Starship Asterisk*

APOD Robot wrote: ↑Sun Sep 18, 2022 4:05 am It is not known if the placement of the Callanish Stones has or had astronomical significance.

johnnydeep wrote: ↑Sun Sep 18, 2022 7:50 pm Well hello there again to my old nemesis the Analemma. I'll probably be dead before I really understand thee. Caveat lector: no need for helpful others to venture to explain it further: there have already been many who've tried in vain before you. You will not fare any better. What might help is an app that allows you to change the angle of the tilt of the Earth's axis and the eccentricity of the Earth's orbit to see how the shape of the analemma changes as a result.

johnnydeep wrote: ↑Sun Sep 18, 2022 7:50 pm that allows you to change the angle of the tilt of the Earth's axis and the eccentricity of the Earth's orbit to see how the shape of the analemma changes as a result.

johnnydeep wrote: ↑Mon Sep 19, 2022 3:49 pm Thanks for trying. Apparently, the only case I am capable of understanding is if the Earth had a perfectly circular orbit, a 0 degree axis tilt, and was tidally locked so that one side faced the Sun at all times (such that the rotation rate matched the yearly orbital period). In that case, the position of the Sun in the sky would never change at all for any observer anywhere on the Earth, and the "analemma" would be a single point.

Now, let's see, if the orbit was allowed to be a proper Newtonian ellipse (all other aforementioned hypotheticals remaining in effect), would the analemma still be a point? Yes, right?

Chris Peterson wrote: ↑Mon Sep 19, 2022 4:13 pm

johnnydeep wrote: ↑Mon Sep 19, 2022 3:49 pm Thanks for trying. Apparently, the only case I am capable of understanding is if the Earth had a perfectly circular orbit, a 0 degree axis tilt, and was tidally locked so that one side faced the Sun at all times (such that the rotation rate matched the yearly orbital period). In that case, the position of the Sun in the sky would never change at all for any observer anywhere on the Earth, and the "analemma" would be a single point.

Now, let's see, if the orbit was allowed to be a proper Newtonian ellipse (all other aforementioned hypotheticals remaining in effect), would the analemma still be a point? Yes, right?

Maybe you can slowly devolve your "perfect" system.

If the Earth had no axial tilt with respect to its orbital plane, and an orbital eccentricity of zero (i.e. a circular orbit), the analemma would be a point. The Sun would appear in the same spot each day at a give time. (Noon is easiest to understand, with the Sun right on the meridian and its altitude determined only by the latitude of the observer.)

Now lets give the Earth some axial tilt. This basically just introduces seasons, so the Sun is higher or lower in the sky depending on which part of our annual orbit we're on. So now the analemma has become a line perpendicular to the horizon. The Sun is still always on the meridian at noon, but its altitude depends both upon the observer's latitude and the position in our orbit.

Now lets make our orbit a little eccentric. That means our orbital speed isn't uniform. Per Kepler's Second Law our orbital speed will be greater when we're closer to the Sun. So this makes the Sun a little before or a little after the meridian at noon, depending on where we are in our orbit. If we had no axial tilt, the analemma would again be a line, but in this case parallel to the horizon. With both tilt and eccentricity, we have deviation vertically and horizontally, which is why real world analemmas look like closed roundish or figure-eight paths.

johnnydeep wrote: ↑Mon Sep 19, 2022 8:51 pm

Chris Peterson wrote: ↑Mon Sep 19, 2022 4:13 pm

johnnydeep wrote: ↑Mon Sep 19, 2022 3:49 pm Thanks for trying. Apparently, the only case I am capable of understanding is if the Earth had a perfectly circular orbit, a 0 degree axis tilt, and was tidally locked so that one side faced the Sun at all times (such that the rotation rate matched the yearly orbital period). In that case, the position of the Sun in the sky would never change at all for any observer anywhere on the Earth, and the "analemma" would be a single point.

Now, let's see, if the orbit was allowed to be a proper Newtonian ellipse (all other aforementioned hypotheticals remaining in effect), would the analemma still be a point? Yes, right?

Maybe you can slowly devolve your "perfect" system.

