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Hipparchus' Celestial Globe

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Bianca
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« Reply #30 on: December 08, 2007, 10:05:33 am »







                                         TABLE 5. Positions of stars off the circles.





On the Farnese Atlas  Star Position in 125 B.C. 



#  Position  Star  α(°)  δ(°)  λ(°)  β(°)  α(°)  δ(°)  λ(°)  β(°)  ∆λ(°) ∆β(°)
 
48  Argo’s s. rudder tip  α Car  77.4  –49.4  62.7  –71.9  84.4  –52.7  75.7  –76.1  –13.0  4.2 

49  CMa’s mouth  α CMa  76.3  –17.6  72.8  –40.5  77.7  –16.8  74.6  –39.9  –1.8  –0.6
 
50  Corvus’s beak  α Crv  152.3  –16.6  161.0  –26.2  155.7  –13.2  162.8  –21.7  –1.8  –4.5
 
51  Middle of Orion’s belt ε Ori  60.6  –2.3  57.9  –22.7  57.5  –5.1  53.9  –24.8  4.0  2.0
 
52  Hydra’s eye  δ Hya  108.8  5.5  109.6  –16.9  100.7  10.7  100.8  –12.6  8.8  –4.3 

53  Scorpius’s sting  λ Sco  229.5  –30.0  235.0  –11.2  228.8  –32.3  235.0  –13.5  –0.1  2.4
 
54  Middle of Sco’s body α Sco  211.6  –23.6  217.6  –10.0  216.3  –19.1  220.2  –4.3  –2.6  –5.7
 
55  Tip of Sgr’s arrow  γ Sgr  234.5  –29.0  239.1  –9.1  238.0  –27.3  241.7  –6.7  –2.6  –2.4
 
56  Taurus’s s. eye  α Tau  46.7  12.2  47.6  –5.3  39.6  9.6  40.2  –5.7  7.4  0.4 

57  Pisces, head of fi sh  γ Psc  315.9  –6.6  316.4  10.0  322.0  –7.4  321.9  7.3  –5.5  2.7 

58  N. tip Cap. rear horn  α Cap  276.8  –12.0  276.7  11.5  274.4  –16.4  274.2  7.2  2.5  4.3

59  Medusa’s head  β Per  29.3  28.6  37.4  15.4  15.5  30.9  26.7  22.2  10.7  –6.8 

60  Aries’s muzzle  α Ari  –5.6  18.4  2.6  19.0  3.5  12.2  8.2  9.8  –5.5  9.2 

61  Andromeda’s head  α And  339.7  17.2  348.3  23.9  335.9  17.5  344.9  25.6  3.5  –1.7 

62  Pegasus’s muzzle  ε Peg  302.5  10.2  307.5  29.7  299.8  1.8  302.4  22.2  5.1  7.5 

63  Delphinus’s head  α Del  287.4  15.3  291.4  37.7  285.2  10.5  288.0  33.2  3.5  4.5
 
64  Ophiucus’s head  α Oph  240.0  21.6  231.7  41.3  239.5  16.3  232.9  36.1  –1.1  5.2 

65  Hercules’s head  α Her  233.0  25.2  222.0  42.9  234.8  19.1  226.6  37.5  –4.5  5.3
 
66  Cassiopeia’s breast  α Cas  333.9  40.2  356.0  46.5  343.8  44.8  8.5  46.5  –12.5  0.1
 
67  South edge of CrB  α CrB  224.5  33.5  207.8  47.7  211.3  35.4  192.6  44.5  15.2  3.2
 
68  Cygnus’s beak  β Cyg  273.0  25.3  274.2  48.9  271.4  25.5  271.9  49.2  2.3  –0.3 

69  Cygnus’s tail  α Cyg  299.5  42.7  319.1  61.4  292.4  39.1  306.2  60.0  12.9  1.4 

70  Centre of Lyra’s shell α Lyr  261.5  38.3  255.9  61.6  261.6  38.7  255.9  62.0  0.0  –0.4 




On each of my pictures of the Farnese Atlas on which a given constellation fi gure is visible, I have measured its position and calculated its corresponding right ascension and declination. I have a total of 67 such positions for all 23 points. I have averaged together all the positions for each point, and the resulting right ascensions and declinations are presented in Table 5. I have also calculated the corresponding ecliptic coordinates,21 and these too are presented in Table 5.

The first column of Table 5 lists a running number, which is a continuation of the numbering from Table 3. The items are ordered by increasing ecliptic latitude. The second column gives a verbal description of the constellation position selected, while the third column gives the modern name of the star at that position. The next two columns give the derived α and δ value in the reference frame of the Farnese Atlas, these being the observed positions. These equatorial positions are converted to ecliptic positions (longitude λ and latitude β) in columns 6 and 7. Columns 8–11 give α, δ, λ and β of the modern stars as precessed back to the epoch of 125 B.C. Columns 12 and 13 are the differences between the observed and the model star positions in ecliptic coordinates.

