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Re: Sine curve to approximate declination
From: Frank Reed CT
Date: 2004 May 20, 14:26 EDT
From: Frank Reed CT
Date: 2004 May 20, 14:26 EDT
Trevor K wrote:
"But in that most extreme case, and assuming a perfectly circular orbit, I think declination would change linearly from solstice to equinox and vice versa -- meaning that the curve would have "narrower shoulders" than a sine curve.
Can you explain how you see the "broader shoulders"? "
(and George H wrote something similar).
Sorry, I should have been more explicit. When I said "imagine what the ecliptic would look like", I was refering to its appearance on a standard star chart. So the difference between these two points of view amounts to a difference in choice of independent variable. If you graph the ecliptic's declination versus time, then you would get a "sawtooth" graph for extremely high axial tilt, just as you've described above. If you graph the ecliptic's declination versus right ascension (or SHA) then you would get a "square wave" graph for extremely high axial tilt. At the Earth's actual axial tilt of 23.45 degrees, the appearance of the ecliptic on a typical star chart (like the one in the Nautical Almanac) resembles a sine curve and is closely approximated by Dec=-23.45*sin(SHA) but it bulges out a bit from a sine curve (giving the "broader shoulders" that I was talking about earlier). If you graph the Sun's declination versus time, then you get straighter sides (narrower shoulders).
There's a basically identical case involving the ground tracks of artificial satellites. If you look at an ordinary map of the ground track of one of the GPS satellites, for example, it will look roughly like a sine wave, but the up and down curves, north and south of the equator, are more "squared off". The more extreme case of the ground track of a satellite on a nearly polar orbit (like the recently-launched Gravity Probe B) makes a square wave on a map of the Earth. But if you graph the latitude of the sub-satellite point versus time, you get a sawtooth pattern. Neither one looks much like a sine wave anymore though, of course, you could add harmonics and eventually get very close to either of those curves (but that's way beyond the issue we've been discussing here).
By the way, I ordinarily read the messages on this list via the archives on i-DEADLINK-com, and I noticed that two of my messages from yesterday do not seem to have turned up in the archives despite the fact that they clearly made it out to list members. Has anyone else seen this happen? Maybe that site's server was down for a little while...
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
"But in that most extreme case, and assuming a perfectly circular orbit, I think declination would change linearly from solstice to equinox and vice versa -- meaning that the curve would have "narrower shoulders" than a sine curve.
Can you explain how you see the "broader shoulders"? "
(and George H wrote something similar).
Sorry, I should have been more explicit. When I said "imagine what the ecliptic would look like", I was refering to its appearance on a standard star chart. So the difference between these two points of view amounts to a difference in choice of independent variable. If you graph the ecliptic's declination versus time, then you would get a "sawtooth" graph for extremely high axial tilt, just as you've described above. If you graph the ecliptic's declination versus right ascension (or SHA) then you would get a "square wave" graph for extremely high axial tilt. At the Earth's actual axial tilt of 23.45 degrees, the appearance of the ecliptic on a typical star chart (like the one in the Nautical Almanac) resembles a sine curve and is closely approximated by Dec=-23.45*sin(SHA) but it bulges out a bit from a sine curve (giving the "broader shoulders" that I was talking about earlier). If you graph the Sun's declination versus time, then you get straighter sides (narrower shoulders).
There's a basically identical case involving the ground tracks of artificial satellites. If you look at an ordinary map of the ground track of one of the GPS satellites, for example, it will look roughly like a sine wave, but the up and down curves, north and south of the equator, are more "squared off". The more extreme case of the ground track of a satellite on a nearly polar orbit (like the recently-launched Gravity Probe B) makes a square wave on a map of the Earth. But if you graph the latitude of the sub-satellite point versus time, you get a sawtooth pattern. Neither one looks much like a sine wave anymore though, of course, you could add harmonics and eventually get very close to either of those curves (but that's way beyond the issue we've been discussing here).
By the way, I ordinarily read the messages on this list via the archives on i-DEADLINK-com, and I noticed that two of my messages from yesterday do not seem to have turned up in the archives despite the fact that they clearly made it out to list members. Has anyone else seen this happen? Maybe that site's server was down for a little while...
Frank R
[ ] Mystic, Connecticut
[X] Chicago, Illinois
