Almagest declinations

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Offline martinchaide

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Almagest declinations
« on: 07/12/2015 21:59:52 »
Helo, this is my first post, i would apreciatte that someone with good command in astronomy read this, as I do not know if I am in a basic error or some circular reasoning,

those interested can read the complete article with the data here: wild wild west wherethalmagestdeclinationscomefrom info

ty in advance

In the chapter seventh of the third book of the Syntaxis Mathematica (a.k.a Almagest) we find 18 declinations measured by Timocharis (circa -290 ) and Arystilus (circa -260), then by Hipparchus (circa -128)  and finally by Ptolemy (circa 135 AD).

The source for the declinations  is Robert Newton’s  article named ‘ in the obliquity of the ecliptic two millennia ago’, but the real positions of the stars had been recalculated without counting refraction, as in this case we are interested not in the real positions, but where the stars were really seen in the sky. Notice that R. Newton explained that six of these declinations where altered by Ptolemy on purpose to demonstrate his erroneous equinox precession rate of one degree by century. For safety, they have been removed from this study so there are just 12 declinations recorded by Ptolemy.

The measurements errors

Declination δ= ϕ-z                                                                                                                                       

ϕ is the observer latitude and z the distance to the zenith. Either if the declinations were measured using a globe or directly into the sky, the error of observation should be independent of the right ascension (α), in fact R. Newton, in his study about the obliquity, considered  the errors in the declinations independent of the right ascension.

In the other hand, if the declinations were not directly measured, but calculated from ecliptic coordinates, as the obliquity of the ecliptic couldn’t be known with perfect accuracy there have to be some systematic errors in the declinations so

Error (calculated) = k cos(α)                                                                                                                    (2)

,in this, k is some constant and α the right ascension for the epoch. If we compare the historical error with the calculated error using the formula above a remarkable similitude is found:

Graph 1. The errors of Timocharis and Arystillus, K= -0,13

Graph 2. The errors of Hipparchus. K= -0,077




Graph 3: The errors of Ptolemy. K= 0,1

This correspondence between the two lines lead to the conclusion that most of the declinations if not all of them were derived from ecliptic coordinates. Let us analyze this possibility:

1.       The chance that the four astronomers independently decided to transform ecliptic to equatorial coordinates that were less useful and much easier to measure seems so absurd that this possibility is out of question.
2.       The possibility that the astronomers before Ptolemy measured the star positions in ecliptic coordinates and then Ptolemy change them into equatorial is also nonsensical, as precisely the purpose of that chapter of the Almagest is to calculate the precession rate. If Ptolemy (and other Astronomers of his time, of course) had the ecliptic coordinates yet, calculating the precession rate would be straightforward, so changing then to equatorial ones to calculate the precession rate in a much complex and misleading way is quite absurd.
3.       The only possibility remaining is that the declinations were all derived from just a single set of observations and then extrapolated to different epochs**. If so, then we are not talking about real observations, but about forgery.

The original source

If the declinations were in fact not observed, but calculated, the original source cannot be Ptolemy declinations as his data are more imprecise than Hipparchus’. It is incomprehensible how someone can calculate older declinations from contemporary data and that the extrapolated coordinates result more precise than the original ones. The source of the declinations, if any, only can be Hipparchus, as his data present the minor standard deviation for the 12 declinations (0,095), compared with Ptolemy (0,124) and Timocharis and Arystillus (0,140). This also is in good correspondence with the bias: Hipparchus data for the 12 declinations of Ptolemy present a bias of -0,013 degrees, Timocharis and Arystillus data present a bias of -0,052 degrees and Ptolemy’s himself +0,043. There is a difference of 0,025/cy between Timocharis and Arystillus and Hipparchus and -0,021/cy between Hipparchus and Ptolemy. Similar rate and opposite signs is what you can wait if Hipparchus was the source.

From the 18 stars, three of them are also used by Hipparchus in his Commentary on the Poem of Aratus ‘Phenomena’. These are alpha Bootis, alpha Geminorum and Beta Geminorum. For alpha Bootis and Beta Geminorum, the values agree with those in the Almagest, so there are chances that they share a common origin. As the declinations stated in the Almagest are dependent on the right ascension, the declinations in the Commentary should be dependent as well. But I do not find such a relation in the declinations stated in the Commentary. The problem with the Commentary is that it is more imprecise and that a lot of big mistakes are present. Many  of these mistakes can’t be measurement errors but scribal mistakes or bad identification of stars. On the contrary to the chapter in the Almagest, that present bright, well identifiable stars, many stars in the Commentary are dim, anonymous stars.

Possible time of composition

It is surprising that Ptolemy calculated the declinations in his epoch from Hipparchus data instead of measuring them by his own.

