Re: Analemma dilema
From: Lewis Mammel (l.mammel_at_worldnet.att.net)
Date: 01/30/05
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Date: Sun, 30 Jan 2005 09:35:17 GMT
Roy Smith wrote:
>
> I've always "known" that an analemma was a figure-8, and never gave it
> much thought as to why it was. Now, I just read that the analemma for
> Mars is a single loop (http://en.wikipedia.org/wiki/Analemma).
>
> Is there something special about Mars's orbit which make the curve not
> cross itself? At first glance, Mars's orbit (eccentricity, inclination,
> etc) doesn't seem terribly different from Earth's.
( Please skip to the end of this for the fun part ! )
On Mars or the Earth, the analemma has the same origin. It is a measure
of the Sun's position at the same time of day over the course of the
year, with the days measured by physical, or true, time.
If the orbit were circular, and the axis of rotation perpendicular
to the orbit, the sun would return to the same spot every day, and
the analemma would be a point.
If the orbit were circular, but the axis of rotation were oblique
to the perpendicular to the orbit ( obliquity of the ecliptic, which
is comparable for Earth and Mars ) the analemma would be a symmetrical
figure eight, due to a purely geometrical effect.
The sun would travel uniformly along the ecliptic, which would be above
the equator in summer and below it in winter. At these times, the sun
would be gaining on the "mean sun", since a degree of longitude is shorter
away from the equator than it is on the equator.
Near the equinoxes, the sun would be losing to the mean sun, since it
is moving obliquely to the equator. However, at the solstices and the
equinoxes, the sun would match the mean sun, since these points
divide the circle of the sun in equal quadrants. Thus the sun gains
and loses against the mean sun, or true time, by an amount approximated
by alpha sin 2t, so that the sun's postition is given by
t + alpha sin 2t
where t is measured in "radian years", meaning an orbit
takes time t=2pi.
Now if the sun moves non-uniformly along the ecliptic, due to an
eccentricity of the planets orbit, we can dsignate its position
along the ecliptic by d(t), so the t + alpha sin 2t becomes
d(t) + alpha * sin 2d(t)
But d(t) gains and loses against true time like beta sin ( t + delta ),
since the planet's orbit is divided into two equal halves by the long
axis of the orbit, so d(t) ~= t + beta sin ( t + delta )
Let's suppose delta = 0, corresponding to perihelion at solstice.
then alpha sin 2d(t) = sin 2(t+beta sin t) =
sin 2t * cos ( 2 beta sin t ) + cos 2t * sin ( 2 beta sin t )
~= alpha sin 2t + 2 alpha beta sin t * cos 2 t
we ignore the term in "alpha beta" as "second order" and we have
the postion of the sun given approximately by
t + alpha sin 2t + beta sin t
********* fun part ***************
Well, that's a long way to go, but the upshot is that we can
regard the two effects as additive for conceptual purposes.
Just don't forget that this is an approximation.
Now here's the fun part. Suppose we make a 3-d wire figure,
with ( x, y, z ) = ( alpha sin 2t, sin t, cos t )
z = cos t is the vertical axis of the analemma, representing
the displacement of the sun from the equator.
If we sight right along the y-axis, we'll see the circular
orbit analemma, ( alpha sin 2t, cos t ) . If we rotate it
slightly so that we are sighting just off the y-axis, we
will see ( alpha sin 2t + beta sin t , cos t )
We can produce an analemma for varying degrees of eccentricity
( with perihelion at solstice ) by rotating this fixed wire figure!
You can see what will happen. Sighted along the x-axis
we have ( sin t , cos t ) ... i.e. a circle. If we tilt
the figure enough we will be looking through the open
loop of the circle, and not at the figure eight formed
by looking at it edge-wise.
If we look at just the right angle, we will see a "cusp"
at the top, corresponding to the case where the gain and
loss of the two effects at solstice exactly cancel. Note
this must be the solstice coinciding with aphelion.
I actually got this idea from looking at your web page cite, as
the Mars and Earth analemmas are very suggestive of this idea.
Lew Mammel, Jr.
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