[math-fun] Correction; Analemma
rwg>Based on [Veit's] earlier analysis, I withdraw my claim about hypercubes.
The effective center varies (slightly) depending on what plane the centroid oscillates in. Wrong again. The moment of inertia of a cube about its center is orientation- independent. The center of oscillation, when hung from a vertex, always divides the major diagonal in proportion 11:7.
(Off list, Veit pointed out that the center of oscillation of a hoop swinging in its plane about a point on its rim is indeed its "south pole", as if all its mass were concentrated there. Absurd. But apparently irrefutable. Slightly less absurd: Turning an ordinary disk shaped pendulum 90 degrees to swing like a paddle shortens its period. (Assuming zero air resistance.) ---------- Doug Dodds at Symbolics used to send the following every year at this time. If you haven't seen one, check out "analemma" in Google images for one of those 365-exposure sky pix. There's even one for Mars!.--rwg --- Received: from ELEPHANT-BUTTE.SCRC.Symbolics.COM by RUSSIAN.SPA.Symbolics.COM via CHAOS with CHAOS-MAIL id 52312; Tue 8-Dec-87 13:52:55 PST Received: from YUKON.SCRC.Symbolics.COM by ELEPHANT-BUTTE.SCRC.Symbolics.COMvia CHAOS with CHAOS-MAIL id 244027; Tue 8-Dec-87 16:18:32 EST Received: from ANNISQUAM.SCRC.Symbolics.COM by YUKON.SCRC.Symbolics.COM via INTERNET with SMTP id 303771; 8 Dec 87 16:11:04 EST Date: Tue, 8 Dec 87 16:11 EST From: Doug Dodds <DODDS@YUKON.SCRC.Symbolics.COM> Subject: Rejoice, the light returneth! To: BBoard@YUKON.SCRC.Symbolics.COM, Fun@YUKON.SCRC.Symbolics.COM File-References: SCRC|Y:>Dodds>analemma.mss.newest, SCRC|Y:>Dodds>analemma.text.newest Message-ID: <19871208211112.0.DODDS@ANNISQUAM.SCRC.Symbolics.COM> The sunset today in Boston is at 16:11 EST, which is Plenty Early. But take heart, today's sunset is the earliest of the year! (Everywhere in the northern hemisphere, regardless of the actual sunset time.) The sunset starts to become later tomorrow, even though the shortest day is still two weeks off. If you haven't yet learned of this fascinating phenomenon, take a look at one of the referenced files. The first is formattable (m-X Format File); if formatting is not available to you, then print the second referenced file. Enjoy the increasing afternoon light! 0,, *** EOOH ***
A puzzle for the computationally minded ... The orbit of Mars is much more elliptical than Earth's. So the analemma effect is much stronger. On the Earth, we simply use mean solar time in our daily affairs, and ignore the slight difference between clock noon and local physical noon. The difference is only +- ten minutes or so, and is swamped by the effect of time-zone quantization. (Never mind daylight savings.) The puzzle: Could future Mars colonists ignore the analemma? The day length is about 24:40, so the natural thing to do is simply live by the longer day. Most of us could use an extra forty minutes a day. But would the Martians sometimes find the sun rising at (clock) midnight? Rich -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Thursday, December 09, 2010 8:43 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Correction; Analemma <stuff about pendula removed> Doug Dodds at Symbolics used to send the following every year at this time. If you haven't seen one, check out "analemma" in Google images for one of those 365-exposure sky pix. There's even one for Mars!.--rwg --- Received: from ELEPHANT-BUTTE.SCRC.Symbolics.COM by RUSSIAN.SPA.Symbolics.COM via CHAOS with CHAOS-MAIL id 52312; Tue 8-Dec-87 13:52:55 PST Received: from YUKON.SCRC.Symbolics.COM by ELEPHANT-BUTTE.SCRC.Symbolics.COMvia CHAOS with CHAOS-MAIL id 244027; Tue 8-Dec-87 16:18:32 EST Received: from ANNISQUAM.SCRC.Symbolics.COM by YUKON.SCRC.Symbolics.COM via INTERNET with SMTP id 303771; 8 Dec 87 16:11:04 EST Date: Tue, 8 Dec 87 16:11 EST From: Doug Dodds <DODDS@YUKON.SCRC.Symbolics.COM> Subject: Rejoice, the light returneth! To: BBoard@YUKON.SCRC.Symbolics.COM, Fun@YUKON.SCRC.Symbolics.COM File-References: SCRC|Y:>Dodds>analemma.mss.newest, SCRC|Y:>Dodds>analemma.text.newest Message-ID: <19871208211112.0.DODDS@ANNISQUAM.SCRC.Symbolics.COM> The sunset today in Boston is at 16:11 EST, which is Plenty Early. But take heart, today's sunset is the earliest of the year! (Everywhere in the northern hemisphere, regardless of the actual sunset time.) The sunset starts to become later tomorrow, even though the shortest day is still two weeks off. If you haven't yet learned of this fascinating phenomenon, take a look at one of the referenced files. The first is formattable (m-X Format File); if formatting is not available to you, then print the second referenced file. Enjoy the increasing afternoon light! 0,, *** EOOH *** _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