The RMS scatter in the differences gives us a reasonable measure of the total uncertainties in the placements. The RMS scatter in the errors in ecliptic latitude is 4.1°. The deviations in ecliptic longitude will vary with ecliptic latitude by a factor of 1/cos (β) due to the convergence of the lines towards the pole. With this, I fi nd that the RMS scatter of the errors in ecliptic longitude is 5°/cos (β). These errors have a good Gaussian distribution: 15/46 = 32% deviate by more than one-sigma, 1/46 = 2% deviate by more than two-sigma, while 12/23 = 52% and 11/23 = 48% deviate less than the average.
« Last Edit: December 08, 2007, 10:06:56 am by Bianca2001 » Report Spam   Logged

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« Reply #31 on: December 08, 2007, 10:08:54 am »








A.2.4. Chi-square Analysis





The best estimate for the date of the observations that were used to construct the constellations on the Farnese Atlas is that year for which the observed positions from Tables 3 and 5 most closely match the real sky. The optimal statistic for all such analyses22 is always the chi-square statistic, χ2.

The χ2 statistic is simply the summation over the data points of the squares of the deviations between the observations and the model in units of the standard deviation for the measurement. Symbolically, this is χ2 = Σ{[(Oi – Mi)/σi]2}, where the summation is over all data items, the subscript ‘i’ itemizes the datum, ‘O’ is the observed value, ‘M’ is the model value, and ‘σ’ is the measurement uncertainty. For example, for the first datum in Table 3, for an epoch of 125 B.C., the observed right ascension of the star γ Ari is 0.5°, the model value is 0° corresponding to the vernal equinox colure, and the typical scatter near the equator implies σ = 3.5°; so the χ2 contribution for this datum is 0.02. For the first star on the equator (item #13 in Table 1), the observed declination of σ Ari in 125 B.C. is 4.7°, the model value is 0° which corresponds to the equator, and the typical scatter is σ = 3.5°; so the χ2 contribution for this datum is 1.8. For item #56 in Table 5, the observed ecliptic longitude of α Tau in 125 B.C. is 47.6° on the Farnese Atlas, the model value is 40.2°, and the uncertainty near the equator is 5°; so the χ2 contribution for this datum is 2.2. For a given date, the χ2 contributions can be summed over all data points to produce the χ2 value.

The χ2 value is smallest for the best model. In this case, the best model would be ideally that the Farnese Atlas represents the true positions of the identified stars for some particular date. By varying the date, the χ2 will vary also, with the date of the minimum χ2 being the best estimate date. The 68% probability error bar (i.e., the one-sigma uncertainty region) is that range of dates for which the χ2 is within 1.0 of the minimum value. With this, we have a standard technique for determining the best date for the Atlas as well as quantitatively deriving the real error bar in that date.

In Tables 3 and 5, I tabulate a total of 70 data points taken from the Farnese Atlas, and these constitute the observations. The model consists of the modern positions for these same stars or target circles as calculated by precession for various years. The χ2 contributions are simply the differences between the observed and model values (divided by the measurement uncertainty) squared. The uncertainties are 3.5° in position for Table 3 and 5° in ecliptic longitude for Table 5 (both with corrections for the latitude as needed). These are summed to give the χ2 value. I have calculated the χ2 for all dates from 400 B.C. to A.D. 200 at fi ve-year intervals.

The date of minimum χ2 is 125 B.C. The χ2 at minimum is 66.3.23 The χ2 rises to

67.3 for the years 70 B.C. and 180 B.C., so this is the one-sigma range for the date (i.e., there is a 68% probability that the real date is between 180 and 70 B.C.). The two-sigma range (i.e., the 95% confi dence region) is that for which the χ2 is lower than 66.3 + 22 = 70.3 and is between 245 and 10 B.C. The three-sigma range (i.e., the 99.7% confidence region) is that for which the χ2 is lower than 66.3 + 32 = 75.3 and is between 305 B.C. and A.D. 50. The total size for the one-, two-, and three-sigma intervals is 110, 235, and 355 years respectively. It is encouraging and expected that these interval sizes are nearly in the proportions 1:2:3. The standard way to present such results is to quote the year of minimum χ2 as the best estimate of the date and to give the uncertainty as half the size of the one-sigma region. Thus, I conclude that the constellations on the Farnese Atlas were based on observations made in the year 125 ± 55 B.C.

