User talk:Jc3s5h/sandbox3

Sources

• More power to your elbow in sorting out this article!

• The existing text shows a small calculational error in the figure for the tropical year according to Newcomb. The Meeus citation gave it in different units, as 365d 05h 48m 46.0s, and this converts to 365.2421991 (to 7 d.p.), a little different from the current text, but this is still not quite right, a discrepancy has arisen because Meeus et al used what was clearly a back-calculation from a truncated derivative of a more authoritative and original figure.

A better reference is Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (London, 1961), which gives (page 99) (and I've checked that the longitude polynomial matches ok with Newcomb (1895), although I don't have that with me right now):

ESAE 1961 (page 98) gives Newcomb's mean solar longitude, relative to the epoch of Greenwich mean noon on 0 January 1900 = JD 2415020.0 (and w.r.t. the mean equinox & ecliptic of date) (where T is interval in Julian centuries of 36525 days from the epoch):
. . . 279d 41' 48".04 + 129602768".13*T + 1".089*T*T.

ESAE 1961 (page 99) then gives length of tropical year (equinox to equinox) (in days):
365.24219878 -- 0.00000614*T
(where T is in Julian centuries of 36525 days from the epoch of Greenwich mean noon on 0 January 1900 = JD 2415020.0).

This figure derives pretty exactly from Newcomb's t1 coefficient of 129602768.13"*T, and it's worth noting that most astronomers of that period quoted their data primarily in arcsecs to a definite number of d.p, 0.01" or 0.001" in Newcomb's case, successive units-conversions to other terms can give rise to noticeable discrepancies relative to the primary data after a few stages of conversion.

As it happens, I have also with me (for a short time) an original copy of the Connaissance des Temps for 1924 (Paris, Bureau des Longitudes, 1922), and so, for the record, here is what it says on the current subject. It quotes solar data of both Le Verrier and Newcomb, both brought forward to the epoch of Greenwich mean noon on 1 January 1900 (JD 2415021.0), where T is now counted from the 1 January 1900 epoch. (1 January rather than 0 January seems to have been a preferred epoch date in 19th & early 20th-c French data. Le Verrier's original data of 1858 was based on an epoch of Paris mean noon, 1 January 1850, later French data starting from some time after the 1884 Meridian Conference began to use Greenwich mean noon, but still 1 January):

• CdT 1924 (page XII): Leverrier for JD 2415021.0:
  

The Earth's mean longitude (+180d to get Sun's):
. . . . 100d 40' 57".05 + 129602768.95"*T + 1.1073"*T*T
Length of tropical year (days):
. . . . 365.24219647 - 0.00000624*T

(which matches the Meeus paper except that Meeus has cited it as if from the 0 January epoch, a negligible difference for the length of year but not negligible for other data.)

• CdT 1924 (page XV): Newcomb for JD 2415021.0:
  

The Earth's mean longitude (+180d to get Sun's):
. . . . 100d 40' 56".37 + 129602768.13"*T + 1.089"*T*T
Length of tropical year (days):
. . . . 365.24219879 - 0.00000614*T

Just for the record in case of use! Terry0051 (talk) 00:11, 23 January 2010 (UTC)

Thanks. I will go through this in more detail later. I do have Explanatory Supplement to the Astronomical Ephemeris and the American Ephemeris and Nautical Almanac (London, 1961) so I will take advantage of it. --Jc3s5h (talk) 00:18, 23 January 2010 (UTC)

Looking at ESAE, it seems they mislabeled the length of tropical year result. It is known that there are differences in the tropical year if it is measured equinox-to-equinox. Newcomb's approach derives a mean tropical year. I checked the calculus; the expression on ESAE p. 99 derives from the expression for mean longitude on p. 98, so the "equinox-to-equinox" description seems wrong. I don't have Connaissance des Temps for 1924. It seems kind of frustrating trying to find one good source that gets all the details right. --Jc3s5h (talk) 05:49, 23 January 2010 (UTC)

Well, all those ESAE 1961 data occur in a section that deals with mean elements, I suggest that in that context, 'equinox-to-equinox' certainly means 'mean equinox to mean equinox', and not 'true equinox to true equinox' (which I agree is more variable). With that reading, it looks to me (subject to another check!) that the given figure is either numerically correct to the # of decimal places given, or at most 1 (unit in the last place) off. I'd agree there's arguably a slight ambiguity in sense, the ESAE author seems to have meant, rather than 'equinox to equinox', something more like 'from passage through one value of mean longitude (reckoned from mean equinox of date), to the next passage through the same value -- whether that value is 0 or some other constant'. But I guess nobody writes with as many words as that.