As long as the clocks are set for 24:00 Mars time = 24:40 Earth time (i.e., each Mars minute would be a bit longer than an Earth minute), then almost certainly, the sun would never rise at midnight. As the planet moves in its orbit, the clock is such that after 1 day of clock time, the planet has actually rotated a bit more than 360 deg, so that it will still face the sun directly at noon, clock time. The extra rotation compensates for the movement along the orbit. If the clock were on sidereal time, noon at perihelion would be midnight at aphelion, and you would have your sunrise at midnight, but, even with an elliptical orbit, the extra daily rotations will compensate for most of the sidereal vs. clock time difference. -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Schroeppel, Richard Sent: Friday, December 10, 2010 8:17 AM To: 'math-fun' Cc: 'rcs@xmission.com'; Schroeppel, Richard Subject: Re: [math-fun] Analemma A puzzle for the computationally minded ... The orbit of Mars is much more elliptical than Earth's. So the analemma effect is much stronger. On the Earth, we simply use mean solar time in our daily affairs, and ignore the slight difference between clock noon and local physical noon. The difference is only +- ten minutes or so, and is swamped by the effect of time-zone quantization. (Never mind daylight savings.) The puzzle: Could future Mars colonists ignore the analemma? The day length is about 24:40, so the natural thing to do is simply live by the longer day. Most of us could use an extra forty minutes a day. But would the Martians sometimes find the sun rising at (clock) midnight? Rich -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Thursday, December 09, 2010 8:43 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Correction; Analemma <stuff about pendula removed> Doug Dodds at Symbolics used to send the following every year at this time. If you haven't seen one, check out "analemma" in Google images for one of those 365-exposure sky pix. There's even one for Mars!.--rwg --- Received: from ELEPHANT-BUTTE.SCRC.Symbolics.COM by RUSSIAN.SPA.Symbolics.COM via CHAOS with CHAOS-MAIL id 52312; Tue 8-Dec-87 13:52:55 PST Received: from YUKON.SCRC.Symbolics.COM by ELEPHANT-BUTTE.SCRC.Symbolics.COMvia CHAOS with CHAOS-MAIL id 244027; Tue 8-Dec-87 16:18:32 EST Received: from ANNISQUAM.SCRC.Symbolics.COM by YUKON.SCRC.Symbolics.COM via INTERNET with SMTP id 303771; 8 Dec 87 16:11:04 EST Date: Tue, 8 Dec 87 16:11 EST From: Doug Dodds <DODDS@YUKON.SCRC.Symbolics.COM> Subject: Rejoice, the light returneth! To: BBoard@YUKON.SCRC.Symbolics.COM, Fun@YUKON.SCRC.Symbolics.COM File-References: SCRC|Y:>Dodds>analemma.mss.newest, SCRC|Y:>Dodds>analemma.text.newest Message-ID: <19871208211112.0.DODDS@ANNISQUAM.SCRC.Symbolics.COM> The sunset today in Boston is at 16:11 EST, which is Plenty Early. But take heart, today's sunset is the earliest of the year! (Everywhere in the northern hemisphere, regardless of the actual sunset time.) The sunset starts to become later tomorrow, even though the shortest day is still two weeks off. If you haven't yet learned of this fascinating phenomenon, take a look at one of the referenced files. The first is formattable (m-X Format File); if formatting is not available to you, then print the second referenced file. Enjoy the increasing afternoon light! 0,, *** EOOH *** _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
My rough calculation based on what I quickly looked up of Mars's orbit eccentricity suggests that solar noon won't be off from clock noon by more than an hour. --Joshua On Fri, Dec 10, 2010 at 8:35 AM, Cordwell, William R <wrcordw@sandia.gov> wrote:
As long as the clocks are set for 24:00 Mars time = 24:40 Earth time (i.e., each Mars minute would be a bit longer than an Earth minute), then almost certainly, the sun would never rise at midnight.
As the planet moves in its orbit, the clock is such that after 1 day of clock time, the planet has actually rotated a bit more than 360 deg, so that it will still face the sun directly at noon, clock time. The extra rotation compensates for the movement along the orbit. If the clock were on sidereal time, noon at perihelion would be midnight at aphelion, and you would have your sunrise at midnight, but, even with an elliptical orbit, the extra daily rotations will compensate for most of the sidereal vs. clock time difference.
