Friday, September 21, 2007

First Day of Fall 2007

The first day of fall this year will be on September 23, 2007. More precisely, the autumnal equinox will be at 9:51 UT that day, which means that the Sun will be crossing the celestial equator from north to south at that moment sharp. Thus, on September 23, the Sun will rise exactly in the East, and set exactly in the West, for all places in the world. Moreover, the time span between sunrise and sunset will be 12 hours, and equal the time span between sunset and sunrise - this equality of day and night is at the origin of the Latin name "equinox", which means equal night [1].

When I was a kid, I could understand fairly well how the seasons come about because the Earth's axis is tilted by roughly 23° with respect to the plane in which it is orbiting the Sun, called the ecliptic by the grown-ups. Playing around with a torch and an apple is good to get a feeling what is going on, and what happens when the days and nights are roughly equal. So, I could make sense of September 23 as the first day of fall - however, I remember I was extremely skeptical when my mum told me that the beginning of fall that year was at, say, 12:21 sharp. What a nonsense, I thought, how on Earth can on know this date so precisely, to the minute? Today, I know that there is an easy answer, but that, in fact, this answer just hides the question I had asked myself more than 25 years ago.

Relative orientation of the Earth and the Sun at the first days of spring, summer, fall, and winter, respectively, The first day of fall corresponds to the position of Earth in the foreground. (Source: Wikipedia on the Equinox)

The easy answer as to the exact moment of the equinox goes as follows: If we want to introduce a coordinate system on the celestial sphere, in order to denote the positions of stars and planets, we can use longitude and latitude as we do on a globe: The North Pole is the point around which the night sky seems to rotate - close to Polaris - and the equator, where, for example, the stars of Orion are located, is the great circle corresponding to the equator on Earth. On the sky, these coordinates are called right ascension (corresponding to longitude) and declination (corresponding to latitude).

Now, because of the tilt of the Earth's axis, the declination of the Sun changes during the year (the right ascension changes also, giving rise to all kinds of astrological complications, but that doesn't play a role here). In the summer of the Northern hemisphere, it has a declination of up to +23.5°, which means that it rises high in the sky, and doesn't even set below the horizon at polar latitudes. In the Northern winter, declination will be negative, down to -23.5°, meaning short days and low culminations at noon. Thus, between summer and winter, the declination of the Sun has to change smoothly from about +23.5° to about -23.5°. Of course, there should be one precise moment when the declination is exactly 0.00° - that's the equinox. This year, this will happen at 9:51 UT - more precisely even, at 09:51:08 UT, according to the data of the US Naval Observatory.

The declination of the Sun since March 1, 2007. At the solstices, the declination takes extreme values: the Sun reaches its highest, respectively lowest, position in the sky. At the equinoxes, the declination is exactly zero. The curve looks similar to a sine curve, but it is more complicated function. (Data from the US Naval Observatory)

Of course, this easy answer isn't a real answer at all. There are two obvious questions left open: How can one, first of all, define a coordinate system in the sky that is that precise that it makes sense to define the exact moment when the Sun will cross the celestial equator? And then, how precisely can the actual position of the Sun be known? These are, in fact, difficult questions, and to answers them was one of the main tasks of the big National Observatories, such as Greenwich, or the US Naval Observatory.

Defining coordinates in the sky is a subtle task, because of the problems to identify some really fixed points. The right ascension for example is measured along the celestial equator starting at the vernal equinox. However, the tilt of the Earth's axis is not fixed in space - due to the torque of the Sun and the Moon, the Earth is precessing, and for this reason, the point of the vernal equinox wanders around the celestial equator once every 26,000 years. There are other problems with the motion of the Earth's axis. Even the ecliptic is not completely fixed with respect to far-away stars, because not all planets are exactly within one plane, thus causing so-called secular variations of the inclinations of the orbits. And then, as astronomers in the 19th century had to notice, even the "fixed stars" are not fixed: Some of them have proper motions so large that they cannot easily be used as references to define a coordinate system.

Defining celestial reference frames today is achieved by sources much further away than stars: Quasars are radio sources in cosmological distances, and thy do not move in the sky. Fortunately, combining radio telescopes by Very Large Baseline Interferometry allows an extremely precise determination of the position of quasars, which is used to fix a coordinate system. The orbits of the planets, and the coordinates of the Sun, can be calculated and measured very precisely with respect to this coordinate system.

