Preamble
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Today's topic is a response to @McNasty's challenge[1]:
"chrimony claims that he can calculate exactly how far a star at 500 light years away should appear to move throughout the night"
But the real origins of the question are the star trails that can be seen from Australia[2]. Two poles on a spherical Earth, one pole in Flattard Land.
McNasty tries to cope with his failure to explain half the night sky by claiming[3] because we don't see parallax in star trails, then "If the stars show no movement at all relative to each other, then it means those stars are at the same exact distance from the observer. So basic observation and by the laws of perception, the stars in the night sky are a blanket."
Whatever the fuck it means for the stars to be a "blanket".
My response to this is, "If the perceived parallax for a star is 0, then 0/2 = 0". Meaning a star twice as far away will show no parallax if we can't detect parallax in the closer star.
The closest star is Proxima Centauri, at about 4 light years away. If we can't detect parallax for that star in a 12-hour night sky star trail, then we sure as fuck won't be able to see it for a star 500 light years away. So to give the flattard every advantage, I'm going to calculate the parallax for a star 1 light year away over a 12-hour period.
The Math
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#Right-angle trigonometry.
#Do flattards know SohCahToa?
parallax angle = arctan( displacement / distance )
#Distance to the star.
#Flattards ignore how big this number is.
distance = 1 light year = 5.88 x 10^12 miles
#The lateral motion of an observer on Earth.
displacement = orbit displacement + rotation displacement in 12 hours
#The Earth spins.
rotation displacement = diameter Earth = 7.93 x 10^3 miles
#How far, in degrees, the Earth moved along its orbit.
#It moves 360° in 365 days.
#360°/365 in one day.
#And 360°/(365x2) in half a day.
orbit displacement angle = 360° / (365 x 2) = 0.493°
#Convert orbital degrees to lateral motion.
#Some more right-angle trigonometry.
#Radius of Earth's orbit = 93 million miles.
orbit displacement = sin(0.493°) x 93.0 x 10^6 miles = 8.00 x 10^5 miles
#Total displacement (orbit + rotation).
#Note the orbital displacement dominates the displacement due to Earth's rotation.
displacement = 8.00 x 10^5 miles + 7.93 x 10^3 miles = 8.08 x 10^5 miles
#Final calculation!
parallax angle = arctan( displacement / distance )
= arctan( 8.08 x 10^5 miles / 5.88 x 10^12 miles )
= 7.87 x 10^-6°
Or in English terms, approximately 8 millionths of a degree!
Conclusion
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I don't care how nice the smart phone on your camera is, you aren't going to see any parallax on a 12-hour time lapse video of the night sky taken from your backyard.
Citations:
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[1]:
https://www.upgoat.net/viewpost?postid=65f9e88d087db[2]:
https://www.youtube.com/watch?v=56ZMZtq0qfY[3]:
https://www.upgoat.net/viewpost?postid=65f3085348086