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Schroeppel, Richard Sent: Friday, December 10, 2010 8:17 AM To: 'math-fun' Cc: 'rcs@xmission.com'; Schroeppel, Richard Subject: Re: [math-fun] Analemma
A puzzle for the computationally minded ... The orbit of Mars is much more elliptical than Earth's. So the analemma effect is much stronger. On the Earth, we simply use mean solar time in our daily affairs, and ignore the slight difference between clock noon and local physical noon. The difference is only +- ten minutes or so, and is swamped by the effect of time-zone quantization. (Never mind daylight savings.)
The puzzle: Could future Mars colonists ignore the analemma? The day length is about 24:40, so the natural thing to do is simply live by the longer day. Most of us could use an extra forty minutes a day. But would the Martians sometimes find the sun rising at (clock) midnight?
Rich
-----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Bill Gosper Sent: Thursday, December 09, 2010 8:43 PM To: math-fun@mailman.xmission.com Subject: [math-fun] Correction; Analemma
<stuff about pendula removed>
Doug Dodds at Symbolics used to send the following every year at this time. If you haven't seen one, check out "analemma" in Google images for one of those 365-exposure sky pix. There's even one for Mars!.--rwg --- Received: from ELEPHANT-BUTTE.SCRC.Symbolics.COM by RUSSIAN.SPA.Symbolics.COM via CHAOS with CHAOS-MAIL id 52312; Tue 8-Dec-87 13:52:55 PST Received: from YUKON.SCRC.Symbolics.COM by ELEPHANT-BUTTE.SCRC.Symbolics.COMvia CHAOS with CHAOS-MAIL id 244027; Tue 8-Dec-87 16:18:32 EST Received: from ANNISQUAM.SCRC.Symbolics.COM by YUKON.SCRC.Symbolics.COM via INTERNET with SMTP id 303771; 8 Dec 87 16:11:04 EST Date: Tue, 8 Dec 87 16:11 EST From: Doug Dodds <DODDS@YUKON.SCRC.Symbolics.COM> Subject: Rejoice, the light returneth! To: BBoard@YUKON.SCRC.Symbolics.COM, Fun@YUKON.SCRC.Symbolics.COM File-References: SCRC|Y:>Dodds>analemma.mss.newest, SCRC|Y:>Dodds>analemma.text.newest Message-ID: <19871208211112.0.DODDS@ANNISQUAM.SCRC.Symbolics.COM>
The sunset today in Boston is at 16:11 EST, which is Plenty Early. But take heart, today's sunset is the earliest of the year! (Everywhere in the northern hemisphere, regardless of the actual sunset time.) The sunset starts to become later tomorrow, even though the shortest day is still two weeks off.
If you haven't yet learned of this fascinating phenomenon, take a look at one of the referenced files. The first is formattable (m-X Format File); if formatting is not available to you, then print the second referenced file.
Enjoy the increasing afternoon light!
0,, *** EOOH *** _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
Looks like you got the calculation pretty close. An expert on the subject[1] says that the equation of time on Mars varies from +40 minutes to -50 minutes relative to the mean. Here is a plot: http://www.giss.nasa.gov/research/briefs/allison_02/analemma.GIF and the page that talks about it: http://www.giss.nasa.gov/research/briefs/allison_02/ and a nice photo on the Mars surface with multiple Sun images added in to show what it looks like compared to the horizon: http://apod.nasa.gov/apod/ap061230.html Note those Mars Sun images are a little smaller than the Sun that we see from Earth. So the analemma is actually a bit smaller than it might appear if you are taking the size of the Sun's disc as the scale cue. - Robert [1] Allison, M. 1997. Accurate analytic representations of solar time and season on Mars with applications to the Pathfinder/Surveyor missions. Geophysical Research Letters 24(16) pages 1967-1970. (97GL01950) PDF here: http://pubs.giss.nasa.gov/docs/1997/1997_Allison.pdf On Fri, Dec 10, 2010 at 15:25, Joshua Zucker <joshua.zucker@gmail.com>wrote:
My rough calculation based on what I quickly looked up of Mars's orbit eccentricity suggests that solar noon won't be off from clock noon by more than an hour.
-----Original Message-----
A puzzle for the computationally minded ... The orbit of Mars is much more elliptical than Earth's. So the analemma effect is much stronger. On the Earth, we simply use mean solar time in our daily affairs, and ignore the slight difference between clock noon and local physical noon. The difference is only +- ten minutes or so, and is swamped by the effect of time-zone quantization. (Never mind daylight savings.)
The puzzle: Could future Mars colonists ignore the analemma? The day length is about 24:40, so the natural thing to do is simply live by the longer day. Most of us could use an extra forty minutes a day. But would the Martians sometimes find the sun rising at (clock) midnight?
