Next message: Frank Reed: "Re: Sine curve to approximate declination"
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
irbs.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
