APOD: Opportunity's Mars Analemma (2014 May 16)

Comments and questions about the APOD on the main view screen.
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neufer
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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by neufer » Sat May 17, 2014 12:48 pm

Nitpicker wrote:
Thank you alter-ego. Nice work.
The amplitude of the eccentricity component is indeed 3 times that of the obliquity component (as shown by alter-ego); however, cross over is determined by the first derivative (day to day change) functions which, in fact, are comparable (i.e., 3:2 resulting in a teardrop shaped analemma).

For the Earth the first derivative (day to day change) functions are far from comparable/closely matched (resulting in a figure eight shaped analemma).
Nitpicker wrote:
These two components of the more circular Earth orbit, are much more closely matched in magnitude (but the effect of the inclination has twice the frequency).
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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by Nitpicker » Sun May 18, 2014 12:06 am

Neufer, that strikes me as a rather obfuscated explanation. (At least I know it is really you.)

Assuming a planet is only orbiting one star, and assuming the only significant components to the analemma are eccentricity and axial tilt:

1) A planet with a circular orbit and no axial tilt, will have an analemma in the form of a dot, and a sundial on the planet would be accurate throughout the year.

2) A planet with an eccentric orbit and no axial tilt, will have an analemma in the form of a line along the Celestial Equator.

3) A planet with a circular orbit and some axial tilt, will have an analemma in the form of a symmetrical figure-8. It is only the tilt component which can give an analemma any area in the sky, and without any eccentricity, the analemma will always cross itself.

4) A planet with an eccentric orbit and some axial tilt, will have an analemma which may or may not cross itself (oval, teardrop, peanut, or figure-8 shaped), depending on whether the component of eccentricity is dominant enough to overcome the cross-over inherent in the tilt component.

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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by southern cross » Sun May 18, 2014 5:49 am

Thanks for the analemma calculations - seems quite easy as long as you know the orbital shape / equation of time?

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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by alter-ego » Sun May 18, 2014 7:07 am

Nitpicker wrote:Neufer, that strikes me as a rather obfuscated explanation. (At least I know it is really you.)

Assuming a planet is only orbiting one star, and assuming the only significant components to the analemma are eccentricity and axial tilt:

4) A planet with an eccentric orbit and some axial tilt, will have an analemma which may or may not cross itself (oval, teardrop, peanut, or figure-8 shaped), depending on whether the component of eccentricity is dominant enough to overcome the cross-over inherent in the tilt component.
Well, I can't help much with Art's interpretation but it may not be in conflict with your statements. It seems he is relating analemma crossing to EOT derivatives. I don't think he's disagreeing with you that eccentricity does affect crossing. In fact he'd quickly say that the crossing shares a coupled dependence between e and ε. However, I do find his discussion overly abstract. My contribution below is not easy to understand either, but you may find the result helpful.

Anyway, I decided to try to express analemma crossing mathematically in terms of the key parameters. This is not an easy problem. My reasoning was founded in an equivalence condition between EOT 1st and 2nd derivatives, but likely not related to Art's reasoning. By looking at the threshold condition for crossing in terms of the derivatives, I realized that one might be able to express an upper bound for crossing, i.e. given an axis tilt, the eccentricity must be < xxx for analemma crossing to occur. I soon remembered another significant factor that would have to be modeled. I call it Δ and it is defined as the angle between the Line of Apsides and the Line of Solstice (a similar line connecting the planet's summer and winter solstice orbital positions). Δ is a parameter that describes the apparent analemma tip. For Earth, if Δ = 0° then our "high noon" analemma would appear truly vertical (which it doesn't). Since the perihelion dates are ~13 days after the solstice, |Δ| ≈ 13°.