-- Robert Munafo -- mrob.com Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - youtube.com/user/mrob143 - rilybot.blogspot.com
Are clocks set so that noon matches solar noon in the middle of a time zone at aphelion (or perihelion)? I.e., when does the eq. of time give zero correction? Bill -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Robert Munafo Sent: Saturday, December 11, 2010 12:16 PM To: math-fun Subject: Re: [math-fun] Analemma Looks like you got the calculation pretty close. An expert on the subject[1] says that the equation of time on Mars varies from +40 minutes to -50 minutes relative to the mean. Here is a plot: http://www.giss.nasa.gov/research/briefs/allison_02/analemma.GIF and the page that talks about it: http://www.giss.nasa.gov/research/briefs/allison_02/ and a nice photo on the Mars surface with multiple Sun images added in to show what it looks like compared to the horizon: http://apod.nasa.gov/apod/ap061230.html Note those Mars Sun images are a little smaller than the Sun that we see from Earth. So the analemma is actually a bit smaller than it might appear if you are taking the size of the Sun's disc as the scale cue. - Robert [1] Allison, M. 1997. Accurate analytic representations of solar time and season on Mars with applications to the Pathfinder/Surveyor missions. Geophysical Research Letters 24(16) pages 1967-1970. (97GL01950) PDF here: http://pubs.giss.nasa.gov/docs/1997/1997_Allison.pdf On Fri, Dec 10, 2010 at 15:25, Joshua Zucker <joshua.zucker@gmail.com>wrote:
My rough calculation based on what I quickly looked up of Mars's orbit eccentricity suggests that solar noon won't be off from clock noon by more than an hour.
-----Original Message-----
A puzzle for the computationally minded ... The orbit of Mars is much more elliptical than Earth's. So the analemma effect is much stronger. On the Earth, we simply use mean solar time in our daily affairs, and ignore the slight difference between clock noon and local physical noon. The difference is only +- ten minutes or so, and is swamped by the effect of time-zone quantization. (Never mind daylight savings.)
The puzzle: Could future Mars colonists ignore the analemma? The day length is about 24:40, so the natural thing to do is simply live by the longer day. Most of us could use an extra forty minutes a day. But would the Martians sometimes find the sun rising at (clock) midnight?
-- Robert Munafo -- mrob.com Follow me at: mrob27.wordpress.com - twitter.com/mrob_27 - youtube.com/user/mrob143 - rilybot.blogspot.com _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
On Sat, Dec 11, 2010 at 2:19 PM, Cordwell, William R <wrcordw@sandia.gov> wrote:
Are clocks set so that noon matches solar noon in the middle of a time zone at aphelion (or perihelion)? I.e., when does the eq. of time give zero correction?
I don't know, but I would guess one of the solstices or equinoxes, rather than aphelion or perihelion. The reason days on earth are more or less than 24 hours is primarily related to the tilt of the earth's axis, rather than the ellipticity of the earth's orbit. The easy way to see this is that on a planet with an axis perpendicular to the plane of its eccentric orbit, the analemma is an oval, while on a planet with a circular orbit and a tilted axis, the analemma is a figure 8. The earth's analemma is a figure-8, with one lobe slightly larger than the other because of the slight eccentricity of the earth's orbit, while the moon's analemma is oval shaped, with some irregularity because the axis of mars is tilted, too. Andy
Apparently, it's due to the tilt of the earth's orbital plane and to the ellipticity of the orbit. There are 4 days a year when the correction is zero (since it's a figure 8), and one of them is Christmas. Is this a coincidence? Bill -----Original Message----- From: math-fun-bounces@mailman.xmission.com [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Andy Latto Sent: Saturday, December 11, 2010 12:39 PM To: math-fun Subject: Re: [math-fun] Analemma--related question On Sat, Dec 11, 2010 at 2:19 PM, Cordwell, William R <wrcordw@sandia.gov> wrote:
Are clocks set so that noon matches solar noon in the middle of a time zone at aphelion (or perihelion)? I.e., when does the eq. of time give zero correction?
I don't know, but I would guess one of the solstices or equinoxes, rather than aphelion or perihelion. The reason days on earth are more or less than 24 hours is primarily related to the tilt of the earth's axis, rather than the ellipticity of the earth's orbit. The easy way to see this is that on a planet with an axis perpendicular to the plane of its eccentric orbit, the analemma is an oval, while on a planet with a circular orbit and a tilted axis, the analemma is a figure 8. The earth's analemma is a figure-8, with one lobe slightly larger than the other because of the slight eccentricity of the earth's orbit, while the moon's analemma is oval shaped, with some irregularity because the axis of mars is tilted, too. Andy _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
participants (6)
-
Andy Latto -
Bill Gosper -
Cordwell, William R -
Joshua Zucker -
Robert Munafo -
Schroeppel, Richard