It is appropriate here to note that this derived date is a very confi dent result. The techniques in the Appendices are all standard, straight forward and defi nitive. The deviations from the simple model are all Gaussian with zero outliers. The selection of various subsamples of the data still yields the same date (to within the quoted error bars), so the result is not simply due to some number of localized errors of any type. I cannot think of any astronomical effect or error nor of any historical vagary or error that would artificially produce the derived date.

In all, to within the quoted error bars, the derived date of 125 B.C. is of high confidence.
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« Reply #32 on: December 08, 2007, 10:12:20 am »






                                                    ACKNOWLEDGEMENTS





I thank Maurizio Paolillo and Guy Consolmagno for help with access to the Farnese Atlas for photography; the National Museum in Naples for granting me access to the statue for taking photographs; Roger Macfarlane and Paul Mills for access to their translation of Hipparchus’s Commentary before publication; and Eric Seidel for help in translations.








                                                            REFERENCES





1. The best pictures of the Atlas appear in H. Stern, “Classical Antiquity” in the entry titled “Astronomy and astrology” in Encyclopedia of world art (New York, 1960), ii, cols 48–57 and plate 23. Many short descriptions of the statue have appeared, of which the best are G. Aujac, “The foundations of theoretical cartography in archaic and classical Greece”, in History of cartography, ed. by J.

B. Harley and D. Woodward (Chicago, 1987), 142–3; J. Evans, The history & practice of ancient astronomy (New York, 1998), 78–79; M. Fiorini, Sfere terrestri e celesti (Rome, 1899), 8–25;

C. Gialanella and V. Valerio, “Atlas Farnese” in Cartes et figures de la Terre (Paris, 1980), 84;

E. L. Stevenson, Terrestrial and celestial globes (New Haven, 1921), 1–25; G. Thiele, Antike Himmelsbilder (Berlin, 1898), 27–45; and D. J. Warner, The sky explored, celestial cartography 1500–1800 (New York, 1979), 278–9. A detailed history of the historical writings on the Farnese Atlas appears in V. Valerio, “Historiographic and numerical notes on the Atlante Farnese and its celestial sphere”, Der Globusfreund, xxxv/xxxvii (1987), 97–124.

2. Here is the complete list of constellations visible on the Farnese Atlas in the constellation order as given in the Almagest: Dra, Cep, Boo, CrB, Her, Lyr, Cyg, Cas, Per, Aur, Oph, Ser, Aql, Del, Peg, And, Ari, Tau, Gem, Cnc, Leo, Vir, Lib, Sco, Sgr, Cap, Aqr, Psc, Cet, Ori, Eri, Lep, CMa, Argo, Hya, Crt, Crv, Cen, Lup, Ara, and CrA. (Stevenson cites 42 constellations while Gialanella and Valerio [and hence Aujac] cite 43 constellations.) There are a number of constellations from Antiquity that are missing. Ursa Major is missing as it is located entirely within the hole in the top of the globe. Canis Minor is entirely positioned under one of the hands holding the globe. Piscis Austrinus is completely hidden by the shoulder of the Titan. Of the older Greek constellations, the only ones missing without cause are Ursa Minor, Triangulum, and Sagitta.

A short, simple line segment near a wing of Cygnus could conceivably be Sagitta, but this has the strong iconography, the wrong orientation, and is 15° misplaced, and so I conclude that the line segment is not Sagitta. The later Greek constellations of Equuleus, Coma Berenices, and Antinous are all missing and would certainly have been visible if depicted. The only addition that is not part of the set of old constellations is a curious small rectangular structure with three internal lines that appears above the Crab. This is certainly not any known constellation, and the only explanation suggested is that is this represents the comet known as “Throne of Caesar” or a throne of Zeus (Stern, op. cit. (ref. 1); Thiele, op. cit. (ref. 1)).

Stern, op. cit. (ref. 1), col. 49.

B. E. Schaefer, “Latitude and epoch for the origin of the astronomical lore of Eudoxus”, Journal for the history of astronomy, xxxv (2004), 161–223.

Aratus Phaenomena, transl. by D. Kidd (Cambridge, 1997). Another widely available translation is by G. R. Mair and appears in the Loeb Classical Library series titled Aratus (Cambridge, MA, 1921) with many subsequent reprints.

R. T. Macfarlane and P. S. Mills, Hipparchus’ commentaries on the Phaenomena of Aratus and Eudoxus, manuscript, 2003. This translation is the first in English and the first in modern times.

7. T. Condos, Star myths of the Greeks and Romans: A sourcebook (Grand Rapids, MI, 1997).

G. J. Toomer, Ptolemy’s Almagest (Princeton, 1998).
That is, one of the primary statements by Aratus on the cardinal points of the ecliptic is true for a time almost a millennium earlier. This immediately tells us that Aratus (and hence Eudoxus) include lore that was already ancient for their times. A detailed analysis of the lore in Aratus and Eudoxus (in Schaefer, op. cit. (ref. 4)) shows that all the lore is from many centuries before even Homer and Hesiod. A detailed analysis gives a date of 1130 ± 80 B.C.

