We're told that the sun disappears over the horizon for the same reason that airplane contrails reach the horizon (which ironically is due to them following the curvature of the earth). So let's consider perspective at night. According to a Flat Earther I asked, Australia is about 10k miles from the US in a Flat Earth model.
We know that:
$ \theta_{ele} = atan(h/d) $ Where h is the height of the sun off the flat earth and d is the distance to the sun's nadir point (directly below the sun).
When the sun is over Australia, it's dark in the united states.
It's 10,000 miles from US to Australia.
Flat Earthers generally report that the sun is 3000 to 4000 miles in elevation. This is probably because if you tried to triangulate the sun (at approximately infinity miles away) from a curved surface you erroneously believed to be flat, you'd find that it appears to be about $ r $ miles away.
But how can that be?! $ atan(3000/10000) = 16.7 \degree $ !!. That's a pretty good elevation. About one and a half fists above the horizon! That would be pretty obvious!
Making the small-angle approximation, these angles and distances scale approximately linearly. To get as low as half of a fist above the horizon, we need 1/3 the height (let's say 1000 miles!). To get as low as one degree (still 2 sun widths) above the horizon, we'd need to be just 100 miles up. That's just at the edge of space. The sun could be hit with an amateyr rocket!!
With that, I've got no way to try to rescue this theory. The angles don't make sense. Anyone got a way to make this problem work?
I keep getting asked for pictures of satellites by #FlatEarthers I'm talking with. Rather than find them each time, here's a collection. The gif was from reddit.
Amateur Videos Showing Satellites
I'm not sure which satellites these are, but they're geostationary over Switzerland. This sort of imaging is surprisingly easy to do (this guy does it VERY well!). Just get a camera, a tripod, and take 5-10 sec exposures all night of the plane where the moon, sun, and planets pass. Stitch them all together and you'll notice that some don't move. Those are geostationary satellites. You can figure out which ones they are with a bit of extra math. Geostationary satellites in the Swiss Alps from Michael Kunze on Vimeo.
This guy wasn't even really trying, but when he co-adds frames (go make the streaking effect), it jumps right out.There's a clear GEO satellite near the top of the frame, about 40% in from the left.
This photographer isn't taking exposures long enough and has too much city light to see the satellites clearly. The "flashes" are likely glints off the solar panels or an aliasing artifact in his video conversion. Longer exposures could cause streaking, but would bring the satellites out more clearly. Geostationary Satellite Flashes, Night of March 7-8, 2013 from Ken Musgrave on Vimeo.
I'm also often asked why we don't see satellites take pictures of other satellites. To someone naive about otbits, this might seem like an easier task. You're closer, right? No. Space is very big, and satellites moving 17,000 mph, not all in the same direction! Nevertheless, It's been done. An imaging satellite named Pléiades 1A took a picture of SPOT 5.
I finally found a video which helped me understand the flat-earther model of a sun circling a flat earth. Thanks to p-brain for helping me out here.
Setting aside his disastrous misunderstanding of perspective, he has a decent point regarding how distant points converge at long distances. The "plane" that the sun would orbit within would indeed appear to approach (though never quite cross) the horizon.
I suspect this claim falls apart when we start examining distances necessary to accomplish this. Clearly the Sun isn't flying along at 30,000 ft like the contrails p-brain uses as an example. Rather, it would need to be flying a minimum of 2-3x higher. My intuition is that the angles will come closest to working if we set the sun at an altitude equal to the radius of the earth (4000 miles) .
So let's compare the two approaches. For the sake of simplicity, I'll assume the curvature of the sun's path is so slight that we don't notice it curving northwards. That's right, I'm going to give the Flat Earthers a pass on the fact that we don't see the sun curving to the north! No "Where's the curve?!" from me.
So anyways, the distinction is really easy to make. If you believe in a Flat Earth, simply measure the angle to the sun throughout the day and compare the following plot of arctan(1/x):
If it's straight, the angular rotation is constant, which matches the spherical earth model:
If the solar angle matches the plot of $ arctan(1/t) $, then it's a flat earth.
If the solar angle is a straight line, it's a sphere.
Note: The angle you need to measure is called the "Right Ascension" (RA). Align a pole to point to the North Star at night. Measure the angle to the sun about this pole. This could be done with a protractor oriented perpendicular to the pole on the back side (away from the sun). Note the angle where the shadow is cast. Here's my attempt to draw the experimental setup.
I'm so tired of repeating myself to Flat-Earthers. It's like they revel in doing their math wrong, but there's not enough characters in twitter to send them the proper equations. So I'll address a few classic pics here to explain how it's done.
The distant mountain picture
This is a favorite among flat-earthers, and it's pretty easy to show they get the math wrong. Someone told this guy that he can use a simple linear fit for earth's CURVED surface (8 inches per mile or some such tripe). I'm not sure where this approximation came from, but it obviously doesn't fit a curve very well. Anyhow, here's an example:
Here, we have the classic flat-earther example of a "mountain that's too far away to see." First, let's check the facts. After a bit of digging, I figured out that they're claiming this is Mt. Denali. That's 140 miles away, with an elevation of roughly 20,000 ft.
It's actually 700 ft at the summit of the Hilltop Ski Resort, but it turns out that won't matter in the end.
Anyhow, let's assume they're right and it's Denali. First, calculate the distance and angle to the horizon from an elevation of 700 ft. I'll let WikiHow explain how to do it. I'm using the arccos formula:
Suppose the picture were taken from the parking lot of Hilltop instead of the top. The horizon is still 27.5 miles away, well within the range of Denali's summit.
