, to me anyway, just a little questionable.

The SR71 pilot casually says he was directly over Tucson AZ and could see the skyline of Los Angeles CA out the window. Which seemed, um, improbable.

So I hired an 11th grader to work out the trigonometry of the situation, and make a plot of (distance to the horizon) versus (flying altitude). Result is below.

To my surprise, Google Maps says there are 484 miles of Interstate road (which is certainly NOT a perfect straight line) from Tucson to Los Angeles. While the plot below says the horizon is about 500 miles away when you're flying at a comfortable spyplane altitude of 16 miles. So it's quite possible. Golly!

Note: earth is assumed to be a sphere of diameter 7900 mi.

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PRR

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...I hired an 11th grader...

How much do you have to pay?? Is it worth it?

I surveyed the internet. Assuming LA is zero or 1 foot tall, it is pretty consistently 356 miles. I left LA a loooong time ago. Granting it may be 1200 feet tall now, the number is near 400 miles.

{mild over-estimate: Wilshire Grand Center is 1,100-foot (335.3 m), the tallest building in Los Angeles.} {I left LA just before AT&T Switching Center 448' (137m) went up in 1961; we thought City Hall was tall.}
 

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How much does it cost to hire these consultants?

2 large pizzas + 3 liters of soda + ice cream sandwiches = approx 65.00 USD

Not only did that buy the answer, it also bought the systematic step-by-step trigonometric identities which inexorably lead from the givens to the conclusion.

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Sorry for the US-centric terminology; "11th grade" here is two years before university. 11th grade students are typically 16 years old at the beginning of the school year. Another US name for them is "High School Juniors"; 12th graders are "High School Seniors."

Math curriculum in ordinary city-operated, open to everyone, tuitionless "public schools" is typically

  • 9th grade (14 years old): Algebra
  • 10th grade (15 years): Geometry
  • 11th grade (16 years): Trigonometry and Analytic Geometry
  • 12th grade (17 years): College Math, which may or may not be Calculus, depends on the school

"Private schools" in the US, which do not admit all applicants, and which typically charge tuition to attend, often teach these subjects far sooner. Some of them offer Calculus-1 and Calculus-2.
 
In Germany back in the early '80s we began differential calculus in the 2nd half of the 9th grade and integral calculus in the 10th.
I went to a state school because there is no private school culture there but we do stream at age 12 into 3 different schools (Main, Middle and Grammar) according to ability.
You need a Grammar school education to go to university.

Age varies because if you fail any two subjects (out of 10 or 12) in any given year you have to repeat that entire year ie all the subjects.

In British terminology I have 12 A levels but I have no idea what that means in American schools. Usually you require 3 A levels for uni.
 
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Surely you MUST remember your trigonometry??? Why not derive a formula for distance-to-horizon versus altitude-above-sea-level, then we can program your formula into Excel and plot it. Maybe you will confirm what the undereducated American kids derived, or maybe you will find that (a) kids were wrong; and also (b) the spyplane pilot was a big fat liar. Maybe he COULD NOT POSSIBLY see Los Angeles from 89000 feet above Tucson. Please confirm or refute this potential myth.
 

PRR

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> Surely you MUST remember your trigonometry???

Nope, I only SIN. Don't need that other stuff.

The sight-line, and the earth radius, intersect at right angle. (The lowest sightline must be a tangent to the circle.) The radius through the airplane is earth radius plus airplane height.

Take 7,900mi diameter, 3,950mi radius.

Take classified-info altitude as 16 miles. (Only two folks know the true number: the US and the USSR/Russia.)

So a Right triangle, 3950 miles and 3966 miles. 3966^2-3950^2, rooted, is 355.888.. miles. Which agrees with Googles. And that is Good Enough For Me until I own my own Blackbird.

This is strictly the length of the sightline. The path on the ground is shorter on the arc but longer on the curve. Proportions suggest less than 1% error. There is another few-miles uncertainty in "directly over Tucson". Downtown? Ballpark? Major freeway intersection? Does he have a plumb periscope? (He has a plumb camera but can't see it.)

As noted in several Google links, there is also refraction. When you are in the desert there are conditions when you really can see beyond the geometric horizon, though often distorted or fleeting.
 
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