After some bona-fide derivation and brute force analysis, I derived a very good approximation for the upper bound eccentricity limit for analemma crossing. I only tested it for Earth and Mars having axis tilts (ε) ranging from 1° to ~70°, and Δ ranging from 0° to 65°. For all cases, the threshold eccentricity for crossing landed within 1% of the analemma models, and the predicted threshold eccentricity ranged from <0.0001 to ~3.3.
Thresold Eccentricity for Analemma Crossing.JPG
I don't tout this to be a general solution for all conditions, but it is surprisingly accurate over a fairly broad test window. I haven't run across this relationship before and I'm probably the only one geeky enough to enjoy this find.
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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by alter-ego » Sun May 18, 2014 7:13 am

southern cross wrote:Thanks for the analemma calculations - seems quite easy as long as you know the orbital shape / equation of time?
You're welcome.
Well, "easy" is a relative thing, but yes, you do need to know the orbital shape and the planet's solar day to obtain the EOT. Of course the EOT then yields the analemma.
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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by Nitpicker » Sun May 18, 2014 8:36 am

I think I'm just geeky enough to enjoy that too, alter-ego. Well done.

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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by neufer » Sun May 18, 2014 12:59 pm

alter-ego wrote:
Nitpicker wrote:
Neufer, that strikes me as a rather obfuscated explanation. (At least I know it is really you.)

Assuming a planet is only orbiting one star, and assuming the only significant components to the analemma are eccentricity and axial tilt:

4) A planet with an eccentric orbit and some axial tilt, will have an analemma which may or may not cross itself (oval, teardrop, peanut, or figure-8 shaped), depending on whether the component of eccentricity is dominant enough to overcome the cross-over inherent in the tilt component.
Well, I can't help much with Art's interpretation but it may not be in conflict with your statements. It seems he is relating analemma crossing to EOT derivatives. I don't think he's disagreeing with you that eccentricity does affect crossing. In fact he'd quickly say that the crossing shares a coupled dependence between e and ε.
Yes...my nit-pick was that Nitpickers' rule of thumb was oversimplified & somewhat misleading and deserved a little more thought (which it now has been given). Eccentricity effects upon the morphological shape of the analemma are less than one might at first expect from an "alter-ego" type plot due to both: 1) the EOT derivative factor and 2) the relative phase factor.

The way I see it:
  • 1) Jupiter has an oval shaped analemma due to dominant eccentricity effects.

    2) Mars and Saturn have teardrop shaped analemmas due to comparable eccentricity/obliquity effects.

    3) Earth has a slightly bloated figure eight shaped analemma due to relatively weak eccentricity effects.

    4) Uranus, Neptune & Pluto have symmetric figure eight shaped analemmas due to dominant obliquity effects.

    5) (Long days on Mercury & Venus do not produce true analemmas.)
See: http://www.analemma.com/Pages/framesPage.html
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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by Nitpicker » Sun May 18, 2014 10:28 pm

True, I hadn't considered planets like Mercury and Venus where tidal locking effects (or whatever the cause) result in very long days.

In plain English, when comparing the two components of the EOT, I would naturally consider the primary indicator to be the amplitude, not the gradient. Regardless, (assuming some tilt, but no matter the phase difference between the line of apsides and the equinox nodes) the amplitude of the eccentricity component may always be increased from zero, to a point where the analemma no longer crosses itself. Therefore, it is accurate and concise to say that an analemma which does not cross itself is caused by a sufficiently eccentric orbit (which is more or less what the original APOD caption is saying, too).

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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by Nitpicker » Sun May 18, 2014 11:30 pm

In other words ... neufer, the only point of difference between us here, is in our comparison of the two components, and how we choose to say one has more of an effect than the other. The tilt component always has about twice the frequency of the eccentricity component. So, if the two components of the EOT have equal amplitudes, the maximum gradient of the tilt component will always be about twice that of the eccentricity component. Fairy nuff.

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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by neufer » Mon May 19, 2014 12:25 am

Nitpicker wrote:
In other words ... neufer, the only point of difference between us here, is in our comparison of the two components, and how we choose to say one has more of an effect than the other. The tilt component always has about twice the frequency of the eccentricity component. So, if the two components of the EOT have equal amplitudes, the maximum gradient of the tilt component will always be about twice that of the eccentricity component.

Fairy nuff.
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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by Nitpicker » Mon May 19, 2014 1:47 am

Lettuce snow if I've made a mistake.

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Re: APOD: Opportunity's Mars Analemma (2014 May 16)

Post by neufer » Mon May 19, 2014 10:08 am

Nitpicker wrote:
Lettuce snow if I've made a mistake.
http://crowdingthebooktruck.blogspot.co ... anley.html
Art Neuendorffer