What leaves me with one question I am scratching my head about: How does one measure to a high precision the position of the Sun on the celestial sphere? After all, it is not a neat small pointlike source, but has quite a large extension. From a quick search at the ADS database, I have learned that in the 1970s, this was still done in the traditional way, using a zenith tube at Herstmonceux, then the location of the Greenwich observatory in England. It may be that also in the measurement of the position of the Sun, VLBI methods have taken over: For example, precise timing of the occulations of quasars by the Sun may provide very reliable solar coordinates. So, there are still enough puzzeling questions to think about on long autumn evenings. Perhaps someone of our readers knows?

There is one good news, at least for those of us living in the Northern hemisphere: Because the Earth is closer to the Sun in winter, with the perihelion reached around January 3, fall and winter are shorter than spring and summer. And in 179 days, we will be again at the vernal equinox, and spring will begin.

[1] Restrictions apply: The Sun will rise exactly in the east on those places on Earth where sunrise is at 9:51 UT, i.e. along a line somewhere on the Atlantic Ocean. Moreover, due to the extension of the Sun's disk and the effect of atmospheric refraction, the Sun is visible already when in fact it is, geometrically speaking, still below the horizon. This means that actually, the day is a little bit longer on September 23 than the night.

[2] Plenty of data can be obtained from the Data Services of the Astronomical Applications Department of the U.S. Naval Observatory. For example, from the table of Earth's Seasons Equinoxes, Solstices, Perihelion, and Aphelion, here is an excerpt covering the next few years:

d h d h m d h m
2007 2007
Perihelion Jan 3 20 Equinoxes Mar 21 00 07 Sept 23 09 51
Aphelion July 7 00 Solstices June 21 18 06 Dec 22 06 08

2008 2008
Perihelion Jan 3 00 Equinoxes Mar 20 05 48 Sept 22 15 44
Aphelion July 4 08 Solstices June 20 23 59 Dec 21 12 04

2009 2009
Perihelion Jan 4 15 Equinoxes Mar 20 11 44 Sept 22 21 18
Aphelion July 4 02 Solstices June 21 05 45 Dec 21 17 47

2010 2010
Perihelion Jan 3 00 Equinoxes Mar 20 17 32 Sept 23 03 09
Aphelion July 6 11 Solstices June 21 11 28 Dec 21 23 38

2011 2011
Perihelion Jan 3 19 Equinoxes Mar 20 23 21 Sept 23 09 04
Aphelion July 4 15 Solstices June 21 17 16 Dec 22 05 30

Moroever, there is a very nice feature to plot maps and views of Earth that show day and night across the globe.

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  1. another post from the 'global warming is great!' crowd! there are some of us who wish the northern hemisphere fall/winter were longer!!!

  2. Hi Bee,

    however, I remember I was extremely skeptical when my mum told me that the beginning of fall that year was at, say, 12:21 sharp. What a nonsense, I thought, how on Earth can on know this date so precisely, to the minute? Today, I know that there is an easy answer, but that, in fact, this answer just hides the question I had asked myself more than 25 years ago.

    It takes a good dose of healthy skepticism to realize that the easy answer is incomplete. Your question is actually pretty general. In your case, you asked how we could get a fractional day, a precision that seemed ridiculous to you. And you realized that even if we could get a fraction, we can only go that much further and the problem is not solved.

    The same holds true for any measurement. I can define a fraction of a metre, for instance a millimetre, pretty easily. However, physics is not invariant through that change (just ask any bug how they manage to fall from any height without hurting themselves ;-) ) So, having defined the metre with a solid rod (as we did historically), you can define the millimeter with a solid, the micrometre becomes problematic, the nanometre is technically challenging, and the picometre is out of the question (being the order of magnitude for the radius of the atoms of the solid).

    This leads me to another question: if you change the physical phenomenon you use to measure something, does the physics remain the same? For instance, when we decide to use light instead of solid rods to define the metre, is this innocuous?

    The easy answer here is: of course they are the same if you calibrate the two measurements to match. E.g. you say "a metre is the distance travelled by light in S seconds", knowing that S seconds is itself defined as a pretty large number of periods of some radiation.

