Re: Shadow of sundial a straight line on equinox day?

Peter Lewis wrote:
Does the shadow of a simple sundial form a straight line plus/minus a
few hours around equinox (at all latitudes)? And a curve at all other
times?

Heh, you've started a lively little discussion.

Assuming

* I've understood you correctly. :)
* The Sun is a point source.
* Atmospheric refraction is negligible.
* Variation in the Sun's declination during the day is negligible.
* The sundial is planar. (Some have cylindrical or other curved
surfaces, for reasons we can ignore here.)

Then the answer to your question is yes, the Sun's shadow does traverse
a straight line at the equinoxes.

Reasoning: The Sun's path during the course of a day is an arc of a
circle on the celestial sphere. At the equinoxes, this circle is a
great circle, like the equator on the Earth; at all other times, it's
less than great, like the 10 degree north latitude circle.

The Sun causes the gnomon's tip to cast a shadow. We see this shadow as
a point on the surface of the sundial, but it's really a line extended
from the gnomon's tip out to infinity, extending away from the Sun (just
as the Earth casts a shadow out into outer space, which occasionally
strikes the Moon, during a lunar eclipse).

On most days, because the Sun's path is less than great, this shadow
line traces out a cone (just as a line drawn from the center of the
Earth to points on the 10 degree suoth latitude circle would describe a
cone). What we see over the course of the day is the intersection of
this cone with the plane of the sundial. From conics, the intersection
of a cone with a plane is a conic section: a circle, an ellipse, a
parabola, or a hyperbola. If the sundial is level, the path is a
hyperbola.

However, at the equinoxes, the cone degenerates to a plane (just as a
line drawn from the center of the Earth to points on the equator would
also describe a plane). The intersection of this plane with the plane
of the sundial is necessarily a straight line, unless the two planes
are parallel, as they would be for a level sundial at the poles. As
you correctly said, there is *no* shadow for such a sundial at all on
that day.

I'm afraid there I can't think of a much simpler way to explain this
in text. With graphics it would be much easier, so see the Web sites
cited by others.

--
Brian Tung <brian@xxxxxxxxxxxx>
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