(The ESAE data occurs in a list with other kinds of years measured relative to other reference points such as 'fixed star to fixed star', so a bare reference to (mean) 'equinox' (of date) was probably thought to give all of the distinction, relative to the other cases, that was at the forefront of the writer's attention and thought necessary at that moment.)

(Another source of possible ambiguity is the time-scale in use. ESAE on pages 98-99 mentions only 'ephemeris time', but ESAE is written from the post-1960 perspective after the redefinition of Newcomb's formulae was in use; Newcomb clearly indicated mean solar time. ESAE does admit elsewhere (pp.73-74) that Newcomb's formulae were originally written and intended in terms of mean solar time.)

I agree it's troublesome that fully correct consistent sources are hard to find. (Maybe there isn't even such a thing, errors can be found in original sources such as Le Verrier and Newcomb as well as in derivative works, some have been pointed out but others may remain undiscovered......) best wishes -- Terry0051 (talk) 13:38, 23 January 2010 (UTC)

Newcomb's tropical year comes from his Tables of the motion of the Earth on its axis and around the Sun, pages 9–10 (1895/98), which gives

279° 41' 48".04 + 129 602 768".13T + 1".089T² for "the Sun's geometric mean longitude, freed from aberration" and

365^d.242 198 79 − .000 006 14T for "the tropical year", where T is a Julian century of 36525 days.

Surprisingly, the linear term in the solar longitude is equivalent to 365.24219878, terminating with an 8, unlike his explicitly stated value of 365.24219879, terminating with a 9. The former value, 365.24219878, was used for the 1900.0 tropical year in the 1952 definition of the ephemeris second, in the form of 31556925.9747 seconds, as seen in the Explanatory Supplement to the Astronomical Almanac, page 80 (1992).

The linear term in the article, −6.15×10⁻⁶T, has a typo because it does not even match the secondary source, Borkowski (1991), who gives Newcomb's value, −6.14×10⁻⁶T.

The Meeus and Savoie value for Newcomb, 365^d 05^h 48^m 46^s.0, although derived from 365.24219879 days, is rounded to the nearest 0.1 s, whereas it should have been rounded to the nearest 0.0001 s or 365^d 05^h 48^m 45^s.9755 to match Newcomb's value.

Although Ptolemy (and Hipparchus) give the same value for the "equinoctial year" in several ways (Ptolemy's Almagest translated by G. J. Toomer, book III.1, pages 137–140, 1998), usually in the form 365${\tfrac {1}{4}}$ − ${\tfrac {1}{300}}$ days (in words), his equivalent succinct value is 365;14,48^d, which uses Otto Neugebauer's standard sexagesimal notation where the colon ';' is the 'sexagesimal point' and a comma ',' separates fractional sexagesimal positions. Decimally, this is 365.2467 days (with comparable precision) or 365^d 5^h 55^m 10^s, where I've intentionally changed the number of seconds from 12 s to indicate a lower precision.

The tropical year in The Alfonsine Tables of Toledo, page 255 (2003) is 365;14,33,9,59,...d or 365.242546219 days or 365^d 5^h 49^m 15^s.9933.

In summary, Meeus and Savoie ignored the original form of the tropical year given in their sources, which was almost always in whole and fractional days (once in seconds), either sexagesimal or decimal, never in days, hours, minutes and seconds, and they ignored its original precision. — Joe Kress (talk) 23:07, 23 January 2010 (UTC)

It took me a while to figure this out, and I have never seen it stated in this manner, but the mean tropical year is the year derived from an analytical or semi-analytical (with assigned numerical planetary masses) development of the equations of motion of the Solar System, with all sinusoidal terms ignored. This leaves the tropical year in a polynomial form. Often only the constant term is stated (at the epoch used for the development). The mathematicians at the Paris Observatory would call the constant term the mean mean tropical year. Strickly speaking, this is not an average, because the periods of the several sinusoidal terms are not commensurate, that is, no single period with a reasonable length can have a whole number of all sinusoidal cycles. Equinox-to-equinox, mean equinox-to-mean equinox, and a 360° increase in the mean solar longitude are all valid descriptions, provided it is understood that all of their sinusoidal terms are also ignored. The true equinox includes nutation and planetary perturbations, while the mean equinox may ignore those terms, while still including the sinusoidal terms describing the speed of the Sun/Earth in its orbit. This means that over time, all equinoctial and solsticial years change, while the mean tropical year (in its full polynomial form) remains the same (over its period of validity, maybe a few thousand years). — Joe Kress (talk) 01:26, 24 January 2010 (UTC)

I agree that's a good description. Terry0051 (talk) 02:44, 24 January 2010 (UTC)