    But then, the easy answer of a linear law hides the question "is this the same physics". Consider what a gigametre means using both definitions. Any structural engineer would tell you that a 1 million kilometres metal rod is unlikely to follow the same path in space as light. One definition is sensitive to gravity, the other is not.

  3. Hi Christophe:

    I am afraid you've just made my husband incredibly unhappy since it was him who made the effort of writing the above (which, in addition was my suggestion since for whatever reason lately about 200 visitors each day end up on this blog searching for 'First day of Fall 2007'). The question I asked myself as a kid was whether it is painful for the trees to loose their leaves. Well - now you know who is the real intellectual in that marriage and who knows things like 'the point of the vernal equinox wanders around the celestial equator' ;-)



  4. Misner, Thorne, Wheeler have the nice

    "Time is defined so that motion looks simple".

  5. Hi Stefan,

    "What leaves me with one question I am scratching my head about: How does one measure to a high precision the position of the Sun on the celestial sphere? After all, it is not a neat small pointlike source, but has quite a large extension."

    I don't think anyone has a good way to find the Sun's precise position by looking at its outline, especially since it doesn't have a well-defined surface. Probably the best estimate of its position is in the NASA J2000 ephemeris.

    The J2000 ephemeris was derived from many kinds of data on bodies moving around in the Solar System: planets, moons, asteroids, space probes. (The data include optical observations over long time periods, radar ranging to planets, VLBI observations, Doppler and ranging signals from the spacecraft.) You have a mathematical model of how bodies move in gravitational fields, including general relativistic effects, written into a computer program. Then you do a global best fit for the significant masses and the positions of all the bodies as a function of time.

    We get the Sun's position (its barycenter) from what everything else in the Solar System is doing. As described by Pitjeva, the solar oblateness is also determined by a similar method.

    (The Earth-Sun distance, our Solar System yardstick, seems to be changing slowly. Krasinsky and Brumberg find no satisfactory explanation at present. I'm interested in that and other anomalies.)



  6. Stephan,

    Please accept my apologies for the misattribution. I'll be more careful next time. As for Bee's question as a kid, it is a pretty good one too.


  7. Beautiful post! I just
    linked to it from my
    blog. Happy Sunday!

  8. Hi Kris,

    thank you for these explanations, and the links references! That's very interesting and really helpful.

    When searching for "determination equinox" at the ADS database, I had found a few quite recent papers on the usage of minor planet data to find the equinox, and my guess was that the minor planets are used as sort of test bodies to map the gravitational field of the Sun and the Planets, and to get an estimate for their positions and the location of the barycentre of the Solar System. So what you say seems to confirm this.

    It's a bit ironic that the best estimate for the exact position of the most prominent heavenly body is obtained is such an indirect way!

    Best regards, Stefan

  9. Hi Christophe,

    don't worry... in contrast to the anonymous snowboarder, it seems that you have at least taken the trouble to read the post and to try to understand what I wanted to say before commenting.

    As for your question if you change the physical phenomenon you use to measure something, does the physics remain the same? For instance, when we decide to use light instead of solid rods to define the metre, is this innocuous?

    - well, from my understanding this is not innocuous at all, and was only done after many many experiments have made sure such things as the observer-independence of the speed of light and so on. So, in fact, there is a lot of empirical knowledge about nature behind such redefinitions...

    As for Bee's question as a kid [whether it is painful for the trees to loose their leaves], it is a pretty good one too.

    Very true! And a difficult one, given that the much simpler question as to why leaves turn red isn't answered yet, as I've learned from this nice little book (thanks, Renate!) ...

    Best regards, Stefan

  10. Related to the rites of fall, but off-topic otherwise:

    Maybe Bee or Stefan, you can explain why Oktoberfest begins in September?

    Picture here

  11. Hi Stefan,

    Looking at this by Kolesnik and Masreliez, I see it was wrong of me to suggest that direct observations are no longer used to determine the Sun's precise position. These authors are looking in particular at relative motion of its equinox position, where again there is a small anomaly.

    Previously the anomaly was attributed to a possible rotation of the stellar reference system used. The authors point out that explanation is ruled out now by the HIPPARCOS sky map, whose rotation has been established very precisely with respect to distant quasars.