Besides, for extrapolating the declinations correctly you have to know the real rate of precession or at least a very good approximation, but the first known good calculation of the axial precession was attained by the Persian astronomer Nasser al-Din al-Tusi in the XIII century, who measured 51” / year, instead of the real 50.2 “/ year. (This makes (n‘-n) =0,011 degrees by century as the maximum possible accuracy for our calculations). The XIII century seems too late  for editing the Almagest because older dated copies of the book had survived until the present. Maybe a good precession rate was known before that date, but it can’t be much earlier. In any case, the forger should belong to a very late epoch and had to know that the precession rate stated in the Almagest is wrong. There are two possibilities, or either Ptolemy knew a better value for the axial precession that he declares and  for whatever reason he is lying, or well, someone else, centuries later, added those passages to the Almagest.

To answer this question, it can help to know if the probably original source, the declinations attributed to Hipparchus, consist in genuine data or are a late forgery themselves. To know this, we need to take in mind the proper motions of the stars.

Let ∆t be the difference of time in centuries from Hipparchus to the ‘forger’ data and let ∆δ be the real change in the declinations for any star in the same span of time: ∆δ=n cos(α) ∆t +μ ∆t. In this ‘n’ is about 2000 seconds for historic times and μ is the star proper motion in the declination axis. This is not the real ∆δ but is an approximation good enough for the purpose of this article. As the supposed forger had to ignore both the existence of proper motions and the exact value of the precession, his calculated change of  declination, whenever it was his method of extrapolation, would be more or less: ∆δ’ =n’ cos(α) Δt, being  ∆δ’ and n’ his wrong values. So, his ‘error’ of observation will be:

∆δ’ - ∆δ = (n‘-n) cos(α) ∆t - μ ∆t                                                                                            (3)

Now, if we compare the errors of observation with this newly calculated error we obtain a better correspondence for some of the fastest stars: alpha Aquila (1) and alpha Bootes (3), much worse for alpha CMA (4)*, and slightly worse for alpha Aquila (2). The remaining stars are not fast enough to appreciate a remarkable difference.

Graph 4. Hipparchus declinations for ∆t=7  n’-n=0,011 for the four fastest stars

The analysis of the proper motions also can explain the big mistake in the position of Arcturus (Alpha Bootes) attributed to Timochares. If Timocharis declinations were calculated in a very late epoch, as Arcturus is the fastest star in the group, it had changed its position in relation to others stars greatly. The forger noticed it and thinking it was a mistake he corrected its position accordingly to the relative position of the star in his own epoch, far way from where it was many centuries before.

Final conclusion

The declinations attributed to Timocharis, Arystillus and Ptolemy were probably extrapolated from the declinations attributed to Hipparchus , either by Ptolemy or lby someone else in a later epoch.

The declinations attributed to Hipparchus were extrapolated from either a contemporary source in ecliptic coordinates (maybe the lost star catalog of Hipparchus himself) or from a late source in unknown coordinates, unfortunately, the analysis of the proper motions is inconclusive and answering this last question with the data in the Almagest is not possible.

** the error in Hipparchus data fit also the equation Zenith systematic error (arcmin)=6-zenith distance (degrees)/2 that is the same systematic error found in the stars zt the same latitude in the ALmagest star catalog (see the measurement method of the Almagest Stars. D.Duke, for more details) this als points to the same origin of oth sets of data, hat maye the lost star catalog of Hipparchus.

*We need to point out that Sirius (alpha CMA) is a special case as it wasn’t regarded a normal star in the past, it moves along the old Julian Calendar so its heliacal rising was always in the same date, marking the beginning of the new year in Egypt for many centuries. Once the precision of the equinoxes was known, people should realize that Sirius wasn’t still in the sky as the remaining stars were supposed to be but it moved slowly through the heavens. So his proper motions could be known at least in an approximate way. This would explain why if we do not take in mind its proper motion the error of observation of Sirius matches the calculated error pretty well, but when we do count the proper motions then the position of Sirius in the three sets of records is very discordant. 
« Last Edit: 08/12/2015 20:06:19 by martinchaide »


Offline Space Flow

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Re: Almagest declinations
« Reply #1 on: 08/12/2015 01:00:21 »
Martin, I am not one qualified to give you any insight into the discrepancies you seem to have uncovered.
Just the same I had to comment on the impressive amount of research you seem to have put in to this.
When and how those discrepancies were introduced would effect History and phycology as much as astronomy, (There are some very prominent historical figures featured) so I hope your post generates some interesting discussion.
We are made of Spacetime; with a sprinkling of Stardust.
Matter tells Spacetime how to Flow; Spacetime tells matter where to go


Offline martinchaide

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Re: Almagest declinations
« Reply #2 on: 08/12/2015 15:10:37 »
ty for your answer, this is not closed, maybe the consensual dates for the observations are a bit wrong and this is because i obtain errors proportional to the right ascension, i going to try to give a more simple explanation and post it again


