# 歲差 (天文)

## 北極星的變換

 地球自轉軸繞行黃道北極的歲差圈 地球自轉軸繞行黃道南極的歲差圈

## 方程式

$\overrightarrow{T} = \frac{3Gm}{r^3}(C-A)\sin\delta\cos\delta\begin{pmatrix}\sin\alpha\\-\cos\alpha\\0\end{pmatrix}$

Gm = 攝動天體的標準引力參數
r = 攝動天體的地心距離
C = 圍繞地球自轉軸轉動的轉動慣量
A = 任何環繞地球赤道直徑的轉動慣量
C−A = 地球赤道隆起的轉動慣量(C>A)
δ = 攝動天體的赤緯 (赤道以南或北)
α = 攝動天體的赤經 (從春分點向東)

$T_x = \frac{3}{2}\frac{Gm}{a^3 (1-e^2)^{3/2}}(C-A)\sin\epsilon\cos\epsilon$

$a$ = 地球(太陽)或月球的軌道半長軸
e =地球(太陽)或月球的軌道離心率

$\frac{d\psi}{dt} = \frac{T_x}{C\omega\sin\epsilon}$

$\frac{d\psi_S}{dt} = \frac{3}{2}\left[\frac{Gm}{a^3 (1-e^2)^{3/2}}\right]_S\left[\frac{(C-A)}{C}\frac{\cos\epsilon}{\omega}\right]_E$

$\frac{d\psi_L}{dt} = \frac{3}{2}\left[\frac{Gm(1-1.5\sin^2 i)}{a^3 (1-e^2)^{3/2}}\right]_L\left[\frac{(C-A)}{C}\frac{\cos\epsilon}{\omega}\right]_E$

$e''^2 = \frac{\mathrm{a}^2 - \mathrm{c}^2}{\mathrm{a}^2 + \mathrm{c}^2}$

Gm=1.3271244×1020 m³/s² Gm=4.902799×1012 m³/s² (CA)/C=0.003273763
$a$=1.4959802×1011 m $a$=3.833978×108 m ω=7.292115×10−5 rad/s
e=0.016708634 e=0.05554553 $\epsilon\,\!$=23.43928°
i= 5.156690°

S/dt = 2.450183×10−12 /s
L/dt = 5.334529×10−12 /s

S/dt = 15.948788"/a   vs   15.948870"/a 取自威爾斯[2]
L/dt = 34.723638"/a   vs   34.457698"/a 取自威爾斯

