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View Full Version : Sun's Rays: Radial or Parallel?

mac_bolan00
06-27-2011, 05:26 AM
there was the mathematician erathosthenes' calculation wherein he assumed the sun's rays hit the earth in parallel rays. he observed two dry wells located in different cities inside egypt, along the same longitude (syene and alexandria.) at exactly the same time, the sun was hitting the bottom of one well while it was shining on the sides of the other. measuring the difference in angles from vertical, he was able to conclude that the earth was round and even came up with an acceptable estimate of the earth's diameter (google erathosthenes' experiment.)

but i want a reverse experiment: can we actually measure whether the sun's rays hit us at radial or parallel? guys tell me that as the distance from the point source of light to the observer increases, the rays change from radial to parallel. i'm thinking of two friends living several miles away making simultaneous measurments while the sun's shining on all three of us. but online calculators won't even return a figure for the angle if i enter the hypothenus as 93 million miles while the base of the right triangle is, say 100 miles.

Blake
06-27-2011, 06:01 AM
Assuming the question is "can we actually measure whether the sun's rays hit us at radial or parallel?", then the answer is yes. You already explained one obvious way to do it. The reason why calculators won't return an answer is that the the answer is smaller than significance allowed by the of the processor. IOW the rays are indistinguishable from parallel from a practical perspective.

An alternative method is to actually reverse Eratosthenes' experiment. Place a relatively nearby object between one observer and the sun, and check whether it is also between another observer and the sun. If the rays are truly parallel then it must intervene for all observers regardless of how far apart they are. An artificial satellite would be a good candidate. Have your two observers take a series of photographs as it transits the sun, and if the rays are parallel then it will always be precisely the same distance from the edge at the same moment for both observers. Once again, this relies on an ability to place the observers equal distances from the sun, which is practically impossible on the surface of the Earth. This is pretty much the method they used for original determinations of the distances between planets.

If there is another question in there, I don't see it.

MikeS
06-27-2011, 07:17 AM
but online calculators won't even return a figure for the angle if i enter the hypothenus as 93 million miles while the base of the right triangle is, say 100 miles.If this is all that's keeping you from doing the calculations you want, then you should just use a small-angle approximation: sin(x) ≈ x, tan(x) ≈ x, and cos(x) ≈ 1 - x2/2. The idea is that if x is much smaller than 1, then the "approximation" sign above means that the difference between the right-hand and left-hand sides above will be practically zero. The angle x has to be measured in radians for this to work, though.

This means that the angle in the triangle you're describing (if you plug in the numbers) is approximately 1.08 x 10-6 radians, or about 62 millionths of a degree.

DrFidelius
06-27-2011, 07:21 AM
Close enough to parallel for government work.

md2000
06-27-2011, 07:34 AM
If this is all that's keeping you from doing the calculations you want, then you should just use a small-angle approximation: sin(x) ≈ x, tan(x) ≈ x, and cos(x) ≈ 1 - x2/2. The idea is that if x is much smaller than 1, then the "approximation" sign above means that the difference between the right-hand and left-hand sides above will be practically zero. The angle x has to be measured in radians for this to work, though.

This means that the angle in the triangle you're describing (if you plug in the numbers) is approximately 1.08 x 10-6 radians, or about 62 millionths of a degree.

Exactly. The rays are always radial; they never "change to parallel". However, the deviation from parallel is so miniscule for practical purposes, that it might as well be parallel.

Also note the sun is a visible size from earth. That means it's not that simple. Every part of the sun's visible disc acts as a separate point source, so we get a real mess of rays.

kanicbird
06-27-2011, 08:35 AM
there was the mathematician erathosthenes' calculation wherein he assumed the sun's rays hit the earth in parallel rays. he observed two dry wells located in different cities inside egypt, along the same longitude (syene and alexandria.) at exactly the same time,

Wondering how he was able to check at the exact same time. Any thoughts on this :confused:

Blake
06-27-2011, 08:39 AM
Wondering how he was able to check at the exact same time. Any thoughts on this :confused:

It was noon. Nothing tricky about it.

yabob
06-27-2011, 08:41 AM
Wondering how he was able to check at the exact same time. Any thoughts on this :confused:
He didn't actually. He didn't need to measure anything in Syene. He knew from records that at the summer solstice, the sun was directly overhead in Syene. He simply measured the angle of the sun at the summer solstice in Alexandria.

yabob
06-27-2011, 09:00 AM
Of course, I should have said he measured the angle of the Sun AT NOON to be clearer.

