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

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Expand view Topic review: APOD: Opportunity's Mars Analemma (2014 May 16)

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

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

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

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

Lettuce snow if I've made a mistake.

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

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.

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

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.

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

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).

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

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

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

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

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

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

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.

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

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
Thresold Eccentricity for Analemma Crossing.JPG (10.44 KiB) Viewed 2529 times
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.

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

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?

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

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.

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

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).

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

by Nitpicker » Sat May 17, 2014 7:42 am

Thank you alter-ego. Nice work.

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

by alter-ego » Sat May 17, 2014 6:18 am

Nitpicker wrote:
neufer wrote:
Nitpicker wrote:
No, it does not cross itself. This is because the higher eccentricity of Mars's orbit has a much more dominant effect on the analemma, than the inclination of Mar's polar axis to the Ecliptic. 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).
No, it does not cross itself... because the equation of time effect due to the inclination of Mars' polar axis (i.e., the close spacing of time zones near the poles) is closely matched in magnitude with that due to the high eccentricity of Mars' orbit (such that they nearly cancel each other at aphelion).
Not convinced. Show me the "equation of time" graph for Mars, including the two components from which it is formed.

A sundial on Mars would become a lot slower/faster over the course of a year, than a sundial on Earth (compared to an accurate clock). But Mars and Earth have similar inclinations, and both have the line of apsides far from the nodes. The big difference is the high eccentricity of Mars's orbit.
As you stated, the figure-8 shape is lost due to the dominant eccentricity component within the analemma. Per your request, I've calculated the EOT and analemma components for Mars. Instead of using Earth months on the EOT X-axis, I used orbital position angle which works the same. I personally like viewing the analemmas.

Mars Equation of Time & Components.JPG
Mars Analemma & Components.JPG
  1. Edit: At first, I mistakenly annotated the "Eccentricity Only" graph as being a 0° inclination case which is not true. The declination would remain 0° if that were the case. Just think of it as having axis tilt component removed.
Mars Analemma - Half Eccentricity.JPG
  • Edit: I decided to add the analemma for the case where the eccentricity is ½ the actual value. The figure-8 is beginning for form.

As an aside, geosynchronous satellites show the same gamut of analemmas too, and for the same reasons. From nice figure eights (circular orbits with non-zero orbital inclinations) to the pointy type (high eccentricity).

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

by Nitpicker » Sat May 17, 2014 3:25 am

neufer wrote:
Nitpicker wrote:
No, it does not cross itself. This is because the higher eccentricity of Mars's orbit has a much more dominant effect on the analemma, than the inclination of Mar's polar axis to the Ecliptic. 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).
No, it does not cross itself... because the equation of time effect due to the inclination of Mars' polar axis (i.e., the close spacing of time zones near the poles) is closely matched in magnitude with that due to the high eccentricity of Mars' orbit (such that they nearly cancel each other at aphelion).
Not convinced. Show me the "equation of time" graph for Mars, including the two components from which it is formed.

A sundial on Mars would become a lot slower/faster over the course of a year, than a sundial on Earth (compared to an accurate clock). But Mars and Earth have similar inclinations, and both have the line of apsides far from the nodes. The big difference is the high eccentricity of Mars's orbit.

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

by neufer » Sat May 17, 2014 3:14 am


walker1001 wrote:
But sadly no canals, no Martians, no War of the Worlds. :D
They sought it with thimbles, they sought it with care;
They pursued it with forks and hope;
They threatened its life with a railway-share;
They charmed it with smiles and soap.

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

by neufer » Sat May 17, 2014 3:09 am

Nitpicker wrote:
No, it does not cross itself. This is because the higher eccentricity of Mars's orbit has a much more dominant effect on the analemma, than the inclination of Mar's polar axis to the Ecliptic. 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).
No, it does not cross itself... because the equation of time effect due to the inclination of Mars' polar axis (i.e., the close spacing of time zones near the poles) is closely matched in magnitude with that due to the high eccentricity of Mars' orbit (such that they nearly cancel each other at aphelion).

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

by Nitpicker » Sat May 17, 2014 12:21 am

No, it does not cross itself. This is because the higher eccentricity of Mars's orbit has a much more dominant effect on the analemma, than the inclination of Mar's polar axis to the Ecliptic. 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).

See also:
http://en.wikipedia.org/wiki/Equation_of_time (for Earth)
and:
http://en.wikipedia.org/wiki/Timekeeping_on_Mars

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

by Chris Peterson » Sat May 17, 2014 12:06 am

JEH1138 wrote:So forgive me fore being dense, but does the sun in a Martian analemma cross back on itself at all?
No.

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

by JEH1138 » Fri May 16, 2014 11:59 pm

So forgive me fore being dense, but does the sun in a Martian analemma cross back on itself at all? Is it just too hard to see from greater distances? Probably my lack of understanding about what actually causes the shape over time.

--j

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

by Nitpicker » Fri May 16, 2014 11:05 pm

geoffrey.landis wrote:The caption is a little misleading. It's not that the martian sky is "blacked out" around the solar images; what you're seeing is that when we're pointing the camera directly at the sun, we use a neutral density solar filter, which is so dark that the sky comes out black.
I would not have called that misleading. "Blacked out" and "underexposed" are more or less synonymous in this context.

Black sky

by geoffrey.landis » Fri May 16, 2014 9:21 pm

The caption is a little misleading. It's not that the martian sky is "blacked out" around the solar images; what you're seeing is that when we're pointing the camera directly at the sun, we use a neutral density solar filter, which is so dark that the sky comes out black.

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

by walker1001 » Fri May 16, 2014 4:02 pm

neufer wrote:
Nitpicker wrote:
Simulating or modelling an analemma is a fine thing to do. But it is somehow more beautiful to record one directly.

And to see one recorded authentically from Mars ... well, it's a good time to be alive.
We now have the means, motive, and Opportunity.
But sadly no canals, no Martians, no War of the Worlds. :D

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

by starsurfer » Fri May 16, 2014 3:15 pm

neufer wrote:
Nitpicker wrote:
Simulating or modelling an analemma is a fine thing to do. But it is somehow more beautiful to record one directly.

And to see one recorded authentically from Mars ... well, it's a good time to be alive.
We now have the means, motive, and Opportunity.
Opportunity rover:
"If it wasn't for those pesky dust storms, I would have gotten all the frames. Drat and double drat!"

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