## 數值

pA = 5,028.796195×T + 1.1054348×T2 + 高階項，

p = 5,028.796195 + 2.2108696×T +高階項

p = A + BT + CT² + … 只是下面公式的近似 p = A + Bsin (2πT/P)，此處P 是410個世紀的週期。

## 註解

1. ^ Robert Main, Practical and Spherical Astronomy (Cambridge: 1863) pp.203–4.
2. ^ 2.0 2.1 2.2 2.3 James G. Williams, "Contributions to the Earth's obliquity rate, precession, and nutation", Astronomical Journal 108 (1994) 711–724, pp.712&716. All equations are from Williams.
3. ^ IAU 2006 Resolution B1: Adoption of the P03 Precession Theory and Definition of the Ecliptic
4. ^ 引用错误：无效<ref>标签；未为name属性为Astro_101的引用提供文字
5. ^ Dennis Rawlins, Continued fraction decipherment: the Aristarchan ancestry of Hipparchos' yearlength & precession DIO (1999) 30–42.
6. ^ Neugebauer, O. "The Alleged Babylonian Discovery of the Precession of the Equinoxes", Journal of the American Oriental Society, Vol. 70, No. 1. (Jan. – Mar., 1950), pp. 1–8.
7. ^ Milbrath, S. "Just How Precise is Maya Astronomy?", Institute of Maya Studies newsletter, December 2007.
8. ^ Siddhānta-shiromani, Golādhyāya, section-VI, verses 17-19
9. ^ http://episte.math.ntu.edu.tw/articles/sm/sm_18_01_1/index.html 參照曹亮吉教授：印度的數學，翻譯的書名。
10. ^ Translation of the Surya Siddhānta by Pundit Bāpu Deva Sāstri and of the Siddhānta Siromani by the Late Lancelot Wilkinson revised by Pundit Bāpu Deva Sāstri, printed by C B Lewis at Baptist Mission Press, Calcutta, 1861; Siddhānta Shiromani Hindi commentary by Pt Satyadeva Sharmā, Chowkhambā Surbhārati Prakāshan, Varanasi, India.
12. ^ cf. Suryasiddhanta, commentary by E. Burgess, ch.iii, verses 9-12.
13. ^ Rufus, W. C., The Influence of Islamic Astronomy in Europe and the Far East, Popular Astronomy, May 1939, 47 (5): 233–238 [236]
14. ^ Ancient records of astronomy are discussed in Baron Georges Cuvier A Discourse on the Revolutions of the Surface of the Globe (1825) in the chapter titled "The astronomical monuments of the ancients." Part of the discussion in the chapter concerns the date of the Dendera zodiac, which Cuvier suggested is 123 AD to 147 AD page 170 and page 172. The current consensus is around 50 BC.
15. ^ Kaler, James B.. The ever-changing sky: a guide to the celestial sphere. Cambridge University Press. Reprint 2002: 152. ISBN 978-0521499187.
16. ^ The Columbia Electronic Encyclopedia, 6th ed., 2007
17. ^ Ivan I. Mueller, Spherical and practical astronomy as applied to geodesy (New York: Frederick Unger, 1969) 59.
18. ^ G. Boué & J. Laskar, "Precession of a planet with a satellite", Icarus 185 (2006) 312–330, p.329.
19. ^ George Biddel Airy, Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics (third edititon, 1842) 200.
20. ^ J.L. Simon et al., "Numerical expressions for precession formulae and mean elements for the Moon and the planets", Astronomy and Astrophyics 282 (1994) 663-683.
21. ^ Dennis D. McCarthy, IERS Technical Note 13 – IERS Standards (1992) (Postscript, use PS2PDF).

## 參考資料

• Explanatory supplement to the Astronomical ephemeris and the American ephemeris and nautical almanac
• Precession and the Obliquity of the Ecliptic has a comparison of values predicted by different theories
• A.L. Berger (1976), "Obliquity & precession for the last 5 million years", Astronomy & astrophysics 51, 127
• J.H. Lieske et al. (1977), "Expressions for the Precession Quantities Based upon the IAU (1976) System of Astronomical Constants". Astronomy & Astrophysics 58, 1..16
• W.R. Ward (1982), "Comments on the long-term stability of the earth's obliquity", Icarus 50, 444
• J.L. Simon et al. (1994), "Numerical expressions for precession formulae and mean elements for the Moon and the planets", Astronomy & Astrophysics 282, 663..683
• N. Capitaine et al. (2003), "Expressions for IAU 2000 precession quantities", Astronomy & Astrophysics 412, 567..586
• J.L. Hilton et al. (2006), "Report of the International Astronomical Union Division I Working Group on Precession and the Ecliptic" (pdf, 174KB). Celestial Mechanics and Dynamical Astronomy (2006) 94: 351..367
• Rice, Michael (1997), Egypt's Legacy: The archetypes of Western civilization, 3000-30 BC, London and New York.
• Dreyer, J. L. E.. A History of Astronomy from Thales to Kepler. 2nd ed. New York: Dover, 1953.
• Evans, James. The History and Practice of Ancient Astronomy. New York: Oxford University Press, 1998.
• Pannekoek, A. A History of Astronomy. New York: Dover, 1961.
• Parker, Richard A. "Egyptian Astronomy, Astrology, and Calendrical Reckoning." Dictionary of Scientific Biography 15:706-727.
• Tomkins, Peter. Secrets of the Great Pyramid. With an appendix by Livio Catullo Stecchini. New York: Harper Colophon Books, 1971.
• Toomer, G. J. "Hipparchus." Dictionary of Scientific Biography. Vol. 15:207-224. New York: Charles Scribner's Sons, 1978.
• Toomer, G. J. Ptolemy's Almagest. London: Duckworth, 1984.
• Ulansey, David. The Origins of the Mithraic Mysteries: Cosmology and Salvation in the Ancient World. New York: Oxford University Press, 1989.
• Schütz, Michael: Hipparch und die Entdeckung der Präzession. Bemerkungen zu David Ulansey, Die Ursprünge des Mithraskultes, in: ejms = Electronic Journal of Mithraic Studies, www.uhu.es/ejms/Papers/Volume1Papers/ulansey.doc