Vances horizon distance

StanFoster

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Nov 16, 2003
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Paxton, Il
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Helicycle N360SF
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Vance just posted his amazing trip to 12,500 feet msl. His first picture which I posted here had me wondering how far the horizon was as it was so clear and you could see the curve of the earth, assuming it wasnt in his lenses.

Anyway I worked the problem out and posted it in the two drawings. By the way...the drawings angle is exaggerated for clarity. I was waiting on glue to dry, but this only took a few minutes.

First known is the radius of the earth...rounded off here to 4000 miles.

2nd known is Vances height above the earth...12500 feet..or 2.367 miles.

From this one can compute the angle from Vances eye perpendicular to the center of the earth and a line tangent to the earth...which will be the horizon.

The 2nd drawing shows this tangent line makes a right angle with the radius line of the earth.

We know that that line at the horizon to the center of the earth is 4000 miles...and the other side of this right triangle reaching Vance from the center of the earth is 4002.367 miles.

If everyone remembers the sine/cosine/tangent rule.....

Old Hags -------- Opposite/Hypotenuse = Sin

Always Have -------- Adjacent/hypotenuse = Cosine

Old Asses ------- Opposite/adjacent = Tangent


then this triangle can be computed quickly.

4000/4002.367 = Sin of the angle

.9994086 = sin

The angle Vance has to look down from perpendicular to the earths center is 88.02922 degrees.



Now the first drawing breaks down with another right triangle whose base one side points from the center of the earth to Vance....a line perpendicular to this that intersects the horizon visible by Vance.

This time we take the tangent of the angle opposite 88.02922799 degrees and this comes out to and this is 1.97077 degrees.

Tangent of 1.97077 degrees equals 0.034410031

Multiply this by 4000 and 137.64 miles is the answer.



I did this in haste, oriiginally posted an error by takng the tangent of the wrong angle in haste. I edited out the one drawing.


Stan getting back to gluing...
 

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You are amazing Stan.

You are amazing Stan.

I love the way your mind works.

I corrected my post to read that the picture you reference was taken at around 9,870 feet.

I was taking pictures of the instruments on each lap and it identifies the altitude of the picture.

I was being sloppy in my post and I am sorry. I will try to trust my memory less and documentation more.


Hello Brian,

I wanted to know because I have extrapolated a lot of the performance for Mariah Gale from the Predator's performance numbers. I understand how altitude affects engine performance but I am unclear about how it affects propeller performance.

I am going to call Craig Catto today and see if I can make more sense out of it. Up to around 11,000 feet I was able to maintain my propeller RPM of 2,400 by leaning. It was close all the way up except when I slowed down to get a little bit of a step up. I do not understand this relationship.

My ground speed seemed to increase in relation to my indicated air speed as we climbed higher.

Practical ceiling is important in planning the route near the Sierras or the Rocky mountains and ground speed is an important part of navigation and check points as well as fuel consumption calculations.

Thank you, Vance
 
Stan, I did a different calculation, nearly same result.
I applied Pitagoras's theorem, where X2+R2=(R+H)2, therefore X=sqrt(2RH+H2), where the 2 after a term means squared.
Putting R=6371 Km, H=3,81 Km X is 220 Km or 136 miles. The angle is 90-1,981 Degrees.

But a simpler rule is that the range of radio waves in NM is the sqrt of the (height in feet times 1.5). This is also 135 miles for an altitude of 12.500 feet.
Great view from up there!
 
I feel Dwarfed by Genius!

I feel Dwarfed by Genius!

Thank you for making it simple Walter and the practical application.

I wish I could share my memories, the pictures only tell part of the experience.

I am familiar with the haste thing Stan.

I didn’t catch your mistake until you pointed it out.

Thank you, Vance
 
Buddy...

I really like the way you think!!!
 
I suspect that the distortion of the photographic lense (which is largest at small focal widths and near the edges) also contributed some to the perceived curvature in the picture.

-- Chris.
 
First known is the radius of the earth...rounded off here to 4000 miles ... 2nd known is Vances height above the earth...12500 feet..or 2.367 miles ...

Old Hags -------- Opposite/Hypotenuse = Sin
Always Have -------- Adjacent/hypotenuse = Cosine
Old Asses ------- Opposite/adjacent = Tangent
4000/4002.367 = Sin of the angle
.9994086 = sin

The angle Vance has to look down from perpendicular to the earths center is 88.02922 degrees ... blah, blah, blah ...

Stan,

You are just too damn smart.

Tom
 
Stair question.

Stair question.

Stan, others,

You numbers wiz'z are really showing off now. The real question I want to know is if Stan built a stairway to a platform at said altitude, how many stairs would one have to climb ?

How many sheets of sand paper would it take to finish them ?

Once you got that how many gallons of paint would it take to coat them ?

J
 
I suspect that the distortion of the photographic lense (which is largest at small focal widths and near the edges) also contributed some to the perceived curvature in the picture.

-- Chris.

I'm with you on that detail. Long ago I flew on Concorde across the Atlantic at FL600, and managed to convince myself that I could see some small effect of the curvature of the earth up there, but it wasn't extreme even at five times Vance's altitude. Most small cameras won't produce a "flat field", will give some "coma" error toward the edges, and so forth, so you can't entirely trust the optical effects in a snapshot.

Nonetheless, it's nice to get the view up high. I've had my Bell up to 14,000 over the Rockies, and a sailplane up over FL300 near Pike's Peak, and the view is always inspiring.
 
Good point, Jürgen.

More on the light side (pun intended), if one takes relativistic effects (attraction of photons by gravity, or better, curvature of space by gravity) into account, it will be even a bit farther :)
 
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Must be them queer Germans, Walter, I also wondered whether to consider relativistic effects, alas I had no idea whatsoever how to treat that question so I decided to follow a rule my Latin teacher had given us "Si ta quisses, philsophus mansisses" (If only you hadn't said anything people might still think you're a philosopher...... no guarantee as to the spelling, that's always been my weak point ....;-)
 
... "Si ta quisses, philsophus mansisses" (If only you hadn't said anything people might still think you're a philosopher...... no guarantee as to the spelling, that's always been my weak point ....;-)

Ha! Finally something I can knowledgably comment on and in one of Jürgen's posts :)

It is: Si tacuisses philosophus mansisses.

Translating to: If you had kept silent you would have remained a philosopher.

-- Chris.
 
I'm sorry I missed this post until today!

I have to agree that it is most likely is the lense of Vance's camera that is causing the apparent curvature of the earth and not the altitude he was flying at.

The following photos were taking from a Lockeed TR-1 (follow on of the U-2) over Cape Canaveral from 60,000+ feet MSL of a Space Shuttle Launch. The curvature of the earth on these photos is correct from this altitude.

Wayne
 

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