B. E. Schaefer, “Latitude and epoch for the formation of the southern Greek constellations”, Journal for the history of astronomy, xxxiii (2002), 313–50. Roughly, the extinction angle plus refraction correction for the bright stars in Centaurus, Crux, and Carina is close to 2°, while the gap for these stars (the angle above the region of invisibility) is certainly greater than 1° and more likely around 2° (although it could be larger). Thus, the correction to the simply derived latitude is likely 4°, but the value could be 3° or larger.

Evans, op. cit. (ref. 1), 91–95.

Schaefer, op. cit. (ref. 4), Table 7 and Section 6.7.

Stevenson, op. cit. (ref. 1), 1–25; Evans, op. cit. (ref. 1), 78–79 and Figure 5.11.
 
The quote is from O. Neugebauer, A history of ancient mathematical astronomy (3 vols, Berlin, 1975), 277–80; Evans, op. cit. (ref. 1), 103.

K. A. Pickering, “Evidence of an ecliptical coordinate basis in the Commmentary of Hipparchos”, Dio, ix (1999), 26–29.

D. W. Duke, “Hipparchus’ coordinate system”, Archive for the history of exact sciences, lvi (2002), 427–33; J. B. J. Delambre, Histoire de l’astronomie ancienne (1817; repr. New York, 1965), i, 117, 172, 184.
A history of this dispute is given in B. E. Schaefer, “The great Ptolemy–Hipparchus debate”, Sky & telescope, ciii (2002), Feb. issue, 38–44.

H. Vogt, “Versuch einer Wiederstellung von Hipparchs Fixsternverzeichnis”, Astronomische Nachtrichten, ccxxiv (1925), cols 2–54; G. Grasshoff, The history of Ptolemy’s star catalogue (New York, 1990); N. M. Swerdlow, “The enigma of Ptolemy’s catalogue of stars”, Journal for the history of astronomy, xxiii (1992), 173–83; and D. W. Duke, “The depth of association between the ancient star catalogues”, Journal for the history of astronomy, xxxiv (2003), 227–30.

The obliquity of the ecliptic for 125 B.C. is 23.71°, as determined from J. Meeus, Astronomical algorithms (Richmond, VA, 1991), 135–6. The mean obliquity of the ecliptic is 23.76° in 500 B.C. and 23.70° in 1 A.D.




Meeus, op. cit. (ref. 19), 126–8. Proper motions of all stars used in this paper are completely negligible.
The conversion from ecliptic to equatorial coordinates is taken from Equations 12.3 and 12.4 in
Meeus, op. cit. (ref. 19), 87–92.

Technically, the chi-square analysis applies only when the errors have Gaussian distributions. However, in practice, the chi-square technique is still best even for distributions far from Gaussian, even though the derived error bars become more uncertain. Fortunately, the residuals for the analysis in this paper are all good Gaussian distributions.
With 70 data items and one variable fit parameter, the total number of degrees of freedom in this chi-square fit is 69. Then, the reduced chi-square is 66.3/69 = 0.96. In general, a reduced chi-square of near unity implies a good fit. However, this conclusion of a good fit does not apply for this case because the size of the uncertainties were selected as the observed scatter which then forces a reduced chi-square of near unity. If alternatively, we had chosen a substantially larger or smaller value for the sigma, then the derived reduced chi-square would have been greatly smaller or larger than unity respectively. If we had any way of independently knowing the typical total uncertainty (which we do not), then we could have used this value and derived a meaningful reduced chi-square that is useful for evaluating the quality of the model. One comforting fact is that the typical scatters (which yield a reduced chi-square near unity) have reasonable sizes, for example judged based on the typical scatters for the lore of Eudoxus (see Schaefer, op. cit. (ref. 4)). The determination of the best date does not depend on any scaling of the adopted uncertainties. However, the derived error bars for the best-fit epoch will depend on the size of the adopted uncertainties for the individual data points. And, indeed, the process can be turned around so as to derive the typical total uncertainty of a single point (based on the average uncertainty so as to make the reduced chi-square near unity). With this derived uncertainty for an individual observation, we will necessarily get an acceptable reduced chi-square and the derived error bars on the epoch will be good. In all, the exact value of the reduced chi-square is not meaningful since the uncertainties have been taken from the observed scatter in the data themselves, but this choice allows for valid error bars on the date.


http://www.phys.lsu.edu/farnese/JHAFarneseProofs.htm
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