32 + 174 means you can see the tip of Denali from as far as 206 mi at an altitude of 700 ft (assuming there's no mountains in between and neglecting atmospheric lensing, of course). This is obviously not a problem for observation from 140 miles away.
Just for convenience, we could figure out how tall an object needs to be to see it from a distance, x. We simply add the horizon distance to the solution to our first equation for h:
$ \cos( \frac{d}{r} ) = \frac{r}{r+h} $ $ h = \frac{r}{cos(\frac{d}{r})} - r $
This means we should be able to see the top 12,300 of Denali.
Time Zone Solar Models
This is another flat earther hand-waving explanation for time-zones.
Even the most casual thought regarding this picture and you'll realize that since the sun is above a flat earth, you'll be able to see it at all times from any location on earth. Turn this model on its side and draw a vector from any point on earth to the sun and you'll see that it never sets. The figure below shows this and explains the problems.
If there's some other way to interpret the animation above, I sure can't think of it. Send me a comment or tweet to help me understand.
I'm left wondering if flat earthers lack all spatial reasoning. It sure seems that way.
Pilots Would End up in Space!
This one is so bizarre that it takes a little work to wrap your head around. Essentially, the argument is that airline pilots would need to adjust their elevation by 1 degree every 6 or 7 minutes to keep from climbing higher and higher and crash into the firmament or flying into space. There's some specious assumptions here:
That airplanes fly along laser-like straight lines
Airplanes will continue climbing at the same rate for a given angle of attack even as the air thins.
That the pilot has the sensitive equipment to recognize one-degree drift over 6 minutes
In the same way as you might make minor course corrections along a straight road to keep your car centered in your lane, the autopilot (or the real pilot) is constantly adjusting to maintain a reasonably constant altitude and heading. These corrections are happening at a rate of tens to hundreds per minute. To the pilot or autopilot, this feels like maintaining altitude. In reality, it's conforming the flight to the curvature of earth. This publication described the typical roughness of a commercial flight. I've pulled out the "rough cruise" section because the constant readjustment of altitude is more clearly apparent. Same thing happens in "smooth cruise", but more gradually.
Simple computer models can help us understand when and where we'll be able to observe curvature of a sphere. The key factors which affect the appearance of curvature are:
Altitude above the surface
Camera Focal Length (or field of view)
Radius of the sphere
I'm sure someone could write an expression for the apparent curvature versus these parameters, but it's easier to just show you. I'm using an open-source tool called Celestia which accurately presents the position, sizes, and velocities of celestial objects. I highly recommend it. It's fun to play with.
Changing the Field of View
Here's the same view as the Field of View is altered. This is like zooming out on your camera. The FOV is reported in the bottom-right while the location (constant) is presented in the upper-left.
Changing the Altitude
This one is a little more obvious. As you move to higher and higher altitudes, the curvature becomes more apparent.
Note that the Distance is 30 km. That's a decimal place, not a comma.
Radius Matters Too
This one is just for fun. Here's some spheres of different radii from the same distance.
Mythbusters are shills?
Well, as we've shown, the curvature isn't expected to be visible at low altitudes. Here's Adam Savage at a 12 mile altitude witnessing the curvature for himself. So I suppose he's got to be a liar now, eh?
The Moon / Spinning Earth can't be felt!
Moon: The gravitational acceleration of the moon on the surface of earth is given by $ a = G m_{moon}/r^2 $ or roughly $ 3.6 \times 10^{-5} m/s^2 $ compared to the $ 9.8 m/s^2 $ I measured in high school for earth's gravity. That's less than one part in a million. A hard thing to measure.
Rotation: The acceleration of an object on the equator due to circular motion from the rotation of the earth is given by: $ a = \omega^2 / r $. This also tiny at $ 0.034 m/s^2 $. This effect (0.34%) MIGHT be measurable by exceptionally sensitive equipment and a skilled scientist, but these are the sorts of people the Flat Earth crowd seems to consider untrustworthy.
Edit: I fixed my math above. Rotation is actually much more important than I originally calculated. Thanks to @TheOlifant for catching my error:
The mentality of flat-earthers seems to be very similar to that of anti-vaxers and deeply religious. The believer thinks they've figured out that most of humanity is wrong, and that their answer is the right answer. They often tell you to "research it," and couple commands with insults "stupid" or "dummy" or "sheep."
These believers think they've figured out what "they" don't want you to know. The "they" varies between people, but it seems to be illuminati, the government, or the Free Masons. For devout Christians or Muslims, the "they" is Satan, heretics, or demons.
These believers pride themselves in being different. They think they're visionaries for knowing the truth when everyone else has it wrong. Despite having no formal training in the specific scientific claims they reject, they feel sure that all the professional scientists have been deceived by the "they."
What's particularly interesting is that these people seem to blindly follow (IMO obvious) quacks. Some guy with a YouTube channel is seen as more reliable than all the world's scientists. They wave off these brilliant scientists by presuming they've never actually TESTED any of the claims they learned in science text books without seeming to notice that:
The YouTube quack has never tested his flat earth claims. At best their "evidence" seems to be that they find actual physics hard to understand or inconsistent with scripture.
Scientists actually do verify the basics. They build more complex experiments on top of them, so if the basics weren't right, nothing would work.