Offline martinchaide

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Re: Almagest declinations
« Reply #3 on: 08/12/2015 18:20:14 »
i gonna try to explain it in a more comprehensible way:

there are two mean celestial coordinates systems: equatorial coordinates and ecliptic coordinates

the reference for equatorial coordinates is the terrestrial equator projected in the sky, the coordinates are right ascension (west-east) and declination (north -south).

the reference for the ecliptic coordinates is the ecliptic itself, that is the trajectory of the sun in the sky through one year, its cordinates are celestial longitude and celestial latitude.

the origin of the coordinates is the point in the sky where the celestial equator and the ecliptic crosses each other, this point is called the Aries point, and the sun passes this point during the spring equinox


because of a phenomenon called the precession of the equinoxes, this point, as the years goes by, is travelling backwards thorough the heavens, in fact, the Aries point is no longer in Aries, but in Pisces, and very closed to Acuarius.

the precession rates are aprox as follows:

for ecliptical coordinates: latitude=0, longitude=+1.384 degrees  by century

for equatorial coordinates: righ ascension= a very complex formula (it doesn't matter here),
declination= 0.56 degrees * cosine of the right ascension

that means that for stars with right ascension 0 or 180 the rate of change  is maximun (ex. stars belonging to Aries or Virgo) and for stars with right ascension of 90 or 270 the change is minimun. (like Gemini or Sagitarius)


in the chapter 7 of the book 3 of the Almagest there are 18 declinations measured by four astronomers in 3 different epochs. timocharis-Arystillus8 (-290), hipparchus (-128)and Ptolemy (135 ad)those were used by Ptolemy, by comparing the changes in the declnations, to calculate the precession rate.

these coordiantes were used much later, in 1974, by an american astronomer called robert newton to calculate the obliquity of the ecliptic in those ancient times, he considered some instrumental bias in the measurement of the decliantions by the ancients: to calculate the decliantion of one star you have to know perfectly your terrestrial latitude and where the zenit is, and neither of them could be known by the ancients in a perfect way.

newton considered the measurement errors independent of the right ascension, as the accuracy in the measurement of the delciantions do not depend on where the star is in the sky in a sense west-east as expalined before.

however, i have compared the measurements errors of the three sets of declinaions an all of thenm shows a clear relation to the cosine of the right ascension. how can it be that?

the simplest explanation is that the supossed time in which the observations were made is wrong, the fastest a star changes his declination with time, the bigger the mistake of the modern astronomer , if you think that the observations where made too soon you obtain a  bias proportional to the cosine f the right ascension in one direction and if you think they were made too late then there is a bias in the opossite directin , the time in wich the bias shift its sense has to be the real epoch of observation

being said that, the shift in this kind of bias for hipparchus hapen in -140 and for Ptolemy  in 170 AD, aprox. 310 apart but History and Ptolemy himself said that there were just 265 years beetwen both of them. So i do not thing this is the reason of the cosine bias.

the second motive could be that the declinations were measured in ecliptic coordiantes and then changed to equatorial ones, if the ecliptic was baddly measured then it has to be some mistake, the bigger the  farer you are from the equinoxes: the effect of a bad measured ecliptic is similar to the precession of the equinoxes, but is not real, but consecuence of a bad meaurement.

however it makes no sense that the four astronomers decided in different epochs to measure the positions in ecliptic coordiantes that are much more difficult to measure just to change them into equatorial ones

that Ptolemy himself, changed then is also nonsensical as the purpose of the chapter is to calculate the precession rate, if ptolemy had the ecliptic coordinates yet, calculating the precession rate is straighforward, so changing then into ecliptical is absurd and the only possible explanation is to disguise his erroneous precession rate of one degree by century.but if ptolemy had access to the data from hipparchus, Timocharis and aristyllus, then chances are that other astronomers had also those data, so this explanation is also very unlikely

and a last explanation: all the coordinates were extracted from a singles source stated in ecliptical coordiantes and extrapolated to equatorial ones for an unknown reason, this original source, if any, only can be hipparchus because his data is the more exact of the three sets

it is very extrange that ptolemy calculated the declinations from ancient data instead of observing then by himself, besides, to falsifie observations of past centuries you need a good knowledge of the real precession rate, and it wasn't know until 1000 years later.

I think that there are only two possible explanations:

1- ptolemy lived almost half a century later than supposed

2- the declianations of Ptolemy and possibly from timocharis and Aristillus were forged in an unknown epoch from some manuscript that is lost (maye the lost star catalog attributed to Hipparchus)

both explantions are unlikely , but i cannot find nothing better...

ty for your patiente an excuse my english
« Last Edit: 08/12/2015 18:31:38 by martinchaide »