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Re: [Phys-L] analemma and equation of time

David's comment was very interesting, even if beyond the scope of my intro
class. Very cool! I feel like this could/should be a TED talk! Will you do
it? :)

I appreciate the computer program outputs about all the planets.

On Tue, Sep 15, 2020 at 10:32 AM David Bowman <> wrote:

Regarding Anthony L's comment:

I recall decades ago there was a photo in *Sky And Telescope* of
the analemma where someone photographed the Sun every two weeks
over the year; I was intrigued by the asymmetric figure-eight

It should be noted that the particular shape of an analemma is not a
permanent or fixed feature, but is a function of a planet's orbital &
orientational/rotational parameters. Different planets have different
shaped analemmas that gradually change size and shape as their orbits
precess and change eccentricity, and as their rotational axes precess and
bob. In a few thousand years the Earth's analemma will no longer be a
figure-eight but a closed non-self crossing figure that is sort of
cashew-shaped. Some thousands of years later it will again be a
figure-eight but its large lobe will be for northerly declinations rather
than for the current southerly ones. Gradually things will drift back
toward the current shape over tens of thousands of years.

One can think of an analemma as a Lissajous figure driven both
horizontally and vertically by two anharmonic but nearly synchronized
nearly periodic signals of nearly the same fundamental frequency. If we
neglect the quite slow precession/Milankovitch cycle effects then the
signals are synchronized from year to year. If the planet's spin axis is
significantly tilted (such as for the Earth) the horizontal (right
ascension) channel is strongly contaminated with a significant 2nd harmonic
term coming from the axis obliquity. If this 2nd harmonic term dominates
over the fundamental term or is at least comparable in magnitude to it the
Lissajous becomes a figure-eight. If the perihelion/aphelion points are
close to the solstice points, like the current situation for Earth, the
northern & southern halves become more mutually asymmetric. If the orbital
eccentricity is high enough then that effect (whose period is of the
fundamental frequency) can dominate over the 2nd harmonic term fro
m the obliquity. In that case the figure remains a single
non-self-intersecting closed loop. This is the situation with the Martian
analemma which currently has the shape of a partially tilted teardrop.
Remarkably, the obliquity effect, besides driving the horizontal channel's
strong 2nd harmonic, also sets the overall amplitude of the
vertical/declination channel. If the orbit was a perfect circle, but with
a significantly tilted axis the analemma would be a perfectly symmetric
figure-eight and look like a vertically oriented lemniscate.

A few months ago I wrote a computer program that calculated an analemma
for any planet once one inputted the appropriate orbital elements and axis
orientation/rotation information. I used it to make analemmas for all the
major planets. For some reason I didn't apply it to minor planets like
Ceres or Pluto. I suspect the program might give an incorrect analemma for
Pluto because it doesn't account for perturbative effects of the planet's
satellites, and Pluto's satellite, Charon, is so big compared to Pluto that
their mutual barycenter is out in space between them a significant way
toward Charon. And the Pluto/Charon orbital plane is significantly tilted
w.r.t. their barycenter's orbital plane around the Sun. But maybe Pluto is
so far from the Sun that its significant local dancing on its way around
the Sun doesn't matter much. In any event here is a description of the
shapes of the current analemmas of the major planets:

Mercury: Non-self-intersecting closed loop that looks like a somewhat
tilted cross section of an aircraft wing.
Venus: Non-self-intersecting closed loop that looks like a somewhat
tilted ellipse with just a very little ovoid eccentricity.
Earth: Figure-eight with southern lobe significantly larger than the
northern lobe and a smaller east-west asymmetry.
Mars: Non-self-intersecting closed loop with a somewhat tilted teardrop
shape with some bi-lateral asymmetry.
Jupiter: Nearly symmetric tilted ellipse with negligible ovoid
Saturn: Actually an extremely asymmetric figure-eight with a very tiny
northern lobe and a relatively huge southern lobe. The northern lobe is so
tiny that a fuzzy version of it looks more like a cusp with the whole thing
shaped more like a teardrop a la Mars.
Uranus: Somewhat flattened figure-eight that has sort of an hourglass
appearance. (This one surprised me.)
Neptune: A somewhat tilted figure-eight/lemniscate.

I should note that the calculated analemmas described here are *not* for a
series of snapshots of the Sun's actual position at the same mean solar
time for each sol throughout the year. This is because that would not work
out well for very slowly rotating planets like Mercury and Venus. On
Mercury one solar day is exactly 2 of its years and thus its analemma
(neglecting any precession effects) would be just a single point if a
snapshot of the Sun's position was used. Venus has nearly 2 solar days per
year, but they are not comensurate, so over a time scale of very many
Venusian years a sol-wise series of snapshots of the sun's position would
eventually trace out the whole analemma figure. To avoid such issues my
computer calculations and descriptions above are simply for a plot of
differential right ascension horizontally vs. solar declination vertically
over the course of one orbit of the Sun. Also, the differential right
ascension was measured in degrees rather than a minutes/
hours (which vary quite widely over the range of planetary rotation
rates). Also the graphs for the descriptions above were not necessarily
scaled with a common degree scale for both the declination direction and
the differential right ascension direction because a common scale would
hide the much of the shape features when there was a huge mismatch between
the obliquity angle and the maximum/minimum right ascension shift angle
where a common scale would sometimes tend to show a very thin figure in one
direction and stretched in the other oriented along either the north/south
or east/west directions.

Perhaps if there is significant interest I could find a way to post the
calculated figures as labeled graphs.

Now that I've been motivated to revisit the issue I may just calculate
analemmas for Ceres & Pluto if I can track down their relevant orbital &
spin axial parameters. Other dwarf planets may be cool to do, too, but I
don't know if such things have the needed direction of their spin axes well

David Bowman
Forum for Physics Educators



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