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

[David Bowman <David_Bowman@georgetowncollege.edu>]

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
> pattern.

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 from 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 eccentricity.
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 characterized.

David Bowman