If the Earth had no axial tilt with respect to its orbital plane, and an orbital eccentricity of zero (i.e. a circular orbit), the analemma would be a point. The Sun would appear in the same spot each day at a give time. (Noon is easiest to understand, with the Sun right on the meridian and its altitude determined only by the latitude of the observer.)

Now lets give the Earth some axial tilt. This basically just introduces seasons, so the Sun is higher or lower in the sky depending on which part of our annual orbit we're on. So now the analemma has become a line perpendicular to the horizon. The Sun is still always on the meridian at noon, but its altitude depends both upon the observer's latitude and the position in our orbit.

Now lets make our orbit a little eccentric. That means our orbital speed isn't uniform. Per Kepler's Second Law our orbital speed will be greater when we're closer to the Sun. So this makes the Sun a little before or a little after the meridian at noon, depending on where we are in our orbit. If we had no axial tilt, the analemma would again be a line, but in this case parallel to the horizon. With both tilt and eccentricity, we have deviation vertically and horizontally, which is why real world analemmas look like closed roundish or figure-eight paths.

Thanks. My 3D sense has mostly vanished from when I had it as a math major 40 years ago. Or maybe I never had it at all when it comes to astronomical motions. What does it mean to take a photo of the Sun at the "same time" each day? How is a day defined here? Using solar time as opposed to sidereal time I suppose? I feel like I'm regressing to the point of needing a course in remedial astronomy...

Chris Peterson wrote: ↑Mon Sep 19, 2022 8:58 pm

johnnydeep wrote: ↑Mon Sep 19, 2022 8:51 pm

Chris Peterson wrote: ↑Mon Sep 19, 2022 4:13 pm

Maybe you can slowly devolve your "perfect" system.

If the Earth had no axial tilt with respect to its orbital plane, and an orbital eccentricity of zero (i.e. a circular orbit), the analemma would be a point. The Sun would appear in the same spot each day at a give time. (Noon is easiest to understand, with the Sun right on the meridian and its altitude determined only by the latitude of the observer.)

Now lets give the Earth some axial tilt. This basically just introduces seasons, so the Sun is higher or lower in the sky depending on which part of our annual orbit we're on. So now the analemma has become a line perpendicular to the horizon. The Sun is still always on the meridian at noon, but its altitude depends both upon the observer's latitude and the position in our orbit.

Now lets make our orbit a little eccentric. That means our orbital speed isn't uniform. Per Kepler's Second Law our orbital speed will be greater when we're closer to the Sun. So this makes the Sun a little before or a little after the meridian at noon, depending on where we are in our orbit. If we had no axial tilt, the analemma would again be a line, but in this case parallel to the horizon. With both tilt and eccentricity, we have deviation vertically and horizontally, which is why real world analemmas look like closed roundish or figure-eight paths.

Thanks. My 3D sense has mostly vanished from when I had it as a math major 40 years ago. Or maybe I never had it at all when it comes to astronomical motions. What does it mean to take a photo of the Sun at the "same time" each day? How is a day defined here? Using solar time as opposed to sidereal time I suppose? I feel like I'm regressing to the point of needing a course in remedial astronomy...

You can work backwards to time. If we had a perfectly circular orbit, and measured the time from one meridian crossing of the Sun to the next (solar noon to solar noon) that's the time we're interested in. Solar time. A synodic day. And that's what we continue to use once we have tilt and eccentricity.

johnnydeep wrote: ↑Mon Sep 19, 2022 9:07 pm So if sidereal time was used, would the shape of the analemma change much? And how about if the Earth rotated the opposite way (at the same rate)?

Chris Peterson wrote: ↑Mon Sep 19, 2022 9:13 pm

johnnydeep wrote: ↑Mon Sep 19, 2022 9:07 pm So if sidereal time was used, would the shape of the analemma change much? And how about if the Earth rotated the opposite way (at the same rate)?

If you used sidereal time the Sun wouldn't even be in the sky for about half the year. I'd say that's a change to the analemma!

Mathew32df wrote: ↑Thu Oct 27, 2022 4:08 am

APOD Robot wrote: ↑Sun Sep 18, 2022 4:05 am Analemma over the Callanish Stones

Great picture, can I use it?