    Best wishes,


  12. Thanks for the nice post Stefan, and of course Happy Equinox to all!

    My 2 cents:

    You forgot to mention that the torques due to the Sun & Moon on the Earth oblateness not only produces the precession of the Earth's axis, but also produces the nutation of the Earth's axis (a small amplitude wobble following the precessional motion) with a period of ~18.6 years (the period of rotation of the lunar orbit's nodal line).

    A nice anecdote: The point at which the Sun crosses the celestial equator northward (i.e., the start of Spring in the N. hemisphere) is called the "1st point of Aries"...and yet it is now located in the constellation Pisces!

    Regarding the comment by kris krogh, there are several "NASA J2000" ephemerides, the most recent of which I believe is called DE400.

    Oh, in the astrodynamical community, the coordinate system containing the R.A. and Declination , with the x-axis pointing towards the vernal equuinox (at a given Epoch) and the Z-axis along the Earth's axis of rotation is called ECI (Earth-Centered-Inertial)



  13. Hi Changcho,

    thanks, great! I wasn't aware that the nutation period of the Earth's axis is the same as the precessional period of the lunar orbit, and that both effects are actually coupled!

    Best, Stefan

  14. Hi Kris,

    thank you for the reference to the paper "Secular Trends in the Mean Longitudes of Planets Derived from Optical Observations" by Kolesnik and Masreliez - in the table they mention the Herstmonceux observations of Sun, Mercury, and Venus 1957-1982 I had read about.

    By the way, according to this story of the Royal Greenwich Observatory at Herstmonceux, "To the north of the castle lay the Spencer Jones Group of Meridian instruments. These consisted of the Photographic Zenith Tube (PZT) which was used for time determination and for measuring latitude variation; the Danjon Astrolabe, also used for time and latitude determination; and the Cooke Reversible Transit Circle, used for determining star positions and planetary positions and motions."

    Which brings me to the Danjon Astrolabe - I had stumbeld on that type of instrument when I searched the ADS database, but I thought it was a historical instrument and not used any more since long... completely wrong: The Kolesnik and Masreliez paper cites this recent paper "Astrometric observations of the Sun at Santiago: 1998-2000" by F. Noël, Astronomy&Astrophysics 375, 614-616 (2001), which analyses optical observations of the Sun with the Danjon Astrolabe at the Cerro Calán observatory of the Astronomy Department of the University of Chile (here is a photo of the instrument) 14 km northeast of Santiago.

    The usage of the Danjon Astrolabe is discussed briefly in the paper "Analysis of Solar Astrolabe Measurements during 20 Years" by P. C. R. Poppe, N. V. Leister, F. Laclare, and C. Delmas, The Astronomical Journal 116 (1998), pages 2574–2582, and in much more detail on the page on the astrolabe solaire of the Définition et Observation du RAYon SOLaire group of the Observatoire de la Côte d'Azur at Calern in the Provence.

    So, optical observation of the Sun seem still to be in use!

    Best regards, Stefan

  15. Hi Changcho,

    I think the current version of the J2000 ephemeris is DE405.

    Hi Stefan,

    You're right. It does seem direct observations of the Sun's position are still useful. I think Kolesnik and Masreliez are using them to try to avoid a potential problem in the current method of constructing the ephemeris: It assumes you already know exactly the correct equations governing gravitational motions. If not, using the direct solar observations might give hints to the correct ones.

    (Masreliez and I have both developed alternative cosmologies. Neither was formulated with the Pioneer anomaly in mind, but both accurately account for that effect. Most attempts to explain it using new physics have run into trouble, because they predict planetary orbits which are not observed. I believe I've shown that problem does not happen in my theory.)



  16. Hi Kris, changcho:

    have just seen that the current version of the JPL ephemeris seems to be JPL DE-406.

    And here is some more interesting stuff:

    According to this explanation, the JPL ephemerides have followed this evolution [advent of much more accurate celestial measurement techniques and fast computers for coordinate conversions] with the DE100, DE200, and DE400 series, which are based on the B1950, J2000, and IERS coordinate systems, respectively, where the B1950 system is tied to the sky by star coordinates in the FK4 catalog, and J2000 is tied to FK5.