His methodology was good, but he got lucky. He assumed that Alexandria was directly north of Syene. There's a degree or two difference in longitude, actually. His measurement of distance was likely even more inaccurate - it was an estimate of the overland distance obtained from the drivers of camel caravans. His tools for measuring the Sun's angle couldn't have been that precise. Any of those things are of much greater magnitude than the deviation from parallel of the Sun's rays. We aren't sure exactly what length he meant by a "stadion", either. There were several different "stadia" in use in antiquity, and we may be assigning him the best one after the fact.

kanicbird
06-27-2011, 09:31 AM
His methodology was good, but he got lucky. He assumed that Alexandria was directly north of Syene.

OK thanks, I was thinking more like these cities were east-west of each other, north south makes a lot more sense.

yabob
06-27-2011, 10:55 AM
OK thanks, I was thinking more like these cities were east-west of each other, north south makes a lot more sense.
BTW, I just looked it up. Syene (Aswan) is at 32 54 E, Alexandria at 29 55 E. At 24 5 N latitude, Syene is also about half a degree off the actual Tropic of Cancer, too. Using naked eye observations, the Sun still appeared directly overhead on the solstice, and would for any location within a sizeable band around the tropic. Another source of experimental error in his calculation.

Chronos
06-27-2011, 11:08 AM
Quoth Blake:
An alternative method is to actually reverse Eratosthenes' experiment. Place a relatively nearby object between one observer and the sun, and check whether it is also between another observer and the sun. If the rays are truly parallel then it must intervene for all observers regardless of how far apart they are. An artificial satellite would be a good candidate. Have your two observers take a series of photographs as it transits the sun, and if the rays are parallel then it will always be precisely the same distance from the edge at the same moment for both observers. Once again, this relies on an ability to place the observers equal distances from the sun, which is practically impossible on the surface of the Earth. This is pretty much the method they used for original determinations of the distances between planets.
This doesn't work. All this would tell you is the distance to the obscuring object, not the distance to the Sun. That's fine if you have some other known relationship between the distance to the other object and the distance to the Sun, as for instance if the obscuring object is Venus. But it won't work for an Earth-orbiting satellite.

OldGuy
06-27-2011, 12:28 PM
Since the diameter of the sun is substantially greater than the diameter of the earth there is a rays of sun impinging on every point of the earth that is precisely parallel to a ray of sun impinging on any other point on the earth. There are of course non-parallel rays as well. (This of course assumes Euclidean geometry ignoring the space curvature effects of General Relativity.)

md2000
06-27-2011, 12:54 PM
Yes, a ray from the sun at point "A" striking the earth at point "B" will be parallel to a ray of light from the sun starting 2 feet left of A and hitting the earth 2 feet left of B (My left, not yours).

scr4
06-27-2011, 01:14 PM
This means that the angle in the triangle you're describing (if you plug in the numbers) is approximately 1.08 x 10-6 radians, or about 62 millionths of a degree.

That's 0.22 arcseconds. Should be measurable under the right circumstances. (it would be very easy if there was a star nearby for reference; doing it for the sun would be more tricky.)

mac_bolan00
06-27-2011, 06:01 PM
thanks fellas. i particularly liked two replies: the small angle approximation and the fact that the sun is a big visible disc that radiates light from all points in that disc. even if we assume a single point-source, the resulting angle will still be too small to be measured using earth-bound survey equipment (much less home-made set-ups.) that, even if the two observers were at the geographic north and south poles at the exact time the axis was perpendicular to the central ray.

i was suggesting this to my nephew as his high school physics experiment but it doesn't seem feasible anymore.

Manduck
06-27-2011, 06:48 PM
Of course, I should have said he measured the angle of the Sun AT NOON to be clearer.

His methodology was good, but he got lucky. He assumed that Alexandria was directly north of Syene. There's a degree or two difference in longitude, actually. His measurement of distance was likely even more inaccurate - it was an estimate of the overland distance obtained from the drivers of camel caravans. His tools for measuring the Sun's angle couldn't have been that precise. Any of those things are of much greater magnitude than the deviation from parallel of the Sun's rays....

He wasn't trying to measure the deviation from parallel of the sun's rays - he was assuming they were parallel and using that to measure the curvature of the earth.