    Interestingly, problems with both stellar Fundamentalkataloge 4 and 5 were detected by classical astrometrical methods: observing the stars at the meridian, etc. - and by observations of the Sun with Danjon astrolabes: "Corrections to the FK5 reference frame from Sun's observations with the astrolabes at Abrahao de Moraes (1988-1994) and Calern (1988-1991) Observatories.", by the already mentioned P.C.R. Poppe, N.V. Leister, F. Laclare in Astronomy and Astrophysics 315 (1996) 633-637.

    The problem is the proper motion of the stars, and it was solved by switching to the IERS (International Earth Rotation Service) reference frame, which is essentially the J2000 system except that it is tied to the sky by the published positions of 228 radio sources, roughly 23 of which are monitored by several VLBI networks to determine the day to day changes in the orientation of the earth. These radio sources are very distant compared to stars and should not suffer any of the proper motion problems. The DE403 and IERS celestial frames are tied together to an accuracy of a few milliarcseconds.

    And, yes, direct observation of the positions of stars, planets and the Sun is independent of calculations which have to assume Newtonian and PPN dynamics, and cannot include potential unknowns in the gravitational law...

    By the way, I have found a good description of the Danjon astrolabe by André Danjon himself: "The contribution of the impersonal astrolabe to fundamental astronomy (George Darwin Lecture)" Monthly Notices of the Royal Astronomical Society, 118 (1958) 411.

    Somehow I have learned a lot more about how the exact position of the Sun in the sky is determined :-)

    Best, Stefan

  17. Hi Stefan,

    Here is more info on DE405 and DE406 from NASA's top expert on the ephemeris, E. Myles Standish:

    DE405: (includes both nutations and librations) ... Our latest ephemeris; it is based upon the International Celestial Reference Frame (ICRF).

    DE406 : the New "JPL Long Ephemeris" (includes neither nutations nor librations) ...
    This is the same ephemeris as DE405, though the accuracy of the interpolating polynomials has been lessened (interpolation on the 64-day mesh points remains exact, however). For DE406/LE406, the interpolating accuracy is no worse than 25 meters for any planet and no worse than 1 meter for the moon.

    DE406 is the same thing, but covers a longer time period (past the year 3000) at less accuracy. The ephemeris and associated tools are available to the public here. If you want to know exactly what is where, and when, it's the place to go!

    Cheers, Kris

  18. Oops. I see you already provided a link to the NASA Horizons on-line system for the ephemeris. This country lacks public health care, and they're starting to charge for national parks, but you can use the ephemeris for free!

  19. Hi Kris,

    thank you for pointing me to the JPL ephemerides in general, and to the link to the IAU ephemerides commission page!

    It's amazing indeed how exactly the positions of the bodies in the Solar System are known, and that this information is available "for free"... Well, I do not know of anyone who could actually use it ;-), unless for an exact determination of the equinox...

    Remotely related... Browsing around today, I came across another paper by Fernando Noel (Solar Astrometry with the Astrolabe of Santiago; Lecture Notes in Physics 599, 2003), where he writes that between 1975 and 1990 it was carried out [using the Santiago Danjon astrolabe mentioned in the comments above] a program of observations of Uranus as a long term contribution to the research of its orbital motion. However, a short term application was made by JPL when the orbit of Voyager II spacecraft was corrected according to the results of Santiago, prior to its successful encounter with the planet in 1986. - not so long ago, classical optical astrometry still ruled to pinpoint the planets in the sky! But that's probably still the only way to check the positions of the outer planets, where radio echo tracking and transmission from space probes are not available?

    And I learned that even quasars do not necessarily make for fixed points in the sky.

    Some quasars of the VLBI derived reference system seem not to be so static after all - there are small unsystematic motions measurable for some quasars: Ample evidence suggesting that the apparent positions of the distant radio sources used to construct the fundamental celestial reference frame undergo significant changes at a level of 1 marcsec (mas) has been accumulated in recent years. ... Many of the quasars from the ICRF catalog that were among 212 defining radio sources were found to be also astrometrically unstable. For example, five quasars whose positions are particularly subject to changes were specially pointed out in the latest version of the ICRF-Ext.2 catalog. (O. A. Titov: Apparent proper motions of radio sources from geodetic VLBI data, Astronomy Letters 33 (2007) 481)

    Certainty gained, certainty lost...

    At least I know better now how the time of the equinox is determined!

    Best regards, Stefan


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